Copyright © 2013

Avello Publishing Journal ISSN: 2049 – 498X

Issue 1 Volume 3: Principia Mathematica

Newton's Diagrams

Philip Catton

University of Canterbury, New Zealand.

Isaac Newton — theologian, alchemist, raging recluse, icon,

mathematician, practical man, magician, measurer extraordinaire,

and most profoundly, progenitor of physics as exact science — comes

into coherent focus for us as he was in himself only when we

recognise why his diagrams in Philosophiæ Naturalis Principia

Mathematica are made by him to be not merely pedagogical, but

instead indispensable to the reasoning itself.

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§1. Diagrams and natural philosophy.

When in his published writing Newton reasons out for us what he terms

the mathematical principles of natural philosophy, he does so with diagrams.

How essentially? Supposedly, not essentially at all. Allegedly1 Newton had

actually developed Principia by roundly analytical means, which then he hid.

And indeed, the following facts are scarcely in contention. Roughly two

decades previous to the publication of Principia, Newton had secured for his

own use calculus. When Principia was first published, the calculus was a secret

art as yet largely private to him; in the use of the calculus, Newton was

prodigiously adept. If Newton in the Principia had made direct use of the

calculus as we know it now, then this would have simplified the mathematics

there, and reduced the function of diagrams to mere pedagogy; and,

mathematical contemporaries of Newton’s were accustomed to making out the

diagram as not merely pedagogical, but rather as indispensable part to the

reasoning itself. Moreover, taken all together, these facts precipitate the

rumour that Newton sought in Principia as simply the best way to show his

mathematical genius off to his contemporaries. That alone supposedly is why

diagrammatic reasoning is left essential within that work.

1Niccolò Guicciardini (1999, pp. 110 ff.) argues that Newton himself promulgated this

view, in order to claim that Principia evidences his priority over Leibniz in the discovery of the calculus. Guicciardini’s scholarship is roundly magisterial, yet I will resist subtly, below, his perspective on this point.

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In fact, however, the rumour that Newton faked his dependence upon

diagrams nourishes a litany of confusions, confusions that one will endeavour

here to identify and redress. It is true that in Principia Newton stays, and stays

quite deliberately, the inchoate symbolism of his pregnant but still private

calculus. Instead of producing there any calculative, algebraical, symbolically-

styled reasoning about what is infinitesimal, Newton instead literally draws out

for us in diagrams what is momentary about motion. Yet these executions of

his are to his mind analysis at its surest. Newton in no way hides analysis from

us in Principia. Rather Newton practices analysis there boldly, in the

diagrammatic form he is deeply disposed to think most apt.

The understanding that the surest way of reasoning mathematically is

essentially diagrammatic is deep-lying within the tradition of mathematical

inquiry coming down from the Ancient Greeks.2 Newton made a difference to

what mathematics can comprehend, not by departing from the Greek view of

the diagram but by compounding commitment to that very view. For, Newton

added to Greek practices in diagrammatic reasoning his own new flourish, of

drawing out what is momentary about motion. Within the diagram-based

tradition of mathematical thinking, there had long existed an understanding of

analysis. Newton cites in this connection the Greek mathematician of late

antiquity Pappus of Alexandria (c. 290 – c. 350 C.E.). When Newton represents

within diagrams what is momentary about motion, he departs not at all from

Pappus’s view of what analysis comes to diagrammatically. We connect with

Newton therefore when we note the following. Analysis, on Pappus’s

2For a more thorough study of this point in both philosophical and historical perspective

see Catton & Montelle 2012.

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understanding of it, is a creative process, often even productive of new

concepts. Pappus-styled analysis is not merely symbolical or calculative, and

does not reduce what is complex to its simpler elements; instead it builds a

novel insight. Pappus writes in his Mathematical Collection (c. 340 C.E.) as

follows about analysis and synthesis:3

Analysis … takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we admit that which is sought as if it were already done and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards.

But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what before were antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought; and this we call synthesis.

Moderns consider with some bewilderment what Pappus could possibly have meant by these words, but in the context of diagrammatic reasoning his meaning becomes plain. In this connection, the way in which, to our ears, ‘analysis’ links with the logical, calculative, algebraic, or symbolical, becomes spurious. Bearing in mind that in Pappus’s day algebra was not yet invented and so had in no way been brought to geometry, it is essential for us to think that Pappus means by ‘analysis’ neither more nor less than he says he means by it. Analysis concerns diagrams. To consider Pappus’s meaning, here is an illustration:

3Quoted, as it (very commonly) is quoted, as in Heath’s Euclid [1956], volume 1, p. 138.

For more, see Pappus of Alexandria [1986], pp. 82 ff.

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The illustration is the simplest that I can think of to make Pappus’s

meaning clear, and concerns the Euclidean demonstration (Elements I.32) that

the internal angles of a triangle sum to a straight line. Synthetically the

proposition is demonstrated for a given triangle by “producing” one of the sides

and constructing the parallel to another, as above, observing then the equality

of the exterior angle with the sum of the two opposite internal angles and thus

that the sum of the three internal angles is a straight line.

By analysis however one could discover more or less directly how such a

synthesis should be accomplished. The trick, as Pappus says, is “to take that

which is sought as if it were admitted” i.e. to assume that the exterior angle is

the sum of the opposite internal angles so that the three internal angles will

sum to a straight line:

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One then quickly finds oneself inserting the parallel to the opposite side, and

then the needed synthesis becomes clear.4 One might even say that from the

phenomenon that the internal angles of triangles do sum to a straight line, by

analysis one deduces the cause of that’s being necessarily so. (Pappus himself

in the above quotation even introduces causal talk to us this way.)

Every phase of analysis in Newton’s Principia can be thought of as

deducing-from-a-phenomenon-its-cause. The converse phase of discussion,

synthesis, deduces from-a-cause-the-resulting-phenomenon. Principia contains

in spades both these phases. To illustrate this, let us consider the very first

propositions of Book 1 of Principia, Propositions 1 and 2 (which are converses of

one another, and so respectively consider the converse phases of synthesis

and analysis). These propositions and their accompanying diagram are

perhaps more often discussed than any others of Newton’s. Yet our present

purpose is to link them to Pappus and that purpose is novel. We need the

Pappus distinction between analysis and synthesis in order to understand

Newton fairly. Newton writes:

4Admittedly, how quickly depends on whether it is as a first or second step that one

inserts the angles as depicted, which ordering alone is helpful towards the discovery of the starting point for synthesis.

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• Proposition 1 The areas which bodies made to move in orbits describe by radii drawn to an unmoving centre of force lie in unmoving planes and are proportional to the times.

• Proposition 2 Every body that moves in some curved line described in a plane and, by a radius drawn to a point, either unmoving or moving uniformly forward with a rectilinear motion, describes areas around that point proportional to the times, is urged by a centripetal force tending toward that same point.

Newton reasons to these conclusions in the following diagrammatic fashion:

Consider first the degenerate case where the sun-centred (S-directed)

accelerative force is zero. This, degenerately, remains one way that an

accelerative force could obtain in the direction of S — the force in question is

simply zero in magnitude. In that case a body (a planet, say) that begins at A

and sweeps to B across an initial moment will carry on inertially and sweep to c

in a second equal moment. Notice that in this reasoning we are led to draw out

for ourselves what is momentary about motion. We reason that AB and Bc are

equal, the motion of a body from A to B and then from B to c across successive

moments being an inertial one. We thereby draw into connection with the

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momentary significant geometrical structure.5 Proceeding as discussed within

the present diagram, we readily recognise the equality of the areas swept out

about S during the two successive moments of time. For triangles SAB and SBc

can be made out as having equal bases (viz., AB and Bc) from each of which

the perpendicular height to S is the same. Therefore, triangles SAB and SBc

are equal in area. Yet on next considering what will happen if from one

moment to the next the planetary body is deflected by the sun (at S) — say

from B not to c, but rather to C — we then make out geometrically the

following. The planetary body, thus deflected, will sweep out in the second

moment an area equal to that swept out in the first moment, if and only if the

triangles SBC and SBc are equal in area. For, we already know that the area of

triangle SBc is identical to that of triangle SAB. And yet as the triangles SBC

and SBc share the base SB, they are equal in area if and only if they are

constructed between parallel lines, that is to say, if and only if Cc parallels SB.

