Some Uses of Proportion in Newtons Principia, Book I

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EMILY GROSHOLZ

SOME USES

OF

PROPORTION IN NEWTON'S

PRINeIPIA, BOOK I: A CASE STUDY IN APPLIED

MATHEMATICS

DURING the 17th century, a gradual transition occurs in which proportions

between ratios

:

B

::

C :

D

are treated more and more as equations between

quotients B

=

C

D

In this essay, I will examine the significance of

this transition for an understanding of the relation of mathematics and physics

in the period, illustrating my arguments with a detailed analysis of

Propositions I, VI and XI from Book I of Newton's Principia, In particular,

I want to urge

that

Newton's use of proportion-notation was appropriate and

indeed advantageous to his mathematical physics; and that it reveals a complex

interplay of mathematical

and

physical elements which illuminates the vexed

philosophical problem of how to understand scientific idealization.

I

In her essay entitled Compounding Ratios ,

2

Edith Sylla argues

persuasively that the seventeenth century inherited two distinct traditions

concerning ratios and proportions. The first, arguably Euclidean tradition,

views a ratio as

a

kind of relation in respect of size between two magnitudes

of the same

kind, and

a proportion as an assertion of similitude between

two such relations. This view is expounded in Book V

of

the Elements; it is

applicable to magnitudes of every kind. In Book VI the general theory of ratio

is applied to ratios between geometrical magnitudes (incommensurable as well

as commensurable), and in Book VII and succeeding books to ratios between

numbers (integers).

*Department of Philosophy, 246 Sparks Building, The Pennsylvania State University,

University

Park, PA

16802, U.S.A.

Received

19

March 1986.

'Citations are taken from Andrew Motte's translation of the Principia revised by Florian Cajori

(Berkeley: University of California Press, 1934).

Transformatian and Tradition in the Sciences, Everett Mendelsohn, (ed.) (Cambridge:

Cambridge University Press, 1984), pp. 11- 43.

l The Thirteen OOks Euclid s Elements, Sir Thomas Heath (ed.) (New York: Dover, 1956),

Book V, Defn. 3.

Ibid.,Book

V,

Defns, 6 8

Stud. Hist. Phil.

sa

Vo . 18, No. 2, pp.

209-220,

1987.

Printed in Great Britain.

9



210 Studies in History and Philosophy of Science

The leading principle of this theory of ratio is the Eudoxian or Archimedian

axiom, which states: magnitudes are said to have a ratio to one another

which are capable, when multiplied, of exceeding one

another.

In other

words, one can form a ratio

A

B

(where

A

B

if f

there is a positive integer

n

such that

nA

>

B

This principle illustrates a salient feature of classical

mathematics, noted by Jacob Klein

6

and Ernst Cassirer, in that the terms

(magnitudes) are first in importance and are treated in analogy with

substances. Terms exist as heterogeneous kinds, each with its own peculiar

integrity (numbers; line lengths, shaped areas, shaped volumes;

commensurable and incommensurable magnitudes; finite and infinitesimal

magnitudes). This integrity must be accounted for (as in, e.g. the Definitions,

Postulates and Common Notions of Book I of the Elementst; and respected.

Specifically, when terms are associated in the formation

of

ratios and

proportions, their association must be carefully justified, and may sometimes

be proscribed. Thus, the Eudoxian axiom allows the conjunction of two

numbers, or two line segments, or two areas in a ratio, but proscribes that of a

number and a geometrical magnitude, of two geometrical magnitudes of

different dimensions, and of a finite and infinitesimal magnitude. In a sense,

this enforcing of a certain homogeneity between terms in a ratio entails a

segregation of mathematical magnitudes and prevents various strategies of

generalization which appear only later in the work of Vieta, Ferrnat and

Descartes,

But there is another side to conceiving of a ratio as a relation between terms.

