1I1R PUBl.lSlJERS

B. CMllnrA

B nOrOllE 3A

/{PACOTOlil

D31lA TEll bC'l BO cMO�10JtAR rIJAPJlI1R.

IN THE SEARCH FOR BEAUTY

by

V. Smilga

T,QMlottd from ,h. Run/a" bV

Ceo,." Yankov.kv

.IIIR PUBLI HERS MOSCOW 1970

UDC 513.81(02.1) _ 20

RevIsed from the 1968 nu!Sian Edit.ion

HtJ O.HlAuiklCo.w Jl3tMKt

CONTENTS

Chapter

1. Before Euclid-Prehistoric Timu . 7

2. Euclid. . . . . . . 26 3. The Fifth Postulate. 57

4. The Age of Proof •. The Beginning 81 5. Omar KhaY!lam. . . . . . . . . 92 6. The Age of Proofs. Contillued. . . 129 7. N on·Eu.clidean Ct:()II1t/r!l' The SOI'I'

tion. . . . . . . . . . . . . 155

8. Nikolai Tvanovich Lobaehevsky 198 9. Non-Euclidean CeorMtry. Some /1-

lustrations . . . . . . . . . . . 230 10. New Ideal. Riemann. Non-contra

dictorin .. s. . . . . . . . . . . 246 11. All U M:z;�ted FiTUlle. The Central

Theory of Relativity 269 12. Einstein . . . . . . . . . _ . . 301

5

Chapltr I

BEFORE EUCLlD­

PREHISTORIC TIMES

lru beginning of this story goes back to

Ii immemorial.

Where was it, when and how did geometry

DI into being? Where, how Dlld when did it

taL: sbape nnd become a science? Who was the

ery first to propose the axiomatic structure 01 geometry?

We do not know, and most likely never will.

It i generally believed that he was 0 Greek.

But perhap tbo glori lied pric ts of Egypt or

lh renowned chaldenl1 magi are tbe true fath­

rs of science. However all that may be, geomotry arrived

in Greece in tbe seventb century before tbe

Christian era. It wos tbere and then that the Greeks, admir-

ers ot cold logic and the cxqul ite elegance of

pore intellect, lovingly polisbed to a brilliance

(or perhaps originated) one of the most beauti­

ful creations of hum6Jl tbought-geometry.

Elegance indeed, yet actually tbe matter was

far more involved and intriguing. One thing is

certain, and that is tbat geometry sprnlll: from

practical needs. Tb.e development of logic (nod consoquenLly

geometry as woll) 'l:as.iJlOuonced 1..0 �om utenl

by tile. Greeks' dovo.Lion to law and llratQry,

But in Egypt, too, geometry was important t

1

men of the practical world-very important. And as for endle s litigations and court proceedings the Greek.. were far behind the country of th� pharaobs.

In .a word then, a serious analysis of this queshon would take us 100 far astray· let us �e sati�6ed with. th� fact. Geometry ha's estab­hsbed �tself. Thl� IS the start of a gripping, dramatiC conlest III pure logic that bas conti­nued for two and a half thousand years

. The history of the fifth postulate goe� back ius.t ?bout a� many years. It is as dramat.ic as It IS Instruchve, a detective story with an Unex­pected but happy ending.

Now for the story . Geometry, we beliove, hegan with the Ionic

school. To be more precise its founder was �hales of Miletu! wbo was

' believod to havo

lived close to a hundred years (eilher from 640 to 540 or 640 to 54.6 B.C.).

We don't seem to know very mucb about him. We know for sure he had tbe tiUe of One of

�ho Seven Sages of Greece; we also know that In accordance with tbe established rockoning he ,,:as the first philosopher, the lirst mathemati­Cian, . tbe /irst. astronomer and, generally, the first In all sClOnees In Greece. We migbt say that he was to tho Greeks wbat Lomonosov was to tbe Russiaos: THE FIRST.

As a yOUDg man he most likely made his ",

:ay to Egypt On affairs of trade, for be hegan

his caree; as a m�rcha�t. Here, the pbaraoh Psammehchus had Just lifted the "iron curtain" and was beginning to allow foreigners into his country.

8

Ddt! remained in Egypt for a good number

_ studying in Tbebes and Memphis. La-

U re nmed to Greece and founded a school

·J05Ophy. Obviously, be appeared more as

•• c�u,:��'tr of Egyptian wisdom than as an

a thinker. view is tbat bo brought with bim geo­

and a tronomy.

_ t any rate, there is one thing that all philo­

-::�: can learn from him-and tbat is con­

c· . Legend has it that hi completo works

.Illch naturally were all lost) consisted of only

about 200 poems. \ e c.on only conjecture what he accomplished

in eometry, though Greek aUlhors altributed a

at deal to him. For instance, Proclus Diadochus (we will bo

ling him ag io) claims that it was Tha-

-no other-tbat proved tho theorems

t: (a) vertical angles aro equal;

(b) the angles at the base of all isosceles tri­

angle arc equal; (c) a diameter divides a circle in balf . . ..

And some others. Assuming oven tbat the hi lorians of science

wrote the exact trutb, we still do not know wbe­

ther Thalos himself arrived at the e theorems

or simply repeated ideas of tho Egyplians.

Perbaps th ouly definitely estahlished fact

of the scholarship of Thales of Miletus is his

prediclion of the solar eclipse of 585 B.C.

But legends grew up around him in basts.

And this in itself iudicatea tbat 110 was a /!Cho­

lar of stature.

9

One of �be stories is particularly dear to men of learning. It is Aristotle who relat""s it:

"When Thales was reproached Cor his poverty, since, as they said, stndies in philosophy do not create any profit, it is said that ThaJes, foreseeing a rich harvest of olives on the hasis of astronomical findings, advanced during win­ter a small sum of money he had accumulated to tho owners of all the oil-presses in Miletus and on the island oC Chios. He waS able to eo­gage the oil-presses cheaply, for there was no competition from anywhere. When harvest time arrived, there was a sudden demand by many people for the oil-presses. Thales then rented out the oil-presses at prices that he himself desired.

''Thales thus accumulated a great deal of mo­ney and proved in this manner that it is not difficul� for philosophers too to becomo rich, the only thing is, however, tha� lhat is not �be subject of their interest.s."

We do not know what Thales did with tho money he made in this snccessful practical ap­plication of astronomy. We hope he spent it as a true philosopher would.

His pupiJs and followers apparently paid pro­per attention to geometr'y in their philosophic�l deliberations. However, the C<lntrai mathematI­cal school of the 6th and 5th C<ln�uries B.C. was the Pythagorean school.

The authentic biographical iniormulion about Py�hagoras boils down, in ".sence, to a {Q\� s�ories. In tbis respect, he is mucb like Thales of Miletus. The obscurities begin wilh his origin.

R�rtrand Russel sums the maUer up by Slly-

10

_ me believe him to he �he SOn of II ".Js.Ir.y ciUten named Mnesarch, others the 500

god Apollo, and adds that the reader his pick.

further believed that Pythagoras lived long a life as Thales-something in tho 'wio<iJail� of one hundred years (perhaps 569 to

.\gain Tbales, he spent some twenty years Egypt imbibillg wisdom, but later (here he

_rp."'-!ed Thales) ho lived about ten years in ylonia adding still more to hIs store of

ledge. It is also claimed tha� he travelled India, but nobody seems to believe it.

Boxers claim tbat Pythagoras took boxing laurels in the Olympic games, but the source of ueh claims is nover indicated. I have ooth-

10 support them either. M in the case of al , the exciting thing is �he unexpected mbination of philosopher, mathematician and

boxer. Pythagoras may not have done much in box­

ing, but in politics he did, and very actively, though not at all successfully.

The cititeos of the Sicilian tOWD 01 Crotona, where he fouoded his school after his wande­rings in distant lands and also go� the town involved in a.n exhausting war, finally asked him to leave together with his school. Which he did in rather much of a hurry, which was a reasonable thing to do.

As a mathematician and scholar he was a giant, bu� nevertheless he does no� call forth great admiraUon. His Pythagorean order of phi­losopbers and mathematicians is In uch too J'O-

1 1

mioiscent of a barracks and Pythagoras bimself suspiciously resembles a fUhrer, tbough much mOre cultured than any of those of the twen­tiet,h century.

It is precisely Pythagoras himself-most likely in a campaign to build up bis authority-who built up and popularized the idea that his lov­ing father was the fair-haired eITulgent Apollo. Actually he became the true rather of tbe pre­seotly popular custom of attributing to himself the scientiOc results of his pupils. Thero, the matLer was quite official. There existed a liat according to which the author of all the mathe­matical studies of tbe school was to be named Pytbagoras.

Though one might ropeat that such things Me done right aod left loday, the pass ago of 25 centuries has greatly soltened and civilized tbo customs. The eSSence is the same, but tbo form bas become enobled.

Pythagoras is tbe unsurpassed leader bere be­cause be handled matters so tbat his faithful pupils claimed him author of work done long after his death. Quite understandably then-that being the state of affairs in the Pythagorean school-that the mo t cogent of all argumonts was a simple reference to The Authority Him­sell.

Tbat is exactly how the wording went: "He said so Himscll". After which any discussion was totally out of place-evon dangerous.

He and his dear pupils also held in secret their methods for solving mathematical prob­lems. Too, ho compiled lor the members of his order a long list of taboO!!.

1 2

qlIole Irom the rules of good manners of gentlemen of the �yt,hagoreS? Club:

&1. Restrain from USlDg beans ID your food. ·1. Do not pick up what has fallen.

3. Do not touch white roosters. -4. Do not take a bite from a whole bread. a;;. Do not walk on a highway. M6. When removing a pot from. the lire, d?,

BOt leave traces in the a hes, but ID'X the asbes. The list could bo extended. IL was this bun�h

that rO c to I)ower in one Greek town, then "' another, implanting the cult ol

,Pythag�ras ao�,

accordingly, demanding compliance WIth their tatutes. With melancholy, Bertrand Russel re­

late tbat those wbo were not reborn in the new laith thirsted for beans and so sooner or later ro e up in arms. .

It is also told that he preached to the arumals, for he made little distinction between them and human beings.

But the Pythagorean scbool advanced geo­metry and mathematics in general. Very much sO in fact. All of this taken togetber is not a b;d illustration of the danger 01 idealizing rep­resentatives of the exact sciences and of the intellect generally.

Incidentally, to us, Pythagoras is . mainly a mathematician. Yet he himself and h,s contem­poraries look the view that his profession w�s that of a prophet. That was of course thelf business, they were closer to ev�nts. But, �s we know, every prophet must be ID part mag'-cian, demagogue and charlatan.

, Pythagoras was apparently past master '? each

lield. The pupils tried hard too. AccordlDg to

\3

One storr, one of his hips was 01 gold, to another tbat rehable people saw him at two different places at the same time, to a thlrd that when he was wading across a stream, the water over­flowed the banks crying "Long Live Pythago­ra J" True, the Greeks had a goodly number of reasonable people. ?Ce�ophalles, the well-travelled , realistic, free­thl.n.kmg and malicious-tongued philosopher and writer,

. spoke of Pythagoras in a rathor differ­ent vew: On? of his epigrams went: Pytha­goras witnessing a I)UPPY being beaten said· "!?o not hit him, it is the soul of a f riend or mIne. I reco�nized it when I heard it cry out." The teachlllg of transmigration of souls is one

.of the basic elemenls in tho overall con­coptlon of Pythagoras, and X enopbanes , 8S the reader can see, had a pointed thing Or two to say On that score.

Horacli��s wa� very trict in his portrait of Py_ thagoras, m ultlJlle knowledge without reaSOn ".

1 4

leave Pythagoras, but before do�ng so, ne more curious story by one of hiS hOD­admirers. How devious indeed are tbe path­of science. Quite naturally, geometry, like -

branches of knowledge, was most carefully ealed from the com moo people by tbe Py­

reaos. Who knows, perbaps to tbis day no would know of geometry (outside the Py­

gereaos) if it weran't for .... But here is lhe legond as to how the Pythago­

... ans account for the spread of geometry. One of them is to blamc, for he lost tho money of the community. After that calamity, the com­munity permitted him to carn the money by teaching geometry, and geometry waS given tho name "the legend of Pythagoras".

A curious thing is that there seems to have been a geometry textbook by that name.

As to the story itsolf, if there is a grain of truth in it at all, then, though I do not consider mysell a malicious person, 1 would bo pI ased to learn tbat the truant Pythagorean had not lost the moncy alter all but had spcnt it in a spree in the local port tavcrn s\�i�ling wino, eating a while roostor With boans, b ltll lg a whole roll of whi te hread and si nging drunken ongs on the high way.

Another man contributed greatly to geometry, and again to my taste he was an unpleasant character.

Ris namo was Plato (428 to 348 B.C.). I n his views, in his methods of setti�g up a

school and in his love of self-advertisement, Plato 'much resembles Pythagoras. But befo�e I S8Y why I do not like him, let me elplaJD

15

t .IUs most significant contribution to geo­_uy is. He is considered-and perhaps justly so, for I am not a specialist ill the field-one of the greatest philosophers of Greece. Indeed he did a great deal for the development of mathema­tic and valued it highly. At the entrance to his Academy he had, hewn in stone, the inscrip­tion: "Let no one destitute of geometry enter my doorsl" The point is that Plato helieved that "the study of mathematics hrings us closer to the immortal gods", and educated his pupils in this spirit, adding mathematic where it Was needed and where it wasn't. Some of his pupils became brilliant geometers. Plato bad numerous pupils and they naturally spread numorollS stories praiSing the teacber.

It was apparently Plato who first made t,he explicit demand that mathematics generally and geometry in particular b constructed in deduc­tive fashion. To put it differently, all the pro­positions (theorems) must be rigorously logi­cally deduced from a small number of basic statements callod axioms. This was a momentous step forward. By tho time Plato arrived on the scone, geo­melry had developed extonsi voly. A multitude of extromely complicated pro­blems had been solved and highly involved theo­rems proved. What was apparently lacking was a clear-cut general scheme of construction. As is froquontly tbe case in science , tho developmcnt of geometry was spurred tremendously by three problems that adamantly refused to suc­cumb.

16

:::ince we have gone this fur, I will staLo tbe

It It'ms:

reqUired, witl, lhe aid .of compass a�d

.M-edge alone (no olher lostruments -

I, �i'��de a given anglo into three equal

(trisecting the anglo); h :!) construct a s!Juare of afea eq�al to t e -

of a given circle (squaring the �ltcle); f (3) construct a cube of volume tWice t��t 0_

a given cube (duplicating the cube, the Del phian proIJlem "). .

til H Was only at tho end of the Ulneteen cen­tury that it was proved that, thus ]losed

l, n�1

of tbo pl'ohlems is solvable, thoug 1 . a I ��:e() aro readily rosolvable if othor gcomft"I C�

e instruments are omployed. Thoy .call a so . h�ndled by utilizing arcs of a ell'cle or lOCI dilferent from a st,raight Iille. .

But the Greek rules only permitted compass and straight-edge. . .

t b Plato even substantiated Ihls reqw: eme� th� some sort of reference 10 the authon ty 0 gO�hat is why not one of the problems �rs

solved, but in tbo elfort geomotry was grea y

2-1987 1 1

expanded. Too bad we have no place or time for the numerous exci ting stories that go alollg with these problems. But we will recall a legend to show that we are objective in onr attitnde towards Plato. Qne of tbe versions of this story makes him out a very reasonable man.

Eratosthenes relates tbat once, on tho island 01 Delos an epidemic 01 plague broke out. The inhabitant.s of the island naturally turned to the Delpbian oracle who orderod to duplicate tho volume of the golden cubical sacrificial repository to Apollo withont altering its shape. Plato was asked to advise.

He did not resolve the problem but interpre­ted the oracle as meaning to say that the gods were angry with the Greeks for tho endless internecine wa.·s and desired that tho Greeks should give up warfare and engage in the scion­ces, particularly geometry. The plague would then vanish.

Legends or no legends, Plato as philosophc,' and man is in my opinion extremely unpleasant. It is not even the fact that he was supporter of tho 1Il0st rabid idealism and on every occa­sion appealed to the gods. What is worse, he built up a thco,'y 01 the state taking as his mo­del nearby Sparta-a real haven of fa <lism. Too, tho bnsic planks 01 his utopia fully con­form to the demands of nazism.

He spent his whole life fighting tooth aud nail against democl'acy in political life and against materialism io spiritual life.

He not onJy scourged the materialist-thinking philosophers abstractly io his philosophical writ­ings, but, demonstrati og a very practical ap-

1 8

'::! to matters, olten employed political de­li! liOIl-8 beloved weapon io all ages-to

!cientific opponents. is eVOD a story that be bought up the

of hi bitterest enemy Democritus so as oy them.

Dulocritus is a special topic of discussion. U one agrees that the source of our modern

'cs is to be sought among the Greeks (and is most likely the case), then the distaoce

e ered is great ltldeed-something like two tho-d years. From Aristotle to ewton. The

ioMIr primal elements of Aristotle-air, water, h and fire-marked ooe of the first attempts

define the concept of the "elementary partic­D of physics, True, the Greeks did not know physics in

&he modern sense of the word, At the heart .f matters wore speculative argumonts, oot ex­perimont. But this is not so important to us BOW.

Perhaps it is the almost total absence of ex­periment tbaL brings out the utteriy amazing conjecture of tbe sly philosopher Democritus of Abdera.

Roughly half a century before Aristotle, he believed that all substances consisted of minute indivisible particlos-atoms-and that tho diffo­rent properties of ubstances were determined by lhe dWerent qualities of the atoms themselves. In a given substanco, however, all atoms were identical and devoid of any individuality,

These views are so closo in spirit to modern conceptions that one of the founders of quan­tum mechanics, Erwin Schrodinger, took great

,. 1 9

pleasure in startling his listeners with tho ele­gant parado : ''1'he first quantum physicist was not Max Planck bllt Domoeritus of Abdora."

Most Iikoly, Demoeritus would hav been most amazed to hoar this flattering comment, yet one must agree that Scllriidinger surely has certain rights when it com to discussing quantum theory.

The fate of Demoeritus ' viow is remarkable in yet another two ways. FirsLly, not a single one of his writings has come down to us. Ei­ther Plato indeed succeeded in his neat IitLio methods of sciontific discussion, or simply tho books woro lost througb tbe age; at any rate. to our misfortune, the idea of one o[ tbo fir t materialists in the world can be judged only on tho basis of extract and later retellings.

econdly, the first Ilopillar-scionce treatise (and Ilopularizers 01 cience should never forget this) waS devoted to a di cussion o[ his ideas.

What is more, the book in question set a world record, lor the poem is of extreme length. I am of cour�o alluding to tho poom 0" the Nature 0/ Things (De rerulII natura) by Titus Lucretius CartlS, which was wrilten somo threo hundred years after tho death o[ DemocrHus­two thousand years ago.

By tho way, J)cmoeritus had it ralher good neverthel ,because traces of many otber scho­lars (particularly among tho materialists) have been I t completely. For in tance, thero is still great doubt about whother DemocriLus' teacher Leucippu over livod. Then o[ course iL is en­tirely conjecture whether Leucippus was eo-au­thor Or author of the ideas of atomism.

20

we have the version that the teaching oeriLus was borrowed from some ehaldean

. granted to hi father by the Perslan king Xaxes

And if we may permit ourselves a bit of mora­. ,it is worth noting that in science ideas incomllarably longer-lived than the memory

.. tb who ongender tbem. Incidontally most nti t in any brauch of knowledge can

' grasp

.. t anything oxcept this not-too-unexpected u-

But whoever was tho founder of tho atomi tic eory, and wbeth r quantum mecbanics has its orce ill DemocriLu or tbo cbaldeans the views

f the atomistic school aro roughly �s [ollows. The world consist of atoms and void. The

atoms are unitary and indivisible. Thoy are elementary and qualitatively invariablo. Atoms do nol succumb to aoy kind of out ide actiOn �hatseeve�. they �re not generated and they are lndestructlble. r .. mordial distinctions exist bet­.....een atoms, and theso di tinctions dotermine the variety of proportie of all things.

21.

What we today regard as elemontary particles represents entities that are far removed, as to properties, from the atoms of DemocrHus.

They appear and disappear, they eon vert from one into another, and they are readily acted upon-in a word, wo must say that the Greeks were much more logical iu their concopt of an elementary particle than are the physicists of the twentieth century.

There is a reUable statement made by Archi­medes which strongly suggests that Democritus was a marvellous geometer. It would soom that bo was the One who computed tho volume of a cOne (IUd a pyramid. That was a brilllant achieve­ment, but unlortunately Dot m(lny (letHi Is are known. Be that as it may, of the forerunners of tho integral calculus tho first was apparently Demoeri tus.

Another complieating eircumstance is tbo fact that practically our only ourco is a book by Proelus Diadochus. Sinco Proelus was a follow­er of Plato, ho hardly made any mention of hls scholarly opponent.

Quite naturally, Domocritus was enemy num­ber one and was first to bo banished from his­tory.

The picturo is praetically tho ame wi th re­gard to Anaxagoras. We know hardly anything about the geometrical studies of that remarkablo philosopber who was one of the first materia­lists. The only thing We do know is that in the dungeon whoro he ro idod because of bis views, he investigated the problem of the squa­ring of the circle. His prulosophical views do­finitely merit a good word.

21

lacidentally, this was best done by Plato. One of hls works we lind a dialogue be\.woon

_ thenian (Plato himself) and a Spartan. This ow Plato handles Anaxagoraa.

Athenian: "When we seek to obtain proof of existence of the gods and refer to the sun, mOOn and the stars and the earth as divine

cnatures, the pupils of those new sages object t all tbese things aro only tho ground and

e stones and thoy (th.e stonos, that is) are quite unable to take eare of the affairs of human

ings." Obvious, then, that Anaxagoras and rus pu­

pils urc simply a product of murky Tartarus. The Spartan straightway perceives the heresy

and cries out with indignation: "What awful harm is this for the family and for the state Ihat flows from sucli attitudes of the yoang people! "

That is Plato and discussion. T would be very pleased if his contribations

to tho development of geometry tarned out to be greatly exaggerated.

But as things stand today we must admit that bis school bronght forth a galaxy 01 brlli liant matbematicians, and hls is the first men­tion of the axiomatic method.

To summarize then, in the fourth and thlrd centuries B.C. geometry was a fully developed science. With traditions, with fully elaborated methods of solving problems, mighty achievo­mcnts and even a number of textbooks and schools of t.hought.

Thero is no need or place to go into tho story of all the geometers of the pro-Euclidean period.

23

Sulfiee it to give a list of the ma�hematical giants of that period that preceded Euclid­Thales of r,liletus, Aouimander, Amerist, Mand­riat, Eonipidus, Aoaximedes, Democritus, Ana­xagoras, Pythagoras, Hippias, Archytas, Hippo­crates of Chi os (no relation to the physician), Antiphon, Plato, TheaeteLus, Eudoxus of Cnl­dus (these last two are Lowering figures, especi­ally Eudoxus, who lived between 400 and 337 B.C. and is believed to have also been an astro­nomer, physician, orator, philosopher and geog­rapher).

Menaechmus, Leodatus, Deinostratus, Arista­eus, Eudemus, Theophrastus, Theudius and yet another couple of dozen names.

And also Aristotle. Aristotle, beyond any doubt, is ono of the

greatest minds in the history of humankind. True, when balanced, the harm done by his

works almost tips the scales against tho good. Aristotle is hardly at all to blame for this, bnt in the Middle Ages, his works, pared down and purified to the point where they could nO longer engender fresh thinking, became tbe prin­cipal"weapon of reaction.

But an appraisal of his works is a whole history in itself. The only thing tbat need bo said here is that he was definitely and deeply Interested in geometry. Note tbat be paid spe­cial attention to the theory of parallel lines.

What is moro, be contributed two extremely important propositions to this field. True, they do not appear in the works that havo como down to us, but all succeeding mathematicians unani­mously attribute these statements to Aris_tolle.

24

-_ping out ahead of our story, we may

. that tbe cleverest proofs of Euclid '5 fifth

nulate are based on Aristotle's "principle".

..w come back later to what Aristotle said

t the properties of parallel lines. In tbe

�time ....

,.

Clroplu 2 EUCLID

Enough about forerunners let us begin the count with Euclid. '

. He Ii ved and worked in a ti me that is curious In the oxtreme. III tho year 323 B.C., as a result of an acute fover Or of immoderate drinking, or simply due to a '!oodly portion of poison, the king of mOr­tal �llIgs, Alexander of Macedonia, though a I'elatlvely young man of thirty-three years yet ,,"-orn and weary, departed for a meeting with hIs father Zeus.

The demigod was hastely disposed of, Cor aHairs of stal? .demanded attention. The empire had to be divided, and this was no ordinary empire Within a mattor of ten years, lands had bee� conquered that exceeded the tiny poverty-strick­en Macedoma by hundreds of times. . How and why this came abont is not of great Interest to us here. There were many reasons. One of which, as it will be recalled, was that Aloxander of Macedonia was a hero. . Bo that as it may, tho world had changed I? ten years. Its boundaries had expanded four times �ver, and !lOW came the time to digest the spOIls .. Ono tblng was cloar, it was too much for one hell'. And it would ha vo been ridiculous to have givon it all to the infant SOD of Alexan­der who was born several months aCter the fath-

26

.' death, or to tho second heir, Alexander's ile step-brother, And 50 It came about

l the empire was ripped to pieces by those oved geoerals that Alexander had not had

to execute. They concluded an eternal pellCo,

. pled�e� lust

eternal a friendship, drank h avlly reJoICI?g, clasped each other's hand in a firm .mascuhne

'e and went their ways . .. to hogm slaugh-&ering and lighting among themselvos. . Tho Limes were exciting. Kings grow up like anshrooms alter a rain und wore wiped out just as quickly, The lawful heirs, with n� m?re guilt than thoir Ol'igil�, were by the begtnntng of tho second decade olther cut down or strang­led. The dynastic wars and slaughter continued for a few more decades.

That was bow the extremoly intel'osting era of Hellenism got under way.

In this fracas, luck was on lhe side of the circumspect Ptolemy, who sliced off Egypt as his share,

He rather successfully inlel'fered in the quar­rels of the Diadochi (hoirs) , more or less reason­ably held in rein (within the confines of Eg:ypt) his desperate Macedonians whose sword-polO Is kept bim in power. He did not opposo lhe wor­ship of black cats and crocodiles so dear to the hearts of the local scholars, and he himself became a god in accord with the position he hald (after all, he was a pharaoh). He plUJldered the country, and bo plundered effiCiently. True, nothing could roally surpl'ise the EgyptIans. Ho encouraged trade, killed orr-on .n small sc"!e­those dissatislied wilh the way tlung s were gOlDg,

27

and he pampered tho bureaucratic a I �e got to liko tho banks of tho Nile� ������:iiy o towrn

Alto Whl�h Alal(ander gave tho novel uame 0 exandna. His heirs gradually settled down and the d nasty took firm root. What is hlore it av

Y­!�e ";.orld Crleopatra , alld literature had a

go e�c�� �� °PIC or two thousand yean!

t Iho very first Ptolemy, called' Ptolemy So­er, and .11 succeeding Ptolemys stand out P"

htrons Of. Ibe sciences. I t is hard I; say tOda

ay

. w Ilt motl\'OS lay b b' d I' tbat Ib Pt I C ID. t lis sudden intorest e . 0 emys took 111 tho sciences. I t

PerhhPs; t Was a .kJDd of intellectual coquetry. mIg t a that In attract ing mathematic' and philosophors, Ptolemy r was aping Al lans

der-art II AI < exan­d h

er t, oxander was a pupil of Aristotlo an 0 va ued men of learning (Iruo his Iiki look Very peculiar forlllS) 'J'h r' nil'

• . en 0 course it rna}.

even be assumed that thero wa 110 e of puttlOg .Ihe wiseman 10 some killd of pragUcal nse. ThIS was rathol' doubtful tho h LI el

r us put aside guc work andg

n�te facts

on y acts. '

28

10 the third and econd cenluries B.C., Ale­Ddria had become the pl'inciral ceotr� or learn-

of the Hellenistic world. And tho most ..gnificen! institution of learning was the cel­Hrated M u urn of Alexandria ,,�th it." famous library. Unfortunately, it was plundered many times, and to complete matters, all 70,000 scrolls perished in a fire started i n the so\'enth century by some furiou8 Ara bian caur.

I ncidentally, it seems that the calif \Vas really Dot sO much to blame. The first one to hllve a band in it was the great Caesar-Gains lulius Caesar, a fairly decent writer of prose and al 0 and mainly a general and political demagogue with boundless am bition.

Too, there are eX tremely weighty reasons to belinve that in the main the work was t.hat of the early Christian church (at that time, extre­moly tolerant of other faiths), which got out ahead of the simpleton calif by about two bund­red or sO year . All the calif bad to do was clear away the remain . However that may be, the very hest work of the Ptolemys awaited an unplea ant fate.

At any rale, if we are to remember tho Plole­mys for what good lhey did, it is for their patronage of learning.

Human history has known many kingdoms and more kings. It may be that historians will trace t.he relationships of the doings of one and another salrap and subsequent evonts. But the living memory of tho people carries along a neg­Iigiblo percentage of all tb is crown-boaring horde. And what memory there is, is m o t often bad.

Those that sIand out most-lheir luck-are

29 1

cut-throats and adventutists like TamerlaDe or apoleoo. But tho role thoy play today in Our lifo is

practically nil. Since I bave delved ralher deep in these an­

cient variations on the topic of the frail ty of earthly kingdoms and their glory let me COn-clude with a parable.

'

Some few decades prior to the invasion of Mexico by tho Spanish, a certain Aztec leader wi lh � totaily

. unpronounceable name (let us

call hlln X) uUlted all the tribes into a king­dnm, thus to some exlont eliminat.ing lhe feu­dal fragmentation of tho land. It Was naturally thought that tho kingdOID and bis dynasty would last for long centuries. X hinlsell rulud long and happily.

But Moxico was SOon visited by the gangsters of C<Jrtez, and all that was left of the Aztec empire was the ruillS of what were onco magni­fice�t cities. But .lbat is only half tho story.

KlDg X (the caslka, to be pl'eeise) quito natu­rally had a harem, for King X adored lhe fe­male sox.

He ,�as indeed an extraordinary man, a lalen­ted Il�"lcal poet. Most naturally, he wrOle pootry for his nu�ero lls wives in between allcniling to

. the a(falrs of state. It is his ongs that can

stIJJ-today-be heard in Lho viUages of Mexico. We may rejoice onco again that genuine works of �rt ru'O always more lasting than aoy empi re.

I t is probably worth recalling the name of tho poet, but ala I only romember lhat it is very long and hard 10 pronounce.

30

Ptolemy the First, Soter, invited Euclid Alexandria. There Euclid wrote the Elements,

'que book unparalleled in human history. .\pin r have \0 ad mit that practically noth­

II finite is known ahout Euclid lhe man. � course, a couple 01 apocrypha we have.

Il i3 said that at lirst Ptolemy hiDlSel[ wanled .aster the intricaeie of geometry. But he

found that the Sludy of matbematics was onerous a burd II for a pbaraoh . Then he

"fit d Euclid and asked him (oh, surely, as gentleman would anothor): "Is there nob

me easier way of grasping all the secrets of Inrnillg?" To which Euclid, the slory goes, r p­lied proudly and 1I0t sa politely : "There is no royal road to geometry." We do not know wheth­er Ptolemy continued studying geometry. Most likely he found comfort in the busine s more

nitahle to kings (receptions, hunting, drinking

and his harem). The other story is that Euclid was approached

by one youog pragmatist, who asked: "What is the practical use of studying 1110 Elements?" Whereupon Euclid, toucbed to tho quick, called a slave and said: "Give him threepence, since he JOust make gain of what lie learns".

True, one is not inclined to believe either

story if one bears in mind the vi w tbat Lhe

Greeks took of wi emen and mathematics. The first story may be very pleasont to the

modern ear, but tbe second is rather objectio­

nable. One has to take into coo ideration that

the Greeks thought ... well, all the otbe,· hand,

I guo s I just dOD't know what the Greeks re­

ally thought. Some, 1 'm afraid, thought one

31

way, and othel's another. We know (that is, we t hi n k wo know) that they despised nil IlI'acti­cal application of mat homatics. And i t would indeed 80em that the philosophical works of those age (particularly among the followers of Plato) corroborate as much. Repeatedly, in fact. That may be. Aud i t may be true. But tho grea­test genuis 01 matbematics of antiquity-Archi­medes-was a phy.ici t, a n experimenter and not a theoretician. There is more. He was also a urst-class military engi neer who spent many years and much energy in building an i mpreg­nable fortress out of his home town 01 Syracuse.

01 course, pl utarcb, taking upon himself to justify Archi medes, explained shamefacedly that all these thi ngs were tOys, intellectual bau­bles of the phi losophor. One however docs not need to be perspicacious to realize that to plan the delence of your town oquipped - wbat is more-with weapons of your own invention is more than imple recreation. 1 relleat, an ab 0-lutely-for t hose ti mes-impregnablo system of delence. Just One little aside: Archi modes and his work is a beautiful instance demonstrating tbat in th080 distant naive Li mes physics and otber sciences played just as important a role i n altair of war as they do today.

As to the actual attitude of tbo Hellenistic world to tha pract ical utilization of mathemati­cal knowledge, we are not sure.

Generally speaking, sweeping statements abollt that long-past epoch are always a bit irritating. We know 0 l ittle; far too Iragmentary and accidental are the facts of that past for US to speak definitely about the psyche and the cus-

32

• or �hose people. I fonr I am wa l k i ng on iJa ice myself taking np a discllssion of mat­

ith which I am not so very familiar. But returning to geometry and Euclid, 1 will

"I my elf just one remark, i t is so tempting. ra seem to be two extreme trends in ap­

'",Is of the ancients. Either the Gr oks (the Helleni tic world, i n

'cular) are idealized, and the protagonists this view lament hitterly the decline of mo­

over the past 2,500 years and the forever­_t days of tbe childhood of mankind when

�ple wero pure, nai vo and devoid of guile lhis is very popular among sophisticated intel­lectuals with a humani tal'inn slant}.

Or-to take the other ex L ,'olllo-one need only "",;tcb on the vacuum eleaner Or tolevision at to realize modern man's total moral supremacy over roprc entati ves of any earlior civilization. That often is the reasoni ng of technologists, the military and other exact professions (you wi l l excu e me fOr not including physicists i n this group).

As is so ofton the case, the d isciples of the oppo ing camps are, essentially, at 000 in their lack of any desire to i nvestigate tho matter eriously, and tbey rely almost completely on

haphazardly 3ma sed i mpresSions. There is i n addition a sceptical school of

thought whoso adherents claim that human be­ings have been tho same over Ihe ages and that man's intellect and moral qualities have not changed substantially duriog tbis measly period of only 2,500 years.

The anthor sides more with this latter view,

3- 111187 33

though j udging by hat he-the author-has read , bumankind bas, ovcr the pa t 2,500 years, beon improving slowly but surely. One would like the advance to be somewhat more acth'�. But that is a dif!(lJ'cnt question.

It is now probably lime to �xplain to the tired reader why a book devoted to geometry digre55Cs time and again into discu sions about everything bllt geometry.

I will do that and then we will return to Eu-clid.

This is in lieu of an introduction. Y 5, what follows is going to bo about non­

Euclidean geometry Ilnd about th general the­ory of relativity, th origination of wbich ma�, without stretching tho point very far, be consI­dered the logical culmination o[ the whole story of the fifth po tulate. . . ,

But what strikes me as most mtercsltn.g . In

this story, is nOI geometry or t�e relatiVIty theory. Ultimatel)', the entire ep�c about the lifth postulate is just as m�c� wltn to the power of human tbought a It IS to tbe remar­kable almo t fantastic narrow-mindedne of math�maticians. No wonder, incidentally, that fax Planck permitted himself tho perhaps ove­

rly categorical but, generally, correct stateme�t that "in comparison with the theor� of relatl­vi ty, the construction of . nO,n-Eucl\?ean geo­metry is nO moro than cblld s play . Let us, howover, not be too juhilant. The i mportant thing is somulhi ng olse.

The most important thing, tho most in t.'u­clive thing, and if you like,

.tho most touchl

.ng

thing is that this slory, whIch we nOw begIn,

34

..bolic. It is an i l Iustralion o[ one of the qualities that mark 01T human beings from

other primales and unite all races inlo a Ie species. The reader has guessed whnt tho or is about: he sings the praises of the endeav­

to find out what the world is like in which live, how our universe is constructed. And

.. finds that the internationalism of earthlings, the internationalism of epochs, countries and peoples will et roally sland against lhe just as

ternal coalition of narrow-mindedness, the broth­erhood of satraps, go-getters, conquerors, clim­bers, grabbers, and the worst portion of sports fans.

I f one could imagine for a momont the fan­tastic picture of Euclid, Omar Khayyam, Gauss, Lobachevsky and Einstein a l l in One room to­gether, it is hardly likely that ikolai Loba­chevsky would feel the need to seek out acquain­tances or, {or lack of a topic of conversation, to say, "bow about a couple of jokes . "

B u t o n the other hand, one h a s t o admit, albeit grudgingly, that the jokes of Euclid's time (with slight modi lications for local colour,

s· 35

of conrse) a l m ost fll l ly exhaust the spiritual arsena l of very many of OUI' cOlllcmporal"ies.

I ncidentally, i t is nOL worth idea l i zing eiL Io�r learning or the priests of learning. H undred upou hundreds of brilliant minds have t urned out to be quite amoral personages.

And perhaps one ot tho most attractive fea­tures ot this whole story is that just as non­Euclidean geometry l ogica lly culminated in the general tboory of relativity, so the galaxy of mathematicians-as a rule, not only remarkably talented but bumanly i nteresting people-onds with Einstein.

But let 115 return to Enclidl To begi n with, a few words- t.he stronger,

the botter-about all the beasts that liqUidated the A lexandrian l i brary. If it had not been des" troyed, we would now know scores of ti mes more about tho Greok and Roman worlds than We do.

We would probably know a bont Euclid as well. But, unhappily, as of today practically the most flmdamentai sonrce on Euclid is Pro­clUB Diadochus of Constantinople, a geometer who wrote an exceedingly detailed Commentary 00 the lirst book of tho E lements. Si tlce we are rMerring to sources, a slight remark will not bo amiss.

When we turn to tho history of antiquity, the effect is somewhat l ike that of regarding a chain of mountains from an aeroplane. E very­thing is smoothed over, distances contract, and small features vanish. Only the general overal l picture remains.

Involuntarily we look upon all Greek mathe­maticians as almost contemporaries. ote, then,

36

Proclus (412-4.85 A.D.) lived seven hundred after Euclid, a span 01 lime much greater I at which separates us from Ivan the

--�"'Lle. Quite obvious then that the facts at PndllS' disposal concerning the life of Euclid

fragmentary and haphazard . re i another author who Ii ved a few deca­

before Proclus. He was tho A lexandrian ematician Papp us. He wrote of Euclid des­aiilli.ng him as mild, modest and, at tho same

, independent. Both relate the incident with lemy. "Exact" b iographica l daLa are mostly

on the remarks 01 an unknown Arabian _thematician ot the tweHth century: "Enclid,

of Naucratus, tbo SOn of Zenarchus, known lhe namo of GeolneLo'·, a scholar of olden

"III , of Greek origiu , lived in Syria, born in yra . . . . JJ That is all. The man dissolved ill tho ages without a trace.

What remains is his work. We repeat, the E lem.enls is a book wi thout

parallel. For over two thousand years it \Va the prinCipal and practically solo manual 00

37

geometry lor scholars of boLh the Occidont ani tbe Orient. As late as tho end of the 19th cen­tury, many English schools taught geometry OD the basis of an adapted edi tion of Euclid's E lemen/so There can hardly be a more eloquen' witness to its popularity. In this sense, only the Bible can compete with tbe Elements 01 Geometry. But unlike the Bi ble, tbe Element. are a rigorous system of logic. To be mOre pre­cise, Euclid ever stlived towards such a system.

We can presuDle tbat Euclid was a fol lower of Plato and Aristotle. Plato, as you recal l , deman­ded a strictly deducti ve construction of matbema­tics. A t tho fOlmdation were axioms: tbe basic propositions that were accepted without proof; from thon on, everything had to follow with utmost rigour from these axioms.

Tbat was the ideal that Euclid attempted to accomplish. Attempted, because from the view­point of today l iterally his whole axiomatic" is unsa tisfactory.

But that is ell5Y to say now, after 25 centuries of investigations. I n its day, Euclid's logic left an overwhelming i mpression.

Attempts had been made before Euclid to describe geometry on the ba is of an axiomatic method. Not bad attempts, eitber. But we can assuredly say that Euclld's work was the most successful, as witness the unprecedented popu­larity of bis book already i n aneient limes-a popularity that brought the book down through the ages to us.

000 can say all kinds of harsh (a.ad truo) things about Euclid's axiomaLics. But one should ne­ver forget that the scheme i lsolf boca me, since

38

me, the canonical model for COllStrllct.ing branch of mathematics. And of course

I15t nevor forget that the E lernents prosent elIent piece of writing by a ski l led mast�r,

icacious scholar and a magni liccnt tea­That explains and justi fies the universal

.a:f:::!liJ .. lion of mathematicians for Euclid and IDMnts. Let 118 add tbat tbis book brought to

6eld of mathemat.ics scores of yolmg men who became the world 's greatest mathomaLi-

. Tho ellect of Euclid bll5 been amazing througb­

tbe ages and throughout the world. Take of tbe most prominent mathematicians of Renaissance, Cardauo, who, it mu t be add-

• was a rabid adventurist (not to say scOun­) but thore is no getting around bis mathemn­

talent and cul ture. Here is how he admired Enclid's Elements.

"The irrefutable strengtb of their dogmas and their perfection are so absolute that not a single

ork can justifiably be compared with tbem. • a consequence. thore is such a light of truth renected i n them that, llpparently, only he is capable of d istinguishing tbe true from tbe false in the intricate problems of geometry who has mastered Euclid . "

I n the middle o f tbo 19th century an outstan­ding geometer had this to say: "There bas never been a system of geometry, which, in its essen­tials, has differed from the plan of Euclid; and until I see such wi th my OWll oyes, T wil l not believe that such a system can exist. "

True, i t must be said that in the middle or the 19th century, that geometer could hn vo rea-

39

soned more progressi vely and tbese words, 3 ide from worship of Euclid, demonstrate tbo au­thor's own hidebound conservat ism.

We could go on citing numerous other writ­iugs in the same vein, but we will confine our­selves to what is probably the most brilliant demonstration of the err ct tbe Elemen ts had On literall y a l l fields of thought. Bonedict Spi noza, celebrated philo opher of the Western world, bor­rowed tbe entire plan of his basic work, Ethics, from Euclid.

Perbaps tbo autbori ty of Spinoza is not con­vincing enoug h to some readers. I f it isn't, let 100 ment ion Isaac ewton.

His fundamental work, tho Prillcipla (The Mathematical Prillclpies of Natural Philosophy) copi es Euclid botb i n title and out l in e . Axioms form the starting point from whicb all else follow . Tbe simi larity may bo conti n ued be­cause Newton 's axiomatics turned out to be j ust as epbemeral as did Euclid's.

Ono final piece of i nformation . By the year 1880, tbe Elemellts had appeared i n 460 edi­tions.

Perhaps a word is in order, at this point, about the axiomatic method itself.

It was only at the begi nning of this twentioth century that we achievod a perfectl y clear and rigorous understanding of ded uctive schemes. In the maio, morit for tbis goes to tbo great Ger­man mathematician Hilbert.

I n a rough and greatly Si mplified form, tbe matter stands as follow5. We confine ourselves i n what follows to the concrete material of geometry so as to avoid too many abstractions.

40

Stago 1. A List of the Basic Concepts

Foundat ion- Basic Concepts (basic clements). Tbese arc the r"sul L of a prolonged experimen­tal study of nature, a study bolh i ntricate and confused and nebulous and more.

Stem ming therefrom is a certain abstract re­flection of actualit.y, resulting in the Basic Con­cepts. NOlhing at aU is said of them in Ihe rudo­malic •. They como ready-made, as you might say.

Tbis is nalural enougb. To define tbe BaSic Concepts or notions, one needs otber, fresh nO­l ions, whicb i n turn with tbe aid of. . . and so on ad in finitllm. One has to start somewhere. As tbo French say, "in order to make a dish of rabbit stew, olle at leasl has to find a cat".

So we have the Basic NOlions. Mal homalicians havo a delightful way o f putling il : these are elementary entHies thaI are 1101 defined, they are si mply stated. A sligbt supplomont, by tbe way. I n tbo modern axiomatics of geometry, the Basic Concepls al'o djvided inlo Iwo groups:

(a) basic i mages; (b) basic relations.

4 1

Generally speaking, loday there are at Jeast two essenLially different axiomatic schemes. In what follows we wi ll use the scheme in which the Basic I m ages are as follows:

(1) point, (2) straight lioe, (3) plano. ow Jet us see what tbe Basic Rollltions are.

They arc formulated as: ( 1 ) to belong to, (2) to lie between, (3) moLion. The Hasic Concepts have been established.

We can now slart the second stage.

Stage 2. Basic A xioms.

For our Basic Concepts we make a set of a scrtions tbat arc accepted without any proof. Those are axioms. Speaking in strictly formal fashion, i t is only tho axioms that fill our Ba­sic Concepts wi lh li vi ng content. Only they i mpart life. Wilhout the axioms, the Basic Con­cepts are devoid of any con lent. They are noth­i ng. Amorphous ghosts. The adoms defin� the rules of the game for these "ghosts". They out­line a logical ordor. The malhematician call say only one thing about his Basic Concepts, that they obey such and sucb axioms. That and noth­ing elsel And all because tbe mathematician does not know what be is talking about. He demands only ono tbing: that his axioms bo satisOed.

That and nothing elsol Whon the axiomatic method has been elabo­

rated to perfection, geometry, speaking formally, is converted into an abstract game of logic.

Tho notions of point, straight lioo, plane, molion can mean anytbing, any entities.

42

Let us constroct a geometry for them. We will then call our goometry Euclidean geomet ry i f the axioms establislled [or the "I'ou l " geo­metry of Euclid are fullilled.

For example, one, and only one, straight line call be draw" through two distinct paints. This is nn axiom formulated in ord i na ry languago.

I f we were to ad horo trictly to tbe termino­logy just introd uced, we would have to make the statement:

only on. straight line ciln belong to two diffe­rent point •.

And so on i n the same spirit . On the basis of this axiom, a good exercise is to prove tbe theorem: ''Two straight lines have only one pOint in common.»

At the present ti me, live groups of axioms aro disti nguished i n Euclidean geometry. They oro:

(1) axioms of connection; (2) axioms of order; (3) axioms of motion; (4) the axiom of contionity; (5) the axiom of parallel Ii nes. There can hardly bo any use i n enlllDerating

ali these axioms, we wiII put them i n the appen­dix, for, 85 Herodotus Once said , nothing gi�es such weight and dignity to a book as an Appendlx.

We sball have occasion to return to the axioms a number of times. Meanwhile, we take up Stage 3.

St.�gc 3. Th. Basic DejinWons Enumerated.

With llle aid of the Basic Concepts wo con­struct moro complicated ones. For in laoce, an

43

angle is a figure formed by two hal/-lines (rays) ema/MUng from a single point.

A careful reading of this phrase will make it clear at once that one complox concept (na­mely, ray, Or halI-lino) is used in the definition o( an angle.

Obviously, we should have given the defini­tion of this notion earlier with the help of the Basic Concepts. This is rather easy to do. The reader cau check to see how much he is now i m bued with the spirit of deduction SlId armed with a list. of axioms, can try t� solv� the problem.

II it t urned out that i n employing the Basic Concopts, it was i m possiblo to define a ray, thon One would have to place t bis notion i n the category of Basic Concepts.

In general, all remaining notions and defini­tions aro introd uced wHh the aid of the Basic Concepts, and also (take note I) of those axioms which are estabU hed by us (or the Basic Cou­cepts.

There remains the last,

Stage 4. Statement 0/ Theorems. Pro% / The­orems.

With regard to our concepts (basic and non­basic) we express propositions, theorems, which wo prove.

That, properly speaking, fOl'ms the subject malter of geometry.

[ should like to repeat onco again that when stated in those terms, geometry is converted into an ahsolutely abstract game, like, say, chess.

44

'rhcre, too, we have Basic Concepts, called chessmen. The axioms a re t he collection of ru les o[ t he gamo. Finally, t be.'e are theorems. Actually, only One theorem: bow to checkmate the opponent.

In solving this "theorem ", a p layer proves dozens o[ lemmas (auxiliary theorems) i n the course of a game, each time selecting the best (in his opinion) move in a givon position.

Incidentally, there is a difference botwoeD games and g ometry. It consists ill the lact that the partners very often produce incorrect proof. In chess, (or example, no strict logical criteria for evaluating every movo or posi tion have yet been evolved. I n geometry they have. Here, i t i s always possible t o establish whether 8 newly formulated theorem r,ontradicts earlier theorems, and hence runs countor to still earlier ones, aud consequently . . . Unravelling the roll to the end, we arrive at two possi bilities: either we have erred in our reasoning, or the theorem just formulated is erroneous.

The formar possi bility is of little intarest to scionce: the only thing it shows is that we have handled the mathomatics poorly. "But i o the latter case there is o(teo a definite and very i m portant result. I f we have become convi nced that our !lypotbe is (theorem) is wroug, then other theorems are right, namely tbose which contradict our OWII. I f thero is only one sucb contradicting theorem, then wo have proven i t b y our reasoning.

This last paragraph , though perhaps rather nebulous and abstract ill form, is an explana­tion of a scheme tha� is very common in geo-

45

metry (and mathematics generally). I t goes by the name oC reductio ad absurd um, Or indirect proof.

Coming down to earth again, lel US lake a speci fic case to prove.

Let there be two perpendiculars dropped onto It straight l i oe. Using radian measure of angles

aod writing � in place of 90 degrees, wo lind

two, and only two, varianlS: either they meet al some point C or they do not intersect at all. Let uS prove that the second theorem is correct. We do so by the mothod of reductio ad ahsurd um.

Assume that tbo first supposition is ful filled and that the two porpe odicular l i nes intersect. Then wo have a triangle A BC (for trianglo we will uSe the symbol 6, Cor aogle the symbol L). The remarkable thing hore is that the exterior L B is equal . to lhe ioterior L A. And of course tbe exterior LA is equal to the i ntorior LB.

But there exists a t heorem (we will take it to be true): "An exterior angle of a triangle is always greater thnn any interior aogle not ad­jacen t to i t . "

Our triaogle docs not satisfy this theorem . Hence there can be no such triangle. Consequeot­Iy, wo a.re in error.

A check of the reasoning shows that everything is correct. I 1ence, the error was made at the very beginning, when i t was assumed tho perpendi. cular lines intersect.

Thus, perpendicular lines do not intersect. That has bcon proved rigorously. Euclid called nOniotersectiog lines parallel lines. For the time being wo too will II!!O tnis terminology.

46

e

To summarize, then, we have foulld lhat two straight lines perpendicnJnr to a common straight line are parallel. We shollld also prove that these straight lines do oot intersect in the lo­wer haH-plane either. But that would simply bo repeating the preceding proof, and our time is limited.

r n carrying Ollt the proof we relied 00 the theorem of the exterior angle of a triangle. The alert reader will of COII1'l39 see thal tho whole example is very i mportant for what is to fol­low, and so without any more digressions we will prove this theorem too. I t is oC ullimat.e importanco to us, and to the entiro story invol­ving the filth postulato.

