## Geometria organica: sive descriptio linearum curvarum universalis.

London: for William and John Innys, 1720.

The ‘organic’ construction of curves in the title of Maclaurin’s book refers to the kinematic generation of curves by continuous motion subject to mechanical constraints. Probably the best known example is the description of an ellipse given in Descartes’ Géométrie (1637): if A and B are fixed points and another point X moves in such a way that the sum of the lengths of the line segments AX and BX is constant, then X describes an ellipse (the constraint could be realised by looping a closed piece of string around A and B, and holding a marker X against the string so that it remains taught).

“This was an important topic, since in order to determine the point of intersection of curves in the construction of geometrical solutions, it was natural to think of the curves as generated by a continuous motion driven by some instrument. It is the continuity of the motion generating the curves that guarantees a point of intersection can be located exactly. Descartes had devised several mechanisms for generating curves. In De Organica Conicarum Sectionum in Plano Descriptione, which Newton had read in Exercitationes mathematicarum (1657), van Schooten had presented several mechanisms for generating conic sections” (Guicciardini, Isaac Newton on Mathematical Certainty and Method, p. 6). Newton’s notes on Scooten’s book, probably made in late 1664 (when he was 21), can be found in Papers, I, 1, §1. Later (probably in 1667 or 1668) he carried out his own research on the organic description of curves. “It had been Schooten’s aim to construct any given conic systematically by a uniform method and in one continuous, uninterrupted motion. For the individual conic species (ellipse, parabola and hyperbola) he had been lavishly successful, giving a proliferation of simple, elegant constructions of each, the gist of which Newton duly noted; but the final uniform mechanical construction of the general conic eluded him. It was Newton’s present triumph to succeed where Schooten had failed: in one of his most brilliant geometrical moods he frames a universal organic construction-method for all conics, including the degenerate line-pair” (Whiteside in Papers, II, p. 8; see also II, 1, §3). Newton also gave generalisations to curves of higher degree (cubics and quartics), but without proofs. Twenty years later, the substance of Newton’s researches on conics was incorporated into Principia, Section V, Book I (Westfall, Never at Rest, p. 197).

Newton’s work on the organic description of conics was one motivation for Maclaurin’s work in the Geometria organica. Another was Newton’s treatise Enumeratio linearum tertii ordinis, composed around 1695 and first published in the Opticks. In this work, Newton sought to classify cubic curves, in a manner analogous to the classification of quadratic curves (conics) into ellipses, parabolas and hyperbolas (and some degenerate cases). Newton identified 72 species of cubic curves, mostly classified in terms of the properties of their diameters and asymptotes.

The Geometria organica is the first detailed study of higher plane curves, i.e., curves defined by an equation involving powers of the Cartesian coordinates x and y higher than the third. “It is divided into two parts. In Part I, curves of all orders are described by the sole use of constant given angles and fixed straight lines.

“Chapter I is devoted to the Newtonian organic description of conics, of which the whole treatise is a development. The conic is obtained by rotating two constant angles POQ and PO’Q round two fixed pints O and O’, and restricting P to lie on a straight line, when Q, in general, generates a conic passing through O and O’ … The generality of this construction is established by applying it to trace the conic through five given points [this shows that every conic can be described by this construction]. The species and asymptotes of a conic are also determined.

“Chapter II gives a method of tracing curves of the third order that possess a double point [a point where the curve intersects itself] … Let R be any point on a straight line l; RPQ an angle of constant magnitude, whose vertex lies on a line l’, while RP passes through a fixed point O, RO’Q a constant angle rotating around the fixed point O’. Then Q generates a cubic with a double point at O’ … He discusses a great variety of particular cases, and gives a general method of tracing the rational circular cubic, treating especially the curve now known as the Strophoid … many of the rational cubics supposed to have been invented in the nineteenth century may be found here.

“Chapter III deals with the description of quartic curves with two or three double points, or with a triple point, also of cubic curves that have no double point …

“Chapter IV generalises the results of previous chapters. Let OP1P2 … PnQ be a broken line which, on deformation, retains unaltered the angles at P1, P2, …, Pn, while the lengths of the segments of the lines may vary … Let O be a fixed point, and let P1, P2, …, Pn lie on fixed straight lines. Then, if O’ is another fixed point, and O’Q makes a constant angle with PnQ, the locus of Q is a curve of degree n + 2, possessing an (n + 1)-ple point at O’ … The fundamental propositions of this chapter Maclaurin proves by the Cartesian geometry …

“In Part II, the linear loci are replaced by curves Cm, Cn, etc., of degrees m, n, etc.

“Chapter I contains the generalisation of Newton’s organic description of conics, as summed up in Prop. III:

‘In the quadrilateral POQO’, O and O’ are fixed points and the angles at O and O’ constant. If P lies on a curve Cn, Q, in general, generates a curve of degree 2n, having n-ple points at O and O’, and also at a third point O”. If, however, O, for example, is an r-ple point on Cn, then the locus of Q is a curve of degree 2n – r’ …

“In Chapter II, we may note the following generalisations of theorems in Part I.

Prop. V. In the quadrilateral PQO’R, O’ is a fixed vertex, PR passes through a fixed point O, and the angles at P and O’ are constant. If P and R lie on curves Cm and Cn, then Q generates a curve C2mn.

