8 The Analytical Method of Fluxions

The chief Principle, upon which the Method of Fluxions is here built, is this very simple one, taken from Rational Mechanicks; which is, That Mathematical Quantity, particularly Extension, may be conceived as generated by continued local Motion; and that all Quantities whatever, at least by analogy and accommodation, may be conceived as generated after a like manner.

— John Colson, 1736

Newton’s first attempts to codify the method of fluxions date from the October 1666 tract.¹ This chapter, however, deals with the analytical version that Newton fully developed in De Methodis (1671) and in De Quadratura (1691−1692).² The method is divided into a direct and an inverse part. Newton considered the techniques of the direct method as having been brought to perfection in his 1671 treatise. After 1671 he sought both to improve the algorithm of the inverse method and reach a better conceptual foundation for the direct method. Newton kept on working on these issues until the early 1690s, when he composed De Quadratura, a work that provides the most advanced refinement of the method of fluxions.

According to Newton’s standards, analysis had to be followed by synthesis. Therefore, from the early 1670s he attempted to develop a synthetic method of fluxions (see chapter 9).

While the method of series was developed by Newton by deriving inspiration from Wallis’s work, in the method of fluxions he followed the steps of Barrow, even though the influence of Barrow is less manifest. Thus, some features of Barrow’s work that most probably were important for Newton are considered first.

In 1663, when Newton was a young student in Cambridge, Isaac Barrow, a theologian and a mathematician highly esteemed by his contemporaries, was appointed to the newly instituted Lucasian Chair of Mathematics.³ The value and nature of Barrow’s mathematical research has been the object of much debate since the inception of the Newton-Leibniz controversy. It is evident that some of his results are related to what is identified in the literature as the infinitesimal calculus, a term that neither Barrow nor Newton ever used. The extent to which all this can be taken as proof of Barrow’s contribution to the calculus, however, is unclear. Barrow proved some geometrical results concerning the drawing of tangents and the squaring of curves that were later identified as equivalent to the so-called fundamental theorem of the calculus, that is, as the statement of the inverse relation between differentiation and integration.

In 1916, Child defended the thesis according to which Barrow was the first inventor of the calculus:

Barrow was the first inventor of the Infinitesimal Calculus; Newton got the main idea of it from Barrow by personal communication; and Leibniz also was in some measure indebted to Barrow’s work, obtaining confirmation of his own original ideas, and suggestions for their further development, from the copy of Barrow’s Lectiones Geometricae that he purchased in 1673.⁴

Child’s claim, however, cannot withstand the careful reconstruction of Newton’s and Leibniz’s independent paths to discovery—respectively, of the method of series and fluxions and of the differential and integral calculus—which most notably emerges in the seminal studies of Hall, Hofmann, and Whiteside.⁵ It is now clear that neither Newton nor Leibniz was exclusively indebted to Barrow, since both drew from a large body of mathematical literature; besides, their original contributions proved momentous.

Even though the question of Barrow’s priority often proves misleading, it is difficult to deny that Barrow’s presence in Cambridge must have shaped Newton’s mathematical ideas to some extent. As Feingold has shown, Barrow and Newton might have been in contact to exchange ideas on mathematics during Newton’s formative years. In his retrospective memoranda Newton always attributed a major role to Barrow as his mathematical mentor. Further, it is through Barrow that Newton got in touch with John Collins in London; the Lucasian Chair was conferred to Newton thanks to Barrow’s recommendation; and in 1670, Barrow asked Newton to edit his Lectiones Geometricae.⁶ All these factors substantiate the hypothesis that between Barrow and Newton a close relationship existed at some time. Arthur has pointed to notable similarities in language between Newton and Barrow on themes such as absolute time and space, and the generation of magnitudes by motion.⁷ After all, Newton himself, in 1712, recognized that “it is from him [Barrow] that I had the language of momenta & incrementa momentanea & this language I have always used & still use.”⁸ It is not my purpose to explore the matter of Barrow’s early influence on Newton and Newton’s indebtedness to Barrow any further. In the years 1664−1669, Barrow and Newton lived close to one another; if the two men were ever in contact with one another (as seems most likely), they did not leave any traces for future historians. Certainly, from 1669 onward Newton became deeply acquainted with Barrow and his mathematical work: from this date at least, Newton’s own work shows signs of Barrow’s influence. My aim in the following pages is to highlight certain aspects of Barrow’s mathematics that bear some resemblance to Newton’s early method of fluxions as expounded in De Methodis.⁹

Both Lectiones Mathematicae and Lectiones Geometricae, which Barrow delivered during his tenure at Cambridge from 1663 to 1669, clearly state that the object of mathematics is geometrical magnitudes generated by motion.¹⁰ This was to become a basic tenet of Newton’s fluxional method as well. The continuous motion of a point generates a line, the motion of a line generates a surface, and the motion of a surface generates a solid.¹¹

Conceiving objects as generated by continuous motion presented two advantages that Newton appreciated. The first is that the limiting procedures deployed in calculating tangents and areas can be grounded on the continuity of motion, that is, it is possible to claim that the limits determined by such procedures exist because of the continuity of the generating motion. Further, the continuity observed in physical motions allows mathematics to be envisaged as a language applicable to the study of the natural world.

In this context, Barrow, proving himself an innovator rather than a conservative, did away in Lectiones Mathematicae with the traditional distinction between pure and mixed (or concrete) mathematics, by stating that since continuous magnitude is the affection of all things, there is no part of “physics” that is not reducible to geometry. In conclusion Barrow went so far as to claim that “Mathematics is … co-extended with physics,”¹² a statement that paved the way for the legitimation of Newton’s “geometrical philosophers” (see chapter 2). By stating the generality of his kinematic geometry, Barrow broke both with the limitations of the Cartesian canon and with the traditional Aristotelian disciplinary taxonomies.¹³

Barrow’s aim in Lectiones Geometricae was to “study and display the affections of curves which emerge from the composition of motions.”¹⁴ In the first lectures Barrow explained in detail how curves can be generated by composition of motions in many different ways. One could think, for instance, of a point sliding along a line which itself has a translational or a rotational motion (think about the generation of a Galilean parabola through the composition of uniform and accelerated motion, or the generation of an Archimedean spiral through the composition of uniform rotation and uniform rectilinear motion in (§1.3)). Another way of generating curves is by the concurrent motion of two lines so that their intersection traces a curve (think about the generation of the quadratrix in (§1.3)). As Mahoney observed about Barrow’s mathematics, “[T]he properties of concern to Barrow follow directly from the curves’ mode of generation.”¹⁵ Most notably, in the sixth lecture Barrow studied how the subtangent to a curve generated by motion could be determined “without the trouble or wearisomeness of calculation” in function of the generating motions.¹⁶ In 1665, Newton devised a kinematic method for determining tangents to “mechanicall lines” (§1.3). It is therefore possible to discern Barrow’s influence here.

