Harmonia mensurarum: sive analysis & synthesis per rationum & angulorum mensuras promotæ: accedunt alia opuscula mathematica … Edidit et auxit Robertus Smith.

Cambridge: n.p., 1722.

First edition, the fine Macclesfield copy, of the first publication of virtually all of Cotes’ mathematical work, including the ‘Cotes-Newton formulae’ for approximate integration, ‘Cotes’ factorization theorem,’ ‘Euler’s identity’ (stated by Euler a quarter-century later), and an anticipation of the method of least squares (credited to Gauss and Legendre at the beginning of the next century). Cotes (1682-1716) was a brilliant mathematician who assisted Newton with the preparation of the second edition of Principia, and wrote its preface. His early death aged 33 caused Newton to lament: “Had Cotes lived we might have known something.” “His only publication during his life was an article entitled ‘Logometria’ (Phil. Trans., 1714). After his death his mathematical papers, then in great confusion, were edited by Robert Smith and published as a book, Harmonia mensurarum (1722) [which] gives an indication of Cotes’s great ability … This is shown most powerfully in his work on integration, in which long sequences of complicated functions are systematically integrated, and the results are applied to the solution of a great variety of problems. Cotes first demonstrates that the natural base to take of a system of logarithms is the number which he calculates as 2.7182818. He then shows two ingenious methods for computing Briggsian logarithms (with base 10) for any number and interpolating to obtain intermediate values. The rest of part I is devoted to the application of integration to the solution of problems involving quadratures, arc lengths, areas of surfaces of revolution, the attraction of bodies, and the density of the atmosphere. His most remarkable discovery in this section occurs when he attempts to evaluate the surface area of an ellipsoid of revolution. He shows that the problem can be solved in two ways, one leading to a result involving logarithms and the other to arc sines, probably an illustration of the harmony of different types of measure. By equating these two results he arrives at the formula ix = log(cos x + i sin x), a discovery preceding similar equations obtained by de Moivre (1730) and Euler (1748). The second and longest part of the Harmonia mensurarum is devoted to systematic integration. Cotes proceeds to evaluate the fluents [integrals] of no fewer than ninety-four types of fluxions . . . His calculation was aided by a geometrical result now known as Cotes’ theorem . . . The third part consists of miscellaneous works, including papers on methods of estimating errors, Newton’s differential method, the construction of tables by differences, the descent of heavy bodies, and cycloidal motion. There are two particularly interesting results here. The essay on Newton's differential method describes how, given n points at equidistant abscissae, the area under the curve of nth degree joining these points may be evaluated . . . A modernized form of this result is known as the Newton-Cotes formula. In describing a method for evaluating the most probable result of a set of observations, Cotes comes very near to the technique known as the method of least squares. He does not state this method as such; but his result, which depends on giving weights to the observations and then calculating their centroid, is equivalent. This anticipates similar discoveries by Gauss (1795) and Legendre (1806)” (DSB). Cotes’ work on least squares was referenced by Laplace in Théorie analytique des Probabilités (1812, p. 346).

Provenance: The Earls of Macclesfield (South Library bookplate on front paste-down, embossed stamp on first three leaves of text). Although not signed by him, it is likely that this was William Jones’ copy. Jones (1746-94) was a friend of Cotes and the two enjoyed an extensive correspondence, particularly on questions relating to interpolation procedures – two of Cotes’ tracts in the present volume are on this subject, and they are based partly on Newton’s Methodus differentialis (1711), which was published by Jones. Jones’ great library of mathematical books inherited by the Macclesfield family on his death. Sold Sotheby’s, June 10, 2004, lot 565, £1,680 ($3,070).

The work is in two parts: the first is the ‘Harmonia mensurarum’ proper, the second is a collection of six ‘Opera miscellanea’. The ‘Harmonia mensurarum’ is itself divided into three parts. Part I is ‘Logometria’, as published in the Philosophical Transactions, vol. 29, no. 338, for March 1714 (pp. 5-45); the methods used in part I, notably including extensive tables of integrals, are explained in part II; part III contains a further selection of problems to which the methods apply (parts II and III of Logometria were published for the first time in the present work).