Prop. VI. O and O’ are fixed points, P and Q lie respectively on Cp and Cq, and the angles OPT and O’QT are constant. If T lies on a curve Cr, the intersection R of OP and O’Q lies on a curve of degree 4pqr

“Chapter III contains Maclaurin’s theory of pedals, and is one of the most interesting chapters of the whole book. [The ‘pedal curve’ of a curve C with ‘pedal point’ P is the locus of points X such that the line PX is perpendicular to the tangent line to C that passes through X.] … He gives a very full account of the pedals of a circle (Limaçon and Cardioid), of a parabola (including the Cissoid of Diocles), and of the central conic, the latter giving rise to the rational Bircircular Quartic … He points out that the pedal of a parabola is a straight line, and the pedal of a central conic is a circle, when the pole is at a focus … Regarding the rectification of such curves, he gives a beautiful theorem, which has been strangely overlooked, but which surpasses in elegance the other theorems known regarding the integration of pedals …

“Chapter IV is concerned with various applications to Mechanics [see below]. Chapter V is of great importance, and furnishes the foundation of the theory of higher plane curves.

“He starts with the lemma, which he cannot yet prove in its generality, that two curves Cm and Cn intersect in mn points. [This was stated by Newton in his proof of Lemma 28 in Book I of Principia; the first reasonably satisfactory proof was given by Étienne Bézout in 1779 – it is now known as ‘Bézout’s theorem’.] [Maclaurin then proves the following propositions.]

If, of the points necessary to determine a Cn, (nr + 1) lie on a Cr (r < n), then must Cn degenerate into a Cr and a Cn-r [this means the Cartesian equation of degree n that defines Cn factorises into the product of an equation of degree r and an equation of degree n – r.]

A curve of degree n is, in general, determined by ½ (n2 + 3n) points. Two curves of degree n cut in n2 points. Hence 9 points may not uniquely determine a cubic, while 10 would be too many. This is what has come to be known as ‘Cramer’s Paradox,’ although Cramer expressly quotes Maclaurin as his authority for the statement.

A curve Cn cannot possess more than ½ (n – 1)(n – 2) double points without degenerating.

The concluding propositions solve the following two problems:

Prop. XXV. To draw a Cn through 2n + 1 given points, of which one is a (n – 1)-ple point. The curve is unique.

Prop. XXVI. To draw a curve C2n through as many points as determine a Cn and three other points, each of which is an n-ple point on C2n. There may be several solutions.

“The production of this brilliant piece of geometrical research on the part of the youthful professor in Aberdeen would alone be sufficient to render Maclaurin’s name immortal” (Tweedie (1915), pp. 139-142).

In Chapter IV of Part II of the Geometria organica, Maclaurin “gave some theorems on central forces ‘in order to show the use of curved lines in natural philosophy’. Here Maclaurin used dotted letters to represent ‘momenta’, i.e., infinitesimal quantities generated by motion. These pages [120-135] are completely separate from the rest of the work since no attempt is made to link the geometry of Newton’s ‘Enumeratio’ with the analysis of ‘De quadratura’ [in the latter of which Newton gave his first published account of calculus; both were published in 1704 as appendices to the Opticks]” (Guicciardini, p. 37).

In Chapter IV, “two of the most eminent problems in mathematical philosophy are solved. In the first, the centripetal force, by which a body describes any curve, is investigated after an easy manner; and a simple construction of all those curves that a body would describe, if projected with the velocity that it might acquire by falling from an infinite height, in any hypothesis of gravity, is demonstrated. In the second, it is found that if any body describe a curve in a resisting medium, the resistance is always as the moment or fluxion of a quantity that expresses the ratio of the centripetal force to that force by which it would describe the curve in vacuo, multiplied by the fluxion of the curve. It is also demonstrated that if a body describe any curve in a resisting medium which in vacuo could have been described by a centripetal force, proportional to any power of the distance, the density of that medium will be reciprocally as the part of the tangent intercepted between the point of contact, and a line perpendicular to the radius at the center of the forces. This theorem is applied to several curves; and then the 10th Prop. of the second book of the Principles [Principia], and all its examples, are demonstrated from it. These propositions are treated of here, not only because they show the use of curves in philosophy, but because more simple ideas of the descriptions of some curves may be drawn from them, than from any other method; and because this is the method by which Nature herself describes curve lines” (Review of Geometria organica in Philosophical Transactions, 31 (1720-21), pp. 38-42).

A child prodigy, Colin Maclaurin entered the University of Glasgow aged 11. At the age of 19 he was elected a professor of mathematics at Marischal College, Aberdeen, and two years later he became a fellow of the Royal Society of London, and there became acquainted with Newton. On the recommendation of Newton, he was made a professor of mathematics at the University of Edinburgh in 1725. In 1740 he shared, with Leonhard Euler and Daniel Bernoulli, the prize offered by the French Academy of Sciences for an essay on tides. His most important work, Treatise of Fluxions (1742), was written in reply to criticisms by Bishop George Berkeley that Newton’s calculus was based on faulty reasoning.

ESTC T94366. Guicciardini, The Development of Newtonian Calculus in Britain 1700-1800, 1989. Tweedie, “A study of the life and writings of Colin Maclaurin,’ The Mathematical Gazette 8 (1915), pp. 133-151. Tweedie, ‘The Geometria Organica of Colin Maclaurin: A historical and critical survey,’ Proceedings of the Royal Society of Edinburgh 36 (1916), pp. 87-150.

4to (231 x 182mm), pp. [xii], 139, [1, errata], including imprimatur leaf before title, with 12 folding engraved plates (bowned due to paper quality), woodcut device of the Prince of Wales on title-page with inscription ‘ICH DIEN,’ woodcut initials, head- and tail-pieces (plates browned, light spotting confined mostly to endpapers). Contemporary full vellum with gilt lettering to spine, red speckled edges (some soiling to boards, corners slightly rubbed). Overall, a very good, wide-margined copy.

Item #5542

Price: \$5,500.00