Barrow also had a method for finding subtangents ex calculo.¹⁷ He discussed it at the close of Lecture 10 following the advice of a “friend”: the young Newton, who was then rather enthusiastic about algorithmic methods. Barrow’s method is based on the limitative assumption that there is an equation relating the abscissa and the ordinate of the curve. Barrow proceeded as follows (see figure 8.1):

Let AP, PM be straight lines given in position (of which PM cuts the proposed curve at M), and suppose MT to be tangent to the curve at M and to cut the line AP at T. Now, to find out the quantity of this line PT, I posit the arc MN as indefinitely small. Then I draw lines NQ parallel to MP and NR [parallel] to AP. I call MP = m, PT = t, MR = a, NR = e; the remaining lines determined by the special nature of the curve and useful to the proposition I designate by names. But MR and NR (and by means of them MP and PT) I compare to one another by an equation expressed in terms of calculation [ex calculo]. In doing so I observe these rules:

1.

In the computation I reject all terms in which a power of a or e occurs or in which these are multiplied by one another (because these terms will count for nothing).

2.

After the equation has been set up, I reject all terms consisting of letters designating known or determinate quantities, or in which a or e does not occur (because these terms, when brought to one side of the equation, will equal nothing).

3.

I substitute m (or MP) for a, and t (or PT) for e. From this finally the quantity PT [the subtangent] itself is determined.

And if an indefinitely small part of any curve should enter the calculation, substitute in its place a suitably chosen small part of the tangent, or any line equivalent to it (by virtue of the indefinite smallness of the curve).¹⁸

As Mahoney noted, there is nothing radically new in this method. Pierre de Fermat and many others after him had deployed similar techniques to calculate subtangents. Barrow’s technique depended upon the assumption that an equation relating the abscissa and the ordinate is available. Newton adopted techniques for drawing tangents that resemble both Barrow’s kinematic and algorithmic methods. What remained unclear in Barrow’s formulation was how to deal with equations in which radicals occur and, more important, how to deal with mechanical curves generally. There is no doubt that Barrow found these two cases burdened by “wearisome calculation” and that he gave preference to the kinematic method, similar to the one already developed by Gilles Personne de Roberval, over the algorithmic one precisely because it was unclear how the latter might be extended to nonalgebraic curves.¹⁹

Another aspect of Lectiones Geometricae that shows resemblances with Newton’s fluxional method is the fact that Barrow organized his work around two related problems: the finding of tangents and the finding of curvilinear areas. At the opening of Lecture 6, Barrow stated that he would pursue two goals: “the finding of tangents without the trouble or wearisomeness of calculation” and the “ready determination of the dimension of many magnitudes by the help of tangents which have been drawn.”²⁰ Here Barrow showed clear awareness of the fact that the determination of the dimensions (i.e., the areas and volumes) of curvilinear figures can be achieved thanks to theorems that concern the finding of tangents. A fundamental relation between two apparently disconnected problems was thus identified in Lectiones Geometricae. This lesson is likely to have polarized the attention of Barrow’s gifted student, who broached these two problems in the direct and the inverse method of fluxions, respectively.

How tangents and quadrature problems are related is explained by Barrow in a number of propositions. One can cite Proposition 11 from Lecture 10 (figure 8.2) and Proposition 19 from Lecture 11 (figure 8.3):

Proposition 11, Lecture 10. Let ZGE be any curve [see figure 8.2] of which the axis is AD; and let ordinates applied to this axis, AZ, PG, DE, continually increase from the initial ordinate AZ; and also let A I F be a line such that, if any straight line EDF is drawn perpendicular to AD, cutting the curves in the points E, F, and AD in D, the rectangle contained by DF and a given length R is equal to the intercepted space ADEZ; also let DE : DF = R = OT, and join DT. Then TF will touch the curve A I F.

For, if any point I is taken in the line AI F (first on the side of F towards A), and if through it IG is drawn parallel to AZ, and KL is parallel to AD, cutting the given line as shown in the figure; then

$LF:LK=DF:DT=DE:R,$

or

$R×LF=LK×DE.$

But, from the stated nature of the lines DF, PK, we have R × LF = area PDEG; therefore LK × DE = area PDEG 〈 PD × DE; hence LK 〈 DP 〈 LI.

Again, if the point I is taken on the other side of F, and the same construction is made as before, plainly it can be easily shown that LK 〉 DP 〉 LI.

From which it it is quite clear that the whole of the line TKFK lies within or below the curve AIFI.

Other things remaining the same, if the ordinates, AZ, PG, DE continually decrease, the same conclusion is attained by similar argument.

Proposition 19, Lecture 11. Again, let AMB [see figure 8.3] be a curve of which the axis is AD and the BD be perpendicular to AD; also let KZL be another line such that, when any point M is taken in the curve AB, and through it are drawn MT a tangent to the curve AB, and MFZ parallel to DB, cutting KZ in Z and AD in F, and R is a line of given length, TF : FM = R : FZ. Then the space ADLK is equal to the rectangle contained by R and DB.

For, if DH = R and the rectangle BDHI is completed, and MN is taken to be an indefinitely small arc of the curve AB, and MEX, NOS are drawn parallel to AD; then we have

$NO:MO=TF:FM=R:FZ;$

therefore NO × FZ = MO × R =, and FG × FZ = ES × EX.

Hence, since the sum of such rectangles as FG × FZ differs only in the least degree from the space ADLK, and the rectangles ES × EX from the rectangle DHIB, the theorem is quite obvious.²¹

These propositions can be immediately understood (perhaps too optimistically) as a statement of the fundamental theorem of the calculus. Child’s claims about Barrow’s priority over Leibniz and Newton are mainly based on these propositions. I do not wish to enter this discussion here, but I refer the reader to Mahoney’s study, which concluded that “what in substance becomes part of the fundamental theorem of the calculus is clearly not fundamental for Barrow.”²² Indeed, as Mahoney observed, Barrow did not give particular emphasis to these two propositions and did not relate them to one another; they occur in two separate lectures and seem to play independent roles. Further, Barrow did not translate these propositions into an algorithm for determining areas in function of antiderivatives. There is much wisdom and historical sensitivity in these cautionary remarks. But I fear that in subjecting Barrow’s Lectiones Geometricae to evaluations that are polarized by the purpose of refuting Child’s wild claims one risks failing to understand his mathematics in its own terms.

Barrow was not interested in developing an algorithm for broaching the problems concerning curves to which he devoted Lectiones Geometricae, since he was convinced of the superior generality of geometry over algebra, an altogether justified position given the fact that algebraic techniques for dealing with mechanical curves had yet to be invented by his young protégé. What Barrow wished to do, it seems to me, was to demonstrate general relations between propositions concerning curvilinear areas and propositions concerning tangents. Such relations had already appeared in the literature on specific curves. For Barrow, geometry was the means to prove that such relations are quite general and hold for any curve independently of its algebraic representability. Geometry offered Barrow a language appropriate for expressing general theorems concerning the tangents and areas of curves.

Whatever Barrow’s awareness of the centrality of his theorems on areas and tangents, it is a fact that in 1665 Newton based his first demonstration of the inverse relation between area-problems and tangent-problems on a proposition that is strikingly similar to Barrow’s Proposition 11 from Lecture 10 (§8.2.6 and figure 8.7).²³ Further, in 1670 he turned to Barrow’s theorems on quadratures (as those surrounding Proposition 19 from Lecture 11) to find a synthetic construction of his analytical algebraic method of quadratures (§9.1).