But this will hold if and only if the deflection that the planetary body has

suffered is in the direction of S, the sun.

Thereby, in one fell diagrammatic swoop, Newton convinces us, on the one

hand, that if — or to the extent that — Kepler’s Second Law (the law that

planets sweep out equal areas in equal times) is true, then the planets

participate in an acceleration always directed to the sun, and thus are

subjected continually to a sun-centred force; and, on the other hand, that by 5

We give effect, in other words, to a transcendentalist faith. By a drawing out for ourselves, grossly, of something in fact infinitely finer than we can draw, we admit the grossness of our practical powers and the finitude of our intellect. Yet we assume that nevertheless, by our reason we are made in the image of God. Just as an unavoidable human grossness means that we can only ever deal approximately with breadthless lines or dimensionless points, so also, when we draw out what is momentary about motion, we do so grossly. The expectation however is that the ideal to which we are beholden in these practices can itself be innocent of this grossness of ours. The ideal itself can be risen higher than any possible accomplishment by us.

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being subject to a sun-centred force, the planets are required to sweep out

equal areas in equal times about the sun, and thus satisfy Kepler’s Second Law.

The first phase of this is analysis: it deduces from a phenomenon (that of

Kepler’s Second Law holding) its cause (the force upon the planets being sun-

centred). The second phase is synthesis: it deduces from the cause (the force

upon the planets being sun-centred) the phenomenon to be explained (that

Kepler’s Second Law holds). Synthetic reasoning was

traditionally received as mathematically superior, in as much as synthetic

understanding is the endpoint. Perhaps this is why Newton orders synthesis

first (as Proposition 1). A larger but related reason for Newton’s prioritising the

synthesis could concern God. Prowess in analysis is not far separated from

being able to fathom what is so very ultimate in the order of things that it

requires a supernatural cause. That would mean that prowess in analysis is not

far separated from being adept in magical arts, a point that we will pause to

consider again, later. Be that as it may however, Newton is notably completely 9

up-front about delivering within Principia the analytic phase (as Proposition 2),

not only the synthetic one (as Proposition 1). Newton does not hide the

analysis, but simply performs it diagrammatically.

Newton makes diagrammatic performance essential to the reasoning itself.

Present-day mathematicians and philosophers struggle to comprehend that this

is possible. They consider diagrams to represent mere visualization. Diagrams

seem to them at best of pedagogical or heuristic value. This perspective

reflects their commitment to symbolical reasoning. Formalism or logicism as

one’s philosophy of mathematics are pinnacle expressions of this commitment.

The age of computers just is an age of formalising and logicising and to that

extent of formalism and logicism. This is not to say that such philosophies pan

out. We contemporaries need in any case to recall that our perspective is new

and even rather peculiar. At one time the practice of mathematics made

diagrams or diagrammatic performances essential to the reasoning itself.6 And

to participants in of mathematical inquiry back then, diagrams certainly did not

seem a mere form of visualization.

The visual cognitively pales compared to the practical. A diagram relates

to the practical. In Euclid, insight is produced when practice becomes intuitive,

when reason commends as necessary every step of a sequence of steps

through the overall performance of which one achieves an end. Quite typically,

propositions in Euclid are things that it is proposed to do, that is to say,

practical ends that Euclid forms for mathematicians to make their own. And

then the demonstration is a rationally perfect execution of diagrammatic steps

that most excellently achieve the desired end. Just as ‘to diagram’ is a richer

6Catton and Montelle 2012 marshals the case for this contention.

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verb than ‘to picture’ and embodies a practical connection, geometers garner

insights not merely visually but by practically drawing out into diagrams the

conditions for a kind of perfection of agency. From what alone allows the

relevant manual practices to achieve their ends rationally harmoniously arise

constraints upon the form of visualisations. The conditioning is not the other

way around, from the form of the visual upon the possibilities for the practical.

It is very much from the practical upon the visual. (We must mature

significantly as agents before we truly see.) To make rationally perfected a

practical grasp of space is to fathom constraints upon possible motion. We are

brought by Euclid to understand spatial structure in terms of constraints upon

possible motion. This is what prioritises the practical to the visual, and argues

that the visual can have its form only by virtue of practical conditions upon the

motions that are possible.

§2. Vicissitudes of the diagram and of analysis.

Many philosophers expect that reason more or less centrally is logic. Logic

is the study of the calculative function of reason. In the context of calculation,

a diagram is but a heuristic aid. Any diagram simply helps initiate the thinking,

thinking that will then be entirely borne by the symbolism. The diagram is then

for pedagogy; it becomes propaedeutic to the reasoning rather than any

principal part of it. We live in an age that much links analysis and calculation.

Given this identification, which does however step us far away from Pappus, or

thus far away from the whole ancient mathematical culture of diagrammatic

reasoning, and thus far away from Newton, we may regard logic as the study of

the analytic function of reason. Few present-day philosophers would complain

against this understanding. But recall that logic leapt forward late in the 11

nineteenth century or even early in the twentieth. Logic only lately has

achieved the power that it now possesses. By the discovery of just such power,

the potentialities of calculation became fully charted. It is by the discovery of

their scope that we have been propelled into a computer age. Computers

epitomize the new power that logic has attained. Newton, we shall see, stands

as a signal influence upon this development. But he was himself of the earlier

age.

For a while (early in his career), Newton himself helped to make out his

methods of mathematical analysis as calculative. Later, he returned himself

determinedly to diagrammatic forms of rumination and professed them to be

superior. Newton participated to the extent that he did in the development of

calculative methods because just such a shift in the culture of mathematical

rumination was on the make. Within it eventually issues of the rigor or calculus

would be addressed. The rigorizers of the calculus would clarify the mode for

logical reasoning itself. Via their accomplishments humans would be propelled

into a computer age. To the extent that Newton’s mathematical analysis is

calculative, it concerns infinitesimals. In what he called the ‘analytic method of

fluxions’, Newton introduced a symbol for just such quantities. Yet

infinitesimals have seemed doubtful. Empiricist philosophy in particular

undermines any conviction that the term ‘infinitesimal’ is even meaningful.

Was there anything really to which Newton’s symbol might refer? In decades

not long after Newton, empiricist philosophers such as Berkeley and Hume

were significant in sounding alarm concerning infinitesimals, though neither of

these figures was himself even remotely a mathematician. Neither remotely

possessed strong enough logical theory to license what each himself proposed

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should be understood by ‘mathematical analysis’. As empiricists these

philosophers had an understanding of analysis that differed from Pappus’s.

Analysis to them was about breaking things down to elements. Analysis was

not the creative diagrammatic performance that Pappus describes. For

empiricists, analysis is about finding original elements that enter our situation

merely as givens. The contentfulness of such elements would need to be from

how they are given to the mind. Yet in that case, infinitesimals are entirely

suspect. Newton’s symbol for the infinitesimal seems liable not really to refer.

For how could what is infinitely small have any content before our minds?