Since the terms maintain their distinctness and dominate the relation, they do

not disappear into the relation, and the relation may be thought of as a virtual,

not

actually completed, operation. Thus, the Eudoxian axiom is also famous

for allowing a mathematics of ratios to include incommensurable or irrational

magnitudes. The ratio of the diagonal of a square to its side can figure in the

framing and solution of problems even though the operation : cannot be

carried out as it can in the case of the ratio 4 : 2. This tradition can therefore

accommodate a certain heterogeneity among terms within ratios, and so

facilitate certain strategies

of

generalization. Indeed, in the work of Newton,

as we shall see, it accommodates the introduction of non-Eudoxian (or non

Archimedean) magnitudes.

Since terms are preeminent in this tradition, terms are strictly distinguished

from relations or ratios between terms, and the heterogeneity of terms may

Ibid.

Book V,

Def n,

4.

Oreet: Mathematical Thought

n

the Origin 01Algebra

(Boston:

M.LT.

Press, 1968),

Chap.

9. _

7Substance and Function W. C. and M. C. Swabey (trans.) (New York: Dover, 1953), Chap. 2.

Heath. op. cit.



Uses of Proportion in Newton s Principia

211

infect, as it were, the ratios which they constitute. Thus the unification of

ratios in proportions, on the one hand, and through the operation of

compounding, on the other, also requires careful justification and

proscriptions. Proportions, for Euclid, are not equations between ratios but

assertions of similitude between one kind of ratio and another, e.g. between a

ratio of two line segments and a ratio of two volumes.

Euclid does not say that the proposition

A

B

e :

D

can be constituted

when A x D B x e because A and D and Band C, potentially

heterogeneous, may be such that it does not make sense to take the product of

the two terms. In classical mathematics, multiplication is not a closed

operation; the multiplication

of

two line segments, e.g., produces an area.

Thus there is no natural interpretation for the product of two volumes, or two

areas. Instead, in Book V, Definition 5 of the

Elements.

Euclid generalized

the Eudoxian axiom, stating that proportions between non-continuous ratios,

A : B ::

e:

D

can be formed

if f

for all positive integers

m

n, when nA

mB

then correspondingly, ne

mD.

Note that this definition excludes

infinitesimal or infinite magnitudes, and tolerates incommensurable

magnitudes, as terms; and allows for the possibility that (mutually

homogeneous)

A

and

B

may be of a kind different from (mutually

homogeneous) e and D

The compounding of ratios in this tradition is carefully restricted; this is

Sylla s main point in her exposition of it in reference to

Elements

V,

definitions 9, 10, 17 and 18, Boethius

De institutione arithmetica

Bradwardine s

De proportionibus velocitatum in motibus

and Oresme s

De

porportionibus proportionum. The operation of compounding was only

performed on continuous ratios, in pairs or series (e.g. A B B:

e: D .

Since continuous ratios share terms, this condition insures that all terms and

hence all relations between terms (ratios) will be homogeneous. Such series of

ratios are compounded by forming the ratio of the antecedent of the first term

and the consequent of the last

A

D

in the example given above). This

operation is considered a kind

of

addition, as if the ratio

A

D

were composed

of A B, B

e and e :

D,

since small components form a whole additively.

Sylla notes that in the first edition of the Principia and even afterwards,

Newton refers to compounding as addition, and adheres to the first tradition

of compounding in many of his calculations.

To conclude, the first tradition takes terms as preeminent; allows for their

possible heterogeneity and accordingly puts restrictions on how they are

unified in ratios, proportions and compoundings; subordinates relations to

Ibid.

Sylla, op. cit. pp 18 22



212

Studies in History and Philosophy

of

Science

their relata, and proportions to the relations they relate; and may treat : as

a virtual operation.

The second tradition appears to originate with Theon, a commentator on

Ptolemy s

lmagest

and is transmitted in the Middle Ages by Jordanus

Nemorarius, Campanus and Roger

Bacon.

