theory and mathematical practice in the seventeenth century

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nuthcmatical development to raise important philosophical question\ con-

ccr-ning the natui-c of niathcniatics.’

Nevcrthelca. intinitc4ima14 arc ot

alfficicnt significance in the hi5torq’ of mathematics

and the

history 01’

philosophy that ;I study of some 4C’~C’iitCCiith-ceiitur)

attitude5 tou;ird the

infinitcsiniaI 4wiiis well worthwhile.

One

rather widely accepted account of 4C~eiiteC’iith-CeiitLii- niathcniatics

run4 roughly

as follo~vs: the

m~lthcm~rtician~ OF the lxriod happil , rcliccl

LI~OII

infinitcsinial methods in the solution of important prohlenis (cywciallv those

l~rohleni~

raised hy the “ iii~ith~iiiati~~rtioii of nature”)

and the

champions ot

infinitesimal techniques cared littlc

ahout the methodologic~il or episte-

niological clucstions raised by such nicthock. According to this i;torv. the

niatheinatic~il dc\~elopnicnts of the pc’riod prcwcded according to their own

dynamc and tlic

matlicm~iticians rcgai-clccl philosop hical rcsci-vati about

the USC of’ int’initc simals

as

irrelevant to the task at

hand. Philip Kitchcr-

plainly takes this view of the matter in his hook Tlrc) ,V\irrtrtw o/’ :Irr/lrt,rlrrrtic.trI

kk~wlcYlgL~. xkiing that such Iack of coiicci-ii with justifyins infinitcsinial

methods was ;I thoroughly pxd thing. A\ Kitchen WCS it. questions of I-igor

need not Iw ~iddrcsscd if ii ni~itheiiiaticxl tcchniquc is \ucccssful in solving

important problems.

2nd ;i dcmnnd for rigor- heconies rational onlv in c;isc~

where the available mathematical theories

c;iilnot

solve outst~inding lxwhlenis

without an initial clarification of central concepts. Thu\. iu Kitchcr’s

estimation. proponents of infinitesimal methods are to be applauded Ear their-

cagcrness to disr-cgard traditional standard4 of rigor and estcncl the frontiers

of ni~itheniatical

knowleclgc. gaining iicw results while leaviny tlic iiic5sb

husincsc of rigorous proof fo i dnother. hleahcr day.’

Of course. Kitcher’\ is omcthing of ;I minority \ieu. blow traditional

historians of nuthcmatics vim. the scventecnth century more hardily

and

regard

the

I-ccourw to infinitesimal methods as an unfortunate lapse

which

was

later

corrcctud.i

On

thi5 account. niodcrn niathemitics appears 3s

something conceived in the sins of seventeenth-ccntur-v tlccadencc hut later

redccnied hq the

sacrifices of

(‘a~icliy and Wcierstrass. I find both accounts



unconvincing because they depend upon the untenable thesis that rigor was

not a matter of importance in the seventeenth century.

I hope to show that questions of rigor were of great importance to some of

the leading mathematicians of the seventeenth century, and that these men

were neither philosophically naive nor unconcerned with the epistemological

and methodological problems raised by infinitesimal mathematics. My

attention will be directed toward two important theories in the development

of seventeenth-century infinitesimal methods. namely the “method of

indivisibles” of Cavalicri and the Leibnizian differential calculus.

1

want to

argue that Cavnlieri. Leibniz. and others who defended the new methods

were concerned to show that the important new results could bc obtained in a

manner consonant with reigning philosophical principles. I should stress that

in arguing that there

was ;I

widespread concern with questions of rigor in the

seventeenth century I am not claiming that every practitioner of infinitesimal

mathematics was deeply interested in problems of rigor. However. there is

ample evidence of serious conflicts between the accepted account of

mathematical reasoning in the seventeenth century and the proof techniques

introduced by the proponents of infinitesimal mathematics. Moreover. these

conflicts were not generally resolved simply by ignoring the traditional

criteria of rigor or by repudiating the philosophical principles which

underwrote them.

I begin with ;I discussion of the seventeenth-century background and the

reccivcd view of geometric reasoning, concentrating on the finitistic nature of

such classical proof techniques as the method of exhaustion. I then go on to

consider the most important infinitesimal approaches of the scventecnth

century and the philosophical problems they raised, especially such problems

as arise when the traditional standards of rigor are contrasted with the new

methods.

I

The best way to approach the philosophical problems posed by the

introduction of infinitesimal methods is to begin with a brief account of the

generally accepted methodological background to seventeenth-century math-

ematics. It was a commonplace among philosophers and philosophically-

minded mathematicians of the era that the object of mathematics was

“quantity in general”.

and further that such quantity could bc either

continuous or discrete. Discrete quantity (i.e. any quantity composed of

a

number of distinct units) was taken to be the proper object of arithmetic.

while continuous quantity (that which cannot be measured by a collection of

units) was declared to be the object of geometry. As our concern is almost



It is in the theory of ratios of magnitudes that the finitistic nature of

traditional geometry is most clearly manifest. The classical codification of the

principles which govern magnitudes and their ratios is Book V of Euclid’s

Elermwts. where the general theory of proportions is developcd. The most

important aspect of the classical theorb) is contained in definitions 3. 4. 5 and (3

of the fifth book. They read:

3. A rrrrio is a sort of relation in respect of size between two magnitudes of the

same kind.

4. Magnitudes are said to /ICILY N rutio

to

one another which arc capable. when

multiplied, of exceeding one another.

5. Magnitudes arc said to hr itt tlw .sutttc rrrrio. the first to the second and the

third to the fourth. when. if any equimultiples whatever be taken of the first

and

third, and any equimultiples whatever of the second and fourth, the former

cquimultiples alike exceed.

are alike equal to or alike fall short of. the latter

equimultiples respectively taken in corresponding order.

6. Let magnitudes which have the same ratio be called l,r~~-opOrfiotttrl.i

The significance of these definitions is that they provide the means for

comparing magnitudes within each species by the formation of ratios. and

then comparing ratios across species of magnitudes by constructing propor-

tions. The finitistic character of the classical theory should hc apparent.

especially when it is understood that the multiplications referred to in

Definitions 4 and 5 are finite multiplications. To compare two magnitudes (1

and \ 3 in a ratio (1 : /3. it is necessary that continued multiplication of one

will make it exceed the other. This explicitly bars division by zero (or its

geometric equivalents).

and it prevents ratios from being formed across

species because there is no multiplication of ;I line which will allow it to

exceed an angle or surface. But proportions can be constructed from ratios

whencvcr the criterion in Definition 5 is satisfied. so it makes sense to say that

the ratio between two given lines is the s;mic as that between two given

spheres. cvcn though the lint and sphere arc incapable of direct comparison

with one another. This theory of proportions is put to use throughout classical

geometry. with the standard form of a problem being that of finding the ratios

and proportions between various geometric magnitudes. One of the most

significant kinds of problems approached with thcsc methods was the

problem of constructing ;I square whose area is equal to that of a given figure:

such problems were known as quadraturcs. and the difficulties encountered in

solving them led to the development of infinitesimal mathematics.



One important style of proof licctiscd hy the classical conception ot

magnitudes is the “method of exhaustion”

in which ati ~~tik~iowti geometric

quantity (or the ratio between two ut~kt~owt~s) is detct-mined hy considering

qucnces of know~n quantities which can hc ~mde to clifi’cr from the ~~t~kt~owt~

l7y an arbitrarily small amount. This technique set the atand~~rd of rigor for the

scvelltcetlth cclltlll’ ‘.

and

it is itiipot-tatit that we survc’\~ it hriefl\, hcforc

emharking on

;I

study of ~e~etiteetith-century methods.

