634 PHILOSOPHICAL TRANSACTIOKS. [aNNO 1742-3. 



cone of the same height with the frustum ; and in other figures it is reduced to 

 the difference in the cone. 



In the remarks on the method of the ancients, the author observes, that they 

 established the higher parts of their geometry on the same principles as the ele- 

 ments of the science, by demonstrations of the same kind ; that they seem to 

 have been careful not to suppose any thing to be done, till by a previous pro- 

 blem they had shown how it was to be performed : far less did they suppose any 

 thing to be done, that cannot be conceived to be possible, as a line or series to 

 be actually continued to infinity, or a magnitude to be diminished till it be- 

 comes infinitely less than it was. The elements into which they resolved mag- 

 nitudes were always finite, and such as might be conceived to be real. Un- 

 bounded liberties have been introduced of late, by which geometry, wherein 

 every thing ought to be clear, is filled with mysteries, and philosophy is like- 

 wise perplexed. Several instances of this kind are mentioned. The series 1, 

 2, 3, 4, 5, 6, 7, &c. is supposed by some to be actually continued to infinity ; 

 and, after such a supposition, we are puzzled with the question, whether the 

 number of finite terms in such a series is finite or infinite. In order to avoid 

 such suppositions, and their consequences, the author chose to follow the anci- 

 ents in their method of demonstration as much as possible. Geometry has been 

 always considered as our surest bulwark against the subtleties of the sceptics, 

 who are ready to make use of any advantages that may be given them against 

 it ;* and it is important, not only that the conclusions in geometry be true, but 

 likewise that their evidence be unexceptionable. However, he is far from 

 affirming, that the method of infinitesimals is without foundation, and after- 

 wards endeavours to justify a proper application of it. 



The grounds of the method of fluxions are described in chap. 1, book ], and 

 again in chap. 1, book 2. In the former, magnitudes are conceived to be 

 generated by motion, and the velocity of the generating motion is the fluxion 

 of the magnitude. Lines are supposed to be generated by the motion of points. 

 The velocity of the point that describes the line is its fluxion, and measures the 

 rate of its increase or decrease. Other magnitudes may be represented by lines 

 that increase or decrease in the same proportion with them ; and their fluxions 

 will be in the same proportion as the fluxions of those lines, or the velocities of 

 the points that describe them. When the motion of a point is uniform, its 

 velocity is constant, and is measured by the space which is described by it in a 

 given time. When the motion varies, the velocity at any term of the time is 

 measured by the space which would be described in a given time, if the motion 



• See Bayle's Dictionary, Artide Zeno. — Orig. 