True, the postulate i tself has not yet beeo formulated in any way, but tho whole story of the fifth postulate started with this very theorem.

Let thero he a 6 ABC. Lookl The exterior angle C"" is clearly indicated by tbe arc. We

47

�hall provo th.L it is greater than any interior angle not adjacent to i l; that i� lo say, WCHle/' lhan LA and g/'ea ler than L B, We slart wilh B,

Divide side BC by the point D into two equal parts and draw a straight line through A and D ,

O n this line, mark of! a segment D E equal to AD and conned points E and C by a straight line.

The triangles A BD and DEC aro congruent. Indeed, segments A D =DE and BD=DC as gi­von in the COli truction. The anglos CDE and A DB are equal because they are vertical anglos.

Hence, tho triangles are congruent on tho basis of a familiar critorion .

But then L B (or angle A BC) is equal to angle BCEI And notel Angle BCE is only a part of angle C,,,.

Thus, the entire angle C .. , is greater (natu­rally great.er, for the whole is always greater than One of its p arts) than angle B.

Some doubt remains about angle A. It i s immediately felt that our construction will not bo of any p articular help, since i n the figure angle A is cut into two parts. I t would be good to put it in the position of anglo B . Perhaps we should draw a straight line from the vertex

48

B and repeat our construction and proof. But then angle C .. , will be located othemise.

A complete analogy would result if we p ro­longed the side BC and regarded the new angle N.

Angle N is of course greater than angle A . Wo have just proved as much.

An inspirationl Angle N equals angle Cm because they are vertical angles.

That is all. An exterior angle of a triangle is greater than

any Interior angle not adjacent to it. We havo proven this and we can now cross out the doubts we had on page 46 about the validi ty of the theorem.

I f we go over the path traversed with excee­ding care . . . And if we check to see which axioms have been utilized in the proof of the theorem of the exterior angle. . . To do this we would of course have to verify the axioms that were used in proving tho theorems of the congruence of the triangles and the equality of vertical aogles.

Now if all that were done, we would find thaL we have utilized practically all of the axioms.

4-1M7 49

But nowhere have we taken advantage eilher

of the very nolion of nonintersecting (parallel)

straight lines, nor (all the more sol) of theo­

rems Or axioms concerning such straight lioes.

The reader can ea:rily veIify this by taking

the list of ax.ioms and analyzing all the Con­

cepts that are needed for the t,heorem of an ex­

terior angle aod for all auxiliary theorems.

Our detour has been too long and it is t ime

to return to the axioms.

First, let us figure out what logical require­

ments tbey m ust satisfy.

Only two: (1) completeness and (2) indepen­

dence. The first signi 6es tbat tbere must be a suf­

ficient number of axioms to prove or disprove

any possible a sertion concerning our pri mary

Basic Concepts or the more Complex Concepts

built up from tbem. The second implies that we did not take too

many axioms. We have just exactly the number

we need. And Dot a si ngle one of Lhe axioms cno

be proved or disproved with the aid of the

otbers.

50

. Both these demands may be formulated in a sIngle statement. The axioms must be necessary and sufficient.

Nocessity is a requirement of completeness. Sufficien7y is a requirement of independence. To put It very roughly, the requirements of

necessity and sufficiency signify t.hat there must be exactly the number of axioms as is needed neither more nor less.

'

Now [or one very important refinement. From the independence of the axioms there

foUows straightway their consistency. Indeed, if in our development of geometl'y we at SOme stage arri ve at a theorem that contradicts the rest, this "ill be a clear unpleasant indication that there is something wrong in the foundation. Namely, that one axiom (or several) contradicts the rest. And if there is an inconsistency that means they are not independent.

'

Actually, all these logical arguments are ex­tremely simple. But in a lirst reading they may appear rather Involved. My Suggestion is for the reader to go over them once more.

For tho present I would emphasize once again !hat the requirement of independence of axioms IS stronger and morc rigid than the requirement of consistency.

The axioms may be consistent but from- this consislency it does not yet c1ea�ly lollow

Athat

One of them might not be a corollary of the others. Perhaps it is a theorem. Naturally when a mat�ematician proposes a system of geo'metri­cal axlOIDS, be IS obliged to prove their indepen­dencel Let us stop our chain of rea oning at l itis paint. There will be time and opportunity to

c· 5 1

retllrn to them again. We will not miss the op­porttmity and will not lose time either, of that I am certain.

. Although everything that has iu;;t. been wrIt­ten is rather simple, and I am posItive the rea­der thinks sO tOO, Ellclid did not know any of it. Intuitivaly ha felt it all, �ut he could not formulate it in a clear-cut logical scheme.

Now a rigorous statemant of the �roblem . of the independence of axioms or tho flgorous 1 11-trod uction of tho Basic Concepts was generally beyond the ken not only of the Greeks . but of mathematicians in all ages and peoples fight up to the f9th century. .

Both the axiomatics and tha proofs provIded by Euclid are actually a rather varkol

.oured

mixture of intuition and logical lacunae-If ona regards them from tho standpoint of today.

Yet on the otber hand, E uclid advanced so far add so crucially along the road to rigorous logic that all othe� text�oo�s, and all other "elaments" current In antIquIty palad comple­tely when compared with the E lemen/so . When the Greeks spoke 01 Homer they sImply said the "poet", and when the Greeks recall�� Euclid they said the "maker of the E lemenls .

All �redecessors on the deductive pathway of geometric constructions were forgotten . .

There remained the Elements and their crea-tor Euclid. .

A lthough the thirteen books written by E uclid are believed to contain mainly the

. re�ults of

others, and lor this reason the questIOn IS often debated as to whether ho may be classed �s one of the greatest mathematicians, ho was WIthout

52

doubt a t<lacher of tho Iirst magnitude. We may also add that he was apparenLly an inspired and versaLiIe scholar, for in addition to the Ele­ments he also wrote E ieltUlnl$ of Music, Optics, Catoptrica, Data Phaellornena (a work on astro­nomy), Introductio harmonica; then also works that came down to us aod d isappeared: the Po­rlsms (in three books), Conics (in four books), Perspective (in two books) , Sur/ace-Loci, On di­visions and a Book 0/ Fallacies.

A very impressive list. Most of the books, it is true, make no original

contributions, but the output of work is tremen­dous. Incidentally, the Data was highly valued by Newton, which is a rather solid recommen­dation. Euclid apparently advanced substantial­ly the highly complex and exciting division of Greek geometry devoted to the teaching of co­nic sections. Howover, he did not include these results in the Elements, since there was a cnr­rent view that this branch was unworthy of "pure mathematics, whose aim is to bring man closer to god n.

53

It was again Plato who decided why precisely the theory of conic sections did not bring one closer to the divine. The point was that Plato viewed as heresy the use in geometry of any in· struments other than tho compass and the straight-edge, or-what is Iho same thing-the use of loci other than the circle and the straight line (which loci were needed in the study of conic sections). Plato passionately denounced the bril­liant geometrician Monaechmus (incidentally his friend), who demonslrated that the solution of the notorious problom of duplicating the cube, also thaI of trisecting an angle, is found rather simply if use is mado of new geometrical instru­ments.

PlalO maintained that all of that "spoils and destroys the good of geometry, for geometry thus strays away from incorporeal and mentally per­ceivable things and moves towards the sensorial ,

making use o f bodies that are needed i n the ap­plicatioo of instruments of vulgar handi­crafts".

Obviously this rebuke frightened poor Euclid, and his work on conic sections vaoished without a trace.

There would seem to be something in the E Ie­ment. dealing with regular solids (polyhedrons)

that belongs to him. I n the thirteonth book it is

proved that there exist only five differenl types

01 such solids. This is a brilliant, unexpected,

celebrated . . . classical result. Generally speaking, there is much in the Ek­

ments other than geometry. They contain certain

essentials of the theory of num bers and the geo­

metrical theory of irrational quaotities. The three

54

l�s� �oo� arc devoted to solid geometry. Every d i VISion IS preceded by axioms and postulates. �rope�ly speaking, plane geometry is ex­plained 10 the first six books, the very first he­ginning with axioms and post ulates.

Mathematical historians are still not in agree­ment as to how Euclid distingui hed between axioms and postulates.

Generally, to Euclid, axioms (which he calls "general attributes of our mind ") are truths that refer to a�y en�ities (oot only geometrical). For example, If A IS equal to C and B is equal to C, theo A is equal to B. Here, A and B may be nu mbers, segments of lines, weights of hodies tria ogles, etc.

'

Po.stlllat";!' On the �ther hand, are purely geo­metrical aXioms. For I nstance, Euclid's first pos­tulate: "A llllique straight linc can be drawn from any point to any other point. "

Euclid also has Basic Concepts (common no­tions).

There is hardly any reason to gi vo his entire system of axioms-we have said that a dozen ti mes if once-because it is quite unsatisfactory. There are, properly speaking six axioms in Euclid 's plane geometry, and �'e shall not men­tion them. But the postulates are worth noting. Here are tho first four.

I t must be req uired : I . That a straight line may be drawn between

any two points. I I . That any terminated straight line may be

produced indefinitely. I I I . That about any point as centre a circle

with any radius may bo described.

55

JV. Tbat all right angles be equal. For the timo being we shall not stro what is

bad in these postulates. As Nikolai Lobachevsky

once said, forgi ve Euclid and the Elements all

their "primitive shortcomings". The important

tbing for us at presont is that all (our postu­

lates are very elementary in content. Hero Eu­

clid postulated absolutely natural, comprehen­

sible truths that are part and parcel of our con­

sciousness and our intuition. All is well and

good, and . . . then we come to the fifth postnJate.

Chop!" 8

THE FIFTH POSTULATE

Tho fifth postulate reads: If two lines are cut by a transversal aM the

sum of the Interior angles on one side of lhe I,rans­versal Is less than a straighl angle (2d, or 180°), the two lines will meet if produced aM will meet on that side of the transversal.

That's a formulation for youl First of all, what a lot of words. Secondly. what a lot of geometrical concepts. A person poorly familiar with the fundamentals of geometry will find it hard to understand anything. The postulate dif­fers radically from all tho others. I t sounds mOre llke a theorem. And not a simple one either. There Is quite obviously something strange here. Before we go any farther, allow me to bow w Euclid.

Though I myself have no proof, naturally. J am convinced that the fifth postulate was pur­posely formulated i n this extremely undesirahle

57

form. Therein lies the great wisdom of the "r,I"C3-tor of the Elements".

O f all possible stalements of the fifth postu­late, Enclid chose the most i ntricate and cum­bersome one. 'Vhy? To answer, let uS see how he conslructs geometry.

After tbe axioms aod postulates, Euclid na­turally proves theorems. He proves 28 theorems straight 011 without once using tbe fifth postu­late. I t is ool nceded. All 28 are indifferent to tbe fifth postulate, lor, as they say, they rofer to absoluto geometry.

Among tho twenty-..ight there is also a theo­rem of the exterior angle of a triangle. In Eu­clid 's list it is No. 16. The list terminates with, as you can easily i magine, No. 21 aod No. 28. These theorems contain the so-called "direct the­ory" of parallel lioes. We shall prove them to­gether.

Let two straight l i nes be intersected by a third at points P and P ,.

It is asserted that il angle A equals angle A " the straight lines are parallel.

....... f-. � " " -'0 ' c ... ...... � '" ';e , "0 p .A • __ ... �- -. . . .• . -

~ 58

Working hy the reductio ad absurd um method, we first assume that the straight lines intersect at poin t C. Then we get a triangle P P, C, whose exterior angle A , is equal to the interior angle A not adjacent to it. But this is impossible. The theorem "An exterior angle of a triangle is always greater than any i nterior angle not ad­jacent to W' does not allow this to occurl I lence, tbo straight lines cannot i ntersect when produced LO tho right.

There is a second possi bility. The straight li­nes i ntorsect at point C ,. Then we get the trian· gle PP,C, for which angle B is an exterior ang­le and B I is an interior angle not adjacent to B.

But L B = L A ; LB,=LA " they being ver­tical angles.

But LA = LA , (by hypothesis); benco, LB= = LB,.

Actually that completes the proof. For the hypothetical triangle PP ,C l' angle B

is an exterior angle and B , is an interior one not adjacent to it. And tboy are equal. Which is i mpossible. Consequently, the triangle PP,C, cannot exist. Henco, tbo straight lines do not intersect i n point C; eitber.

That completes the proof of the theorem . It is obvious to tho roader that B and B 1

were introd uced so that for the hypothetical trian­gle PP,C, we could completely duplicate the situation that im mediately arose for triangle P PIC (tho lirst triangle).

Now, so as to completely repeat Euclid, let us introduce four more angles into our drawing. A glance at the figure will indicate which ones.

59

From the equality -L A = LA 1 there straight­way follows a whole family of equalities.

1 . LB=LA ,; L C = L D , ; these angles are called "oppo ite exterior angles".

2. LA = L B,; L D = LC,; these are called "opposite interior angles".

3. L D = LD , ; LC= LC, ; LB=LB, and, naturally, LA = LA I' All these angles afe called corresponding angles.

LDt LB, = ", L A LC, = ", LC LA, = 1', LB+ LD, � ".

Here, we have interior and exterior angles on One side.

Obeying the generally accepted order of things, I listed�all twelve equalities and now regret it. So many can easily obscure a clear matter. Any single ooe would suffice. Tha other eleven are immediately obtained if even one is valid. We

60

started with the equality LA = LA , . But any other ooe would ba ve beeo perfectly suitable.

We proved that i f aoy one of the twelve equa­lities is ful filled, then the straight lines ara pa­rallel. This is the essence of Euclid's two the­orems, No. 27 and No. 28.

Jncidentally, it is worth recalling at this point that the theorem about tbe parallel oature of two lines perpendicular to a common straight line-tha first theorem proved in this book-is a special case of Our theorem of parallel lines.

Upoo proviog a theorem, the geometer always investigates the converse. In tha converse, one proceeds from that which is proved in the direct theorem, aod, naturally, the attempt is made to prove what is already given in the direct theorem.

One of the most common logical mistakes of beginners is connected wiLh direct and converse theorems. It Is casually thought by many that the converse of a theorem lollows directly from the theorem itself.

To disprove this, let me cite the familiar rea­soning of Captain Wrungal of child.hood fame which I have kept in my memory all these years lor just such a case.

Direet lheore,m

Any berring is a 6sb

Converse theorem (Tbe tbeorem of Captain

Wrung.l) Any fisb Is a berring

In keeping with certain traditions of popular­science literature, one adds at this pOint that the above example is just a joke. But I won't bother to do that.

61

Examples taken from geometry (Euclidean): Direct theorcOl8

I . If In '·be lri.ngles ABC and A,B,C, the ,ides A B = A,B,: AC -- Aiel a.nd LA = LA,. the. " ABC = " A,B,C,.

I I. Two linos perpendi. cular to 8 common straight. lino are pa.­raUci.

I l l . II " ABC i. similar to b. A, D,C •. then

AB AC AlB, = AlGI'

Converse theorems I . If " ABC= " A,BIC"

then t.be sides AB = A,B,. AC = A,C, and

LA = LA , .

I I . ]( two panllel slra· Igbt liDOS aro cut by a transversal. tboy arc perpendicular t.o It.

m. II I be proportion AB AC

A IBI - Aiel

hold. lor tbo triang· Ics ABC and A IBICl. then the triangles are simi lar.

In Example IV, we shall co1ebrate by combi· ning lour diffcron� theorems into one.

IV. If " ABC 18 an i .... • I . II in " ABC eelos trianglo II ) LA = LC: (AB - BC), Ihen: 2) lb. a1li1udes or

II ) LA - LC: the mediaD •• or Ibe 2) tbe altitudes or bisectors 01 I bo aug·

Lhe medians. or tho les A and C are equ-bisectors of Lbo ang- ai, then Lbe trianglo les A and C arc oqu· ABC Is .n isosceles aI triangle (AB = BC)

In these examples, all the d irect theorems are correct. I t is lelt to the reader to figure out wheth­er the converse theorems are also valid.

It is a curious Iact, incidentally, that vory olten, though the COn verse is qui te correct, it

62

is Iar more complicated to lind its proof than the proof 01 the direct theorem. Naturally, there is such 8 case in our examples as well.

Theorem 2 (Example IV)-the equality of bi­sectors in an isosceles triangle-has a simple prool, whereas the converse (which is an absolu­tely correct theorem) is somewhat 01 a tricky geometrical problem.

With the theorem 01 parallols Ilroved, lot us try the converse. We lormulate it as lollows:

Dlr •• t theorem 01 paranela

1/ tteO U"�I au cut bv 0 third 4IId the Tend, il ..:.A + ..:.C, _ K (or a.v one Df th� 12 eqUGlittt. gfven <arller I> fulfilled) , lh�n. lhe Unu au parallel.

Con,",*, tbeore", 01 panlU.1a

II two Unc. dre parallel, n thIrd 11M ,"terud'". them will prod"". LA + LC, = lt (oranv of the. 12 equalitie, liven .arller will be ''''l'fi.d).

The convorse theorem of parallel lines was laken by Euclid os the Fiflh Postulato, though Euclid's lormulation 01 tho fiIth po tulate is somewhat different.

Recall tho definition given at the start of this chapter. I t is well worth the trouble. Here it is.

63

Postulate V. 1/ two lines are cut by a trans­versal and the sr.m 0/ the illterlor angles on olle side 0/ the transversal is less than a slraight an­gle (thaI is, the sr.m L A + L C 1 is less lhall 2" (180,,), the two lines will meet i/ prodlLCed and will meet on Ihat side of Ihe Iransversal.

Both the purposefully cumbersome way in which Euclid introduced the fifth postulate and the fundamental 28 theorems which preceded it and which were proved quHe independently of it, all go to demonstrate the amazing intuition of Euclid or of the one he borrowed the idea frOOl (if that person existed).

I shall try to explain myself and suhstantiate my claim. This is all the more pleasant a task, since it will be quite impossible to refute what I have to say. There are no facts at all, thus opening wide all opportunities for an historico­psychological investigation.

Let us examine the initial data. By the time the Elements were writteo, geo­

metry had already grown into a mature, well­elaborated science.

Behind it lay three hundred years of develop­ment and dozens 01 intricate problems solved, aod several tough uoresol ved ones like the du-

..

64

plication of tho cube. Thanks to Plato and Aris­totle, the deductive scheme was established, had gained recognition and was llourishing.

The historian or geometry could already revel in two score names of celebrated mathematiciaJl!l. 1 give this number meaning those scholars whose names have come down to us. For each one of them there are undoubtedly at least ten geo­melers of lesser magnitude whose names never roached us.

Practically all were in agreomont that geomet­ry should develop 00 the basis of axioms. Ob­viously, the majority were io full accord with Aristotle in that axioms and tho basic notions should satisfy the requirement of being obvious. As Aristotle put it, the formulation of the axi­oms themselves is a matter of too great a res­ponsibility to entrust to mathematicians. I t is the supreme problem.

Naturally, then, only the most worthy were admitted to resolve it.

Philosophers, in other words. Whether tbo geomelers belioved Aristotle or

not, is not tho point; the point is that with Aris .. totlo ono agrees.

There can bo no doubt that before Euclid '5 lime attempts bad boen made (and numerous ones) to provo the converse 0/ the theorem 0/ par­allel lines. And 1 personally think tbat by Eu­clid's time it was clear that two solutions exis­ted:

1 . To prove the converse theorem of parallels on the basis of the remaining postulates of geo­metry, and, by the rules of the game, wi thout the introduction of any additional postulates.

65

The adherents of tbis school mu.�t have pre­sumed that the converse theorem of parallels was nothing more than a complicated theorem that followed unavoidably from the other pos­tulates.

2. To the (our postulates it is possible to add a fifth such that the converse theorem of paral­lels would readily be obtained with its aid. And this additional postulate might be formulated in such manner that it would appear natural and obvious in the extreme.

It is hard to believe that the predecessors Md contemporaries of Euclid-all brilliant geome­tricians of the age of flourishing learning-could not conjure up a whole galaxy of equivalent and "obvious" statements of the fiftb postulate. It is bard to believe for the simple reason that some of them pracLically beg to be stated.

Taking the Grst path, it is quite natural that nO success was acWaved either at that time or during the two thousand years following Euclid. Today, thanks to Lobacbevsky, we know that success was out of the q uestion. But that is what we know today.

All the more aUuring was, most apparently, the second possibility: to propOso an equivalent but simple and natural postulate-to smear over and co\'er up the unpleasant spot and cal m down.

Numerous commentators of Euclid who dealt with tho fifth postulate did just that explicitly or in veiled form.

I t is impossihle to beliave that such an out­standing mathematician as Euclid who pro!ound­Iy researched the problem of the fifth postulate (and tho entire construction of the first book of

66

tho Elements is witness to this particular atten­tion with respect to the fifth postulate), it is impossible, I insist, that he did not come ac­ross a number of equivalent and rather natural formulations 01 the fifth postulate. For instan­co, if We combine the diroct theorem on paral· lels and the filth postulate in Euclideao lorm, we immediately get:

A now formulation of the filth postulate, Through a paint C lying outside a straight line A B in a p laM A BC, it is possible to draw only One Une that does not meet A B.

TWs statement is usually attributed to the En­glish mathematician Play(air (18th century), but, naturally, it was proposed by very many com­montators of Euclid many centuries before Play­fair's time.

"Playlair's axiom" docs look much more na­tural and attractive than Euclid's postulate, doesn 't it?

Here is another formulation. It is usually at­tributed to Legendre, though it too was emplo­yed earlier by Eu.ropean and Oriental geometers.

Legendre's postulate. A ltne perpendiculllT to, and a line inclined to, a common secant A B, lo­cated In the Same p lane, definitely meet. (Natu-

.' 67

rally on the side of the sceant where the inclined line Jorms an acute angle with the sc­cant.)

Again a very pictorial assertion. I n place of lhe Euclidean postulat� we have a special case. It wil l readily be seen that this is quite sltffi­cient to provo the fi ftl! postulate in the Eucli­dean form (the converse theorem of parallel lines). I ncidentally, for those who are making their first acquaintance with geometry, tltis is a worthy and rather involved problem that merits ome attention. I will gi ve a few hints and leave

the rest to the reader. Those who are not particularly excited about

this proposition can simply skip the mathematics. But wo will accept the Legendre postulate-a line perpendicular to, and a line inclined to a common secant meet-and will prove the fifth postulate in the Euclidean form, which is the converse of the theorem of parallels.

First let us prove an auxiliary theorem, a lemma.

Let two straight lines 1 and 11 be intersected

by a third so that LA< ; and the sum LA+ + L C 1 = Tt . Then, b y the direct theorem we know that these li nes do not meet, for they are pa­rallel.

Let us again investigato the proof of the di­rect theorem.

From point C drop a perpendicular onto tho straight line 1.

This can always be done. The appropriMo the­orom was proved witbout a word a bout parallel lines_

68

Prove, given our condition (LA<; ), that the perpendicular CB is located as shown in tbe drawing.

�rove by moans of reductio ad absurdum and ullllze the theorem on the exterior angle of a triangle.

We thon have LD+LN=LC •. N is tbe un-known.

Tben we have LA+LD+LN=". (Recall the hypothesis!) Now consider t::,ABC. There are three possibilities for the sum of

its angles.

LA + LD+T� " Note: we cannot use the theorem that the sum

of the angles of a triangle is equal to T:. This theorem is a corollary to the parallel postulate.

First examine the hypothesis: LA + LD+� > 2 >- . " .

� ------��-----<.

I --f..Je.. __

i --��-oN

[_-p;.�t�_

69

Compare this inequality with the equality

L A + L D +LN =r. and obtain LN<;.

Now employing Legendre '8 postulate you get tbe straight Ii oe I and II meeting on the right of point B. This contradicts the hypothesis. Con­sequently, the hypothesis is wrong.

Consider the hypothesis LA + L D + LN<". Tn exactly the same way sbow tbat in this

case the lines I and II meet to the left of poi 0 t B; then reject this hypotbesis as well.

You have proven two i mportant theorems at once:

1 . The sum of the angles of the triangle A BC is equal to 1t.

2. The a ogle N is equal to 90'. Now prove the converse parallel theorem by

employing the following auxiliary construction.

70

Gi ven: wben I and II are cut by a third line,

let LA +LC,<" and LA<� .

1 . Drop a perpendicular onto the line I from point B.

2. Draw through B a line parallol to II that is a straight line that satisfies the "direct

'theo­

rom of parallels". Prove that i t will pas as shown in the drawing.

Think for a moment and then again make use of Legendre's postulate to prove that the line II will intersect I.

You have thus proved Euclid's postulate. But do not forget that you made use of an equiva­lent postulate.

If you were somewhat embarrassed by the cOn-

dition LA < 1-, convince yourself tbat i t does

not restrict the generality of your reasoning. Now check through to be sure tbere are no

errOrs in your roasoning. The a bove p roof has at least two noteworthy

features. First of all , we prove i n passing that as SOOn

as we took Legendre's postulate (the equivalent of Euclid 's postulate) we found a triangle the sum of whose angles is equal to ".

Secondly, I have never rcad a bout this proof. I thought it up in a couple of min utes. I write this not because I am ambitious and hope to gain th.e admiration of the reader for my mathe­matical talent.

The equivalence of the postulates of Legendre and Euclid can be proved much more simply and elegantly, in just two lines. All one needs

71

to do is take the fifth postulate in the form of Playfair's axiom. (Through a given point only One line can be drawn parallel to a given straight line.)

So, as you see, our theorem is unwieldy and unneeded. Its sole justification is that it suggests another one which is indeed an important theo­rem: If the sum of the angles of a Iriangle is equal 10 ". then the jif Ih postulate is valid. What is more, it is useful for exercise. Still more im­portant-most important-in my opinion is the fact that such "investigations" demonstrate how the very !irst naive steps take us directly to ever new equi valents 01 the fiftb postnlat .. . And of cOW'sa there can be no doubt tbat our simple chain of arguments was tried out by any numher of commentators of Euclid.

But now convinced bow easy it is to simplify tbe statement of tbe filtb postulato, we unwiL­tingly ask: why did not Euclid do tbis himself.

I ' m sorry hut ! cannot help my el!. Tbe situati· on demands a series of rbetorical questions, like:

Can it be that Euclid d id noL try 10 prove his tbeorem?

Is it possiblo thnt a scholar of tbat magni­tude, sucb a perspicacious analyst could not ohtain a lew elementary corollaries and choose for the po tulate tho more Datural and obvious one?

How can i t be Ibat be, a follower of Aristotle and Plato, let such an opportunity pass by?

How is it possible tbat be ruined tbe whole harmony of geometry Ibus bri nging upon himself the ire of tbe im mortal god of Olym­pus?

72

Can i t be that any one of a host of commenta­Lors was able to penetrate deeper and better into tho problem than he?

The absurdi ty is so obvious. . . The most likely version is tbe following.

Euclid. like bis predecessors, undoubtedly did attompt to elevate tbe fifth postulato to the rank of a tbeorem and prove it without invol­ving any supplementary assumptions.

Taking into account tbe exceptional position of the fifth postulate In the Elements and also the notorious 28 tbeorems that preceded it, one can conclude with . urance tbat tbis problem worried Euclid and that be paid very special attention to it.

Recalling that all the methods 01 elementary geometry were fully elahoraLed in Euclid's day, recalling for instance tbat studies in the tbeory of conic soclions were i mmeasurahly more com­plicated than most of the reasoniog involved in t.he filtb postulate, recalling (once again) tbat tho fiftb postulate-in the form that Euclid in­vested it-is a challenge to tbe demands of PIa· to and Aristotle, outright effrontery, recalling that Euclid, judging by everything we know. was a truo follower of. . . . and, finally. recalli ng that Euclid was a brilliant geometrician . . . Re­calling all Lhis, we arrive at one and only ooe conclusion.

In the process of ,'ain attempts to prove tbe rutb postulate. Euclid most likely found seve­ral equivalent formulations. Simple ones. Ob­vious ones. But Euclid knew wbere to stand.

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On the one hand, he clearly understood that it would be impossible to prove the postulate without i nvoking some equivalent assumption. On the other band, not one of the equivaJent forms of the fifth postulate satisGed-to his li­king- the requirement of being solf-<lvident. And so he concluded that the situation was very sad and the problem remained unresolved. And, like an honest geometer, he decided to emphasize the lact that the filtb postulate was a rejected, despi­cable monster i n the closely knit family of axioms. That being the case, there is every justi fication for choosing the most complicated form. It is as i l Euclid purposely nudged his colleagues: do not cberisb any vain hopes, do not seek consola­tion i n the pleasanter equivalents of my postu­late, do not attempt to hide the blemish. You will never allah the desired self-<lvident nature that we require 01 axioms. This postulate is no­thing other than tbe "converso of the parallel theorem ". It bas to be proved with tbe aid of the other postulates, Or the beauty and harmony of geometry will be ruined. I could not demote this postulate to the rao.k of a tbeorem. You try.

To put it brieDy, I presume that Euclid had a more profound grasp of the situation tban did most of bis commentators. Eitber they were hy­pnotized by their OWJl analyses and convinced tbemselves tbat tbe postulate was proved, or they attempted to formulate some equivalent and "more natural" postulat-e. Now Euclid most likely ciearly understood that he bad not been able to resolve tbe Grst problem, and to seek self-evideot statements would mean simply to aggravate the illness.

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10 this ratber balanced version of matters tbere is a weak spot (01 course). I f tbere were some kind of investigations, tben why didn't Euclid publish them? That is Dot clear to me. Possib­ly be lelt some inconvenience i o putting forth theorems that did not lead to any results. Per­haps, like many great scientists, ho did not like to make public uncompleted studies. Take Gauss, who did not publish his investigations into non-Euclidean geometry I But maybe there was a manuscript alter all.

That is my strong point: there is very little information to prove or disprove anything in this matter.

Actually the best source of antiquity on the history 01 the filth postulate is Proclus' commen­tary on Euclid. This, as the reader sbould bear in mind, was in the filth century A.D.

Here we take leave of Euclid. In parting, al­low me to say a few warm words.

Euclid was a good, a brilliant, mathematician. He was a great teacher. One wants to believe

7S

that he was just as good a man and that he lived a long and happy life in his sunny Alexandria, drin­king wi th friends the sweet wine of Chios or the pungent wine of Cyprus-diluted of cOurse be­cause i nebriation is a sin of tbe Scythians but not of the Greeks-joking tolerantly about Pto­lemy, instructing his pupils, reading Homer and working to the vory end of his days. We hope that he praised tho gods of Olympus for making him a geometer.

Tbat is tho accopted way of thioking, aod since, for lack of facts, no one can disprove it, wo will thlok so loo.

And with that, farowel l to you, Euclid. Tho problom has been posed. Let us see what bappend then.

The Appendix Tbat I Promised. A List of the Axioms of Plane Geometry

Six Basic Concepts are considered, namely, Three Basic I mages (entities): point, line, plane. Three Basic Relations: belonging (incidence), lying between (for points), motion or coincidence.

I. Axioms of connection.

1. OM and only OM straight liM can be drawn through two point •.

2. A straight 11M contains at least two points. 3. There exist at least thr •• paints not located

on OM .traight line.

II. Axioms of order.

1. A mong any three paints of a straight line, there is always one and only one that lies be­tween the other two.

76

2. If A and B are disltnet points 0/ a straight line then there is at least aIle point C lhat lies between A and B.

8. 1/ a liM intersect. One side of a triangle (that Is, contains a point lying between two ver­tices) , then it either pas ••• through the �Ylrtex of the opposite a.ngle, or intersects arwther side of the triangle.

By employing tho axioms of order it is pos­sible to define vcry i mportant notions that will be needed later 00. Namely: the concepts of "line-segmont", "balf-line" (or ray), and "an­gle " .

Ill. Axioms oC motion.

For ma �hematieians, motion is a basic (pri­DIary) concept. The properties of mathematical motion are defined by the following axioms.

1. For a gi vell trans/ormation 0/ mOlioll (call it D) any point A of tile p lane undergoing trans­formation passes into a single definite point A ' .

2. For a gi !len trans/ormation 0/ motion D , a certain point A 0/ our p laM passes into any point A'.

3. For a given transformation of motion D, distinct points A and B are carried into distinet points A ' and B'.

These threo axioms demonstrate that motion is a one-to-one transformation of a plane into itself.

4. A sequential execution 0/ any two trans/or­mations 0/ motion D 1 and D . is also a trans­formation of motton. We sM.ll call It D " D ,.

5. E very motion D has an in verse motion D-I, such that the product D-l . D is a mot/on that leave.

77

all point. 0/ the plane unchanged, that I., it i. a so-called Itkntieal tranI/ormation.

J n view of Axiom 4 it is obvious that an iden­tical transformation (rest) should be regarded as a special case of the transformation of motion.

These are followed by axioms which demon­strate tbat in motion there docs not occur any "deformation» of tbo plane.

6. J / nwtion trans/arm. the end. 0/ a line-seg­ment A B into the endl 0/ the line-segment A ' B', then any interior paint 0/ A S I. carried Into an interior paint 0/ A 'S'.

• ow comes a most i m portant axiom, without which i t would be impossible to establish the concept of the congruence of figures.

7. J / A , B and C are three point. 0/ lOme figure that do not lie on a sing Ie straight line, then this figure may be tran.lated 10 that:

(a) point A coineUU:. with any preaSligned paint A ' 0/ the plane;

(b) ray A B coincides with anr preassigned ray A ' B' emanating from point A ;

(c) point C colneltk. with some pOint C' in any prea18lgned half-plane resting on the rail A 'B' (there are naturally two rw:h half-planes).

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Following this, no further nw�ment 0/ the figure Is possible.

And, finally, an axiom which shows that mir­ror reflections are a special case of the transfor­mation of motion.

8. There are motions that carry segment A B Into BA and angle AOS into angle BOA .

These eight axioms defino all tho properties of motion, and it is now possible to introduce rigorously the notion or the equality, or, to be scientific, the congruence of figures.

" Figure S is congruent to figure S' i f i t can be made to coincide witb figure S' by means of motion. »

I t is nOW easy to prove tbe following theo­rems:

1 . Figllre S is equal to itsel1. 2. If S is equal to S', then S' is also equal to

S. 3. I I S is equal to S', and S' is equal to S",

then S is eq ual to S". The axioms of plane geometry are Dearly ex­

hausted . What remains are:

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tv. Axiom of continuity (Oedekind's axiom). I I all lhe points 0/ a straight LIM are partitio­

ned into two classes-I and II -such lhat any point 0/ Class II lies to the righl 01 any point of Class I, then either In Class I there is a right­most poinl and then in Class II there is no /el t­most point, or conversely, Class II has a le/l­most pOint, and then Class I I does IWI ha ve a rtghtmost pOint.

To put it crudely, this axjolD implies that there are no gaps or empty spots in the straight line.

ft is necessary to introduce lhis axiom so as to bo able to construct a rigorous theory lor mea­suring line-segments.

And linaUy:

v. The parallel axiom.

Only OM line (all be drawn parallel to a gi­ven line through a gl t"," poillt 1I0t on this line.

We might jump ahead in our tory for a 100-mont to say that lho axiomatics of Lobacbov­sky's goometry dillOn! from E nclidean axioma­tics solely in this last axiom. All the other axi­oms of both geometrics coincide.

Chapter .t

THE AGE OF P ROOFS.

THE BEGINNING

We begin wilh a short l ist of names. The pro­blem of paraliel lines was attacked by Aristot­le, Poseidoniu., Ptolemy, Proclus, Simplicius, and Aganis in the ancient world; by Al-Hasan, at.-Gusi Ah-Shanni, an-Nairizi, Omar Khayyam, Ibn al-Haisan, Nasir-ud-Din, in the East.

By Clavius, Wallis, Leibniz, Descartes, Play­fair, Lagrange, Saccheri, Legendre, Lambert, Ber­trand, Fourier, Ampere, d 'Alembert, Schwei­kart, Taurinus, J acobi in Europe.

And b)' scorcs of known and several thousands of nameless mathematicians as well .

The problem of the fifth postulate wrecked so many minds it would be possible to fill a good­sized psychialric hospital.

That is no exaggeration either. Many spent thoir whole life in vain attempts at a proof, win· di ng up in mystical terror or a psychiatric ward.

One of the most unexpected indicatious of the exceeding popularity of the problem lies in a remark made by St. Thomas Aquinas.

St. Thomas Aquinas was a most prominent theologian 01 tho chrislian world. In one of his researches he found it necessary to solve a pro­blem of exceptional di fficulty: "Wbat is beyond the capability of God?"

He pointed out a number of items in this class.

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According to St. Thomas Aquinas, God cannot drastically upset the fundamental laws of na­ture. For instance, he cannot turn a humao be­ing into a donkey. ([t might be worth adding that most people handle that problem daily with­out any divine aid.)

To continue, God cannot tire, be angry, sad or take away man's soul, and the like.

The list also contains an item that states that God Cllnnot make the snm o[ the angles o[ a triangle less than two right angles.

J am almost cOnvinced that this example is not accidental. St. Thomas Aquinas could have

.chosen any other more seH-evident theorem. It is very likely that he chose this One [or the sim­ple reason that bo was famiUar with vain 3t­tempts to prove tho fi!th po tulate and with the fact that the assertion that "the sum of the angles of a triangle is equal to two right angles" is equivalent to the fifth postulate.

I t is ordinarily supposed that this theorem became knolVn in Europe in the 1 8th centDrY. St. Thomas Aquinas lived i n the 13th centDry.

But wo nlnst also say that Arabian mathema­!Jciens fundamontally investigated the problem of paraliel liDOS and, among other things, ob­tained that result as well.

Many works might have been knOIVO in the early Middlo Ages that were subsequently lost.

Today it is hard to realite just how hopelessly confused was tho wholo theory of parallel lines prior to Lobaehevsky. . Today any good mathematics major in col­lege would fleed no more tbal] two or three weeks

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of cal m work to prove the theorem: if the SllOl o( angles of a triangle is equal to ", then the fifth postulato holds.

And he would prove it even if ho wore almost totally unfamiliar wi th non-Euclidean geometry and, consequently, formally in the same posi­tion 3S geometers of the past.

As recently as tI", 18th century this theorem was considered-aod rightly so-to be ono of the greatest attainm Dts of science. I do not in the least wish to defend tbe obviously pleasant tbe­sis that "people arc morc talented today". That is not the point at all . Simply, in cientific work, confidence in the ultimate result, a clear­cut knowledge that tho approach is correct proves to be all almost decisive lactor.

An A merican physicist is reported to have said that as soon as the atomic bomb was OXI)loded the production of it ceased to he a secret. Tlii m"y be a slight. exaggeration, but in principle it is correct.

I am sure the reader will recall how mucb 5i m­pIer it is to solve a problem Or to prove a theo­rem i f the answer is already kno\vo.

Now in tho whole prohlem of parallel lines, only olle guiding idea is needed: tbe fifth postu­late of Euclid is independent of all the others. With just that knowledge, any mathematician today would readily repeat most of Lobachev­sky's results i n a very short ti mo. But be would remain 811 ordinary mathematician. He would know only one thing: "you have to dig here. " And that would solve al most everything.

I think a case from chess caD oller enough supporting evidence. Take any chess puzzle which

•• 83

states that white has a winning move. The uSllal requiremenl, in such a position is to find an ele­gaut cOI.ubination of moves. Any dncent chess­player can resolve 90 per cent of such problems in an hour or so. Yet in 90 cases out of a hun­dred he wouJd never see such a combination in an actual game.

These remarks are to forestall any stupid feel­ings of superiori ty Over mathematicians 01 earl­ier ages. I t is true that most of the theorems in­volving proof of the fifth postulal<l are quite ele­mentary in their logic, and quil<l accessible to grade-school students. What is more, the logi­cal errors of those who thought they had proved the fifth postulate are also very elementary. But tbe elementary nature is ovident ouly today. In the very same fashion, twenty years hence ma­ny of the problems that plague scientist<l nowa­days will appear ridiculously simple and naive. That is wbat SO ofl<ln happens in physics.

Alter tbis heavy dose of general discussion, it is high time to returo to the filth postulate.

I have time aod again repeated (the reader will have to excuse mc-I admit I 'l l havo to do i t again), th.at aU attempts at a proof were mo­tivated actually by a single factor: a certain lack of elegance, a lack of beauty, as the artist would say.

I t rankled and it rulned the aesthetic feelings of scholars by its complexity. Tho reaction to it was the same in ancient Greoce, in Persia and in Europe.

How delightful was the indignaLion of one of tho greatest mathematicians of Lhe Arabic world, Omar Khoyyam.

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" . . . Euclid thought that the reason for the in­tersection of straight lines was that the two an­gles (iol<lrior angles on One side-Smilga) are less than two right angles.

"In so believing be was right, but it can be proved only with the aid of supplementary argu­ments. (Kbayyam holieved thaL he had proved the fifth postulal<l-Smilga) . . . But Euclid accep­Led this premise and proceeded from it without proof. I swear upon my l ife . . . that here we Deed tho aid of reason, and that is its right... "

"How could Euclid have permitted himself to enter this statement in the introduction (which meaDS choosing it as ao uiom.-Smilga) whe­reas he proved far more simple facts . . . "

Lot us see how the struggle went with the fifth postulal<l. There were throe canonical approa­ches.

1 . A postulate equivalent to the Euclidean ono was opouly proposed. These authors formed a group called the "modest" or "pessimistic" trend.

2. Reductio ad absurdum is one of the most elegant and powerful of logical methods of sol-

85

ving mathematical problems. Here, no new pos­tulatM were introduced.

A theorem was formulated contrary in mean­ing to tho filtb postulate or to One of its equiva­lents; tbis was followed by the elaboration of diversified corollaries in the hope that SOOner or later all this would lead to a contradiction, which would ipso facto prove tbat tbe fiftb pos­tulate followed from the other axioms, and tbe problem would be solved.

This is tbe optimistic, presumptuous trend. 3. And, finally, we bavo tbe group of "eclectics". They proved some theorem equivalent to the

fifth po tulale. And they proved it with the un­witting employment of some other equivalent of Euclid's postulate.

Trend No. 2, the opti mists, had the hardest timo. They kept stringing out the chain of their tbeorems, floundering more and more in tho co­rollaries, and still linding no contradictions.

From the vantage point of today we realize that tbis gronp of mathemalicians actually were proving the initial theorems of non-Euclidean

86

geo�etry, and tbat they were on the most pro­miSlDg pathway, for only in this way one could come to realize that tho Euclidean postulate was independent of aU the others. But that did not make things easier for Ibem. As a rule, they either lost heart or went over to the eamp of "eclectics".

On� must nole that many of the proofs of tho eelecltc group are magnificently witty.

To Pllt the actual history in a rather crude form, ono might say that, in tho main, attempts were made to prove two basic varielies of the fiflh postulate:

1 . A perpendicular line and an inclined line meet.

2. The sum of the angles of a triangle is equal to 1t.

In this way, soveral very pictorial equivalents of the fifth postulate were found. At ti mes the authors realized that they had found an equivalent; at other Hmes they were deluded into thinking bhat Ihey had proved tbe fifth.

Here are a few ersalz· postulates: j . The locus of points equidistant from a gi­

von straight line is a straight line. 2. Tbe distance between two nOnintersecling

strnight lines romains bounded . • • 3. Tbere exist similar figures. 4. I f tbe distance bolwoon two straight lines

first diminishes upon molion in some direction

• In roronulaling the equivalents or tho nIth pqs­tuJnte, 1 will always prcswnc t.hat. everything OCCurs. in one plane .

•• TbJs is tl less strict demand Lhan No. 1 .

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along tbese straight lines, it cannot begin to io­crease until tbe straight lines meet.

And so 00. I n all tbere are about thirty stich statements. For the amusement 01 the reado.r, I give seve-

ral "proofs" of tbe fifth postulate without any critical commentary. If he so desires, he will he able to figure out what postulate was used eacb time in place of tbo fiftb.

1 . Tbe proof of Proclus. One of tho very first, one of the simplest and One of tbe cleverest.

Proclus starts out with Aristotle 's a ertion: When we produce two straight lin.s /rom a pOint 01 Intersection, the distance between them In­creases without bound.

He takes tbis to be an axiom. Actually, it is a theorem. What is more, i t is

a tbeorem that is quite independent 01 the filth postulate. So wo can rely fully on this theorem. I t belongs to "absolute geometry" and, bence, as we understand matters today, it holds true both in Euclidean geometry and in the geome­t ry of Lobachevsky. But the postulato-the equi­valent of ProclUS-is different.

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Here is the proof, actually an outline of the proof. I will not hold rigorously to the formal scheme of any proof. That would be a little too much.

Draw two definitely parallel straight lines; that is, such that L A + LC,="

Draw a third straight line. How? It is shown i n the figure as a dashed line.

The distance between the dashed lino and the upper lino (when moving leftwards) increases without bound. Consequently, thero 'viII come a time when it will exceed the distance between the parallel lines.

Well, aod then it will be clear that tho dashed lino will cut the lower line.

We suggest the reader formulate matters io rigorous form and also state which postulate Proclus employed implicitly.

2. The proof of Wallis. We will prove that a perpendicul ar and an

inclined line to a common secant intersect. From point B drop a perpendicular to a se­

cant, producing a triangle ABC. Take a similar triangle such that its side corresponding to A C i s equal to A D .

I n view of its importance, make a special drawing. This is triangle A ID , F,.

Superi mpose the dashed triangle on our t,ABC so that A D 1 l ies On A C. 'rben A IF I win lie on our inclined line and side D ,F, will lie on our perpendicular.

Actually, the proof is complete: there are only a few formalities Iefl-. I 108 ve tbem to the reader.

Let us not got too invol ved in examples. A more interestiJlg point is the following. Dozens

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of mathematicianl!, people of farflung cultures, separated by centuries of time and frequently nOL ovon knowing the existence of one another reasoned i n almost identical fa hion, repeating one another in al most the samo words.

Prior to tho 18th contury, proofs of the fifth pOSLulate via tbe method of reducLio ad absur­dum did not sIring out the chain of corollaries too far and did not delve deeply in analy,is. A t some moment they would say: there it ist there is the conLradiction. Actually, the contradic­tion turned out to be an equivalent of the fifth postulate.

But since matters did noL go very lar, there were more hunters tban rabbits. There were more matb maticiaos working 00 tbe 61tb postulate than tbere were distinct modes of prool. Almost all the greatest mathematicians of tbe world en­gag d tb fifth po tulate. Thero is one about wbom 1 want to say a bit more. Not because his investigations into tbe theory of parallel

90

lines are something exceptional. 0, not at all. His most interesting results were obtained in tho field of algebra. He did not advance much beyond any of the others in the theory of paral­lels. In this sense we will be gi vi og hi m more than his due of attention. What is more, we will noL even speak abouL his proof of tho fifth pos­tulate. True, the proof he offors is extremely cle­ver. Truo, again, his in nuonce Oil subsequent stu­d ies of Oriental mathematicians is definitely felt. Finally, the techniqu which he employed Was extremely suiteble and was in advance of West­ern mathomatici8nl! by six hundred year.!. (We shall touch on this matter somewhat later.) But, really, it is not the fifth postulate thaL interests us so much in this book.

The exciting thing abont this man is thaL he is a casc which iIIu trates beautifully how little are the differences between peoplo of all naLionl! and a 11 ages.

J now speak of the mathematician that is known as tile poeL Omar Khayyam.

Chapt.,. 5

OMAR KHA YY AM

The full name is Ghiyathuddin Abulfath 'Omal ibn I brahTm al-KhayyamT. To Europeans he is simply Omar Khayyam.

Tbo East, as we all know, is the East, in con­tradistinction to the West, which is the West.

Tho E ast, tho Orient-to most people it sig­ni fies tho usual collection of harems, sultans, the Islam, califs, emirs, mosques, minarets, m uez­zins, burning sun, fountains, Genghis Khan and tho shade of plano trees. The stifling heat of the high-noon sun, and lazing in the shade.

That is tho Orient of tho past, at loast, the way somo peoplo pictnre it.

All these things could be found, sul tans, ca­lifs, emirs and the rest. And even many of them are sti ll found in the East.

Notwithstanding . . . there never was any East. There were and still are dozeDS of countries

and over a thousand million human beings. These millions upon millioDS of people are quite di­\'ersi 6ed.

000 might presume that their inner world is tho samo as that 01 d wellers in tho West.

InCidentally, K i pling, who coined the famous phrase about tbo East and tho West, thought so. Such is tho idea that is advanced in his colobra­ted ballad, of which people usually remomber

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only the iirst line (such, alas, is the late 01 many a brilli:.tnt poet).

Since this chapter will be permeated with the "atmospbere of poetry ", let uS take a few linos of Kipling's poem, all the more so that they are indeed beautiful lines.

"Oh, East is East.. and \Yest. is WOSl., aud never tho twain shall mec!.,

Tilt Earth and Sky 'band preoenUy ab God', great 1udg­ment Scat.;

Out. there is neither East. nOr West. Border nOr Breed nOr Birth,

. ,

When two .trong men 'L�nd race to race, tho' they come Irom t.he end. 01 tb. earthl"

There is no use quoting any further because what follows is pitifully bad. The poetry is still excellent, but the topic and its rosolution is a terrible let-down, hard.l y better than a routine Hol lywood film about tho Wild West.

Kipling coo lined himself to a hymn i n hooour of the spiritual unity of warriors, heroes strong i n body and spirit. Taken at face-value, these warriors are something in tbe nature of a pre­i mage of the noble bandits of Hollywood. But if one ignores his choice of boroes, he can fully agree with Kipling. Gangsters throughout the world find a common language with just as much ease as humanists in the world of science.

Unfortunatoly, Kipling sang tbo praises of the former aod gave to them his amazing poetical taloot.

This whole discussion is very much to the point if 000 recalls that we are speaking of Ghiya­thuddio Abulfath 'Omar ibn I brahIm al-KhayY3mT of Nishapur.

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Ghiyathuddin means "the help of rait�" n�d is a traditional title lor al l scholru's, si nce III those days the h ierarchical ladder of scientific knowledge was apparently not so involved. Abul­lath means the falher of Falh.

Khayyam was born io ishaput, which was olle of the chief cities of glorious Khorassan.

Khayyam-wh:lt we have taken as the last name-meaJll! tent-maker. Most l i kely his father or grandfather was so engaged. .

I bn I brahim is Ihe SOn of Ibrahim. Finally, Omar, is the given namo. In short Omsr Khayya m, who conquered tbo

West in the 19th century and conquered it as a poet.

He was lirst translated into English and camo out in 25 edi tions last contury. In England and America admiration for Khayyam developed i n­to 3n epidemic. He was quoted and praised, and clubs named alter bi m sprang up everywhere. Willy nilly we shall h3\'e t� delve i n�o �he li­terary side of Khayya m. HIS poetry 15 I

.odeed

beautiful' but his 0 ceptional populanty IS due possibly

'to a certain "marvellous revelation".

It turned out that a thousand years ago, some­where i n Turkey, or India, there l i ved a man whose thoughts and emotions excited people liv­ing in the modern age 01 the 19th century. More he cast tbe thougbts arid emotions i n mag­nific�nt poetical form , which was i ndeed ama­zing.

True, ill his home land he was bardly at all known as a poet.

Thus arose two Khayyams. In the West was the poet.

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In the Enst, the mathematician, astronomer and pbi losopher. Oh East is East aud West is West.