The

foundation of the method of eshaustion i\ Proposition

I of Book X 01

Euclid’s E‘let~rc/r~s. This proposition t’ollows imtncdiatel~ from lkfinition 4 of

Book V. and its

115;~‘

s cssetitial in the ccjursc of an exhaustion pi-oaf

wheit ;I

sequeticc of ~rpproximations is shown to differ from a given niagnitudc by Icss

than

any

assign4 amount. The proposition

r-cads:

The general ptxxxdurc in at1 cshnustioti proof is to begin \vith

uppct- and

lower bounds for an unknown magnitude and then to provide ;I tiiethod foi

systematically improving these h~utids. In the cxc of ati exhaustion proof to

determine the cluadraturc of ;I figut-c.

the initial I~~~~ncls will hc gi\,cn in the

form of inscrihcrl and circumscribed f‘igures. I‘hcti ;I nicthod for improving

these bounds must hc cxhihited. tgically hy inscrihin,

0 and circuniscribiti~

two new figures which reduce the retiiaitidcr hetwccti the hounds and the

u~iktiowti hy

niorc than half. If the

tnethocl c;iti bc

itet-ated. it gctict-atcs ;I

sequcncc of impi-ovai ~il’I~roxitii~itioiis which [ t3y Euclid (X. I )I will dift‘er

from the LIII~I~OWII by Icss than

any

gi\,cn magnitude.

As an

cuample of

this

ptxndurc. the iircii of

;I

circle

can

lx

l~o~indcd above

and hclow hv inscribing and circutiiwrihitig squat-es. (Figures I and 2.)



Philosophy of Maths in Seventeenth Century

221

If we double the number of sides in our approximating figures, we can

reduce the difference between the area of the circle and the area of the

approximations by more than half. Moreover, by continuing to double the

number of sides in our approximations we can form two sequences of

approximations such that successive terms of each sequence reduce the

remainder by more than half.

Thus, by Euclid (X. 1). the difference between the area of the circle and the

area of the inscribed and circumscribed polygons can be made as small as

desired. When such a “compression” of the value of an unknown between two

sequences of known quantities is attained, the classic exhaustion proof is

rounded off by a double

reductio ad uhsurdum

which shows that the unknown

value can be neither greater nor less than a specified amount.

Two points should be stressed here. First, there is no need to consider

infinitely small quantities in the course of an exhaustion proof. Throughout

the course of the proof we make reference only to the finite differences

between finite magnitudes, and the procedure of “exhausting” the area of our

unknown requires only a finite number of steps.7 This avoidance of infinitary

considerations is rooted in the definitions of magnitude given in Book V of

the Elements and is characteristic of Greek geometry. Not surprisingly, it is

on this account that the mathematical analysis of the seventeenth century

differs most substantially from the Greek model. The second point that

should be clear is that a fully worked out exhaustion proof is a very

cumbersome (not to say torturous) chain of argument. In full dress, an

exhaustion proof requires the specification of a method for generating

sequences of inscribed and circumscribed figures which provide successively

better approximations to the unknown, and it is in general difficult to achieve

this. Moreover, the required double reductio ud uhsurdum can make the

proofs of even the most elementary results unmanageably long and intricate.

These two features of the method of exhaustion were widely acknowledged

by mathematicians of the seventeenth century. who agreed that exhaustion

proofs were paradigmatically rigorous but complained that the technique

was both cumbersome and difficult to apply to any but the most simple cases.

Indeed, the main attraction of infinitesimal methods was that they could be

applied with ease to a whole class of problems whose solutions could not be

‘That only ;I finite sequence of approximations is requirud for an exhaustion proof follows from

the fact that the process of approximating the unknown can he thought of as heginning with the

specification of :I dcgrec of accuracy within which the value of the unknow~n is to be calculated.

By Euclid (X, 1). this degree of accuracy an he attained. and when we reflect upon the fact that

the level of accuracy in our approximatton is arbitrary. the proof by reclwtio irtlU~.WU~IUNUI

be generated. But in no case does the “exhaustion” of the unknown by ;I sequence oi

approximations require the completion of an infinite proccxb,



obtained through the traditional exhaustion pro~f.~ In fairness to the

champions of infinitesimal mathematics,

it must be acknowledged that the

traditional formulation of the exhaustion method renders its application all

but impossible. In particular, the requirement that a method be given for

constructing sequences of inscribed and circumscribed figures makes the

technique difficult to apply generally. It is straightforward (if laborious) to

determine the nrca of the circle by exhaustion. and a similar situation obtains

for other simple figures, but the cast is essentially hopeless if WC try to take

the strict formulation of the traditional exhaustion proof and use it to

determine areas or arc lengths for more complicated curves. The frustrations

encountered in trying to apply exhaustion techniques to more complex

problems eventually inspired the development of infinitesimal mathematics.

which ranks as one of the most philosophically interesting episodes in the

history of mathematics.

Having seen some of the problems associated with the traditional

exhaustion proof, let us now consider the basic features of infinitesimal

mathematics. The main idea behind infinitesimal theories and the

principal motivation for their introduction into seventeenth-century mathe-

matics can bc gathered from the consideration of a relatively simple problem.

that of determining the area of a circle in tams of its radius and

circumference. Take a circle with radius r and circumference C. We begin by

observing that the area of a circle can be interpreted as the sum of the areas of

the equal sectors 0,. (r2. qj.

o4 (Fig. 3). By re-arranging the sectors, we

can construct a “pseudo-parallelc,grnm”

whose arca c n bc approximated as



rc. If we then increase the number of sectors into which we decompose the

circle, our pseudo-parallelogram will more closely resemble a true parallclo-

gram with area VYC.

Now it seems natural to assume that if we could take an infinite number of

sectors, we could get an exact result for the area, since our pseudo-

parallelogram would become a true parallelogram in the infinite case. In

doing this, we effectively treat the circle as composed of an infinite number of

isosceles triangles, each of which has an infinitely small (or irrfirritesimrrl)

base (Fig. 4).

Ei,q. ‘l.

One fundamental question to be asked here is: what is it for a triangle to

have an infinitesimal base? The answer is not at all obvious because we want

to deny that the base has a zero length, but at the same time want to deny that

its length can be measured by any positive real number. The reason for

denying that the base has a length of zero is that such an admission forces us

to regard the circumference of the circle as an infinite sum of the form

0 + 0 + 0 + 0 + . which is equal to zero - a most unwelcome result when

we consider that we have assumed the circumference to be non-zero. But to

admit that the length of the base can be measured by any positive real number

forces us to deny our previous claim that the circle is composed of isosceles

triangles, because there will be a tiny area E which is left over from each of

our sectors if the base of the triangle has length o (Fig. 5)

It thus appears that we are led to regard infinitesimals as quantities greater

than zero but less than any finite real number. This characterization may

seem bizarre and inconsistent, arousing suspicion that talk of infinitesimals is

simply incoherent.” There was no universally accepted theory of infinitesimals

“As it turns out. a consiatcnt theory of infinitesimals can be developed by the techniques 01

contemporary model theory. In the modern theory. infinitesimals appear as “hypcr-renl”

numbers in certain non-standard models of the axioms for the real numbers. The relcvancc of the

modern theory to issues in the scvcnteenth century is minimal. although Abraham Robinson (the

founder of modern “non-standard

analysis”) took

his theory to bc a vindication of seventeenth-

century doctrines. See Abraham Robinson. NW-S/rr~cltrutl A~r/vt;.s (Amstsrdam: North-

Holland. 1966).



in the seventeenth century. although several authors did spcnk of taking

“infinitely small” quantities in the course: of solving problems. One

way to

describe the infinitesimals in this period was to characterize them as

magnitudes which stand in

the same

ratio to any finite number as any finite

magnitude stands to infinity. The intent of this definition is to capture the two

properties WC’noted above as characterizing the infinitesimal - being greater

than zero and less than any positive real number. Treated as ratios of finite

magnitudes to an infinite magnitude, infinitesimals can be

regarded as greater

than xro (bccausc they are

ratios of

posiri~~c

magnitudes) and at

the

same

time less than any

positive

real number (because any positive real number

can

hc

cxprcssed

as

the

ratio of finite magnitudes). The main difficulty with this

attempt to define infinitesimals is that it t-quit-es

us to make

Sense of the

notion of ratios between finite and infinite magnitudes. Given the I-estrictions

placed upon the theory of magnitudes in Book V of the f3etwttt.v. it is clear

that such ;I theory would require ;I substantial break with the classical

tradition as well as the development of a positive theoretical justification for

the use of infinitesimals.