This tradition countenances the

compounding of non-continuous ratios, and associates with each ratio a size

or denomination such that in compounding, denominations are multiplied

together. Thus, it tends to treat ratios as numbers (the size

of

the ratio) and

proportions as equations between numbers. In other words, terms are

subordinated to (swallowed up in) relations, and relations to equations. This

development is strongly associated with the rise of analytic geometry and

infinitesimal analysis, and forms an important strand in what Cassirer calls the

functionalization

of

mathematics and science.

In general, this tradition homogenizes the magnitudes occurring in ratios

and proportions, treating them uniformly as numbers. The assumption of such

homogeneity played a central role in the systematization and unification of

algebra and geometry in analytic geometry and later in infinitesimal analysis.

At the same time, at least in the initial stages exemplified by the work

of

Oresme and Descartes, it tended to limit the conception of number to the

rationals by requiring that division be an operation which could actually be

carried out in familiar ways.

In a deeper sense, too, it buried certain important questions and possibilities

of conceptualization particularly germane to the mathematization

of

science.

The distinctions between finite and infinitesimal magnitudes, between

numbers and geometrical elements, between geometrical and physical

elements, and among lines, curves, areas and volumes were, as I shall argue,

still quite significant in the context of the application of geometry to physics.

Newton worked with and exploited these distinctions, in part because he

retained an allegiance to the first tradition. More generally, philosophers of

science cannot afford to forget certain fundamental questions which the

second tradition suppresses. What

justifies

the claim that physical magnitudes

(like distance, or time) stand in relations to each other similar to relations

between numbers? Or that geometrical magnitudes do likewise? Or that

instances of one kind of physical magnitude stand in relations to each other

similar to relations between instances of a wholly different kind of physical

magnitude? Or that we can form ratios between two wholly different kinds

of

physical magnitudes (like velocity and time) to form new kinds

of

physical

magnitudes? These questions cannot be answered in a trivial way, nor without

appeal to the history of mathematics and science.

Ibid pp .

22 - 26.



Uses ofProportion in Newton s Principia

213

e

-

·· ·t

v i

E.

./

.

..

: :

s ~ .

Fig.

I.

Proposition XI of Book I of the Principia is central to Newton s project.

It states If

a body revolves in an ellipse; it is required to find the law

of

the

centripetal force tending to the focus of the ellipse and concludes that the

required law is an inverse square law. The statement of the problem and its

solution depend in essential ways on two prior results, Propositions ID and

VI 4 which I will discuss first in order to better exhibit certain features of

Proposition XI.

Proposition I is Newton s version

of

Kepler s Law

of

Areas: The areas

which revolving bodies described by radii drawn to an immoveable centre

of

force do lie in the same immovable planes, and are proportional to the times in

which they are described. Newton s

proof

of this claim is illustrated by Fig. 1.

S is the center of force. A body proceeds on an inertial path from to in an

Newton op. cit., pp.

56-57.

Jbid.,

pp.

40-42.

One might debate whether Newton countenanced actual infinitesimals.

Without wanting to reopen the debate here,

I

would observe that his ways of using evanescent

magnitudes are Leibnizian enough to support that contention, despite the Archimedean air of

Principia I, Lemmas and his reservations about indivisibles in the following Scholium.

Ibid. pp.

48-49.



214

Studies in History and Philosophy

of

Science

interval of time; if not deflected, it would continue on in a second, equal

interval of time along the virtual path

Be

However, Newton continues,

when

the body is arrived at

suppose th t a centripetal force acts at once with a

great impulse so that the body arrives

not

at c, but at C. Then cC

V

represents the deflection

of

the body due to the force; indeed, as we shall see,

cC RV becomes the geometrical representative of the force. The perimeter

ABCDEF represents the trajectory of the body as it is deflected at the

beginning

of

each equal interval of time by discrete and instantaneous

impulsions from S. Newton then uses the Euclidean theorem that triangles

with equal bases and equal elevations have equal areas, to show that area

~ S

area

~ S c

area b SBC; this equality extends to triangles SDC

SED SFE . . by the same reasoning, so that equal areas are described in equal

times. We have only, Newton concludes,

to

let the number of those triangles

be augmented,

nd

their breadth diminished in infinitum for this result to

apply to a continuously acting force and a curved trajectory.