Who is this Omar Khayyam? Since I lean mor to tho oriental version, let

us begin our story of the honourable wise ma.n and i mam Omar al-Khayyam of Nishapur, may Allah sanctify his dear soul.

"In the name of the gracious and merciful Allah, praise Allah, the lord of the worlds, and blessing unto all his prophets. "

Thus did Khayyam , bound by a rigid tradi­tional form, begin his marvellous "Treatise on tbe Proofs of Problems of Algebra and al-MuI­qabalah", a mathematical work tbat was roughly five hundred years in advance of the mathema­tics of the Occident.

This work of the "greatest geometer of the East", as that remarkable encyclopacdi t of the Orient, the Arab l bn-Haldun wrote of him la­tor, contains the first systematic theory of thiro­degree algebraiC equations. It was well known

95

among Arabian maLhematicians and undoubted­ly exerted a tremeudous orr cL Oil the develoJl­ment 01 mathematics in the East. I n EuroJle, the first and rather nebulous reference to it Oc­curs only in the year 1 742.

The historian actually only says that it would seem , by the title of the manuscript, which is in the Leyden Museum, that one may suspect that it contains something about equations of tho third degree, but ... "It is such a pity that none of tbose who know Arabic has 8J)Y Laste for mathematics and none of those who have mastered mathematics has any taste for Arabian literature. »

When the treatise of Khayyam wus finally road, it was found that his results were repeated (and in many respects surpassed) by no other than Descartes. I ncidentally, it is possible tbaL in yet another treatise that has been lost irre­trievably Omar Khayyam woot much fartlior. Who knows?

We know of yet anotber treatise of Omar Khayyam, to wit: "Commentaries On tbe Di ffi­culties in the Introductions to the Books of Euclid . " Tbis composition of Lhe most glorious sheikb, imam, Of tho Proof of Truth, of Abnl­fath 'Omar ihn I brahTm al-KhayyamT is in tllree books.

Again, this treatise, in the beginning, lacks originality: U [ n the name of Allah, so gracious and merciful, Praise Allah, tho lord of grace and mercy, alld peace be unto his slaves and in pa.rticular unto Muhammad, the lord of tho prophets, and UJ1to all his pure clan."

All this ritual breaks off suddenly just a

96

I ine down: "The study o f the sciences and the comprebension of tbem by means of true proofs is necessary for him who seeks salvation and happiness. "

Tbat's enough. He '''ho was eager to understand did. Already too much was said. On went th� soul-salvaging ritual.

"And especially (of course, most naturally) this refers to the general notions and laws to which one resorts io studios of the hereafter, proof of the existence of tho soul and its eter­nalness, comprehension of the qualities tbat are necessary for the existence of tbe A l m ighty and Ius magnificence (Khayyam is worried beyond reason ahout the magni ficence of Allah), the an­gels, the order of creation and proof ol the pro­phecies of the lord, the prophet ( 1uham mad, that is), to the orders and prohibitions of which bow i n obedience all creatures (incidentally, tbere was a time-in Medina-when Muhammad introduced a very rigorous order and the best of tbo creatures of Allah were ever at attontion) in accord with the pleasure of the A lmighty Allah and the power of man."

What a Dawless piece of writing, it would seem.

Yes, it would seem, for the entire paragraph is one solid heresy, extremely dangerous to any orthodox preacber of I lam.

Let the worshiper of Aristotle smooth over his writing with hypocritically pious phrases, for he will be understood by those 01 the same views and those of other views as well.

Omar's luck tbat, in general, Islam was a morc tolerant religion than Christian ity. On tho ave-7-11:187 97

rage, that is. There was no burning at tbe st�e. But one could e x poct, wh.en needed,

.s SWIft

plUJIge of tbe dagger. Very much so, In fact. Even for just a tiny bit of heresy. On the oth r hand, one could get around tbat too.

, Then follows the treatise p,·oper. (We shall havo moro to say about it later on.) A l l the way along, however, Omar put i n. the proper pro­portions of glory to the Al mIghty Allah, and to his greatest creation, Muham mad, and to tbe whole lineage of Muhammad, to the great belp of Allah and more and more.

Praise the LordI How merry and nice it was ror his creations.

His creatures I mean. ote however that the merciful servants of tbe merciful Christ pushed the morciful AUah into the backgrouud .and again We begin "in the name of the gracIOus and merciful Allah".

We I .... ,OW hardly anything at all about Omar Khayyam, only a few fragmenta'1;'

, bit" here. an?,

there. By way of complicated .astronom�cal

computations on the basis of lDd�roct rmdlDgs, tbe dates of his life are, approXlmately, fixed at 1048 and 1 131. Or from 1040 to 1 122. Or from 104.8 to 1 1 22.

Ho was born in Nishapur. At that time, the city was located in the emirate

.of Khorassan.

Today, Nishapur is on tho terrttory of Ir?n . Omar wrote his verses i n the litera,"}, PersIan language, and his learned studies in Ara�ic. Since, as linguists explain, both lD:0dern Per�lan and Tajik developed out of medIeval PerSIan, we may justifiably say, today, tilat Kbayyam is a Persian poet and a Taji k poet.

98

A few years prior to tbe birth of Omar K hay­yarn, that "egion of tbe "calm and lazing" Orient was the scene of bitter battles, and the leaders of the nomad Seljuks (Turk mens) first routed the earlier sultans and then set up a collosal empire alld a nice fresh dynasty of Seljukiall sul tans.

What followed was rather standurd. Fighting for the tilrone among tbe aspirants. Tho sultans fighting feudal lords aud frenzied attempts of tbe Leudals to rule by themselves, indepeudently. I u about one hundred and twenty years the em­pire fell to pieces completely. But that period of time, which to history is minuscule, to a human being is quite enough.

Khayyam lived in tile empire of tbe Seljuks and lived q uietly for a long time, for he had 3 patron. A strong protector.

The great vizier izam-al·Mulk. Nizam-al-Mulk was possessed witb the idea of

a strong state. And he furthered it in many ways. He apparently believed tllat culture and learning would strengthen his empire and so, like those dear Ptolemys of antiquity, be patro­nized his cholars in many a way.

He himseU was not above literary forms and wrote a ratiler serious, fundamental and very interesting-to historians-work entitled the Book oj Govemmellt-a sort 01 handbook ror sultans who needed training (thoy certainly did). In this 1V0rk of popularization be engaged the serviceS of his scholars and in particular those of Omar Khayyam.

BuL before Omar entered into tbe service of Nizam-al-Mulk ho had endured much indeed.

,. 99

When a sullen is setting II» all empire, tbe inlla­hiluots do not havo it easy at all .

There is practically no information about tbe youth of Omar, other than that he may have studied in 'ishapur.

The tory goes that "at the age of seventeen years ho allained profound knowledge in all fields of philosophy".

It is said that ho was "a proroundly I.." ow­lodgeable man in li nguistics, Muslim law and history " and was a rollower or Aviconna (Abu­Ali ibn ina).

It is also rolated that he had a marvellous memory and that on one occasion he learned a whole book by heart after reading i t seven times.

Some said that he was a " ago with extensive knowledge in all fields or philosophy, especially mathematic ".

I n u word, then, all source (and al 0 the writings of Khayyam himself) describo a man wilh encyclopa die knowledge and a mind of exceptional gifts and per picacity.

At the beginning, however, all these good points worked more against him than for him. He wa compelled to leave Rhors n, and we lind Omar Khayyam i n Samarkaud.

Q ui te naturally, a patron was needed. And Omar found him. We do not know how, but he did. This "marvellous and incomparable judge of judges the i mam A bu-Tahir, may Allah con­tinue his rise and may Allah ca t aside those who are envious and wish him evil".

To put it simply, this was lhe chief judge of Samarkand, a high-placed official. But only Al­lah realJy knows whether he possessed evon a

1 00

mi�ute porlion o f the merils that Omar 80 pains­taklDgly and sweet-singingly described i n his algebraic treatise. A bit earlier, in the introduc­tion to the same treatise, Omar wrote sullenly and bitterly:

" . . . I was deprived of the opportunity of en­gaging regularly in my studies (of algebra-8m;l­gal and I could not even coneen trate on med i le­tion about it because of the reverses of de tiny that plagued me.

"We wero wilness to the death of learned mon of whom there remains a small and suffering group. The harshness of fate in the ti mes prfr vents them rrom giving themselves wholly to refi­ning and deepening their learning.

"Most of those who today have the aspect of a scholar drc truth in falsehood, without going beyond imitation i n science and only pretending to knowledge.

"The knowledgo which they havo amassed is used for base purposes of the Desh. I f thoy en­counter a man that seeks the truth and loves the t�uth, if ho attempts to reject fal hood and hy· pocnsy and gi ve up boasti ng and d ceil, they mao ke him tho object of their contempt and mockery."

When reading an excerpt liko this, one no longe� wishes to relate the story of Omar Khay­yam In tho cool and slightly irOnical tone of the objeetive ob rver. There is nO place for words about the great and merciful AlJall. Here lifo is harsh and cold. These bi tter Ii Des were wriLLen by a very young man. A t that lime he was hard­ly more than twenty-five years of age. Such a desire vanishes completely WhOD wo recall that four centuries latcr al most the vory same thing

101

was writt<ln by GaliIeo, and within another five centuries by Einstein.

I am not sure what Omar wanted to say, but tbe next sentence ("Allah helps us i n all ca!!!!s, he is our refuge ") followed by an extremely long paragraph prai ing tbe henourahle judge of Sa­markand reads l i ke a savage, vicious, rll2or­sharp taunt.

Let us not stray. Omar was lucky. He found a patron. What is more, one "whose . . . presence opened up my chest and wbose society levated my glory, my work expanded due to his Iigbt and m y back was strengthened becauso of his good deeds".

So you see how wonderful everything was. Yet that was only the beginning. Allah is never grudging i n his generosity.

Omar Khayyam is honoured (glory be unto Allahl) by tho friendship of the khakan of Bukha­ra himself. What this title signi fies, I do not know, nor have I tried to lind out. A t any rate, be was some kind of minor king of sorts. And an historian (a contemporary of K hayyam) reports, with an undorstandable U nge of envy, that " . . . khakan Shams-al-Mulk ol()vated hi m greatly and seated tbo i m a m Omar on his tbrone".

But the good deeds of Allah are indeed inex­haustible. And in the year 1074 Malik-Shah him­self (tho khakan is only a va al of the hab) sllmmonod Omor to his court i n Isfaban and­rejoice oh ye faithfnll- makes hi m his nadim.

You would probably liko to know wbat a oa­d i m IS.

A ratber strange po t. A sultan is always In need of i nterlocutors, coofidallts, body-guards.

102

Those are tbe duties of the oadi",. He has his meals witb the ruler, converses with him and engages him, thinking up all kinds of things in order to ki 1 I t i me. A nd of cours he shows his admiration ror the wisdom of tbe ruler, the courage, the beauty, the poetical gilts of tho sultan, for his steed , his eagles and his concu­bines. Trn , I do not know whether be demon­strated the most beautiful Dowers of his harem or not, buL . . . .

o need [or this amateuri h talk, we give the noor to the radiaot patron of Omar Khayyam. NiZli m-al-Mulk himself.

We quote from the Book 0/ GO tV!r"nment (Sia­set-NllIMh).

"The benefits of the nadim oro severa l : One is that ho is a close friend of tho sovereign, another is that since he is wi th tho sovereign day aDd night, ho acts as a body-gnard, and i

.n

case of necessity-do not allow i t , oh, Allah-If there is �omo kind of danger, he sacri fices hi

1 03

body by using it as a shield against that danger; and fourthly, a thousand kinds of words can be said to the nadim rather than to those who perform the duties of the amils and the ollici­als of the sovereign; the fifth benefit is that they report, l i ke spies, on the affairs of the kings; tbe sixth, that they converse i n all manners without compulsion about good and evil, wheth­er inebriate or sober, thus bringing about much lhat is useful and purposeful . "

o you see , six distinct benefits. Few indeed can occupy such an honourable post. Very few.

" I t is necessary that the nadim be gilted by nature, virtuous, good-looking, of p ure laith, a guardian of secrets, well-mannered; he must be a narrator 01 stories, a reader of what is merry and what is seriOUS, he must remember many legends, he must always be ready with a good word, a reporter of p leasant news, a player of nardy and cbess, and if be can play some musi­cal instrument and band Ie arms, all the beUer. The o.adim must be in accord with the sovereign. To everything that takes place or that lhe so­vereign ut.ters, he must answer: "Excellent, mar­vellous"; he may not instruct tbe sovereign with words "do tbis, do not do lbis, why did you do this? " He must not so speak because the sovereign will tben be depressed and will reject him. It is proper ror the nadim to arrange all matt­ers pertaining to wine, recreation spectacles, friendly congregations, hunting, tbe playing 01 chougan and tbe like, for that is what they are needed lor."

Tbat is all.

1 04

Thus preached iz:lm-al-Mulk, who presented Kbayyam to hlik-Shab as nadim.

Without donht, an amazingly p leasant post. Historians console us somewhat. One group

thinks it bigbly improbable that Kbayyam was honoured so greatly, and they believe that the biographer exaggeraled. Perhaps he wished to elevate a scbolarly colleague in the eyes of tho reader and allowed lor some exaggeration, a bit of boasting. Others feel that Khayyam was in­deed a nadim but, tbey say, of a somewhat dillerent kind.

For Nizam-al-Mulk conlin ues: "Many sove­reigns havo made pbysicians and astrologers their nadims so as to know the opinion of eacb 01 tbem as lo what should be done by them, what by the sovereign, wbat needs to be done to preserve naturo and tbe healtb of the sOve­reign. Astrologers observe the ti me and the hour; lor every matler lhat is pleasant thoy give nO­tico and select . favourable hour. "

1 05

In general , then, there is a faint hope Lhat Khayyam did not have to arrange the drinking spre of Malik hah and locate concubi nes for hi m. But who know ? On thing we can bo sure of is that he had to do everything that came into tho head of tho rulor.

At any rate, he definitely delved into astro­logy, though just as definitely he believed H to be nonsens .

As astrologer, Omar Khayyam was an indispu­table authorHy, but it is his secret how it came • bout.

And with what professional skill one had to cringe i n the courts of the East! Ti me wHhout numberl

On the whole, then, this l i fe which to mauy was so pleas�nt, was thoroughly disgu ting to Omar Khayya m .

There were a lew things i n oxchange, though. Firstly, the court sag of Malik-Shah, his

confidential agent, almo t pal, was inacc ible to all servants of the Koran, who were, oh, so cager to make Omar toe the Ii ne.

ccondly, Omar was wen provided lor. True, he did not have a family, but the position of a scholar in thoso days was precarious, SO much SO i n fact that it was i mpossihlo to exist without a patron. 0 better 3 shah than somo kind of sma l l fry.

Thirdly, perhaps mo t i mportant of all was the po ibility to work. Omar had at his dis­posal what at that time was a first-cla obser­vatory, the I lahan Observatory. And probably the shah reasonably as umed that his wi man should have some pare timo for meditation. At

1 06

any rato, Khayyam did a great doal d uring hi years at court. Three years alter hi arri val hore he had already completed his "Commentaries to the Difficulties in t he Introduction to tho Book of Eu l i d ", whero among certain other correc­tions he proved-so ho thought-the filth postu­lato.

Ho wo busy in the observatory and obtained oxcellent result . Actually, he was the founder of the obsorvatory, for which he constantly reque­stod monoy of Malik bah lor building purposes.

Again a routino si tuation . o ono was i nterested in his a tronomical work.

Ho compiled a calendar that was marvellousiy accurate, but the calondar was neVer accepted though his astrological studies were view d a� undoubtedly valuable.

A numbor of centuries l a ter, Kepler, who Va­luod astrology like Khayyam did, tread Lhe same pathwa!. H was solely through astrology that he ac�eve.d a po ilion i n society, tho means for dally hIe and the opportunity to engage in scientific studies.

Omar did not believo in astrology. Historians havo not yet decided what his boliof was. There seems to bo oue and perhaps the most i mportant symbol of his faith: a person should study scien­ce and learn about how the world is made. But b.ero too tho situation is complicated, so it is �Imo to return to his verses. SI)eaking generally, I f we knew exactly which verses wore indeed written hy Omar, they would be an extremely valuablo document.

He did not consider himself a poet. Most like­ly ho wrote lor himsolf and was naturally Jess

1 07

secretivo than i n his philosophical treatises, in which he always had to be extremely clU"eful, cautioUlliy interpolating minusculo deviaUons from orthodoxy. Meanwhilo specialist.s in li­terature IU"8 still fighting over which verses are genuinely his. . . . r.

Tho canonical text IS clat med to contalD 2;>2 rubs'Ts (quatrains). But hore too tho debate con­tinues. A total of about 1 ,000 quatrains aro as­cribed to Omar.

We shall take it that the ver are genuine. Novertholess it is rather d ifficult to dotermino exactly the philo ophieal world view of Omar Khayyam. Even the specialists are uoable to roach a single opinion, which, incidentally, is how things UlIually stand.

Some of the vef1!es are magni ficent oven io translation and IU"O better in the original, so they say. Truo, Omar's topics are rather restric­ted; frankly speaking, twenty to thirty verses fully exhaust overy thing that Khayyam wanted to say.

Now so that tbo reader can rest with some good p�ose and uice poetry, I will give a rather UOUllual analysis of Khayyam '8 works and then a fow of his quatrains.

O 'flenry, probably irritated by the Omar Khayyam crate, put the matter in story form as follows.

The bero o f bis slory "Tbe Hand book of Hy· men» Sanderson Pratt, cowboy, was caught in a snow �torm in the mountains and had to sit it out with another cowboy, Idaho. It was most li ke­ly a case of psychological incompatibility: tra­gedy was averted when tbey found two books.

1 08

One wus a II andbook 0/ Indispensable hI/orma­tion and the other was Omar Kbnyy"m. I n a card game, Idaho WOn and chose Khayyam, and

anderson got tho hand book. Over tho monoto­OOUll weeks each studied hi book.

At last released from the snow, tbe two cow­boys returned to normal l ifo and began paying court to a cbarming wealthy widow, each d is­playing his nowly acquired culture and utili­zing to the fullc t what he had read. Idaho's poetic guide-Omar Khayyam-was defeated roundly by tho bandbook , and tbe bappy mar­riage of Sanderson Pratt was tbe worthy reward of the bearer of common sense. As anderson Pratt put it:

"I sat and read that book for four hours. All the wond r of education was compressed i n it. I forgot tbe snow, and I forgot that me and old Idabo was on tbe outs. Ho was silting still on a stool reading away with a kind of plU"tly soft nnd partly mysterious look shining through his tan­bark whiskers.

'''Idaho, ' says I , 'what kind of a book is yours? '

"Idaho mUllt have forgot, too, for he answered moderate, without any slander or malignity.

"'Why , ' says be, 'this hore seems to be a \'0-lume of Homer K. M . '

'''Homer K . M . what?' I asks. "'Why, just Ilomer K . 11-1. , ' says he. "'You're a liar,' ays I , a little riled that

Idaho should try to put me up a tree. 'No man is going 'round signing books with his initials. 1 1 it's Homer K. M. Spopondyke, or Homer K. M . McSweeney, or flomer K . f . Jones, why don 't

1 09

YOIl say so like a man instead of biting orr I.he end 01 it like a calf chewing 011 tho tall of a shi.·t on a clothes-line?'

''' r put it to you straight, Saody, ' says Idaho, quiet. ' I ts's a poem book, ' says he, 'hy Homer K. M. r couldn't get colour out of it at lir t, but Ulere 's a viow if yOu fol low it up. I wouldn ' t have missed this book for a pair of red blankets . . . '

The new Omar Khayyam conv •• ·t, T dabo, t hen gives an analysis of the poet.

, ... . . He seems to be a kind o f a wine agent. His regular toast is "nothing dOing", and he seems to havo a grouch, but he koops i t so well lubricated with booze that his worst kicks sound like an invitation to split a quart. But i t 's po­etry, ' says Idaho, 'and I have .sensations of scorn ior that tn.ck of yours that tnes to convoy sense i ll foot and inches. When it comes to explaining the instinct of phi losophy t hrough the art 01 nature, old K. M. has got your man heat by drills, rows, paragraphs, chest mea uremenL, and average annual rainfal l . ' "

Pratt wasn ' t 000 t o give i n easily. "This Homer K. 1.1., from what leaked out

01 his l ibretto through Idaho, seemed to me to bo a kind of a dog who looked at life .like it was a tin can tied to his tail . After runntng h. mself half to death, he sits down, hang. his tongue out and looks at the can and says-

'Oh well sioce we can ' t shake the growler, let's get it

' filled at the corner, and all have

a drink on me. ' "Besides that, it seems he was a Persian; and never hear of Persia producing anything worth

1 1 0

mentioniog unless it was Turkish rugs and Mal­tese cats. "

Though Jovers of Omar Khayyam will be in­dignant, we must admit that the hasic topic was grasped "ather "eatly by the two cowboys.

True, one never knows what exactly O ' Henry is driving at.

H might woll be Uwt as a true admirer of Omsr Khayyam, he sim ]lly wished to illustrate the ancient but sad theme: forget poetry if you want to acbieve success with a charming lady, forget or give up all hope. EspeciaUy if the woman is the owner of a two-storey hOUM ill a neat little provincial town.

Now for Omar Khayyarn's verses. Crudely, we might divide them into three group : ( I) tho love and wine cycle; (2) tho philosophical cycle; and (3) civic lyrics, th.e q uatrains in which Omar describes more or less straight forwardly his at­ti tude towru-ds his surround i ngs.

1 1 1

Since I have boon cons�antly balancing on the brink of soiving psychological enigmas, let uS tbis time try to figure out to wbat extent Omar's verses convey tbe true i mage of the writer him­self.

Perhaps in tbis sense the most mealli agfnl are tbe verses o f the third cycle: irritated, fuji of gall, defini tely vicious.

Of all 252 verses, tbere is not a single one that says something decent about the thinking croations of Allah. Everyone gets it in tbe neck, but Omar is particularly bitter towards tho cler­gy.

Oh come with old Kbayyam, ond lCR'\'o the Wise To ' talk; ol1e thing is certain, that. Lire . nics.'

Onc thing is certain, and Lho Real. IS L i es ; The Flower that ouce bas blown for ever dies.

It is quite natural now to go on to tbe merci­ful Allah himself. Omar does not get along so well with tho Lord in verses as he docs i n treati­ses.

Oh Thou who didst Pitfall and witb Gin BC5et Lh� Road 1 was 1..0 wander in,

Thou wm not with Predestln 'd Evil round I nmesh, aud tlJen impulAl my FaH to Sin! Oh Thon, who Man of baser Earth dldsl make, and ev'n with Paradise devise tho Snake:

For all the Sin wberewith tbe Face of Man Is blackeo 'd-Man's forgiveness give-and takel

This entire cycle may very logically be con­cluded with quatrains in which Omar explains the situation in which he is compelled to live and work.

One Moment.. in Anllihilal ion'8 Wasl(�. Que Momoll", of tbe Well of Ufe to lasl.c-

1 1 2

Tbo Stars are selling alld the Caravan Slarts lor tb. Dawn of Nothi�-Oh, make b.stel How l.ong. how long, in in fimt.e Pursuit Of Th,. and Tha� endeavour aod dispute?

aotlAlr be merry witb tho fruWul Grape Than sadden alter none, or bit.Ler, Fruit.

The writer of such "radiant " verses is definitely not a man with an optimistic turn of mind. Complete spiritual isolation and nO breaks i n the gloom.

And tbat invorted Bowl we call The Sky, \Vereunder crawling coop's we livo and die

Uft not your hauds to It for holp-for It As impotently rolls as you or I . 'Tis all a Cbequer-board 01 Nights and Dais Whe r� Destiny with Men for Pieces plays:

Hither a.nd Lhit.ber moves, and maLes and slays And ODe by one back in the Closet lays.

'

Again, not a single bright spot, not evon a hopeful hint. In tho first cycle thel"1l appear to bo cerlain prescriptions for arranging life O ' Hen­ry's beroes ( I can repeat) grasped tbo

' gist of

the maller qnite precisely. Incidentally, the first English translation by Fi lzgerald paid special and exceptional aUention to this particular trend.

l!lto Lhis Uni.\'crsc, and �hy nol knowing, Nor whence, hko \Valer wllly-niUy Oowing'

And oul 01 it, 8$ Wind along th. \Va;to I know not. U'htt�r. "" iUy-nilly blOWing. ' You know, my Frlonds. bow long since in my House For a new Marriago I did make Carouse: Divorced old barren Reason from my Bed, And took tho naughter 01 tbo Vine to Spou,",_ Ab, fill the Cu�:-wbaL boo15 it to rope.t How Time is slipping underneath Oll' F .. �:

Unborn To-morrow, and dead YesLerday, Why fret about tbem II To-day bo sweetl

8-1P81 1 1 3

And fioally Ah with tho Grape 111)' rading Liro provide, And wash my Bady whence the Lire hIlS died.

And lay me. shrouded in the living Leaf. By some not unfrequented Garden- ide. That. ev'n my buried Ashe.s 5uch a Snare Of Perrume shall Ding up Into the Air.

As not. Il True Believer passiJ.lg by But. shall be overtaken unaware.

Not so cheerlol , I would say. Bot there isn't so much to buoy up onc's spirits.

For lack of any other hypotheses, let US take it that the writer of these quatrai.ns waS indeed Omar Khayyam. At least half of them, that will be enough.

The portrai t of the man who wrote these verses would seem to be clear. A clover, gifted skeptic aod misanthrope. Definitely cultured, but to­tally lackiug io any kind of intellectual i nter­ests all his days and nights spent with COnCu­bin�s and wine, in the company of drinking revellers; and in a rare sober moment he writes marvellou , but deeply pessi mistic verses. He values nothing more i n this world than the opportunity to carouse, and does SO to the l imit of his strength and money.

An indigestible blend of a Byron hero, a low­class patrician or Rome, Goethe's Mephistophe­les, the debauchery of a R ussian merchnnt or a French aristocrat.

Omar's ideas are by no meanS new. There have boen skeptics and Ilessimists thro­

ughout the ages, and their Weltanschauung does not call for admiration.

Omar, at times, appeal's close to spontaneous materialism. At any rate, he abuses Allah often

1 1 4

enough. But on tbo other hand thero are a goodly number of partly mystical quatrains; and, what IS there t.o ad mire in them?

In �v�ry.

8¥e and period of buman history, material! lie Ideas have inspired many think­ers.

In the case of Kbayyam tbore is nO need, how­ov�r, to make allOwance for tbe intellectual n,8lvete of that ago compared with our OWn. No need at all to pat past conturies On the back goodnaturcdly. 1 (, however, we speak as equals and judge

by ve�ses alone, tho imago of Omar Khayyam tho

. thinker loses much of i ts lustre. There re­maIns a magnificent poet, but not a very like.

�ble. or profound persOn. We can understand and lustlfy but we cannot agree.

Literary critics do not speak so frankly, per­haps, because tbe pootry of Omar Khayyam is firmly placed along with the greats of world cuI ture and so also is Khayyam the man-canon­Ized.

But if I �oJy knew Khayyam the poet, I lVould, alter a perIOd 01 enthusiasm fol' his pessimism, botwoon the agos of 15 and 25, agree with 0 'Hen­�y, tho.ugh paying full due to his supremo poet­Ical skill.

However, the charm lies in the iact that our h.ypothetical image is but a caricature, and lop­

SIded at that. Because Khayyam was not a poc� by profession. He was a scholar. His buslOess was learning. Verses? Only for I·ecrea. tlOO.

Houris and wine? If Omar had but i mbibed a hundredth part of tho wine that flows through "

1 1 5

his verses. I f his harem had contained n lenth

of tho beauties whose praises he sang-he would

not have strength left for anything else.

Yet all his contemporaries-well-wishers a�� i ll-wishers alike-are of one opinion: the hall'

imam Omar was one of the greatest of learned

men of the East. J ust what was he? He was a - . • Mathematician. Probably the g�eatest In. �rl-

ontal history. That, at any rate, 15 the oplru�n

of many mathematical historians. T?e algebr�'c

works of Khayyam llI'Il-no harm In repeatlDg

it-brilliant. Ho made a thorough study �f the

mathematical legacy 01 the Greeks. That , n It­

self is quito some llldertaking requiring years

of work. . Astronomer. Recall the years he spent settlOg

up the Isfahan Observatory. You remem�er the

constant prolonged astronomical observatIOns he

carried out, the reform of the calendar and the

newly devised system of chronolog.y. . Part physicist. He produced a hIghly cunous

treatise on Archimedes' celebrated problem of

King Hiero's golden crown, the problem that

gave rise to Archimedes' law and the tradem�� of th() Soviet Union's "Molodaya Gvardla

("Young Guard") publishing Hous? . . Yet that is not a l l . From Omar s works It IS

evident that he had a fundamental knowledge

not only of Arabian philosophy but Gr.eek as

well , particularly the philosophy of . Arlstol.ie.

Khayyam was oven too openly e�lfa\,lshed WIth

Aristotle. This is most evident In �hc v:ay he

refers to Aristotlo- brieDy and lacking lD any

1 1 6

emotion. In place of the name, he writes "phil­osopher".

Philosopher and no oriental compliments. Omar cO�d

, use epithets. whon he wanted to.

But he didn t here. Ho did not want embelllsh­ments, the inOation of which he felt so keenly' he did not want falsely honeyed phrases to sLick to llamcs that were really dear to him.

The philosopher was enough. Generally, wbon Omar gets down to business

t�e pootical, wurtier, oriental style \'anishc� WIthout a trace. Between the traditional bows to Allah, . M�ammad and tho current patron at the beglomng and end of each piece of writing we find a restrained and reserved text.

'

Re.ferences, arguments, draWings, formulas. Euchd is simply Euclid, and not the prince of mathem�tici.ans. Or the beacon of knowledge. ApolloDlus IS SImply Apollonius. Ptolemy jusL Ptolemy. A touch of editing here and there and the style is that of the twentieth century. Aris­totle is the philosopher.

We have strayed a bit. What is interesting hero is something else. Rocall that "Tho phil­osopher" .wrote in a very turgid confused style. �ny detailed study of his writings is an excep­tionally difficult job. I ' m sure that today there are not many specialists i n the history of phil­osophy that have worked through al\ of Aris­totle '5 le�acy in the original Greek. Perhaps only a few phtlosophers specializing in the l i fo and work of Aristotle. ow thore is no doubt that 0 n,'ar Khayyam studied aU of the works of the phIlosopher. Yet Aristotle is ouly a small part of tbe philosophical legacy of Lhe Occident and

1 1 7

Orient that Omar studied, as is so eloquently witnessed to by references to dozens of divers­ified fundamental writings.

Speaking of tho volume of digested literature, Khayyam is tho envy of any academiciau in philosophical scionce.

Phi losophy does oot exhaust Omar. He was also knowledgeable in the Koran and fuslim law.

This is not all . He was also an astrologer. We have already

said that Omar knew the true value of astrol­ogy, but a good dose of information has to be absorbed in order to gra p its rules.

By the way, one of the stories of Omar's astrological feats makes one think that he was familiar with the essentials of meteorology.

The recollection is that of an- hami as-Sa­markandi :

" . . . the Sultan sent to Merv to the great hajji (tbis is followed by a tremendously long name) to ask the i ma m Omar to predict lhe weather and find out, if they go hunting, whether there will bo snow and rain on those days."

Khayyam thought for two day , i ndicated tbe time, and tben "wont and put the Sultan On horseback" .

From then on, the action in an- hami 's story develops liko a standard movie. 0 sOOner was the Sultan ol!, than ''black clouds appeared over the land, the wind blew and snow began to fall, and a fog enveloped the eartll. Thero was geo­eral laughter, and the Sultan wanted. to return, but tho bajji imam (Khayyam, that IS) told the Sultan not to worry, for tbere would be no mois-

1 1 8

ture !n the course .of five days. The Sultan went on hiS hUntIng triP, the clouds dispersed, and [or live days there was no moisturo, and no one saw any elouds. "

At the end, the narrator adds that Khayyam as far as be, the narrator, knows had no faith whatsoever in astrology. But I;c had to be able to forecast tho weather, because that was one of t�e s�andard demands made by sultans upon tbelf wlsemen. Consequcntly, he had some knowle�ge of meteorology. (I suppose this would be. tbe rIght place to draw some parallels between orlOntai sages and 20th century weather bure­aus, but I won't.) So let us add meteorology to the list. He v:as, tinally, a physician. His biographers have .tlme and again pointed this out. Besld.es, Omnr busied himself with the theory of musIC. An.d �esides all else, he translated from the ArabIC I lltO Persian. Last o� al l , recall tbat it was llis duty to per­�orm, dally, a host of minor dntie for the Sbah III the nature of forecasting the woather or in­terproting dreams. O� l;'csl We almost forgot, he was also a poet, a brllhant poet.

. Now comes. the question of whon be found hOle for dalliance with his beauties. About women I am not suro but about wine he definitely was in the know. Suffice it to recall t�e highly profeSSional analysis of a variety of WIDes that Omar gi ves in tho treatise uNauruz_ amah ".

ow i f all his duties are I'ccalled, one is forced

t 1 9

to the conclusion that he had mUe time indeed

to indulgo i n the worship. of Ba�chus. �h , ho

sinned of course, no quesllon 01 It. He smned,

but not excessively. In any ease his interests arc im measurably

broader than one might think it one focusses

only on his quatraillS. . The amazing thing, howevor, IS that Omsr

never says allyl.hiog about science in his verses.

He wrote an autobiography in lyrics, a confes­

sion, you might say, yet not a w�rd �bou� w�at

was truly the most important thing In his Ille.

One might think tbat such themos were o�t­

side the traditioDs of oriental poetry. Yet WIS­

dom and sages were "ery of ton praised. Al.�

, in poetry Omar did Dot care much lor tradltl·

ons if he haudled the almighty merc!ful Allah

in such rough fashion. The only thiog 10 h,s

poetry that can bo regarded as referring to scien­

ce is soma skeptical remarks OD att<lmpts at

1 20

learning tho meaning 01 being. Omar Khayya m's world view is by 00 meanS so miserable and gloomy.

Tho only way to tie things together is to pre­sume that Omar was simply showing off to llim­sol[ by rojocting all and everything and by not Iinding a Single good word even for mathematics. Such coquotry is encountered much more fre­quently than some are inclined Lo think. Partic­ularly in tho case of poets. There is nO reason to be teo trustlng when i t comes Lo skepticism.

Perhaps more credence can be given to his third cycle of "civic lyrics". Omar seems to have been somewhat of an irritable type, with a rather low opinion of those about him. But try to he calm and gOOd-natured when surround­ed by knaves, mountebanks, money-grubbers . . . i f overy single day you lear for the future, i f i L is only your high po i Lion a t the court that holds in check a pack of thick-skulled scholast­ics ready to devour you in a moment 01 weak­ness, if the position you hold can disappear at any time because 01 a simple slip of the tongue, or an uncalled-for smile.

Try to bo merry and respect those about you if every morning you are not suro how tbo day will ond, if you cannot be like others and if you have to lio every minute, every second and watch others round about you doing the same with ovident pleasure. Try all this, and note too that you have nO one ill whom you can cOn­lido, for to share such thoughts is tantamount to a self-imposed exile at best. Try all these things, and if YOll have the talent of a poet, jUllt see wbat kind of verses you will produce.

1 2 1

But If, while clearly r allting all these thjngs, you can continue working intensely, remailling a pessimis�, a cyilic and a drunkard only i n poems, hut in real l ile spending your time, energy and oerves i o bili ldiog an observatory, iovestigating equations of tbe third degree. writ­ing commentaries on Euclid, studying Aristotle and working wHh pupils... If you are capable of doing all tbis, �heLl I will read your verses with pleasure. E pccially i f they arc written in your old age and i f loving pupils remain after you.

The year 1092 was the beginiling of hard times in the life of Omar Khayyam. In tbat year, Nizam-al-Mulk-his main patron-was kjlled.

The killing was probably carried out by feu­d.l lords. The murderer was a member or one of tbe darkest, most fanatical and strange sects in human history: the Ismailians. I recall this for tbe reason that th ro is a very c urious bllt obvionsly unauthentic legend to the effect that Kbayyam, Nizam-aJ-Mulk and the founder of tbe Ismailisn Sect Hasan Sabbah all studied at one school and were childhood friends.

I n the same year, Malik-Shah with whom Omar had been so close also died.

The situation was very had undor the succes­sors, but lalor he was able to arrange his I i le. A good deal of money was needed for lho ob­servatory, but tho subsidies were stopped. so Omar had to Dlake requests here and tbere. Ho even had to wrile a historico-didaclic lreatise, "Nauruz-Nameh n , whtwe, illnong a host of anec­dotes and tale. of eagles, heautifni visages, steeds, aod wino i lbe persistent refrain that

1 22

J "MaIik-Shab provided the money for the ob­servatory, and he patronltcd Dlen of I aarnjng".

But. 1 repeat, things worked out alter all. First the soo and then tho nopbow of iu,m­al-Mnlk became viziers. Probably by force of habit they continued lO support Omar.

Meanwhile. the clergy were keoping a keen eye 00 Khayyam. That he had strayed very far from orthodox Islam was long since evident. Occasion­ally. the sullen hostility cooled off, but it in­variably boiled up anew. Omar bad to go io for writing semi-loyal treatises, but that did not help very much.

At ti mes he was intolerant. When he should have kept quiet, he entered into discussions and told sheikhs and imams to their face what he thought of them. Towards old age his temper grew worse, ho was sharp-tongued, and still, despite hjs glory and high-placed patrons, be had to make the pilgrimage to Mecca, the hajj. "And from bis hajj he returned to his home town, where morrung and eveiling be vi ited the placo of prayer, hiding his secret which will inevit­ably come to light. He had nO equal in astronomy and philosophy; in these fields he was prover­bial. Oh, i f only he had been given the gift. of avoiding i oobedienco to god . "

Thu did tho loyal muslim Djamal ad-Din ibn al-Kifti write regretfnlly in his Histortes 0/ tM Sages.

I t is likewise said that towards old age he ceased laking pupils and "grudged writing books" .

Doring the last teo to fifteen years he no longer J ived at tbe court. He somehow displeased the

1 23

new Sul tan and ei ther was asked to resign or was simply dismissed. Perhaps he left of his Own accord DOt wishing to be asked to go. He bad no family. The old man was lonely, and the greater part of his gloomiest verses were apparenlly written during this period.

His pupils were, as belore, glad to see him, but he did not seem inclined to receive thern.

To aU this add the fact that Khayyam was concoited, and "�lh the years his conceit grew; for people of that sort, old age, particularly a luckless old age, is bard to endure.

That he had a very high opinion of himself is acknowledged by his biographers. And his own treatises tell the sarna story. Evon by orient.­al standards, he would appear to have overdono the selI--elovation of his person.

This is how one of his treatises begins: "These are the rays that emanate from the throne of the king of philosophers and the all-inundating pure light of wisdom of the enlightened, skil­led, outstanding, elevated, sagacious great ce­lestial, glorious, worthy lord of the

' Pro�r of

Trulh and Conviction, the victor of philosophy and faith, the phi losopher of both worlds the 10l:d sage of bolh Orients Abulfath 'Orna; ibn I brlihYrn al·KhayyarnT . . . "

Fourteen titles, self-imposed. After that the beginning of anothor treatise is a model of �lod­esty: " . . . the honoured lord , Proal of Truth philosopher, scholar, seat of faith, king of phil: osophers of the East and West . . . "

A rather decent description, title-wise, is given at th beginning of the treatise " anruz-Namch " whicb was writtell, as you recal l , for the succos-

1 24

1

sors of Malik-Shah: ..... the learned hajji, phil· osopher of the ago, chief of investigatol"S, king of scholars . . . "

But i t is curious to note that all "special"­mathematical and phy�ical-treatises of Khay­yarn begin in a restrained, dry rnanner.

Glorification appears i n treatises of a general nature. I t may be that, 10 put i t into modern lingo, for purposes of publicity he tried to build up his image when the treatise could be read by those that held the strings of power. Natur­ally, such stratagems added yet another humilia­tion to the long list of those that Khayyam had to bear. All the more unpleasant to him was this self-advertisement. The last piece of ill luck was that towards the end he experienced real money dil!iculties.

It is doubtful whether he actually lived i n poverty, as some o f his modern biographers write. Over th& years he held high ollices and most likely had some reserves. And even to the very end, despite all tho attacks of the c1er-

1 25

gy, be remained the recognized "king of learned men" . Also, his numerous pupils could uJlport him if the necessi ty were real.

So my view is that Omar did not starve and probably l i ved as prosperously as any small trader. B ut expendi tures had to be cut. At any rate, he complains in a number of quatrains of poverty and of tbe hard time l ife was giving bim:

Ab, Lovel could you and J with Him conspiro To grasp this sorry Scheme of Things elltire, \VouJd nor.. we shat.ter it. to bit.s-and then

R ... mould it nC."'r to the Heart's DesI",I The aged �mar was apparently not very happy,

the only thing that remai ned were books. It is said tbat be died wi tb 8 book of his beloved Abu Ali ihn-Sina in hi hand.

One need not tbink that he Was always sigh­ing and grioviJlg, but he was a broken man. Ap­parently he did not work during tho last twenty years of his l i fe, either because he had no strength, or no desire. Life was at an end.

He died in 1 128 and even this dato was hit upon by accidont, thanks to a story reIn ted by his pupil an- izami 8 amarkandi. I give i t here i n full, for it is far mOre important for an understanding of Omnr tbe man than all the conjectures of bis contemporaries.

An-Nizami 8 amarkandi relates: " I n 506 ( 1 1 12-1 113 A. D.) the hajji imam

Khayyam and tbo bajji Muzaffar IsfazHri wero al the court of the E mir Abu a 'da in the quar­ter of slave-traders in Balhn. We met at a merry meeting. There ( heard that the Proof of Truth Omar said: 'My tomb shaU be in a spol where

1 26

the north wind will twice e�ch y ar scatter flowers upOn it ' .

" I wondered at tho words he spake, but I kJJew that his were no i d le words.

"Whon in 530 ( 1 1 35/36 A .D.) 1 arrived in ishapur, several years had already pa sed since

thal groat man covered hi vi ago with the cur­tain of dust, the world was without him. He was my teacher. On Friday 1 went to his grave and took a man with me to show me it. He led me lo lhe graveyard of Hira. 1 turned to the left and at the foot of the garden wall I saw the grave. Apricot and pear trees of the garden tretched their branches over the wall and sprink­

lod his grave with so many of their nOwers that the ground was completely covered. Then J re­called the words tbat I had heard him speak in Balha and I weeped, for nowhere in the whole world , from One end to tho other, have I seen the equal of bim . "

We may be quite SlI.ro that au-Niz8mi was absolut<lly sincere. H woul d be hard to believe tbat thus recalling Omar Khayyam he desired to elevate his reputation in the eyes of the min­isters of J lam. But when a man is thlts remem­bered by hi pupi ls, olle believes that be was a good man. That apparently was the most im­portant tbing. One must believe an-Nizami, for of all tho stories of Khayyam. this one is the story of a friend. Only in tbis way can we judge tbe attitude of those who spiritually were close to him.

Very gen rally, Omar strikingly resembles Ga­l ileo in temperam nt, in views and in many features of his l ife. It is as if two close relatives

1 27

lived at di llorent corners of the world soparated by an int rval of 500 yenrs.

r shall oot try to justify this parallel. Anyone with a litlle pains can do it fOt" himself. As for me, they are as of one kin. With Kipling r can repeat that E ast is East . . •

Unlike the West, which is the West.

Chapter 6

THE AGE OF PROOFS.

CONTINUED

They wefe many. Very many. No less than a tboosand.

One way or another, earlier or later, fortune throw them into company witb the fifth postul· ate and they plunged into the luring labyrinth of theorems.

Not a single one found a way out. Some were confused from the start, others

ad vanced some d istance, but the end was in­variably the same.

Some spent their entire l ives, others retreated early. Still others went on until nervous break­down, mysticism , despair overtook them, and yot others philosophically dispatched their sheets of scribbled paper to the wasto basket. The end was invariablo.

A number followed the mirage aod they were happy in the eonviction that they had escaped. But the end was still the same.

They had covered the ground of these that came before, without knowing that they were travers­ing the same falso pathways. Hope would Dare up at times, and Ooe decisive thrust would seem to have heon enough. But again tbe end was the same.

Dilettantes, professionals, naive medioeri ties and hrilliant mathematicians; Gree.ks, Arabs, Persians, E uropoans; those that stumbled after

I-UIIl 1 29

the first few steps and those that fought on pers­istently and inventively-ror ovel' two thousand years. They all met the same fute.

The fifth po t\llate was invincible. It was one of those problems that seemed too hard for tho human mind to resolve.

It would appear that mathematicians follow­ed to the letter the moUo cut on the grave of Captain Scott:

To StriW!, to Seek, To Find and Not to Yield

Like the snowy wasles of the north, the fifth postulate devoured one after the other.

Most left no traces after them. But there were Borne who perished nohly, leaving much to rem­emher them by.

In tbe graveyard of victims of the "fifth" there is one of exeeptional bonour, Henri Logendre.

Legendre was probably the greatest of tbe matbematicians hypnotized by tbe fifth postul­ate. He was engagod in the problem for many long years, attacking the monster from one side and from anotber. He found evidence and then had to reject it, he proposed proef after proof, passing from confidence in success to despair, still hoping fer luck, but at tbe end he had to admit tbat no exact solution had been found. The acknowledgement is found i n the very title of his sum marizing work tbat he published at the eud of his life (1833) "Meditations on Va­rious Methods of Proof of the Theory of Parallel

1 30

Li noS or the Tbeorem of the Sum of the Angles of a Triangle".

As often happens io science, this cautious, extensive, and ultimately pe intistic investi­gation appeared when a sol ution had already been found and published in the Vestnik Ka­zanskovo uni Wlrsiteta ( TM Herald 0/ tile Kazan UnI Wlrsity)-tbe first published work of Lo­bache" kyo

Actually, there should he no cause for sur­prise. But the fact that exactly twenty years later, the Russian Academician Bunyakovsky, who at aoy rato should have beon acquainted with tho works o[ Lobachevsky, published a similar study . . . this is indeed a sad com montary. Note-I wish to stress this once again-note the ridiculous oature of this event. But we will come to that a bit later.

In his numerous attempts through Lbo years to prove the GlLh postulate, Legendl·C displayed both persistence and remarkable ingenui· ty.

Firstly, he proved jn elegant fashion a number of theorems of "absolute geometry". Secondly, in proviog the fifth po tulate via reductio ad absurd urn he actually found a series of theorems in Lobachevskian geometry. He did not attempt to prove tbe fiftb directly, but rather an equi­valent, or "lhe sum of tho angles of a triangle is equal to ";ttl .

He first tried to pro\'o the equivalence. Even in our home-grown tbeorem, when the

postulate "a perpendicular and inclined line meet" is investigated for equivalence with the "fifth", one could already feel how closely tied

." 1 3 1

in the fifth was with the theorem of the sum of angles of a triangle.

We of course did not give proof of the equi­valence of this theorem and tbe fi lth postulato.

The complete proof of tbe equivalence of any two assertions contains two parts.

1 . One first proves "if assertion A is assum­ed , then assertion B lollows from i L " .

2. Then One proves the converse: "If asser­tion B is assumed, then from i t follows asser­tion A ". r n our case we have to prove that if tho fifth bolds, tben the sum of the angles of a triangle is equal to ".

This first part of the proof is a familiar theor­em found i n all school textbooks of geometry. Tbo second balf of the problem was solved by Legendre, and solved with a flawless techniquo. Let us see bow ho operated. First be proved that:

(1) The sum 0/ tile angles 0/ a triangle cannot be greater than ".

The proof is rigorous. And of COurse does not involve tbe fifth postuJate. fie even gives two

1 32

versions of the proof. Both are correct. Tho method is the tried and tested reductio ad abs­urdum. I t is assumed that thoro oxists a trianglo the sum of angles of which is <"+(1) and it is demonstrated that in this case we invariably arrive at a contradiction. The proofs are rather simple.

I do not repeat them, for lovers of geometry wiII then have the pleasure of obtaining the res­ult themselves.

Then follow a few auxiliary theorems and he proves a very i mportant proposition:

(2) l/ tile sum 0/ tile angles In any one triangle is equal to " tilen /I Is the same in any other tri­angle as well.

AJI proof is given without inveking the fifth postulate. By means of absolute geometry.

Now everything has been read ied for the last tbeorem of this series-proof of equivalence:

(3) J / the sum 0/ tile angles 0/ a triangle Is equal to Ti, tilen Euclid's postulate holds. Gener­ally speaking, if we accept the fU'st two asser­tions, then the equivalence is i m mediately prov­able with lhe aid of "our" theorem. I leave i t to the reader to verify this by himseU. Incident­ally, tbat is roughly the way Legondre himself proved it. There is only one thing left to obtain:

(4) Tile sum o/ the angles of a triangle cannot be less than 7t. This, nothing moro and tho fifth postulate is provedl

Legondre thon proceeds to prove it. The proof he offers is magnificent. Elegant. Simple. Unexpected. h contains everything that makes us admire

mathematics. With one sole exception.

1 33

I t is not correct I Still and al l , i t deserves our attention. The metbod is again tbat of red uctio ad abs­

urdum. We bave a triangle ABC. This is tbe most important thing and Our starting pOint. Aod tbe sum of it� angles, by bypolhe is, is equal to (,,-a.).

Produce tbe sides of angle A to infinity (some­thing wo shall need 8 bit later).

Now 80 auxil iary construction. On tbe side BC construct one �ore triangle, an exact copy of the first one. I t is depicted i 0 tbe figure­this is triangle BCD. It is so built that BD=AC and CD=AB. It is easy to see that tbis can al­ways be done. So far tbe tbeory of parallel lines does not come into our reasoning in tbe least. Now from pOint D draw a straight lino. We make only one demand: that lhe lin/! should illtersect bolh arms of the allgle A . I t would seem to be quite obvious that we could find uot one but many straight li nes Lilat would satisfy tbat con­dition.

'l'bat is enough . The problem is solved . Tbe fifth postulate is proved . 'l'he rest is simply a

1 34

matter of uncomplicated technique. Take a look at the figure. The Sum of t.he angles of tbe tri­angles CDF and BED is invariably less than " . Indeed, Theorem 1 prohibits i t from exceeding :t , while Theorem 2 plus tbe existence of trian­gle A BC precludes tbe possi bility of its being equal to ".

How much smaller is quite i mmaterial to us. More, the only thing wo aclually need is that the sum of tbe angles in these triangles should not exceed 1t. What remains are tri fles. Take a look at the large trianglo A EF. Find tbe sum of its angles. Tbis can bo done in a rather cir­cui toIlS way.

We bave a tolal of fOllr small triangles. Tbe sum of all their angles is equal: 2("-Cl)+("-r) +("-0) �41t-2a.-T-a.

Ow note that the same sum may be writlen somowhat diITerently. Out of the angles of tbe small triangles, at points C, B and D three angles cao be arranged that equal " i n each case. Tben tbere are angles at the vertices A , E and F. But tbe 8UOI of these angles is precis­ely the sum of tbe angles of the triangle AEF.

And so: tho sum of the angles of triangle AEF+3,,=4r:-2a.-r-8.

And so the sum of tbe angles of the triangle AEF +3,,=4r:-2a.-r-o.