“In Logometria, Cotes: defined logarithms as measures of ratios and established the functional equation f(xn) = nf(x) which these measures satisfy; explained how different systems of logarithms were related through the modulus (a scale factor, in fact the logarithm of e, the base of natural logarithms, to the base concerned); calculated the value of e and of 1/e, giving the result to 12 decimal places; demonstrated a very neat method of computing Briggs’ logarithms, avoiding almost all use of series; showed how to convert logarithms from one system to another; explored at great length the relationship of logarithms to various quadratures of the hyperbola; described the properties of the logarithmic curve; and showed the logarithmic properties of the logarithmic spiral. In illustrative scholia, Cotes: obtained series for ln[(z + x)/(z – x)] (z constant, x variable); extended this to the series for ln(1 + v) and ln(1 – v); developed the continued fractions for e and 1/e, for generating rational approximations to these quantities; applied the results from the hyperbola to vertical ascent and descent in resisting media; investigated the density of the atmosphere at any given altitude; and showed how to calculate the change in longitude for a given change in latitude, along a loxodrome [a curve, used by navigators, which cuts the lines of longitude at a fixed angle]. The paper ends with a long Scholium Generale, in which Cotes gave his solutions to a number of problems of then contemporary interest, the problems being: the rectification of the parabola and the Archimedean spiral; the rectification of the reciprocal spiral, and to find the volume of the cissoidal solid of revolution; the area of a conchoids (between its branches), and to find the volume of the conchoidal solid of revolution; further quadratures of the hyperbola, and centres of gravity; the surface area of the hyperboloid of revolution; the surface areas of the prolate and of the oblate spheroids, and the corresponding volumes; a note on cubic equations, noting that the solutions depend either on ratios or on angles; the gravitational attraction of spheroids; spiral central orbits described under the inverse cube law; the oscillation of a cycloidal pendulum in a resisting medium; the density of air at any altitude, allowing for gravitational variation; and the division of the meridian on the Mercator chart.

“This is an impressive list for a first tentative publication. We know that Cotes read widely in the current literature; references to, for example, Briggs, Huygens, Leibniz, Halley, De Moivre, Newton, Wallis, occur in his writings. J. E. Hofmann goes further and says of Logometria: ‘Cotes completely worked through and used the whole rich contemporary literature.’

“The problems dealt with are not new; the interest lies in the solutions, which Cotes gives, and in the methods of arriving at those solutions, which Cotes does not give … the solutions are, for the most part, given in geometrical form, although usually requiring the drawing of a line equal in length to a given logarithm or arc length. These lengths have first to be calculated, and Cotes had developed methods of integration applicable to a wide range of problems, enabling the calculations to be performed relatively simply. In Cotes’ posthumous works, published as Harmonia mensurarum (Cambridge, 1722), by Robert Smith, Cotes’ cousin, successor in the Plumian chair [of astronomy at Cambridge] and literary executor, details of the integrals appear as Logometria, part II” (Gowing, pp. 21-22).

“Quite early in his career, in fact whilst still at school, Cotes had an interest in ‘the squaring of curves’, and his first letter to Newton (18 August 1709) concerned corrections he had made to Newton’s fluents [integrals] as published in De quadratura curvarum (London, 1704). From Robert Smith’s Editoris Notae to Harmonia mensurarum, it is clear that Cotes had developed these tables before 1714 … Cotes developed eighteen tables of integrals, and associated reduction formulae (called by him ‘Continuation Formulae’) … They did not appear in print until the 1722 publication of Harmonia mensurarum, and one agrees with De Morgan’s remark that they ‘represent the first substantial advance in the development of integration techniques applied to logarithmic and trigonometric expressions.’ Some of the problems in the Scholium generale arise from Cotes’ work on Principia, second edition (Cambridge, 1713), notably those on spirals, and gravitational attraction of spheroids; others, familiar problems of the day, were chosen to demonstrate the superiority of the methods, as, for example, the rectification of the parabola and of the logistic (logarithmic) curve. The final problem of the Scholium Generale shows a neat construction for the length of the meridian on the Mercator chart, effectively the integral of sec θ between variable limits.

“It is clear then that the link between the principles set out in Logometria, and the problems to which they are applied, is provided by the tables of integrals. These are much more than a collection of more or less ingenious integration techniques. They epitomise Cotes’ perception of that ‘Harmony of Measures’ between measure of angles (trigonometrical quantities) and measures of ratios (logarithms)” (Gowing, pp. 34-35).

Indeed, the tables of integrals involve a quantity R which, to use our terminology, can be either a real or an imaginary number (i.e., a real number multiplied by i = √-1). The value of the integral involves either trigonometric or logarithmic functions, according as R is real or imaginary. As Gowing shows (pp. 37-38), this dual nature of Cotes’ formulae implies the equation

ln (cos ϕ + i sin ϕ) = iϕ.

This famous equation, which was obtained later (in the form, e = cos ϕ + i sin ϕ) by Abraham de Moivre in Miscellanea analytica (1730), and by Leonhard Euler in Introductio in analysin infinitorum (1748), expresses concisely Cotes’ ‘harmony of measures’ between the trigonometric and logarithmic worlds.