There

re

two related points which I want to make about this diagram (Fig.

I). First of all, it is thoroughly ambiguous, and must be read in two apparently

incompatible ways, as a collection of finite lines and areas (where the

perimeter is composed of rectilinear line segments) and as a collection of

infinitesimal as well as finite lines and areas (where the perimeter is a curve).

The first reading allows the application of Euclidean theorems to the problem;

the second reading makes the problem applicable to the kind

of

motion and

force Newton is interested in. For this reason, the diagram, whose meaning

and intent cannot be deciphered unless it is read in both ways, could not have

arisen in Euclidean geometry.

Secondly, the diagram would not have arisen in Euclidean geometry alone

even if it is read as a finite configuration. The theorem about triangles of equal

area provides no motivation for studying triangles of equal area multiplied and

linked in this particular fashion. Neither does this resemble a problem of

quadrature, for in that case (a) the perimeter would be a known curve, (b)

there would be no reason to single out the point S and (c) there would be no

reason to study the line segments Be and cC BV In other words, though this

diagram can be treated by Euclidean means, no Euclidean geometer would

have formulated it, or been interested in it if it were presented to him. This

diagram is a problem about force, motion and time, distances and areas. It

arose within physics, and geometry enters as an auxiliary means to help solve

it.

The formulation which I have just given of the status of this problem in

physical-geometry, however, does not yet do justice to the complex interplay

of geometry and physics in this context. I will refine it as I discuss Propositions

VI

nd

XI,

nd

their.attendant diagrams

nd

proofs. The general train of my

argument will be the following. In these propositions, Newton is dealing with




Fig. 2.

215

problems which arise within physics (the problem of motion in general and

planetary motion in particular; the problem of force) but cannot be solved

within physics alone. The sterility

of

Aristotelian physics in the seventeenth

century testifies to this. Geometry is brought in as an auxiliary field to help

solve the problems. However, physics has already been penetrated and

structured by geometry (in the work of Kepler, Galileo, Descartes and

Huygens) so that what counts as a physical problem for Newton is already to

an extent formulated in geometrical terms. Moreover, geometry itself is

already in a process of revision and expansion, in response to the problems

concerning motion and force which physics has presented to it. That is,

there has already been a long process of assimilation between physics and

geometry. Although each retains items and problems alien to the other, the

common ground between them has steadily increased, with the shape of each

field altering accordingly. And Newton profits from his exploitation of this

common ground, the hybrid techniques, items and problems of

physical-geometry.

Proposition XI is a special case of the general result which Newton works

out in Proposition VI, l7 where he shows that for any kind of revolution Q

of a body P around a center of force S the centripetal force will be inversely

proportional to the quantity

sp

2

QT

2

QR

In Fig. 2, R is the virtual inertial trajectory the body would have followed if it

had not been attracted by S, and by Law I it is directly proportional to the time

t. By Kepler s Law

of

Areas (Proposition I) the curvilinear area SPQ is also

j don t

mean to exclude here the mathematization

of

physics already accomplished in the work

of

Euclid, Ptolemy, Archimedes, and the author

of Mechanical Questions.

I ·For a full-dress defense of this claim, see Francois DeGandt s Mathematlques et realite

physique au XVII<siecle , in

Penser tes mathematiques

(Editions du Seuil, 1982), pp. 167

-194.

My exposition here is heavily indebted to an unpublished essay by Francois DeGandt, Le

traitment geometrique des forces centrales dans les

rincipi

de Newton , forthcoming in the

Graduate Faculty Philosophy Journal

New School for Social Research.