This is followod by a chain reaction. Repeat­ing in l iteral fashion our construction for tbe triangle A EF, we bui l d a triangle with tbe sum of its angles Ie s than (,,-40.). Tben we con­struct a triangle wit,h tho sum of its angles less than (,,-Set). I n short, no matter how small a. is, we can build a triangle such that tbe sum

1 35

of its angles is negativo. But this is an obvious absurdity. Our assumption has led US ad abs­urdum. Which completes tho proof of tho theor­em. Tbe sum of the angles of a triangle cannot be less than ". The proof is indeed beautiful. I n professional terms, it could be written down in three lines. And ooly two operations in the auxiliary constructions.

But to presume that through a point inside an angle i t is always possible to draw a straight line that meets both sides signi fies that in place of the fifth postulate we have introduced its eqnivalent. And Legendre realized that. But it is such a pity to give up a beautiful solution. So, quite humanly, and somewhat plaintively, he explains that the angle chosen for LA is

that which is less than 600( i-). Then i t is easier

to believe his premise. I t certainly is easie.r to believe, but that does not alter matters be­cause it is not po sible to prove the assertion without invoking the fifth postulato. So in the end Legendre had to give up his proof.

'i'here is nlore. Let LA bo arbitraril y small. Less than aoy

preassigned number. Less than, for Instance, - 1 0

10-10 second of nrc. Evon in this case i t would be i m possi blo to provo Legendre's assumption. I f that were possible, the filth postulate would be proved straightway. I t is of course possible to prove Legendre 's bypothesis rigorously for points i nside angles tbat are sufficiently closo to the vertex. But only for close-lying points, whereas now, in our construction, a contradic-

tion is obtainable ooly when we go farther and farther away from the verto�.

If tho analysis is continued II In Legendre, nu­merous curious eqnivalents 01 the fifth postul­ate come to light.

Actually, it is thus possible to obtain a largo number of tbeorems of oon-Euclidean geometry. Hero is a problem for recreation. I n an analySiS of Legendre's premise, demonslrato the follow­ing: let L C be an angle at the vertex of a fam­ily of isoscoles triangles ACB, A 'CB', A "CB" and so forth.

A ssum/ng that in this family Ihere will always be a triangle with altitude greater than any pren.s­signed number, we will pralle Ihe fifth postulate. A rather unexpected-weuldn't you say-and quito nalural, at first glance, equivalent of tho fifth I It emerges rather simply in analysing Legendre's proof. Running ahead 01 our story, i t may be ooled that in Lobacbevsky 's geolll­etry the opposite theorem is correct.

e

137

Most of the other workers did not go so far as Legendre. They became entangled at the very beginning.

But tbero were also more i nteresting works. In tbe year 1 889 , the Italian geometer Bel­

trami found a forgotten work of his com patriot, the lesuit Girolamo Sac cheri , who as early . 1 733, anticipated and surpa.-'lSIld all tbe results of Legendre.

Up to that Lime i t was believed that namely Legendre had demonstrated that:

(i) Witbout resorting to the 61th postulate of Euclid, by mealls o[ tbe remaining axioms, it is possible to provo that the sum of the angles of a triangle cannot be g.'eater than two right angles (greater than 1 80', >,,).

(2) I f tbe fifth postulate holds, then the sum of the angles in one triaugle at least is exactly equal to 180" (to 1t).

'Vhence the conclusion: If the fifth postulate is not true, then tbo sum

of the angles in all triangles is less than 180" « 1t) .

Legendre wanted t o believe that he had ref­uted this possihility as well, but-well , we have already spoken about that.

It turned out that Saccheri had obtained al l t hese results much earlier. What is more, his investigation, his chain of theorems stretches much Carther thall tbat oC Legendre. True, his starting point was somewhat different. He be­gun with a quadrilateral, not a triangle, just as Omar Kllayyam had done a Cew centuries be­Core.

Tho construction was as follows:

1 38

1 •

III TXI

� tt! s. ee, D

1 . Take a line segment A B. 2. Erect perpendiculars at tbe extreme points

A and B and lay off On them segments A A ' nnd BB ' of equal length.

3. Connect A ' and B' with a straight Une. The resul t is a quadrilateral.

4. Take the midpoints of the bases C and C' and join them with a straight line.

5. Take the "second identical copy" of the quadrilateral AA 'B8' the quadrilateral A ,A ' . B ,B ' , and superimpose it on the lirst so that tho side 8,B' , l ies on the side A A ' .

I t i s then casy to prove that angle A ' i s equal to angle B ' , and the straight line CC' is per­pendicular to both bases. Tbe reader can finish the rigorous proof of trus theorem, nnd he can also obtain tbis result in a sligbtly different way-by proceeding on tbe basis of symmetry.

1 39

For angle A ' and angle B' tbere are three pos-sibilities:

(i) tbey are equal to 90' ( = i-); (2) tbey are acute, that is less tban 90' ( <;); (3) they are obtuse, that is great<lr than 90'

(>i). First of all, Saccheri demonstrates that iI

aoy of these possibilities are realized in any quadrilateral, then it will he accomplished in all possible quadrilatorals of this type.

He then submits proof that: 1 . If tbo "hypothesis of the obtuse anglo"

holds, then the sum of the angles of any triangle is greater than 1t.

2. I f the "hypothesis of the right angle" holds, then the sum of the angles of the triangle is equal to 1t .

3. I f the "hypothesis of tbe acute angle" holds, then the sum of the angles of the triangle is less than ".

He then proceeds to prove that the "hypothes­is of the right angle" is equivalent to Euclid 's postulate.

Consequently, in order to prove the firth pos­tulate it is necessary to refute the other two hypotheses.

Sac cheri handled the "hypotbesis of the ob­tuse angle" with speed and C<lmplete rigour.

Thore remained tbe "hypotbosis of the acute angle." It tben transpired that all this was only an introduction, for the real story only nOw hegins.

1 40

1 On over a bunclred pages Saccheri invostigated the consequences of this truly titanic "hypothos­is of tbe acuto aoglo".

He ohtai ned one theorem alter the other, each more terri ble than the preceding one, but he clearly understood that so far tbere was no inner contradiction. Then he thought he had it , the proof, the divine spark that would reduce this hypothesiS to ashes.

,ufhe hypothesis of the acute angle is a bsolut­ely false, for it contradicts tho nature of the straight line. "

Here it was that the enemy of humankind caught Girolamo SaccherL He was in error. Crud­ely.

But no, do nOL hurry witb conclusions. Sac­cheri was sti M unsure. He fol t something out of order and wrote:

"I could calmly stop at this poiot, hut I do not want to give up the attempt to prove that this adamant hypothesis of the acute angle tbat I have already uprooted is io cootradiction with itself. "

The game was thus resumed. Saccheri again sought proof, hut tbis time in

another direction. He wished to prove that if ono accopted the

"hypothesis of the acute angle", it would turn out that tbe "locus of points eqnidistant from a given straight line is a curved line".

And this is rigorously proved. Note that tbo conclusion would appear to he so absurd as to compel One to halt. But Saccberi reali!ed that this was not yet suIncient.

At this point let us take leave of Saccherl aDd

1 41

recall our honourable Ghiyalhuddin Abulfath 'Omar ibn I brahTm al-KhayyamT. I t is time Lo deliver the goods we promised and relate what ho did in attempts to prove the filth postulate.

Omar began his proof of the fifth postulate with a critique (as was usual with all others) of all predecessors. He disproved the efforts of Hero, Eutoxis, al-KIIasan, ash-Shanni an-. airizi. Also he refuted Abu Ali ibn-al-Khaisam who had taken an extremely curious and novel pathway.

Ali ibn-al-K haisam proceeded from tho hy­pothesis t,hat 8 l ine doscrib d by tho upper end of a perpendicular of given length is al 0 a stra­ight line H tho lower extremity is moved along the given straight line. (The figure shows a stick On a roller and a dotted straight line. That is how I attempted to portray the postulate of Abu Ali ibn-al-Khaisam.)

Abu Ali ibn-al-Khaisam himseU tried to sub­stantiate this assertion by reasoning about tho properties of motion.

1 42

I •

That is precisely what caused certain indig­nation on the pari 01 Omar Khayyam. He at­tacked Abu Ali [or introducing motion into geometry. This is where Omsr was mistaken.

But Abu Ali was l ikewiso in error, Actually, in his proof bo utilized an equivalent of the Euc­lidean postulate, to wit "the locus 01 points equi­distant from a straight line is also a straight line.» But he had hoped to prove it, not postul­ate it.

Howevor KIIayyam was also punished by Allah for his arrogance. It was here that he finally fumbled the problem. UnWittingly, he too em­ployed the very same equivalent of the fllth po tulate that Abu Ali llnd. We shall not go into Omar's proof, for it does not stand out a mong the others. Wo need only say that all this was included only to permit ollrselves a tiny lyrical interlude- sfter all, mathematicians reason rather wel l , whother in Greece, in KIIo­rassan or in Italy; no matter that tbey seek belp from Zeus, Allah or J esus Christ-tboy

143

strive towards nawlcss logic and i f they err, it is On a very high level. And many o( them fully realized that tho assertion that "tho locus of equidistant points from a straight lino is a straight lino" had to be proved.

The 01'1'0 ite vorsion may sound strango, but there do not oem to be any inner contradictions in i L ; tho bypothesis will be reluled only when its consequences are reduced to an absurdity.

So Saccheri renewed the slruggle. He analysed the "curve of equal distances"

with extreme care, quite rigorously, until-that moment came-the devil led him astray and ho . . . found the proof. A straight line. And again ho was mistaken. But Saccheri did not see the trap and he was Sllre lhe proof was at last ac­complished.

That would have seemed to be all, tho work was finished, the fillh postulate was proved, and the book could go to lhe press.

It did. That is, the book appeared n few months after his death (1 733) under the sensational title o( "Euclide., ab omni naevo vindicatus . . . " ("Euc_ lid ,'indicated o( all flaws, or nn experiment establishing the very first principles o( a uni­vorsal geometry").

But the con.science of the scientist was, appar­enLly, still agitated . He wrote in conclusion: "I cannot help but pOint to tho difference here between the above-given refulatiolls of both hy­potheses. I n the case of the bypothesis of the obtuse angle, tho matter is as clear as day . . . while 1 have been unable to disprove the hy­pothesis of the acute anglo otber than by prov­ing . . . "

144

In a word, then, Sacchcri was not satisfied. That is clearly felt.

The las� !.rick the dovil played with him was vicious indeed. His work remained practically unl.:nown until t889, at which time it was of puroly historical interest.

Actually, Girolamo Saccheri had brilliantly proved several dozen theorems of non-Euclidean geometry, but his starting pOSitions failed him, for he was always certain that he wouM soon prove the filth postulate.

Without knowing of the work 01 Saccheri, the German mathematician Lambert (t728-t777) went deeper. He can by rights be considered a direct precursor of non-Euclidean geomolry.

Lambert began his analysis by employing a somewhat d i fferent quadrilateral. I refer you to the drawing. In it lh ro are three right an­gles-A , A ' , and B: ncgarding angle B ' thero can be three hypotheses. That it is aCllte, right, obtusc.

�' '..,..--�. 110"

10-11111 1 45

Lamhert rather simply liquidated the "hypo­thesis of the obtuse angle." We have no time to say how this is done.

But that is not all. Lambert realized this and statod t.hat the "hypothesis of the obtuse angle" was justi fied on a sphere, if one ascribes to circumferences oj great circles the role of straight lines. This is an exceedingly interest­ing aod profouod ob ervation.

The point is that both Saccheri and Lambert refuted the "hypothesis 01 the obtuse angle" by rigorously proving that if i t is accepted the straight lines AA' and BB' arc found to in­tersect in two points.

But this runs counter to a familiar axiom: one and only one straight line can be drawn through two different points.

I ncidentally, it suIfices to prove that AA ' and BB' intersect in one pOint lor one to reject the "hypothesis of the obtuse angle ".

The reader can a m use himself by verifying the latter assertion.

Now On a sphere wbero tbe arcs of a great circlo intersect at two poi uts the "bypothesis of the obtuse angle" holds true.

After this slight departure, Lambort returned to tho plano. Ho demonstrated tbat tbe "bypo­thesis of tbe rigbt angle" is oquivalent to Euc­lid 's postulate. Once agai n i t is necessary to verify and refute the "hypothesis of the acute anglo".

Lambert began the analysis in the hope of arriving at absurdity and he extended his chain of thoorems beyond the point reached by Sac­cberi.

1 46

He proved one of the most remarkable and strange (at first glance) theorems of the geom­etry of Lobachevsky.

The area 01 any triangle is proportional to the difference between 180· and the sum 01 Its angles:

S = A (,, - 1:) Here, A is a number that remains constant for all triangles, and 1: is the sum 01 the angles of a triangle.

From this it im mediately follows that tho area of allY triangle cannot exceed

Smo.x: = A 1t'

Tho optimal case for us is when the sum 01 the angles of a triangle is zero. In turn, it tben follows i m mediately that one has only to as­sume the exist nee of a triangle of arbitrarily large area and the postulate of Euclid is prov­ed.

It is again clear at once that given the "hy­pothesis of the acute angle", or, simply, given Lobachevsky's goometry, there are no similar triangles, because thore cannot be two incon­gruent triangles with equal angle .

So tho theorom that Lambert proved may be used to propose two new formulations of the fifth postulate.

1. There exists a triangle whose area is greater than any preassigned number.

Or: 2. There exist at least two similar triangles,

that is, triangles such that the areas are differ­ent and all the angles are correspondingly equal.

1 47

(True, as you will recall, this equivalent 01 the fifth postulate was employed much ear­Iier. )

Both stalements are extremely natural and obvious.

Thoro can be no d oubt that the elemenlary consequences of the theorom on areas were clear to Lambert. However, he did not succumb to the sly aod delusi ve charm of the obvious. Quite the contrary , he wa.s enticed by the unmanago­ab le "hypothosis of lhe acute angle".

" I am even inclined to t.hink tbat tho third hypothe is ("the hypothesis of the

, acuto

. an­

gie" . -Smilga) holds truo on some kind of , ma­ginary sphere, for tho,'o must be. s�mo reason , as a resul t of which o n the plane I t IS so obd ur­ate to refutation, wheroas tho second hypothesis is so amenable."

That is absolutely correct. Indeed, consider­ing Euclid's geomotry to hold on th� plane,

one can i ndicate sucb surfaces that Will fully accom modate tho plane geometry of Lobachev­sky.

These go by the name of pseudospheri�al sur­facos and wero discovered by Beltram I . (We shnll have occa ion to oxamillo such surfaces, but meanwhilo let us seo what else Lambert has to say.) . ,

His princi llal task is Lo prove that Euchd s geometry holds true O n th

.e plane. ,!,be remark

concerning pseudospheres IS a subs,d,ary con­clusion.

And Lambert fluly real i.ed-one simply must admire the logic of t.his man-that he had not p,'oved 8nythi ng.

1 48

"Tbe proofs of Eucl id 's postulate can be car­ried sO far that what appal'eotly remains is but a trine. However, a thorough analysis domon­strates that the whole esseoce of the matter l ies io this apparent trifle. It ordinarily contains either the proposition being proved or the pos­tulate equivalent to it .. "

That is his conclusion , and it is a flawless, precise one.

Without a doubt he disentangled tbe problem better than any of his predecessors, be carried the analysis farther and enumerated a number of absurd (from the viewpoint of Euclidean in­tuition) conclusions to which tho "hypothesis of the aCllte angle" led, but he did not find a logic­ally flawless proof. And "arguments called forth by lovo or ill-wil l " as he classi fled them are not tbe arguments of a geometer.

What is-more, deep wi thin h i m Lambert ne­bulously snspected that perhaps the fifth pos­tulalo was, in general , unprovable. He discus-

1 49

sed the possible truth of the "hypothesis of the acute angle".

In his enthusiasm for the unwinding chnin of bis theoroms, he unwittingly broke away from his academic style. "There is something enchant­ing here that even makes one wish that the third hypothesiS be true.

"And still, despite such an advantage, I sho­uld like this not to be, for i t would involve a whole series of other i nconveniences.

"Trigonometric tables would then become in­finitely extended, similarity and proportionality of liguros would di appear al together, not a single ugul'o could thon be represented other than in nbsoluto magnitude, and astronomy would find matters very d i llicult . . "

Tho words "despite such an advantage" refer to a remarkable conclusion of oon-Euclidean geometry-the existence of an absolute unit or length.

As we see, Lambert was in possession of this concept too. (We shill come back to the absolute unit of length later on.) Unfortunately, the work of Lambert was l i kewise overlooked by mathe­maticians. To the very end of his days, Lobach­evsky knew nothing of it.

lt is not clear, however, whether one should regret this or not. I f Lobachevsky had known about La mbert 's work, i t might have saved him a couplo of years or work, but it also might have q uenched tho interest in the problem, for he might have convincod himself that all the initial results had boon already achieved.

Be that as it may, ho did Dot know of this work.

1 50

There was very little distance to cover for Lambert to become the a uthor of non-Euclidean geometry. Actually, only one thing had to be done.

And that was to state firmly that the "hypo­thesis of the acute angle " stood equivalent to the fI(th postulate.

Neither the fifth postulate nor Its countersta­temeot (tho "hypothesis of the acute aogle" in Lho terminology of Lambert) 101l0w from the other axioms. Tbey 8re quite independent. Which ooe is accomplisbed in our universe is simply a question of experiment.

One had only to formulate clearly these, one would think, simple thoughts and believe Lhat that is exactly the way things stand, and the rest would have been a simple matter of tech­nique, so to speak.

A mathematician with the endowments of Lambert could have rolatively simply proven a few dozen mOre theorems and could have, with just a little effort, systematized them and thus constructed tbe entire system of non-Euclidean geometry.

Let U$ stop hero for a moment. The laws of scientific creativity aro hazy in­

deed. Discoveries aro made in a variety of ways; some accidentaUy, others appear to crown the efforts of years of exc.ruclaUngly intense work. Aoything is possible. But one law is unalter­able. Any ultra-brilliant provision �hat is in­compreheosible to contemporaries, appears after the passage of fi.fty years (8 hundred, at the most) natural, simple and al most !<ri· yia!.

v'fv \ : I \ I "

" ,

I n order to appraise a piece of wOrk properly, one has to attempt to shed oneseff of the range of knowledgo tbat bas since accumulated and mentally picturo tho epoch under study.

Let us try to conjure up a picture of a geometer of the end of the 18th century or the beginning of the 19tb century i nvestigating the fifth po -tulate.

From an early age we are told that the geom­etry of Euclid is tho most perfect creation of Ihe human mind. We are not only taught that but we ourselves. as the years pass, succumb more and moro to the enchanting logic of the proofs, sinking deeper and deeper into the cold beauty of the draWings, the lem mas and the theorems, into tho illusive kingdom of the in­tellect.

We live in a closed world , and the only laws governing our thinking processes are the laws of this world. Geometry has long since changed

1 52

from what it was in ancient times-"the science of the measw'ing of the land". The problem of its reality, of its practical accompli hments in our world was solved so long ago tbat today not a person gives any tbought to it any more.

Geometry has so long since risen from the sinful earth to the mountain peaks of tbe ideally abstract . . .

The very idea that geometry still can and must be veri lied by experiment, that geometry is actually only ono of the di vision of physics cao never enter our mi nds, for at the very be­ginning of our days at school we learned tbat geometry has been in ma.n·s faithful service for sevoml thousand yeBI·S.

True, in recent L i mos the entire system of axioms has been undergoing a certa i n critical review.

1'rue again, the notorious fifth postulate is a shock to our aesthetic feelings. But that is a l l .

There can bo 00 doubt wbatsoever of the truth of tbe IlCth postulate. The only thing we are doubtfnl about is whether it is 8 postulate or not. We simply suspect that a theorem has found its way ioto the axioms.

To suspect the fifth postulate a. such, would mean to put the whole of geometry in doubt. A od if tbat were so, tben there would be just as many grounds to suspect, say, the axiom that "one and only one straight line can be drawo tbrough two pOints". Or any olber axiom . Then one would have to revise tbe notion of l ines. And the axioms of arithmetic. Then tho ideal structure of ancient- proportions will turn into a shapeless conglomeration of fragmonts. That

1 53

is possible. But that is the work of a barbarian, a vandal, not a mathematician.

Tbere is nothing more perfect in the world t.ban geometry, and hero there is only one min­ute blemish that embarrasses us-the Iifth po t­ulate.

As for the other axioms, tbey aro so obvious tbat no serious problem could ever arise. Sligbt modi fications and more polished formulations? Yes, those are possible. But of no interest, when One comes down to it. That is how we think, that is how mathematicians have thought for the past 25 centuries. To give up this faith is to give up everyUling we bave.

We strive towards beauty and harmony in our Euclidean geometry, and to,�ards ao ultim­ate finish to tbe edifico it is. Least of all do we contemplate destruction.

And we aro convinced thut to think one could chango a single axiom in Euclid's geometry without arriving at a horrible absurdity is to explode the whole system.

l ust OnO thought is needed, one phrase, but that thought is such that will change our entire world view.

I Chopter 7

NON-EUCLIDEAN GEOMETRY.

THE SOLUTION

In 191 1 the bibliography on non-Euclidean geometry totalled 4,200 works. Today, that num­ber has rison to between 20 and 25 thousand.

Not less than a thousand of these are studies of an historico-biographical nature.

This is only an esti mate, of course, but it is based on such definite data that the actual numb­er of works should be substantially greater. We shall take it to be one thousand . Probably at least two hundred books and articles have been d()voted exclusively to Lobachevsky.

So why put out one more? That is exactly tho question tbat bas p lagued the author since he began-even before he began his work-and it still stood after the book was finished. One conso­lalion of course is that such problems arise in any field. Just about in the year 1968 B.C., a forgot­ten a.ocient Egyptian pessimist and skeptic com­plained bitterly: "If I could only say something that has not already been said many times before ' "

That i s slight comfort, a l l the more so since tho amount of writing done in tbe past foUl' thou aod years has nearly drowned humanity, though if we are to believe the cIa iCB, a truly gr at book is wriLten once in a hundred years. But a rcasonnbl() person of middle ag() (the autbor, by the way) cannot think in such terms. So we are left with the question I Why?

l S5

Indeed, what can J add to the many many volumes devoted to the Ilistory of geometry i n general and non-Eucli dean i n particular and to the general theory of relativity in still more particular?

First of all , we can say that the book is su­perficial. I t is, and could not be otherwise.

Even aside (rom purely special questions, about two years of hard every-day work would have to be spent in spading up and lOOking through the more important hiograpWcal sources. But that i n i t<Se1f is not yet sufficient. A conscien­tious biographer has to make a tborough study of all tbe works of the person in qu tion and investigate Jlainstakingly the re ponse of t he scienti fic col leagu 5 that were acquainted with Wm and/or his works. He should . . . there is even mOre that he should do.

Incidentally, Lobachevsky has such a biogr­apher. It is Academician V. F. Kagan . He wrote a magni ficent and profound hiography of Lo­bachovsky. A Ii ttle too profound, perhal' . It is not very easy to understand .

As a dilettante i n mathematics (and for a va­riety of otber r 850ns) I reaHzed that I could not comp te in those respects with Kagan. Nei­ther could I compete with many other biograph­ers and investigators of Lohachevsky and of other scientists that have been and will bo ment­ioned in this book.

That brings us straight back to why I wrote tbis book. I t is important to know, otherwise tb se pag S would not have been written (maybe tbaL would have been tbe best version).

1 56

My idea was lhaL nobody had yet \VriLten ahouL Lhese heroes as human bei ngs , not as outstand­ing mathematicians, mon of genius, buL as norm­al (or almost normal) people.

That is what I set out 10 do-wri te a r al hook ahout real men . About strong men, brilliant men, celebrated men, great mell,-but more im­portanL ahout the human-interest down-to-earth doings of these peop le, as people.

o to me and yon, they are-here-ord inary people and not geniuse . The very novel idea tbaL J bave up my sleeve is that a person should, abOve all, be a person, a man, a human being. And even such a triO as a bad temper, a d isagreable disposi tion and a di fficul t nature can di porse any kindly feeli ngs stemming from his work.

Starting in tWs key, I lind i t hard to decipher my own feeling with respect to J Onos Bolyni.

His gi£ts were amazing. An inoxplicably hril­Iiant talent. His style alone proves thaL he was a mathem tician by tho grace of God . I t was only later, in the 20th cenLury thaL works On mathematical logic began to be written il l his style. NOL a si ngle exira word, ultimately com­pact, flawless logic, exceptiona l clarity of reas­oning. I n tbe cenlral pl'ohIem-that of tho cons­istency of non-Euclidean geometry, he advanced farther than Gauss and Lobacbevsky. Actually be was very close to the basic idea of proof. l Ie did not find it, but he clearly realized the direction i n which iL was to he sought.

nero he was ahead of a l l the rest. It is quite possible that, for hi mself, he form­

ulated the ideas of non-E uc lidean geo metry so-

1 57

mewhal earlier than did Lobachevsky. A bout 1823.

True, his work was published two years later than the first work of Lobachevsky (t831).

But, generally speaking, let lovers of prior­ity debate that issue.

For that matter, still earlier a German lawyer (at One time professor of law at Kharkov Uni­versity), Ferdinand Schweikart had mastered the basic elementary conceptions o[ non-Euc­lidean geometry. True, he never published any­thing, but his nephew, Taurinus, whom he got interested in this problem, put out a booklet.

Though an incomparably weaker mathemati­cian than any in this story, Taurinus came very closa to a solution. He developed non-Euclidean geometry in rather some detail, solved a large number of subtle problems, but he did not bave a clear notion of the matter. In the end he arrived at tho same point tbat investigators of tbe filth Jlostulate had-an attempt to prove i t and, COn­sequently, the truth of Euclidean geometry.

This is all the more surprising since at the same time he would seem to havo an excellent grasp of the consistency of hi non-Euclidean con-structions, yet. . .

.

We have already mentioned the fact that ac­tually only one single idea was needed for the construction of non-Euclidean geometry. Any­one who stri ved to prove the fifth po tulate by reductio ad absurdum invariably camo to theo­rems of non-E uclidean geometl'Y. Lobacbovsky himself, writing of Legendre, said:

"I find tbat Legendre ti me and again took the pathway that I had so luckily chosen. "

1 58

. But it was the basic idea that Legendre la­cked. It was this sale idea that was absent io mathematics for over two thousand years.

I t was rU'st expressed , but was not fully reali­zed, by Lambert; it was stated nebulously by Schweikart and Tanrinus; Gauss had been incli­ned in tbat direction for a long time without actually mentioning it. I t lVas only Bolyai and Lobachevsky wbo formulated it clearly.

As to rigour and profundity, the first (and only) work of Bolyai exceeded ali otber .

Later all, working intensely, Lobachevsky in­vestigated non-Euclidean geometry much more hroadly and in far greater detail, hnt i f we com­pare the first works, the more brilliant is tbat of Bolyai.

The brilliance of his talent WIlS evident iu all things.

He was not only a mathematician of genius, he was an extremely gifted musician. At the age of ten he had already written a number of com­positions. Later he b�amo an accomplished vio­linist.

This does not exhaust the talents that Bolyai posse d. Apparent ly, he was one of the best fencers of the country. This is no i mple matter in any country, but particularly so in Hungary.

Finally, his social views make him closer to us tbaD any of the otber persouages. Ue waS ho -Lile to all nationalism; Il n ardent supporter of the Hungarian revolution of 1848, he thought in­tensely and profoundly on problems of social being. His ideas were akin to those of utopian communism. Towards the eDd of his lifo ho got tbe idea of constructing a mathematical model

1 59

of an ideal state with tho aim 01 finding a per­lect blueprint lor universal happiness.

The "theory" was called "The teaching of uni­versal good ".

In mathematics he combined the cold reason­ing of the fencer with the poetry and i nspira­tion 01 the musician.

But there is one thing that hopelessly spoils this charming i mage. Bolyai had one lUIldamen­tal flaw-his jealou , touchy, egotistical ambi­tion coupled with a very UIlpleasant tempera­ment. That is what d termined the course 01 his life. In the end it ruined him.

Trne, I am afraid to be too categorical i ll such cases, and quite naturally all that has nothing whatsoever to do wilh any aplll'alsal of his work, but it is important when discus ing his attitude towards his lellow men. And Bolyni , 1 believe, belongs to that category of people who apply es­sential.ly different criteria to th mselves and to others about them. That i why 1 do (lot finrl him very pleasant. 1 would like nothing better than to learn that I am wrong.

As to mathematics, his place in mathematical history is clear. Together with Lobachevsky he

1 60

enjoys full rights 3S the creator of non-E uclidean geometry.

True, there was yet a tbird person. And here it is that we enter upon that arduous

pathway of priority litigation, though, in my opinion, such questions merit hardly a hun­dredth of the attention that tbey so often claim. But the history of non-Euclidean geometry is 01 exceptional interest from a puroly human stand.

The first to como to the ideas of a non-Eucll: dean geometry was the GOttengen genius, the prince of mathematicians, the colossus, the tI­tan, the first mathematician of the world, no other tban Carl Fl'iedrich Gauss (1 777-1855). Tho­se were only a few of the nu merous titles that ho boro during his liletime, and-tbere are no two ways about it-they are all deserved.

Gauss was unique among geniuses. As a ma­thematician he was, without any doubt, far abo­vo Bolyal and Lobachovsky. He was simply a scientist 01 a different category.

So it was Gauss who wrote time and again that the basic ideas of non-Euclidean geometry were clear to him oven at the ond 01 the 18th century.

I am positivo that he wrote the pure truth. But he did not publish his resuIts elther at that time or at any time later. The results Gauss ar­rived at can only be coniectured from his letters and diaries that were published aftor his death.

Why did he not publish his investigations? Tho reason seems to bo known, for he himsell gi ves it a number of ti mes.

U-1Je.7 1 6 1

For example, an excerpt Irom a letter to the celebrated German mathematician Be I. It was written alter Lobacbevsky had published his work. True, Gall had not yet heard 01 it.

"Most likely 1 hall not be able very soon to propare m y exten ive invesligations into this problem so as to hav them published. I t may even be that I hall relrain from doing so lor 1 fear the "Geschrci der Bootier" (tho cries of the Boeothian ) that ,,�ll rise up when I express m y views . ..

So earl Fri drich Gau was afraid 01 tho "cri­es 01 tho Booothiaos".

I n this day and age, cia icism has to be de­ciphered. Whether justly 0 or not, I do not know, but th inhabitant 01 Booothia were con­sidered i n ancient Greece to be the most dull and thick-skulled 01 all, and in tho age of Gauss and Lobachevsky, tho age of cIa icism, quota­tions from the classics were much in vogue.

1 havo always been rather dissatisfied with Gauss' explanalion.

It very well may be that he stated one 01 the reasons, perhaps OVeD the basic ono. But there wer undoubtedly otbers.

Gauss waSll 't the kind to hush up a discovery of such exceptional, unparalleled signi ficanee for

foar of losing his authority. All the moro SO tbat bo was risking very littlo, lor his authority was so bigh i n the world 01 matbomaticians that if Lobachevsky's memoir bad appeared with his signature, all the "Boeothians" woul? have ac­clai med it and applaudod non-Euclidean geo­metry, bo"�ng once moro i n reverence to tbe go­ni us of Gauss.

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Incidentally, sometbing ot tbat nat uro actu­ally took IJ laco. Lobacbcvsky's works attract d attontion only after Gau ' death, when Gauss' attitude towards non-Euclidean geometry beca­me known. The n w ideas th n in tantaneously became under tood and recognized. I t tho writ­ing had been that of Gaus thero would havo been no doubts whatsoever.

I t was quite obvious tbat Gall did noL in tbe least underestimate hi po ilion in tho com mu­nity of mathematicians. I • m uro that, l i ke tbe prince of matbematiciaus tbat he W8S, he could call bis vassal to rdcr if there Were any unrest, so that tbe "Geschrei der Bootier" taken all by itself could hardly havo frightened Gauss that much.

Tho crux of the matter lies et where. Whether Carl Friedl-jch Gauss was a good man

or bad has been under d iSCll ion by his biogra­phers lor a full century, but one thing is certain: Gaus gave his whole Iile to mathematics.

To him, mathematics was aLI. It was just as necessary for hi m to solvo problems as to breathe, eaL and drink. I t was an in tinct. Ther were no sucb things a unat tractive problems to Gauss. He could spend months on tbo most rou­tine, monotonous computational job. He could compile table for weeks On end, and with tbe greate t 01 plea ure he would do work that in tbis enlightened age is handled by technicians, l ike listing weary colum or figllres-Ior Gauss they wero apparently ini mitably alluring.

There i not a di vision 01 mathematics tbat is witbout certain fundamental contributions made by Gau . A simple enumeration would cover se­veral pages of texL

II' 1 63

He is amazingly like Isaac Newton in tempera­ment, tYIIO o( clIO racIer MId way of life, and iL seems no accidont that Newton was his favourite hero. Like Nowton, Gau was xlromely am­bitious. Yet this was not the ambilion that burnt up Joinos Bolyai.

The first requirement was that ho hi mself must appraiso his work, he must be positive, and he must be able to say to himself: "Gauss, that is good . "

So i t was that numerous studies awaited publi­cation for tbe sole reason that they were not fini­shed , and there was much to do. Gauss elimina­tod from his Ufe everything that could in any way distract him from his work. Gauss prayed in the temple of a cruel god, he believed with (anatlcal intensity and, like every fanatic, ho was limited.

He was barsh, even cruel, in his attitude to people, though from bis own standpOint he was just. But this freeziog condescension is quite jus­t ifiably perc ived as indifference hordering on rudeness. His was a complicated nature, a try­iog person, such that can call forth one's admira­tion, worship, but never love.

Abel, Jacobi, Bolyai are some of the brilliant mathematicians cruelly hurt by Gauss.

But he did not try to offend, and there is no reason why people write that he was a consumma­te egotist and that he sllffered when som One el­se obtained outslandiag results. Tbat is not so. I t is definite slander. Gauss always paid full due to the genJus of bis brothren. But it was not his fault that their results so often simply coincided with what he himself had achieved but had not

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yot pllblishod, and he had not yet published them for tbero was much still to be done-tho work was so ofton not finished.

Gauss has been reproached, and sorely so for his review of tho \York of Abel. Why?

'

�e wrote: ''The works ef Abel are above my praIse, fer they are above my own studies.»

How can it bo that people think that Carl Gauss simply lied? That he never took up simi­lar problems and did not obtain similar results? Or is he supposed to play the part of a noble father? Is it not enough that teos of fllodameo­tal theorems which he had proved but, for a va­riety o( reasons, had not published, were pllb­Iisbed, by others so that the fame of discovery had to be divided?

Gauss did not read the papers sent to him (or roview aDd did DOt allow his friends to give him the memoirs of other scientists to read.

He wished te serve his god in such (ash ion that no oDe (aDd above aU, he him If) could entertain the slightest suspicion of otber people's phrases io his teachings.

His love (or mathematics was inseparable from jealollSY. This was the lovo of a man the love of a 1uslim. And he was cruelly hurt i f One o( his many "conclihines" should so much as smile at aoyone else. But he also knew that only the deserved entered his harem, and this consoled him somewhat. He was always ready first to re­cognize the merits of a rival. But it did not givo him jey.

Thus Gauss lived an even, quiet, monotonous life, while in his brain thero continuously rose up and vanished marvellously magnJficent, im-

1 65

measurably more beautiful worlds than that in which be existed.

I t is wortb repeal ing tbat Gau deservos wor­ship, hut i t is very hard to love him. In fact, if it wero not for Archimedes and Einstein one might have to accept the fact that a g.ni

'us of

mathematics cannot be other than tbat. A hundred or so years ago, I think it was

E merson who said a very curious thing to the effect that each may take what ho wants and pay the full price.

The price of Gau s and Newton was extremely high. Einstein and, as far as I can judge, Archi­medes, too, recei vod everything that those two had, and got around paying for it.

Another mau of the same mould was Nikolai Lobachevsky. Although he was brilliantly gif­ted, he was a scientist of a diflerent class than this quartet, but to my mind ho was much more ploasant than Gauss.

I must repeat that I would helieve Gauss to be a superior being, a man of tbe future or a descen­dant of a Martian sage, if i t were not for Eins­tein.

One of Gauss' loves was non-Euclidean geo­metry. What was i t that dissatisfied Gauss and wby did he Dot publish his studies? Here we again enler onto the slippery path of psycbo-de­tectivo analysis, but it is too late to give Ull. !"irs! 01 all, the lacts.

1 . Gauss wrote in Ws private letters-and thoro is no reaSOn to douht that what he wrote was the truth-tbat tho basic ideas of non-Enclidean geometry were clear to him as early as the end of tbo 18th ceutury. At that time Lobachevsky

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had not yet begun studying i n the gymnasium (secoodary school), and Bolyai hud not oven beon born.

2. The exceptional signi ficanco of the problem itseU is obvious. I t is inconceivablo tbat Gauss could have underestimated it.

3. I t is a fact-and we shall come back to this again-tbat Gauss made several attempts to mea­sure tbe sum of the angles of a triangle formed by tbe vertices of three mountain peaks. Conse­quently, he allowed for the possibility that the geometry of nature might be non-Euclidean.

4. An investigation of Gauss' archives after his death revealed only very meager sketches, and nothing i n the way 01 a systematic conside­ration of non-Euclidean geometry.

5. After reading the works of Lobachevsky and Bolyai, Gauss-in both cases-stressed the fact that thore was nothing essentially new that he could find for hi msel f.

True, there is a slight complication here. The point is that Lobacbevsky gave an incomparab­ly broader view of tbe possible consequences of non-Euclidean geometry tban Bolyai did. In tbis sonse, their works cannot be compared.

For instance, Lobacbevsky carried his inves­tigations to a stage that demanded the apparatus of mathematical analy is. One of his works is specially devoted to the application of "i magi­nary geometry to the computation 01 definite in­tegrals".

In tbe fragments tbat Gauss left, there is not even a bint that he had reacbed such problems. Nevertheless, one is led to think that Gauss was perfectly sincere in his letters. I f he did not de-

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velop non-Euclidean geometry so fully as Loba­chevsky, there can be no doubt that he could have very easily . . . if he had wanted to.

He of course foresaw, in principle, all the routes of non-Euclidean geometry into analysis. It is very likely that he could have, without any trouble, doveloped the scheme of non-Euclidean geometry much more profoundly and fully be­cause his genius and matbematical range were unparalleled.

This last statement is beyond the shadow of a doubt.

6. Be that as it may, Gauss did not invest his ideas in any kind of finished form and did not publish anything. It is only his JeLters that show he possessed a great deal.

Let us try to figure out WHY. We reject Gauss' ovm explanation, which is

about as convincing as the statement of a ship's

o

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commander to the effect that he failed to carry out an important assignment for fear of the ad­verse reaction 01 some fishing boats that might be lingering on the horizon.

Well, that may be going too far, but uno SPOC­tre could have pursued Gauss. To accuse bim of mediocrity, as Lobacbevsky was accused, is out of the question. No ono would have dared to. But the suspicion that Gauss might simply have gone mad is a possibility, lor One should not un­derestimate the conservatism of mathematicians (scientists in general , for that matter).

The whole story of non-Euclidean geometry is the best instance of this nature. Even so late as the seventies of last century, when it was al­ready clear and the noncontradictoriness of non­Euclidean geometry had been proved, when its ideas had seen brilliant development and were support"d and strengthened by the authority of all the greatest mathematicians of the world , there were sWl professional mathematicians, some in the rank 01 academicians, that continued to propose all manner 01 proofs of the fifth pos­tlllata and even refused a serious and objective consideration of the geometry of Lobachevsky.

Jncidontally, one of Lho most consistent, im­placable opponents of the now ideas was B uuya­kovsky, who in 1853 completely ignored the works of Lobachevsky.

However, there is no need to overemphasize the conservatism of mathematiciao . Gauss rea­lized full well that the best scientists, tbe yOIlIl­ger ones first, would grasp and properly appraise the new ideas. Too, he was not the kind to re­treat in the lace of possible unplen antness.

169

Firstly, the most prominent featufo of his bo­ing was a striot, demanding pride, even arro­gance. Seoondly, ho never betrayed mathematics, for he worshipped it with the frigid passion 01 a puritan. He would do anything for mathematics, so no speotres would have stopped him.

The next suppo ilioo, to the effect that "Ga­IISS did not coo ider the problem so very signi fi­cant and for tbis reason he simply did not have the t i me to investigate non-Euclidean geometry further" is just as absurd.

But that would i mply that Gauss was just a med iocre mathematician devoid 01 much 01 what is called mathematical culture.

'VIlat is more, Gauss' numerous leUers that bring in the topic of non-Euclidean geometry, constantly treat i t as a problem of the first rank, central to all mathematics.

So why indeed did Gauss not turn his energies and his amazing unparalleled talent to this pro­blem? Why did he remain silent for so many years allowing, in the end, Lobachevsky and Bo­Jyai to outstrip him?

To get thi ngs illto better perspective, let me give a picture of the whole problem of nOn­Euclidean geometry.

As you recall , when speaking of axioms and axiomatics, we agreed that only two demands are impo cd on the axioms of any mathematical theory-completeness and independenco. Thecom­pleteness of a system of axioms im plies that any conceivable assertion relative to t.he primary notions can be IJroved with their aid.

Axioms permit investigaUng everything. Let lIS not go too far into abstract logic, a few con-

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crete examples will better servo our pur­pose.

Suppose two chess players have studied the game from a textbook that by accident failed to mention a situation in which one of the pla­yers cannot make a move without infringing the rules and his ki og is not under attack. This si­tuation is conveyed by a Single chess term-sta­lemate. Our players would not know what to do. The game could not go on, and they would simply have to introduce another rule, another axiom . In chess this sitnation repre ents a draw, in checkers the side that ini tiates the stalemate wins.

But some new axiom has tu be chosen. Their system of axioms proved to be incom­

plete, for it did not provide for ali possible situ­ation .

One could t.ake football with its elements of 1 1 players, the ball, the referee, goal, field, etc. And again the axioms (rules of tbe game) have to be fOl'mulated SO as to be able to judge unam­biguously about any possible situation of tbe elementary enUties.

That accounts for the constant arguments and fights that break out in scruh games where the partici pants do not I'ave a full code of rules­hellce the danger of neglecting axiomatics. Though, as a rule, the teams first come to certain modifications of the terms about the game as applied to the local field; setling up a complete system of ax ioma tics eVen concerning such a simple game as footba ll is by nO means an easy matter. Whence, agai n, all the trage­d ies.

1 7 1

Or, to take n final case, the criminal code should in principle provide a complete system of axioms governing all possible situations hazar­dous to society.

The requirement of completeness would seem to be clear enough now. Would seeml I f only things wore as simple as I have pictured them here, mathematicians would be in ecstasy.

I f I may be allowed a few naive suggestions . . . . A s)'stem of axioms relative to n given group

of basic (primary) notions is complete if for any general propos! tion A (any theorem) relerring to the given primary motions, We can resolve the following question on the basis of these ax.ioms: "Is A true or false? "

Now think over what has just boon said. To verify the completeness of the axioms we must do no les than prove or refute every concei vable theorem. If that is done, then any mathematical discipline would be exhausted to tho end. Ex­hausted in the same way that the game tic-ta.c­toe has been.

Our demand is obviously unrealistic. Even in such a comparativo;ly simple system

a.s chockers, we cannot precisely investigate the basic theorem and answer the question: what re­sult should an ideal game gin?

Still less do we know of the situation in chess. And less still can we provide for and analyse

all the theorems of geometry, arithmetic and, in general, any mathematical discipline.

'l'hat is the reason why the whole problem 01 the completeness of (I system of axioms must be formulated quite dil!ol"ently.

We cannot here delve too deeply into tbe

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depths of higber mathematical logic and so wo sball not give in full the problem of the comple­teness of a system of axioms. Perhaps a beautiful and incomprehensible phrase will suffice: a sys­tem of axioms is complete if any two interpre�a­tions of it containiog real content are isomorphic. Let us now examine this splendid statement.

The idea of isomorphism was introduced by Hilbert, and is one of the most elegant Iinds of thi� century.

But we will not spe(tk about isomorphism. An inst(tnco in which a system of axioms was

incomplete has already been given, and most likely the reador can thillk up a few more cases.

The requirement of independence (or lIoncon­tradictoriness)· would at Iirst glance seem to be clearer. Let us phrase the independence require­ment rigorously.

Let there be a group of axioms 1: (this letter, called sigma, is ordinarily used to designate a sum).

Let there be some kind of assertion A. And the opposite assertion is i\ .••

• I n Ch. 3 we wroto that th o requirement of 000-cool..rad ictorilloss JeollSISl..OnCy) of axiolUS is a special caso of that of in cpcndcnce. Somo textbooks, howe\Tcr t slaw t.b.at a syslom of Dioms must satisfy both the re­quirement of consistency and that. of independence. The point is that what is needed I practically .speaking, is consistency of tho axioms. It is even convenient at limes to chooso somo of tho axioms as independent 3.xioIllS. Therefore, tho requirements of consislency and independence are frequenUy separated.

• • 1'be bar on top or tho letter is a mat.hematical symbol to denote tho contrary assertion.

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Then A is independent of the group of axioms 1: if neither A nor l\ contradicts tbe group of axioms. In other words, both the assertion A and the contrary asset·tion }l. are compatible with the group of axioms r .

All of this i s rather elementary logic, though i t is probably a bit unusua l , and so appears to be complicated. That is why we shall explain everything for the case o[ the filth postulate.

We wish to prove that tho fifth postulate is independent of all Lbe other axiom o[ Euclid's geometry (here, the liIth postulate is aD example of Our assertion A). We expre an assertion that is contrary to lhe fifLb postulate (assertion }l.). For instance, we state that through a given pOint al least two parallel straight lines can be drawn to a given trnight line. (To simplify mat­ters, we shall writo the postulate, which is con­trary to the fifth, up ide down, like this A 91uID1sod).

We now prove that A 01Ulnl od does not con­tradict the remaining axioms of geometry. This means that no matter how far and wide we de­velop the Ilossi ble conse" uences, we shall never come to a logical contradiction.

So far so good. Now be careful. How is One to be sure that Lbere will never be any contra­diction?

Suppose we have proved twenty noncontradic­tory theorem . This is nO gnarantee that a con­tradiction may not appear in Lbe twenty-first. A [ter proving One hundred, we can expect a fai­l ure in the 0110 hundred and first. The same goes for the thousandth. It is quite clear thal in this way wo will never obtain a rigorous proof of

1 74

consistency. But we mllSt, [or otherwise the pro­blem will remain unsolved. It would seem to be a hopeless task. There do not appear to be any concei vable pathways, other than what we have described. A bsolntely hopeless.

Let us stop here again and cOllcentrate for a moment.

[n the laLlor balf of the nineteenth century, rougbly 20 year after tbe deaths of Lobachev­sky and Gauss, a rigorous proof was gi ven of the noncontradictorincss o[ !lon-Euclidean geometry. The proof was unexpected, im probable. We will relate it a bi t later.

The point is that neiLber Lobach vsky nor Gauss even sllSpected possibili ties o[ this nature. Remember one thing: the very possi bility of fun­damentally new ideas tbat would belp to prove the noncontradictoriness of non-Euclidean geo­metry was in tho days just as inconceivable as tbo po ibility of determining the chemical com­position of a star. I ust as inconceivable as over­throwing tho mechanics of Newton. I ust as nn­thinkable as a thermonuclear reaction.

There was, at tbat timo, still nO clear concep­tion of axiomatic •. Ther was complete chaos i n all definitions and axiom o[ geomotry, the di­sarray that was tho legacy of Euclid.

Mathematicians had not yet formulated for themselves practically anything of what has just been written.

It was only the brilliant Bolyai who was gro­ping in the right direction. f am afraid that even Gauss was not ful ly recepti ve to his ideas. There was only a semi-intuitive conception about tbe notions of independence and consistency.

1 7S

But then- Well , then it i. clear that it is al together i m possi ble to prove logicnlly tho "in­dependence of tbe fifth postulate". 0 matter how long the cou is tent chain of theorems obtai­ned by meaus of A 01\l1" 1sod , there will always be the possibility that the contradiction is con­cealed stiU deeper. There wil l be a feeling that we simply have not yet reached it .

' n despair, of course, One could resort to ma­uipulations that are totally alien to mathema­tics-experiment. For if it were found-some place in the universe-that oon-Euclidean geo­metry is accomplished, theo the problem of non­contradictoriness would ipso facto be resolved.

You recall that Gauss allempted to verify what tbe Silln of the angles of a triaogle is equal to. Quite independently of him, Lobacbevsky as­ked tbat similar measurements be carried out. Lohacbevs1 .. y chose a better object. At his re­quest, astronomers at Kazan observatory measu­red the angles of a triangle whose vertices were three stars. i n hoth cases, the sum of the angles proved equal to pi (r.) to within experimental error.

This result did not refute anything because even if Euclidean geometry were not accomplished in our world , any deviatioo from pi might be very slight.

As to proof, thero was even less of that; noth­ing in fact.

So what havo we? Reasoning in accord with rigorous logic, 000 thing remained, and tbat was to conclude that tho question was open. And will probably remain so fer ever. 'fhat, in errect, is what Gauss once said. (In a private letter, na-

1 76

•

turally.) Here is what he wrote: '" incline 1ll0ro nnd more to tho conviction that the necessi ty of our geometry cannot be proved rigorously. At any rate, by the human mind for tbo human mind. "

This is open to tho following interpretation. I do not see any cooceivable possibility of pro­viog tbat a postulate cootrary to the fifth postu­late (A 91uln1sod) does not contradict the other axioms of geometry. And although intuition of course hints to Gauss that tho correct anSwer is "oon-Enclidean geometry is just as consistent as Euclideao", there is no proof.

The prohlem remains unsolved. And if that is the way things stand, it is en­

tirely i n the spirit 01 Gauss not to publish his resul Is. He could not risk his reputation and publish a paper of wbich he was not one hund­red per cent positive. Ho did not possess the idea that would permit cutting the knot and resol­ving the matter. So what next? At this point,

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factors enler whicb a,'e not d i rect ly conn�clcd with pUl'e science.

One a fter the o t her, bis correspondenls (. chwe­ikart, 'Paurinlls, Boly"i) sent. him let tN which contained a more 01' less b"oud hint lhut it was impossible to prove the Wtb postulate and that the contrary postulate did not run counter to the other axioms of Euclid.

As far as Schweiicart and Taurinus were con­cernod , the idea was nebulous ariel sta led in un­wieldy fashion . Gauss saw the matte,' in a clearer l ight .

Picture Gauss for a moment. I t is not so easy to give a direct and honest answer. I t is not so easy to present one 's ideas to a Schweikart and give up completely tbe hope, in one's beart of hear�s, to resolve that accursed problem, o�pl.in the situation, and to advi e: develop your ar­guments as fully as po ible, and with tho grea­test pos.si ble care, for tbe more di versi fied the corol laries and theorems you get, on Lbe basis of tbe postulate conll'ary to the fifth, the more s cure will your inner rai th be tha t it is noncon­tradictory. Examine non-Euclidean trigonome­try, try to compute the length of curves in non-E uclidean geometry. Get, for example, an expression for the length of a ci rcumferen· ceo . Gauss knew wha t tbe length of a circumference would bo in non-l�"clid an geometry. He gave the formula in one o( hjs letters. But our "ideal Gauss" would o[ course not write about such a Ihlug to his correspondent . .

