RHEOMETRr. 337 



CHAPTER V. 



RIIEOMETRY. 



RHEOMETRY, or the Differential and Integral Calculus, is that 

 branch of Mathematics which treats of the correlative increments of 

 quantities that are mutually dependent ; and of the relations of these 

 increments to each other, and to the quantities from which they are 

 derived. For this branch of Mathematics, we venture to propose 

 the name Rheometry ; from the Greek f<o, I flow, suggested by the 

 name Fluxions, applied to this science by Newton. The word Cal- 

 culus, is the Latin for a small stone, or pebble ; and as the ancients 

 used pebbles to assist them in numbering or reckoning, the word was 

 hence applied to the method or means of numerical calculation. 

 Leibnitz conceived the dependent quantities to receive infinitely small 

 increments, the sum of which would make up the quantities them- 

 selves : hence he proposed for this branch the name above given. 

 Newton considered all quantities as generated by motion ; a line by 

 the motion of a point ; a surface by the motion of a line ; and a solid 

 by the motion of a surface. This idea of magnitudes moving, or 

 floiving, led him to propose for this new science, the name of 

 Fluxions; which is now, however, for the most part superseded by 

 the name proposed by Leibnitz. 



Kepler was the first, among the moderns, who applied the infini- 

 tesimal method to geometrical figures ; and he considered the circle 

 as composed of infinitely small triangles, formed by the radii. The 

 method of indivisibles, first published in 1635, by Cavalieri, or 

 Cavallerius of Bologna, regarded surfaces as made up of mere lines; 

 whereas Roberval, his contemporary, regarded them as composed of 

 infinitesimal areas ; and applied this method to the measure of the 

 cycloid. Fermat's method of finding maxima and minima, im- 

 proved by Descartes, in his method of tangents, and still farther 

 extended by Wallis of England, in his Arithmetic of Infinites, on 

 the quadrature of curves, as also by Huyghens, in his theory of 

 evolutes, was among the successive steps which led to the invention 

 of the Differential Calculus. This invention has been claimed both 

 for Newton, and Leibnitz ; but the question has never been fully 

 decided. Newton is said to have invented his method of Fluxions 

 as early as 1672 ; but he made no publication of it till that in his 

 Principia, in 1686. Leibnitz claims to have invented the Calculus 

 in 1676 ; and the first publication of it was made by him in 1684, 

 in the Leipsic Acts, under the title of A New Method for Maxima 

 and Minima ; but it contained no demonstrations. Leibnitz gave the 

 first ideas of the Integral Calculus, in two small tracts, on the quadra- 

 ture of curves, published in 1685. 



From this time, the new calculus made rapid advances, in the 



hands of its inventors and others ; and its great utility was shown in 



its successful application to many of the more difficult problems in 



physical science, which had never before been resolved. The first 



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