1929 a sourcebook in mathemathics - david eugene smith

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Sourcebook in Mathematics

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SOURCE BOOKS IN THE HISTORY OF THE SCIENCESGregory D. Walcott General

Editor

A SOURCE BOOK

IN

MATHEMATICS

1

SOURCE BOOKS

INEditor

THE

HISTORY OF THE SCIENCESGregory D. Walcott General

7^0 w ready

A SOURCE BOOKHarlow Shapley

IN

ASTRONOMY

Dnector, Harvard Observatory

Harvard University

ANDHelenE.

Howarth

Harvard Observatory

A SOURCE BOOK

IN

MATHEMATICSUniversity

David Eugene Smith Columbia

OTHER volumes TO BE

announced later

Endorted bj the American Philoiophicjl Aisocuition, tht

Am^rxan

Auoc\Atum for Ac AdvancementSociety.

of Science,

and

the Htsicny of Science

Alto by

the

Ameucan Anihropological Association, th Maihc

Assoaatum of America, the American Miiihanerceive not.

your great Clerk-Iinesse by the I will let that passe now, because it toucheth not the matter that I intend, and I will shew you how Arithmetick doth profit in all these somewhat grosly, according to your small understanding, omitting other reasons moreperceive

may

ordering of your Sciences: but

substantial!.

you reckon them) Musick hath not onely great help is made, and hath his perfectnesse of it: for all Musick standeth by number and proportion: And in Physick, beside the calculation of critical! dayes, with other things, which I omit, how can any man judge the pulse rightly, that is ignorant of the proportion of numbers? And so for the Law, it is plain, that the man that is ignorant of Arithmetick, is neither meet to be a Judge, neither an Advocate, nor yet a Proctor. For how can hee well understand anotherFirst (as

of Arithmetick, but

mansor of

cause, appertaining to distribution of goods, or other debts,

summes

of money,

if

he be ignorant of Arithmetick?

This

oftentimes causeth right to bee hindered, when the Judge either delighteth not to hear of a matter that hee perceiveth not, or cannot judge for lack of understanding: this commeth by ignoranceof Arithmetick.

Grammer, me thinketh you would not doubt in what it needeth number, sith you have learned that Nouns of all sorts. Pronouns, Verbs, and Participles are distinct diversly by numbers: besides the variety of Nouns of Numbers, and Adverbs. And if you take away number from Grammer, then is all the quantity of Syllables lost. And many other ways doth number help Grammer. Whereby were all kindes of Meeters found and made? was it not by number? But how needfull Arithmetick is to all parts of Philosophy, they

Now,

as for

may

that do read either Aristotle, Plato, or any other For all their examples almost, and their Philosophers writings. It is the saying of Aristotle, probations, depend of Arithmetick.

soon

see,

that hee that

is

ignorant of Arithmetick,

is

meet

for

no Science.

7

RECORDSAndPlato his Master wrote alittle

1

sentence over his Schoolhouseis

door, Let none enter in hither (quoth he) that

ignorant of

Geometry. Seeing hee would have all his Scholars expert in Geometry, much rather he would the same in Arithmetick, without which Geometry cannot stand.

And how needfull Arithmetick is to Divinity, it appeareth, seeing many Doctors gather so great mysteries out of number, and so much do write of it. And if I should go about to write all theso

commodities of Arithmetick in civill acts, as in governance of Common-weales in time of peace, and in due provision & order of Armies, in time of war, for numbering of the Host, summing of their wages, provision of victuals, viewing of Artillery, with other

Armour; beside the cunningest point of all, for casting of ground, for encamping of men, with such other like: And how many wayesalso Arithmetickis

conducible for

all

private Weales, of Lordsail

and

all

Possessioners, of Merchants,all

and

other occupiers, and

generally for

estates of men,

besides Auditors, Treasurers,

Receivers, Stewards, Bailiffes, and such like, whose Offices without Arithmetick are nothing: If I should (I say) particularly repeat all such commodities of the noble Science of Arithmetick, it were enough to make a very great book. No, no sir, you shall not need: For I doubt not, but Scholar. you have said, were enough to perswade any man to this, that

think this Art to be right excellent and good, and so necessary for

man, that

(as

I

think now) so

muchfarre

as a

man

lacketh of

it,

so

much heeMaster.

lacketh of his sense and wit.