Barrow employed infinitesimal magnitudes in his proofs. He often stated that such proofs could be reframed by means of more “apagogical” procedures. The term apagogical was used in scholastic logic to designate a reasoning that demonstrates a proposition A by proving the impossibility of the negation of A. These ad absurdum proofs were generally lengthy, and Barrow claimed that it was only for the sake of brevity and perspicuity that he used less rigorous proofs where infinitesimals occurred. In the second appendix to Lecture 12 of Lectiones Geometricae, he wrote,

Having regard for brevity and perspicuity (mainly the latter), the preceding results were proven by direct arguments, by which not only the truth is cogently enough confirmed, but also their origins most neatly appear. But for fear anyone less used to this sort of arguments had difficulty, we shall add the following short notes. With them the said arguments are secured and with their help apagogical proofs of the preceding results will be easily worked out.²⁴

In the appendix Barrow developed an apagogical demonstration of the assumption that a curvilinear area can be equated with the summation of an infinite number of areas of infinitesimal parallelograms. He did so by means of ad absurdum reductions reminiscent of the Archimedean method of exhaustion. Barrow considered the curvilinear surface shown in figure 8.4 and conceived it as being subdivided into an infinity of parallelograms, whose bases ZZ are infinitesimal.

Barrow aimed to prove that the assumption that rectilinear circumscribed and inscribed figures are greater or smaller than the curvilinear surface leads to contradiction. The gist of his argument consists in showing that their difference (which is equal to rectangle ADLK) is infinitesimal, that is, its area is “less than any given magnitude.”

As Malet observed in his careful study on infinitesimal techniques in the seventeenth century, Barrow

was not concerned about the use of infinitesimals and did not make an attempt to get rid of them. What did concern him was to show that the difference between an aggregate of infinitesimals each one being not truly identical with a part of the whole surface, and the surface is less than any finite magnitude.²⁵

Newton also offered proofs based on infinitesimals “for the sake of brevity” and attempted a more rigorous foundation of them along the lines of exhaustion techniques (see chapter 9). While Newton’s approach to similar demonstrations was different from Barrow’s (based as it was on limiting procedures), he retained Barrow’s diagram (see figure 8.4) and the idea that the difference between the circumscribed and inscribed figures is equal to the rectangle ADLK (see figure 9.3).

After this brief introduction to Barrow’s Lectiones Geometricae, I now turn to Newton’s De Methodis Serierum et Fluxionum. Several features of Barrow’s Lectiones surface in Newton’s method of series and fluxions.

While De Analysi (see chapter 7) is a short tract mainly devoted to series expansions and their use in quadratures, De Methodis (written in 1670−1671) is a long treatise whose aim is to deliver the applications of an “analytical art” useful to the study of the “nature of curves.” In this masterpiece Newton systematized his early work on tangents and quadratures by reworking and greatly extending the October 1666 tract on fluxions.

Newton’s method of fluxions is deeply intertwined with his method of series. Indeed, in the opening lines of De Methodis Newton incorporated and expanded De Analysi by presenting his methods of series via long division, root extraction, and the resolution of affected equations (§7.5). Then he wrote,

So much for computational methods of which in the sequel I shall make frequent use. It now remains, in illustration of this analytical art, to deliver some typical problems and such especially as the nature of curves will present.²⁶

This statement defines Newton’s object of study and method: De Methodis is a work on analysis applied to the resolution of problems concerning curves. Synthesis, with its constructions and demonstrations is given little space. Newton, however, retained the structure of the analytical/synthetic canon. He always began from analysis: he considered a geometrical problem, translated it into algebra, and manipulated symbols until he achieved a resolution of the problem. He then briefly turned to synthesis, that is, he provided a geometrical construction and a geometrical demonstration that this construction is what is required in order to reach a solution of the problem considered. The last stage, however, which was Barrow’s main concern in Lectiones Geometricae, is only briefly touched upon. Newton developed the synthesis more fully in an Addendum composed in 1671 and in a treatise entitled “Geometria Curvilinea,” written around 1680 (see chapter 9).

In De Methodis, Newton made it clear that the objects to which his analytical method applied are geometrical quantities generated by a process of flow in time. For instance, the motion of a point generates a line, and the motion of a line generates a surface:

Therefore, consider a point that flows with variable speed along a straight line (figure 8.5). The distance covered at time t is the fluent, and the instantaneous speed is the fluxion. The indefinitely or infinitely small parts by means of which the fluent increases after indefinitely or infinitely small intervals of time are the moments of the fluent quantity.²⁸ Newton further observed that the moments are “as the speed of flow” (i.e., the fluxions).²⁹ His reasoning was based on the idea that during an infinitely small period of time the fluxion remains constant, and hence the moment is proportional to the fluxion.

Newton warned the reader not to identify the time of the fluxional method with real time. Any fluent quantity whose fluxion is assumed constant ($x˙$ = 1) plays the role of fluxional time (x = t + C). The choice of x as the time variable is arbitrary (and further if y = kx also $y˙$ is constant) and in general dictated by computational convenience. His language here is reminiscent of Barrow’s Lectiones Geometricae (§8.1.2).³⁰

Contrary to Barrow, Newton developed in De Methodis an algorithmic approach extensively. His notation, however, is not particularly handy. For instance, he might employ a, b, c, d for constants, v, x, y, z for fluents, and l, m, n, r for the respective fluxions, so that, for instance, m is the fluxion of x, and n is the fluxion of y, etc. The indefinitely (or infinitely) small interval of time is always denoted by o, so that the moment of y is no. This notation is very confusing; in what follows, I always employ a notation that Newton invented much later.

In fact, it was only in the 1690s that Newton introduced the now standard notation according to which the fluxion of x is denoted by $x˙$, and the moment of x by $x˙$o. Fluxions themselves can be considered fluent quantities. In the 1690s, Newton denoted the second fluxion of x with $x¨$ (whereas he had previously employed letters, so that, for instance, q was the second fluxion of y). Multiple points or numbers placed over the fluent symbols can denote higher-order fluxions.

Newton did not use a consistent notation for the area of the surface under a curve.

Most often, he used words such as “the area of” or (as recorded in one instance) a capital Q before the analytical expression of the curve.³¹ In some cases he used “a/x²” for “the area of the surface under the curve of equation y = a/x2.”³² In the 1690s Newton also employed x to denote a fluent quantity whose fluxion is x. The limits of integration were either evident from the context or explained in words, not symbols.

In the De Methodis, Newton applied the method of series and fluxions to several problems. The main ones were how to find maxima and minima of varying magnitudes, how to determine tangents and curvatures of plane curves, and how to calculate curvilinear areas and arclengths. Thanks to the representation of quantities as generated by a continuous flow, all these problems can be reduced to the following two:

Problem 1. Given the length of the space continuously (that is, at every time), to find the speed of motion at any time proposed.

Problem 2. Given the speed of motion continuously, to find the length of the space described at any time proposed.³³

The problems of finding tangents, maxima and minima, and curvatures are related to Problem 1, the problems of finding curvilinear areas and arc lengths are related to Problem 2.