216 Studies in History and Philosophy of Science

proportional to t; since

2SPQ

SP x QT it is proportional to

SP

x

as

well. (The fact that

SP

x QT is an area, and an infinitesimal area, is thus

physically significant. Kepler s insight, so important for the development of

astronomy, was that the area swept

out

by

SP

serves as a reliable uniform

measure of time. That the area is infinitesimal signals that here Newton is

deploying a mathematics developed to treat the problem of motion, non

uniform as well as uniform, through an analysis involving instantaneous

velocity, time and force.) The segment

QR

represents the virtual trajectory of

the body as a result

of

the centripetal force exerted by S;

it

is thus directly

proportional to the force. t

is also directly proportional to 2 , by Lemma X S

in which Newton generalizes Galileo s result that in free fall the space

traversed is proportional to the square of the time, to hold for all cases

of

a

body attracted by a constant or continuously varying force provided

that

one

considers only the first instant of motion. Then since

F

2 F

QR

QR

er X

SP

x

QT

Sp2 X QT2

or, as Newton prefers to write t so

that

the ratio has three dimensions,

Sp

2

x

QT

2

Fa

QR

Newton uses the diagram in Fig. 2 to find a way of representing the

centripetal force at S, and concludes

that

t is inversely as

Sp2

x

QT

2

QR

Note that very little in either the diagram or the reasoning about it is of

Euclidean provenance: only

that

SP is a line segment, PY is the tangent to the

curve at and the area of a triangle is half the product of its base and

altitude. (But the application of the latter theorem is curiously unfiuclidean.)

What the diagram is for and how it should be read is determined for the most

part

by the way it represents a physical situation. Why QR and SP x

QT

should be chosen as particularly significant, and how and why they are related

can only be explained by theorems

of

physics developed by Kepler, Galileo,

Descartes and Newton. Of course, these theorems also geometrize physics,

providing techniques for representing physical magnitudes by lines and areas,

and

therefore Newton can set forth his results in such a diagram. But the

diagram of Fig. 2 could only have arisen, and is only intelligible, within the

context of physical-geometry. Finally, to reiterate my earlier point, Newton s

expression of his result as the centripetal force will be inversely as the solid

Sp2 x QT

2

QR

10

Ibid. pp. 4 35.




Fig. 3.

217

allows him to exploit the proportion-idiom of the first tradition to relate and

yet discriminate heterogeneous physical magnitudes lines and areas and finite

and infinitesimal magnitudes.

Since Proposition VI is a general result and since the expression for force is

given in terms of infinitesimal magnitudes Newton must apply it to cases in

which the trajectory PQis specified in such a way

that

he can replace

SP

2

QT

2

QR

with another expression derived from the specific trajectory of and

involving finite magnitudes. In Proposition XI he chooses the case where the

trajectory is an ellipse clearly a crucial step in the application of his physical

principles to the solar system in Book

ll

Figure 3 combines the physical-geometrical schema

of

Fig. 2 with the p ure

geometry

of

the ellipse but

is instructive to examine the combination in

detail. The latus rectum L

=

and the diameters of the ellipse BC S

DK and PG have no physical import. But the presence

of

the elements of Fig. 2

impose physical import on other parts of the ellipse: the perimeter PBDGKis

also the orbit of a revolving body S is also the center of force SP also the

distance

of

the revolving body to the center of force. By contrast the auxiliary

lines PH

IH

and PF like the ellipses s diameters enter the reasoning only

insof ar as they are geometrical. Thus. though Fig. 3 contains more purely



218

Studies in History and Philosophy of Science

geometrical elements than Fig. 2, the problem

and

solution which it represents

are thoroughly hybrid, both geometrical and physical.