Neither Berkeley nor Hume himself possessed even the remote beginnings

of a workable philosophy of mathematics. The worries of Berkeley and Hume

about infinitesimals generalise easily to breadthless lines and extensionless

points, and thus to the conditions for Euclid’s geometry to have become exact

science. How could the notion of a truly breadthless line, or a truly

extensionless point, have any content before our minds? To answer this

question positively one must look away from empiricist accounts of meaning,

and recognise a connection between meaning and ideals for practices. Truly,

meaning consists not in a content or in a way of that content’s being given to

the mind. Meaning links to ideals — for example, to ideals to which our

practices with straight edge and compass are beholden, when we fashion

geometry an exact science. What words such as ‘point’ or ‘line’ mean depends

not on the images we create on paper or in our heads, but rather upon our

practical directedness, and what perfection in those practices would be.

Whether mathematical analysis possesses integrity as Newton deploys it in

Principia reduces not at all to whether Newton’s symbol for the infinitesimal

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has a referent, and depends not at all on whether of this referent a clear and

contentful image can be delivered to the mind.

By the nineteenth century however, confidence about spatial intuition was

much unravelled for a variety of reasons. Critical concerns about the

meaningfulness of mathematical language by empiricist philosophers were but

one set of such reasons. Mathematicians became concerned for many other

reasons besides to reduce to zero any reliance upon intuition in their reasoning.

Thus I have scarcely begun to detail what pushed the diagram from the

fore of mathematics to the rear, or rather, from being indispensable to the

reasoning itself in mathematics to being but a pedagogical aid. However, for

my present purposes it suffices to observe that, two centuries after Newton, his

calculus itself spurred reconsideration of the very nature of reason. (Two

centuries is a long time, and yet, even by the present day, three and a half

centuries after Newton first developed the calculus, the ruminations are by no

means concluded.) Nineteenth-century efforts to rigorise mathematical

analysis culminated, around the beginning of the twentieth, in the

advancement in logic above discussed. After this great advancement in logic

there followed bold reinvention of the philosophical spirit itself. Not as a

coincidence with this but rather as a consequence of it, the possibility of the

computer was also unleashed. Linking in no small measure back to Newton are

reasons why we live in a computer age, and also why we have just had a

century that was (most especially in English speaking lands) heyday for

analytic philosophy.

Newton himself I make out as closer in his conception of reason itself to

rationalist philosophers than to empiricists. Rather than linking reason with

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logic (which is an empiricist trait), Newton significantly resembles Plato, in

instead linking reason with rational self-determination in practice. Euclid, for

example, mobilizes reason to render what is practical as in its every step

harmoniously necessitated. Let me briefly mention some consequences,

philosophically, of philosophy’s by now having stepped so far away from this

view. Today’s philosophers are poles apart from Plato when (as is utterly

common) they call necessity — for example, mathematical necessity —a

modality. For they mean by that, that necessity is a mode of being-true.

Indeed with different emphasis we better expose their perspective: they mean

to say that necessity is a mode of being-true. These philosophers of the

present day rather overlook the practical, so that, in pondering necessity, they

consider only the theoretical. The so-called ‘analytic’ orientation in philosophy

especially encourages such forgetfulness of the practical and a resulting

overconcentration on the theoretical. In order to limit necessity to the

theoretical, a philosopher needs what is assertoric to take over the universe of

reason (as it were). Analytic philosophers typically willingly stage just such a

take-over. Necessity is but a way-of-being-true that some assertions have and

others lack, they claim. What is imperatival rather than assertoric, and thus

what is practically rather than theoretically necessary, they would simply like to

abjure, or explain away. Yet one thing we can learn by considering Newton, and

his mathematics, and indeed the fit in his mind of that mathematics to physics,

is that the mere theoretical conception of necessity (as a modality) cannot

justly illuminate necessity by his lights.

In any case if we consider diagrams merely as visualizations, then we fail

to bring out a certain duality of the diagram that much fascinated the Greeks.

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Euclid purposes the diagram to portray a practical deed. Yet the diagram also

epitomizes timelessness of form for perfectly executing that deed. If,

fundamentally, a proposition for Euclid is something practical that it is

proposed to do, and moreover rational demonstration for Euclid means moving

straightedge and compass perfectly through specific ideal steps in order to do

what has been proposed, then for Euclid proposition and demonstration both

involve motion. But the diagram epitomizes timelessness. Therefore the

diagram sports a profound duality between temporality and timelessness.

In order for proposition-and-demonstration mathematics, such as Euclid’s

in his Elements, or Newton’s in Principia, to be science, it must be rationally

perfectable, and in order for it to be rationally perfectable, it must sport an

ideality that transcends what humans can possibly accomplish (and that rather,

in its transcendental finality for all cognisers, rises quite beyond what is merely

immanent or human). The Greek route to such ideality is through that which is

drawn. Newton adopts the same path as his own and steps even further along

it. Inhering in the actual materiality of straight-edge, compasses, and

geometer, there are doubtless impediments to performative perfection in the

execution of diagrams. But, to make geometry a science, you ignore the

impediments. You ignore, for example, that if two drawn curves truly intersect

punctually, then it will be infinitely involving just to push a straight edge up, or

set down the point of a compass, exactly there. Over against the actual

impossibility, you just do it, ignoring the imperfections that must actually

inhere in your practice. By drawing out for us what is momentary about

motion, Newton redoubles that kind of ideality that the diagram must have, but

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at the same time also redoubles the connection, via the diagram, between

insight, and the grasp of a timeless constraint upon motion.

Thus Newton states at the outset in Principia that whereas “mechanics is

distinguished from geometry by the attribution of exactness to geometry and

of anything less than exactness to mechanics … [y]et the errors do not come

from the art but from those who practice the art”. Consequently, in the limit of

a transcendent perfection of the artisan, geometry and mechanics merge as

one single science after all. “Anyone who works with less exactness is a more

imperfect mechanic, and if anyone could work with the greatest exactness, he

would be the most perfect mechanic of all” —namely, God (Principia 1999

[1687], Author’s Preface to the Reader; p. 381).

Newton’s own human efforts with diagrams in Principia concern of course

how the world moves. Yet for Newton that is a delving not just down towards

the exactitude of which God alone is capable (although it is that). Also it delves

down towards the actual on-going work of God. By diagrams themselves

humans fathom formal timelessness within the manner by which material

things change. As God is geometer, He brings timeless form to how the world

goes.

§3. Analysis of continuous change: Newton’s artful rumination upon forces.

Some background: in some further reasoning of Newton’s that I next

briefly discuss, Newton significantly uses Galileo’s law of free fall (that in freely

falling motion ignoring resistance the vertical distance that a body traverses

from rest is proportional to the square of the time taken, irrespective of the

body’s mass or state of horizontal motion). Newton significantly uses Kepler’s 17

three laws of planetary motion. These are: First, that secondaries orbit their

primaries (e.g. the planets orbit the sun) in elliptical orbits, with the primary at

one focus; Second, the law of areas, above described — that secondaries

sweep out about their primaries equal areas in equal times; and Third, that

among the secondaries of a common primary, such as among the planets

relatively to the sun, R3/T2 is a constant, where R is a secondary’s average

distance from its primary, and T its orbital period. Newton also very

significantly uses his own three laws of motion: First, the law of inertia, Second,

F = ma (which Newton himself hardly expresses thus algebraically!), and Third,

the law of the equality of action and reaction. Newton, Lucasian Professor of

Mathematics, in that capacity also stands deeply immersed in the tradition of

diagrammatic reasoning that propagated down the centuries from Euclid. It is

true that Viète and Descartes had produced symbolically-styled forms of

geometric reasoning and Newton helped compound the power invested in

these inchoate symbolisms. I call the symbolisms inchoate because they long

antedate any adequate theory of a numerical continuum, yet they are boldly

beholden to some such future success. To Newton’s mind,7 symbolically-styled

reasoning is untrustworthy, whereas diagrammatic demonstration is by

contrast entirely apt to the qualities of mathematical necessity and insight.