Newton demonstrated the fundamental theorem in some of his early manuscripts and in De Analysi. Nowadays the fundamental theorem is understood as a statement that the two central operations of calculus, differentiation and integration, are inverse operations. An important consequence of this is the possibility of computing integrals by using an antiderivative of the function to be integrated, a consequence that Newton fully appreciated and deployed in De Methodis (§8.4.3).

The fact that the calculation of curvilinear areas (in Newton’s terms, the problem of quadrature) can be reduced to Problem 2 is implied by Newton’s proof of the fundamental theorem. Newton demonstrated that if z is the area generated by the continuous uniform flow (x = 1) of ordinate y, then y = z (figure 8.6). Note that the conception of quantities as generated by continuous flow allowed Newton to conceive the problem of determining the area under a curve as a special case of Problem 2.

The reduction of arclength problems to Problem 2 depends on the application of the Pythagorean theorem to the triangle formed by the moment of arclength s, the moment of the abscissa x, and the moment of the ordinate $y:s˙o=(x˙o)2+(y˙o)2$. Therefore, $s=x˙2+y˙2$.

Proof (1665) Newton discovered the fundamental theorem in 1665.³⁴ His reasoning, which strongly resembles Barrow’s Proposition 19 from Lecture 11 (§8.1.6), refers to two particular curves z = x³/a and y = 3x²/a. However, it is completely general; the only property that matters is that y be equal to the slope of z (see figure 8.7, where z is drawn above the and y is drawn below the x-axis).³⁵ More precisely, y is defined as

where bg is an ordinate of the curve y, βm and Ωβ are infinitesimal increments of z and x, and dh is a unit length segment. It immediately follows that the area bpsg (= Ωβ · bg ) and the area μκλν (= βm · dh) are equal. It was commonplace in seventeenth-century mathematics to consider the surface subtended by a curve to be equal to the juxtaposition of infinitely many infinitesimal rectangles such as bpsg. It follows that the curvilinear area subtended by y, for instance, dυn, is equal to the rectangular area dhσρ. A knowledge of z then makes it possible to “square” y, since the area under y (the derivative curve) is proportional to the difference between corresponding ordinates of z.³⁶ In Leibnizian terms, Newton proved that the integral of the differential of z is equal to z, namely, ƒ dz = z. A proof of the fact that d z = z can be found in De Analysi.

Proof in De Analysi (1669) A proof of the inverse relation of area-problems and tangent-problems is given at the end of De Analysi. Newton proceeded as follows. He considered a curve ADS (figure 8.8), where AB = x, BD = y, and the area ABD = z. He defined Bβ = o and BK = v, so that “the rectangle BβHK (= ov) is equal to the space BβSD.”³⁷ Further, Newton assumed that Bβ is infinitely small (infinite parvam).

Given these definitions, Aβ = x + o and the area Aδβ is equal to z + ov. At this point Newton wrote, “[F]rom any arbitrarily assumed relationship between x and z I seek y.”³⁸ He noted that the increment of the area ov divided by the increment of the abscissa o is equal to v. But since one can assume “Bβ to be infinitely small, that is, o to be zero, v and y will be equal.”³⁹ Therefore, the rate of increase of the area is equal to the ordinate. The mathematically trained reader will notice that several assumptions that tacitly operate in this reasoning (most notably, the existence of v) only received attention and were systematized in the nineteenth century.

I now turn to the two problems into which Newton’s method is subdivided, in this section the direct Problem 1, and in subsequent sections the more difficult inverse Problem 2.

Problem 1 is stated as follows:

Given the relation of the flowing quantities to one another, to determine the relation of the fluxions.⁴⁰

Newton presented the basic algorithm for Problem 1 by providing some examples.⁴¹ First he dealt with polynomial equations. Then he considered equations in which “fractions and surd quantities are present.”⁴² Last, he considered the case of “quantities which cannot be determined and expressed by any geometrical ratio [nulla ratione geometrica], such as the areas and lengths of curves.”⁴³

In De Methodis Newton dealt with the equation of a cubic curve:

His (inaccurate) prescription is as follows:

Arrange the equation by which the given relation is expressed according to the dimensions of some fluent quantity, say x, and multiply its terms by any arithmetical progression and then by $x˙$/x. Carry out this operation separately for each one of the fluent quantities and then put the sum of all the products equal to nothing, and you have the desired equation.⁴⁴

This is basically the rule explained in one of the appendices to Descartes’ Géométrie by Johan Hudde. The application that follows illustrates what Newton meant far more clearly. He obtained:

This equation gives the ratio of the fluxions of, $x˙$ to $y˙$, as

Note that in the example the rules for the calculation of the fluxions of the sum x + y, the product xy, and the integer positive power xⁿ are simultaneously stated, respectively, as x + y, xy + yx, and nxⁿ⁻¹x. Naturally, the fluxion of a constant quantity is equal to zero.⁴⁵ It was unclear how to extend this rule to nonpolynomial equations until Newton provided an answer.

In De Methodis, Newton wrote,

Whenever complex fractions or surd quantities are present in the proposed equation, in place of each I put a corresponding letter and, supposing these to designate fluent quantities, I work as before. Then I suppress and exterminate the letters ascribed.⁴⁶

Take $y2−a2−xa2−x2=0$. Newton set$z=xa2−x2$ and so obtained y² − a² − z = 0 and a²x² − x⁴ − z² = 0. Applying the direct algorithm for polynomial equations, he determined $2y˙y−z˙=0$ and$2a2x˙x−4x˙x3−2z˙z=0$ He then eliminated Z, restored $z=xa2−x2$ and thus obtained $2y˙y+(−a2x˙+2x˙x2)/a2−x2=0$ as the sought relation between y and x. In this first example proposed by Newton, of course, the radical could easily be eliminated. But Newton’s procedure is a very effective method for the calculation of fluxions in more complicated cases. In practice, by substitution of a variable it is possible to eliminate radicals (and quotients, of course) and thus apply Hudde’s rule, which is valid for polynomial equations.

Another example will help to illustrate Newton’s procedure. Let the relation between the fluents be

Set

with

and

Applying Hudde’s rule to these polynomial equations,

and

Of course,

as Newton stated (§8.3.2). By applying Hudde’s rule to $y12=z$ and $wy22=1$

Substitution for z, w, ż, and ẇ delivers the sought ratio of the fluxions (ẏ to ẋ):

I have shown how Newton dealt with the calculation of the relation between the fluxions when the relation between the fluents is expressed by an equation involving quotients and radicals. In De Methodis he considered a more difficult case:

To be sure, even if quantities be involved in an equation which cannot be determined and expressed by any geometrical technique, such as the areas and lengths of curves, the relations of the fluxions are still to be investigated the same way.⁴⁷

The relation of the fluxions can in fact immediately be calculated by applying the fundamental theorem.⁴⁸

One of Newton’s examples in the De Methodis is

where z is the area of the segment ABD of a circle whose diameter is a, abscissa AB = x, and ordinate $BD=ax−x2$ (see figure 7.5). From equation (8.14) one gets

From the fundamental theorem, the fluxion of the area ABD is equal to the length of the ordinate BD times the fluxion of the abscissa:

Thus, for the relation of the fluxions $x˙$ and $y˙$,

Note that in (8.17) binomial expansion is necessary in order to determine z.One must expand $ax−x2$ in the right-hand term of (8.16) and integrate (in Leibnizian terms) term-wise.