He would keep sil ot about his own results, :)Jl d would outlioe an eXlensive programme of re-

\ 18

""arch , giving encom'agement and support lo bis young colleague. He would writ :

" f myself wa, attracted to this Idea, hut, alas, nO matter how far YOII develop your theorem�, the question-ultimately-of the noncontradiclo­dness of non-Euclidean geomelry is a question of fai th. It is impo sible to obtain a rigoro us proof. Ooe can ollly rely On one's intui t ion .

"The probabil ity of e,·,·o,· w i l l always remain. You are young. Your name is not eanolli.ed , you cnn alford to write silly tbi ngs. I insistently advise you to devote all your energies to tbis problem. Awa iting further lotters, . . . ..

Aren '( wo oxpecting too mucb of Gau . ? A lot, but not too much. Science knows of such peoplo and sucb

cases. Tho pbrase "you are young enough to WI'ito

siUy things" was actuaUy wriLten once-by a remarkable man, teacher aud physicist, Ehren­fest, to two young men, Uhlonbeck and Gouds­mi t , wben they bad wanted to withhold publi­cation of a paper tboy bad seut to a journal. Later, it turned out to be their chief contribu­tion to science. Incidentally, they got the most fundamental reasoning from Einstein, wbo gave i t unsol fisbly , caring not a whit about bis own priority in the matter.

But Gauss was not tho ideal of scienti fie dis­interestedne . 1'rue-this wo must say-he ne­ver permi ttod himself any improper actions either. He was always scrupulously honest . Well, nearly always so.

Because in the case of non-Euclidean geometry he never explained bimself ful ly and nevor gave

n' 1 79

the trllO roa on 101' nol wallti Ilg to pu"l is" his work.

Tn all his leUel s he childish ly io iSled on hi. lenr of tho "Oeschroi der Bootier". These Boe­otllians, like l i lesa,'ers, turn up in almost eve­ry leLler dealing with non-Euclidean geometry.

T ad m i t even lhat Gauss himself finally be­g," to bel ievo his pet excuse. But does tbat change anything? Nothing at .11. One of tbe most suhlle, convinci ng and wi despread type.� 01 lie is that which you yoursel f have corne to be­lieve i n .

Falth is llCeded; that precisely is what COIl­vinces otbers.

Non-Euclidean geometry is likewise a product of fai lb .

Bolyai and Lobacbovsky bel ieved . Strictly speaking, i n tho most hlndamontal problem, tbo crucial question, tboy reasoned as poets reason, and not as worshippers of rigorous logic.

"This is correct lor it is beautiful" would seem to be their chi f argument. f t is worth going into. I said "reason d as poets reason". It would have been better and more correct to havo said "like mathematicians", Or more precisely still, "like people endowed with creative tbought" .

The nature of the creative process is unitary i n its basic and decisive features. MaUlOmati­cians, physiCisLs, poets, art.ists, engi neers, mu­sicians differ among themselves to a far smaller degree than is generally thought.

Incidentally, tbe anciont Grecks reasoned more exactly i n this matter, for they hardly at all distinguished the nature of the different types of crea tivily. They may bave overstepped the li-

1 80

mils when they clai med that a musician needed professional training in philosophy and ma the­malics. Bu t this exaggeration grew up on a ba­sis that wns sounder than that 01 Lhe opposite view.

True, it mu t be noted that a sharp demarka­tion between tho exact sciences and the al·ts can­not unconditionally be COn idered the sland of our century. I t is simply a vcry common view, One beld mostly by those who have no contacts with any area of creativity.

Quite naturally, to explain to such people tbe nattlfe of the creat'; vo process is an extremely di fficul t task, the difficulty progressively i ncrea­Sing with the official standing of the person wi th whom you are arguing. It is just as hard as to explai n to a lover of ballet that a magni fi­cent footballer is no less worthy of admiration tban a bri l l iant prima balerill3. And if one adds that, essentially, tho artistry 01 our centre for­ward and of the pri rna is of a si milat nature, uni tary in its very essence, in its objecliY s and resolls, tho intel lectual ballet lover will most likely walk out of the conversation. I ncidental­ly, conversing with a football fan, you wOIIJd get tbe answer: "Football is not ballet", plus some unprintable variations On your mental sta­tus.

All III more reason for w;ping out this dismal, settled narrow-mindod ness-i t is very wido­spread.

Let us return to geom Iry. One of the cbieI cri teria of any lype of art is, as we well know, beauty. The search [or beauty permeated the whole lifo story of the filth postulate, fro m Eu-

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cUd to .Lobachevsky. The ugliness of Euclid's postulate predetermined Ihe futile two-thousand­year attempts to provo it.

And the elegance of the constructions of non­Euclidean geometry WOn the heart of Lambert, almost coovi uced Gauss and compelled Bolyai and Lobachevsky to declare: this is so beautiful that it has as much right to l ive as the geomotry of Euclid.

By rights, Bolyai occupies lirst place when it comes to faith and eflthu iasm. H is work ent i t­led modestly "Appendix containing the science of space that is absolutely true and independent of the truth Or falsity of tho X I th axiom of Eu­clid, which, a priori, can nover be proved . . . .. is most unconditional.

A curious train of events followod this floria­ted tille.

The work was pul>1i hod as an appendix to a toxtbook of geometl'y wrilten by his father, Far­kas Solyai. As was natural in those days, the book was ,witten in classical Latin, tho laoguage of scholars and philosophers. Of the long tillo, only the word "Appendix" remains when tho work is quoted. That is now the tille we kllow it by.

I t is curious and symbolical that at the cradle of non-Euclidean geometry there clashed three human and scholarly temperaments, and three scientific modes of thought.

OppOSite stands were taken by Gau and Bo­Iyai.

Carl Friedrich Gau s. Gauss the cautious rea­li t. He was undoubtedly tho mo L logical of the three. The most academic. To him tho problem

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had noL bocn solved to tho end, and ho could Qot allow himself the luxury of following bis intuition, to have faith without proof; that he could not do. He had a clear conception of tbe matter and, given the desire, he would probably surpa s Bolyai and Lobnchevsky. Ho knew it but he did not believe in it enough. And he lost out.

[t matters little what historians will write la­ter. I t matters hardly at all that in all his let­ters he insistently repeated "I have known this [or forty years already ". Alone, by himself, Ga­uss admitted that ho had been left behind. What is more, ho was honest enough and soverely strict i n bis attitudo towards himself to admit this unconditionally. He had lost the game.

Janos B�lyn i. Bolyai was a romanticist, struck hy beauty and elegance, carried away by his own talont, onthusiastic beyond measuro. ''This was done by Hnos Bolyai". His (aitb was r -warded. I t was the prime mover of his life. J n his £ir t work he grasped the problem more pro­foundly than anyone bitherto. I ncidentally, he never made any more headway. Po ibly becau­se for him everything was solved. Subconscious­ly, perhaps, hut solved.

He achieved his goal , he was a geuius. That much he had proved.

Nikolai lvanovich Lobachevsky. [n our story he is close to tbe ideal scbolar. Combine in equal measure the scienli lie enthusiasm of Bolyai and tbe skoptical cautiousno of Gauss, and te this add a por i tence bord ring on stubbornness, an almost instinctive inner cOltviclion o! the irre­proachahlo trulh of his ideas . . . . Also make the demand that this scientific intogrity IIOt waver

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during twenty years of a complete lack 01 un­derstauding by his colleagues, a lack of com­prehension that at ti mes took the form of open mockery, and you will have an approximate pic­ture of Lobachovsky creating Lhe foundations of non-Euclidean geometry.

He believed nnd he veri6ed his holiefs. I t is quite fair that tho non-Euclidean geo­

metry of which we are speaking is always called Lohachevsky '. geometry.

We shall return to Gnuss and Lobachevsky, but first let us take up Bolyai.

T have already said that as a person Bolyai wa not very pleasant . That remains my feeling. But, in brackots, we might add a trivial fact: he was a brilliantly gi fted mathematician. This he magnific ntly demonstrated and there is no maybe about i t . But apparently Bolyni the man was a difficult case.

He was of the species of "geniuses". Every school has two or three "Newtons"-talented youngsters, sharp-witted, far advanced, towering Over a l l the other childreu, and with sparkling lightning-fast minds. A l l too ofteu, recognition of this intellectual superiority speils them and brings them to a !dnd of iettscheallism . They are temperamental, intolerant, egotistical, trust only themselves, and regard all others as the gray mass, the rabble whose job i t is to hoist their bero ento a pedestal.

W ithout a doubt, t h�e arc ti mes wben they are kind, respensive aod charming, hut subcon­sciously (aod, later, even consciously) lboir phi­lo ophy is that of "Ioadrrs" and "masses". Thi kind of development 01 gifl d children is sad-

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doning, but it i s natural perhaps because the education o[ an inner cultul'O is a much more lengthy, complicated and subtle process than oven the Dowering of a talent. And the conDict b twoon a talent and the cultuIe is the moro acute and lacking i n compromise, the SOOner the suporiority of tbe child becomes evident. I f T may permi� my ,If SOme pbHosophiziug, one is inclined to think that most ef the tribulations of mankind are associated with complacency, self-satisfaction, which, alas, is a practically ina­lienable feature in most people. And if a man is gifted and also ambi tious, life becomes arduous either te himself or to those about 1Jim, or to both.

Bolyai '. destiny was of a third kind. His ta­lent appeared early and in a diversi fied fashion. He was an extreme representative of the kind of temperament that is usually descri bed a8 "artis­tic", "poetical". Elegant, i m pulsive, sciutilla­ting.

The supreme proof of bis mathematical talent and intuition is that by the age of 21 to 23 he had already mastered the fundamentals of non­Euclidean geometry and, what is mo t i mportant, was apparently (ully convinced of the truth of bis ideas. When Farcas, his father and a pro­minent H u ngari an mathematician, and , inciden­tally, a school-day friend of Gauss, learned that. his eighteeu-year-old SOn was captivated by the tho­ory of parallel l ines, be wrete in desperation te his son, pathetically imploring him to give up that mad venture.

The let lor is wl'iLt('n in such high-flowing sty­le as to irritate Lhe modern reader and cause him

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to doubt the sincerity of the writer. Of particular interest, in my opinion, is that it gives an excel­lent picture of the relatioos that obtaioed in tbo Bolyai family .

"I i mplore you oot to attempt to surmount tho thoory of parallel lines; you wi ll waste all your time On i t aod slill not prove the propo i­tion . Do not try to overcomo tbo theory of pa­rallel lines either by tho metbod you speak of or by aoy otber method. I bave studied all avenues to tbeir ends and have not eocount rod 3 single idoa that I have not developed. I bavo passed through the wholo hopeless darkoess of tbat night aod have buried i o it every beacon, every pleas­ure of life. For God's sake, I i mplore you, leave tbis maHer alone, lear i t no less tban sen unl passions, for it is capable of depriving you of all your time, your health, peace of mind, the eotire happiness of your Ii fo. Tbis bopeless dark­ness . . . will nover be clarified here 00 earth and the miserable humao raco will never wield any­thing perfect eveo i n geometry. This is a great aod eternal wound in my soul . . . . ..

Incidentally, Farkas-i n his youtb-did stu­dy tbe tbeory of parallels and even sent Gauss somo proofs of the fiftb postulate. Thero cao be 00 doubt that tbe latber was sinceroly upset about 18nos. Strange to say, starting from i ncorrect premises bo correctly foresaw the final result: the theory of parallels was indeed destined to be the curse of J a nos Bolyni's Iile, though for quite different foaSOnS than his father suppo80d .

When thero is a d vilisb obsession, so there must be an evil spiri t . Gnu S lVas the evil geni­us for J anos Bolyui from oarly cbildhood al-

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most to the end of his days; though-suhjecti­vely-Causs lVas bardly to blame i n any respeot.

H all tarted when lbe father began to harbour the ambi tious dream of sendi ng his talented SOn to Giillingen to completo his mathomatical edu­calion under the guidance of Gauss. Farkas wrote to his kind old friend asking h i m to receive his son. Ho wa naturally prepared to pay all the expenses.

The answer was silence. Causs may have had a variety of reasous, some

very weighty, to refuse, and he can onl y be reproacbed for a lack of tact with regard to Farkas. Admittedly it is " cry difficult to jud­go. Farkas' lelter was somewbat impertinent. Some 01 the questions were reasonab le enough , but 000 can easi ly understand Gau too. "Is your wife an exception to the whole f malo sex? . . . Does her mood change like a weather-va no?" The point was that Hnos would have to live there in her house, and so Farkas wanted to know how 18nos would get along. Quite natural­ly, Gauss must have winced at such sweet inge­n uousness .

However, I am not interested here eitber ill Gauss or in Farkas Bolyai . As rar as ooe can gather, Gauss would not have taken an unknown boy even j[ the letter had been written witb the diplomatic eleganco of a Taleyrand. That is bis right. AU this go ip is of interest only in thai i t once again demOllstratos bow IiUle and how poorly adults undor tand children. Both grown­up in this story aro to bl3mo.

I magi no a high- trllng fourteen-year-Qld boy in wbom his eITusi ve father had undoubtedl y io-

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st i l led great hopes. Tho boy did not know much about the I'olationship between his [athOl' and Gauss. He had nO idea of what could offend Gauss and why. The only thing he knew- and you can be suro the father poke of iL several l i mes 8 day- was that as students, the father and the great Gauss had been the best of Inends and that they had cvon solemnly sworn to eter­nal Iriendship.

So, naturally, the father is convinced tbat Carl will answer the very next day. Fourteen-year­olds believe their falbers. E p cially i f your rather is also your toacher and is a talented , versa tile interesting person. One must add t hat Farkas was a profoundly gifted mathematician. In his textbook of geometry, he clearly formula­ted for the first time the demand that axioms be independent . He doubtlessly deserves full CI'e­d it for J anos' deep understanding of problems of axiomatic. at the age of twenty.

The boy could not hel p respecting his fathor. He was confident and he waited. He-a boy fr'lm a backw'lods provi nce of Eu­

rope-alrandy saw himself a student of t he great Gauss, and perhaps, later, hi aSS'lciate i n scien­ce. F'lr months on end bo wailed expectantly for tb postman cbecking the days, adding 'ln when too lOany went by, waiting for Gauss' answer, tbinking up fresh reasons for doluys, again waiting and hopi ng; still hoping when his fatber took h i m t.o Vienna to a military engi­neering academy, Cor i t had become clear that Gauss won ld 1101. roply. Gauss simply did not wnot to. Yet t. here l i ngered lbe hope that, por­haps, an unknown messenger would cOlOe ri-

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d i ng at their heels with the long ovel'd tte loller. _�o leU('r e\'cr came.

I must say that though I hav absolutely nO facts to go by and I do not know how all this affected 1 8 nos, I can easi ly so h'lW a month or two of waiting like that could totally derange the nerv'lu.s system of n high-strung fourteen­year-<Jld boy. Particularly if the b'lY was girted, excitable, deeply sonsitive.

But let u.s not be overstrict in judging Gauss. He might easil y ha ve been offended. ADd to worry about tbe lIerves of some unknown young­ster, as we so frequently d'l today . . . . Let us n'lt ask for to'l much.

The years as a student and especially, the years of military service i n outlying garrisons of H ungary were years of d ismnl al'lneness for J a­nOS Bolyai. True, he had a couple of friends at the academy, brought together by their love for mathematics. Nothing morc. Afterwards, there was no One.

I do not think that provincial officers offered bim appropriate company. Apparently, he not only failed to conceal his haugMy disdain for the wh'lle crowd, he went to lengths to stress it. Tbe result was constant quarr lling and du­els. He saved hi mself by his skill at wordplay. He most likely was right in his attitude to his "colleagues". Yet during all those years he could have found a lew docent fellows, even lhough somewhat lacking in education and intelligen­ce. Of that there can be 110 doubt. He ohv iously presumed that thoy would be of no use to him. He waS mistaken. But h was uot mistaken when i t came to the theory o[ parallels. Before his re-

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signatioll (again t he r�. "It oC .'!Ome kind of scan­d a l) he had wrillen up his invest igat ions i n the form of the celebrated "Append i x " . The work is writt�n in extremely compact form and makes d i l£icuJt reading .

, That i n genera l was the poor luck of non­�u.c "dean �eometry: Lobacbevsky 's papors are \ 1f! llCO baZlly, and I f One judges from the stand­

pOint of a scientific edi ter, they aro si mply no good . Numerous essential ly si mp le matters are t angled uP

. beyond measure. For such th ings

ma�hemaltclans have tbe aphorism : "the repu­tallon of a mathomatician is determined by the number of unwiel dy proofs thnt he has concoc­t-ed ".

Tho poi nt eems to be tbat path-makors as a rule , do not find the sim plest and most el'egant pathway. They slash through the trees culting a road becauso tbey bave to advance. Those who come later bring elegance, beauty and polish. �here are exceptions, but they are truly excep­tIOnal.

Be al l Ihat as it may, Farkas Bolyai did not understand the work of his On at a l l . Since i t

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was tbought to be publi hcd as a n appendix t o the geomelry t('xtiJook which farka had wri t­t�n, tho conniet reachpd its ap� x .

I t i s h�r�, "flrr rofl��n YI'nr. , 1 ho L Farkas aga i n wroto to GauS.'! asking h i m 10 act a s judge. (This was in 1 8.32). "My son l'espects your opi­nion lIlore than that of tl,o whole of E urope", he wrote.

Tlus ti me Gauss replied . True, 3 mon th later. But he read l8nos' paper carefully and favourab­ly. What ever else may be said of hi m , he va­lued talent. And i n others too . Al most the nex t day he wrote to a friend of his: "A few days ago I recei ved from Hungary a small (laper on non­Euclidean geomelry; in it I found all my own resulLs carried out with marked elegance."

Well? Such wore tho facls. Almosl tbe actual facts. We havo no right to blame h i m , al most no right.

Then tbo father and son recei ved his reply. The u ual introductory remarks and generalities, and then:

" ow a bil about the work of your son. I f [ begin by saying that I ought not 10 praise his work, you will of COurse be amazed for a momenl, but I cannot do olherwise, for to pra­ise it would mean to praise myself. The entire contents of tbe composition, tho palh that your son has tilken, and tho results that he bas ob­tain d al most cornpleLcly coincide with my own attainments, which in part are a lready 35 years old. I am indeed extremely amated. My inLcn­tion , regarding my own work, which incidental­ly has but slightly been (lut to paper up to the presont time, has been not to publish anything

1 9 1

during my l ifetime. Mo t people do not take the proper view of lbe Ilroblems di cu cd bere. r have found only a few peopl that evinc spe­cial inlere L in what I had to tell them On this subject. I n order to be i n a tate to master lhis, One ha to feel with great vitality that whicb is, properly speaking, lacking here. 'ow this is not clear at all to most people. However, my intention has beell to write all this down, in good limo, and i n sucb form tbat tbese ideas should not perish with mo. Thus, I am exceo­dingly surpri d that this job has been taken from me, and I am plea cd i n th extreme that it is procisely the son of my oId friend who has anticipated me in this remarkable fasbion."

To say thot Ja.nos Bolyai was distressed is to say nothing. He was nraged, obliterated, crushed. He was convinced tbat Gauss ' whole leller was one Ii trom the first word to the last. A lie, the sole purpose of which was to arrogate Janos' bril l iant idea.

This second blow from Gauss was heavier than the first. Ho, Hnos Bolyai, had reacbed what be bad sought. He had become a mathe­matician. Ho had grasped what hundred of the greatest geometers had failod to understand ovor the past two thousand and more year . He alone in the whole universe knows the ans­wer (but he did Ilot know that somewhere on the boundary line between Europo and A ia a certain Lobacbev ky had alread y published a paper). And this arrogant old man wanted, so bo thought, to snatch lip Ihe work of hi wholo tif , hi glory, and to bury his genius.

Yet ono should not reproach Gauss overslri·

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ctly. He wrote the truth. Al most the truth. He dissembled only when he tried to explain why he had refrained from writing up his re­sulls and publishing them. Too, there can be no doubt that Gauss sinned both before mathe­matics and before Bolyai, and also before him­self in that he did not express any opinion in printing concerning the work of 1 anos, for in this ho would not ri k his good name, he risked nothing. This, ei ther consciously or sub­consciously, was the logic of ambition. Though Hnos' rage was unjustified in many ways, he keenly perceived that Gauss was manoeuvring in some way, that there was an unpleasant, false note in all his reasonillg.

We have some notes that convey lanos' ro­action to this evont, and we can agree nncondl­tionally to the wbole toxt. The words about scienco and the ethics of the scientist are good and proper. Here, bis accusations levelled aga­inst Gauss are fair in full measure.

" I ll my opinion and, i ' m convinced, ill tbe opinion of any unprejudicod person, all tho ar­guments given hy Gauss to oxplain wby during his IHetime he does not want to publish any o( his own works on the subject at hand are com­pletely impotent and trivial, for in science, as in everyday IHe, t he problom is precisely that of il luminating sufficiently nee ssary and gene­rally usefui things, particularly those which are not quite clear yet, and of awakening i n every possible way the still deficient or even slumbe­ring awareness o( the truth . . . . To the general detriment and misfortune of al l , an understan­ding of mathematics is unfortunately the lot of

13-11117 1 93

only a few; and On thoso grounds and lor those reasons, Gau.s could have kept to bim If a still more sub Lantial portion o( bis plendid studies . . . . An extremely unpleasant i mpression is created by the fact that Gau , instead 01 expressing, relative to tho "Appendix" and the whole ,ul'cntamen", a direct and honest ack­nowlegment of their bigh value . . . so as to think of means to open the way wide (or a good under­taking-in place o( al l tbis, Gauss strives to avoid tho direct pathway and hastens to pour (orth pious wishes and regret concerning the in­sufficient education o( people. That, of course, is not what life i • . . . "

But alone by himself, Bolyai did not reason so broadly. He su1!ered, aspiring to fame and recognition. Recognition is what he wanted. He wanted the whole world to see that ho, 1 anos Bolyai , was a "geometer of genius 01 the first rank" (that was how Gauss described him in one of his lelters, but nOl in a letter to Bolyai and not on tbe pages of a journal).

Wbat Gauss' letter resulted in was a nervous breakdown for 1 anos Bolyai. He even suspected his own father of betrayal.

I can 't say that I am particularly delighted with the reaction o( J a nos. Ooe caD of course understand him, but ono finds i t hard to agree with and justify h i m . I f he bad paid heed to his own words about science, his conduct and futuro lifo would have been dilferent. Bo­I yai waS then no longer a boy, he was thirty yoars old and he could have taken all thcso things liko a man. He could have. But, too, leL liS noL judge Bolyai harshly. l ie was not

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yet eru hed. He COnLinued working on the same problem that, a few thousand kilometers away from him, Lobache,' ky was engaged in. He was constructing the whole of geometry on a new foundation.

B u� the intM. ity 0.1 bis work had dropped.

Ho stIli took a l I vely Illtere t in a great variety of prohl ms. Togethor wi tb his father he drea­m�d o( constructiog a universal language; he tned his hand ill other divisions of mathomatics' he tried other things too, bllt none o( the� was really normal serious work-only a mor­bid desire Lo do something out of the ordinary, to prove to the world that he was indeed a ge_ nius.

Meanwhilo his relations with his father had become eXLremely bad. Obviously, Bolyai the SOn was not capable 01 being a co·author. Tru • B?lyai the latber was lar from a paragon of WIsdom aod good wiIl. Mutual scientific jea­lousy and an assortmoot of m uddled alfairs cul­minated most unusually. On one fino day, tho rovorent son challenged his father to a duel. Later still, ] 8 nos became a nerve-patieot in the full clinical senso of the word.

Tho decisivo blow was deal t by that (once again) accursod Gauss.

III 1 84 1 , on Gauss' sugg StiOll, Farkas Bolyai ordered a booklot by Lobachevsky published in German and olltiUod GeoTMlric In veIl/gallons in 1M Tlteorv 01 Parallels. Recalling 1 aoos in con­nection with Lobachevsky's work, Gauss was possibly making amends (or his long disrogard and, undoubtedly, was guid d by tbe very bost of intentions.

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But Janos' morbid mind viewed all this 3S a MachiaveUian intrigu On the part of Gaus • He was convinced that this mythical Ru. ian pseudonym simply concealed ono 01 Gauss' myr­midons, i f not Gauss himself.

1 anos Bolyai subj cled to analysis every com­ma of this tiny picco 01 writing; he did it thor­oughly, punctiliously; with an il l will he sub­jected it to a thorough cavilling criticism.

He was scientist eoough to appreciate the work, but he was glad of every lault and regar­ded the author as his personal eoemy.

He was then thirty-niae. l a his prime. He was destined to live another twenty years.

But he was already broken and crushed. IIis i l lness was a form of nervous disease. He was haunted by the theory of parallels. Those twenty years were awful years both for him and those close to him. The rupture with his father was complete. Their only correspondence lVas on scien­tific topics. They corresponded, though they li­ved in tho same town. And it was mathematics that finally set them at loggerheads. For the last time, the retired captain, J anos Bolyai, came to life in 1 848 during the Hungarian rG­volution with which J anos sympathised comple­tely. But he was ill. That was one thing. The other was that he did not wish to be a rank-and­file participant-only a leader. By the way, one can believe that he could ha" e been a splen­did military leador. But he was unknown. And so he remained at home. The defeat of the re­volution was yet another blow. His i IIne s tortured him, and he nO longer worked.

During the remaining years of his l ile be did

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practically nothing, only busying himself with utopian ideas. What a remarkable thing that the brilliance of his talent continued to shine even in this production of his affected brain. One of the last of Bolyai '5 passions was the construction of an ideal mathematical theory of the stato and his hope, i n this way, of leading humanity to universal good. Of course, he was unable to do anything of real value here, but the idea itself was very close to modern concep­tions of cy OOrneticians.

The end was close. He was morose, suspecting, and though he

loved humanity at large, he could hardly got along with his closest friends. He left his wilo; his children coased to interest him. Once moro, for the last time, he quarrelled with his old and dying father. But now, at 54 he himself was an old man.

He would have beon happier if he had died earlier.

He was a brilliant mathematician, no ques­tion of it . But what he valued above all else was not science, but hi msell in science. And cruel though i\ may sound, I am afraid thaI he himself was the maker of his fate.

Chapl" 8

NIKOLAI IV ANOVICH

LOBACHEVSKY

By tho start of this century, Nikolai Loba­chevsky had already been canonized. He was the pride o� R ussia� science. He was the grea­test talent 1 0 tbe lustory of mathematics des­pised by his compatriots who did not I:nder­�tand h i m: He was . the victim of a bigoted, bcaurocrallc academIc clique. He suffered the whole of his life and died i n poverty, an unre­cognized genius.

Sucb, i n brief, were the broad outli nes of the cheap melodrama that so often comes to l i fe on the pages of popular journals and books. The most remarkable thing about all this is that i t i s essentially t rue, t hough exaggerated.

One thing is unquostionably true, and that is that Lobachevsky is indeed the pride of Rus­sian cience. The reader would do well to read V. F. Kagan 's marvellously detailed and pro­found biography of Lobachevsky. I highly recom­mend it. For our purpose here, a fow highlights of his l ife will suffico.

I n the year 1856 and tho month of February, alter a protracted il lness, mathematician Ni ko­lai Ivanovich Lobachevsky died. Shortly before, due to i l l hea lth, he had left his post of trusteo of the Kazan chool Di trict. For many years he had been Rector of tho I m perial niver ity, honoured professor of pure mathemalics, Corros-

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ponding Member of tbe Gotlingen Royal So­ciety, honorary member of tbe I mperial univer­sities of {oscow and Kazan, and also of many scientific societies, occupying tbe high post 01 Councillor of State; he was bearer of the orders of St. Stanislav, Third and First Degrees, St. An­ne, Second Degreo, St. Anne, First Degree ador­ned with the Emperor's Crown, and the Order 01 Prince Vladimir, Fourth Degree and Third Degree, repea ted l y noted for outstandingly zea­lous service and especial efTorts by tbe Supreme Grace of I he Monarch.

Such was hereditary nobleman ikolai Iva­novich Lobachevsky.

The funeral wa8 solemn and beautiful, for ho wa8 loved and revered i n the city. The speakor said: "His noble l ife was a living chronicle 01 the university, of its hopes and strivings, its growth and develop ment.»

Tho Kazan Gu�rnia Vedomosti, tbe local newspaper, gave a brief obituary in moderalely solemn style as befits such an event.

One apeaks well of the departed or one docs nOt speak at all. And after a hort enumeration of bis merits: " l I is work and attainmonts i n the field o f cienco, which "re now in the chro­nicles of tho scientific world , will without doubt find a worthy judgo. We, for our part, are hap­py to be able to adorn in tbese fow lines the m mory 01 the deceased ill laking leave of our eloquent profe!!Sor.»

One speaks wel l or one does not speak at all . The w\'ile. of the obituary most l ikely was

inccre i n congra tulating himself for tho clever

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rhetorical figure that saved him from the grea­test danger of all. Everyone i n Kazan knew that professional people and authoritative critics re­garded the works of Lobachevsky as the product of sick mind. For many years, tbe reply to an enthusiastic students query "Is it not true that our rector is the first mathematician of Rus­sia?" was professorial silence. An awkward, sul­lenly embarrased silence in the case of well wishers, and a sarcastic silence in that of his opponents.

The late professor was, undouhted ly, a most worthy citizen of the city of Kazan. He was an excellent administrator. He was paternally strict with the students, friendly with his col­leagues, a skilled diplomat with the mighty of the world, a highly esteemed teacher, an oxtre­mely erudite mathematician, zealous in his run­ning of t.he university, its founder and its pride.

Yet there was one blemish. His ridiculous works, the monstrous belief, over so many years, in those mad i deas of his. One could only tact,.. fully remain silent.

For those that knew, the obituary /lotice left open a tiny deoply concealed ambiguity" . . . will without doubt find a worthy judge".

Was this not a hint on the part of t.he writer? ot a well-wishi.ng hint either, for surely eve­

ryone knew the true worth of the deceased's works, which had been appraised by such out­standing personages as academicians.

Unfortunately, we must admit that this un­pleasant rock still projocted above the surface. Neither was Bolich, who pronounced the brief funeral oration, able to circumvent it. A pro-

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fesser of philology. he very properly confined himself to a single smooth phrase, ".. . It is not for US here to speak of his independent scientific studies ill mathematics that brought him renOwn and glory . . . . ..

All the rest was said cordially, simply and well , and the sincere, well-wishing educated spea­ker concluded in elevated moving words with even a touch of poetical fervour.

But again all this was equivocal, even unplea­santly ambiguous, for his renown in the world of science was of a joking kind. God save me from such glory.

Nikolai Ivanovich had indeed stumped his friends, for they had to say something (after all , he was a mathematician, not ju t SOme of­ficial), but what?

Bulich was simply unl ucky with his speech. In some marvellously strange way, by some superior seose, tho archpriest perceived a crime in the funeral oration, a crime against censor­ship, against morality-atheism to put i t simp· Iy.

How he perceived it is not clear. He was probably indignant that nO word was said of divine affairs, lIot a word about God was men­tioned.

And so of course he was duly reported to the authorities in very high spheres. Bulich wrotc to friends imploring them for help and assuring them that he had not said anything unlawful "except the Il'uth rogarding the deceased, ex­cept respect for thinking and science that are so natural today, and except unavoidable rheto­rical ligures".

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Luckily, Ihere were benefactors in St. Peters­burg and Ihe affair was hu hed up. That was i n the winter of 1 856, when, as BuHch put it, Kazan accompanied their pride

thoir great citizen "10 the de rted road to eter: nity",

Only a year and some latcr did one of Loba­ch vsky's pupils, A. F. Popov, write an obi­tuary and solve tru. difficult problem in the be t fashion. And again a single sentence to cover a l i f lime of work: "The loctures Loba­chevsky del ivered for a select audience in which ho devoloped rus lU!W foundations 0/ geometry mu t in a l l truth be termed profound. n

Acl.lIully noHling sald, yet no ad vorse innuen­does eith r.

One could nOw cease talking of tho tragedy of Lobachevsky 's life. The atmosphere of his funeral and the obituarios tbat followed it ex­plain moro tban does allY collection of exclama­tion marks and tragic phrases.

Let liS forget for a moment that he was a hrilliant mathematician. Let us appraise the initial (and terminal) conditions wi lh the un­demanding yardstick of tbo prulistine.

ikolai Lobachev ky was born on ovember 20, 1 792, in a ratber i rnpoverished family of tbe registrar, I . M. Lobuchovsky. Tbis post, in tbe table of ranks of the R lI ian Empire, wns equi­valent to that of second Ii utenant. Wrote One of rus conlemporaric , with tbe modish romantic melancholy of tho Ii mes, "Povorty and want hovered over the cradlo of Lobachovsky . "

There were tbree bey i n tho fa mily, and IVhen, in 1 797, the br adwi nner Ivan Maksimo-

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vich died, th still young twonty-fonr-year-old bardly litorate motber wa on the brink of a catastrophe.

By whaL means and ways she was able to send all three to Lho Katan Gymnasium, and aL government expenso to boot, whaL all this cost her really, what toars and whaL devious dealings, we shall never know.

All tha t romains is an application wri Lten for her either by a kind soul or for a gla of spirits by some heavy-d rinking advocate, of wbom there wer many i n saintly old R ussia . The form was p rfect, dictated mo t l i koly by an expe­rienced hand. It contained worthy want, due respect, the moderated grief of an unlortunate widolV, and iL oncl ud d w i th the mosL loyal feeli ngs for th sovereign, and Ule signaLnre of Praskovya Lobachevskaya written in two I illes, thus exhibiting extra propriety and extreme res­p ct. But who was this Pra kovya, holV and by what means did she mako end meet? 0 ono knows.

o i t was that on November 1 7 , 1802, the three boys, Alexander aged 1 1 , Nikolai 9 and Aleksoi 7-aU Lobachevskys-were matriculated at the gymnasium at governmenL expense.

Careers were thus opened Lo the Lobacbev ky hoys.

Thero were not many ill the old R u ian Em­piro tbat acrueved as much as ikolai Lobachov­sky did. Of course there were brilliant caroers made i n and around the royal family and occa­sional skyrocketing from peasant to Privy Co­uncil lor despite genc810gy and, liko under the Emperor Paul, frolll valet to counL-buL for a man of science, tbo administrative career of

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Lobacbovsky was extremely outstanding, though not unprecedented. And i f we add that he pro­ceeded by unsullied pathways, did not dissem­ble ?r seek, hard�y cringed or flattered for pro­motIOn, he was indeed a rarely lucky minion of fortune.

True, he was no angel. He was a forward­thinking person of tho times, nothing more. There were thi ngs he was ashamed of, and it �as nO easy �ask to serve and still to be morally lmmac�late in those days. He lived a compli­cated l ifo and earned in lull measUl'O what is al lotted to humankind in this world.

For the most part, ao untroubled carefree �outh, yot there wore grievous losses too. The JOY of success and tho delight of creativo work yet dangerous unpleasantness in his studeni y.ear •. A radiant, cerulean-to begin with-seieu­tdle career, ret vici?us, humiliating, taunting attacks of IllS enemlcs. Varied administrative and social activities, and the intrigues of his colleagues. Exalted praise for his administrative work, and pi n-pricks to his self-esteem. Reco­gnition by Gauss himself and the pleasant vani ty of �wards, yet the bitterness of offences. But again the love emanating from a happy family life.

At the end, destiny delivered old age, trouh­les at work, the death of his beloved son, the nervous breakdown of his wife, illness, blind­ness-yet, to the very last days of his life an incomparable joy derived from his studies. '

Chronologically, he spent 1802 to 1 804 at the gym �asium with obligatory studies including R US8lan gram mar and li torature, history and geograpby,

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8l'ithmetic, algebra, geo metry, trigonometry, mechanics, physics, chemistry, hydraulics, sur­veying, civil architecture,

logic, practical philosophy, and the foreign languages-German, French,

Greek, the inevitablo Latin and Tatar as wel l.

Then came military studies, which included tactics, artillery science and fortification.

That was not all. Law, followed by such required items of society lHe as fencing, draw­ing, dancing and music.

Such was the course pursued in three years (not eight or ten I). ot overyone could cope with it. The Lobachevskys did. Apparently, they knew that for them there was nO other way up. What is more, it was easier for them, for they were three brothers together.

Nikolai was tho mischievous one. He was good at his studies. An ordinary capabJe child, the only difference [rom other, softer, easier­going boys of the nobill ty being his harshly practical approach to aU things. I t might have been the adult realization of tho necessity to get ahead.

A mong their teachers were cultured, talent<ld people, some even outstanding. The mathematics teacher, KarLashevsky, was excellent, brilliant.

Tbon came J anuary of 1807. AHer some unpleasantness with Latin, Loba­

chevsky was accepted into the university. He was 14 years of age.

The first heavy blow came in J uly of 1807 whon his beloved elder brother Alexander IVas drowned.

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The result was a nervous breakdown for i­kolai, hospitalization, and a firm ro olve to become a doctor. For over two years he studied medicine. True, he was the lirst in mathematics at the uni versity, but his firm deci ion was that mathematics was not his vocation. The boy wavered betweon "duty" and vocation (re­member he was only fi[teen), deeply depressed by the death of his brother. He was obstinate hard to get along with, though he was quit� normal, very decent, and rigorously adhered to the student code of honour. He dolved in all things that students do: fancy balls, the theatre lights, and just pranks (for instance, the Um� he rode up to the university building On the back of a cow; that is the episode that so many of his biograph�rs claim indicates a spontaneous protest against reaction).

Actually, the situation did cbange for tho worse at the uni versity during these years, and Lobachevsky's personal life was poisoned by a capable but raUIOI' unpleasant, unprinCipled and ambitious Kondyrov.

But Lobachevsky-it must be said-was not above ordinary boyi h stupidity either, the same lond so of ton met wilh i n hundreds of thou­sands of ordinary ilTi table, arrogant, quick-tem­pered boys SO snre of themselve , positive that they can got out of any fix.

K.ondyrev nearly ruined him. Lucky that the

foreign professors Bartels, Li tlrow and Bronner who had beon invited to teach at Ihe university, were able to rescue the boy.

I say 'rescue ' becau e the issue was one of sending him off to the army as a soldier. I n the

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best traditions of people of that sort, Kondyrev accused Lobachevsky of atheism and nlllwst sub­version of the estahli hment. It is not clear whether Nikolai was actually an atheist, but we do know that he never liked hypocrisy and the clergy.

10st likely, at that timo and later Lobachevsky was "moderately progre sive", with largely hu­manitarian views.

To extricate himself he had to repent. ]I'lake a speech, express loyal sentiment, ad!D it his mistakes and condemn t.hem, and promIse that in future he would not . . . .

So much for pranks. But i t was dnring these years, that Lobachevsky finally made up bis mind concerning his futuro: he would become a mathematician. He succeeded greatly i n this field. He was the first mathematician of Kazan University, and Bartols was always glad to poiut out his attainmenl.s and talents.

I f one recalls that in those years tbe whole of Russia had several thousand students, it isn't too much to say that Lobachevsl..-y was already known throughout the country. I t is almost like S(lying today, th.at Lobachcvsky is the mo�t promising young scientist i n the Siberian DiVI­sion of the Academy of Sciences of the USSR ( Ka.an U niversily was then an i mportant cenLre o! learning o[ the H ussian Empire).

I n August of the year 18'\1 , at tbe age of eighteen Lobachevsky received his �as�er's de­greo. His first success and the beglDDlDg of a number of very good yoars of inten:i ve work. Socially, too, he was a success, accepted in the "best society" of Kazan. A young man of the

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world, quite a man about town, always well­dressed.

The war with Napoleon hardly touched him. His youoger brother, Aloksei, tried to ruo away to the army, but was returned. True, Nikolai was worried sick uotil they found him. Loba­chevsky, i t must be said, always had very strong family feelings.

His moral code was already set-it waS that of decency, decency in the moaning of the no­tions of that time.

Bartels, a very culturod teacher but a mediocre mathematician made him study the classics of science. Lobachevsky's only serious drawback, i t would seem, was his excessive excitability and conceit. He was characteriled once as "ex­cessi vely self·centred ". On the other hand, Lo­bachevsky clearly and soberly saw tbat he was far behind the biggest mathematicians of his day.

I n March, 1814, he was made junior scientific assistant (about equivalent to associate profes­sor of today) in the field of physico-mathemati­cal sciences.

20B

l Je began delivering his own lectures. I n J uly 1816 he was installed as extraordi llat'y

professor. This at the age of 24. His career had begun. Meanwhile the university was a beehive of intrigue with chaDges occurring constantly. For a short time, tho reactionaries were on top. Tben the "progressivists" got the upper hand. Trustees changed. I n a word, thon, life at tbe university was on "an even keel".

Lobachevsky had enemies in the reactionary party and he had influential patrons.

This was the period when Lobacbevsky began to get interested i n the problem of parallel l ines. Tho beginning was standard. He attemp­ted to lind a proof. I n 1815 at lectures ho even described to his students the proofs he had found. But ohviously he soon fouod his mistako. A rather elementary One too.

A big change came in 1819. This was the period of reaction throughout the country. I t affected Kazan as well. The new trustee Magnit.­sky was a clever but unprinCipled, cruel , cold eli mber-ge-getter. . Ho waS one of those people wbo fight their way to the top elbowing, climbing, pushing others out of tbe way and trampling on those who have fallen. His one and only aim was to roach the lOp. If reforms were needed, ho would carry thorn through; i f extreme obscurantism was the word, he would be the extremist. But he was a rather clever man with a flair for ad­ministration. He made his first appearance as an inspector summing up the situation in these words: close the Kazan University because of the free-thinking and general moral degradation.

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Alexander T , however, decided nol to destroy but to rc tify lhe s ilu"tion, find he pUL Magnit­sky in charge.

Those wero d,wk days for the I Ini versily, h"t Magnitsky was kindly disposed towards Loba­cbevsky at first. He was possibly thinking of making him one of I,is prolege . During the years 1819 to 1821 Lobacbevsky was On the upswing, elected dean, bead of tho l i brary and membor of the constructiou commi tteo. Posts and titles came one after tho other.

In February of 1 822 he was elected professor in ordinary. Tbese were years when Lobachov­sky acted against his con cience. True, with a person liko Magnitsky there did not seem to be any otb r way out.

Bear in mind, too, that Lobachevsky was a.n i ndependent· thinking person, q uick-LO III pored and , simply, a hard person to got along with. Also his convictions were far removed from tho 0 of Magnitsb;y. Tbat is, under tbe situation of that time, because if suddenly Lobachevsky's views were approved Ull above, by those in power, tben . . . Magnitsky would turn out very progres­sive indeed. All this boils down to tbe fact tbat in 1822-1823, Magnitsky was nO loogor kindly disposod.

In 1 823 came the first major trouble in Iino of duty. His newly "Titten textbook Geometry was rejocted by Academician fo'u . I t may be that �'uss was on tbe wbolo noL righL, although serious investigators agroe that thero were es­sential dofects ill tbe book and some of FlLSS ' remarks were quite truo. Lobachevsky was stung to tbe quick and haughtily refused to reply to

2 1 0

any o f Fuss' rom arks, or Lo correct any 0 1 tbe [aulls of the book, or even to take tho manus­cript back. His arrogance ceca ionaJy made mat­ters worse for him. Howe,·er, he continuod to work i ntensely, alld d llring t.hese years he be­came fully convi nced of the impossibility of woving the fifth postulato, and fully con inced of t.ho equal rights of a non-Euclidean sysLem of geometry.

A fow pleasant even13 occurred in 1 825 and 1 826. Lobachovsky was put ill charge of lhe construction com mittee o[ the university, he was also elected Chie[ Librarian of tho U Ili ver­sity. His salary was raised to [our thousand roubles a year. Very good money in tho 0 days.

Then, Decembor 14, 1 825, MagniLsky was r -moved. He had nol beon able to lat.hom the /lew situation a[ter the death of Alexander I , he had risked everything 011 a big jump up, figu­riog that this was tbo time, but failed. He had placed everytbing on Konstantin, wbereas Ni­kolai won out. And then it was that tbe old memorandum came to ligbt whore ho had, it turned out, complaiued of the li beralism- no Ie s-of ikolai Pavlovich, then Grand Prince. Only in Russia could a paradox like that take place.

Quito naturaliy, an invesLigation was ordo­red. Certain sums of mODel', i t appeared, had disappeared. I t wasn't as i f tbe barracks set-up at the university was alien to the spirit of Tsar

ikolai; simply Magnitsky had overplayed his hand, and wbM is more important, he had sim­ply not been able to guess the events of Decem­bor 1 825. He bad lost. First discharged from

... 21 1

an posts and then exilod to nevel with ao ad­ditional investigation assigned into the money that had vanished.

Of course, the university, and Lobacbevsky as wel l , r joiced.

This is ti me to stop. February 23, 1 826. Up to this point we have wi tnessed the career

of an interosting, gifood, pleasant provincial mathematician, though one not devoid of draw­backs. Wo havo looked kindly on hi clim b. There bas been no exciooment, and we bave not been unduly enthusiastic. A very decent career, where the hero was promoood from rung to rung of tbe ladder. He was not indilIerent to his advancement and with the years Ilis worldly wisdom grew, and tho desperadoes of his youtb, ridiculous wild protesting 01 the malcontent, all remained in the past. Gradually, bit by bit, the common sense so usual in succe rul men accumulated. At the age of thirty-four he was a moderate man of fashion, a bit condescending i n manner. Further advancements were in Sight. Within a year he would be appointed rector (J une 30, 1827) . . .

As o f February 23 , 1826, a U these had beco­me mere tri nes of IHe, rather essential, but not over much so, and of course not decisi vely so.

That was the day the great mathematician gave his talk 00 Don-Euclidean geometry to an indifferent, bored audience who understood noth­ing at all. Of course, if an angel had appeared and i f there had b en some sign from heaven­"This is the Man "-things might have been different. I t might havo been forgotten even that two days earlier an investigation at the

21 2

university had begun. But at this junctnre of events, the very last thing that could have aroused his audience was undoubtedly a dis­cussion by tbo very revered ikolai [vanovicb on the theory of parallel lines.

Only Lobachevsky himself realized at that instant that this was the moment of triumph.

Tho lecture was forwarded to a commission for a reviow to decide on whether it should be published or not. The commission did not un­derstand anything and, apparently, did not ex· press any view at all . Either they did not want to endanger the well-being of a colleague, or tbere was some other reason. Anyway, the work was not published.

Then came 1827. The new trusooe was the tyrant and ignora­

mus Musin-Pushkin. But Lobachevsky had long been acquaJnted with him, and, judging by al l things, seemed t o be just tbo person to rehabi­litate the university t,bat had fallen so low under Magnitsky.

On Musin-Pushkin's suggestion, Lobachevsky was elected rector, which post he occupied until 1846.

He was re-elecood six times, first by slight majorities and then by overwhelming majorities. That was something, if one recalls the atmos­phere of constant intrigue within the university. There can be no question that he was a magni­ficent rector, who put a great deal of energy and love into his work, and a forward-thinking and very skillful admil ustrator. He actually foundod tho university. With great IlrOrcssional 8ki I I , be beadod tbe construction work, !'Ilt ul'

213

a library, organized the regimo of the students nnd adjusted relationships between the Russia� "

.nd German professors teaching at the univer­

sity. It i.s hard to sec whon he found ti me to devote

to SCience. Yet all his hasic scientilic results were obtained during these very years of admi­nistration 88 rector.

The year 1829. �he Kazan VI!,/nik (Kazan no-a/d) publ ished his memoir "On the Princi­ples ?I �eomotry". This Was the lirst systematic doscnptlOn of non-Euclidoan geometry.

The year 1830. In this year Lobachevsky be­came the hero of Katon. Choiera bit the city. That was the terri hIe epi demic thaL swept across L�e whole �f R ussia and of which tbe poet Push­kin, then In Boldin, wrot The Feast During the Plague. Later, PusH;n admitted that he couldn 't dialingui h between cholera and pla­gue. The epidemic was extromely sevore. In those days no ono knew how to protect oneself .gainst the contagion. The common peoplo set great storo by the "bite of three bandits".

Lobachcvsky arrogatod unto hi mself tho ftu­thor!ty 01 dictator. Tho whole staff of the uni­verSIty, to(,'Otber with their families, were iso­lated from the rest oC the world wIthin the walls Of. tho univ rsity buildings. Food Was deliverod wILh great �are. Out of 560 perSOns, ouly 1 2 wor t�en III and th�y were im mediately iso­lated. fhe r suIt was J U L over 2 per cent who conlracted th disease. Bril liantl

Thon came Ihe year 1 832, when ho married a nico young girl by tho name oC Varya Moi­secva. Love was mutual, though on his part a

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tiny bit too unrufned a n d somewhat too ra­

tional. Outwardly, the yoars 1 827-1834 woro very

lucky onCS for Lobachevsky. Fortune was On his

side. Ho was maturo and everything was tur­

ning out the way he wanted it.

} J is activities during the epidomic were mar­

ked by higher titles and by the Tsar himself.

Lobacbevsky, though a civilian, displayed the

efficiency and courage of a military leader; these

wero things Tsar Nlkolai valued. He simply must be rewarded. And "his ma­

josty", his "Imperial Iligbne " rewaIded him

for his efrort -witb the diamond ring, tbe titlo

01 Councillor oC State, the Order of St. Stani­

slav and tb personal gratitude of the sovereign

bimself. Vory good indeed. Tbe personal enU­

meration 01 deed. of, now, Councillor of State

. I . Lobacbevsky wa brilliant. The Iirst severe blow came in 1832 wben

Kazan University sent Lobachevsky 's memoir

"00 the PrinCiples 01 Geometry" to tbo Academy

of Sciences for a reviow. The oral review was

2 1 5

LO he given by Academician Ostrogradsky. He took his time about it, and then stated: "What is true is not new, what is new is not true. The memoir is not worthy 01 the a\,lention of the Academ y of Sciences. "

From that time onwards, Ostrogradsky became a Sincerely vicious and implacable scientific ad­versary of Lobachevsky. Timo and again he gave blistering reviews of Lobachevsky's work, because to him one thing was clear: that Loba­chevsky was a provincial charlatan who must be driven out of science i m mediately.

Ostrogradsky was a good mathematician in the full meaning of the word, though his merits have been blown up unduly. He cannot, of course, be compared with such Russian mathema­ticians of the 19th century as Chebyshev, Mar­kov, to say nothing of Lobachevsky. But if he really had wanted to, he could have mado SOnse out of Lobachevsky's memoir. True, Lobachev­sky bimseU was partly to blame. The style of his paper made reading it an arduous task. Not only is it concise beyond measure, but not clear­cut in tho least. True, Ostrogradsky should have been ablo to grasp the main idea. But bo didn't, he was enraged and did not coniine himseU to an oral response.

I n 1834, a well-known journal put out by Faddei Bulgarin , entitled The Son of the Fa­therland carried an article in which both Loba­chevsky as a scieotist and his work were slashed to pieces. Today it appears to be firmly estab­lished thaL this "roviow" was i aspi red by Ost­rogradsky. Howevor, I beliove that the al'Ucle is of 'Illite an indep�ndent alld extremely

21 6

instructive value. It deserves Our close stu­dy.

Tbe awful thing about it is tbat for the non· professional and even for the professional it carries great conviction. It would be hard to fmd a better instance of tho demoniacal power of demagogy, the force of conviction not via logic or reasoning but by implication, by into­nation, sophistry and dishonest tricks of rbe­toric.

