638 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1742-3. 



demonstrated in this chapter, and again in the last chapter of the second book, 

 is, that when the first fluxion of the ordinate, with its fluxions of any subse- 

 quent successive orders, vanish, and the number of all these fluxions that 

 vanish is odd, then the ordinate is a maximum or minimum, according as the 

 fluxion of the next order to these is negative or positive. The ordinate passes 

 through a point of contrary flexure, when its fluxion becomes a maximum or 

 minimum, supposing the curve to be continued on both sides of the ordinate. 

 Hence the cotnmon rule for finding the points of contrary flexure is corrected 

 in a similar manner. Such a point is not always formed when the second fluxion 

 of the ordinate vanishes ; for if its third fluxion likewise vanishes, and its fourth 

 fluxion be real, the curve may have its cavity turned all one way. The s^me 

 is to be said, when its fluxions of the subsequent successive orders vanish, if 

 the number of all those that vanish be even. Other theorems are subjoined re- 

 lating to this subject. 



The J 0th chapter treats of the asymptotes of lines, the areas bounded by 

 them and the curves, the solids generated by these areas, of spiral lines, and 

 the limits of the sums of progressions. The analogy between these subjects, 

 induced the author to treat of them in one chapter, and illustrate them by one 

 another. He begins with three of the most simple instances of figures that 

 have asymptotes. In the common hyperbola, the ordinate is reciprocally as 

 the base, and therefore decreases while the base increases, but never vanishes, 

 because the rectangle contained by it and the base is always a given area, and it 

 is assignable at any assignable distance, how great soever. Tlie points of the 

 conchoid are determined by drawing right lines from a given centre, and on 

 these produced from the asymptote, taking always a given right line ; so that 

 the curve never meets the asymptote, but continually approaches to it, because 

 of the greater and greater obliquity of this right line. The third is the loga- 

 rithmic curve, wherein the ordinates, at equal distances, decrease in geometri- 

 cal proportion, but never vanish, because each ordinate is in a given ratio to 

 the preceding ordinate. Geometrical magnitude is always understood to consist 

 of parts ; and to have no parts, or to have no magnitude, are considered as 

 equivalent in this science.* There is, however, no necessity for considering 

 magnitude as made up of an infinite number of small parts ; it is sufficient, that 

 no quantity can be supposed to be so small, but it may be conceived to be di- 

 minished further ; and it is obvious, that we are not to estimate the number of 

 parts that may be conceived in a given magnitude, by those which in particular 



• See Euclid's Elements, Def. 1, lib. i. — Orig. 