The proof of Proposition XI proceeds by establishing proportions between

line segments

and

products

of

line segments by means

of

theorems

about

similar triangles, isosceles triangles

and

ellipses, and culminates in an elaborate

compounding of these ratios. Newton first proves that

EP :;

AC using

auxiliary lines HI and HP; H is the other focus. Since MHS is similar to

MCS

SE :;

El

and EP :;

Yz PS Pl or Yz PS PH since tJlIH is

isosceles.

PS PH :; ZAC

by the nature of ellipse-construction, so

EP :;

AC.

Then Newton begins to establish certain extended proportions. L

X

QR

:

L x Pv :: QR : Pv, This seems to be a straightforward bit

of

reasoning about

ratios,

but

note

that

all the magnitudes involved except L may be construed as

infinitesimal, because they are being used for the analysis of motion. Newton

is profiting

from

: as a virtual operation to manipulate non-Archimedean

magnitudes. QR Px

:

Pv

:: PE :

PC because

tJlxv

is similar to tJlEC.

This is a Euclidean result employed in a highly non-Euclidean way to relate a

finite to an infinitesimal triangle. Finally, PE : PC :: AC : PC by the first

result, so

L

x

QR : L

x Pv

:: AC: PC.

Next, Newton asserts L x

v

Gv x Pv

::

L

:

Gv Pv here is infinitesimal

and

that

Gv

x Pv :

v

::

p : CIY

a fact

about

ellipses, except

that

Pv

and

Qv

are infinitesimal

magnitudes,

Actually, it is only at this point in the

proof

that

Newton says explicitly let Q-P. One might then take the foregoing

reasoning as about, not infinitesimal, but small finite quantities, so that the

application

of

Euclidean results is straightforward. However, as we shall see,

the final compounding of ratios evenhandedly combines ratios established

before and after the step where

Q-P.

Newton depends on the ambiguity of his

diagram: read as finite, it allows the application

of

Euclidean results; read as

infinitesimalistic, it provides a mathematical schema for motion.

When points P and Q coincide, Newton claims that Qy2 ;

Qx

z

and

so Qx

z

Qvl

:

QT

2

::

Ep

z

:

pp2

since the infinitesimal triangle

QxT

is similar to

tJlEF.

Then

Ep

z

: pp2::

CA

z:

ppz by the first result,

and CA

z:

PP:: CIY:

cj by a previously established result

about

ellipses. Thus Qx

z

Qvl

:

QT

z

::

CIY: ce. Newton is now ready to carry

out

the final compounding, which

Sylla summarizes in the following perspicuous array. 20

L

x QR :

L

x Pv :: AC : PC

L X Pv : Gv

x

Pv

::

L : Gv

Gv x Pv : Qy2 .. p :

CIY

Qv

2

:Qx2

.. 1 I

x

:

Q P

cir

:

ce

19 Ibid. Lemma XII,

p.

53.

2IlSylla,

op. cit.,

p. 16.




219

Sylla notes that Newton has set up the left-hand ratios as a continuous series,

and compounds them according to the first tradition, taking the extreme terms

and forming the new ratio L x QR

:

QT

2

• The right-hand ratios he

compounds according to the second tradition, by multiplication:

AC

x

L

x

PC

1 x x .

Gv

et» x I x Cg or (substituting

Bc:aIACfor

and cancelling) PC

:

Gv.

Our prior investigation into the geometrical

physical nature of Newton s diagrams and procedures sheds further light on

this process

of

compounding. For the magnitudes on the left-hand side have

physical import and are largely infinitesimal. That is, they are just the kind of

magnitudes which ought to be treated in the first tradition, which respects the

heterogeneity of terms and, given the virtual nature of : allows for the

manipulation of infinitesimals. The magnitudes on the right-hand side, if we

recall

that

Gv

GP

as

Q- -P,

are all constant finite geometrical line lengths

(some of which are squared) with no physical import. Having no reason to

think of them otherwise than as numbers, Newton can handle them according

to the second tradition, multiplying them like quotients.

The compounding thus yields the proportion,

L

x

QR

: QT

2

PC

:

Gv.