Tbe crude, understandable, cheap humour that permeates tbe article is so convincing, act" so surely on the suhconscious that it compels ooe to believe that this Lobachevsky is an ignorant self-satisfied nonentity. It is al most as much as spelled out in full by the author, who was without doubt a giIted writer. One linds it hard to find a more brilliant instanco of the complete triumph of solf-confident superficialHy and idle twaddle over genius.

How journalistically professional it sounds, how keen, how seintillatingl

"There are people who read a book and say it is too simple, too ordhary, it contains noth­ing to think about. To uch readers I recom­mend the Geometry of Mr. Lobachevsky. Here indeed is something to think about. Many of our fiIst.-rate mathematicians hu,-e TOad it, thought about it and still do not see tho pOint. Aftor that I hardly need say that I , baving thougbt over this book Ior some time, could think of nothing to say; in other words, I bard­Iy understood a singl' idea . It is oven dj[ficult to uuderstanu holY Mr. Lobachcvsky was ablt} to concoct out of the sim rlc"t and cl m'ost chap-

21 7

ter of mathematics that we know geometry to be-how he could build such an abstruse, m urky and i mpenetrable theory, if it were not that he hi mself helped us by saying that his Geo­metry differs from the comnWII kind that we all studied and which, most likely, we cannot un­learn, and is only an tm.aginwy geometry. Yes, that makes things clear indeed .

"Just try to picture what a lively, yet mOn-trous, i magination can conjure up! Why, for

instance, not try to i magine black to be white, rouod to be quadrangular, the sum of all tho angles in a right triangle to be less than two right angles and ono and the same definite in­tegral to be equal first to ,,/4, then to 001 Very very possi ble, yet to normal reason it is meanin­gless" .

How subtly journalistic . How neat that "li­vely yet monstrous i magination". But that was only by way of introduction, tbe heavy artillery was to come later. And the most powerful wea­pon of all- quite naturally-was tho rhetorical question.

"But one asks why write such ridiculous pban­tasies, wby bave thorn published? I admit tbo query is hard to a nswor. The author did not once even hi n t at why he was publishi ng bis composition, so we perforce must conjecture on our own. True, at one point ho states clearly that, as ho claims, the drawbacks which he had detected in the geometry so far in uSO compelled bim to compose and publish tbis nOw geometry; but thiS, quite obviously, is untrue, and, i n a l l l i kelihood, was said only to conceal better tbe true aim of his composition."

218

After that arti llery barrage of sarcasm, we are ready for tho direct assault.

"And to this allow me to add a few words about the man him elf. How can one think that

fro Lobachevsky, professor of mathematics in ordinary, would wrile a book of any depth that could hard ly be an honour to the lowest village school t acber? Every teacher should have common sense cven if be does nol have much learoing. Yet this new Geometry is devoid pro­cisely of common senSO. Taking all of this �o­gether, I Dnd it highly l i kely that the true 81m of Mr. Lobacbeysky in composing and printing his Geometry was a joke, or better, a satire on scholarly mathematicians, or perhaps on all scbo­iarly writi ngs of tbo present time. Thereby I do not merely assume, with a high degree of pro­bability, I am fully convincnd that the insane passion to write in a bizarre and obscure mannor, which is so common of late among many of our writers, and tbe impudent desire of certain people to discover something new when their gifts are bardly enough to properly grasp what is old, are the two defects which our author wished to depict, and which he depicted with consummate skil l . "

Now that i s what r call real writiog; i n com­plete keeping "ith tbe style and traditions of Faddei Bulgarin, himself a dashing daring "gan­gster of the pen". But ono should not overdo it; it is time to display somo kind of scientific approach. One should not al low tho reader to have aoy doubts at the decisive turn in tbe baUle. The operation begins with a somewhat

219

risky admission. But the experienced warrior is apparently quite Sure of himself.

"Secondly, the new Geomeh'y, as I have al­ready had occasion to state, is written so that the reader cannot understand anything. Wishin" to get mOre closely acquaintod with this com'" posi!ion, I concentrated all my attention, fo­cusslJlg evory effort on every sentence, every word, every. letter even, and for all that I dis­polled so IJttlo the murk that

. envelopes this

composition so completely that I a m hardly in a state to relate to you what the matt;:,r is about, to say nothing at all about what is said . . . . "

. This is only an appareot retreat, for tho ques­�Ion comes naturally, " I f you understood noth-109, thon why do you undertako to reasoo and judge? " Nol He did understand what the maLler was, but this was simply a manoeuvre to demon­strate to bis co-readers how hopeless and mon­strously disfigured was tbe construction of the cnemy. And also to show his objective appro­ach. J ust lookl "Would you like to see for your­self what the original is like?" Then followed a long quote from Lobachevsky's memoir. He kno,�s how effecti voly precise is this manoeuvre.

ASIde from the fact that the memoir was written in a ponderous complicated style to comprehend Lobachovsky 's ideas required a I ugh level of mathematical culturo and a concentra­ted and unprejudiced effort on tbe part of the reade.'. What is more, no isolated quotation pennlt. one to Judge tho merits of a scient; fie work. Moro yot, aft",· .lIch a lI.ychologic:01 IJUl ld-up, au oxcerpt Jj rted out of contoxt cau 110

220

completely disarming. That was a sure move to capture victory. One last offort.

"But I must apologise, I simply cannot copy every word 01 it, ror I have already said too mucb. And r cannot relate this matter i n brief, ror that is where the most incompreilimsihio begins. It would 800m that alter a few dolini­tions, composed with the same art and the same precision as the prec ding ones, the author says !lOmethi ng about triangles, about the dependen­ce of tho angles in them upon tho sides-there­in lies the di fference botwoon his geometry and ours-he then proposes a new theory of parallels, which-and he admits as mnch-no­body is capable of proving whether it exists in nature or not; finaUy, this is followed by a consideration o[ how, in this i maginary geo­motry, one determines the magnitude of curved lines, of areas, of curved surfaces and volnmes 01 solids. And all 01 this, I m ust repoat once agai n, is written so that nothing at all can be understood . . . . "

The amusing thing is that though the author mastered the titles of Lobacbevsky's theorems, he was not up to grasping the fact that Loba­chev ky's geometry "differs [rom ours" solely in 1M theory 01 parallel lilies. But why indeed should One need to understand Rny· thing? The enemy is ill retreat, completely rout· ed, all that is needed is to consolidate the victory.

... . . Mr. Lobachovsky doserves to be praised for taking upon himself the labour of explaining, On the one band, the arrogance and shameless­ness of pseudO-inventors, and, o n the otbor hand,

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Lhe naive ignorance of the ad olirers of thair pseudo-inveuLiolls. However, realizing the [ull 'alllo oE Mr. Lohache\'sky's composition, I call­

not r .. lrain f"om hlaming him slighLly lor not gi villi!' his hook a pl'Oper litle and compel­ling uS to cogitato so wastelully for such a long time. Wby, i nstead of the titJe 0" th. Princip­les 0/ Geometry, eOllld he not have named it , say, A Satire On Geometry, A Caricature On Geometry, or something of tbat nature? Tben anyone would i mmediately see what tbe book was about and tbe author would have avoided a host of unpleasant i n!<lrpretalions and argu­ments. I t is lucky that I have been able to penc­trato to tbe tfllO purpose 01 this book, or heavon knows what people would think about it and its aulhor. Now I think and am evell convinced tbat the worthy author will feol greatly obliged to me for having demonstrated the true pOint of view that ooe In ust take wheo reading his composi tion . . . . U

This lampoon is quoted more or less in full in overy biography of Lobachevsky. However, though the biographers are indignallt aod abuse tbe writer in every i magi nable way, they usu­ally 10 e sight ol the most i mportant thing-tbe fact that it is a very cogent piece of writing. I am not in the Jeast i ntercs!<ld ill tho one (or several) who wroto it; tbeoretically One can as­sume that he was Sincerely lighting for tho pu­rity of cionce.

But, too, olle can readily sco what tbe reaction was of the readel's aod also what tbis article cost Lobachevsky.

After reviews of that kind, people taka to

222

their beds, gi vo lip work al together, or even cornmi L suicide.

On the background of t his pamphlet, Gauss ' letl�'· 1.0 Bo'l)'ai is t hllt of a I ndel', loving sol i­citous ratbur. Taul'inlls-uuother ono o f (;auss ' uvicLims"-burned his pape.r (or the sote rea .. Son, tbat Gauss, olTendod , d,ropped the corres­pondence.

Outwardly, this story seemed not 1.0 have

involved Lobachovsky at a l l . Ho .. eacted wit h

a n amazing Jack o f spirit. The .. e were a few

questions, and a year later ho published, in tho

transactions 01 the university, a very calm and

restrained reply. Too, an extremely cool anS­

wer was sent to the Son 01 the Fatherlalld. Fad­

doi 01 course, never IllLblished it. And Loba­cho'vsky seelDed not to care. He never tried to

i IIsist. That wa� tho cod 01 that. H would bo wrong to think that LobachevskJ'

\Vas not a man 01 actioo. His wholo lile and his 19-yoar-long !<louro as rector demonstrated

quito t.he contrary. Apparently, in this parti­

cular iustance he considered it below his dig­

nity to eoter into a discussion. I n general,.

he \Va surprisingly indifferent to any popularIZa­

tion of bis ideas. This is a psychological riddle

becau e i 0 aU other things bo was an extremely

practical man. What is more, he had it in his

power to put an end to the outpourillgs of his

adversaries. I n 1840 he p u blished one of his works i n

German. And already i n 1842 be was electod­

ou tbe suggestion 01 no otber than Gauss h i m­seU-to eo ... sponding Membership in the Got­

tingen Royal Society.

223

Gauss read Lobachev ky's pap r and was car­ded away by it. True, carried away in bls Own particular way. There followed opinions fuJI of admiration expressed ill letters to his friends; then very sharp replies wi th respect LO a reviow of Lobachevsky 's work gi von in a Germao jour­nal. Essentially, this review was of the same nature as the pamphlet pubJi hed in the Son. 0/ the Fatherland, and Guass' descriptioll of the reviewer was very har h. Finally, in letters to Ws R=ian correspondents he constantly inqui­red about Lobachevsky and even asked to con­vey his greets to the Russian mathematician.

But there was not a word in Lbo press, not a single leiter to Lobacbevsky Wmsolf, witb the exception of the strictly official correspondence pertaining to Ws election. True, he bad wanted to write aod ask for reprints of Lobachevsky 's works. That is, he was on the verge of doing it, but he never wrote.

Well, all right, Gauss bad Ws own reasons. But how are we to account for .Lobachevsky 's sileoce?

Alter being elected Corresponding Member, be was of course quite po itive that Gauss had read bis paper and approved of it. There can be no question that such recognition was extre­mely important to W m and very beartening. What would be more natural than to send Ga­uss Ws papers or at least to write him a letter asking for an appraisal of his ideas?

And tbis is to say nothing 01 the fact tbat if Lobachevsky had ever received such a lotter, then the professors of Kazan University and the whole Academy of Sciences would straightway

224

repudiat.o all earlier attacks and would rejoice in recognizing Lobacbevsky as the greatest ma­thematician of Russi a . .

Suppose he was totally indifferent to t�e OpI­nions of those around bim , though tbat IS very hard to imagine. Even so, he himself should surely have been interested in a detailed ap­praisal of Ws work by Gauss.

He never wrote such a letler to Gauss. Why'l Modesty? Pride? The fear of appearing to be importUJlate? I do not know.

It may he that he was deeply offended be­cause of Gauss' altitude, for surely I.he gr�at man could have writlen a couple of encouragIng words to a Corrosponding Member of the Got-tingen Society. I t may be. . . . .

I simply cannot fiod a satisfa�tol'y vel' 10!1, oven ever so slightly. The only tblng to be saId is that tbis mysterious chain of events demon­strates wbat a complex and uncommon man was Lobacbevsky, for beginning with ovemb�r. of 1842 he undoubtedly realized that recogmtlon as a mathematician in Ru in eould come to bim at any moment that bo himself . de�il·ed. He did not write. When 3 matt.er of Ws h fe IS at stake be is so chastely restrainedl Lobachev­sky the

' mathematician was quite a different

man from Lobacbevsl;y t.he university loctor. The mathematician was impractical, reserved, pbilosopWcally plaCid. . .

All these years be worked intensi' Jly str.lvlng

to find a rigo.rous proof of noncontrarll�lorl11e�s. Quite separatel

.y lr�m t�s nowed IllS �utl�S

at work, Ws famIly h fe, hiS ups and do,ms In day-to-day Ii Ie. His wife proved to be 01 • se-

15-1981 225

riolls turn o[ mind, and Quibbling and opon scandals occurred fairly often in Iheir home. And through it all he was tho model stoic. "Oh, my dear Varvara Aleksoovna . . . " with all respect-and thon he wonld disappear into his fortress, b.is study. Or he would sink into si­lence puffing at his pipe.

Tbero wero a lot of children in the family. He seemed ratber indifferent to the girls but he loved the boys with a kind of jealous, harsh, carpiog love. Particularly Aleksei, the eldest. So capable, so much like himself in childhood.

Meanwhi Ie lhere was no end of ad ministra Ii vo duties, which bo porformed in model fashion, running the uni vorsity o£liciently. And do oot forget the d ifficulties of the ti mos. The govern­ment and the Tsar woro salisfied.

For zealous service His I m perial Majesty had elevated Lobachevsky to his excellency the Councillor of State. And in the oWng lingered the still higher post of Privy Councillor.

Money malters were nol always in the bost of order, but he was still full of energy and not too old.

There wero endless intrigues and smearing and muck-raking among his colleagues. But so al­ways is the case. He took them in his stride, became severe, reticent, pedantically composed. But such trails arc commOn to aging men. He was ordinary in all lhi ngs and habits. His ox­cellency was a good host and knew a thi ng or two about cooking.

At the club thero was card-playing-be liked preference. But his recreation morc oft,cn COn­sisted in translating from the Greek and Latin.

226

He loved his university, and the students loved him. His work occupied him complotely.

Everything waS typically Russian. His bro�h­er Aleksei was a heavy drinker. A relatn'e of his wife was a gambler who lost a large sum of Lobachevsky 's money. His sons wcre grown up now, students. His favourite one gladdened his heart, the younger one didn't; he was so ohviously no mathematician.

What was On his mind all these years? What

gave hi m the strength to pursue the study of

his geomol,ry so persistenll)'? How was it pos­

sible to carry on with his goometry through all

the vicissitudes of a lifetime and not to turn

i nlO the most ordinary of councillors of slate?

Whence the will power? What buoyed h i m up

all these years? What were his thoughts whe.n

he was a lone iu his tudy? What were Ius

dreams? And hopes? I have no answer, and DO one else has either,

[ ' rn afraid . Nikolai [vanovich Lobachevsky ap-

227

pears to me as ono of the most mysterious men i n tho whole history of sciellce.

I n tho opinion of many of the most cultured peoplo of that period, Lobachevsky was, on the wholo, a very respected official. He was also "an eccentric practically out of Ills mind ", "the mad man from Kazan".

or course, his real life began behind the doors of his study. Quite naturally. But what main­tained him, what concentration or will-power, what driving forco? What wa ho guided by­love, batred, bope, uperciliousne , babit tur­ned to instinct? f cannot say. I ' m afraid no one can say. Because all tbe riches of the archi­ves add nothing about this second life 01 Ills wbicb was the most i m portant 01 all, the life that began inside bis tudy when be was alone ,,;tb Ills computations. Perhaps there is, alter all, just oue thing tbat opens up a crack.

In tbe y ar 1853, bis most dearly loved boy Aleksei died. Within a fow montbs ikolai Lo­bacheYSky was a ick man, broken. He began to lose bis sight, and the illne progressed ra­pidly and implacably.

He bad tbree more years to Ii vo. His routine went on and h sti ll performed his duties, bllt Ii Ie was a lready gone.

Let us recall his efforts to mako his SOn study mathematics; how, thougb sell..:ontrolled and calm 1lI0st of the timo, he would shout abuse when the boy was lazy, or would rejoico majesti­cally when he camo to his room to find the boy celebrating a successfully passed exam with his friends: "Continue, g ntlomen, 1 shall not bo­ther you . " Recalling all these tbings, one may

228

conclude that this harsh, unsociable man was

kept alive by a single romantic dream-that

of his son conti nuing his Geometry.

The death of A leksei meant that he IllmseH

W8 dead. Misfortuue does not come unaccom­

panied. During tho last three years of Lobacbev­

ky's life, OnO calamity followed another.

Perhaps be was now to some extent Immune,

for the end had already come. Tbero remained only one thing, his Geometry. .

Already blind, with only a few days 01 lile lolt. he dictated tho last of his works.

Chopter 9

NON-EUCLIDEAN GEOMETRY.

SOME ILLUSTRATIONS

Le� us look into the curiosi ty shop o( nOIl­Eucbd�an �metry: Our intuition, firmly root­ed as It is 10 Euclidean notions, will not avail us (or long and will constantly be in connict with the geometry o( Lobachevsky.

. To co,,!pel our scnsations to vote (or t heore­t lell l eq�lvalcnce of the two geometries, One has to put Intensc and sustained ollorts into the study of the geometry of Lobachevsky. Only then, that which at first superficial glance al�fl ared absurd , paradoxical will begin to sJlino Wlt� tho calm cold beauty of logic and truth. Si nce we speak of beauty, let us recall an analogy from the arts.

The canvases of the i mpressioni t painters that gi.ve us such great enjoyment today were derided WIth guffaws and shouts o ( disgust when first displayed in th art salons at the end of Ia t century. This reaction was of the same nature as that of Lobachevsky's contemporaries to his w�rks. Generally, we must note-to our great misfortune-tho (ollowing simple idea: I t is still a revelation to most people that one should "try to understand the issue at hand before giving an opinion". Too often, fragments of distorted twisted in­formation that reach us by accide�t are taken as sufficient grounds for authoritativo assertions ,

230

no matter whether they are for good or for bad. Incidentally, once again the geometry of Loba­chevsky was l uckless I n a most amusing fa hion.

Quite some number of years ago, I came across the following phrase in the works of a very well-known writer: "Lobachevsky proved that lines which according to Euclid are parallel in­tersect at infinity." This was then followpd by a round of clever, sweeping generalizations. I don't remember exactly what abOut. Al most about what I am now writing.

By the same token of that penchant ror su­perficial reasoning that I have just noted, T de­cided that the author bad never really heard of Lobaehevsky's geomotry. But this same phrase cropped up so constantly in articles and books by other writers that f realized, one fair day, tbat the subject was parallels in the mea­ning of Lobachevsky . . . . 1 ust half a psge from now we shall see that these lines are by 110 meanS Euclidean parallels. They arc relat d, roughly, like th. sea pilot o( a ship of the Middle Ages and an air pilot of a ship of today. The ingle term used to denote two dillerent notions crea­ted confusion in the minds of p ople removed from matbematics. Perhal)! they do not deserve to be harshly censured, yet neil her do they warrant any encourage mont.

To end the parable, 1 may add that what is apparently the primary sourco o[ the "literary version of Lobachevsky geometry" has been fouod.

The culprit Is, it turns out, Fyodor Dostoy­evsky, who wrote some very remarkable things indeed. I n his Brothers Karamazov, Ivan ex-

231

plains his moral and phi losopbical credo to Aly­osha and, among other things, has this to say: "Dut you must note this: if God exists and i f He really did croate the world, then, as we a l l know, H e created i t according t o the geometry of Euclid and the human mind with Lhe con­ception of only three di mensions in space. Yet tbere have been and still are geometl'icians and philosophers, and even some of the most dis­tinguished, who doubt whether the whole uni­verse, or to speak more widely the wholo of being, was only crented in Euclid's geometry; they even dare to dream that two parallel l ines, which according to Euclid can never meet On earth, may meet somewhere in infmity. [ have come to the conclusion that, si nce I can't Ull­derstand evon that T can't expect to under­stand about God. I acknowledge humbly that I have nO faculty for settling such quostions, I have a Euclidean earthly mind, and how could r solvo problems that al'O not of this world ? "

I d o not think o f identifying Oostoyevsky himself with Ivan Karamazov, and we are not hero dealing witb the problem of tbo ex istence of God. But wben i t comes to geometry, tbis is the reasoning of Oostoyevsky himself. And tho fact that it is a magni ficent piece of wriLing just goos to show how a shallow, superficial intui tion on tbe part of a superficial dilettanto is so unwittingly elevated to all absolute prin­Ciple. Strictly speaking, there is not a singlo correct idea in tbo wbole passage. Tbis is all tbe more exciting since tbe magnificent and p urely analytioal mind of tbe author is also. apparent in every word.

232

Ivan tben brings his intell<lctual eccentricity relative to geometry to its logical cuIminat

.ion

aud even extends it to tbe rea l m of PbYSICS: "Even i f parallel l ines do meet and I see i t

myseU, I shall see i t and say that they ' ve met, but still I won't accept it ."

Quite naturally, 1 do not intend to draw any far-reaching conclusions (any at alll) about 00-stoyevsky's writing from these excerpts. Ivan K aramazov had nO i nter sts i n geometry of course. For him the wbole thing was simply an illustration of his ideas.

But for us this can serve as an illustration of a distorted concoption 01 science and of Iight­minded reasoning about uncomprehended thi ngs (and of course tbe obvious ob curantism of Tvan Karamazov).

One migbt, however, justily Oostoyevsky at least i n the seuse that he did not perbaps make any actual mistake. The names o( tbe geome­ters are not given, and so one might hope tbat Ivan was describing elements of R iemannian or projecti vo geometry.

Howover, since on the one band, the words

"non-Euclidean geometry" are associated with

the name of Lobachevsky, and, On the other hand, all cultured wri ters have undoubtedly read Dostoyevsky witb care, the words of I van were consistently oxtrapolated to Lobacbevsky himself.

All these literary-psychological explorations are uselul, aside from general ideas of a didac­tic nature, in that they belp us to grasp the intellectual courage of 80lya1 and Lobacbev­sJ..-y.

233

Now that we have calmed down, let us return to Our Curiosity Shop.

We will naturally confine ourselves to only a rew theorems and will not at all talk about solid geometry. Throughout we will agree that everytWng occurs in a single plano.

First, of course, the postnlate 01 Bolyai-Lo­bachevsky-the great antagonist of Euclid's fiILh.

"Through a gl lien paint it is possible to draw to a given straight line (in addition to 'Euclid's parallel') at least one mOre straight line that does not meet the given straight line.»

Whence i t foUows immediately that one can draw an inJinity of such straight l ines.

Look at the figuro. A perpendicular is drop­ped from point A to tho straigbt line I. Euc­lid's parallol-line EP-is naturally perpendi­cular to this perpendicular.

The dashed line (LP) does not intersect I. By means of symmetry reasoning (bend the

drawing along the perpendicular A BI) it is clear that there will be another straight lino of exact­ly the samo kind . I t is also dashod . Further, i t is clear that any of an infinitude of straigbt lines drawn tbrougb II i nside tbe angle between

' .

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the straight l inos E P and LP will not i ntersect th!l straigbt lino I eitber. We thus have: "Through a given point it Is possible to draw an infinity of straight lines that do not meet a given straight line. tt

But oue can naturally also draw an infinity of straight l ines wbich do Illoet the givon lino. They may be drawn to any point (of the straigbt Iino) arbitrarily distant from tho base. Let us take any point B' and join i t by a straight line to A . TWs can always bo done On the basis of a familiar axiom.

And so we have a straight lioe passing tbrongh bolll A and B'.

nowever, due to the continuity of tbe bundle of traight lines, thoro must be a boundary line that separates tho two classes. TWs is ei tber the last straight line ("intersecting") that meet" the straight lino BB", or tho first "nonmecting" line. I t is readily seen that thero can be no last "intersecLing" linc. Jndeed, uppose i t exist . Let i t be A B' in our figure. But then i f we take B" beyond B' and connect i t with tho point A , we get a flew straight l ine lying beyond B' and meeting (i ntersecting) the straight lino I.

235

Consequently, the bOllndary straighl lino is the first one that does not meet the straight line [.

There are nalmally two such straight lines: ooe for each direction. Within the anglo formed by these straight l ines, we can draw an i n finity of straight lines that do not meet the line 1 ; those w i l l also includo Euclid ' 8 parallel.

IJobach vsky gave tho namo parallel to these lwo extreme nonintorsccting straight lines.

As you see, thoy do not havo 8DY relation to the parallel a understood by Euclid.

Stretching Ihe point 8 bit, we may say thal they, 8S it were, intersect the given straight line BB" at i n finitely distant points. However, i t is not al all clear what is meant by "infinite­ly distant points", so it is better Dol to use that phrase at a l l .

I n LobachevskJ" s terms, a l l straight l i nes within the angle "diverge " from the straight line I.

To summarizo, then, r lalive to a given straight line there are three types of straight l i nes that may be drawn through any poinl.

1 . Contll!Fglng (intersecting); thero is an infini­ly of such line .

2. Paralkl. Thero are two. Of each we say: parallel 11 is parallel to tho straight l i ne I i n the direction BB'; parallel III is parallel to [ in tbe direction B' B. Tho meanillg of these words is clear from the figure.

3. Di verging straight l ines. Tbese comprise tbe i n finitude of lines \\1thin the bundle, one of which is the "Euclid's paralle l " .

236

Tbose are Ihe terms. 'ow lot u look into the theorems.

With regard to parallels, Lobachevsky demon­strated tbal they apporoacb without bound a given strajght line (without ever meeting it) and recede, without bound, On the other side.

So lar, this is nob such a strange resul t. But the next one is like a thunderbolt. Two di verging straighl l ines always bave a

com mon perpendicular wlllcb is tbe shortest dis­tance between the m . They recede wi thout bound on both side of tbe perpendiclllar. Tills naturally holds true for the special case of "Euclid's pa­rallels" a weU.

Thus, a perpendicular d ropped from any Iloinl of the straigbl l i ne II onto tho straight line I is, firstly, greater tban the mutual perpendicu­lar A B and, second ly, dnes not form a right angle with the straight l i ne 11.

This is indeed strange. But the proof is i m ma­culate.

Accordingly, the locus of points equjdjstant from the straight line tmlls out to be a curved lino.

These are only the first stellS.

�" -......;�TAor;;;;;; __ "

----I;-1!r+I

237

A t this point, Lobachevsky introduced a new and very iroportant concept: the paralkl angle.

This is the acuto angle between the straight lino parallel to I and drawn through point A , and the perpendicular A B dropped froro lhis po­iot onto the straight line I . Thus, the parallel angle is CA B. According to Euclid it is

n naturally always equal t o "2 -

I t wil l readily be seen that this angle depends on the distance between A and the straight l i ne I, and diminisbes with increasing distance.

Indeed, take a point A ' on thQ prolonged per­pendicular AB and draw froro this point a "Euc­Ii dean parallel" to tbe straigh t line A C. I t wi l l inlersect the perpend icular AB at the saroe ang­le as the straight Iino A C.

LDA'B = LCAB But we know that from A' it is also possible

to draw n straigbt l i ne A'C' parallcl to AC in the sonse of Lobachevsky.

The angle C' A ' B is also obviously less than the angle DA 'B.

I L is obvious tbat if lhe straight lina A 'C' does not i ntersect A C, it will definitely not in­tersect the straight l i ne I. H wi l l either diverge [rom i t or be parallel to it. ( From here onwards I will no longer say "in tbe sense o[ Lobachev­sky". We ,,�ll adhere to his geometry and to his definitions.)

Lobachevsky actually proved tho theorem: "If two straight lines are parallel to II third

ill one directioll, they are paralkl 10 each olher in the same directioll". And so tit aogle C'A 'B

238

i

is lhe parallel augle to the straight line 1 at the point A ' .

Tbe parallel aogle i s a function 0 1 the distance to the straight line. Lobachevsky de­noted this functiOn as l1(x); x is the distance, that is, the line segment A B.

We have already seen that tbis lunction di­minishes Mth increasing x. Lobachevsky inves­ligated its behaviour wilh decreasing distance x and showed that tbe parallel angle IT (x) tben tends without bound to a right angle. Symbolically, scientifically, this looks like

l im IT (x) � �. But i f we recall that a right pa­"-+0 rallel aogle corresponds to Euclidean geomet­ry, then it is clear that a t small distances tbe geometry of Lobachevsky is practically ind is­ti nguishable from the geometry of Euclid.

Clear so far. What is nol clear, however, i what we mean by " mall distances".

'fhe words usmalP' or "large" have meaning only il we know what is beiog compared. Without that knowledge tbey are devoid 01 any content. There should obviously be some kind of length, or standard that can be used for purposes 01 comparison.

239

How docs such a standard enter here? I t is well worth recalling Legendre at tbis point. He too di covered that the parallel angle de­pends upon the distance. Actually, all tbat needs be done (as We have already mentioned) is to analyse his proof with regard to tho sum of the angles of a triangle . The very fact tbat a relation­ship like this appears se med to Legendre so absurd that at one time he declared it the desired ab­surdum that proved tbe filtb postulate. Legend­re's reasoning was i ngeniou . lie argued more l i ke a physicist than a matbematician .

Actually, he employed a very strong method of qual itati ve analysis of pbysical problems cal­led th di mensional metbod. Brought up to date, b is reasoning might look like tbis .

We see that tbe para l lel angle is a function of only one line-segment, tbe di tance from tbe straigbt li ne. 0 other Lin ar d imensions enter i nto tbe problem. We wri te 'I'=II(x).

Now let us St.'C what we have written_ Any angle 'I' is a dimensionless quan tity. ( I n rad ian measure, an angle is tbe ratio of tbe arc of a unit circle to the radius.)

240

On Lhe lefl we have a d i mcnslon le 5 quan tity . I t remain tho same, no matter what units or measurement Ilre used, whether centimetres, met­res. inches or whilt h�IVO you.

On the right, however, the [unction is lbat of a dimensional argument. I t makes DO di ffe­rence what form it has. The i mportant thing is that no maller what i l is, it'! numerical va­lues wi l l vary with the unit of measurement. I f

�ay, r I (z) = :. , then for x - I metre, r I (x) = 1m-�. But if the unib is 1 cm, tben

I l (x) t = 10-' em-! IOO� ernt

Tbis is obviously nonsense. The relation we bave suggo ted is impossible. Consequently, the nfth po Lulato is proved .

The chain of reasoning is absolutely correct , but tho conclusion i s nOl o The conclusion bas to be d i fferent. From t he same arguments of di mensionality it is cloar that in our form uJa there should be a nondi meflsional quantity On the right in tb a.rgument of the function. This is what tbe equation should look like :

l' = J l (:) wbere k is some segment wbicb we st i l l do not know. Tho question is wbere do we lind tbe segment k? Tho point i thal tho wbolo of our analysis shows that tbe para l lel angle 'I' de­ponds solely on One distllnco, the distance of the point from the straight l ine.

There is only One way out. We bave LO as­sume thal i n tho flew geometry there is a pacific,

16-1M: 241

nature-given consl ,"t unH. A kind of constant length tliat determines all otbor lengths.

This is strange but not entirely absurd. Fu,' i nstanco, the two-dimensional Euclidean geomet­ry of a sphere has such an isolated length. I t i s the radius of a spherical surface. And so when employing the formulas of ordinary Euclidean spherical geometry for a geodetic mapping of Mars we wi ll bave to bear in nilnd that some of the "constants" of our terrestrial tables wi l l undergo appreciable change.

Lobacbevsky was not embarrassed by tho ap­parent paradox and introduced a constant seg­ment k and found the equation for tho paral­lel angle. It is so simple tbat wo give i t here:

x 1 k

col T 'f = e

wbero e is tbe base of natural logaritluns. From tbis equation we immediately see tbat

"' O b 1 ° 1 ..!..,,, r. d when � , t en cot :(1''''' ' = , or 2 '< "" T an

'f"" �. Wben '1' =90·, we bave Euclid's geometry

to a high degree of accuracy.

But i is close to zero wben xtt.k.

Now what we said just a momeot ago about small segments has taken On precise meaning.

I f the distance from tbe point through which we draw a parallel to a given straight line is much less than the constant segmeot k. then the geo­metry of Euclid is ful filled in -approxjmate fasb­ioo.

242

I n tbe linilting case when k= , Euclid '5 geo­metry is always Iulli.Ued and witb absol ute pre­cision.

Tbe first que tion that naturally confronted Lobacbevsky was how to find the segment k.

Aod here i t turned out that his geometry was in it certain sense "better" than Euclid 's. No theoretical arguments help to define k. J t is what pb ysicists tid I a "constant or the theory" . I t can ooly be found experimentally, by means of concrete pbysical measurements.

I t is of course i III possible to measure the par­allel angle directly, but it is for instance possible to measure the slun of angles of a triangle_ The "defect of tbe sum " i n a given trianglo depends on the valne of k.

You remember tbat botb Lobachevsky and Gauss urged such measurements but nothing came of tbem.

Generally speaking. Lobachevsky never said that it was precisely bis geometry that dcscri-

,.'

.. """,8 , fa cP " "" ., "

cP e�� -,---"" IJ,

243

bes the world . Qui te the contrary, ho incl ined tow" .. ds the view t hat in this world, i t is Enc­l i d 's geometry that is accomplished .

But that is not 0 important. The remarkable thing is that from the very mst steps the new geometry was clo Iy tied i n with physics and that it was i nconceivable to disassociate it from ox peri menlo

This natllrally put forth the salient problem of the relationship o[ goometry i ll general to the real world , the possi bility o[ di fTerent geo­metries in the real world .

As we have already said before. this WaS sug­gested earlier but [or two and a half Ulousand years mathematicians took a d i m view of i t, regarding the whole matter as futile and absurd .

Willy-ni lly nOll-Euclidean geometry generated tho p" obl m of experimentation. Are we indeed so sure tbat God made the earth i n accord with the laws of Euclidean geometry, as Ivan Kara­mazov would have us believe?

Th.cro is always beauty i n abstract formulas engineeri ng tota l ly unexpected ideas, which even the discoverer never su pee ted whon bo deri vcd his formulas.

A l l these couc1usions aro so charmingly elo­gant that one can understand Bolyai and Loba­chevsky who had faith in tho logical rigour of their system.

Ole also that we have d i cussed here only ono of the conclusions of Lobachovsky's very first work, his paper of 1826.

Ho immediately developed this scheme in depth and the other results were no les beautiful . But in mathemalics, faith is not a decisive factor.

244

There were no guaranteos that a logical contra­diction might not pop up in the future.

Lobachovsky spont the rest of his l ife in per­i tent attempts to find tbis proof. He strived

to demonstrate witb completo rigour that his system was nawless. On tho way he worked out a great diversity of tho most unox l>ecled con­sequences of his geometry, penetrating ever dee­per.

III this respect, he is without a doubt head and sboulders above his contemporaries, for nei­ther Bolyai nor G auss covered the ground that he did.

He did not find the proof, though he was ra­ther close to the basic idea.

Lobachevsky the man, his persistent, never swerving truggle toward a single goal is worthy of our adm i ration.

Chap I" 10

NEW IDEAS. RIEMANN.

NONCONTRADICTORINESS

No, this will not be a chapter about things of startling beauty. I ,,� Il be bonest with tbe reader. At least the first half will be rather dry mathe­matics.

First about tbe theory of surfaces. Tbe pro­genitor was again the same old Gau .

Let us imagine that On some kiod 0 1 whimsi­cal l y bent surface there re ide i n telligent beings of two di mensions (not three). Wbat wiU their geometry be like? Secondly, bow wiU they be able to seo that tbeir surface is curved?

At first glance, tbe second qucsUon may ap­pear quite naive. The reader may be recalling proof 01 the sphericity of tho eartb given in grade-school geography books. Don't hurry, re­member that we are three-dimen ional beings li­ving on a two-dimensional surface.

To rid ourselves or tbe i l l usion that this is simple, think over the question: How can one find out that our throo-<li mensional world i. cur­ved, and what in general doe. this so Irequently employed phra mean after all?

The throB- and four-di mensional world wiil be looked into later on, lor tbe present let us ro­turn to surfaces.

Causs began by i ntroducing n marvellous quan­lily lhal dofi n�s t hl' geometry 01 a surface. I t is called Gaussian curvnlurc. Tho fun�8nlClllnl

246

property of Gaussian curvature is: it remains constant under any bending of the surface 80 long as no stretcblng occurs. I t is intnitively clear what this means, hut a rigorous formula­tion is better: if in tbe bend iug of a surfacc there is nO stretching, then, first of all, the lengths 01 all curves drawn on tho surface remain unchan­god; secondly, the angles between tbem remain the same too.

This can be stated somewhat differently. Take a sheet of paper. Bend i t . Then measure tho Gaussian curvature at some point. ow you can do wbatever you want to tbis sheet (except stretching or tearing it), like twisting i t into tbe most bizarre forms, and the value of the Gaussian curvature at that point will not change.

The Gaussian curvaturo is so i mportant a eon­cept that wo will delino it in more rigorous fa­shion. To do this, we will bave first to find out what are radii of curvature at a given point 01 ft surface.

We consider some point of a surface and draw a line normal to i t. What is a normal? To ex-

247

plain we will need One more concept, that of a t angent plane. We give a n almo t rigorous defini­tion. We consider all possible curved lines lo­cated On the surface and pa"sing througb a po­i nt P.

It turns out tbat the la"geuls Lo all these curves lie C D ODe plane. This is 1I0t evident 'It first glauco, bu t i t ClUI be proved rigorously . H is lho ontire coUection of tangent lines that forms a langent plano.

For the case shown at tbe bot tom of page 250, the location of t he l'ngent plane is rather ob­vious. But someti mes the tangent plaue is lo­cated more i ntricately relat i ve to tbe surface (see the figure on page 2/.7).

Now let us define precisely tllo notion of a normal. The normal is a straight line perpendi­cular to the taugent p lane. We can now defioe tho coocept of principal radii of curvature. Pass a plane through the normal. There are cloarly an i n finitude of such planes. We take any oDe to hegi n wi th. A plane curve is formed by the i [1-terseclion of tho plano and the surface. One can al ways choose a circle that is contiguous to this curve near t·he point P. I shall not explain the exact meani Jlg of these words in the bope that your intuitiou will suffice to create tbe proper i m age.

The radius of this contiguous (tangent ial) circ-10 R is called the radius of curvature of tbe plane curve. Since ao in finity of plunes can be passed through the normal, wo get an i n finitely largo n u mbOl' o( r"adii o[ cUl'vatul'e, among which t hero is a greates� and a smallest OUt) ill i'h:oluto v"luo. I t ClIlI be proved that plano CUI'VOS to

248

which the least aud grealest radii correspond are mutually perpendicular at the t)oint P. These two radii, R, and R., are called the pri!,cipal radii o f curvature o f our surface at the pOIDt P.

Likewise we can prove that the centres of tbe circles are al ways located on Ule Jlormal .

I f the centres of curvature lie on one side of the surlace, the point P is called elliptical. I I lhey lie OJl different sides, then it is csUed hyperbolical. I n this case, one of tbe principal radii must be considered negativo.

Finally, there are parabolical points. They are points, where one of the principal ra�ii of cur­vature is equal to infinity. The GauSSIan curva­ture at any point 01 a smlace is defiued as:

" I .n = RI/l,

749

Now wo cao set our fiodiogs out io a table:

In Ibe elliptical point In tbe byporbnlieal point In Ibe ptUabnlical point

K > O K < O K = O

Now let \IS see what properties the surface as a whole can have. Imagine som surface aod try to cover i t with a piece of closely adhering cloth. Tho rules of the game are: that the cloth caonot be cut or stretched , and has to cover the surface without any folds.

I f a lady coo[rootAld a tailor with such demands, she would be dismissed without further ado, and he would be right in doing so.

The reader would do well at this pOint to stop reading and try to picture the properties

250

that the figure of our hypothetical lady of fash­ioo should pOSSC8S. A fter what we have fouod out about tbe properties of Gaussian curvature, the answer is imple. The piece was plane at first. Which means the curvature was zero at every point. Bending without strot.ching does not chan­ge the curvature. This means that a plane piece of cloth may be bent ooly into a surface whose curvature at every poiot is strictly equal to xc-ro.

A cyli nder is oDe instance. f t is easy to see that the Gau ian curvature is strictly zere 00 the latAlral surface of the cylinder. Or, in other words, every point of the surface is parabolica!. If you have mastered the concept of curvature, then it will readily be seen that the second examp­le 01 a suitable surface is the coDe.

ow we cannot bend the plane onto a sphere as required. Tho curvature of a sphere is cons­Ian t and posi ti ve. I t is precisely this circumstan­ce that causes cartographers so much trouble.

251

We must observe, rather tardily, that a l l along we have had in view only "good" surfaces. To put i t cnldely, "good " surfaces are those that have nO sharp points or edges. The vertex of • cone, for example, is a "bad" point.

Also, when we speak of bending one surface onto auother, we have in v iew, strictly speaking, the bending 01 a sulijcienUy large piece, but not tbo whole surlace. To take an example. the entiro lateral urfaco 01 a conc can be developed onto a plane only i f We make a cut along the generatrix. The last term we ha,'o to ex plain is a geodetic line. A geodetic is a curved line drawn On a surface between two points so tbat any other curve is longer. Tbis definition is one of those "almost rigorous" ones, but I have hopes that only non-mathematicians will read this chapter and so there will be no one to criticize me.

Hypothetical beings of two d imensions who live 01\ such a surface will say that the geodetic line is the shortest distance ootween two points. I ncidentally, three-dimensional beings (like we are) would say the same thing i f we impose t,he condition that thol' should not leave tho surface.

To uS earth d wollers living On a pbore, the shortest distance between two points on the earth is an arc 01 " great circle. It is precisely along the arc of a great circle that navigators sail their ships in making the briefest voyages. Now let us look into a very c urious problem. We said that a plane may be beot olllo a surlace whose curvature is constant nlld equal to zero. Or-what is tbe same thing-that such a slllface may be developed OlltO a plane. Any figure drawn On the plane wi ll turn into a similar figure 011

252

"ur surlace. The anglos betweell lines do not Ch81l�'" during tho hellding process. The ghol'lest I ioes on the plano-straight lines-wi ll pas inlO geodetic lines on the surface. 'fhere/ore, for a cylindrical triangle, lor instance (its sides are naturally formed by curved l ilies). the sum of the angles remains U,e same as in the plane triangle. We an go on reasoning in the same vein. To overy geometric concept on the plane we can conelate a correspondi ng i mage on the surface.

I t is rather easy to sec that all the theorems that hold for the plane can be carried over without change to the surface. The only tbiug that we must ooar in mind is that those thoorems now hold true for "images". I f Euclideau geometry is ac­complished on tho plano, then it will be accomp­lished on a cylinder Jor the "images" as wel l .

We have noW tOllched 011 one o f the most remarkable and beautiful aspects of all mathe­matics. So long as we are not i nterested i n any practical applications, i t is all the same to liS what ou"r theorems speak about. We only want them to satisfy the demands of logic. What is more, we do not even know what we are talking about. It is only the physicist that has to know what is "actuall y " taking place, what his world is roally like.

For the physicist, a straight line is a ray of light. For UIC mathematician, it is one of the basic undefined concepts. There is nO way of disti nguishing between the straight lines on a Euclidean plane and the geodetic lines 01\ the surface of a cylinder i f they are compared solely from the point of "iew of ax ioma tics.

253

Let us conjure up a fantast.ic picture. Two two-d ime, •• ional world�. One plalle, tho other on the surface of a cylinder. Intelligent beings I i " . in

. both worlds. Suppose they have set up

some kind of communication. The two-di mensio­nal "plane" matbemalician and the two-dimen­sional "cylindrical " mathematician would assert with great satisfaction Olat their geometries are lho same.

I f the sy�tem of axioms were contradictory On the E ucbdean plane, we would know i m me­diately that it WaS contradictory on the cyUnder 8S wel l .

One could eXI)lain to tho other the theorems he bos .developed,

.and Ole laUer could accept

them w.thout makIng any modifications. They could work together without tho slightest ["ic­tion. Now the physicists in the two worlds would be in conDict from the very start. They would claim, each in his Own world, that the laws of nature are difterent in the other world.

I.ncid.eotally, eveo if a ray of light in tho

cyhndrtcal world followed a geodetic line, they would not be able i mmediataly to detect any difference.

The reader has by this time guessed that we are rather close to the problem of noncontra­dictoriness i n non-Euclidean geometry. I f we were able to find,

. i n ordinary Euclidean space, sur­

faces on which Lobachevsky's geometry is ac­complished . . . if theso surfaces could be made so that the whole of a Lobachevskian surface could be mapped onto them, then the problem would be sol voo.

The first "if " is satisfied. Such surfaces (cal-

254

led p-eudospbercs) exist. ThoM tlte surfaces with

a COn tant /loga live CUJ"\'atw·e. But Ihe st'CODlI

condition ha us stumped. The entire surface of a

pseudosphere cO\"fesponds to only a piece of a

Lobachev kiall surface. Let us forget noncontradictori lless for a moment

and say a few words about Riemann. Tn tho ycar 1 854 this morbidly shy youth opened up

fresh vistas in mathematics. And now let us hurry back to the Gaussian

curvature, this timo invested i o a purely ma­

thematical language. We consider two arbitrary

families of CllrVes on a surface. We repeat, the

families can bo quite arbi trary. Together tho two

fam ilies form a coordinate grid. Now suppose

wo want to find the d'istance between two very

close (otherwise, completely arbitrary) points X, and :r .. .

Gauss considered tho following oxpression:

6Sn � gu(x,x.)llx', + 2g .. (x,xJt.x1t.x. + + g .. (x,x,) ll.:l: .

I t is called the basic metric form. For the oon­mathematician this formula is rather formidable

255

in appeArance . 0 need Lo fear. w wi ll not llSO i t . Only two remarks.

1 . The "physicaL " meaning of this e pre ion is very simpLe. It is the square of the di tanco between tbe points x, and x,.

2. K I I(x,x,). g,,(x ,x,) and g .. (x,x,) na turall y vary Irom one point of the surlace to another. We pllt x , and X, in brackets so as to sbow that all Lhe expre ions K I I . K " and K" depend on the po ilion on the surface.

Tbe i mporLant thing here is • r suit that Gau obtained. Ho demon Lrated that the cur­vature of a udace is completely definod by tho numbers KI I (x,x,). K i t (x,x,). Kn (x,xJ. This is not all. Ho proved thaL no !Uatter what sy t­ern 01 coord inates is choson. tbe curvaturo does not change. This i not self-evident in the least. Indeed. all the numbers g i l ' g l t. gu. speaking general ly. change when we switch to a new coord­inate grid. But the Gaussian curvature is built up out of these numbers in such fashion as to remain unchanged. I n other words. the Gaussian curvature is completely independent 01 the man­ner of description .

n is an inner property of tbe surface. And so for plane surfaces the entire geometry is determ­ined solely by tbis relationship or the hasic metric form. This form depended on two va­riable . Knowing tbe coefficient , we could com­pute the Gaus ian curvature of tbe surlace at any pOint.

Riemann's idea can be conveyed i n just two words. [ n a purely lormal fashion let u examine similar expr ions for three. four and /I varia­bles. We will say that these metric forms define a

256

��omelry oC • lhree-, four-. and n-dimell ional world. Forma l ly, we call compute the Gau ian curvature for such worlds. We will be a blo to . ay exactly what geometry will be accom pli hed in each one.

[I the curvature is different [rom zero. we say that tbe world i curved. And we would notice that without eVOD leaving a singlo point. All we have to know is the curvature at that point.

The geometry 01 11 "world" c"n be 01 any kind. A t this junct ure, i t docs not oven matter very much what geomet,·y is used. n ioms"n's theory provide lor 1111 concei va ble cases.

That. roughl y spel1king. is n i l . I t i simply u g nNal i zutiun o f tbo Gau�.ian

theory 01 su ,laces to I he Cllse 01 mony variables. At the begi n n i ng oC this twclllieth c ntury. i t t urned out that i t i s precisely n iomalll\'s geom­etry which we need to descri be the actual world wo Ii,·. in. And not lor three but Cor four d i men­sion • tbe lourth di 11lcn�ion ooing li l1le.

\ e leavo n i mann. My ta k 110 i to relrain from shouling hur­

rah. Througbout tbe wbole of mathematics there

are hardly a dozen ideas equal in sheer beauty to the proof of the non-contradictori ness of Lobachevskia n geometry.

The whole structure re 1s on the fact that tbe mathematician cares not a whit about what lies behind bis Basic Concept -so long as the axi­oms are satisfied.

Up to a point. geometry is hardly more than a game in logic. Tbe straigbt l ine . the pOint. tho plane. molion are i mp ly pieces used in tho

257

game. The only thing tho mathematician knows about thorn is his 8.xioms, tho rules of the game involving the pieces.

At this stage, geometry i just as useless to tho physicist as che s or dominoes. I t is only wben the physicist finds out experimentally that his real straight l i n s, points and so forth Mn be very precisely described by mathematical abstractions, only when he sees tbat tbe axioms o[ mathematics do indeed describe the behaviour o[ quite r(lal lines, points, plan s, etc., only then does geomotry become one of UIO chapters of phy ics, tbe science whieh tudies the world about u . p to that point, geometry is a game of logic.

But it i just this unexpect d ituation that enables ono to prove the noncontrad ictory nature of the geometry o[ Lobachevsky.

Here is tho problem. Thoro ar two games: Euclid's g ometry and

Lobachcvsky's geometry. Let US attompt to demonstrate that if in the

rules of OnO o[ thorn there is a hidden internal contradiction, thon it "il l inevitably occur i n tho rules o f the other one.

The rule of the game-I repoat- are the a1i­oms.

You will e that we havo somewhat changed tbe statement o[ the problem. , e rcaUte that it is a hopelc s undertaking to attempt to provo rigorously tho problem of noncontradictorincss.

o matt r how many millions of theorems we prove, there will nover be complete confidence that the next theorem wil l not contain a con­tradiction.

258

We shall now prove that if the geometry 01 Lobachevsky is contradictory, then the geom­etry of Euclid is unaVOidably contradictory as wel l .

A t Iirst glance there i s no clear way out here either.

'The rules o[ the game (the axioms) are diUer­ent. True, they di ffer onJy in one axiom, that o[ parallel lioes, but fundamentally the situation remains the sarno. The games are diU rent, and it is not clear at all how one can bridge th gull between them. Still and all , there is a way.

J am afraid that the variety of analogies bro­ught in to illuminate the problem will only obscure it the more, and 80 I will start on the proof directly. The man who gave us this proof was ono of the greatest mathematicians of the 19th century, Felix Klein. He was an interesting man, of great complexity, but unfortunately we cannot go too far into history. J wish to recall only one striking fact.

Klein lived a long lire. I f we take onJy the papors he wrote after the age o[ 30-35, he would be a magnificent versatile scientist by any stand­ard. An active, ubtle, fertile mathematician, and a brilliant expert in tl! history o[ his sub­ject; he was one of the best toachers in the whole history of mathematics.

He made a harsh, categorical statement once. He aid that after the age of thirty, because of a nervous breakdown brought on by tho investi­gation o[ a certain mathematical problem, he wa never again capable o[ creative activity. This was not coquetry, eitbor. It was exactly

17' 259

what he thought . I like such people. [ t is ((uilO n di lfcrCIlt. question as lo wbelher t hat makes theil' lives easier [or them or not.

So here is the proof. Firs� we p lay Euclidean geometry. Consider

an ordinary circle. Drllw a chord. 'fake a point not lying on that chord. It is evident that ono can draw through the point any number (an infinity) of other chords that will nol intersect our chord. They will make up a l l the chords that lie between the two chords that i ntersect ours at tho end-points (where i t cuts tho circle).