This assessment by Newton of the situation in his own day was in fact perfectly

apt, and fair.

7See for example Newton on geometry and algebra in Universal Arithmetic, tr. J.

Raphson, London, 1769 (2nd edition), pp. 465–470.

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Consider then as far as this sketched background makes possible Newton’s

reasoning, early in Book 1 of Principia, surrounding the diagram for his

Proposition 6 there:

Here, Newton draws out as PR what the momentary motion would be of a

planet that is not acted upon at all by the sun and instead moves inertially, and

draws out as PQ what, by virtue of the sun’s action upon the planet, that

planet’s motion actually comes to in the self-same moment. QR is taken to

visualize the sun’s deflecting impulse upon the planetary body at P, and,

because sun-centred, is thus parallel to SP. Under constraints imposed by (i)

Galileo’s law of free-fall, which becomes accurately or very nearly true in the

momentary since then the distances to S are approximately all the same, (ii)

Newton’s own laws of motion, and (iii) the knowledge from Proposition 1 that

the area SQP is proportional to the time taken, Newton walks us through

various geometrical considerations that together imply that Kepler’s First Law

obtains if and only if the force that QR helps us to visualize is inversely as the

square of the planet’s distance SP from the sun. Thus in their analytic phase

these reasonings deduce from the phenomenon of ellipticality the cause of that

very ellipticality, namely that there is a sun-centred, inverse-square force. In

their synthetic phase these reasonings deduce from the cause (viz., there being

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a sun-centred, inverse-square force) the phenomenon (viz. that planets’ orbits

are elliptical, as stated by Kepler as his First Law).

Orthodoxy holds that rigor was brought to Newton’s calculus only by the

imposition of a theory of limits. It is even said8 that Newton points us in

Principia to just such limit-theoretic understandings. I am cautious about this

view. Limit theory’s whole purpose becomes, in the nineteenth century, to

suppress intuition in mathematics and heighten the purchase of logic. Dealing

limit-theoretically with motion is in my view the involute of Newton’s approach.

Newton draws out for us what is momentary about motion, whereas limit

theory instead carries what it analyses back away from us, to just beyond the

horizon of our explicit attention. In this way it much nullifies intuition and

heightens the purchase of algebra and logic. The fruit of this effort is the

nineteenth-century delivery of a coherent, cogent theory of the numerical

continuum. The numerical continuum must contain all its limit points, that is to

say, any limit of a sequence of ratios of numbers must also be a number. Then

the workings of the calculus can be made out as secure. This way to ‘rigourise’

mathematical analysis has significantly dominated the mathematical attention

of the twentieth. Again, the patterns of mathematical attention that dominate

in the present day form obstacles to our understanding Newton. Significantly it

was Newton’s calculus that the nineteenth-century mathematicians sought to

rigourise; but this is one instance of many where the enormity of Newton’s

effect on us makes problems for our today understanding Newton as Newton in

his own century was.

8Among others, by Guicciardini (1999, 2009), whom I discuss below.

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Guicciardini’s in-most-ways-magisterial 1999 is plain in its pronouncement

that Newton himself treats of limits. Commonly the diagrams of Newton’s that

I have above introduced are in the present day thought unreadable except in

such a way as implies the ‘taking of a limit’. A first, apparently verbal move

against this conception which I will argue is actually robustly telling, is to note

that Newton writes not of ‘limits’, but of ‘first and ultimate ratios’. There is

something further to spot: the ratios are not numerical, but rather are of

geometrical quantities. I shall next quote from Guicciardini, 1999, about this

(pp. 43-44):

Newton … points out that the method of first and ultimate ratios rests on the following Lemma 1:

Quantities, and also ratios of quantities, which in any finite time constantly tend to equality, and which before the end of that time approach so close to one another that their difference is less than any given quantity, become ultimately equal.

Newton’s ad absurdum proof runs as follows:

If you deny this, let them become ultimately unequal, and let their ultimate difference be D. Then they cannot approach so close to equality that their difference is less than the given difference D, contrary to the hypothesis.

This principle might be regarded as an anticipation of Cauchy’s theory of limits, but this would certainly be a mistake, since Newton’s theory of limits is referred to a geometrical rather than a numerical model. The objects to which Newton applies his ‘synthetic method of fluxions’ or ‘method of first and ultimate ratios’ are geometrical quantities generated by continuous flow.

Here I dispute not Guicciardini’s resistance to the idea that in Newton there is

anticipation of Cauchy’s theory, which resistance is perfectly cogent, but rather

Guicciardini’s crediting to Newton any ‘theory of limits’ at all. My objection is

subtle. The quantities of which ratios, first and ultimate, are to be considered,

are as Guicciardini points out geometric. (Even in some ways by Guicciardini

himself, translations of Newton’s Latin expressions for chord lengths, areas etc.

21

ease us over to thinking numerically, as for us is the familiar way to think,

when in order to grasp Newton’s meaning we need to be thinking

geometrically; so, holding on to Guicciardini’s insight here that Newton is not

Cauchy can be difficult for us.) The “ultimate ratio” of geometric quantities is

still geometric, yet in a way that no human can depict; and it remains a ratio.

Cauchy helps us towards the conception that the limit of a convergent

sequence of numerical ratios just is, not a ratio, but a number. That is a

different point of view. What is ultimate for Newton is not thus numerical, but

is rather irreducibly a ratio of geometrical quantities, quantities infinitely

smaller than we can draw or depict, yet geometrical for all of that. We can

draw out only grossly what is thus ultimate, but that is no more to say that the

ultimate is other than what we draw out, than we would call it impossible to

draw a geometrical line, or to diagram a geometrical point. Properly to grasp

Newton’s intent about the drawing out, is I believe to see in limit theory but the

involute of Newton’s approach.

Now, as is well known9, the Proposition 6 reasoning of Newton’s above,

concerning the way that (with other considerations) ellipticality of orbits implies

an inverse-square distance-dependency of the force, is the less compelling so

far as human knowledge is concerned, because astronomers in Newton’s day

could not attest empirically that Kepler’s First Law is in all respects “accurately

or very nearly true”. To determine empirically the exact figure of a planet’s

orbit relatively to the sun is prodigiously difficult. So while the astronomers of

Newton’s day could empirically attest to the high accuracy of Kepler’s Second

9See Curtis Wilson, 1970.

22

and Third Laws, in respect of the First Law they could attest to the high

accuracy only of a special implication of the First Law, viz., that planets will

approach the sun nearest always on the same side of the sun with respect to

the fixed stars. So in the order of our own knowledge, Newton quite

deliberately privileges, later in Principia, some more special considerations

which (following W. Harper, but interpolating my own emphasis on analysis and

synthesis) I spell out as follows:

23

24

Note incidentally the remarkable consilience within Newton’s reasonings:

by two very different measurement inferences from quite utterly disparate

astronomical phenomena (quiescence of perihelia, harmonic law), Newton is

able to deduce one and the same theoretical conclusion, viz. that the distance

variation of the sun-centred force acting on the planets is inversely as the

square.

Given that these later considerations are, within the order of our own

knowledge, more telling, why does Newton set out so early in Principia the

diagram

with which (by diagrammatic analysis) to link ellipticality and the inverse-

square character of the force? In Newton’s order of exposition, an analysis

occurs early that is scarcely as telling for our knowledge as are the exacting

demonstrations that occur in Principia later on. Newton therefore departs in his

order of exposition from the order of human knowledge. Why?