In part III of Logometria, “Cotes presents a delightful collection of 12 problems, demonstrating not only the range and effectiveness of the methods, but also the details of the proofs. In a brilliant display of ingenious coordinate systems, allied to integration techniques, he develops a geometrical analysis to which traditionally-difficult problems yielded easily. Abandoning his former cautious concealment of proofs, he sets out the details in an expansive style clearly intended for a wider audience … The 12 problems selected by Cotes are: I. The quadrature of the tangent curve; II. The quadrature of the secant curve. III. The volume of the solid of revolution formed from the tangent curve. IV. The properties of the enclosed, or spiral, tractrix, and of its negative pedal (named by Cotes, the lituus); V. The surface area of the paraboloid of revolution; VI. The surface areas of cissoidal solids of revolution; VII. The surface area generated by revolution of the logarithmic curve about its asymptote. Scholium on finite and infinite surfaces of revolution; VIII. The angular velocity of a sphere and its circumscribing cylinder about their common axis; IX. The attraction due to a sphere, under the inverse cube law; X. Motion under a centripetal force varying as the inverse cube of the distance (the Cotes spiral); XI. The vertical motion of a heavy body in a resisting medium; XII. To construct a catenary” (Gowing, pp. 55-56).

The first part of the work is concluded by Smith’s Theoremata tum logometrica tum trigonometrica. Datarum fluxionum fluentes exhibentia, per methodum mensurarum ulterius extensam. This included further tables of integrals which Cotes had found after he had completed Logometria (parts I, II and III) in 1712. Most importantly, however, it contains ‘Cotes’ factorization theorem,’ which expresses xn ± an as a product of linear and quadratic factors.

“Cotes’ interest in practical astronomy led him from instruments to a concern for the accuracy of observations, and to a serious attempt to assess these errors and to minimise them. His ideas are set out in a short Latin tract, Aestimatio errorum in mixta mathesi. The tract is one of several forming Cotes’ Opera miscellanea, usually bound up [as here] in one volume with Harmonia mensurarum. Astronomical observations commonly require the solution of plane or spherical triangles and, in his tract, Cotes presented 28 theorems, in which were derived differential formulae relating the small variations in two elements of a triangle, two other elements being held constant” (Gowing, p. 91). “Aestimatio errorum concludes with the interesting statement that when a number of slightly different observations are made of a particular quantity, the most probable value of the correct observations can be found as follows:

‘Let p be the place of some object defined by observation, q, r, s the places of the same object from subsequent observations. Let there also be weights P, Q, R, S reciprocally proportional to the displacements arising from the errors in the single observations, and which are given by the limits of the given errors; and the weights P, Q. R. S are conceived as being placed at p, q, r, s, and their centre of gravity Z is found: I say the point Z is the most probable place of the object.’

“It has been claimed that this method of Cotes’ for determining the most probable value, from a number of readings, each subject to small errors, is equivalent to the method of least squares (used by Legendre in 1806). This claim is discussed by J. B. J. Delambre in Histoire d’Astronomie au Dixhuitième Siècle, p. 455, and by D. T. Whiteside, in The Mathematical Papers of Isaac Newton, vol. 6, p. 51, note” (Gowing, p. 107).

The next two tracts, De methodo differentiali Newtoniana and Canonotechnia sive construction tabularum per differentias, deal with interpolation procedures, approximation integration, and the construction of tables. The first of these gives, for the first time, a proof of a result stated by Newton in Principia, and then adds some results of Cotes’ own. Book III, Lemma V of Principia (1687), which stated ‘To find a curved line of the parabolic kind which shall pass through any given number of points,’ was the first publication of Newton’s method of divided differences, for fitting a polynomial to a number of tabulated points. In this lemma, Newton presented the interpolation formula which bears his (and Gregory’s) name and extended it to the case of unequal tabular intervals. Cotes added a postscript, having seen Newton’s Methodus differentialis, published by William Jones in 1711. “In this, following suggestions made by Newton, Cotes developed what became known as the Cotes-Newton formulae for approximate quadrature. These are derived by considering the area under a parabola through three points at equal intervals on a given curve (Simpson’s rule), a cubic through four such points (the three-eights rule), and so on up to a curve of order 10 through 11 equally-spaced points” (Gowing, p. 118).

Canonotechnia is concerned with systematic interpolations, i.e., sub-tabulation … The method consists in deducing, from the entries in the central row of a difference table (equally-shaped pivotal values), the new entries which would result if interpolated values were present in the table. From these new differences, the interpolated values can be found. Cotes’ method, then, computes interpolated values from the entries in the difference table, just as modern interpolation methods do. As has been pointed out by Whiteside (Papers, vol. 4, p. 61), formulae equivalent to both the Bessel and the Stirling approximation formulae occur in Canonotechnia. I can only agree with Dr. Whiteside that at least one of these formulae should be named after Cotes” (Gowing, p. 124).

The volume concludes with three short tracts on dynamics, De descensu gravium, De motu pendulorum in cycloide, and De motu projectilium, which were reissued together in 1770, followed by Smith’s Editoris Notae.

ESTC T100930. Gowing, Roger Cotes – natural philosopher, 1983.



Two parts in one volume, 4to (245 x 186mm), pp. [20], 249, [3], 125, [1], with one folding table (a few leaves lightly browned). Contemporary polished calf, covers triply ruled in gilt with gilt corner fleurons, spine gilt in compartments with red lettering-piece, red speckled edges, marbled endpapers (head of spine slightly worn, joints starting). A large and beautiful copy.

Item #5492

Price: $5,500.00