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First edition, rare, of this work which contains the first publication of virtually all of Cotes's mathematical work. 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 untimely death accentuated his being one of the very few British mathematicians capable of following on from Newton's great work. 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. His style is somewhat obscure, with geometrical arguments preferred to analytical ones, and many results are quoted without explanation. What cannot be obscured is the original, systematic genius of the writer. 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 1 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 (pp. 27-28) 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 differnt 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 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's 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). 4to, pp. [20], 249, [3], 125, [1], with one folding table (some light browning and foxing). Contemporary polished calf, spine richly gilt with red lettering-piece (head of spine chipped, corners a little worn). A very good copy in an attractive, unrestored contemporary binding.
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