Much magic, not a little theology, and yet at the same time the ignition of

science itself, all devolve upon the answer that is needed, here. Rich clues to

what is going on with Newton, Newton himself delivers to us in the very Preface

to Principia. Here, Newton mentions Pappus six words in. The opening

paragraph of the Preface is as follows:

25

Since the ancients (according to Pappus) considered mechanics to be of the greatest importance in the investigation of nature and science and since the moderns rejecting substantial forms and occult qualities have undertaken to reduce the phenomena of nature to mathematical laws, it has seemed best in this treatise to concentrate on mathematics as it relates to natural philosophy. The ancients divided mechanics into two parts: the rational, which proceeds rigorously through demonstrations, and the practical. Practical mechanics is the subject that comprises all the mechanical arts, from which the subject of mechanics as a whole has adapted its name. But since those who practice an art do not generally work with a high degree of exactness, the whole subject of mechanics is distinguished from geometry by the attribution of exactness to geometry and of anything less than exactness to mechanics. Yet the errors do not come from the art but from those who practise the art. Anyone who works with less exactness is a more imperfect mechanic, and if anyone could work with the greatest exactness, he would be the most perfect mechanic of all. For the description of straight lines and circles, which is the foundation of geometry, appertains to mechanics. Geometry does not teach how to describe these straight lines and circles, but postulates such a description. For geometry postulates that a beginner has learned to describe lines and circles exactly before he approaches the threshold of geometry, and then it teaches how problems are solved by these operations. To describe straight lines and to describe circles are problems, but not problems in geometry. Geometry postulates the solution of these problems from mechanics and teaches the use of the problems thus solved. And geometry can boast that with so few principles obtained from other fields, it can do so much. Therefore geometry is founded on mechanical practice and is nothing other than that part of universal mechanics which reduces the art of measuring to exact propositions and demonstrations. But since the manual arts are applied especially to making bodies move, geometry is commonly used in reference to magnitude, and mechanics in reference to motion. In this sense rational mechanics will be the science, expressed in exact propositions and demonstrations, of motions that result from any forces whatever and of the forces that are required for any motions whatever. The ancients studied this part of mechanics in terms of the five powers that relate to the manual arts [i.e., the five mechanical powers] and paid

26

hardly any attention to gravity (since it is not a manual power) except in the moving of weights by these powers. But since we are concerned with natural philosophy rather than manual arts, and are writing about natural rather than manual powers, we concentrate on aspects of gravity, levity, elastic forces, resistance of fluids, and forces of this sort, whether attractive or impulsive. And therefore our present work sets forth mathematical principles of natural philosophy. For the basic problem [lit. whole difficulty] of philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces. It is to these ends that the general propositions of books 1 and 2 are directed, while in book 3 our explanation of the system of the world illustrates these propositions. For in book 3, by means of propositions demonstrated mathematically in books 1 and 2, we derive from celestial phenomena the gravitational forces by which bodies tend toward the sun and toward the individual planets. Then the motions of the planets, the comets, the moon, and the sea are deduced from these forces by propositions that are also mathematical. If only we could derive the other phenomena of nature from mechanical principles by the same kind of reasoning! For many things lead me to have a suspicion that all phenomena may depend on certain forces by which the particles of bodies, by causes not yet known, either are impelled toward one another and cohere in regular figures, or are repelled from one another and recede. Since these forces are unknown, philosophers have hitherto made trial of nature in vain. But I hope that the principles set down here will shed some light on either this mode of philosophizing or some truer one.

Armed, in part, with the above thinking concerning Pappus and

diagrammatic reasoning, One gathers from this Preface conclusions that are

largely novel to Newtonian scholarship. It is true that in recent decades

Newton scholarship has significantly far-redressed an earlier resistance to the

idea that Newton could have produced deductions from phenomena at all.

Philosophers of science (who evidently had not read their way even the tiniest

distance into Newton’s Principia) for some decades treated with staged derision

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Newton’s concept of deductions from phenomena, signaling that very concept

as indication (and they effected to find others as well) that Newton was

philosophically feeble. Yet by now Principia has been better investigated.

William L. Harper, George E. Smith, and many others, have far advanced our

understanding of both the mathematical integrity, in Principia and elsewhere,

surrounding Newton’s deductions from phenomena, and the exemplariness as

science of just such measurement inferences. Yet within this new literature I

know of no-one who duly emphasizes Pappus, or the diagram with its practical

connection, or the theological suggestion that Newton advances in his Preface,

that, by his diagrams in Principia, Newton delves as deeply as a human might

into the working out by God of the movement of the world. Thus Proposition 6

achieves as synthesis deep-going explanation: the cause of the orbit’s

ellipticality is the inverse-square distance-dependency of the force. As Pappus-

styled analysis, that yields the starting point for just such synthesis, the

reasoning treats as given that the orbit is elliptical, and deduces the cause,

viz., the inverse-square distance-dependency of the force.

Many contemporary scholars of Newton seem convinced however that

Newton suppressed analysis from his Principia. To draw as they do a blank

about analysis in Principia and the relation that the Preface tells us is there

between analysis, deductions from phenomena, the movement of the world

and God, all represents a grave misstep in my view. How large a misstep, I

believe can be measured by Newton’s rages.

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§4. Magic, natural philosophy and God.

Newton rages like no other, at precisely those ordinary men in his midst

who dare distrust his delvings into the ultimate nature of things. Who is Robert

Hooke (for example), to question Newton’s penetration mathematically of the

phenomena of light, or Newton’s mathematical priority in the discernment of

inverse-square centripality of the celestial force? Newton’s rage against Hooke

is the stuff of wizardly combat. A magician delves deep into hidden

connections in the universal order. This confers upon the magician powers that

ordinary mortals do not possess. Question the magician’s deep-going

discernment and you question the very art. Newton is, and aims to be.

thoroughly a magician of the dark arts of alchemy and the occult. His rages tell

us all. But Newton at the same time redefines aspirations for knowledge, in

ways that have precipitated us into a scientific age. To this, mathematics is

key. A scientist delves deep into hidden connections in the universal order.

This confers upon the scientist powers that ordinary mortals do not possess.

Truly we need not to read the magic out of Newton’s Principia, but rather to

read magic into the subsequent science.10 Remove the dross and render it

serious, and magic takes the form that we today recognise as science.

Mathematics is key to this process. And similarly we need not to read religion

out of Principia, but to understand instead simply that, in its every diagram,

Newton seeks after a closest human approximation to the intellection of God.

Moreover, something like the transcendent standard that Newton presupposes

is presupposed still by science in the West.

10To write these words, I electronically transferred files between home and office and

depended on LCD screens and laser printers. Such technologies are for all of that pretty profound realisations of the ambitions associated formerly with the magical arts.

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Why people understand Principia as serious science is much because of its

mathematical brilliance. To call Principia exemplary is both true and deeply

ironic. Scarcely anyone can even read the book, let alone directly emulate it.

Yet it shifted fundamentally what humans would consider epistemic knowledge

even to be. Its direct influence is less on the understanding than on the

understanding of the understanding. Principia profoundly reshaped humanity’s

aspirations for thought. Indeed, the work, despite its scarcely ever being read,

is often picked, rightly as I believe, as the most influential single work of its

millennium.