Newton’s procedure for Problem 1 is algebraic. In De Methodis he presented his method as a series of algorithmic procedures that are explained by particular examples. The style is didactic, heuristic, and algebraic. This is perfectly in line with the seventeenth-century tradition of the analytical school embodied by Oughtred and Wallis. However, in a section of De Methodis, Newton included a demonstration of these rules based on a reasoning that was strongly reminiscent of Barrow’s determination of tangents ex calculo (§8.1.4). Newton wrote,

The moments of fluent quantities (that is, their indefinitely small parts, by addition of which they increase during each infinitely small period of time) are as their speeds of flow. Wherefore if the moment of any particular one, say x, be expressed by the product of its speed $x˙$ and an infinitely small quantity o (that is, by $x˙$o), then the moment of the others v, y, z, will be expressed by $v˙$  o, $y˙$  o, ż o …. Consequently, an equation which expresses a relationship of fluent quantities without variance at all times will express that relationship equally between $x+x˙o$ and $y+y˙o$ as between x and y; and so $x+x˙o$ and $y+y˙o$ may be substituted in place of the latter quantities, x and y, in the said equation.⁴⁹

Let us reconsider equation (8.2). From what has been said, it is permissible to substitute $x+x˙o$ in place of x, and $y+y˙o$ in place of y. Next, Newton deleted x³ − ax² + axy − y³ as equal to zero, and after division by o he obtained an equation from which he canceled the terms with o as a factor. These terms “will be equivalent to nothing in respect of the others” since “o is supposed to be infinitely small.”⁵⁰ This procedure leads straight to Hudde’s rule.

This demonstration is achieved through two steps. The first step assumes that it is possible to substitute $x+x˙o$ in place of x, and $y+y˙o$ in place of y. Here Newton meant that the relation valid for the fluents x and y, expressed by an equation, continues to be valid for the values $x+x˙o$ and $y+y˙o$ obtained after momentary increases. In geometrical terms, if the point (x, y) is on the curve, then the infinitely close point ($x+x˙o$,$y+y˙o$) will also be on the curve. The latter step is a rule of cancellation of higher-order infinitesimals (equivalent to Leibniz’s x + dx = x). According to this rule, if x is finite and o is an infinitesimal interval of time, then $x+x˙o=x$. Newton set out to justify the use of infinitesimals in an Addendum to De Methodis that he drafted in 1671 (see chapter 9).

The algorithm for Problem 1 allows the resolution of several geometrical problems: the determination of maxima and minima (Problem 3), the determination of tangents (Problem 4), the determination of curvature (Problems 5 and 6).

How does the algorithm work for tangents? Newton assumed that ED is a given curve and that an equation relating the abscissa x = AB to the oblique ordinate y = BD is given (figure 8.9). Let the ordinate BD “move through an indefinitely small space to the position b∂ so that it increases by the moment c∂ while AB increases by the moment Bb equal to Dc.”⁵¹ The momentary increases Bb = Dc and c∂ are indicated in the figure. Newton stated that the straight line that prolongs the momentary increase D∂ of the arc ED cuts the axis of the abscissae in T and that this straight line (namely, the tangent) will touch the curve in D and ∂. Without further explanation, Newton stated that the triangle Dc∂ is similar to the triangle TBD, where TB is the subtangent.⁵² Therefore, he deduced that

Given this premise, the subtangent will be found by application of the algorithm for Problem 1 to the equation that defines the curve ED. Indeed, the algorithm allows the determination of the ratio between the fluxions of x and y. Since the momentary increases are as the fluxions (§8.2.2), it is possible to conclude that the sought subtangent TB is given by⁵³

Note that Newton was deployed well-established practices in handling infinitely or indefinitely small quantities. His use of the indefinitely small triangle Dcd is very similar to Barrow’s (§8.1.4). This calculation of tangents is like Barrow’s in character, as is the notion that magnitudes are generated by motion.

Already while composing De Methodis, Newton was aware that some firmer foundation for infinitesimal techniques had to be sought. He found it in a theory of limits that he termed the “method of first and ultimate ratios.” Newton developed this method in the 1680s, but its roots were already discernible in the Addendum to De Methodis.

I now consider an example of Newton’s method for tangents: the conchoid. Let ED be the conchoid. G is the pole, AT the asymptote. Recall that given a line AT and a line bundle passing through the pole G, the curve is constructed by placing at both sides of AT a distance LD = L∂ on all lines. The two branches of the conchoid are the loci of points D and д (figure 8.10).

Let GA = b, LD = c, AB = x, and BD = y. Because the triangles DBL and GDM are similar, LB/BD wrote,

Now he applied his algorithm for “surd” quantities (§8.3.3) and set $z=c2−y2$. This leads to the following system:

By application of the algorithm for Problem 1, one obtains

Elimination of ż leads to

therefore the ratio between the fluxions can be expressed as

From equation (8.19) one gets

As previously noted (§3.2.2), for Newton’s contemporaries a geometrical problem was solved by a geometrical construction, not by an algebraic formulation. Accordingly Newton interpreted this result in geometrical terms:

Newton applied the direct algorithm for Problem 1 to other problems concerning tangency and curvature, as he continued De Methodis by considering different coordinate systems (e.g., polar and bipolar coordinates) and developed the theory of curvature in great detail. The determination of the radius of curvature of plane curves was of great importance for Newton in his study of trajectories in the Principia, since he made use of the fact that the normal component of the force [F_N] acting on a mass point is proportional to the square of speed [v] and inversely proportional to the radius of curvature [F_N ∝ v²/ρ].

Problem 2 is worded as the inverse of Problem 1:

When an equation involving the fluxions of quantities is exhibited, to determine the relation of the quantities one to another.⁵⁴

Problem 2 was often referred to by Newton as the problem of the quadrature of curves or squaring of curves (§8.2.5). In Leibnizian terms, Newton posed the problem of integration.

Problem 2 is of course much more difficult than Problem 1. Here Newton stopped teaching his method to discentes (learners) and addressed the artifices (skilled practitioners). The distinction between the parts of his method within reach of the learners and those accessible only to skilled practitioners was quite clear in Newton’s mind.⁵⁵ As with his treatment of Problem 1, Newton explained how to deal with Problem 2 via examples. His strategy was fragmentary and his style that of the craftsman seeking to make a novice become used to increasingly complex cases. Newton’s main techniques for Problem 2 are the following three methods.

The first method was discussed in chapter 7. In Leibnizian terms, it consists in expanding the integrand into a power series. Newton deployed his algorithmic techniques of series expansion by long division, root extraction, and resolution of affected equations.

In a simpler case (Case 1) one has an equation in which “two fluxions together with one only of their fluent quantities are involved.” In Leibnizian terms, Newton was here considering ordinary differential equations.⁵⁶ From Newton’s examples,⁵⁷ one takes

Applying his technique for the resolution of affected equations (figure 7.11), Newton obtained

which can be squared term-wise, thus obtaining the relation between the fluents:

This is one of the basic techniques of series expansion employed in De Analysi.lt should be noted that the approximation is valid for x ≈ 0: Newton, that is, obtained a local approximation of the fluent (the integral, in Leibnizian terms).