The prohibitions against the use of infinitesimals which arc so fundamental

to the classical conception of magnitudes flow quite naturally from the

philosophical account of geometric reasoning which

was

widely held in the

seventeenth century. According to this standard view. geometrical knowledge

is obtained by demonstrations, and such demonstrations

have

clcarlq

specified criteria of rigor. To count as rigorous a ticmonstration must proceed

synthetically and begin with axioms which ;IIW transparently true. Addition-

ally, the objects employed in the demonstrations must be cledy conceived

and

the theorems must he derived by truth-preserving rules of inference. Such

an account is. of course, something of ;I commonplace and is well summarized

by Barrow at the conclusion of the fourth of his

M~rfl~ctnntic~nl Lcc~tlrrcs:

But I am afraid you

will Ixgin to grow

weary

with the Ixngth anti l%ollslty of

thk

C’ompC~on. From whence

notwithst~lnding it

may

in ~nic Sort appear what

Method of I)erllo,l.Ft,.trtio,l is

t~scdhy

Mathem~ltici;unz: which is such. that the); only

take those Things into Consideration. of which they have clear and distinct

Ideas.

designing them by propel-, adequate.

and invariable Names. and prcmissing only a

few Axioms which are most noted and cct-tain to investipte their Affections and



Philosoph_v of Maths in Seventeenth Cermq

22s

draw Conclusions

from them, and agreeably laying down a very few Hypotheses

such as are in the highest Degree consonant to

Reason and not to be denied by any

one in his right Mind. In like manner they assign Generations or Causes easy to be

understood and readily admitted by all: they preserve a most accurate Order, every

Proposition immediately following from what is supposed and proved before, and

reject all Things howsoever specious and probable which cannot be inferred and

deduced after the same manner.“’

This account of demonstration renders the use of infinitesimals problematic

precisely because an infinitely small quantity is hardly the sort of thing that we

can conceive clearly and distinctly. since our conceptual powers do not readily

lend themselves to the task of framing an idea of a magnitude that is greater

than zero but less than any positive real number. Of course. there is a long

and powerful tradition which declares the infinite to be incomprehensible and

bars the use of infinitely great or infinitely small quantities from mathematical

reasoning. This tradition has classical roots. but expressions of the same

attitude can be found in the writings of many figures from the scventcenth

century.” Because the standard for rigor in the seventeenth century thus

requires that theorems be presented synthetically as the consequences of

transparently true axioms. and such axioms as those in Euclid explicitly bar

the use of infinitesimals, it then becomes a matter of some interest to

determine how the proponents of infinitesimal mathematics justified their use

of new methods. It is this matter which will concern us for the remainder of

this paper.

II

Although infinitesimals present some rather difficult conceptual problems.

their use was of great importance in the development of seventeenth-century

mathematics. By relying upon infinitesimal proof techniques, mathematicians

of the period obtained an astonishing wealth of new results. even though the

attitude toward infinitesimals was frequently one of ambivalence. This

ambivalence can be seen clearly in the work of Buonaventura Cavalieri. one

of the leading mathematicians of the seventeenth century.

“Galileo cxpresscs the opinion that the inlinircly large ancl inl’initely hnx~ll arc inherently

incomprehensible in his Ttvo Nebv .Scio~~~.~, Stillman Drake (trans. and cd.) (Madison.

Wisconsin: University of Wisconsin Press, 1974). pp. 33-36. In a similar vein. Arnauld lists

among his “important axioms which can scrvr as the basis for great truths” the axiom that “The

p. 324. Barrow’s attitude is typical when he declares in the ninth of his Marhemuricml Lecfrms that

we cannot comprehend the infinite and that the only magnitudca which are admissible in

mathematical demonstrations are finite: he attributes thebe opinions to Aristotle and relies upon

his authority to settle the matter. Barrow. Mtr/hrr~~cr/icrr/ Lec~urm. pp. 142-143.



Cavalieri’s major work, C;co/~ctr.irl irltli~‘i.si/,ilihlrs COH~HIIO~~~I fto~‘o

qi1dom rrttiorlc~ p~~moltr lh.?i).”

empoyed

an infinitesimal tcchnicluc

known as the method of indivisible in the solution of classical problems of

finding areas and volumca. The key to C’avalicri’s Ireatrnent of cluadraturc4 is

his concept of the indivisiblcs of I given figure. which hc introduces in the

second book of the Gc~r~c~ir/. If we take the figure ARC‘ with 1x1s~ AH and

then pass ;I line parallel to the lint OY

(called the wglrltr of the

figure in

Cavalieri’s terminology) toward C’, then the individual lines produced by the

intersection of the line and the

figure A ( arc called *‘the indivisiblcs of the

figure”. (Figure 6.)



intent on presenting his theory in a manner that will take advantage of an

intuitive understanding of continuous motion and thereby sidestep some of

the more difficult questions concerning the nature of infinite magnitudes.

Nevertheless. it is not obvious how WC arc to go about comparing the

indivisibles of two figures or how such comparison can help to solve

quadrature problems without implicitly assuming that the figure is literally

composed of an infinite number of infinitely small parts

Cnvalicri was by no means unaware of the conceptual problems posed by

his non-classical approach to geometric problems. but he hoped to resolve the

difficulties by developing a treatment of quadrature that would work success-

fully under any resolution of the philosophical problems involving the

infinite. One of the most important features of Cavalieri’s method is that he

treats the indivisiblcs of a figure as an entirely new species of geometric

magnitudes. What he proposes is that the classical conception of magnitudes

contained in Book V of Euclid’s

E1emet~f.s

be expanded by introducing the

indivisibles of a figure as a species of magnitudes on a par with lines, angles,

arcs, surfaces, and solids. True to this proposal, Cavalieri’s principal line of

attack in the solution of quadrature problems is to compare the ratios

between the indivisibles of two figures and then to use these ratios of

indivisibles as a means of establishing a proportion which will determine the

ratio between the areas of the figures.“’ Thus. a fundamental part of

Cavalieri’s program is to state principles which will allow the indivisibles of

a geometric figure to be fitted into the general theory of magnitudes pre-

sented in Book V of Euclid’s Elements.

Cavalieri’s eagerness to present his theory of indivisibles as an extension of

the classical conception of magnitudes becomes clearer when we observe that

the very first theorem he proves after introducing the concept of “all the

lines” of a figure is a result which declares that the indivisibles of ;1 figure are

magnitudes which have a ratio to one another.” His desire to overcome

“Indeed, in a corollarv to Theorem III of Book II . he dcclarcs “It is clear from this that when

WC want to find what ratio two plane figures or two solids have to one another. it is sufficient for

us to find what ratio all of the lines of the figures stand in (and in the case of solids what ratio

holds hetween all of the planes). relotivc to a given r~grtla. which I lay as the great foundation ol

my new geometry.” In the Latin: “Liquet ex hoc. quod ut inveniamus. quam rationem hnbeant

inter se duz figure planz. vel solida. sufficict nohis repcrire, quam. in figuris planis. inter sc

rationcm h&cant earundcm omncs linetc. et. in figuris solidis. carundem omnia plana. juxta

quamvis rcgulam assumpta, quad novae huius me2

Geomctriz. veluti m;iximum iacio

fundamcntum.” G~omrtriu, p. I IS.

“The theorem is Theorem I o Book II of the C;ror~~ricr and reads “All the lines ~cri ~nrr~i/rrs

of any plant figures. and all the plants of any solids. al-c magnitudes having a ratio to one

another.” In the Latin: “Uuarumlihet planarum figururum omncs lincae rccti transitus. et

quarumlibet solidarum omnia plana. sunt magnitudinea inter se rationcm hahentcs.” GrornctCtr.

p. 10X. The proof of the theorem begins with the postulate that the indivisihles of congruent

figures are congruent and then proceeds to take up casts of non-congruent figures hy considering

remainders and forming sums. In this respect. the rc\ult is rather question-bcg+ng. because the

postulate that the indiviGl>les of congucnt figures arc congruent already ;~ssumcs that indivisible\

can be compared with one another and ordered by ;I Icss-than relation.



traditional strictures against infinitesimals is made explicit in the scholium to

the theorem:

Perhaps one could doubt this demonstratior~, not properly undcrstunding how an

indefinite number of lines or planes can bc ,judgetl to he a kind of thing% which

I

call

aII the lines of the figure or all the plants of the solid. or how AI the lines of figures

cm he comp~md with one another. Becanse of this. it seems that I diould indicate

that WIICII I consider all of the lines or all of the plumes of some figure. I do not com-

pare their number. which we do not

know. hut

only the magnitude. which is

ecp;~I

to

the ;mx occupied by these

same lines and

is congruent to them. Because this

area is containd in boundxies. so therefore the magnitude of aII the lines is con-

taned in the wnw boundaries. and therefore they can tw added or subtracted.

although we do not know their number.