Shocking to the world it was therefore to discover Newton’s ‘magician’

side. Scholars have nonetheless discovered in Newton’s papers vast evidence

of alchemy and passionate religious intellection. Even among the scholars who

have most studied these dimensions of Newton however, and certainly among

those who base their impression of Newton mostly on the published works, the

effort that has been shown is not quite to consider equally parts of serious

science by Newton his prodigious alchemical researches, or the Biblical

exegesis and endeavour to reinterpret religion which were far vaster still.

However serious Newton himself may have been about them, these elements

of his genius are relegated by some to the status of having been frivolous, or

by others to the argument that the 17th century was not yet anywhere a

century of science.11 Even those scholars who study Newton most intently (and

are consequently in greatest sympathy with him) seem inclined to pass with

11Thus, J. E. Maguire and P. M. Rattansi in their pivotal 1966 argue (p. 138) that “Isaac

Newton … was not a ‘scientist’ but a Philosopher of Nature. In the intellectual environment of his century, it was a legitimate task to use a wide variety of material to reconstruct the unified wisdom of Creation.” Very true: and yet by his diagrammatic analysis as a magician’s art with theological intent, Newton nonetheless sets the defining example for the ignition of science.

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some embarrassment for him over these far vaster reaches of his intellection,

and simply to thank goodness that they did not prevent Newton from producing

for us his Principia.

Clearly therefore the aspirations for knowledge by which Newton is taken

to have redefined his millennium are understood to connect neither with

alchemy nor with religion never mind that these pursuits together were well

over half of that man. The epistemic aspirations that we are understood to

have acquired from Newton relate to mathematical brilliance in the

investigation of nature and concern Principia chiefly and little else that that

man was about. Of course this attitude implies that Principia comes apart from

both alchemy and religion. When this attitude is countered by E. Maguire and

P. M. Rattansi (1966), B. J. T. Dobbs (1975, 1991), and even in some ways by R.

S. Westfall (1971, 1980), their point is not so much as I would wish it to be, that

Newton brings out his overall concerns brilliantly within the mathematics that

he prosecutes. The proper appreciation of diagrammatic reasoning is key to

our taking this needed step.

To study Principia in the spirit in which it was written is almost impossible

today, for the mathematical culture it is roundly a part of is significantly alien

to our own. Yet Newton’s many dimensions together may be the basis for

surmounting the problem. For example, his theology can illuminate the high

literal-mindedness that he not only appreciates in Euclid but practices similarly

in his physics. The intellection of the West is regulated by the thought that,

while you can never actually manage to be God (since God is perfect beyond

possible human accomplishment), yet, ideally you would be God. The West

lays upon your shoulders the search for The View from Nowhere.12 High literal-12

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mindedness is the essence of this search. For example, Newton lays out in the

Scholium to the Definitions in his Principia reasons to be literal minded about

temporal durations.13 Treating of duration as universal in its significance and

possessed of an objective measure involves, however, in our own efforts at

measurement of duration, a transcendental finality, a limiting goal that

transcends what we humans can ever accomplish practically. Any material

process short of the dynamical unfolding of the cosmos as a whole will be

interfered with gravitationally if not in other ways by other material processes,

so that any material process to which we might look practically as a clock, if

adopted as our standard for duration, would lead us to conclude inconveniently

that order everywhere is subtly defiled. We would discover in the wider

universe processes sometimes speeding up a little and sometimes slowing

down a little, for no physical reason that can be discovered in those processes

themselves. Consider then the measure, the adoption of which as a standard

for duration is consistent with finding that every other process ever only

speeds up or slows down for an identifiable physical reason. Newton showed

definitively why we must admit that this measure is transcendent of our

possible practical grasp. Nothing short of the cosmos as a whole, or God’s all-

encompassing insight, can pull into view quite that perfect measure of

duration. Consequently those ‘who confuse true quantities with their relations

and common measures’ both ‘do violence to the Scriptures’ i.e. violence to the

glory of God and ‘no less corrupt mathematics and philosophy’ (Newton 1999

Thomas Nagel’s 1986 by this title is in my estimation one of the great books of the West.

13The emphasis on literal mindedness is my own, but see DiSalle 2002 in relation to the

following.

32

[1687], p. 414). We cannot practicably ground in measurement our own literal-

mindedness about duration, but in order to possess exact science, we must

nevertheless help ourselves to the notion (as given by God).

It is hubris perhaps for the West so far originally to have wrapped the

infinite into its cultural fold, but we do at least have burgeoning exact science

to show for it. Consider, then, what cultural conditions there are for our

possession of the very notion theoretically of temporal duration. And behold

the vaulting commitment to reason-in-the-world that lies within that cultural

form. That there is a measure (however transcendental) of duration, the

adoption of which as a standard for duration is consistent with all processes

everywhere ever speeding up or slowing down only for a reason lying within

those processes themselves, is an idea that palpably involves vaulting

commitment to reason-in-the-world. In order for there to be temporal duration

in such literal respects, every last process must be exquisitely rationally

harmonized with all the other processes; everything must unfold as it does for

reasons, reasons across which there is likewise harmony or coherence. We are

a hair’s breadth from theism in the making of such a vaulting assumption.

Cross, moreover, a philosopher or a scientist or a mathematician, on the

question whether literal-mindedness is itself an acceptable commitment, and in

the glowering agony of their response, you will see how palpably you do insult

to an article of faith. I am myself disposed to view our vaulting commitment to

reason-in-the-world “critically” in Immanuel Kant’s sense of that word. The God

concept may, to deep-going degree, be regulative of Western culture including

Western philosophy and science, but I frame no belief with it on that account of

anything beyond the natural. Only on account of my culture’s rationalism is

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the very concept of nature or the natural fully available to me, so that in its

very culturally emergent quality, the concept of nature will not be able to cover

quite everything. Some commitments stand as conditions for the possibility of

that cultural form, and to wrap into philosophy or science the task of

evidencing or otherwise warranting those commitments will on that account

fail. However strongly disposed I am, given my culture, to think that nature

completes my situation, still I will struggle to fit meaning itself or mindedness

or ethics or mathematics, or anything of which the original touch of infinity is

defining, quite under its fold. Naturalizing intentionality, naturalizing ethics,

naturalizing mathematics, all to me seem fraught philosophical projects,

however fully the impulses resonate with me that draw other philosophers into

such pursuits. Yet the agony here is to me no inducement to have truck with

the supernatural. Rather I see it as an original instability of my culture (and yet

a creative one). Some aspects of the vaulting investment in reason itself are

much like faith, and God as metaphor is even helpful in some degree as

explanation of their directedness. Yet the condition in question, being

thoroughly cultural, seems to me not in the end to tell about the world as it is

in itself. It tells at most to tell about us, who are in the circumstance of western

philosophy, and science.

Among the difficulties that stand in the way of comprehension of Newton

by contemporary Newton interpreters is the relation he believes he possesses

to God. Here, however, recognition of the ideality of mathematical practice can

be of some help: we can at least remind ourselves usefully, that in the West,

and as a seeming necessary condition for the full blossoming of exact science

here, humans have relinquished expecting that the ideals to which human

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practices are subject are even possibly realizable by humans. Whereas the

East has conceived the ideal as humanly realizable (albeit ever so infrequently,

e.g. by a Buddha or a Confucius), and whereas the East sports little or no truck

with the monotheist’s God, in the West the ideal for human practice has been

made out as transcendent of possible accomplishment by humans. Newton

may deny the literal divinity of Christ, but insofar as this rivalry (of Newton with

Jesus himself) concerns who stands the closer to God, still the transcendence of

God is admitted. That there is, beyond the humanly practicable, such a

transcendent standard, is as needs be, Newton maintains. That the ideal for

human accomplishment is transcendent of possible human accomplishment is

necessary, he maintains, in order for mathematics, in Newton’s understanding

of that word, to fall out as it does as exact science. While we may balk in other

ways about Newton’s theology, we must concede that he is correct on this

point. A standard that is transcendent of possible human accomplishment is

also necessary in order for natural philosophy to admit of its mathematical

principles. Newton understands this clearly, and if we want for ourselves his

own exact science, again we must concede the correctness of this view of his.