In slightly more complex cases, an equation is given in which either two fluxions x and y together with both the fluent quantities x and y (Case 2) occur, or more than two fluxions are present (Case 3). Newton here implemented an algorithm of successive approximations (figure 8.11), where again the aim was to express $y˙/x˙$ as an infinite series.

Two further approaches to Problem 2 were at Newton’s disposal:

Hitherto we have exposed the quadrature of curves defined by less simple equations by the technique of reducing them to equations consisting of infinitely many simple terms [Method 1]. However, curves of this kind may sometimes be squared by means of finite equations also [Method 2], or at least compared with other curves (such as conics) whose area may, after a fashion, be accepted as known [Method 3]. For this reason I have now decided to add the two following catalogues of theorems constructed … for this use with help of Problems 7 and 8.⁵⁸

Newton therefore distinguished between three quadrature techniques:

The first technique is treated in De Analysi (§7.4) and further developed in Problem 2 (Cases 1, 2, and 3) of De Methodis (§8.4.2).

The second approach is studied in Problem 7 of De Methodis, and its application translated into a first catalogue of curves.

The third approach is studied in Problem 8 of De Methodis, and its application translated into a second catalogue of curves.

The second approach consists in applying the algorithm of Problem 1 to “any equation at will defining the relationship of t [the area] to z [the abscissa]” (figure 8.12). One thus obtains an equation relating $t˙$ and ż, and so “two equations will be had, the latter of which will define the curve [whose ordinate is y], the former its area.”⁶⁰ Following this strategy, Newton constructed a first “catalogue of curves which can be squared by means of finite equations” (figure 8.13). In modern terms, one might say that Newton was aware of the fact that antiderivatives are related to definite integrals through the fundamental theorem of calculus and provide a convenient means for tabulating the integrals of many functions.

Curves “which can be squared by means of finite equations” are an exception: infinite series remain an essential tool for calculating many curvilinear areas. Most often these series are difficult to interpret geometrically and provide only a local and algorithmic approximation. Recall that for Newton the result of an analytical process is best interpreted geometrically. This is why, in Method 3, Newton considered transformations of variables to reduce the calculation of a curvilinear surface to the calculation of the area of a conic surface. Conic areas can be evaluated by binomial expansion and term-wise quadrature, as Newton explained in De Analysi. Therefore, series are still necessary. However, the areas of the conic sections can be considered to be accepted as known, not only because their areas are given by well-known logarithmic and trigonometric tables but also because the conics are geometrically constructible following methods already established in Antiquity.

In Problem 8, Newton took two curves FDH and GEI, in which variables x, v, s and z, y, t denote the abscissa, ordinate, and area of the two curves (figure 8.14). Suppose one knows how to square the curve FDH. The problem here will be to square GEI. Newton introduced two equations, the first relating the abscissae x and z, and the second relating areas s and t. Newton proceeded by examples where the curve FDH is a conic section (which can be squared following the procedures of De Analysi). A few simple examples follow.

Let the curve FDH be a circle whose equation is v² = ax − x². Assume that areas s and t are related by

and that the ordinates are related by

By means of the algorithm of Problem 1, under the assumption that $x˙=1$, one gets

and

Therefore

and “this when $ax−x2$ is substituted in place of $s˙$ and z²/a in place of x” becomes

One begins then with a curve (in this case, a circle) whose area is assumed as known and by suitable transformations of variables obtains equation (8.37) of a curve whose area t is related to the area s of the circle by equation (8.32).⁶¹

Thanks to this technique Newton could develop a “second catalogue of curves related to conic sections” (figure 8.15). The first row states that the area t under

is equal to

where s is the area under

The second column prescribes a substitution of variables:

assuming $x˙=1$, one gets

From the third column

Therefore, from the first column⁶²

Newton applied the second catalogue of curves to several examples. Example 3 concerns the quadrature of the cissoid. The problem is how to square the cissoid AeE, that is, how to determine the area of the surface ACEeA, where ADQ is a circle (figure 8.16).

Set the abscissa AC = z, the ordinate CE = y, the circle’s diameter AQ = a. Because of the defining property of the cissoid, CD, AC, and CE are in continued proportion.⁶³ Thus, the equation of the cissoid is

In order to square the cissoid, reference must be made to the third species of the seventh order of the second catalogue of curves (figure 8.17).

On setting ∂ = 1, ϵ = −1, and f = a, the curve listed in the first column is

which is the equation of the cissoid for η = −1.

The transformation of the abscissae (second column) is z = z⁻ⁿ = x (therefore x = AC).

The conic ordinate (third column) is $v=ax−x2$ (therefore v = CD), and s is the area of the segment ACDH of the circle.

From the fourth column one gets that the area t under the cissoid is

consequently, the area ACEeA of the cissoid is 3(ACDH) − 4ΔADC⁶⁴

Newton added some equivalent formulations: “Or what is the same, 3 × segment ADHA = area ADEA, thatis, 4 × segment ADHA = area AHDEeA.”⁶⁵

To conclude, the calculation of the area t of the cissoid is reduced to the calculation of the area s of the circle, which can be evaluated through the power series expansion of $s˙/x˙=v=ax−x2$ and term-wise quadrature.

Until the publication of the Principia, Newton circulated his mathematical ideas via manuscript exchange and correspondence. This publication practice exposed him to the risk of not having his discoveries recognized (see part VI). In 1685, John Craig published a short treatise on the quadrature of curvilinear figures in which Newton’s contributions were just mentioned in passing.⁶⁶ More dangerously, David Gregory was claiming for himself a theorem on quadratures that Newton had privately communicated to Leibniz in the epistola posterior, dated October 24, 1676, and to Craig, who had visited Newton in his rooms at Trinity in 1685. Through Craig the theorem had passed into Gregory’s hands. In 1688, Gregory’s associate Archibald Pitcairn had published the theorem attributing it to Gregory.⁶⁷ In 1691, after having being elected Savilian Professor of Astronomy at Oxford, Gregory wrote a letter to Newton in which, rather obliquely, he tried to secure the authorship of this important result.⁶⁸ Newton reacted by writing a short account of his discoveries on quadratures. He soon changed his mind and set out to write a full-fledged treatise, whose composition probably occupied him in the winter of 1691−1692. This was to become Tractatus de Quadratura Qurvarum, eventually printed in 1704 as an appendix to the Opticks.

It is worth considering Newton’s method of quadrature, as communicated to Leibniz in 1676 and to Craig in 1685. In the epistola posterior, he wrote,

For any curve let dz^θ × (e + fz^η)^λ be the ordinate, standing normal at the end of z of the abscissa or the base, where the letters d, e, f denote any given quantities [N.B d is a constant!], and θ, η, λ are the indices of the powers of the quantities to which they are attached.