I say

that this is enough to

make them

conipmble to

one

another. for otherwise neither could the

sm~e

areas of the

t’igures bc comprable to one another.“’

The general thesis proclaimed here is that the method of indivisibles can he

vindicatccl by bringing it within the purview

of the clnssical thcorv ot

magnitudes. Cavalieri returns to this thesis whenever hc addresses the

question of the reliability and rigor of hi4 method. Thus. in the preface to

Book VII of the

Gcottzrtricl

hc addresses philosophical reservations about his

method hy referring hack to Book II. Theorem I. claiming that by treating the

indivisihles of ;I

figure as ;I

species of magnitude he has avoiclcd the problem

of the composition of the continuum and has not had to treat the indivisibles

of a figure as infinite ratios.”

AS WC shall XC, Cavalieri was not the only

mathematician of the seventeenth century to argue that infinitesimal methods

could be shown to bc rigorous in this way. hut his willingncas to attempt to

found his method on the classical theory of magnitudes shows that hc was

certainlv not prepared to repudiate the classical standard of rigor.

In addition to this approach.

Cavalicri explored an indirect method for

,justifying his technique. This second approach is to solve ;I number of

traditional quadrature problems with his method of inciivisibles and compare

these results with those obtained classically. By observing that the same

results arc obtained. Cavalicri argues that the method must be sound:



It seems worthwhile to observe in confirmation of what we have assumed how

many things which have been shown by Euclid. Archimedes, and others are equally

demonstrated by me, and my conclusions agree completely with theirs. which can

be an evident sign that I assumed the truth in my principles. Of course.

I

know that

truth can be deduced sophistically from false principles, but

I

should think it absurd

that this could have happened to me in so many conclusions demonstrated by the

geometric method.‘”

Such an attempt at justification will not convince anyone who is deeply

sceptical of infinitesimal methods. but it did help to gain acceptance for

Cavalieri’s methods and was a style of presentation favored by others who

were anxious to minimize the differences between the method of indivisibles

and classical proofs. I”

Another way to see how concerned Cavalieri was with the rigor of his

method is to note the sort of justification he does tmt give for his most

fundamental results. A firm believer in infinitesimals who had no qualms

about the soundness of his technique or about the repudiation of classical

standards of rigor could simply assert that quadrature problems could be

solved by forming infinite summations of infinitesimal elements, but we do

not find Cavalieri taking this approach. As an example, consider the principle

known today as “Cavalieri’s Theorem”, which states:

If two figures have equal altitudes. and if sections made by lines parallel to the

bases and at equal distances from the bases are always in a given ratio. then the

areas of the figures are in this ratio as

well.‘”

We begin by taking the figures ABC and DEF with regulu OY (Fig. 7). We

then consider an arbitrary line X,,Y,,, parallel to the regulu and intersecting

the figures in two line segments K L and M N . The theorem states that if

the ratio

K L : M N

is constant for all lines in the two figures, then

ABC :

DEF = K L : M N . The reckless proponent of infinitesimals could prove

this theorem by arguing that because the figures are sums of indivisibles.

when we are given that the terms in both sums have a given ratio, then the

sums themselves must stand in the same ratio. Cavalieri, however, explicitly

‘S”Non inutilc autt‘m mihi vidctur e~,c animadvcrtcrc pro huius confirmntione. hoc pro vcro

suppositio, quam plurima. quz ab Euclide. Archimede. et aliis oatcnsa aunt, ;I me paritcr I‘ukc

demonstrata. masque conclusioncs ad ungucm cum illorum conclusionibus concordartt. quad

cvidens Signum rssc potest, me in principiis vcra assumpsisse. licet sciam. t2t t‘x falsis principiis

sophistic6 vera aliquando dcduci posse, quad tamen in tot. et tot conclustonihus. methodo

geomctricn demonstratis mihi accidiasc ahsurdam putarem.” C‘;ivalicri. Geon~orirr. p. I I?.

“‘For example. Torricclli Collows a similar path in his treatise DC dirm~sior~r p~~hok. when

hc gtves numerous solutions to the quadrature of the parabola. including an argument tram

indivisibles. See Evangelista Torricelli. DC dirmwsiom~ pnrholw. in 0~~~r~1 ~kornc~lrictr Z vols

bound as one (Florence: 1644). vol. 2. pp. I-M.

“‘I state the theorem in the two-dimensional case for simplicity, hut the extension tc) higher

dimensions should he olnkus. Cavalieri‘s version of the theorem appears in Ckon~~icr, p. I IS.



230

avoids such arguments. He does give ;I prcjof of the theorem in the C;cor?~crr-irr,

but it depends upon ;I more cautious treatment of indivisibles. The details of

the proof need not detain us here. beyond noting that they art’ presented in ;I

manner that is clearly designed to avoid claiming that ;I figure is compos~ci of

indivisihles.”

The final evidence for (‘avalieri’s sensitivity to cluestions of rigor

can he

found by inspecting the \ cventh book of the Gcwnctrirl. which is devoted to

the presentation of an alternative foundation c,f the method of indivisiblcs.

Book VII is ;I late addition to the G’cot?lctritr and is intended to defend the

method of indivisillles against the charge that it is insufficiently rigorous. In it,

Cavalieri proceeds to develop an alternative approach to the central theorems

that will depart less radically from accepted philosophical opinion.

WC can call the method employcd in hooks I-VI of the (;cor~ct~itr the

“collcctivc method”. while the second approach

can he

called the “ciistrihu-

tive method”. The diffcrencc: hctwecn these two approaches lies in the kind of

comparison that is made between the indivisihlcs of two I‘igurcs. Given two

figures, we can compare their inciivisibles collccti~~cly by considering the

whole collection of lines in the first figure and comparing it to the whole

collection in the second: or WC

can

compare them

tli.sraihr,ti~,(,I~, ly

comparing

separately correspondin g lines in

each

figul-c. Such distributive comparisons

permit the dcrivstion of the central t&orcms from Books I-VI of the

Gconlctriu.

and Book VII is &voted to working out the appropriate proofs.

Cavalieri explicitly states that he dcvelopcd the distributive method in

order to show that his results could he obtained without relying LI~OII

principles that arc inconsistent with philosophical scruples concerning the

introduction of infinitary concepts into gcomctry. In the preface to Book VII

he declares:



It is by no mean\ unknown to me that many things concerning the composition of

the continuum and also concerning the infinite are disputed by philosophers. and

this will perhap\ seem to many to lx prejudicial to my principles. in

consequcncc

of which

they will surely be hesitant, because it would appear that the concepts of

all the lines or

all the planes are chimerical. as

if they were more obscure than

inscrutable darkness.

”

He then declares that by introducing the distributive method he will have

obviated the last philosophical objections to his theory. because he

can show

that his results can be obtained without treating geometric magnitudes

as

containing infinitely many indivisible parts. All that is required for the

distributive method is the admission that the ratios of lines contained within

figures can be compared, and this should present no problem for those

worried about the introduction of infinitary concepts into geometry.

Although it is less open to objection, the distributive method plays ;I rather

minor role in Cavalieri’s work, being used to show how key results obtained

by the collective method

can also be

demonstrated distributively. Not

surprisingly, the distributive method is more cumbersome than the collective

method, and Cavalieri relegated it to subsidiary status because it does not

provide the relatively simple approach to quadrature problems that is

characteristic of the collective method.

The broad outlines of the conflict between methodological scruples and the

demands of seventeenth-century mathematical practice can be seen quite well

in Cavalieri’s work. His introduction of a new method helped to solve

outstanding problems. but he was clearly concerned that his solutions would

not meet then-current standards of rigor. But rather than repudiate the

reigning conception of rigor. he tried to accommodate his new technique

within the traditional theory of magnitudes and worked out an alternative

presentation of his method which involved a less radical departure from the

traditional standard. What makes Cavalieri’s work philosophically interesting

is just this kind of tension. He appears torn between the desire to exploit a

completely new proof technique in the solution of long-standing problems

and the desire to adhere to the traditional standards of rigor embodied in

Book V of the Elcmerzts

and

the traditional exhaustion proof. This manifest

concern with justifying his method of indivisibles suggests that he has been

dealt with rather harshly by historians of mathematics who have portrayed

him as uninterested in the question of rigor.‘.’