Newton’s early calculus far further advances the algebraically situated

‘analytic art’ of Viète, rendered into analytic geometry by Descartes, and then

generalized in its application by many subsequent mathematicians, including

some whose concerns were with infinite series. Powerful though Newton’s

advancements were in these methods, by the mid 1670s he began to compare

them invidiously with insights of the ancients. Leading up to this change in his

views, he had deeply studied the seventh book of Pappus’s Collectiones. I

propose to think that Newton as mathematician shifted from symbol back to

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diagram, for the reason that here he better connected in his own estimation

with God. Scholars such as B. J. T. Dobbs have shown that from the early 1670s

Newton connected with alchemy his quest, which was theological, to restore

ancient knowledge. The ancients’ orientation to the diagram was something to

sift and weigh and consider, and doing so promptly paid Newton an inspiring

dividend. By diagrammatic analysis that was newly turned upon motion or

physical change, Newton found that he could discover what is so deep-lying in

the causal order as to seem but the work of God. In the diagram, the ancients

had something right, which the moderns had lost.

Diagrammatic reasoning sorts out how with straightedge and compass one

should as mathematician act, and so its burden is partly muscular. This makes

it apt for physics. Diagrams inform us about constraints on motion. Indeed,

while calculus and nineteenth-century conceptions of rigor chased the diagram

for a long while from the fore in mathematics to the rear, in physics allure for

the diagram has been steadier. Newton as mathematician might seem singular

to have concocted the calculus yet cleaved to the diagram. But we need to

remember that Newton was also physicist. Significantly much as physicist was

Newton seeking special closeness to God.

§5. Plato, Newton, and the diagram.

Developments which are significantly recent, like the nineteenth-century

creation of the theory of real numbers, or the breakthrough discoveries about

logic itself which have made the computer age possible, at the same time

conspire to obscure the vision we might otherwise possess back onto the

culture of mathematical inquiry of which Newton is a part. Yet in all the ways

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by now remarked and more, we need clearly to appreciate Newton’s creativity

within that culture if we are accurately to understand Newton’s place within,

and overall impact upon, the philosophy of the West. This impact is large, in

part because Newton vindicates the ancient Greek Plato, signally. He thus

epitomizes (no less signally than problematically) a certain kind of triumph of

the West. Newton vindicates Plato’s conception that physics or natural

philosophy, if mature, must be mathematical. This accomplishment of

Newton’s also demonstrates how natural philosophy can come together as a

science. In short, Newton begins to transition the West profoundly from one

epoch to another, from an epoch of significantly unfulfilled aspiration for

science to an epoch of burgeoning accomplishment in it. (Before Newton,

western philosophy relentlessly asked the question whether a science of nature

is possible. After Newton, western philosophy asked how science is possible.

Newton accomplished something so significant that, in its light, doubt could

scarcely be maintained whether science is actual.) Yet we are blocked first

from understanding Plato, and thence from properly understanding Newton,

precisely by the scale of that change.

Notably a synthetic philosopher, Plato takes vast inspiration from the

culture of mathematical inquiry in his day. Plato considers the opposition

between presocratic philosophers Parmenides and Heraclitus. Mathematics is

key to how Plato surmounts this opposition. Parmenides makes out the real as

timeless and necessary and utterly unitary. Heraclitus considers with equal

rationality what conclusions to draw about the very idea of order in the real,

granting that, as experience teaches us, change, or flux, obtains. Heraclitus

concludes that change could not but be radically ubiquitous if it obtains at all,

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and therewith the very idea of order in the real seems to Heraclitus forfeit.

These views are thesis and antithesis to Plato’s synthesis. Plato coaxes us to

think that deep-lying rationally apprehensible form in the real might after all be

discoverable by humans, provided (or to the extent) that people can but

fathom formal timelessness within the manner by which material things

change. Necessity of practical imperatives must precede necessity of truths.

Plato considers diagrammatic demonstration of geometrical propositions

actually to involve, and thereby to epitomize, such insight. It is no wonder then

that Plato expects, otherwise wholly presciently, that physics, if mature, must

be mathematical.

The Greek mathematical diagram has as its whole purpose to epitomize

timelessness in the manner in which the instruments of geometrical

construction can most perfectly be moved through certain ideal steps. And just

as Plato’s philosophy of mathematics cannot be separated from the diagram, or

therewith the perfection of practice, so correspondingly, neither can Newton’s

accomplishment. To fathom well why diagrams are for Newton no mere aids to

pedagogy but rather fully essential to his reasoning in the Principia is to begin

to comprehend Newton not in pieces but rather as one whole.

Heterogeneous by present-day lights, the many sides of Newton pose a

challenge to us. Alchemist, theologian, magician, raging recluse, icon,

mathematician, practical man, progenitor of physics as exact science — why

must we struggle so, simply to understand the sides of this singular genius as

forming one magnificent (if estranged, solitary) self? The key to solving this

dilemma is the diagram. For to appreciate the diagram is to fathom many

points all at once of separation of us from Newton. To appreciate the diagram

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is thus to be reminded of many points all at once of how Newton has affected

our thought. The diagram therefore potentially steps us towards connecting

with Newton comprehendingly, of him not in pieces but as one whole.

B. J. T. Dobbs powerfully argues that Newton is to be considered as a

coherent whole. I very much agree, yet I believe I bring the physicist well

forward within this whole whereas Dobbs allows the physicist in Newton to be

much eclipsed. The key to seeing Newton as one whole and as prominently

physicist is the diagram, recalling how it mathematises motion, reveals deep-

lying causes, is practically connected, supports wizardry in deep-going

discovery by those adept in its use, and connects, in the practical ideal to

which it is beholden, ultimately to God. Dobbs (here quoted as by Margaret J.

Osler) contends that Newton’s accomplishment was

not just of ‘the mathematical principles of natural philosophy.’ On the contrary, it was to have been a grand unification of natural and divine principles, and it included a vision of God’s activity not only in this world as we know it but also at the world’s beginning and at its apocalypitic end and renovation. It was a vision in which the Aryan Christ, as God’s ‘Agent’ throughout time, always putting the will of the Supreme God into effect, kept God intimately connected both with the physical world and with humanity: that was Newton’s ultimate answer to the twin spectres of deism and atheism that had always haunted him. It is also a vision that forces one to the conviction that one must give a religious interpretation not only to Newton’s alchemy but to all of Newton’s work, including the Principia and Opticks, since Newton himself was apparently motivated to study ‘the frame of nature’ in order to learn of God’s activity.14

I say yea here to all this, but contend as well that diagram clinches this unity;

about whether the mathematical principles of natural philosophy are key Dobbs

demurs, yet that removes the linchpin of the unity in my view.

14Dobbs 1991, p. 254; my eye was drawn to this in Osler 2004, p. 10. For a broader

picture than concerns Newton’s times alone or the science / theology connection in him alone, see Osler 2010. See also Snobelen, whose unpublished n.d. has helped me here.

39

§6. Attitudes.

When physicists direct their minds to the pinnacle of the human condition,

they of course discover physicists there. Newton is one whom they see there.