Put

$(θ+1)/η=r,λ+r=s,(d/(ηf))×(e+fzη)λ+1=Q,rη−η=π,$

then the area of the curve will be

$Q×{zπs−r−1s−1×eAfzη+r−2s−2×eBfzη−r−3s−3×eCfzη+r−4s−4×eDfzη,etc.}$

the letters A, B, C, D, etc., denoting the terms immediately preceding; that is A the term z^π / s, B the term − (r − 1)/(s − 1) × (eA)/(fzⁿ), etc. This series, when r is a fraction or a negative number, is continued to infinity; but when r is positive and integral it is continued only to as many terms as there are units in r itself; and so it exhibits the geometrical squaring of the curve.⁶⁹

This method of quadrature was proposed to Leibniz as the first of a series of theorems devised in order to simplify the “speculations concerning the squaring curves”; it is thus known in the literature as the prime theorem on quadratures. The prime theorem is a generalization of results contained in the first catalogue of curves of De Methodis (§8.4.3).⁷⁰

More generally, Newton was interested in squaring curves of the form y = z^θ × (e + fz^η)^λ, y = z^θ × (e + fz^r + gz^2η  hz^3η …)^λ, or even y = z^θR^λS^μT^v, where R, S, T denote expressions of the form $∑i=0∞aizin$ (to use modern notation). These theorems were systematized in the De Quadratura (§8.5.2).

There is little doubt that Newton was keenly aware of the significance of quadrature problems. In the October 1666 tract on fluxions, he had already stated the importance of the inverse problem of fluxions:

If two Bodys A & B, by their velocitys p & q describe ye lines x & y. & an Equation bee given expressing ye relation twixt one of ye lines x, & ye ratio q/p of their motions p & q; To find the other line y. Could this ever bee done all problems whatever might bee resolved.⁷¹

Further, in De Methodis, he had underlined the importance of quadrature problems:

Observing that the majority of geometers, with an almost complete neglect of the ancients’ synthetic method, now for the most part apply themselves to the cultivation of analysis and with its aid have overcome so many formidable difficulties that they seem to have exhausted virtually everything apart from the squaring of curves and certain topics of like nature not yet fully elucidated.⁷²

When Newton’s polemic with Leibniz broke out, the exchange of accusations between the two was obfuscated by a different perception of what was of primary significance in the discovery of the new method. While Leibniz focused on the enunciation of the rules concerning the direct method of differentiation and—not without reason—claimed that he was the inventor of a simple and concise algorithm for differentiation, Newton insisted on his superior command of series in quadrature techniques (integration, in Leibnizian terms). Newton—never very receptive toward the importance of advances in algorithmic techniques—saw Leibniz’s rules for differentiation as mere trivialities. The true, difficult problem, Newton reiterated, was the inverse problem of quadrature: it is on this battleground that— again, not without reason—Newton claimed supremacy over Leibniz. Part 6 expands on these themes.

The problem of squaring ample classes of curves had been beautifully solved by Newton in his anni mirabiles by the use of infinite series (Method 1), the fundamental theorem (Method 2), and substitutions of variables (Method 3).

These techniques, didactically presented in De Methodis, constitute a method of solution, an heuristic patchwork of algorithmic instructions. In the 1670s, Newton began a research program on quadratures aimed at transforming his early method into a tractatus, his early rules into theorems. As he explained to Leibniz in the epistola posterior of 1676,

I have tried also to render the speculations concerning [the method of] squaring curves simpler, and have attained certain general theorems.⁷³

This program culminated into Tractatus de Quadratura Curvarum, which Newton wrote in 1691−1692.⁷⁴ It is De Quadratura that Newton chose to print in 1704, not De Methodis, which appeared posthumously, in an English translation, only in 1736.

Preliminaries De Quadratura is a notable work not just for the theorems on the quadrature of curves. In the introductory pages of the work Newton presented a theory of limits that provides a foundation for the method of fluxions, a theory that makes it possible to avoid—so the author claimed—the use of infinitesimals. (The “method of first and ultimate ratios” is discussed in §9.5).

It is in these preliminary pages that Newton also introduced the dotted notation for fluxions ($x˙$,$y˙$) and slashed notation for fluents ($x´$, $y´$). I have adopted this notation here.

After these important introductory pages devoted to foundations and notation, two themes indicative of Newton’s high expectations with regard to this treatise, one finds the treatment of two problems.

The first is Problem 1, on the direct method of fluxions (“having given an equation involving any number of fluents to find their fluxions”). It is essentially a reformulation of Problem 1 of De Methodis (§8.3.1). Here, however, terms multiplied by o are discarded not because they are “indefinitely little” but because they are “evanescent.” Newton took the limit assuming that “the quantity o is lessened indefinitely” and therefore cancels terms multiplied by it (§9.5).⁷⁵

Problem 2 (“to find curves that are quadrable”) is equivalent to Method 2 of De Methodis (§8.4.3). One finds here a statement of the fundamental theorem of the calculus. Newton’s early proof, based on infinitesimals, of this fundamental relation between a flowing surface and the ordinate that generates it was discussed in section 8.2.6. For the reader’s convenience, I present it again by quoting from De Quadratura:

Problem 2: To find the Curves that are Quadrable

Let ABC be the Figure [8.18] whose Area [t] is to be found; BC [y] an Ordinate apply’d at Right Angles, and AB [z] the Abscissa. Produce CB to E that BE may be = 1, and compleat the Parallelogram ABED; and the Fluxions of the Areas ABC, ABED will be as BC to BE: Therefore take any Equation by which the Relation of the Areas may be determined, and thence will be given the Relation of the Ordinates BC and BE.⁷⁶

As in De Methodis, Newton’s strategy consisted in applying the direct method to equations involving z and t in order to determine the relation between z and y, that is, in order to determine the curves that are “exactly quadrable.” In Leibnizian terms, the foundation of this quadrature technique is the inverse relation between differentiation and integration.

A simple example Consider:

where θ and λ are integer or fractional.⁷⁷ Assuming that the ordinate EBC flows uniformly, that is, ż, one has

Now suppose that one must find the area t of the surface under the curve of equation

Equation (8.49) can be reduced to the form of equation (8.48) by setting θ − 1 = 2 and λ − 1 = −1/2, that is, θ = 3 and λ = 1/2. A lucky coincidence results:

Thus one can state that

This curve is indeed “exactly, or geometrically, quadrable.”