‘2”Haud quidrm me latct circa continui compositionem. nccnon circa infinitum, plurim: :I

philosophis disputari, qua meis principijs ohesse non paucis fortasse vidcbuntur. proptcrea

nemp? hzcitantes: quad omnium lincarum. seu omnium planorum. conceptus cimcrijh veluti

ohscurior tcnehris inapprchensihlis videatur.”

Cavalieri. (;c~mVritr. p. 4S2.

“Thus. lor example. Boy-r ~ccms to IX quite far ON the mark when hc imagines that .‘thc lllct

that Cavalieri paid so littlc heed to the demands of mathem;ttic;d rcsor made geometers chary 01

accepting the method ol‘indivi~ihles ax valid in demonstrations, although they employed it readily

in preliminary invcsti~ations”.

Boycr.

T/w Ifitlor:\, of I/W ~‘crlc~rrlu~.

. 123.



111

Cavalicri’s new approxh to the solution ot gconwtric prc~hlcms provoked

rcsponscs which i-ungcd from hostile ci-itique to cnthusktic acceptance,

and

these responses show an interesting interplay of mathematical and philo-

sophica l issues. Although it is beyond the acope of this work to document the

entire history of the method of indivisihlc\ . there arc sonic important thcmcs

which cmergcd in the dccadca after the

publication of the ;cwttwtt-irr

which

arc cspccially worthy of note. C’avalieri’s dcparturc from classical methods

met with criticism from those who upheld the traditional account of geometric

I-casoning and

found his talk of

“all the lines” of ;I

figure to bc inconiprchen-

sihle. Foremost among thcsc critic\ ; was the .lcsuit Paul Guldin. whose

~‘c~~trohr~ui contnincd an cxtendec l attack on C’avalicri‘s methods.”

The tails of Gulclin’s attack on c’avalici-i’s methods

riced not

detain us

here. hut it is important to ohscrvc that his critique clepcnds fundarncn~a~~y

upon the thesis that neither infinitely Iargc nor infinitely small totalities are

acceptable objects of

mathematical investigation. Indeed. Guldin

was

convinced that the

concepts cmploycd

in C’a\,alieri’s approxh to quadratures

were coniplctelv unintelligible and coiiId

never form the basis of a

philosophically

;;cccptahlc

geometry. He

objects to Cavalicri‘s definitions of

“all the

lines” or

“all the

planes” prociuced by the motion of

: I

figure,

declaring:

Surely no grometer can admit these three ciefinitivna.

and

much less the corollaries

which arc connected with such definitions. The geomcter expounds the matter in

a

word

and says that points. lines.

and pl ants in motion arc al ways

in

a

different

pl ace. which the philosophers also concede of

every

mobile thing. and 40 a point

does not l eave behind ; I point but ; I line. ; I

line leaves a surface. and ;I surface 2

solid.”

He also argues that C’avalicri’s attempt to bring

his theory of indivisibles

within the purview of the classical theory of magnitudes is hopclesslq

misguided hecausc it requires us to make sense of

the idea of

; I ratio between

two infinite totalities. Thus. in response to Theorem I of Rook II of the

Geometria (where Cavalieri c laims that the indivisibles of ; I figure arc

magnitudes). Gulclin responds:

““Verum has tt-c\ Dclinitionc5 nullw dmittcre potot Gcomctrki. ct multo minus Coroll;~ri:~.

qua dclinitionilx~b anncctuntut-: Vcrho \ c\ c‘ cupcdlt Cieomctra.

et cllclt. pu”‘tunl. I l ncam

planum, in motu empc‘r c‘\ w in maiori loco \ c’. clued ctiam Philosophi concdunt. dc omnibus

mohilibua. ct sic punctum post sc‘ non relinyuit. punctum ad lineam: linca. superficicm;

supcrficic5. solidurn.”

Gultiin. (‘c,/rlr~,hrr~\ ,c.cl.1’. .Nl.



This proposition is completely false: and

I oppose

this argument to it. All the lines

and all the planes of one figure and another are both infinite. But there is no

proportion or ratio of an infinite to an infinite. Therefore. etc. Both thr major and

minor premises of the argument are clear to all geometers. and so do not need

many words. Therefore. the conclusion of Cavalieri’s proposition is false.“’

In a similar

vein, in commenting upon the third proposition of the second

book of the Gcornetriu (which asserts that the ratio between the areas of two

figures is the same as that between the indivisibles of the two figures), Guldin

writes:

To

which we reply absolutely that figures can have ratios to one another, but not all

the lines or all the planes of one to all the lines or all the planes of another. This is

[a ratio] of an infinity to an infinity.”

These objections to Cavalieri’s procedure amount to a reassertion of the

strongly finitistic standpoint of classical geometry. Guldin and other critics of

the method of indivisibles took Cavalieri’s introduction of the language of

indivisibles as an unwarranted departure from the only acceptable ontology

and epistemology for geometry, and they remained unimpressed by his efforts

to reconcile the new method with the accepted philosophical interpretations

of mathematics. Guldin’s hostility to the method was so strong that he

remained unimpressed with Cavalieri’s “distributive method” of indivisibles.

which he found to be as inadmissible as the “collective method”.

At the other end of the spectrum of responses to Cavalieri were those who

adopted the method of indivisibles unhesitatingly, of whom Wallis is ;I fine

example. His enthusiasm for the method of indivisibles was motivated in large

part by the fact that it could be applied more easily than the classical

techniques, and he treated the foundational issues concerning the method as

having been settled by Cavalieri. Thus, in his 1670 Mc~chnnicu; sive de Motu

Tractutus Geomrtricw he writes:

Definition: It is understood that anv continuum (according to the Geometry of

lndivisihles of Cavalieri) consists of an infinite number of indivisihles.‘”

““‘Hxc Propositio ahcolute negatur; ct hoc oppono argumcntum: Omncs I~ncx et omnia

plana. uniua et itlterius figunr hunt intifiitu et inlinita: bed iniiniti ad infinitum nulla est proportio

sivc ratio. Ergo. Tam major quam minor clara at spud omncs Gcometras. ut pluribus vcrhis non

indigeat. Ergo Conclusio Cavalicrianre Propositionis falsa est.” Guldin. C‘r~rrohnrwu. p. 341.

““Ad quam absolute rcspondemus Figuraa posse haherc ad invicem rationcs. non autcm

omncs lineits aut omnia plani unius.

ad lineas omncs aut omnia

plana

altcrius. hoc e>t infinit i ad

infinitum.” Guldin, C‘c,r~/~ohrrr?c.rr.p. 34.3.

2S”DEFINITIO: Continuam quodvia (secundum Cavalierii Gcomctriam Indivisihilium)

Intclligitur, cx Indivisihilihus numcro inl’initis conrtare.”

John Wallis.

Mzc~hrrr~ic~tr:SII ’C e M orrr.

Trtrctcrt~~ ~w tnrt r ic~~c.s

n:

JoI~N~I I I I~ tr l l i c S. T. Ll . O~C~I.N trr l wmrriur 3

vols (Oxl’ord:

lhY.i-1609). vol. I, pp. 571-1063, p_ 645.



He clahorates upon this definition and links the method of indivisibtc\ with

the classical tcchniquc of proof hy exhaustion when he declares:

A4 from 11

infinity of point4. ;i line; ;I sul-facc

from a11 nfinity of lines. and a v)licl

fi-oiii ati infinite numhcr of

surface: v) also from

a11

Infinity of

~eiiilx~ral momenh.

tinic. etc.

That is. Cram homogeneous prficles. infinitely small and infinite in number.

and equal in at least one dimension

Thk

mems

according to mathcmaticat rigor) there

can at teaxt tx

inscribed

OI-

circumscribed. OI-otherwise adapted, vmcthing co~~~poscxi of partides of thi\ sort

which wilt differ from the given thing hq a11nfinitely small quantity. OI- ;I qumtit):

tern than any given quantity.