Clear so 1aI". But what has this circle to do with the geometry o( Lobachevsky?

Here is the miracle. Klein's idea was to convert the trivial circle

into a model o[ a Lobacbevskian plane. Tbe point is-we repeat-thnt the mathematician is quite indiITol"Ont to wbat his Basic Concepts re­fer. The ultimate thing is that his axioms he satisfied. Now we can start "laying the double game. We define a circle as a Lobachevskian plane, any chord i n tile circle as a Lobachev­skian straight line, and a point as a Lobachev­skian point.

Quite IIll1.llrally we have to add some fresh notions liko "relation", ULo lie between", "lo belong" and "motion". With lheso new concepts at our disposal we can p lay "Lobachevskian geometry" using tho elements of Euclidean geometry.

To be able to do Lhis, we have to check through our list of axioms and see whether our elem­ents satisfy the axioms of Lobachevsky's geom-

260

otry. I t is comparatively easy to see that every­thing is in order with most o[ the axioms. Even­marvellously so, in fact-with the parallel axiom, which is thO only one that distinguishes Loba­chevsky's geometry from Euclid '5. "One can draw through a given point to a given 'straight lille' an inunity of 'straight l i nes' that do not intersect it . '!

r give stl'sight line i n 'Iuotes, but all we hav6 to do is prove that for our concepts all the axi­oms of Lobncbevsky's geometry al"O fulfilled and tho quotatioll marks call bo removed.

Do not (orgel that we arc playing a double gamc. All the t imo we havo to t ranslate fmm the langllage o[ Euclidean geometl'y into that of T.ohach.vskia ll geometry. Aud vice versa.

Everythiug is well with the notions o( "to belong" and "to lie betweon". They romain the samo in both languages. The di fficulties begi n whoo we go over to motion. The concept "molion" has to satisf�' the cnti..., group o( axioms o[ motioo.

261

We have stated that our circle is tho Loba­chevskian plano. Well and good. We can de­line motion in the Lobachevskian plane. Such motion most satisfy all the required nioms. (Glance through them. They are at tho end of Chapter 3.)

This fits too. But ono thing is not clear: is it po sible to formulate tho concept of motion of 8 non-Euclidean plane in the language of Euc­lidean geometry?

I n our case, the non-Euclidean plano is a circle in Euclidean language. Motion, it will be re­callod, i 3 ono-to-one mapping (transformation) of II plano into itself. Tllis mealls that in Eucli­doan laoguago we must fiod some kind of trallS­formation of the circle into itself.

Ono cIa of such transformations insist ntly claims our aLl ntion. These are simple rotations of tho circlo abont its centro. However, it is easy to see that these tran formation cannot be uaed as candidates for "non-Euclidean mo­tion'" .

I n rotations, i t is not po ihle to transfer any given point of the circlo to any other proass�gned point. For example, the centre of the clfcle. I n such transformations, iL is always a fixed point, passing into itself. Now tho axioms de­fining motion roquiro that, in tho process of motion, any given point can bo transferred to a different point. So rotations cannot satisfy us.

Yet tho transformations of the circle that wo need oxist. That is the central and most radiant part of the Klein schomo. lie pointed out an i nfinite Dumber of such transformations of tho

262

circlo (they are called projecti vo transforma­tions) which transfer a circle into a precisely identical now circle, any internal point of the old circlo passing into a n internal point of tho now circle. Any point of tho circumferonco of tho old circle remains on the circumfereoce of tho now circle. And the chords of the old circle Jlass into chords of the new cir�le.

. . These transformations 01 the clfcle (proJecltve

transformations, i n EucHdean language) sa� isfy, in non-Euclidean language, all the axioms 01 motion.

For inslance in non-Euclidean languag the transformation

' of chords signilies tbat straight

li nes Jlas., into straight l ines, ctc. Now comes tile last and deciSive stop. Wo rofer to these transformations 8S "motions of a Lobachevskian plane".

Wo cnn summarizo the foregoing in tho form of Lhe K lein model.

In lbe laoguage of EUClid·'1 I n Ibe langu.do of geometry Lobacbovsky·. geomolry

Circle EnUl(! plane Chord Stralgbt lioe Polot Point. "To beloog" "To belong" "To lio bel.ween" "To lie between" "Projeclho transformalion "'MoLion" of a circle inlo itself"

AU the properties 01 projectlvo trno formatlOos

are of course known, but we do not need to know them. All we have to do is accepL the (acl that such transformations exist.

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And so-thi is tho minute we ha.e beon waiting for-if it is 1'0 ible to declare the ci rele a Lo­hachevskian plane (this wo have provod). then the problem is solved.

Indeed. suppose in proving some kind of theor­em in the geometry of Lobachevsky. wo arri ve at a contradiction. But every theorem of Lo­bachevs.ky's geometry is now also some theorem of the geometry of Euclid.

Each theorem may be stated in two languages. H we havo a contradiction in Lobachevsky's geo­metl'Y, we also, at the same time, get one i n Euclidean geometry.

Of course, in Euclidean language the contra­diction will look d i llerently and ,,;1 1 open up in some other th.eorom, but that is quite i mma­terial. Tho important thing is that if io One of tho geometries there is a logical contradiction, it will be in the otber geometry as well.

The geometries are equivalent.

264

This then proves the independence of the fifth postulate of aU the remaining axioms ot Euclid's geometry.

That is al i i But i n science, l ike ill tbo Arahian igb!s, the

ond of one story is but t he beginning of UIO noxt. Proof of the noncontl'Rdictorincss of tho geom­

etry of Lobachevsky signi fied for mathematicians the slar� of a colossal cycle of studies in axio­matics, III creation of a highly intricate, ideally rigorous and .. b ol utely a bst"act apparatus of ma�hematic81 logic, all apparatus lhat was in­finitely removed from lhe slightest practical ap­plication- unt il it was (oulld tbat eleet"ollic computing 1I13cbines . . . . By the wny, lhis is just the time to concludo our discussion.

Let us return to tbe K lei II model to nole a very amusing point. Take two points within our circle. Draw n chord througb tbem. I n the language of Eucli d , the dislnnce he tween these points is equal to the length o[ the segment of the chord. What is the situation i n Ule language 01 non-Euclidean geometry?

Intuitively, We can see that at any rate it cannot be equal to the lengU, of the segment. I ndeed, d istances between two points on the i n finite Lobachevskian plallO can be arbitrarily gmat, while the "Euclidean dist 'IDces" between points of our circlo are restriclcd by i ts diam­eler. It is clear tbat wo have to define a "nol)­Euclidean distanco" in some otber way. But how? Very s imply if we recall how the concept of length is in trod uced i lito geometry.

Roughly, i t is done as follows. Take a scale unit-some segment-and by means of trans-

265

form. tion of motion make i t coincide with the segment being measured. Its length is dewrm­ioed by the number of ti mes the scale unit !its into it. We wil l not go any further. The impor­tant thing to now is that the definition of equal­ity of line segments (and consequently of length) as also, incidental ly, the congruence of any geo­metrical figures, is detcrmjned by means of the concept of motion .

That is the situation both in the geometry of Euclid and i n the geometry of Lobachevsky. But in Our model, motion in the Lobachevsldan plane is, i n Euclidean language, a projective traosfor­mation of a circle. Therofore, it comes out that in tbe language of Lobachevsldan geometry, two line segments are equal if Ono passes into the othor in a projective transformation. Recall ing that the length should not change during trans­formation of motion, we realiw that the "non­Euclidean length " must remain the same in a projoctivo transformation. It must, as mathe­maticians say, be invariant to a transformation . This quantity- tbo invariant-is natural ly known for projective tran. formations of a circle. I f We also tako into account that the length of the sum of two line segments must be equal to tho sum of the lengths of these segments, i t turns out that tbo "non-Euclidean distance" is determined uniquoly. And of course sucb a dist­anco bohaves normally (that is, it becomes in­fini w) when one o[ the points lies On tho circum­forenco of tbe circle .

The circum ference of a circle corresponds to infini toly distant points of the Lobachovsldan plane.

266

Of course, the somewhat extravagant cbaracter of "non-Euclidean motion" in tho K lein model is also e " i dent in the fact that the size of a "non­Euclidean angle" betwoen two straigbt lines is qmw different from tho rustanco between two chords in the Euclidean language. But these are only details. They are important hut tri nes ne­vertheless. All the essentials have already been staWd.

And now the last poiot. To prove tho noncontrad ictory character of

the solid geometry of Lobachevsky, it suffices to convert the K loin cU'cle into a spbere.

A fow years of tor K lein, the French matbemat­ician Poincon) proposed another model of Lo­bachevskian geometry. A lso on a sphere. I t is perhaps even moro remarkable. Poincaro Oven conjectured a marvellous world of pbysical

267

beings, which, from the Euclidean viewpoint, would hve 1D the restricted circlo of Poincare, but from their vantage point would clai m that tbey were l iving in the i n finite plane of Lo­bacbevsky.

In th is wodd , the straigh I Ii nC" of Lo bachev­sky are, in Euclidean lauguage, arcs of circles perpendicular to the surface of the sphere. The accompanying drawing wi l l give the reader some idea of Poincarc' model. There are a lot of attractive features in tho "Poincare sphc,'e" but wo will have to call a hal l at this IJoi nt, for olber things claim our n 1 tenl ion ,

Chn,plu J J

A N UNEXPECTED FINALE.

THE GENERAL THEORY OF RELATIVITY

We have now come to a ttlrning of tbO ways. Up to this point, we had boeu talking tho lan­guage of elementary school. Wo were able to some extent to convey the essence of proof of noocootradictorioess of the geometry of Loba­chcvsb:y and to impart to the reader some ideas of R iemann. Thi ngs have nOw become complic­aledo To get some foeling of the content of the genera l t heory of relativity, one has to iovesti­gate the special theory. But the anthol' can hardly expect the reader 10 have the deol) know­ledge that this requires and so cannot afford to dwell in detail on tbe special theory.

The most natural thing would be to say noth­i ng. The temptation is great. But that would moan taki'llI the whole sl'mphony of the fifth postulale and throwing away the triumphant, purely Beethoven finale.

Obviously, we cannot do lhat. A l l 1 call do is warn yon tbat what follows is only a bare ouLline, extremely superficial.

The general theory of relativity is based di­rectly 0 11 the idea of the "non-Euclidicity" of space. That is what intol'ests us most. And so let us try to disponse completely with the special theory of relativity. We wil l confino ourselves to ollly a word or two.

269

Geome.lry after the year 1905. The special theory of relah I ty has already ubstantially alwred our views concerning geometry. To begin with, lel US try to grasp tbe connection between geom­elry and physics i n general and also to see what h!ls changed i n geometry as a result of the spe­clUl theory of relati vity.

Before Einstein, the univ rsal and firm con­viction . was that Euclidean geometry reigns su­?,reme I n the real uni verse in which we live. Ihere were no reasons to think otherwise. The theoretical possi bility that our world is des­cribable b! some kind of non-Euclidean geom­etry remamed a purely theoretical One, whi le Lobachevsky's and Riemann's suspicions 011 this score were nO moro than speculations. Tbe situa­tion was as if someone said: "The supposi t ion that so-nnd- 0, Mr. X , is a Martian dweller does not i n tbe least contradict the laws 01 formal lOgic. "

"That ma� be," would be the response, "but aU observahons and experiments point to so­and-so being an inhabitant of the earth. "

o w after the advent of the special theory, there ap�eared real �oubts about the problem 01 the ongln of Mr.X being so crystal c lear . . We must now look into the camp of physic­IS�S .. Let us see what geometry means to mathem­allClons and physicists.

.To .the m athem.atician , geometry, as we havo sa.'d L Ime and. agaIn, is essentially a fOI'mal game WIlh the BaSIC Concepts and axioms chosen for them. I t i s nece ary that tho game obey Lho rules 01 form al logic , and at this stago he docs not care wh ther his geometry can aspire to

270

any relationship with the aclual world i n which wO live.

True, every person was unequivocally coov­i nced that Enclidean geometry reOected the pro­perties of our universe. But that was simply laken for granted. A sort 01 natural property of the human mind. The [act that geometry has a n experi mental foundation wa somehow forgotwn. What is 1lI0re, prior to Lobncbevsky for two thousand years geometry was carelully guarded against the defiling effects o[ experim­ent; it was kept away from any kind of "elll­pirical hasis".

Einstein rather maliciously but precisely r()­marked that what happenod to the axioms and Basic Concepts was similar to the process 01 converting the beroes of antiqui ty into gods. In place of a realistic basis, there arose the "myth of geometry", a rather hazy conception of axi­oms as something "intrinsic to the human mind, to intuition, aud to the spirit". I t is hard to grasp the meaning of the last words, possibly because there isn 't any. However, it mu t be said that the hypnosis 01 a bstraction was so great that i t held the greawst minds spel lbound. Physicists were among tlICm. One can eVOn mention some outstanding names, peOI)lo not without talent-Isaac ewton, to take one in­stance.

His Basic Concepts that aro given in the open­ing chapter of the Principia are fundamentally unobservable and unknowable. ewton's "ab­solute space" and ((absol ute t i me " nre some­thing "intrinsic to the human (and perbaps also d ivine) consciousness". There is no irony here,

271

none in the least. I t precisely conveys the sub­stance o( the notions "absolute spacett and uab. solute time".

So pl.'ysicis!s too wero engaged in "tm'lIillg heroes Into gods". I f we cont inuo tho divi llo analogy, i t will be seen that because of thoir scat ter-braiuedne s, physicists, thollgh theoret­icaUy recoguizing and preaching the religioll of the absolule, actual ly paid no at.tent.ion to it and did not draw any real conclusions therolrom :

The lirst example was set by ewton himself. Ho lormulated ni l the law of his mechanics

101' "absolutes", but straightway employed them in tlw solutioll of quite concrete problems. ince, essentially, the axiomatics did nOI interlere in nllY way, no at teotion was paid to i t .

I n this sense, mathematicians turnod out to be more consistent in tnair attitude. Thoy had already fully analysed the problem or axiomat­ics when phYSicists wero just begioning to take a serious interest in the foundations or their science, tho basis of their concept ions concern­ing sJloce and time.

272

On the other hand, though, they advanced much farther sud at One step. Here almost all the credit goes to One man-Einstein.

It was about this time that the attitude or physicists to geometry hecame clear-cut. In­tuitively, subconsciously they always believed that the ent.ire problem of interrelationships be­tween geometry aod phy ics was rather artilicial.

ow tho situation was substantiated with com­plete rigour. Tbe point was this. The Basic Con­cepts of geometry are abstractions of our concep­tions or actual physical objects. For example, Ei nstein says that rigid bodies with markjngs on them, realize (given due caution) the geom­etric concept of a line segment, and rays of light realize straight lines. He then goes on to say that i f onO does not adhere to this viewpoint in practice, it is impossible to approach the theory of relativi ty.

But i f tbat is tho case, then geometry is 5i m­ply a chapter of physics! Its lirst chapterl

Practically speaking, what we have just said does not change matters much. We have de­throned the axioms and Basic Concepts, we have reduced geometry to a generalizat,ion of physical experiment,s and now see that the truth or fals­it)' of geometry is a question of experiment, but all the specific assertions have remained un­changed.

We recall that, essentially, G auss and Loba­chcvsky and Riemann all thought similarly. They defended the positions 01 the practical physicist.

However, if we consistently develop our views, it will be seen that we have already proved a

11-U.1 273

few thi ngs. Things that at now aud i m portant. What is m ore , our ,.jews suddeli ly lead us to certai ll doubts as to the actual ,'ealizab i l ity of geometry. l Iere tbe attack is [rom fresh posi­tiolls.

One of the principal chapters or any geOIll6try is tbat of tho geometric theory of measurement. In order to dovelop geometry, we have to dellne the concept of length \\�th full mathematical rigour. This was naturally dono by geometers. Their definition of lengt h is hased on two "quite d ifferent whale " .

Wo need: 1 . A lino segmont whoso longth is taken to he

unity. 2. A proced lll'e for measuring, which in geom­

etry amonnts, roughly speaking, to laying off the eale unit Oil the segment being measured and counting the n u m ber of ti mes it takes. The "esUILing f .. actional number of t i mes (it may acCidentally bo an integral number of ti mes) is tbe lellgth of the Une segment.

I n that way, One can, say, measure the length of a side of a triangle . In doi ng so, we taciL ly assume that if the triangle is at rest relative to the uscale unit U Or is ill motion the result will be the same. Now sinco we said that all geornetdeal objects a,'e an idoali zation of actual physical bodies, then the words gi ven above coa to appear so clear.

(f the tdaogle beiog measured is in motion relati ve Lo the seale unit, our procedure for meas­uring is no good a t all. I f we stand on the plat­form of a rai lway slation and wish to measure the length of the doors of a railway coach of a

274

train passing by at bigh speed, we call1lot apply tho scale uni l . 1'0 do tbat, wo would have to he moving along in the same direction and at the same speed as the train (with the scale unit i n our hand). But then both "unit" and "object being measured " would be at rcst r lalive to one nllother, and we "oturn to our origi n a l case.

Obviously some kind of new procedure is need­ed for Dlea uring movi fig bod ies . But i f our pro­cedure is new (it mallers li llie wbat kind, so long as i t is new), then we are not posili ve i n Hny way that our llew "length" will coincidc with the earlier ooe.

Actually, we have introduced a total ly new concept. From the standpoint of formal logic there are no grounds to expect that i t will coi nc­ido with the earlier one. Only experiment can ,'csolve the matter.

LoL uS stop for n moment. A little thinking will make it clear that these

aro very unpleasant words for L1le axioms of geometry.

275

W" assert �hat our geometrical concept gen­eraUy speak i ng, can cbange if the aClusl �olids wbose geometric properties are under tudy are moving relative to us.

We say, "something cnn change in tbe proc­ess". We thus demand that the geometric system of axioms bo supplemented by fresh axioms of a purely physical oa tllre.

CO�istently developing our viaws, we become convlllced that there should be a rather large number of such axioms. Indeed, all our segments (Including of course the scale uniL) sro abstrac­tions of actual solids. But, 8S we know, solids expand wben heated, their length changes. Meas­urements with cold and hot scalo unit.s wi l l change the results we obtain.

Consequenlly, if we want to be absolutely prcc­ise (and that is our a i m) , we must introduce into geometry a "constant temperature of the scale unj t".

However, tem pera ture is not the only thing that aUect .. physical properties. Hence, we will have to specify all the physical conditions. I t then works out t·hat only i f a l l manner of pre­cautions are taken can we hope that tho axioms of "pure geometry" ,,�U describe our universe correctly.

That is the only thing that DOW worries us. Generally speaking, this work has DOL yet been

done in aU its details. Probably it is noL vcry much ncedod, though possibly i t is very much 'loaded . At least twice it has turned out that redefining the physicnl conditions in which tho geometry of tho world was constructed bas com­pletely overhauled our conceptions of nature.

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The !irst ti me this occurr d was when the spe­cial theory of relativHy was introduced. It was fOlmd that the length of a moving segment differs [rom that of a segment at rest.

We will not go into how all that came about and will confine ourselves to a fow general remarks.

i . We are not abashed by the fact that the length of a moving segment comes out dilferent from that of a segment at rest. We realize that determining the length of a moving body in­vol ves a now procedure of measurement, and hence, stricUy speaking, it is a new notion. I t need not coincide "�th the o l d notion.

'..ve also realize that the new notion must in some way or othor be introduced, lor we are no lon�-er playing a game of logic but are creating a tool \vith which to study the actual world. Our concepts must be able to describe this world fully and well . That is the only reason for their existence.

They appear as a re ult of the study of the real physical world. But there arc moving bo­dies in the world. One has to be able to describe them too.

2. It turned out that without employing tbo concept of time i t is impossible to determine, logically and well, the "length of a moving body".

We become suspicious at this point. This is all the mOre disconcerting that a Jlew

and highly important concept-lime-enters ioto our geometry. Up to now geometry had been associated solely with Space.

But-we continue to reason-everything will work out fine and nothing will change if the

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length of the moving scgment coincides exaclly ,,11h the length of Lhe segment at test. Then tho notion of Time will in nO way be associated wilh that of Space.

Now i t experiment shows lhal Ihe length of a moving segment is different, and i f it turns out that this depends on the velocity of the scale unit. for example i f i l diminishes i n ac-

1 thollon cordaoce mtJl the law 1"'1 = • / 1- �' where

II c' v is the velocity of the moving segmont and c is the veloc i ty of light . . • if the veloci ty and, via tho veloc ity. Lhe l i mo too enter b'llomelry • . . . then wo will have t o say: tim" and space lire interrelated. Then in geometry i t will he impos­sihlo to study space independeutly of l i mo.

That is oxactly whal Einstein demonstrated. The length of a moving body is i ndeed depend­

ent on tho velocily: time enters geometry, tlu: properties of time turn out dependent upon tlu: properties of space, and all our earlier views concerning the universe and geometry provo to

278

bo only a rather naive approximation. l L is only when we cOllune oursel ves to sludying cascs when the relative velocit.ios of objects are small that our old conceptions funclion properly and we call regard space as boing independent of time, and time as indep�ndent of space.

I n Utis case. the old and true geometry of Euclid is a line instrument for studying space. Then we can take it that the properties of space do not depend on time.

Such wer the ideas that arose in tho year 1905 as a result of the special theory of re.lativiLy.

The inner logic alld elegance of Einstein 's theory were 0 striking that within three or four years aU tho leading thool'etical physicists were enthusiastic adherents. I n 1 909 Max Planck exclaimed: " It need hardly be soid tbat tho new-Einsloinian-approach to tho notion of lime d mands of the physicist all u l l imate ca­pability of abstraction and an enormous capacity for i magination.

" I n i ts audacity, this theory surpasses every-thing achieved up to this time. . . . . .

" on-Euclidean geometry. by compaflson. IS child '8 play. Yet, i n contrast to non-Euclidean geometry. whose application call seriously be considered only i n p ure mathematics. the princ­iple of relativity has every right to pretend to a roal physical signi /ieaoce.

"fn its depth and consequences, the upheaval wrought· by Iho relati vity principle. . . may �e com Ilared only with that effected by CoperOl­cus. "

Planck was I'ight but he did not know that that was ooly tbo beginning.

279

To summarize, then, together with the special theory of relativity a new concept of four-dimen­sional space-Lime entered physics. But, as be­fore, three-dimen ional space is described by the geometry of Euclid. True, in that arne year of 1909 a very curious lact came to light. It was found that the law of composition of speeds in tho special theory of relativity coincides exactly with the law of composition of vectors in the space of Lobachevsky.

In othor words, the formal space of relativistic veloci ties is Lobachevskian space. But this ap­peared to be a purely formal coincidence. Nei­ther at that time nor later was any profonnd physical moaning found in this analogy.

Still more sensational and startling news fol­lowed.

Physics and geometry (a/ter 1916). Planck was not to blame, because if ono needs an insLance of the most unexpected discovery in the history of science, then this is tho general theory of relativity.

For three hundred years tbo foundations of tho tbeo�y of gravitation were in a state of ab­solute rest. Newton had given the law. And that was all. Actually, there was just one fun­damental formula that lay at the core of cal­cwations of the motion of celestial bodies in all the numberle s volumes of subtle, elefant and ma­gni ficent investigations into celesti a mechanics:

F = T ffllm"

r'

This states that tho forco of attraction of any two bodies iu tho universe is pJ"oportional to

280

the product of their masses and inversely pro­portional to the square of the distance between them.

The quantity T is a constant with the dimen­sions 6.66. 10-8 dynes cm' g-2.

The law of universal gravitation is truly mag­ni ficent! One could go on singing the praises of the "Simplicity" of ewton's ideas, but that is a waste of good lime. The Simplicity lies only i n tho analytical form of the law. This "naive" formula summarizes several nOHo-obvious, sub­tle and-what is more-at first glanco traoge phySical assumptions. I t required a Newton to produce it. Over one hundred years passed be­fore tbe law of gravitation was unconditionally accepLed. Note too that tlte protests came not from ignorant peoplo or obscurantist scbolars, but from tbe g,·eatest and most talented scient­ist-s of tho day. And so the talk of simplicity can reler only to the magnificent barmony of nalure, to the beauty and elegance of her basic laws.

Newton told us how gravitation operates. But no word was said about why it functions in prec­isely the way it does.

By tbe beginning of tbe twentietb century, P 01'10 had almost reconciled themselves to tbis situation. I I was as i f looking at tbe s",ooU.ly polisbed surface of your furniture you find it difficult to im(lgino tho rough uDworked wood that lies underneath.

Incidentally, attempts wnro made from time \0 lime to olfer somo mochanism for the law of gravi­tation, but they "II i uvariably and rapidly came Lo nought. Wben science Dourished, physicists

281

, - -.-�

�. rJ;:� •

:' tf3 II ,

had thei� hands .lull of specific nrgenL problems, and dllflOg penods of declino and qlliesccncc Lhere was neither the enlhnsiasm flor Lbe moral energy to risk invcstigating sucb a cardinal and most certa inly hopeless problom.

1 1 for tho begi? rting. a NewLon was neces ary, then for tho COllllnuatlOn an i ntel lect of perhaps a t ill greator scale was needed.

Most likoly, one m u L agree wi L h Einstein that without h i m the theory of gravitation might noL have boon crealed La this day.

r n scienco (in the arts too, by tho way) the role of a gonius is perhaps greater than in ot her fields. One man is capable of accomplishing more th, m huodreds of hugo research t 'ams. Tho da­cisive iactor is not quanti ty but quality.

So between tho years of 1905 and 1916 Ein­stei n stud! d the problem of gravitation. In 1916 the work was completed. Dnring this same pe­rIOd he. was eJ�gaged i n many other thi ngs and , , n passi ng as I t wore, he obtained fundamental results ill solid-state theory. But al l the time uppermost in his mi nd was tbe general theory

282

of relativity. I t continued to occupy l h is central place to the end of bis days.

Of course, before goil)g to the heart of Lhe mat­ter wo will, as we have been doing al l along,

tart out wHh a few gencral ideas and a story or two. When ooe is dcaling with Einstein and his works this is a l l the more necessary . . . .

Once, reading a hunter's jouJ'na l- I can ' t oven i magine how that ever got into my h.ods- T ca me UPOI) an article about snakes. Tbe author, who bad capLl/rod about 1 ,500 snakes, "eported, among other things, that not one o[ the snakes had ever a t tacked hi III first .

Tbis was ama.ing. I read to tho end. The article was n sorious one writt.en by a profes­sional snake-catcher. He al\8lysod a varieLy of

pecialized probloms, sLressed the value of vo­nom, criticized the situation i n tbe country in tbat respect , and, what was particularly inter­esling, one felt that he l iked all theso poiso(loll snakes and considered them very lIsoful .

The problem of boosting the venom output of a Central Asiao cobra was discussod 8S if one wero talking abouL Kbol mogory cows. There comos to mind a descri ption of a remarkable Hindu lIIathematician , the great nnmber,Uleorist Ra­manujan , of wbom it was said that "evoJ'y po!!­itive integer was ono o [ his personal friends". I hope the parallel is not Ox tended to fractions and tho repti les.

The article begall wi tb the statement that sensational stories abouL snakes do more harm than good. And he listed a [ow of tbe mista kes thaL jOllfnalists make. I gathered that he was truly upset and that be very m uch wauted

283

people to got 8 clear picture 01 this complicated and rather tedious profes ion of snake-catcher, in place of a1\ sorts of "romantic hor­rors",

J recalled this story not because [ wanted to amuse my readers hut because I 3m convinced that people get lar-fetched, highly distorted con­ceptions about things with which they do not personally come into contact.

Unfortunately, the peculiarities of the pro­fession of a scientist (especially that of the phys­icist) is regarded On the samo level as the work of a snake-catcher.

More than anything else, tho theory of relat­ivity (and of course Ein tein himse l f) suffered from sensation stories.

I t was his luck he could dismiSS with calm and indi!lerent irony the end less uproar around his namo that continued from 1919 onwards. One can perhaps only offer prayers of gratitude that all the publicity had practically no effect on his good natmo.

But 0 much nonsense was whipped u(l around the theory of relativity, both genoral and spe­cial, that one feels em barrassed. True, physic­ists themselves are somewhat to blame too: For many years, even i n prole sional circles, it was believed (and still is perhaps) that the ideas of relativity theory are very complicated. Partic­ularly i f one is dealing with tho general theory.

This was quite natural during the first years alter Einstein's work appeared. I t is always the case. From what you have soen in this book, J hope it is clear that such an elementary (i f judged wi thout prejudice) idea as that of Lo-

284

bachcvsky was grasped only with exceptionnl,

unbelievable di fficulty. But forty odd years havo pa sod since .

the

creation of the general theory and some sIxty

years inee tbat of the special theery of relativ­

ity. We should have long since put everything

in its place and realized that tbe fundamentals

of ewton's mechanics are, at any rate, hatier

and, possi bly, more involvod tban the princip­

les 01 the theory of relativity. Even from the most general reasoning it is

clear that it could not be otherwise. In both

casos we deal with the S<1me things-the fun­

damental ideas of space and time. And the farther

we penetrate into the e sonce of the .matter,

the clearer, simple.r and moro harmOnIOUS do

our conceptions become. . . I n building bis gene�al theory, �lll�telU P.'G­

ceeded as he himself said. from a childish, naIve

questi�n that had engaged him ever since his

school days. "What happens in a falling lift?"

Another cleven years of intensive work, sev­

eral dozen faulty versions that had promised suc·

cess and a number of probing invesligations were

needed before the problem was resolved in 1916. However, nO exhaustive result that resolved

the problem, like Newton's law, was yeL ob­

tained. The work was far from completion, but

the foundation had been definitely laid.

Crudely speaking. that �as how.

things s�od.

An oxcerpt from Chaplin s autobIography glves

IlJj a pic Lure of how all this appeared in the minds

of two people who cannot be suspected of the

slightest desire to twist the truth.

285

"As Mrs Einstein had requested i t should be a small .rr"i .. , I in"i led onl" two other friends A t dimler she told me the siol'y of tho morning he conceIved Ihe t heory o[ relativity.

''''I'he Doctor caml) down i ll his dreSSing gown as lI�ual for hreak[ •• t hut he hardly touched " t hIng. J thought sornething was wrong, so r as�ed

" wha t WlIS tron hling hi m. "Darli ng", he

saJd, r hal'O " wouderful idea . " And after drink­ing his coffoo, he wont to the piano and started playing. Now and ngaill he would stop, making � few notes theu repeat: " 1 ' 1"0 got a wonderful ldea, a marvellous idea . ' ''

" ' I said: '''l'llen [or goodness' sako tel l me what it is, don 't keep ll10 in suspense . H '

'''He said: "lI.'s di fficult, I still hal'O t o work i t out. Jp

"Sho told me ho continued playiug the pi<U10 aud makio� nOles for about ha\[ an hour, then went upstaIrs to his study , telling her thaL he

286

did not wish to bo disturhed, and remained t here [or L wO week . 'Each day I sent bim up his meals. ' she said, 'and i n Ihe el·.,ung he would walk a little for exel'cisc, then return to his work again. I

"<Eventually, t she �id, 'he came down from llis study lookillg very pale. "That'8 i t , " he told III • wearily ]Juttil)g two sheuts o[ paper on the table. And t hat was hi theory of relitliv­i t.y. ' H

Most l i kely sornoL lling very much l i ke what is de 'cribed be"e actually look place. It might be l iteral ly tnle. Mr. Chapl i " of course wrote tho way he S8W things. But this chauges nothing at al l . I f that is U,e truth, then it is only a m i n­ule particle of tho truth .

OIV I am about to u ndertako what r myself have SO harshly criticized: a very superficial and therefore unavoidably d istot·ted descri ption of the goneral theory of relativity and its i nter­relationships with geometry.

Ei nstein had two guiding ideas. One did not seem, at r,rst glance, to have allY relation what­soever to geometry. That was the lift. Or, to put it otherwise, the question of the equality of an i nert mass and a gravitational mass. That was the one and only experimental fact upon which the On tire theory was cooslructed .

There is nothing more amazing in the whole history o[ science.

Let \IS try to figure out what the i nert 1na and gravitational mass mean. Everyone should know ewton's second Jaw. However, I suspect that most readers do not have a full grasp of either that law or of the other laws aod, i n gen-

287

eral, of �be fundamontals of classical mechan­ics. U nIOl'tunately school physics only Ilor[orms a few formal manipulat ions with Newton's laws and does not demand much understanding on tbe part of tbe student.

Yet-and I a m prepared to repeat this with­out end-to grasp thoroughly the fundament­als of classical physics is tantamount to fully preparing onesel( for an understanding of the theory of relativity, because as SOOn as the nO­tions of space, lime, force and mass cease to exist as nebulous and p moly intuitively perceiv­ed enti ties, as soon as their exact meanings have been elucidated, then any physical theory will appear ns a consequence of a definite system of axioms. Now any choice of axioms is determi n­ed by experiment.

I admi t tbat this is my sore spot, and since we havon ' t silace enough to give a clear analysis of the basic notions of physics, my suggestion is that the reader con ult a book or two on the subject.

For the pre nt, suppose that the reader is familiar with Nowto n 's second law and oven has fully mastered it.

The proportionality constant betwoon a force and the acceleration of a mass III determines the inertness of tho given body. We shall call i t the inort mass """" , .

Newton's law of uuiversal gl'avitation relers to the gravitational interaction of bodies.

A priori, there arc absolutoly no gl"Ounds to believe, not the slightest hint, that the formula wWch determines the force of interaction must somehow be dependont on the inert mass. For

288

classical physics, this is a still more unexpected and inexplicable fact tllan, say, the depend­ence of the number of weddings in Vladivostok on the weather i n the Antarctic. I n the l atter case, we at least have a logical link-up i n that the Soviet whaling fleet is based 8t Vladivostok. Now in the case of the gravitational and the inert masses there was no clarity up to the time of Einstein .

'l'bere WaS a remarkable experimental fact, and everyone, Newton first, made note of the mar­vellous coincidence. Many experiments were car­ried ont over the years up to the beginning of the twentieth century. The last experiments­those of Roland Eiitviis-were ama.ingly accur­ato. The idea beWnd all the experimont.. was extremely simplo and we hall now examine it. First we wil l write down the law of gravitation.

We will write tho masses as m"'auy, for we do not know whether these masses are the same as miner' . We want to find an expel"i menl, that wi ll demonstrate tbis. So wo have

F _ mlhN"rllmlllMUU - , r' Let us examine the concrete case of a freely­

falling body. The force compelling it to fall (the force of graVitational interaction) is tho lorco of gravity.

On the other hand, if we kl)olV the accelera­tion and the inert mas of the falling body, say a small ball, we can find the force by moans of

ewton 's secood law. We thus have two equations:

1$1-1"7

( 1 ) F - mh .... Mh, ••• - T rl

289

Af ..... is the gravitational IMSS of the earth, and T' is tho distance from our ball to the centre of the earth. Newton established that a massi.o sphere attracts with a force such as i f its entire mass wero concentrated in the centro. That was a purely mathematical problem.

(2) F = min." g where g is the acceleration 01 free fall.

Combi ning the two equations wO get

mintrt JI ht'(U'" g -- = T --m/1o!l1c,", r'

ow if ml",,/ =mheaug for all conceivable bo­dies; if they are equal in the case of steel , wood, gases, Uquids and radioactive elements and po­

M Iymers and so on and on, then g=T ".

In other words, the acceleration of tbe earth's graviLy is the same for all bodies.

This was first established by Galileo. The equality of the inert and gravitat.ional masses,

290

as we have already noted , had been firmly establ­ished in dozens of eX]lcrirnents.

Wi Ih the ad vent of the special theory, when it became clear that every kind of energy pos­sesses an inert mass, experiments wore perform­ed with radioactive substances.

I t turned out tbat in their case too the ioert and gravitational masses were equivalent. Tbat is to say, energy possesses heavy mass as well, which is tbe same as the inert mass. In short, precise experi monts demonstrated the identical equivalence of the inert mass and the heavy mass. However it was One thing to know and quile another to understand. Einstein set out to prove why they are equal.

It may not yet be clear wbat a l l this has to do with geometry, but nevertheless this sole ex­peri mental fact plus the special theory of relat­ivity, plus one more requirement of a purely theoretical character was enough for Einstein to bring about a complete change in our concept­ions of tbe geometry of the universe-the gen­eral theory.

OIV about the theoretical requirement. We can even formulate it in strictly technical lan­guage: "the laws of nature must be generally covariant", or, more. simply, "all systems of reference must be equivalent".

J fully realize that this is not much of an ex­planation, I gave the statements mOre for my OlVn consolation. 'Ve simply do not have the neCe sary time to go i nto the origin of tbe gen­eral theory of relativity. I do not Wish to give only n semblance of an explanation, though that wou.ld be fairly easy to do. The only thing I

to' 291

ask you to take On trlll!t is that tho "equivalence of reference systems" is a demand which largely stems from aesthetics. The inner logic and the beauty of a physical theory were to Einstein One of tho most decisi ve factors.

I t may be that he occasionally overestimated the relative signifi.cance of such arguments, but he believed that the laws of the universe should in principle be very natural and logical, and that theoreticians often distort tbem perceiv­ing things in a crooked mirror. One can, of course, find fault with sucb reasoning-no things exist that do not have weak spots-but tbe fact tbat this mode of reasoning was good is proved by the re ults he achieved.

"Tho theory of gravitational fields construct­ed on the bnsis of the theory of relativity bears the namo of the general theory of relativity. I t was created by Einstein (and formulated in final form by bim in 1916) and is perbaps tbe most beautiful of existing theories. The remarkable tbing is that it was constrncted by Einst in in a p urely deductive fashion and only subse­quently corroborated via astronomical obser­vations." Tblll! wrote Landau and Li! hils in their fundamental course of theoretical physics wWch is considered to be the world's be t in tbat field. I t is the only place in all their ix volumes where tbe autbors display any omotion.

That fact alone speaks volumes. But let us get back to apocrypha. In reply to the query of bis nine-year-old son,

"Papa, wbat is it tbat makes you so famous? " Einstein is reported to bave said quite seriously that wben a blind bug crawls over the surfaco

292

of a ball, it does not notice that th path trav­ersed is curved. Said Einstein, " I , On the con­trary, had the good fortune to notice that."

TWs passage is often quoted, but don't think that it exhausts the content of the general theory.

I t is obvious that Einstein himself believed that the basic result of bi work was a fundament­al cbange i n our conceptions of tbe geometry of the universe.

We have already said that the sllecial tbeory lUlled the idea of the geometric properties of pace being indopendent of timo.

Time bad become a part of geometry. But the properties of lime only alJeeted the

geometry of moving bodies. For bodies at rest, the geometry of Euclid h Id true.

A new pbysical factor app arcd in the general theory of relativity that detormined the geom­etry.

The old result-the mutual dependence of tbo properties of space and time-was naturally ro­tained. But lWs was not nU. I t turned out tbat tbe geometrical properLies of the world a� a

293

given point at a given instant of time are deter­mined by tho gravitational field at that point.

Thls last phrase probably does not mean very much to tbe reader, and so we shall give a few precise statements and then a crude analogy that sbould domonstrate certain tbings.

In the general theory of relativity, the world is described by the geometry of Riemann. Here, when speaking of tbo world and its geometry. we all the time have i n view the four-di mension­a l world. Time is i nextricably woven into tbe geometrical properties of space.

A2. you recall, both Gauss and Riemann reg­arded the curvature of space at a given pOint 85 tbo determining cbaracteristic. Also docis­ive was the Uilltrinsic characteristic of spaceJJ_ the properties of tbe sbortest l ines (geodesics). These lines are physically determined by tbo trajectories traversed by material particles freo of tbo action of forcos.

Accord ing to Einstein, both the curvature at a given point and the properties of geodesics are determined by tho gravitatIonal field. In the general theory of relativity, gravitation oc­cupies an exceptional place of honour. Roughly speaking, it is tho most important of all inter­actions.

I t determines tho geometry of the universe. We call put It differently: gravitation is determ­ined by the geometry. But no matter bow we word it, tbe result is that the geometrical pro­perties of the world are determined by the distri­bution of gra vttating masses.

We repeat again: whenever we speak of geom­etrical properties, wo have i n view a four-dim-

294.

ensional world, so tbat in ordinary parlance one ought to say:

TM geometrical properties and tM properties 01 time are completely determined by the distri­bution of masses in tM universe.

And just like the geometry of the plane is approximately fulfilled for small areas of a two­di m nsional curved surface, so small regions of the four-dI mensional world may be approxim­ately regarded as regions i n whlch the curvature is tero.

Physically, this means that in small spatio­tem poral regions One can exclude the gravita­tional field and pass over to the special theory of relativity.

According to Einstein, geometrical properties appear in space and Limo only when there are rnaterial bodies in the u niverse.

295

That, very roughly speaking, is tbe gist of the ideas of tbe general theory of relativity.

Two remarkable circumstances stand out in the story of the development of this theory.

1 . At first Einstein was not even acquainted with Riemann's ideas. He had wanted to ex­plain the equality of the inert and heavy masse$ and found i ll bis search that Riemann 's geom­etry was the necessary mathematical form for a description of his purely phYSical reasoning.

2. The general theory is probably tbo only instance of a pbysical tbeory created in a purely deductive manner. There was only one experim­ental Iact underlying the whole theory.

Today, the general Uleory has been corrobor­ated experimentally a number of t·i mes and was lust recently verified under laboratory conditions.

Now the analogy which I promised. I magine a piece of cloth stretched taut. This

is a plane. The geodesics On it are straight lines. The curvature is zero. A free material particle on . su.ch a surface will move in a straight line. ThlS lS an analogue of the space-time of tbe spec­wi tbeory of relativity. Now throw a stone into !bo middle. The cloth sinks in the vicinity of l�pact: The sbape will he distorted. The geod­?S1CS wl

.1I no longer bo straight Unes. A particle

ID motIOn on such a surface, Oven in the absence of forces, will bo deflected from a straight-line path.

The farther away from the stone, the less the curvature, and at infinity, the cloth is again lIat. Tbe portion of curved cloth is a rough model of space-time in the presence of gravitating mas­sos.

296

And now tbo last question. What is the actual geometry of the world we live in? .

Experiment has shown that at least in our part of the universe tbe curvature of space-time is positive; crudely speaking, that is, because the question of tho true geometry of the uni­verse is a very touchy one. Physicists have to use their imagination. This is a realm where hypotheses abound.

Pormally speaki ng, the whole problem' cons­isIs solely in determining tbe coefficients in tbe formula that defines the square of the distance in a four-dimensional world: space plus time. That is all!

As of today we have a number of models 01 the world. Several hypothetical Ullivcrses. But we still do not know for sure which one filS tho world in whicb we Jj ve. The portion of tho uni­verse accessiblo to tbo most powerful telescopes (only a paltry len thousand million light years) is far too small.

Of course tbe local geometry of space-timo va­ries from 'Point to point and cbanges very fanc­ifully near gravitational masses.

Let 's try another analogy. Compare our situa­tiOD with a dweller of a mountainous region of the earth attempting, with the aid of geodetic observations, to establish that the earth is a sphere. His region of observations is of course very rest�ricted. Only a Iew kilometres. Quite obviously such a physicist is in no easy posi­tion.

Even if be is able-using his measurements­to find tbaL Lhe meaD radius of curvature of his portion of the surface Is 6,400 IQIQ!Iletres

�97

(the approximate radius of the earth), he will not bo one hWldred per cent confident that the surface of the planet has the same curvature in regions outside his view. And then be ,,�u inev­itably have to do what Isaac Newton so disliked. He will have to fra me hypotheses.

That is the actual situation of down-to...,arth physicists when they are asked ahout the geom­etry of the world in tho large.

Let's stop for a moment at this exciting point. [t is time we did some summing up. There are two main pOints to be summarIzed; both are a direct consequence of non-Euclidean

geomelry. The lirst is the creation of a x iomatics and, subsequently, of mathematical logiC. This w.s

accomplished by Hi! bert. We have already had occasion to mention him, but ollr story was very crude and approximate. Particularly as regards the problem of the completeness of the axioms. I could have done a better job, but only at tho

298

expense of drawing out our story. Anyway, when I sat down to write this book I had no idea of how to tell the story of ax ioma tics precisely, concisely and comprehensibly. Too little has been said oE axiomatics, and most of tbat was not very accurate. The only consolation is thaI I can now add a bit of advertisement.

The whole range 01 problems involving axlo­matics is amazi ngly elegant. Even the state­ments of many of the problems are totally unex­pected. This is particularly truo of tbo problem of completeness. One rcsul� will sulfice as an illustration. Already in the Hl?,o's the follow­ing theorem had been proved.

Suppose you have a cerl.ain logical system. I ts loundation consist.s of tho Basic Concept.s and the axioms. Say, Euclidean geometry. I f this logical system i s "sufficiently powerful" (the meaning of this is of course over our heads), then it will always be possible to formulate theorems which, within the framework of tho system, cannot be proved or disproved.

At /irst glance it would seem that the trouble lies in a lack of axioms. Tbat is not so. 0 mat­ter how many axioms aro taken, no matter how we supplement our system, there will always remain certain assertions about which nothing definite can be said.

Alter this marvellous theorom was proved, tho whole problem of noncontradictori ness took on a different aspect.

Wo were silent on this poinl, just as we did not so much as touch OD the totally wlexpected application 01 mathematical logic in comput­ing machines.

299

We spoke in somewhat more detail about the second line of development that -passes through i;liemanroian geomotry to the general theory of relativity.

One more thing. The whole history of the dev­elopment of non-Euclidean geometry appears a. one of the most brilliant instance. of unexpected turns in tho history of science.

WI.at appeared to be the ult-imate i n abstract speculative and theoretical meditations of mathe­matiCians was in some marvellous way trans­muted into thi ng. of extreme i mportance to practical phy ici.ts and even engi neers.

Chapter 72

EtNSTEIN

The essence and nature of any extraordinary talent are mysterious. That is a trite statement. But the bitter truth is that the mechanlsm, even the rough operating scheme, of that remarkable computing device that is our hrain remains a mystery to science. We cannot make out how, ill the brilliant scheme of evolution, nature fash­ioned some 1 4 to 1 7 thousand million element­ary units called neurons into what is known as the human brain.

We do not even have a suitable answer to the question: "In what way does the human brain differ from that of some other animal?" Wo eithor coniine ourselves to the general phenomenolog­ical reasonings of the biologist or to the scint­illatingly clever but, alas, trivial paradoxes of the writer.

There is even less to be said about how the brain of a genius differs from that of a common­place earth dwoller. More, we do not even have any gronnds to claim thM there are some kind of organic differences of that nature.

I t may very likely be that in every person some exceptional talent wastes away unbeknown to the world. It is a very entiCing and consoling idea, and was developed at One time wi th tbo greatest pleasure by Mark Twain.

301

Thero is of course something very suspicious about it aU. But thore are no objective facls in­dicating any absurdity. I t would perhaps be hard to find a better illustration 01 tho level of our knowledge about the mechanism and bio­logy of thinking. We hardly know anything nnd can only take note of the purely external charac­teristics 01 talen t.

Tho oft-repeated phrase that "talent is work" d lie s one such characteristic. These words are commonly misunderstood to actually mean semo­tlllng; this is done all the more eagerly since the gilted, out of 0 tentatiou modesty and with due r peet lor tradition, though at times quite sincerely underestimating themselves, point to work as the main seluce 01 tbeir excoptional at­lai nments.

Slatements of this kind arc many, but only a portion (and a small one a t thall) is the truth.

Paganini claimed Ills wizard playing came from a supremely xhausting labour Ihat en­abled him to master the potentialities of his in­

slrument. He was wrong 01 course. The writer Lev Tolstoi liked to say that his

gill as a writer wa not at all so great or signi­ficant, tho truly important and valuable things being the moral ideas he preached that were so natural and simple.

I do not think that Tolstoi said what he thought.

Einstein, speaking 01 hi genius, said a remar­kable thing. and we shall come back to it again. But I think ho had in mind something quite diflerent and simply was compelled by circum-

302

slancos (energetic newsmen) to throw • bono to the public.

So my id a is that wo hould not believe ge­niuse on this point. The eternal, mournful in­

dignation of Pushkin 's Salieri (a talonted person, by the way) presents a better and more accurate picture of what a geniUJ! really is.

I L is something incomprehensible. When dealing with a normally endowed per­

son, we can analyse and decipher a fow things.

We can then pick out, more or less clearly, tech­

niques, experience, taste-all that comes as the reward of arduous, exhausting labour.

For example, one can al most alway und r tand what is good and what is bad in Balzac's books.

But wh n you are imperceptibly charmod by the endle , rather clumsy and at limes (hor­rible dictul) imply grammatically incorrect phra-805 of Tolstoi ; when you ceaso to watch the style, the techniques, the images and only follow the story of Ih piebald horse HoI tomer, learniug how ho lived and died, and how many herses there were i n the herd of his la t owner . . . . Whon you call lied dOleos of moro or less suitable ex­planations 01 why uch and such a paragraph was wri ttell and what relative Iitorary merits i t has etc., but canuot grasp how i t could hav come to Tolsto; 's mind to write that way and why you are lolt with that inexplicable convic­tion that that was precisely the way it should have been written . . . .

Then w say that this is a n anomaly which can be r corded as such but cannot be accounted lor.

The curious thing is that quite olten 8 person

303

who is a genius i n one field is by no means a harmoniously endowed personality.

There are paradoxical instances galore. Per­haps tho best one is Tol toi. Tolstoi the philoso. pher was a narrow-minded, biased and capricious personality.

We enn bring this discussioll of genius to a c�oso .by adding that the very conception of ge­OlUS IS oxtremely hazy and subjective, particu­larly when one deals with art, whero objecti ve criteria are still moro nebulous.

In science too, ulti mately, the deciding fac­tors (or, to be more oxact, their absence) are tbe same as In art, and tbat is why very often a first· magnitude star of today becomes noticeab­ly faint tomorrow.

Some cases, incidentally, are unque tionable. One is that of A l bert Einstein. As far as we can judg [rom reminiscences, the

childhood years of Einstein did not in the lea t sugg st that he would be 3D EinsteiD.

He was a quiet, reticent child. Usually chil­dren are full of Iile and energy, noisy, ill a hur­ry, in a hurry to toll the world what th y aro.

But in overy dOZOD there are one or two of the quiet kind. Thoy do not take part in games and keep to thomselves. They 8OO1ll to be occu­pied more by their inner world tban by tho world around them. I t may be that something bas stirred up mistrust in tbeir minds and tbey simply cautiously avoid people, in tinctively be­lieving that it is safer tbat way. Children of this kind are not li ked in the rather merciless kingdom of childhood. Tboy aro continuaUy be­Ing teased.

304

"Si sy", "mamma '5 boy ", uwcakling" are SOme of the international terms lhat often cause more anguish than, in later lifo, a raking down by ono's superior. At any rato, the mark tbey leave in tho person's l i re is deeper.

Ein tein was of the timid kind. His rolati ves recall that he was called "mam­

ma's boy" for his morbid love of tbe truth and lair play.

Another thing. 1-1 did not l i ke soldiers. Nei­tber tho real Ones marchi ng along in bright new uniforms aDd helmet stamping in unison down the quiet stroets of the towns of I,is Fathorland, nor tho pretty tin soldiers that come in nice boxes. He did noL like soldiers.