Physicists are typically very proud that they possess Isaac Newton as one of

their own. To some extent, Newton’s own intellectual purposes play on among

physicists, even now. Yet when, by the enormous force of his mind, Newton

fulfilled his intellectual purposes pointedly, the whole human historical epoch

changed. Physics as exact science ignited. An age of science began. Of

course this iconic example of intellectual virtuosity by a physicist is validating

for physicists. And their satisfaction in Newton the present study in no way

seeks to undermine. But clearly I am attempting nonetheless to broaden, and

in this way to challenge, how physicists think about Newton. Newton was a

creature of a very different century. Twenty-first-century physicists do still

know a profound part of Newton it is true, yet they do not know even a tenth of

what he was. If they consider Newton as at all strictly one of their own, they

fall, I believe, into a mistaken view of how their own scientific discipline first

became possible at all.

Artists, not only physicists, view Newton through the lens of textbook

physics. Artists on that account see a stunted figure. I do not endorse their

understanding of Newton, but I do endorse their conviction that on that

understanding he makes a less than well rounded figure. Hollow yet muscular

and largely empty-headed, Salvador Dali’s Newton possesses a vast personal

energy focused into what is perfectly symbolically describable about the real:

the swing of a pendulum. The pendulum and the pendulum alone commands

Newton’s inside. Seemingly in this artist’s view, nothing about Newton,

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certainly not his paltry genitals, befit a man-of-the-world. Dali’s depiction of

Newton seems to be an essay in contempt. And famously William Blake paints

Newton no more favourably. In his poetry accusing Newton of ‘single vision’,

Blake depicts the man crouched over his diagrams, his back turned to the wider

universe. Muscularity is

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again symptomatic of an ill-focused energy, and the world seems to have been

darkened, not illuminated, by Newton’s effort.

My plea to physicists — to consider the part that they know of Newton to

be not even a tenth of the man — is at the same time a plea to the artists, to

consider Newton a far more enchanting mind by an artist’s own lights than

contemporary textbook physics reveals. Only by dint of many further

dimensions could Newton begin to have the historical significance that he has.

Let us get real, and get over taunts that Newton is somehow fundamentally

feeble. And surely let us understand that Newton is extraordinary in his

practical dimensions: his mathematics is art, its necessity aesthetic, its rational

force a balance of analysis and synthesis.

When philosophers direct their minds to the pinnacle of the human

condition, they of course discover philosophers there. Philosophers: in your

case this foible cannot be helped. On the contrary, this foible is deeply over-

determined by conditions upon the cultural form generally within which the

philosophical activity first emerges. Philosophy requires a high literal-

mindedness such as alone also makes wider theoretical pursuits truly possible.

Once literal minded theorizing is launched, philosophy is inescapable.

Philosophy consists in a deepest-going literally minded effort to consider what

to believe, what to value — and why. To understand that pursuit modestly, as

playing perhaps second fiddle compared to other things that people can do,

abandons prioritizing reflection as highest human calling of all. But that makes

the person no longer a philosopher. Philosophers ask the hardest and most

deep-going questions. They are required to be pinnacle humans in their own

eyes, lest they be left doubting (quite against their calling) whether reflection

42

intellectually is highest. However far symptomatic high regard for reflection is

of a wider orientation of the culture, its pinnacle status is a unique burden for

philosophers professionally to bear. The resulting foible, of over-determined

immodesty, occasionally bends the philosophical intellect into an unfortunately

obdurate state. People talk of a ‘gaggle’ of geese, a ‘flock’ of sheep, a ‘school’

of fish, a ‘herd’ of cattle, and an ‘arrogance’ of philosophers.

Newton is himself key to how philosophy swung wide into an analytic

phase. Certain obdurate qualities follow into the philosophical imagination by

virtue of the analytic philosophical orientation, that quite preclude philosophy’s

currently appreciating Newton’s intellect as it was. The force of Newton’s

discoveries was epoch-making. He precipitated profound intellectual change.

We are left by the effect of his genius the less able to comprehend clearly the

qualities themselves that that genius possessed. In order to grasp Newton we

must understand both our separation from him and in that, a connection. This

is a challenge, and so in order to accomplish it, obdurate about our

contemporary philosophical dispositions is just what we need not to be.

Bibliography

Aquila, Richard. (1983). Representational Mind: A Study of Kant’s Theory of

Knowledge. Bloomington, IN: Indiana University Press.

Catton, Philip and Clemency Montelle. (2012) “To Diagram, to Demonstrate: To Do, To

See, and To Judge in Greek Geometry”. In Philosophia Mathematica (III) 20, 25–57.

Cohen, I. Bernard. (1999). “A Guide to Newton’s Principia”. In Newton 1999.

43

DiSalle, Robert. (2002). “Newton’s Philosophical Analysis of Space and Time”, in I. B.

Cohen & George E. Smith (eds.), The Cambridge Companion to Newton.

Cambridge: Cambridge University Press.

Dobbs, Betty Jo Teeter. (1991). The Janus Faces of Genius: the role of alchemy in

Newton’s thought. Cambridge: Cambridge University Press.

Euclid. (1956 [antiquity]). The Thirteen Books of The Elements. Thomas L. Heath

(trans., ed.). New York: Dover.

Friedman, Michael. (1992). Kant and the Exact Sciences. Cambridge, MA and London,

England: Harvard University Press.

Guicciardini, Niccolò. (1999). Reading the Principia: The Debate on Netwon’s

Mathematical Methods for Natural Philosophy from 1687 to 1736. Cambridge:

Cambridge University Press.

---------------------------. (2009). Isaac Newton on Mathematical Certainty and Method.

Cambridge, MA: The MIT Press.

Harper, William L. (2002). “Newton’s Argument for Universal Gravitation”, in I. B.

Cohen & George E. Smith (eds.), The Cambridge Companion to Newton.

Cambridge: Cambridge University Press.

McGuire, J. E. and P. M. Rattansi. (1966). “Newton and the ‘Pipes of Pan’”, in Notes

and Records of the Royal Society of London 21: 108-143.

Nagel, Thomas. (1986). The View from Nowhere. New York: Oxford University Press.

Netz, Reviel. (1999). The Shaping of Deduction in Greek Mathematics: A Study in

Cognitive History. Cambridge: Cambridge University Press.

Newton, Isaac. (1999 [1687]). The Principia: Mathematical Principles of Natural

Philosophy. I. Bernard Cohen and Anne Whitman (trans.). Berkeley and Los

Angeles: University of California Press.

44

Osler, Margaret J. (2010). Reconfiguring the World: Nature, God and Human

Understanding in Early Modern Europe. Baltimore: Johns Hopkins University

Press.

----------------------. (2004). “The New Newtonian Scholarship and the Fate of the

Scientific Revolution”, in J. E. Force and S. Hutton (eds.), Newton and

Newtonianism. Dordrecht: Kluwer Academic Publishers. Pp. 1-13.

Pappus of Alexandria [1986]. Book 7 of the Collection, Part 1. Edited with Translation

and Commentary by Alexander Jones. New York: Springer-Verlag.

Smith, George E. (2002). “The Methodology of the Principia”, in I. B. Cohen & George

E. Smith (eds.), The Cambridge Companion to Newton. Cambridge: Cambridge

University Press.

Snobelen, Stephen D. (n.d.). “To discourse of God: Isaac Newton’s heterodox theology

and his natural philosophy”. Unpublished manuscript.

Westfall, Richard S. (1971). Force in Newton’s Physics: the science of dynamics in the

seventeenth century. London: Macdonald & Co. History of Science Library.

-------------------------. (1980). Never at Rest: a biography of Isaac Newton. Cambridge:

Cambridge University Press.

Wilson, Curtis. (1970). “From Kepler’s Laws, So-called, to Universal Gravitation:

Empirical Factors”, Archive for History of Exact Sciences 6: 92-107.

45