Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/18

8 INFINITESIMAL CALCULUS Barrow anticipated the methods of Leibnitz and Newton of drawing tangents, so far as rational algebraic curves were concerned. Barrow s researches were delivered in his professorial lectures in 1GG4, 1GG5, and 1666, and were pub lished in 1G70, under the title of Lectiones Mathematical. Roberval. The method of tangents of Roberval is based on the conception of the composition of motions, recently intro duced by Galileo into mechanics, and depends on finding, from the properties of the curve, the different components of the motion of the point at which the tangent is required. The direction of the resultant of these motions determines that of the tangent. This method bears an analogy to Newton s method of " fluxions," but is very limited in its application on account of the impossibility of applying it except in a few cases. Roberval applied it successfully to the following curves the parabola, hyperbola, ellipse, conchoid of Nicomedes, limayon of Pascal, spiral of Archi medes, quadratrix, cissoid, cycloid, companion to the cycloid, and the parabola of Descartes. We thus see that both in England and on the Continent the principles of the infinitesimal calculus were being gradually developed. Their importance was seen and understood, and they were employed in extending the dominion of geometry. Nothing more was required but an appropriate notation to form them into a system. This Newton was accomplished by Newton and Leibnitz, who appeared and nearly at the same time in the field of discovery. In Leibnitz. ascr ibing to each of these great men the full honour due to the merit of the invention of the calculus, it is proper to add that this is a question which at one time divided the opinions of the scientific world, and gave rise to a controversy which was agitated with great keenness for almost a whole century. There never could be any doubt as to Newton being the inventor of the calculus of fluxions ; but the question strongly contested has been, whether Leibnitz invented his calculus independently, or borrowed it from the iluxional calculus, with which at bottom it is identical. Leibnitz, born in 164G, four years after Newton, was also later than Newton in beginning his career of dis covery in mathematics. In 1673, being in London, he communicated to some members of the Royal Society what lie supposed to be discoveries relative to the differences of numbers. It was, however, shown to him that the same subject had been previously discussed by Mouton, a French geometer. He then appears for the first time to have turned his attention to infinite series. On his return to Germany in 1674 he announced to Mr Oldenburg, secretary to the Royal Society, that lie possessed very general analytical methods, depending on infinite series, by which he had found theorems of great importance relating to the quadrature of the circle. In reply Oldenburg informed him that Newton and Gregory had discovered similar methods for the quadrature of curves, which extended to the circle. On June 13, 1676, Newton addressed a letter to Olden burg, for transmission to Leibnitz. It contained his bi nomial theorem, the now well-known expressions for the expansion of an arc in terms of its sine, and the converse, that of the sine in terms of the arc. Contrary to modern procedure, he deduced the latter from the former, by the method of reversion of series, a method called by Newton the " extraction of roots." This letter also contained an expression in an infinite series for the arc of an ellipse, and various other results, accompanied, however, only by brief indications of his method of demonstration. On the 27th of the following August, Leibnitz sent a reply through Oldenburg, in which he requested fuller ex planation from Newton of his theorems and method of demonstration. Leibnitz added that he possessed another method of extensive application in geometry and mechanics, of which he gave some examples. To this communication Newton replied on October 24, 1676, in a letter which occupies thirty pages in Newton s Opuscula (ed. Cast.). As this letter probably gives a more complete account of the order and dates of Newton s dis coveries than is to be obtained elsewhere, it appears desir able to give a brief abstract of it here. He commences by commending the very elegant method of Leibnitz for the treatment of series. He goes on to state that he him self had three methods of such treatment. His first was arrived at from the study of the method of interpolation of series by which Wallis had arrived at expressions for the area of the circle and hyperbola. Thus, by considering the series of expressions (1 # 2 )- , (1 a; 2 )-, (1-a; 2 )^, (1 -# 2 )^, (I - z 2 )*, &c., he deduced from the known values of the alternate expressions, by the method of interpolations, the law which connects the successive coefficients in the expansions of the intermediate terms (1 - x 2 )*, (1 x 2 )% t (1 x 2 )?, &c. Newton thus determined the area of the circle and hyperbola, in infinite series. He adds that this method would have completely escaped his memory if he had not a few weeks previously found the notes he had formerly made on the subject. By following out the idea thus suggested, he was led to the discovery of his binomial expansion. This he tested in the case of (l-x 2 )* by the algebraic process of extracting the square root, as also, in other cases, by direct multipli cation. Having established this result, he was enabled to discard the method of interpolation, and to employ his binomial theorem as the most direct method of obtaining the areas and arcs of curves. Newton styled this his second method. He states that he had discovered it be fore the plague (in 1665-66) had compelled him to leave Cambridge, when he turned his attention to other subjects. He goes on to say that he had ceased to pursue these ideas as he suspected that Nicholas Mercator had employed some of them in his LogaritJimotechnia (1668) ; and this led him to think that the remainder would have been found out before he himself was of sufficiently ripe age to publish 1 his discoveries (priusqiiam ego letatis esscm maturse ad scrilen- dum). Newton proceeds to state that about 1669 he communi cated through Barrow to Collins a compendium of his method subsequently called the " method of fluxions," with applications to areas, rectification, cubature, &c. In his letter, however, he gave no explanation of this method, carefully concealing its nature in an anagram of trans posed letters, thus 6 cc d x 13e ff 7i 31 9?i 4o 4<? rr 4s (H I2v x. 2 At the end of his letter Newton alludes to the solution of the " inverse problem of tangents," a subject referred to in Leibnitz s letter. For the solution of such problems ho says he has two methods, which also he disguises under an anagram. The meaning of this anagram is given in his Opuscula, and, as it throws light on Newton s method of discovery, it is introduced here: "Una methodus consistit in extractione fluentis quantitatis ex requatione simul involvente fluxionem ejus. Altera tantum in assumptione seriei pro quantitate qualibet incognita ex qua caetera commode derivari possunt, et in collatione 1 Newton also states in this letter that, in consequence of the various objections, &c. , which were raised to his theory of light and colour, he felt that lie had been imprudent in having published it, because by catching at the shadow he had lost the substance, namely, his own quiet and repose. This probably may have been the reason why Newton refrained for so long a time from making public his discovery of the method of fluxions, notwithstanding the earnest solicitation of his friends. - It means Data scqwitione quotcunque flucntes quantitatesinvohcnie,