As

Q- -P, PC Gv,

so

L

x

QR: QT

2

: :

I : I, so

L

x

QR

is proportional to

Sp

2

. Sp

2

QT

2

.

QT

2

• Multiplying both these terms by

[QR]

we find that QR IS

proportional to L x

Sp

2,

or, since L is a constant, to SP2. Thus, the central

force in this problem is inversely proportional to the square

of

the distance

SP

IV

What does our examination of these Newtonian propositions reveal about

the much-debated topic of idealization? can help us reject two misleading

models of idealization, and in so doing formulate some positive observations

on the topic.

In Proposition XI, Newton does not begin with a physical problem,

translate it into a mathematical analogue, solve the mathematical problem and

then translate the solution back into physical terms. This account would leave

the unresolved puzzle, how is it possible to find analogical structures in physics

and mathematics? And, how can the analogy itself be justified?

I have said that Newton brings geometry in as an auxiliary field to help solve

problems which arise in physics but could

not

be solved by physics alone, and

21

This view of applied mathematics is presented in Stephan Korner, The Philosophy of

Mathematics (London: Hutchinson, 1960), pp. 176

182.



220

Studies in istory and Philosophy Science

cited the exhaust ion of Aristotelian physics in the seventeenth century as

evidence

for

this incapacity of physics. But notice

that

the interplay

of

geometry

and

physics as a result of this unification of fields is much more

complex

than

the model sketched above suggests. Newton s statement

of

the

problem in Proposition XI in reference back to Proposition VI), while

evidently the statement

of

a physical pr oble m

about

the relation between

motions, times, distances

and

forces, is already couched in mathematical

terms. he mathematical reasoning in the

proof

reveals at every step that

Newton is using a mathematics adapted to and extended by its employment in

the s ol ut io n

of

physical problems,

and the

solution is given in terms

of

physical-geometry. In the overlap between the two fields where Newton is

carrying out his investigation, geometry has reformulated physics, and physics

has changed the shape

of

geometry. The geometrical-physical tools

of

investigation have been developed precisely to solve the geometrical-physical

problems,

and

it makes no sense to try to disentangle the geometrical from

the

physical elements and

then

ask why they match.

or does it m ak e sense to say that Newton developed this physical-geometry

and then

applied it to physical reality tout court so

that

the puzzle

of

idealization is

then

to figure

out

how he could do this. So stated, the puzzle has

no sol ut ion. since physical reality has been g rant ed no cognitive s tructure,

which could

then match

up somehow with the structure of physical-geometry.

Newton obviously intended Pr oposition XI to apply to planetary

motion

around the sun, and just as obviously realized that it was only one stage of that

appl icati on. But the o bj ect of application here is not planetary motion tout

court but p lanetary m ot ion al ready cognitively s tructu red as an o bj ect of

science.

My general po int here is

that

the philosophical problem

of

idealization, in

particular, of how mathematics can be applied to physical reality, sh oul d be

addressed in terms of

the

history

of

science. A series

of

case studies like the

one

contained in this essay would, I believe, provide grounds

for

local

and

piecemeal generalization

and

also reveal striking differences in the nature

and

justification

of

that application f rom one era to the next.

hus

I also believe

that

no interesting global solution to the problem exists.

What

history reveals

in any given era e.g, the seventeenth century) is a physics already partiall y

adapted to mathematics and a mathematics similarly elaborated in the service

of

physics. The application of mathemati cs to physics in a certain case e.g.

Newton s rincipia will then consist in taking that assimilation of distinct but

partially correlated domains one step further. he business

of

the philosopher

is to elucidate the peculiar grounds

and

presuppositions of that assimilation,

and

to cautiously generalize

on

the

basis

of

her

or

his analysis wit hou t

succumbing to philosophy s instinctive desire

for

the last w or d.

Documents

Some Uses of Proportion in Newtons Principia, Book I