For

example:

iii the

pcripheiy of :I circle a cui-vc wilt he

co~iiposu~ of an

infinite

nurnher of arcx and will coincide exactly with the periphery. Hut there cm tw

inscrihcd within the same periphery ;I line cornpod of an infinite number of suh-

tenses.

OI-

circuni~crihed about it

a tine composed of ai

infinite numl~e- of tangents.

\uch that the inscrihd line wilt he tcss than the periphery. or the circumscribed tine

wilt he greater than the txxiphery ty

:I difference

which is less than

any given

al110L nt.~”

It is worth notins that Watlis

has gow beyond

C’avatiel-i

hct-c

when he

cteclares that the continuum is actually

comp~wd of

int’inituly many infinitely

small parts. Whew Cavalieri

had ken

cautious and ottcmptcd to prcscnt his

method in

;I

manner that ~oiild not

clcp~iid ~ipoii the

controversies

concernins infinite divisibilitv and continuity. Watlis tahc\ C‘avaticri to have

resolved aIt of the important questions concerning the foundations of the

method of indivisibles. In fact, Wattis takes the method of inctiviait~tca to hc

equivalent to the classical nicthoct of exhaustion.

In

Chapter 74 of his 7‘rc rtti.w

of Aic yrOr rr.

tic

xscrts that:



And in his Me an . he contrasts the method of indivisibles with the

classical technique of exhaustion, declaring:

And it appears that this doctrine of indivisibles (now everywhere accepted and,

after Cavalieri. approved by the most celebrated mathematicians) rcplaccs the

continued adscription of figures of the ancients: for it is shorter, nor is it less

demonstrative. if it is applied with due caution:”

Although Wallis’ acceptance of the method of indivisiblcs was complete. his

reference here to the “due caution” with which the method must bc used

shows that he was aware of some important questions concerning its

reliability.

In fact, there were objections raised against the method which purported to

show that it delivered the wrong results when applied to the solution of

quadrature problems. These objections were put forward by methodological

conservatives who, like Guldin, found the talk of indivisibles unintelligible

and argued that the method should be rejected not merely because it was

founded upon a false metaphysics, but also because it simply was not a

reliable procedure for solving geometrical problems. A brief account of one

such objection and Barrow’s response to it will bring to a close this section of

our investigation of the problem of rigor in the seventeenth century.

Barrow praises the method of indivisibles in his Maf/~etnuricxl Lccrurcs.

referring to it as “that excellent Metlzod OJ Itzdivisihles. the most fruitful

Mother of new Inventions in Geometry. which was not long ago published

and applied to common use by Cavallerius”,3’ and later speaking of

“Cavalieri’s Method of Indivisibles. which has been praised before, but can

never be sufficiently praised”.”

But in his Gmtrwtrical Lectures. Barrow

expresses some concern with the reliability of the method in the investigation

of quadratures. In his second lecture, he states an objection to the method

which he credits to Andreas Tacquet.j’ The point of the objection is that if we

take the cone DVY with axis VK, we can consider the lines ZA, ZB. ZC. etc.

drawn perpendicular to the axis (Fig. 8).

If we assume that the right triangle VDK is composed of these lines, then

the cone itself will consist of the collection of circles which have these lines as

radii. The objection then states that the surface area of the cone must,

““Atque hanc. de Indivisihilibu~. doctrinam (nunc passim reccptam. atque. post Cava-

lerium. ;I

celcberrimis Mathematicis approbatom) pro Vetcrum continua figurarum Adscrip-

tione. substitucre visum cst; ul hcrviorem; ncc tamcn. minus dcmonsrntivam. si debit;1 cautione

adhibeatur.‘. Wallis. Mrcl~unic~r, p. 646.



according to the method of indivisiblcs. 1x2 cqual to the sum of the

peripheries of these circles. or equivalently to the arca of the circle with

radius DK. But this is not the case. and it secrns that the method of

indivisibles has been shown to product false results and is therefore not to be

employed in the solution of geometrical prohlcms.3i

Barrow does not find this objection convincing. but rather claims that it is

founded upon

a failure to appreciate the

manner in which the method of

indivisibles is applied. He qua:

But I reckon this to bc a wrong

ay of

Entrance

into

the Calculation: for when we

are computing the Periphery of which Superficiea consist. we must not proceed

after the same manner as we do with the Lines of which plant Supert’icics art’ mxi e.

or the Planes of which Bodies xc formed. That is. the Peripheries constituting the

Curve Superficirs generated by the Motion of the line

VO must be looked

upon as

equal in Multitudc to the Number- of Points in the said gcncrative Line: because

there cannot he nmrc than one of these Peripheries, then what can pass through

cvcry Point of it: howsoever remote or near the Axis falls. for it is this alone

according to its various Kemotencss or Nearness and Position. that determine\ the

Mngnitude of the \ aid Peripheries. But the Multitude of Lines of which the Plant

I) VK is supposed to consist.

and of Planes of which the Solid

OVY

i\ made up. is to

he rated by the Number ot’ Points in the Axis VK: t’or there cannot more parallel

right Line5 or Planes perpendicular to

VK

be

contained within the Limits VA’ than

what are equal

in Multitude to those Points. In observing which

Dlffkrcttc~r

(carc-

fully to he minded) we shall avoid all Error.

rrttrl

itr ttty Opittiotr fi’ttd olrf flrc

Strptficir.s gcttcrutcd hv titc Rotcrtiott of

sccc~lt

ike ~‘~trws ~JJ I Wrry ~IIC tto.st r~rr.sy lrrrt

rltc Nrmrrc of thcsc 7’lrittg.s dtttit.s. “’



Barrow’s remarks here are startling indeed. His defense of the method of

indivisibles was designed to show that it is rigorous and can be used to solve

geometric problems, but his line of defense seems to put the method on very

shaky ground because it requires the assumption that the line VK has fewer

points on it than the line VD because VK is shorter than 1/D. This

assumption. however, is deeply problematic and is not likely to satisfy any

critic who finds the method of indivisibles insufficiently rigorous. To assume

that the number of points on a line is a function of its length is to abandon

geometry as we know it. since this assumption requires either that a line have

only a finite number of points on it or that there be a difference in size

between two infinite totalities, each of which has the power of the continuum.

That Barrow can employ such a defense of the method while simultaneously

claiming that geometers “only take those Things into Consideration, of

which they have clear and distinct Ideas, . . , and [lay] down a very

few Hypotheses such as are in the highest Degree consonant to Reason

and not to be denied by any one in his right Mind”“’ suggests that there is a

substantial conflict between his official criterion of geometric rigor and the

methods he found useful for the solution of geometric problems.

This tension is also manifest in his comments in the sixteenth of his

Mathemrrtical Lectures, where he discusses the nature of proportions and the

admissibility of infinitary concepts into the theory of proportions. He notes

that the classical theory of magnitudes does not allow any comparison or ratio

between geometric magnitudes of different species and dcclarcs this principle

to be essential to proper geometric practice, declaring “there is no Fault

greater in Geometry than to seek or assert the Proportion of Heterogeneous

Quantities to one anothcr”.“s But he immcdiatcly observes that the method

of indivisibles appears to violate this principle. and he is at pains to show that

the method is nevertheless acceptable:

It

is true

that

such

Expressions do

often occur

with

those that apply the excellent

Method of Indivisibles to the Solution of Problems or Demonstration of Theorems:

All those Parallel Lines are equal to such a Plane, the sum of these parallel Planes

constitutes such a Solid: but they explain their Meaning, and say they understand

nothing by lines but Parallelograms of a very small, and (pardon the Expression)

inconsiderable Altitude. and by Planes nothing but Prisms or Cylinders of an

Altitude not to be computed: Or at least, by a Sum of Lines and Planes, they

denote no finite and determinate, but an infinite or indefinite Sum equal in

Number to the points of some Right Line. But however, passing by this

Controversy. that Use can be no Offencc to our Doctrine; for this Precept is of

irrefragable Force in the Opinion even of those who embrace that Method with



Here again. Barrow stems to bc caught in an uncomfortable position. lie

seeks simultaneously to maintain that the classical requircmunt of dimen-

sional homogeneity is ;I principle of unshakahlc force. that it i\ sometimes

violated. but that these violations of the principle arc only apparent because

they involve infinite (or. at any rate, indefinite) summations. Moreover. his

account still requires the dubious

assumption that wc’

can coniparc the

number of points in different lines.