True, honesty and fair play are not so rare in children. The question, rathor, lie in the age at whicb it ordinarily disappoars.

Now as to thi in Linctivc dislike o[ soldicrs­that i indeed strange.

There are not many children like tbat, and one might su peet something out of the ordinary in such a child. But no, there doe not seem to be the slightost indication that this "something" will, in fifteen years, Dower iota the theory of relativity.

There were other things that worried Einstein at this age.

I do not know whethor the people around him noticed that at the ago of ten or eleven this boy of well-to-do parents was going through a

crucial internal drama, which i n many ways de­

termined the wbole of bis future life. At least Einstein himself rememb red; at too

age of 67 he wrote:

305

"Even when I was a fairly precocious young man lhe nothingness of the hopes and strivings which chases most men restlessly through Ii fo came to my consciousness with considerable vita­lity. Moreover, [ soon discovered the cruelty of that chase, whicb in those years was mnch more carefully covered up by hypocrisy aud glit­tering words than is the ease today. By the mere existence of his stomach everyone was condem­lled to participate in that chase. Moreover, i t was possible t o satisfy the stomach b y such par­ticipation, but 1I0t man in so far ashe is a thinking alld feeling being. As the first way out there was religion, which is implanted into every child hy way of the traditional education-machine. Thus I came-despite the fact that I was the sen of entirely irreligious (J e,,�sh) paronts-to a deep religiosity, which, howevor, found an abrupt end­ing at the age of 12. Througil tbe reading of popular scientific books I soon reachod the con­viction that much in the stories of the Biblo could /lot be true. The consequence was a positive-

306

Iy faoatic fl'oothillkiog coupled with the impres­sion that youth is intentionally boiog deceived by the state through lies; it was a crushing i m-11I'ossion. Suspicioo against every kind 01 autho­rity gl'ew out of this expel'ience, a skeptical aLti­tude towards the conviction which were alive in any speci foc social euvironment-an altitude whieh has never again left me, even though laler 011, because of a b tlcr insight into the causal connectiollS, it lost some of its Ol'iginal poig­nancy."

This somewhat heavy passage demands more than a hasty reading. It is worth a most de­tailed analysis.

Note, firstly, that Einstein wrote his autobio­graphy as a scientist striving to extract from his inner life with complete honesty ouly what de­sorves attention. He 01 course realized that this was no easy task a t al most 70 years of age. He was eveo academically cautious in the title: "Autobiographical Notes". Mostly he wroto about what he considered to be the ouly interesting thing in his lifo-the formation of his scienti fic outlook. His work.

There is no place for allY tiling else in this self­obituary. There is no attempt to appear better, no ostentatious display of any kind. Actually i t i s a SCientific paper. I n overy lIne one fools the desire to be as truthful and objective as possible in describing how he, Einstein, reasoned.

Such was the lile of ten-year-old Einstein. He did not like school. He recalled school,

picturing his teachers as army sergeants; and tho gymnasium where the instructors wero, to him, lieu tenants. ... 307

Here we have tbe 6rst riddle. One fairly orten meets people who, irrespective or tbeir culture and education, never reach the idea that a per­son needs something more than simple well-be­ing. Some arrive al that conclusion at a mature age, or even at the end o( their l ives.

To one degree or a"other, this striving to­wards the m ysterious "something else" is found in aU children, but mostly in a very intuitive way of which they aro not aware.

Einstein, On the contrary, reasoned wi th rigo­rou

.s logic. As a result he arrived at religion,

wblch was quite understandable, taking into ac­counl the conditions nnder whicb he lived.

So rar there is nothing mucb Ollt of the ordi­nary.

The amazing thing is thaI alter reading a num­hor of popular-science books the boy quito in­dependently canied out a purely logical analy­sis and took a sharp turn away from religion, as a doctrine that is unsatisfactory. He even goes farther, arriving at a c lear-cut conclusion of great social i m port: " . . . youth is intentionally being deceived by the state through lies . . . "

That was a t the age of twelve. And that was the conception that he carried

with him throughout his l i Ie. If that is so, then whet'ein lies his, A I bert Einstoi n 's, "some­thing"?

Very very cautiously, fearful of distorting tbo trufh , he writes that partly conSCiously and part­ly subconsciously he came to the conclUSiOn tbat for him life would be happy if he devoted him­self to science. "The road to this paradise was not as comfortable and alluring as tho road to

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the religious paradise; but it has proved itself as trustworthy, and I have never regrettod ha­ving chosen it. �'

You cao believe him, ho was indeed One of the happiest people of our ago. Perbaps ho would have boon just as happy even if we imagine tbat his work 'was not understood, not rocognized and if he bad to die an unknown eccentric engineer of the Swiss Patent Bureau at Berne, where, as a twenty-five-year-{)Id youth he created the theo­ry of relativity. InCidentally, at tbe end oi his l i fe he experienced something of this kind once again, i n a sense.

Not in the sense of bei ng famous, of course. He was the most recognized and most popular scientist in tbe world. He was al most as well known as Marilyn Monroe or the footballer Oi Stefano. His name had become a symbol of the human inteliect.

But phYSicists did not take much interest i n the works written towards the end of his l i fe. Yet it was only their opinion that carried any woight with Einstein.

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Actually, not too much weigbt, because tbe decisive factor was always tbe opinion of Albert Eiostein.

Why did be choose scieoce? Perhaps if a cartain modical student. had not

suggested tbat be read poplllar-science IitoratllOO be would have been a good musician instead of a brilliant physicist. Einstein played tbe violin from the ago of six and was seriously and sin­cerely in love witb music throughout bis life. Then again he might bave gone into inventing­another one of his passions. But sucb musings are idle.

Ei nstein himself, in later life, always said that if a person was born to be a physicist, if it was in his blood , tben he would be 3 physicist 110 matter how his l i le turned out.

It's hard to say, be most likely was judging by himself. Tnle, on one occasion, recalling his youth, be ox pressed the opposite viow.

Be all this as it may, the existence of all the popular-science literatlltC of the time would be justi fied by the single fact that it had some in­Duence on tbe deeply thinking youngster of twelve who roamed the pictlltCsque outskirts of tho provincial Swabian town of 1m i n 1891,

Soldiers' feet rc.'!Ounded on the streets of Ulm. They were the heirs of the victorious warriors of Mol tke who twonty years before had routed France.

The military traditions of U l m , it seems, went deeper still. I n 1 805-U l m was then a lirs\.­class lortress- a wondorfully equipped Austrian arlTlY surrenderetl to Napoleon ill a most sr.auda­lous lashion, virtually without lightil1g.

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But, first o f all, the army was Austl'inn, which means, formally speaking, not German, and this consequently implies "not at all German".

Secondly, tbe soldiers did not remember the defeats, for their beads wore fiJled with victo­rios.

Defeats were simply regrettable accidents, th. t 's all.

So they marched. That was probably when to the child Ein tein

caDlO hate. A restrained, calm, somewhat cold and rational hatred. A hatred that invariably stayed with him his wholo Ii Ie. He could not stand militarism, war and slaughter. He viowed i t all as the supreme concentration of human stu­pidity.

Thl became clear to him in his early years, alld his viow never changod.

The year was 189 1 . Fascism was a long way orr. The crematoriums of Oswiecirn and Maida­nak were not yet built. They came later.

Germany was still to face the Schlieffen plan. The First World War. Marching armies. Exalted weepi ng WOmen throwing 110wers to their menfolk. Trainloads of soldiers. Ersatz lood products. And the same women weeping, differ�ntly, over tbo endless stream of casualty telegrams from tbo Eastern and Western fronts. Tho final rout, the o,'orlhrow of the Kaiser, tho Treaty of Versailles, io nalion, rnin, hunger, and the epidemic of flu would all come iate,· to tho Gormans. All these thin!:s wUltld come bofol'o tho Fiihrer came.

Truo, L hero were a low tbing . For example, tho hright uniforms lIII,1 the P"ussian general

31 1

slaff, antisemitism, and patriotic military mar­ches, and fraternities, and-probably most i m­portant of all-an unquestioning reverence of titles.

Ci vil.ian or mili tary, it makes no difference. "Herr Privy Couucillor! Ohl Indeed!. . . " Tho great Olympian hi mself, Goethe (aod a

volume of Goethe could of course bo found in every rospectable family), even Goelhe, ladies and gentlemen, was just as preud of his ministeria l post in the miserable Weimar prinCipality as, perhaps, he was of his poetry.

And Hegel? The great "Privy Councillor" He­gel, remember? And his doctrine of the Prussian m onarcby?

In short, tbe German stato was consistent in ,its strivings to wipe out lhe very capability of independent (hence, critical) thought that is part and parcel of every normal human being, Hnd to put in its place ready-made slogans,

rules and traditions. And they did a good job, one mnst admit.

The system was polished to per[octiou by true cra ftsmen i n the art.

The "Wacht am Rhein", the sentimental "Lie­der" of blue�yed girls, and Wagner's operas, and gymnastics at school, the tales of ancient

ordic heroes at the history lesson, and tradi­tional off-colour humour in cbeap editions, and pedantic neatness instilled from early childhood, and absolnte obedience to the bead of the smal­lest unit of the state-the family.

And, finaUy, t.he end Ie m lll liplicity and di­versity of ollicial , scm i-official aud Jlon-official hierarchy of t i tles and r.nks.

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The hierarchy i n the family, i n tho beauro­cracy, i n tho military service; the beirarchy of numberless Veroins, fraternities, llnions in sports, at the factory, in m usic, t.he arts, the sciences, ill li terature and religion; unions of lovers of hunting, lovers of song birds, unions of bee­keepers, yachtsmen, and so on and on.

Al! this crealed and cherished a convontionali­ty that was both self-satisfied and humblo; it created people that forgot that they were capab­le of thinking, people for whom a dictatorship appeared to be the most natllral form of power imaginable for tho reaSOn that each in himself was a dictator on a microscale.

The remarkablo thing about the i n finitely poi­sonous character of tllis whole demoniacal ma­chino was that it fed On v ry decent feelings and aspirations-patriotism, respect for one '8 ei­ders, sports . . . .

But what o f the people themsclves? The same. Whether at the begi nning of the

I Htcenlh century or the beginning of the twen­tieth, or evon during the year. of fascism, they did not dilfor in any way from any other people. There can be no question that a hllndred thou­sand scoundrels can be found in any large coun­try. The historical situation i n Germany at the end of the 1920's was such tbat precisely this group came to power. Possibly, accidental circumstan­ces played an apprec iable role here.

Trne, the prerequisites for this accident wero already prepared. Incidontally, J would not be saying a"ything original 0" new j[ I added l hat roughly tho SalUo prerequisi tes were avai­I a ble in auy of tho largo i n1 poriaHst states.

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This had bcoo made cloar many timos in the books of such wri ters as Sinclair Lewis and H. G. WeUs, where the po sibility of fascism develop­i ng iu tho U uited States or in England was des­cribed in science fiction. Perhaps tho greatest danger of tho domagogy of fascism lies in tho fact that it is not new or exceptional in any way.

I I fascism is a disease of the human race, it is an ancient affliction. States of the fascist type exi ted in all ag . Egypt, Sparta, Rome-all the ancient regimes preachod just about the samo ideology as the 'azis. So Hitlor did not have to concoct anything particularly new. True, ho added a goodly portion of social demagogy, which Egypt got along without but which an­cient Rome found it n ce ary to includo.

And of cou.rse ono of the ba ic axioms of tbe system was nationalism.

otbing very origi nal about tbat eitber. From time immemorial, flattery, Oven of tho crude t kind, ha alway been excellont bait to lure tho bearts of members of th h u man race. I t is al­ways nice to hear that you are bett r than tho next man. All tho more so, whon you yoursoll are not so sure of th fact.

ow if the Dattery is kept up in iatontly enough, ooe begins to helievo.

Every empir huilding state since tho pbaraohs of Egypt bas brought nationalism into play as a means of attracting and uniting tho peoplo.

Tho idea is simple and naive, a truism. The em perors of Rome, Genghis Kban, apo­

Icon, Hi tler have all employed tbe sarno tech­lIique, just as tried alld te ted as compli menting the woman Olle wants to seduce. Towards the

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end o [ bis life, Einstein gloomily remarked that

people learn but little from the Ie ons of his­

tory because ach new act of stupidity appears

to them in a Ire h light. That this sy tern produces results eVOD in our

"enligbtened" age was, unfortunately, demonstra­

ted in tbe cond World War. But we must re­

p at: tbe fact tbat most o[ the G rman people

accepted fascism in one form or anoth r doeg

not, of course, imply that tbo Germans as such

are less responsi ve to tbo generally accepted mo­

ral norms than the R ussians or tbe Fr nch.

ow if the question of the re ponsibility o[

tbo German people as a whole for tbe rise of

fascism comes up, then wiLb just a mncb justi­

fication the qu tion could be addressed to aU

tho capitalist states of our planet, whicb with

comparative calm watch d Hitler advance from

the Beer Hall putscb in Bavaria to tbo furnaces

o[ the concentration cam ps and the mass sboot­

ings in Ru ia, Poland, yugoslavia . . . .

Tbo logic of noni nterference wa just tbe

same . . . . Today, twenty odd years after the end of the

war, today wben i t is possible to judge with re­

lative objectivity, onO shonld hardly throw all

tbe horrible blame onto the German people.

All tho more so since that nation too paid a

sufficiently dear price. Among tbo victims of tbe

ads wore al 0 those B rlin youngsters who du­

ring tbe last April days of 1945, crying from

sheer frigbt, wenl at Ru ian tanks with Faust­

Patronon sincer(lly believing that they w re figh­

l i ng alld dying for their Fatherland .

These urgullIonlg a rc I'roiJahly just lIS true as

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the fact Utat the acti vo SS men aod the "crea_ tive" and initiative Hitlerites of the punitive expeditions and d ath camp should be judged and exterminated today, too, twenty and mor years aftor tbe war's ond; they should be shot calmly and with a elear conscience, "without all­ger and bias", on the basis of tbo very same rea­sooing that professional murderers and recidi­vists are wiped out.

Wo may recall that onco upon a time one of them got the idea of writing "1 0dem das seino" on the gatos to the Buchenwald concentration camp-to each his duo.

Why do r writo this here? Becauso tha t was approximat Iy the way Einstein thought. He ba­ted fascism his whole life.

Humanism and the al l-permeating kind ness that was Einstoin's doe 1I0t appoar to link up with sentimental all-forgivi ngness. which. as a rule, stems from indifferonce and gets along ve­ry well with stone-cold egotism.

This is nicely and precisely described by Leo­pold I n feld i n his recollections.

. Unf�rtunately, in recollections and biographies,

Emsleln very often appears a kind of eccen­tric emanating an endl 58 stream of genUenes and far removed from any thought that there can be meanness, deceit and wickedn ss in the ordinary day-to-day world . Such writings are mostly irritating, for, whether intentionally or not, the authors make Einst in out to bo unori­ginally stupid.

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We will d o better to quote I n feld: " I learned much from Einstein in the real m

of physics. Bul whal I value most is what I Was Laught by my conLact with him in the human raUter than the scientific domain. Einstein is the kindest, most wlderstanding and helpful man in the world . But again this somewhat common­place statement must not be taken literally.

"The feeling of pity is One of the scurccs of human kindness. Pity for the fate of our fellow­men, for the misery around us, for the suffering of human beings, stirs our emotions by tbe re­sonanco of sympathy. Our own attachments to life and people, the ties which bind us to the out­side world, awaken our cmotional response Lo the struggle and suffering outside ourselves. Bul there is alsc another ontirely different scurco of human kindness. It is the detached feeling of duty based on aloof, clear reasoning. Good, clea.r thinking leads to kindness and loyalty be­cause Utis is what make life Simpler, fuller, richer, diminishes friction and unhappine s in our environment and therefore also in Ollr lives. A sound sccial attitllde, helpfulness. friend­liness, kindne , may come from both these dH­forent scurces; to expre it anatomically, from heart and brain. As the years pa d I learned to value more and more the second kind of de­cency that arises from clear thinking. Too of Len I have seen how emotions unsupported by cloar thought are usele if not destructive."

I am only scrry Utat 1 did not write this pas­sage myself. Without sentimentality and pas­sion, witbout melodrama and tragedy and a soul-divesting self-analysis. with the calm logic of

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tho physici�I" InleM has hero form u l a ted tho best towards which every POrsOIl slri ves.

But, as w know, tbe road 10 hell is paved with good i n tent i ons. Tn stTivo does not mpa" 10 accomplish .

That is not aU. I n feld wriLes furtber. "Here again, as r see it, EinsLein represonts

a li miting case. I had never encountered so mucb kindness tbat was SO com pletely detacbed. Though only scientific ideas and phy ics rea l l y matter to Einstein, h e has nover refused to help wben he lelt that his help was needed and could be effective. He wrote thou Iwds 01 letLers 01 recom mendation, gave advice to hundreds. For hours h talked with a crank because the family had written that Einstein was tbe only one who could cltre him. E instein is kind, smiling, undorstanding, talkative with people whom he meeLs, waiting patiently for the moment when he wil l be left alone to retltrn to his work. "

I t is hard to helieve that Einstein found any pleasure in talking to this psycllical\y unbal­anced man. I t would be iust a naive to think that Einstein hoped, through such an encounter, to heal the man. But he probably believed, after weighing and analysing the case, that he might bring about a slight and temporary i mprovement in the state of the patient, and thus alleviate the lifo of the lamily. It was with tb.is purely hypothetical possibility in mind that he consi­dered it necessary to Lear himself away from hi" work-his only god.

And when Einstein arrived , inwardly, at some conclusion, he did not 1ea\'0 it as some specula-

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tive rlogmn; lor b i m , thought signified, o bove

a l l , action coord inaled with thought.

Here I am writing something in the natUl'e of a biography, yet all the time I hear Einstein '$

own cal m remark in his "Autobiographical

otes" that he hi mself could 1I0t bopo to convey

exactly his olVn t.houghts and his i nner world.

QuiLe naturally, a biographer would succeed

even less. Even when oDe is dea l i ng witb a ratber com­

mon personage, this is an i nsurmountable pro­

blem. I t become ab.olutely unresolvablo when

one attempts to write about a mao the stature

of Einstein. Note too that E i nstni n 's own writings on this

matLer are natura l ly very often contradictory,

whilo a biographer's writings are Ullavoidably

subjective. Yet in the case of Ein Lein, it appears, para­

doxically, that some things are si mpler than even

in tho biograpb.ies of some of the long since for­

gotten "i m mortals" of tile French Academy of

Sciences. Tbis may be due, again, to tho fact that i o

his omotional life too he followed with purely

German pedantism the clear-cu� logical criteria

of a consistont and realistic humanist tbat be

bad worked out for himself in his childbood and

early youth. I t was more difficult to shake his convictions

here than in his genoral theory of relativity,

though he hi III sol f.dld not in the least overesLi III ale

his virtues.

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The calm and saddening skeptici m of a mild, clever and kind scholar made completely and uncondi tionally impossible any sort-so com­mon in people o[ that character-of !larrow-min­ded self satisfaction of the righteous man who has learned the truth and is communicating it to the lost world.

Shortly before his death he wrote to Max Born that what every man has to do is be a model of purity and have the courage gravely to maintain etWc convictions in a society of cynics. He add­ed that he had strived for a long time to act in this mannor and had succeeded-to some ex­tent .

Tbese sad and wearied words wero spoken by a man who was-evoryone said-always cbarged witb a natural inner and invinciblo cheerfulness. They indicate that Einstein perpetually felt a beavy inner awkwardness thronghout his Call&­cious life. He was constantly worried 01 being too speculative and too passive in the struggle against basencss and absurdity that stood out so conspicuously in tbe surrounding world. Above all, he was oppressed by the obvious absurdity of what was happening in the world.

How it was and why Einstein decided that social worK was not his business, I do not know.

Perbaps be did not see any real ways out. I t may be that emotions and feelings were decisi vo. To some extent UIlconsciously, obeying tbe oX­hortations of Ws heart, he found Ws identity in physics.

Perhaps some role was played by the i nner ro;­ticence and individuality of Ws thinking. And after the choice was made, nil else waS [lusbed

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a ide and into the background by the cbief pas­sion of bis l ife.

But the surrounding world wa never for a mo­Illent ·witched out of his mind. III actual l i fe he was constalltiy encountering political intri­gues, malice and human paSSions-he could not stalld aside from these thi ogs, for he clearly and fu'mly I'ealized tbat a human being has no right to do so.

This idea is simply a repetition of what wa said carlier; tber ' .1 0 I wrote some sbaq) words about the "Gel'man pedantry" of Einstein. The point here is not one of a definition of pedantry, it need hardly be said lhat by pedantry here was meant a kind of integri ty and ulti mate logi­cality of characler. Since these features are com­mouly tbought to be iutrinsic to the national character of Germans, I used the adjecli vo "Cor­man ".

Howevcr, I do IIOt intu1Id to justify my defL-1Ii lion because-and 1 believe it is wel l wortb sayi ng-Albert Einstein, t hough born a J ew, bad an Amcdcan passport, nod was a consistent and uncolld il ional iutnl'lIalionalist in his cOJivicti· ons, an internationalist i n both mind and heart, was al}\'ays, all his life, a German, fl Ger man in

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language, in culture, in customs und in I. ho hard­ly porceivablo habits, eccentricities and minutiao that, ulti mately, go to Il1nke Ull a nation, patrio­tism, and love of olle's motherland.

He was a German in his rather heavy (particu· la"ly in hls youth) academically dry h u mow. I n later years, the ponderousness retreated nnd hi� pronouncements became polished a.nd aphoris­tic, t hough again thls was tbe humour 01 Heine rather than that of Mark Twain 01" the R llSSiall Shebedri n .

H e was 8 1 0 a Gurman i o his somewhat con­templativo love 01 quiet Jlature anol walking in the countryside, in his household habits, in bis passion for Mozart, his penchant for analysing philosophical problems, and his love of hls mo­ther tongue.

Tho last word be pronounced were i n the lan­guage of hls childhood, GermalJ, and thoy were nOI understood by the nurse wbo was the only one wi Ih hi m wben he passed a way.

After twenty years of l i fe in A merica-it is hard to imagine anything more paradoxical-he was just reachlng the point (said oue historian of pbysics) wbere he could bandle I.be English language "tisfacto" i Iy.

But evon dW'ing the latle" years of his Iile be preferred to speak Genna .. i f his companion spoko that language.

l ie was homesick just like any ordinary bwg­hor wbo m ight bave come to Iho U nited States 011 business '''HI settled down for t be rest of bis life. For-alld I bis illcidentally was the credo of Einstein himself-thel'e are thi ngs and concepts common to all people i rrespectivc of their intel-

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loot and culture. ow in m atters of etbics, in the aspect of norms of hunwn conduct, Einstein was a fully convicted democrat wbo recognized both in word and in deed the compl to it priOl'i equality of human beings.

I feel I mu t stray again and relate a story, which, though it al most sounds like a joke, gi ves a very accwate picture of Biostei n 's stand and style ill bis dealings wi th people.

Tbere seemed to be a vacancy opon at a certain institution, and four dillerent allplicants came to hlm one aftor the otber for letters of recommen­dation. Einstein gave letters to all lour.

To the surprised questions of his friends bo replied calmly that he saw nothing strange or extravagaot in what he bad done, for in each case he gave di fferoot reasons for his cboice of caJldidate and it was, be said, up to the employer to do tho cboosing.

Let us return to 1 891 , to the town of U l m and to tbe twelve-year-old boy who was exporiencing a wonder. I I wa contained in a book on Eucll­dean plane goometry. Euclid was a revelation to Einstein, and it remained 80 to the end of his life. Shortly before hls death he said words to tbe effect that if Euclid's work could not firo one's enthnsia m il l youtb, then that person was not born to be a tbeol·elician.

Einstein's recollection of this wonder on tbe fourth or fifth Ilage of ltis "Autobiographical Notes" is just about the ltlst purely autobiogra­phical recollection.

A few words follow about bis education at tho Polytechnic [n titute of Zuricb, then just in passing 8 remark or two about the syslem 01

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instrllction and . . . rougbl y fifty pages of Ein­stein's idea concerni ng modrs of thillki ng, epis­lemology 81 1d, of COllr�, phy�ic�, 8 nlways.

Hut one bfluld lIot get the idea tbat this way of cOllstructing an a u tobiography is another Ouo of those cuto absurd absent-minded ways of tho a.schetic monk. Don ' t ever try to represent Al­bert Einsleill as a kind of J acll ue Paganel of phy�ics.

A few pages later h. give a c lear-cut and ealm explanat ion of his somewhat extr.vl'gant manner of presenting thi nW!.

"And this is aD obituary?" asks t he puzzled rcader. I feel l ik answering: "Why yes, of cour­se. " Becausc the most i m portant thing i n the Ii! of a man of my ma ke-up is what be thinks and how he thinks and not. what he docs or e x p rienccs. "

That i why Ein lein "ecalls tb wonder of

geometry and docs not even mention his Nobel Prize.

This idea of a "wonder" a of something that tho human mind encou nters Ulot contradicts " I I esta blished notion , i "ery persistently repeated by Einstein throughollt his I i le.

lu reply to a reporter's q uestion as to how it happened that Ei nstein and not somebody else discovered the special theory of relativity, Ein­stein remarked that ho \Va rather late in deve­loping mentally and that for this r'RSOn h sti l l relroined tho perception 01 a child a t the lOge 01 20-25. And so wbell, unencumbered , be medita­ted on the situation o[ thing in physics, be ua­turally was surprised l i ke any normal child would be, but ince bo was at that lime twenty

324

years of age, his i n tellect was mor develop d (this he a d m i t ted) thron throt o[ " normal ten­ycar-old boy and so he wa. able to obtai II re­Sill ts tha t COIllIII'ised t he special t heory of re I 1\­t i v i Ly.

Penotrnting to the kernel of mallers here, we Ond that there is an important and very essential idea behind it all, tltat lhe s ientist should con­

tantly experience 1\ feel i ng of wOllderm nL and regard a l l the phenomena of nntILre i n au unpre­judiced manner; he should rej ct a l l dogmas and authorities . . . . I n shorL, h should thi n k UJld not quote. 1'ru , Lhis wa IIOt an original thought. Plato had already put th idea neatly when Ito aid : ,,\ onder is tho mother of scionce . "

Today this i s such a trui m lbat n o sel f-res­pectillg wriler risks repea l i ng it , yet thero is no second Ei nstei n. Obviously, there must be some­thing more. But, s.,d as it is to admi t , wo are rather i n the po i t ion of a eunuch being told the mealling of lov .

o young Ein.stein ex peri nced One wonder .r­ler another. Between lhe ages of twelve and six­toon he discovered mathematic , 8nd the purely emoLional i mpression that this new world, UIO world of 11I'ecisc logic and unbridled imagination made on him, was exceptional.

At about this l i me Einstein experienced yeL a nother wonder, purely psychological.

"Til fact thal [ neglected matbematics to n certain extont had its cause noL merely in my stronger in lere t in Ule natural cienc.s than iu mathematics buL also i n Lhe followiog strango experience. I saw that JIIathematics wn split up into nUlllerous specialitie , each of wWch could

325

easily absorb the short l ifetime granted to us . . . m y intuition was not strong enough i n the field of mathomatics in order to di Uerentiatc clearly tho fundamentally i rn po.tant, that which is real­l y basic, [(·om the [·ost of the more or less dispen­sable erudition. Beyolld this, however, my in­torest in tbe knowledgB of nalure was also unqua­Ii ficdly . tronger; and it was not clear to me as a student that the approacb 10 a more profound knowledge of the basic principles of physics is tied up with Ihe most in tricato mathematical methods. This dawned upon me only gradually aftor years of i lldopcudent scientific work. 'frue enough, physics also was divided into separate fiolds, each 01 which \ as capable of devouring

a hort lifetime of work without baving satis­fied tbe hunger for deeper knowledge. The mas. of insufficiently connected experimental data waS

overwhelming bere also. I n t.his field, bowever, J sooo learned to sceot out that wbich was able

to lead to fundamentals and to turn aside from

everyth i ng else, from the multitude 0/ t h i ngs which cl utter up the mind and divert it from

Ute e ential . " This i s amazing. I t is not so important whether

E i nstein , at tbe age oJ si x teen to twenty had for­m ulated to himself these tbings or whether tbe

decision was to some extent not consciously relt in his own mind.

The amazing thing is the maturity of such a cboice. ucb lucid critical thinking in general is

very rare , and is somelhing practicaUy unheard

01 at tho age of sixteen to eighteell years.

Indeed, takB a look a t what we have. Here is

a young boy 01 si " teon cart·ied away by mathe-

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lIIalic •. The integra l , tho fundalllontais of ana­lytic g\lometry flre a great source of pleasure, 01 such jOy that nOlbillg else call COlli pare. He of course realizes that he is gifted and that his ta­leot slands Ollt on the general backgl'ound.

He bad every possi bi l i ty of a [reo choice (aud this is most essential). nO circumstance. of l i fe cOlIJ pelled him. Even moro, i f Olle takes into Re­count tho purely external inDuences, lhen there were more points i n favour of mathematics. The Polytechnic I II t i t uto of Ziiric h had a oumber of bri l liant mathematicians such a Minkowski. Thore wcre no outstanding physicists though. Einstei n h i mself said later that up to Ihe age of thirty he had nevel" seell a real tneorelical phy­sici t .

Given statting conditious l i ke these, i t i s bard­Iy possi ble to concei ve 01 a young man giviug up matbematic [or the cognate ubjoct of theoreti­cn I pbysics.

A cballge over 10 poetry or, say. music would have been, psychologic!tl ly speak i ng, more Ull­derstandable.

J feel that tbe problem was resolved hy an amazing feature of Einstein 's character, which, obviously, was already fully matnre in those years, and that is a total a bsenco of intel lectual cOllceit that is so natural among gifted young peoplo.

He a l ways appraised both bis potent iali t ies and his results oberly and ca l m ly. He never played at ostentatious 1IJ0desty and he knew-he said i t openly- that his works repmsent the gl"ea­test I"e ult of twenlietb-ccnt\ll"y science.

At Ihe sa me time he knew (01" he thought he

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knew) tJ .. t be would not become an outstanding matilcmalician.

And so bo gave lip matbematics. Througb!)ut his lifetime, Einstei n 's relation­

ships with mathematics WC"e ratiler complica­ted. On the one hand, i n later Iile, ho timo and agail l regreUed his l'outblllJ self-conudent COn­clusion tbat physics required only the lundamen­tals of mathe",a tics and that tbe more sophisti­cated matters cou ld be left to p"o[e ional ma­thematicians. Ho became 'onvincod of tbi error when he hegan working 01) the general theory or rolativity. During tbo f1"5t stages, he bad to ask the help of his frielld Marccl G"ossmanu in the mathematical port ion.

In later years, Ei nstei n 's views changed. His main works-at least outwardly-aro works or a mathematician.

Neverthcl '58, he Illway' remai ned a physicist in modo of t lJougbt Hud in bis approach to pro­blems.

1 shall not dsk getting into a di cussion "bout the similarities and dillerencos of the tbeornti­cal physicist aud the pm. JJ1athematician. Suf­lice it to say that there is a dilfe,·ence. And a raU.er essential ono, as wi tness tho fol/olVing amUlling exchange 01 wit botween Einsteiu and Hilbert.

I II 1\)15, Hilbert took a l iking to tbe theory of ,-elati"itl' and decidod to t.ry his hand at physics believing iliat substantial progl'e�s wouJd not be made wit hout malh omaLicirulS.

As he ralho.' clevOJ'ly IIUt. it without excessive modesty, "physics is aCLually too dimcult lor tbe physicist". His work was nllturally a t. tbe

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ultimaln mathematical level but somewhat lacking in pbysical content.

In a letter to Ehrenf st, Ein tein rather spi­telully replied for the pbysicists when he descri­bed Hilbert '8 work as the tricks of a superman. Towards the end ol his life, Einstein remarked to the effect that mathematics is tbe only per­fect way of leading yourself around by the nOse.

We will not attempt to draw any moral bere, but will simply repeat that no matter how ma­thematical Ei nstein's works were, he always re­mained a physicist.

It is now time for u to note one import.ant lactor. Tbough Einstein repeatedly said that the response of the community-recognition On the part of his colleagues-was extremely i mportant to h i m , and this was of course true, his own ap­praisal of his work was the decisive factor.

To tbe very end of his days he could not re­concile himself t o the basic ideas of quantum mechanics (which h relegated to the class of ephemeral physics) and t bough he remained alone he never .hangea his opinion.

I n the same way, he was the only physicist i n the world who, without any external prerequisi­t,es and alter having earned fame and recogni­tion, worked for ten years (between 1905 and 1916) On the problem of the gravitational field.

Standing quite out ide the range of interests of tbe phYSics ol that period, be created the ge­neral theory of relatb'ity.

Perhaps duo to a nriety of accidental circum­stances he became the trlost famous scientist in the world. Calmly aod omewbat sardOnically he withstood a virtual avalanche of bonorary

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awards, medals and distinctions (including the title aod attire of honorary cluef of an Indian lribe). And then fol' another 35 years he worked intensely on the general theory of relativity. re­maining practically a lone, actually without any recognition or moral support and appearing i n the eyes of the new generat.ion of quite self­confident theoreticians of the 1930'5 to 1950's something in the nature of an aging monument.

Incidentally, he once mentioned to his wife that tbe resnIts he obtained in the 40 's were I he biggest contributiou that he had ever made.

Who knows whetber he was right, a he almost a l ways waS when Iho subject matter was physics? The only thing to be said is I.hat t here has been an ever increasing i nlerest in Ihe general theory of j'olativity and, in particular, iJl the i nvesti­gations of Einstein carried ont d lll'ing tbe last year of his life. ,

But perhaps tbat too i just a fad which phy­sicists are pro lie to follow like women do fashi­ons. Or it may simply be 8n expression of a cer­lain disappointmen�, a crisis in modern tbeore­tical physics,

Yet perhap the foundatiolls of the physics of the future are indeed to be sought i n Einstein's wOI'k 011 tho unified field t heo,"y, At any "ate, the scienti lie ca,'eer of Einsteill, beginning from his general theol'Y of relativity, is an unparal­leled anomaly in tbe bisto .. 'Y of science.

A lld if one speaks of the purely per onal as­pect of the ma�ter. tbe wbolo story is a miracle that causes more respect than the p urely mat he­matical gi ftedness of Einsteiu, wbich ultimalely scoms beyond tbe scope of human kind.

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I II passing let us add tbat on the side (even if we count from 1920 onwards), Ein tein carried out a range of researches totally unconnected ,vith relativity theory, but of t hemselves quite sufficient to split up among a nnmber of workers alld fill five or six vacancie at all election to the Academy of Sciences,

We may aga i n add that his resnIts ill the tbeo­ry of Brownian motioll and the photoelectric effect (this was in 1905) were i n themselves suflicient to have ensured the author all exceptional place in Ihe history of pbysics.

We migbt also recall tbat the most fashionable and promising trend today in quantum statistic has as its source the theory of the thermal ca­pacity of crystals, which just by the way was proposed by Einstein in 1908,

Finally, Einstein '5 rejection of quantum me­chanics, his paradoxes, yielded so much mate­rial for an el ucidation of the fundamentals of that field that in themseh'es tbey can bo consi­dered Iir t-magnitudo works of cionce. Then, too, he obtained a number of very i m portant resnIts after 1916 in various parts of the quantum the­ory.

But for him all of tbese were onIy a mental game and a pleasant recreation from the main thing-the unified field theory, .

So we have Einstein at the ZUrich Polytechnic I nstitute m ajoring in phySics and neglecting ma­t hematics. He even skipp d lectures- not to spend his time idly but the belle" to utilize it. Before arriving at Zurich together with his fa�ily, he had already visited Milan and had expertenced a nnmber of small unpleasantne , such as being

,,. 331

told to leave the gymnasium at Munich for un­healthy skepticism. Also he failed once in a n examination i n zoology and botany at tbe Po­lytechnic Institute.

. But these events, which for another person m Ight have played a decisive role, wore for Ein­stein merely unpleasant trivia.

Tho die was cast, and his natural bubbling­over cheerfulness and clear-thinking head dismis­sed all

. these and other bumps and scrapes that

came his way. He wrote that he was nevor in a gloomy mood unless he bad a stomachache . . . .

J udging by his letters and the recollections of relative , Ei nstein at 20-25 years of age was a

strong lifo-lOVing young man with a passion for music, �ainting, literature, hiking, with a gift for the Joke, thongh, honestly speaking, his hu­mour was not always up to the mark. He was a b

.it extravagant, a . !riilo forgetful (like forget­

IIDg the �eys to his nat after hi wedding or uSIDg a dOlly for a scarf). But this was all natural �or it stemmed from a strh<iog towards greate; Inner freedom, though-and tbis is i m portant-

Jr. '.

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the�c wa� Dever any hint of tbis constant urgo for IUDcr lIldependence ever bnilding up into ego­tism and a disregard for those about him . This was precluded by an inborn culture and a con­sciously developed mildness .

I n a word, he was a nice well-mannered young man, broad-minded, without a trace of conceit Or morbid reOections. Ooe could readily foreseo his future as 8 school prinCipal or a top-class expert in the patent bureau , where a� that time he WaS only rated third-class. One could see hi m a great lover of music and literature, I'cading S

.ophocles, Racinc, Servantes, discussiog tho trea­

tLses of SplUoza and Hume, whicb he was then reading with a group of friends. One could pic­ture Einstein on a mountain hike animately dis­cussing Mozart, Alexander of Macedonia, Aeschy­lus, Beethoven, Kant, Archimedes, Cleopatra,

ewton, Cuvier. Confucius, Anatole France .. . .

. Later, we might see him the author of progres­

Stve articles on the history of science, or music or pedagogy . . 0 0

I n shor l , his letters and the recollections of people who knew him draw a picture of a very nice young man disturbiogly ordinary.

One linds it hard to believe, then, tbat this was Einstein and oot just some pleasant, educa­ted well-mannered, clever young man.

Perhaps there is one thing, Eistein 's ability to dispense with all externals when the discus­sion turns to philosophy or physics. But no, this was not II vcry oxceptional feature among the young people of those days.

Actually, bowever, an explosion was in tho making.

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And it came in 1 905. J must repeat that any one of three works of

Einstein that appeared i n that year-the theory of Brownian motion, the theory of the photo­electric elfect, and the theory of relati vity­would elevate the author to the rank of exlra-cla8 th oretician.

It remains a psychologica l mystery wbether Einstein himself fully resliwd what he had ac­co m p Iished.

If ho did-and everything about Ein tein and his later pronouncements on this score suggest that that was the case-then we must a d m i t tbat i ntellectually he must have been very much alone, and the pleasant people about llim did not even notice anything out of the ordinary, while Einstein himself, tact£nUy reticent, tried not to suppress his friends whom he liked in a \'ery humsn way. Otherwise how nro we to explain his letter to Habicht, one of hi friends of the Bern period?

This uniqne epistlo begins "Dear Habicht, th ilence between us is sacred and tho fact that r

am i n terrupting i t with lIIere twaddle lIIay seem a profanstion . " And so on in Einstein '5 old­fashioned ponderous play!ul style, cal l i ng l Ia­bicht a "frozen whale" and fanCifully up bra idi ng h i m for not sending his di rtation, which Ein­stein \Va eager to get and read "with plea uro and intere t".

Bul tbe be t joke of all, one quite worthy of Heinrich Heine, i bidden at the very begin­ning of the letter, becanse what is bei ng offered as mere twaddle is tho fOllOWing:

" I n r turn (for Habicht's dissertation.-Smii­gal I promise you four papers, the fir t of whicb

334

J will send oon b cause I am expecting tbe all­thor '5 copies.

"It is deyoted to rad iation and light en rgy and is very re\'olutionary, 3S you yourself wi l l ee, i f you first send me your work.

"Th second paper contains a determ ination of the true i�o of atoms by m 3nS of studying diffusion and internal friction in liquid solutions.

ovl'he third demonstrate that in accordanco with the molecnlar th ory of heat, particle 01 th order of 10-' m m suspended in a liquid x­perience apparent chaotic motion due to the ther­mal motion of the molecules. Biologists have al­ready ob. rved such moLions of suspended par­ticles; th ir term is Brownian molecular moLion.

"The fourth paper is based On th olectrodyna­mics of mO\'i ng bodies and mod i fies tho concop­tion of space and tim : yon will be interested in tb purely kinematic part 01 the work . . . . "

IIabicM certaillly did not lose out i o thi ex­change. I wonder how much is inbred mode ty­Eio tei n's appeal to a scientists of equal stan­ding-and how mucb is merely traditional cour­te y. I t i hard to take seriously the rather ti­mid hope that i n a paper where, in pa i ng as i t were, our conceptions of li me and space 310 overtbrown, there migh t be ometiling of i nte­rest to Habicht. I f re we get a picture of Binst in verging on that of tho vil lage simpleton.

Yet on the otber hand-and tbis is evident from all future lellers, from Einstein's whole life- there is the incel'e l\waren , confidence, cOllviction (wbat have you) thU Habicht is a O1an, a p rsonality and ha tho same valu as he, Albert Ein tei n, and is not different before

335

any law. Above all, before the inner law that Einstein obeyed in his youth, in maturity and in old age.

Most likely the imprc ion of a certain ordina­riness in the person of Einstoin (I speak purposely of his youth wben hi associates and companions could not yet know that he was tbe greatest pby­icist in tho world) was largely due to Einstein's

overriding feeling of democracy, anil an egali 13-rianism just 35 natural to him as his desire to study theoretical physics.

I have already spoken of this, but I want to repeat because to people of the twentieth cen­tury this trait of an outstanding pe.rson is pro­bably the most cherisbed; one is especially at­tracted to a man who, when placed in an excep­tional situation either due to his own merits Or to a more or less accidental set of circumstances, remains democratic and humanistic not only in form but in e sence too.

And note tbat for a scientist of Einstein 's stature, there were not less but perhaps more grounds and conditions to become, at least in the communHy of his associates and pupils, a more unhridled and cruel dictator in the sphere of the inwllect than any actnal dictator has in the sphere of JJolHicaI life.

Self-COnfidence, which expands into capricio­usne , intolerance, and conceit, unfortunately often attends ontstanding (3nd mediocre) scien­tists, who in this respect only fall short of poots and prima donnas.

Such things are not usually written in books yet that is the case.

True, I can judge Einstein only on the basis

336

of biographical material, but tbis case appears to be absolutely clear. Einstein did not have a single one of these traits to even the slightest degree.

That is yet another psychological enigma as­sociated wi th the name of Albert Einstein, and by far not the last in significance.

EinsteIn stood the test of fame in just as easy­going a (ashion-hardly noticing it-as he did his failure at the exams at the Polyl chnic I n­stitute of Zurich.

That, approximately, is the picture I have of Einstein.

One thing remains. It is vcry important. It is the attitude of Einstein to violence and

war. Wi Uy-nilly, from about the 1920's onwards,

when he had become world-famous, and the na­tionalistic, antisemitic fascist scum of Germany had begun victimizing him and his works, to the end of his life he was c losely associated witb political affairs at large.

One cannot say tha t he tried to evade hur­ning political i ues of tbe day. He clearly rea­lized that, firstly, such a thing was simply im­possible (whetber be liked it or not is a different question), and socondly, he felt that he simply had to interfere wherever he believed that SOme good could result.

But here he found himself in a sphere where, from his point of view, very many things were unpredictable, uncontrollablo, and unexplain­ahle.

Because Einstein was extremely perceptive, he could probably picture to hi msel [ and account

337

lor the I' ychology of olficers of the Prussian general staff, but to conceive of a human being reasoning and acting l ike the commandants of extermination camp , like tho men in punith'e expeditions and the hundreds and hundred of thousands of mcn, or to understand how i t came about that tbe leaders of quite a few coun­tries could be morally and int lIectually about on a level wi th those very same S men was something beyond the capacity of Einstein. This wa because he unwittingly OYNe timated the human intellect.

In the '1930' he who wa a convinced and con i tent Ilacifist had to say "now is not tbe time for paci fist ideas", for (this was a natural, immediate conclusion) the only way to halt the spread of fascism is by use 01 mili tary lorce.

I n what followed ho was witno to an i l1\'ol­ved, stupid and dirty political game. H saW poli tician of I he twentieth ccntury ad hering to the old-Ia hioned, nah'o criteria 01 humanitaria­nism to al most the same degree as onghis K118n. He "itnessod the cond World War, and he SItW evonts altel' the war build up into a

rre h threat of yet another war. He wa to some extent reSI)Onsi ble for the making of the atomic bomb, for he had written bis famous letter to R oosevelt.

I n reminisconces of Einstein, wri lers often

speak of the so-called "Bin teiniltn tragedy of the atomic bomb".

To my mind, it was not the bomb. From the standpoint of reason and logic (and

these lactors were always decisive for Ein tein) he was irr I)roacha hIe.

338

He wrote the lotler i n August, 1939, whon there was a direct and immediate danger of H it ler making the bomb and when the only reasonable solntion was to get it beforo fasei m did.

He fulJy roalized that he had bad nothing to do with tbe cold-blooded seosele s murder of tens of thousands of J apanese in Hiro hima and 'agasaki, all the mOre so siJlce in 1945 he wrOle

R oosevelt a king bim not to a l \ow the military use of the bomb.

To Einstein , the atomic bombardment of these cities was in tho way of the last act of hUlllan barbarism, final proof of the hopele posi tion of the sciontist, the absurdity of tbe social struc­ture, the unconditional abnormality 01 buman being in seats of government.

Of course, this gloomy conclusion was aggrava­t d by tbe purely emotional realization that he, A l bert Einstein, was connected with the explo­Sion, bowever indirectly. But tbis was only a n i ncidental factor. More depressing still was the fact that during tho years he at timo lost faith i n the possi bility of any social and moral progre , yet tbis ran count r to everything Ein­stein stood for. However, bere loo be remained true to hi msell, to his manner of outwardly dis­passionate, calm analysis.

lIe learned of the explosion by radio. Einstein's first reaction was one of griof and de pondency. Y"l h .. . eaU""d tlml the tragedy bad nothing to do with tho di covery of tbe chain reaction.

His viow that the discovery of the 6 ion of urani um does not represent a threat to civiliza­tion any more than tbe discovery of matches does, that tbe future development of humanity

339

d pends on it moral code and not 00 the level of technology, was repeated many times.

He VoTote that the world was 00 the varge of a crisis, the whole signilicance of which was not perce. vod by those who have the power to de­cide between good and evil, that the nowly re­leased atomic energy had changed everything, I aving unchanged only our mode of thinking.

"The solulion of this problem lies in the hearts of the people. "

But the fact that he saw all this so clearly did not make things easier. Towards tho end of his li!e his supply of natural cheer was running out, and his depressed state of mind only aggra­vated the mercilessly critical view he took in appraising himsel f and bis work.

"Tbere is not a Single idea wbich I am cOllVin­ced will .. tand the test of time. At times I am in doubt about the correctness of the path I have taken. My contemporaries see i n me at ooce a rebel and a .·eactionary who, to put it figura­l ively, has outlived himself. That is of course a passing fad caused by nearsightedness. The fee­ling of di satisfaction comes, however, from within. "

Einstein' seventieth anniversary was being celebrated when ho wrote this letter to an old friend. Honours never moved him, and now sliLi less. He sadly concluded: "The best that life bas given me is a few real friends, bright and cordi­al, who understand Olle another like you and me. "

One year before his deatb, when he declined the invitation to be present at th fiftieth anni­versary of the creation of the special thoory of "elat i vi ly, he wrote in the same spirit:

340

"Old age and illness do not permit me to take part in such ceremonies. And [ must admit that in part I am grateful to fa te, for everything that is in the least associated with the cult of tbe per­sonality bas always been a torture to me . . . . In my long life I have come to undersland that we are a great doal farther away from a real under­standing of the proce S of nature than most people today realize . "

Thero may have been more optimistic notes at other moments, but in general the last 01 his years were sad. Nsvertheless he continued to work. His cheer at times Ie It him, but never his clear analytical mind, which functioned flawless­ly to the end . He never changed his views or convictions in the least. They merely took On more sombre tones .

As before he was always ready to respond to a letter or to deline hi ideals, though more of­len One would hear h i m say, "people have gone mad", "the world is on the brink of a catastro­phe". During these years 01 the "cold war" the situation in the United States was grave. At such times extremists always come to the surface. The notorious Anti-A merican Activities Com­mittee was active. Tho slightest deviation from official political views was dangerous. Natural­ly, tho intel lectuals-the most Wide-awake por­lion of the nation-were first to come under sus­picion.

Cn Einstein's letters and speeche of this pe­riod, one sees more and more a bitter yet coura­geous stoicism. Not a drop of sentimentality.

As before he was \'ery far a way from a ny kind of complacent all-forgivingness.

341

I n reply to an American teacher, he wrote: "Frankly, I can only see the "evolutionary

W(IY of noncooperation in the sense of Gandhi 's. Every intellectua l . . . must be prepared for jail (Iud economic ruin, in short, lor tho sacrifice of his personal weUare in the i nterest of tho cul­tural welfare of his country.

"If enougb peopJe are ready to take this grave step they will be successflll. I f not, then lhe in­tellectuals of tbis country deserve nothing bet­ter than the slavery which is int nded for them . "

I s not this the same as saying "Yes, tbe people of II country deserve the government that they ha vo"?

He continued to receive letters and no matter what he tbough t , what his mood was, be consi­dered it his duty to belp by writing to lhose wbo lelt they needed his aid. His work suffored of cour. , but what was there to do.

As he put it with a bit of irony just a year be­fore his death, "the time 1 need for meditation and work I have to steal like a professional thief". And yet for all that, he continued to work to the end, whether he was disappointed in humanity or in the level of human knowledge.

Now I call see how poorly I have succeeded in wri ling about E i nstein. '1'0 say nothing of other things, 1 realize that Einstein appears here un­real, 1mbelievable, too good.

But tbat was what he was. Perhaps his greatest weakness was a somewhat

c " uel irony. He acutely saw the weak sides of people and at time ho · overindulged in his hu­ll,our. Of course he was nO saint and would get ini tsted over purely personal miltters. And pro-

342

babl)' a t t.imes unnecessarily so. Particularly i n his yonth.

He was not ashamed of wri ting very bad poo­try, even l i ked to, and he would send his vc,'ses to his friend . He gave concerts eagerly though his violin playing was far from brilliant.

Finally,- true, this is only a su picion I have, based on circumstantial evidence- l think he Was inclined to courting ladies i ll a rather old­fa hioned sort of way. That would seem to com­plete tbe list of his si n .

H i s most salient tra i t \Va that in hi pri vate l ife he strictly adhered to those beautiful prin­ciples that he espoused publicly. People of this kind are rare and the more 0 tbe higher their standing. Naturally, a mall is best tested in the face of death.

From 1948 onward Ei nstein k new lhat at .ny moment his l ife mighL end uddeuly because of a stroke. He had said a number of t i mos that he was not .[raid of death, that the expectation of death would not change anything i n his l ife, and now be proved i t .

I t did not, except perhal)S his diet which he tried to observe. J ust as thirty years earlier, he was calmly sarcastic when speaking about hi possible departure to a beller world and wben in Apt'il 1955 his time came, he remained tbe way he had always been.

ELnstein su(fared greatly, and he knew that be would die. But whenever there was any im­provement he reverted to his beloved irony and stOically awaited events. He d ied in his sleep.

Einstein was probably One of the most l i keable persons in the history of humankind.

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