Theorem 1 More generally, in Theorem 1, Newton considers curves whose area t is⁷⁸

where R = e + fz^η + gz^2η + hz^3η + ⋯. Therefore,

But$R˙=ηfz˙zη−1+2ηgz˙z2η−1+3ηhz˙z3η−1+⋯$. Therefore, the curve whose area is equal to equation (8.52) has ordinate y equal to

Theorem 3 An important result is offered in Theorem 3, a generalization of the prime theorem (§8.5.1).⁷⁹ Let

Further, set r = θ/η, s = r + λ, t = s + λ, v = t + λ, …. Then the area t under the curve

is

where each A, B, C … is the coefficient of the preceding power of z, namely A = (a/η)/(re), B = (b/η − sfA)/((r + 1)e) etc.⁸⁰

The procedure followed in order to prove Theorem 3 is the method of undetermined coefficients (figure 8.19). Newton sought the area of the curve with ordinate (8.56) in the form

Newton next considered the partial areas AzθR^x, Bzθ^+ηR^x, Cz³+²¹  R^x, Dz³+³¹  R^x, etc. From the partial areas (by means of the direct method of fluxions) he calculated their respective ordinates, whose sum must be equal to the given expression (8.56)  z³⁻¹R^X-1(a + bz¹¹ + cz^2r〉 + dz^3r〉 + ···). In other words, Newton obtained two power series that must be equal and equated the coefficients of the equal powers in η (see figure 8.19). He thus obtained a system of equations in e, f, g, … (the coefficients of R), a, b, c, …, and A, B, C, …, which can be solved in order to determine A, B, C,…. Note that it is the inverse relation between differentiation and integration demonstrated in Problem 2 that justifies this procedure.⁸¹

A more advanced example In most cases the application of Theorem 3 leads to a calculation of the area as an infinite series. In a few instances, however, the series terminates. Newton gave the following application of Theorem 3.⁸² Let

this may be written as

In this case one reduce to (8.56) by setting, for the coefficients of S, a = 3k, b = 0, c = -1, d = e = f = · ·· = 0; for thecoefficientsof R, e = k, f = 0, g = -1, h = m, i = l = m = · ·· = 0; and finally θ = -3/2, λ = 1/2, η = 1. The area will be given in finite terms by

As was often the case, Newton assumed that the initial conditions were such that the constant of integration is zero.⁸³ He also noted that the area is negative because it is “adjacent to the absciss produced beyond the ordinate.”⁸⁴ Yet, finite quadratures are by no means the rule. In general, the quadrature will be given by an infinite series.

In concluding this section, I would like to emphasize how general the results on quadratures (namely, integration) reached by Newton in De Quadratura actually are. Newton was clearly aiming at expressing results on quadratures in general symbolical terms. Particularly notable is the use of symbols like R for infinite power series. Newton did not illustrate his rules by means of examples, as in De Methodis, but rather provided general quadrature theorems concerning ample classes of fluents. Newton’s methods allow the integration of all rational functions. It seems to me that in writing De Quadratura, Newton was deliberately aiming to achieve a level of generality and deductive order that went beyond the heuristic level of his previous writings on the subject. While De Quadratura was published as an appendix to the Opticks, the more heuristic De Methodis was left in manuscript form during Newton’s lifetime. Newton preferred to present to the public at large his more general and abstract treatise on quadratures, rather than his rich but unsystematic method of fluxions.

This said, it should be added that the theorems of De Quadratura are statements achieved via Wallisian induction. Newton made this point clear:

[A]t the start of my mathematical studies I first derived particular quadratures and then by induction arrived at general cases.⁸⁵

Newton’s project to transform the method of quadratures into a theory was stillborn. This should not be seen as a failure: the integral calculus was, and still is to a certain extent, a matter of art rather than science, a matter of guesswork rather than of algorithmic deduction. One might wonder whether Newton, always careful to meet the high standards of certainty of the ancient synthesis, would have printed his treatise on quadratures had he not been involved in priorities disputes with Gregory and especially with Leibniz (see part VI).

Since I am dealing with methods for the squaring of curves, a brief mention should be made of the method for the approximation of areas that Newton developed in the mid-1690s in a short treatise entitled “Of Quadrature by Ordinates” in the context of his studies on interpolation.⁸⁶ The idea behind this method is that by calculating the n +1 values y_i acquired by a fluent [y = f (x)] at n +1 isolated points corresponding to abscissae x_i (i = 1, 2, … , n + 1), it is possible to construct “a curve of parabolic kind” [p(x) = a₀ + a₁x + a₂x² +… + a_nxⁿ] which interpolates the fluent [y(x_i) = p(x_i) for i = 1, 2, …, n + 1]. The area will subsequently be easy to calculate by approximation as the area of the surface subtended by the curve of parabolic kind (figure 8.20).

In Newton’s words:

To square to a close approximation any curvilinear figure whatever, some number of whose ordinates can be ascertained.

Through the end-points of the ordinates draw a curve of parabolic kind with the aid of the preceding problems [the interpolation formulas of Methodus Differentialis]. For this will bound a figure which can always be squared, and whose area will be equal to the area of the figure proposed with close approximation.⁸⁷

The Newton-Cotes formula originates from this research. Newton’s work on interpolation dates from 1676 and was partly published in Lemma 5, Book 3, of the Principia ; a full version appeared as Methodus Differentialis in the collection of mathematical essays edited by William Jones in 1711.⁸⁸

Here I consider some general characteristics of the early treatises on the analytical method that were discussed in chapters 7 and 8.

In the time span from 1666 to 1671, Newton produced some well-structured and carefully written treatises on the new analysis. The October 1666 tract on fluxions and even more so De Analysi and De Methodis have the form of publishable texts: they are addressed to readers. In these very early years Newton not only jotted down personal notes or results to be briefly communicated to peers. In the case of these treatises, he rather didactically and systematically elaborated very comprehensive treatments on series and fluxions. He did not assume that readers were particularly advanced in mathematics. Such a mature and didactic style is quite extraordinary for a young man and might be revealing of Newton’s academic ambitions to become a suitable substitute for Barrow, the first Lucasian Professor.

Newton explicitly conceived his treatises as part of a genre that can broadly be associated with the British analytical school of Oughtred and Wallis. The analytical method was presented through a series of specific and increasingly difficult examples. In these early works Newton’s method was not yet a theory but rather a panoply of techniques ultimately justified by their success in resolving problems concerning curvilinear figures. Most of these techniques had no firm foundation. The attempts to provide demonstrations that surfaced from time to time in Newton’s work were far outnumbered by the folios in which the desire to show their effectiveness was given pride of place. An example is the use of power series, which is so important in the analytical method. Neither the binomial series nor the more elaborate methods for the resolution of affected equations were given a proper demonstration. These results were achieved via inductions, analogies, and extrapolations of Wallisian origin. The analytical parallelogram was nothing more than a paper tool explained by testing its successful functioning, that is, by placing asterisks and rulers associated with increasingly difficult polynomial equations. It was a graphic aid that allowed achieving fractional power series expansions whose convergence was to be verified by hand. The same algorithmic approach characterized the extraordinary variety of methods for squaring curves that Newton proposed in the long catalogues of De Methodis.

The man who so carefully and extensively elaborated such heuristic, pragmatically successful, yet ungrounded methods was the same natural philosopher who in 1670 wished to inject certainty into natural philosophy via the use of geometry. Early on in his career, roughly from the mid-1670s, he began to portray himself as an erudite Church historian and chronologist, a theologian and polyhistor whose style was modeled on late-Renaissance philology. Probably in a later period Newton began to mix his anti-Cartesianism with a strong conviction about the superiority of the pre-Aristotelian ancients over the moderns. These cultural orientations, destined to shape Newton’s personality for years to come, increasingly distanced him from the analytical genre of his early treatises on the method of series and fluxions. This divergence between the style of Newton’s early treatises on the new analysis and the style of his nonmathematical researches helps explain his interest in the analysis of the ancient geometers (see chapter 5) as well as his attempts to develop a synthetic version of the method of fluxions (seechapter 9).