Whatever else may hc suggested hy the reactions to the method of

indivisibles, it can

hardly be dcnicd that both the proponents

and

opponents

of this new mathematical technique wc’rc aware of the philosophical prohiems

raised by the departure from the accepted criterion of rigor. Moreover. those

who dcfendcd the new mcthocis did not do so hy dismissing the classical

conception of rigor as inaciequatc. hut sought to find ;I reconciliation between

the accepted standards of rifol-oils proof and the new proof proccdurcs .

Another example of the Inethodolo~ical questions raised hy the introciuc-

tion of infinitesimals is Leibniz’ work on the cd~~ul~r.s Liif~~~rc tlfirrli.s.

Lcibniz

approached problems of constructing tangents and finding quadratures from

a

radically non-classical standpoint. The Lcihnizian formulation of the calculus

proceeds from analytic geometry

and

relics upon the concept of a differential

of

;I variable ~7 - written dy. The dil ‘fercnt ial of ;I variahlc \’ is the diffcr cncc

between two values of the variahlo

which arc infinitely close to one another.

Such differentials at-c capable of further division. 0 that wc‘ can form ;i

sequence of differentials of diffcr cntials, etc. The best way to underst and and

motivate t he Ixibni zian calculus

i\ to Iwgin hy ohser\ ing how finite

differences can hc uscci to give

approsiniatc solutions to problems 01

tangt31cy.

If WC take

the curve tr(-; (Fig. 9) Lvith ordinate _I‘

and abscissa .Y. wc can

approximate the tangent at any point

and the

area under the curve between

any two points hy dividing the

abscissa into ;I finite collection of cqual

suhintervals. say x,. x7. ,

_v,.

The tangent at point 0

can hc

approsi-

mated hy taking the line through the points J’; and

y, corrcspondin3 to the

points x3 and x,. Similarly. the quadrature of the curve can tx approximatcti

by summing up the rectangles whose

hascs arc the suhintcrvals of the

;i bscissa



239

Fig. 9.

Leibniz proposes to transform such approximations into exact results by

taking infinitesimal differences between the variables x and y. In the

infinitesimal case, we will have a “differential triangle” at point b whose legs

d.~ and dy will be formed by the differential increments of the variables x and

_Yat 6. Taken together, these increments will determine the slope of the

tangent at h as dylti.

Using the representation of the curve in analytic geometry. the problem of

constructing tangents is easily reduced to a simple algorithmic procedure.

Take as an example the curve generated from the equation

I’ = x7 + 3x + k

with variables x,~ and constant k and consider its “differential increment”.

Replacing y by (v + dy) and x by (x + do), we get the new equation

(_v + dy) = (X + du)’ + 3(x + dx) + k.

Expanding this equation we get

(y + dy) = x7 + 2x dx- + ti’ + 3~ + 3 - + k.

Subtracting this equation from our original one yields the increment

dy = Zw d_.r + d_? + 3ctu.

Dividing through by dx yields

dy/dx- = 2s + 3 + dx.

But because do is infinitely small when compared to 2x + 3, it can be

discarded, and we obtain the equation dyldx = 2x + 3 as the slope of the

tangent at an arbitrary point on the curve.

It thus appears that the Leibnizian formulation of the calculus is

straightforwardly committed to the use of infinitesimal magnitudes, and the

interesting question is whether Leibniz himself found any difficulty in relying

upon a method which departs so clearly from the classical standard. On this

point it is clear that Leibniz was deeply concerned with the rigor of his



approach

and that

his concern Icd him to explore several alternative lines of

justification when his calculus was challenged.

The most significant challenges to the

calculus raised during

Lcibniz’

lifetime

were by Hernard Nieuwentijt

in 1W4 and Michel Rollc in 1701.

Nieuwentijt’s attack came in the

form of

:I book

entitled C‘rttr.vitlcr-rrtiott.sotl the

pritlcip1c.s c?f‘rrtralysi.srrppiictl to itl /i’tlitelj~sttlrrll llrrrtltiticsrrtttl rttt the ltsc of the

di~~iwritirrl cnlcirllr.s iti

soh~itiL<

pwtlctricrrl pt-ohlcttr.\.“’ Rolle’\ critique was

part of a dispute in the French Academy of Sciences in 1701-1707 over the

foundations of the calculus. In reply to criticisms. Luihniz suggested diffcrcnt

ways in which the reasoning in the calculus could be \ inclicated. and it seems

worthwhile to consider them briefly.”

The first line of defense which Lcibniz adopted was to claim that

infinitesimal reasoning can he made rigorous hy appealing to general

principles which would license the introduction of infinitesimal magnitudes.

This way of justifying the calculus ciominatcs Lcibniz’ published response

to Nieuwentijt. where he tries to show that the calculus can he defended hy

introducing ;I new species of magnituclcs. namely those which arc incompa--

ably small. tic writes:

I think that those things are t qu; ~I not only whose difference is xbv)lutely nothing.

hut also whose difference is incomparably small: and although this difference riced

not he

callcd absolutely

nothing, neither is it ; I quantity comparable with those

whose difference it is. Just ;ts when you acid ;I point of one lint: to another line or a

line to ; I surface you do not increase the magnitude: it is the same thing if you add to

;I line

;I certain line. but one

incomp;lrably smaller. Nor can any increase he shown

by any such construction..‘2



hilosophy of’h4uths in Severlteenth Century

241

Of

course I hold with Euclid (Book V. Definition 5) that all homogeneous

quantities are comparable which can he made to exceed one another through

multiplication by a finite number. Things which do not differ by such [comparable]

quantities are declared to be equal. which even Archimedes assumes. and all after

him. This is what is said to be a difference less than any given difference, and even

in an Archimedean procedure the matter can always be confirmed by a reductio ad

abaurdum. But because the direct method is more quickly understood and more

useful for discovering new results. it suffices that [the direct method) is reduced

once and for all to [the indirect method) and later applied, in which application the

incomparably small quantities are neglected, a procedure which is both sound and

which carries its demonstration with it according to a lemma that was communi-

cated by me in February of 1689.“3

There are clearly quite strong parallels between this strategy and Cavalieri’s

approach to the foundational issues raised by his method of indivisibles.

Both

men were motivated by a similar desire to show that their methods were an

extension of classical techniques but not a violation of the traditional

conception of rigor.

A somewhat different strategy for defending the calculus can be found in

the manuscript known as

“Cum prodiiscr

. . .” - a

study

in the foundations

of the calculus which Leibniz never published but which was written

sometime after 1701.4” Here. Leibniz attempts to provide a foundation for

infinitesimal reasoning by appealing to his metaphysical principle known as

the “Law of Continuity”. In Leibniz’ formulation. the law reads:

In any supposed transformation. ending in any terminus, it is permitted to institute

a general reasoning. in which the final terminus may also be included.”

In its application, the law of continuity allows degenerate cases to be included

in a general account which links the degenerate case with the standard cases.

Thus. the law would sanction treating rest as infinitely retarded motion or

treating parallel lines as lines which intersect at an infinite distance. Leibniz

““Scilicct as tantum

homogencas quantitateh comparabiles essc. cum Euclide lib. 5 dcfin. 5

ccnseo. quarum una numero. sed finito multiplicata.

alteram superare

potest.

Et quz tnli

quantitatc non

differunt. aequalia essc statue. quod etiam Archimedes sumsit. aliique post ipsum

omnes. Et hoc ipsum eat. quod dicitur diffcrentiam esse data quavis minorem. Et Archimedeo

quidem processu rcs semper deductione ad ahsurdum confirmari potest. Quoniam tamen

methodus directa brcvior est ad intelligendum et utilior ad inveniendum. sufficit cognita semel

reducendi via postea methodum adhiberi. in qua incomparabiliter minora negliguntur. quiL

sane et ipsa secum fert demonstrationem warn secundum lemmata a me Febr. 1hXY

communicata.” Leihniz, ” Responsio ad nonnullas difficultates .“. p. 322.