What, are you so

changed

since,

by hearing

these few commodities in generall:

you would be farre changed if you knew all the particular Commodities. Scholar. I beseech you Sir, reserve those Commodities that rest yet behinde unto their place more convenient: and if yee will bee so good as to utter at this time this excellent treasure, so thatlikelihoodI

by

may

be somewhat inriched thereby,I

if

ever

I

shall

be able,

I

will

requite your pain.

Master.Scholar.

am

sith that to learn it

very glad of your request, and you bee so ready.I

will

do

it

speedily,

AndI

to your authorityit

my

wit do subdue; whatso-

ever you say,

take

for true.

That is too much; and meet for no man to bee beleeved without shewing of reason. Though I might of my Scholar some credence require, yet except I shew reason, I do itMaster.in all things,

18

SOURCE BOOK IN MATHEMATICSBut nowit is

you are so earnestly set this Art to some other passion coole heat, and then you leave off before you see the end. this great Scholar. Though many there bee so unconstant of mind, that flitter and turn with every winde, which often begin, and never come to the end, I am none of this sort, as I trust you partly know. For by my good will what I once begin, till I have it fully ended, I would never blin. Master. So have I found you hitherto indeed, and I trust younot desire.sith

attaine, best

to omit no time, lest

were never to the mid way: But I trust 30U will assay, then to shrink and briefly: What call you the Science not do so; therefore tell mee that you desire so greatly. Why sir, you know. Scholar. Master. That maketh no matter, I would hear whether you know, and therefore I ask you. For great rebuke it were to have studied a Science, and yet cannot tell how it is named. Some call it Arsemetrick, and some Augrime. Scholar. Master. And what do these names betoken? That, if it please you, of you would I learn. Scholar. Master. Both names are corruptly written: Arsemetrick for Arithmetick, as the Greeks call it, and Augrime for Algorisme, as the Arabians found it: which both betoken the Science of Numbring: for Arithmos in Greek is called Number: and of it commeth Arithmetick, the Art of Numbring. So that Arithmetick is a Science or Art teaching the manner and use of Numbring: This Art may be wrought diversly, with Pen or with Counters. But I will first shew you the working with the Pen, and then thewill increase rather

then go back.flie in

For, better

it

other in order.Scholar.

This

I

will

remember.

But how many things are

to bee learned to attain this Art fully?

There are reckoned commonly seven parts or works of it. Numeration, Addition, Subtraction, MultipHcation, Division, Progression, and Extraction of roots: to these some men addeMaster.

Duphcation, Triplation, and Mediation. But as for these three they are contained under the other seven. For Duplication, and Triplation are contained under Multiplication; as it shall appear in their place: And Mediation is contained under Division,last

as

I

will declare in his place also.

Scholar.

Yet then there remain the

first

seven

kinds

of

Numbring.

RECORDEMaster.

19

So there doth: Howbeit if I shall speak exactly of the Numbrlng, I must make but five of them: for Progression is a compound operation of Addition, Multiplication and Division. And so is the Extractions of roots. But it is no harme to name them as kindes severall, seeing they appear to have some several! working. For it forceth not so much to contend for the number of them, as for the due knowledge and practising of them. Scholar. Then you will that I shall name them as seven kindes distinct. But now I desire you to instruct mee in the use of eachparts of of them.

Master.

SoI

I

not learn the last so soon as thethat order, as

must be done in order: for you may first, but you must learn them in did rehearse them, if you will learn them speedily,will,

but

it

andthe

well.

Scholar.first in

order:

Master.

as you please. Then to begin; Numeration is what shall I do with it? First, you must know what the thing is, and then after

Even

learn the use of the same.

STEVIN

On Decimal

Fractions

(Translated from the French by Professor Vera Sanford, Western

Reserve University, Qeveland, Ohio.)invention of the decimal fraction cannot be assigned to any single Pellos (1492) used a