“‘l’he manuscript was first published a\ part of the collection Hi.s/or-itr VI Orig~ wlctrli



argues that. when applied to the

calc~dus.

this law yields ;I new kind of

quantity which will provide the foundation for the reasonings which appear in

the solution to gcomctrical problems:

From this pmtutatc ark certain e?iprcAoih which arc gencratty uwd for ttlc sake

of convrniencc. hut 4cuii to contain a11 absurdity. Foi- instance. wc spwli of an

imaginai-y point of interwction ;h if it \vei-c

;I i-cat point. in the wnic in;iiiilcr iis iii

algebra imaginary root5 arc con~iderctl ;I\ xwptcd n~iilhcrS. I leilw. tmwrving

the analogy. wc say that. when [one straight tincl ultimately

hecomcs

p;w;dlcl to

[another], even then it con\‘ei-ges toward it or

mahcs a1 ankle

w+tti it. onlv that the

angle is then

infinitely smalt; simlarly when

;I body

ultimaletv

comes to

rest. it is

stilt

sxid to have ;I velocity. but oiic that i\ infinitety small: ;Inci. I\ hen one‘ straight

tine is ccld to another. it is said to he uikyu;iI to it.

hit that the

ttiffcrcncc ix

infinitely small .-I’)

In the “C’lrrn prodiisrf” Leibniz goes on to derive some of the basic results

of the calculus. but before hc does this he makes a remarkable claim about

the eliminnbility of infinitesimal magnitudes from the calculus. He writes:

For the present. whether such ;I state of instantaneous transition from inequality to

cqwtity. from motion to rat. froin convcryenw to pw~tlctism. or anything of the

sort cm he sustained in ;I rigorou4 oi- metaphvsicat wnw is ;I matter that I own to lx

possibly open to question: hut for him ~vhomwoultt discuss these matter\. it i\ not

iicccssarv to fati hack

up~ii

mctaphysicat controverGc’s. such 2s the compositioil of

the conti~uum. or to mahc geometrical matter4 clepend thereulxui It will he

sufficient if. when u’c‘ speak of infinitetq great or infinitclv smxtt quan-

titics _. it is understood that \cc inc’an quantities that arc indefinitely great 0I

iildcfinitety small. i.e. as great ;I\ you plcasc 0,. ;I4 sIllal 24 you pte;lW. so that the

ci-ror that any one rn;~y assign may Iw lcs5 than ;I certain assigncct quantity. Also.

since in gcnc’-;it it wilt appear that, when any Smillt

ei-roi- is as\igilcd. it can t>C stl0wn

that it should he tcs\. it fottox\ that the

ct-ror is atw~tutt’ty nc)thing.‘7

The most important aspect of this

pass;~gc

is its claim that the calculus

can he

interpreted so a4

to avoid the supposition of infinitcsinial niagnitudc5

altogether. This “reinterprctability thesis”

i4 the hallmark of Lcibniz‘ sccontl

line of response to the critics of

the cacuus It

can be found in the appendix

to his published reply to Nieuwentijt, whcrc he argues that differentials of all

orders

can be

expressed by the proportions of ordinary finite lines, as welt as

in the correspondence between Leibniy am Varignon from February of 1702.

where Leibniz insists that

oilc’ does not ilecd to make mathematical analysis tlepcncl upon mctatAiyical



Philosophy of’A4uth.s in Sevwteetlth Century

243

controversies or to make sure that there are lines in nature which are infinitely

small in a rigorous sense. in contrast to ordinary lines This is why I thought

that to avoid subtleties and make my reasoning clear to everyone, it would suffice

here to explain the infinite by the incomparable, that is. to conceive of quantities

incomparably greater or smaller than ordinary ones. This would provide as many

degrees of incomparability as one might wish, since that which is incomparably

smaller has no value at all in relation to the calculation of values which arc

incomparably grater than it.“”

What emerges from an analysis of the relevant Leibnizian texts is a fairly

straightforward solution to the problem posed by John Earman in his paper

on Leibniz and infinitesimals.‘”

Earman asks how we can reconcile Leibniz’

claims concerning the nonexistence of infinitesimals with his talk of “infinitely

small” angles and infinitely minute differences between two quantities. His

answer is to suggest that Leibniz really had two concepts of the infinitesimal,

and on one conception they do not exist, while on the other they do. The

difficulty with this interpretation is that it makes the Leibnizian doctrine more

confusing and mysterious than it has to be. Rather than postulate an

ambiguity in Leibniz’ writings, we can take him at his word: he is convinced

that infinitesimal magnitudes are eliminable. and the reason for this

conviction is the thoroughly commonsensical belief that the truth of his

mathematical results should not depend too heavily upon the resolution of

metaphysical problems. In the Leibnizian scheme, true mathematical

principles will be found acceptable on any resolution of the metaphysical

problems of the infinite. Thus, Leibniz’ concern with matters of rigor leads

him to propound a very strong thesis indeed, namely no matter how the

symbols “dx” and “dy” are interpreted, the basic procedures of the calculus

can be vindicated. Such vindication could take the form of a new science of

infinity. or it could be carried out along classical lines, but in either case the

new methods will be found completely secure.

I have been arguing that many of the leading figures of seventeenth-century

-IX.

on n’a

point besion

de faire dependre I’analysc MathCmatique des controverses

metaphysiques, ny d’asseurer qu’il y a dans la nature des lignes infiniment pctites B la rigueur. ou

comparaison dcs nostreb. C’cst pourquoy 3 fin d’evitcr ces subtilites. j’av cru que pour

rendre le raisonnement sensible a tout Ic monde, il suffisoit d’expliquer -icy I‘infini par

I’incomparahlc. c’est k dire de concevoir des quantitks incomparahlement plus grandes ou plus

petitcs que le nostres; ce que fournit autant qu’on veut de degres d’incomparahles, puisque ce qui

est incomparahlement plus petit. entre inutilement en ligne de compte g I’cgard de celui qui cst

incomparablement plus grand que luy. ”

Leibniz to Varignon. 2 February 1702, in: C. I.

Gerhardt (ed.) G. W. Lrihniz M uhw zuti schr~ Schr~fierr. 7 vols (Hildesheim: Olms. 1962). vol. 4,

pp. 9-05. p. 91.

“‘John Earman. “Infinities. Infinitesimals. and Indivisibles: the Leibnizian Labyrinth”.

S/udiu

Lr ih~f i r i r rmr 7 197.5). 2362.5 I



mathematics were vastly more concerned with the rigor of their methods than

is commonly supposed. However. this concern with rigor did not produce ;I

resolution of the fundamental tensions which accompanied the introduction

of infinitesimal methods into seventeenth-century mathematical prac tice.

Cavalicri, Barrow, Wallis and Lcibniz all insisted that their approaches could

be brought into conformity with the classical conception of geometric proof,

and there is overwhelming cvidc nce that they took some pains to provide

an

interpretation of their results which would not require ;I departure from the

classical standard. Nevertheless, their efforts on this score remained largely at

the level of programmatic pronoLlncemcnts. and the most fundamental issues

raised by the new infinitcsirnal methods remained unresolved at the close of

the seventeenth century. The two most important of these arc the question of

whether the new methods can ultimately bc presentcd in ;I manner that dots

not require the supposition of infinitesimal magnitudes. and the related

question of whether it is possible to vindicate infinitesimal reasoning fully by

developing a new and philosophically respectable science of infinitesimal

magnitudes.

This failure to resolve such fun mental questions can account for the

somewhat schizophrenic attitude one finds

amon

g some proponents of the

new methods, who simultaneously claim that the new techniques are far

superior to the classical methods and that they are equivalent to them. Fur-

thermore, such “foundational projects” at Newton’s method of prime and

ultimate ratios can be seen as the product of this tension: in this cast, the

conflict between the demands of rigor

and

the prospects of the new nnalysis

can be seen as leading Newton to attempt a presentation of his methods

in a way that did not depart from the accepted criteria for rigorous dem-

onstration. Variations on this theme can no doubt be found by considering

a wider range of figures from the period, but WC

can at

least be confident

that there was philosophical concern with mathematical methodology in the

seventeenth century,

and that such concern was not without its consc-

yuences.

theory and mathematical practice in the seventeenth century

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