VOL. XXIX.l PHILOSOPHICAL TRANSACTIONS. ISQ 



20. Let wg be a line infinitely near and parallel to wg, and terminated by the 

 same lines; and cs, wo-, perpendicular to the meridian; cs intersecting wg in z, 

 and W(r in y. Let cm be a tangent to AC in c; wt a tangent to aw in w; so 

 is CM = o-T. Because of like triangles czc, csm; and wyw, wa-r; cs : cm :: cz 

 : cc : also wo- : o-t :: wy : yw. But wo- = cs; o-t = cm; cz = wy; therefore is 

 yw the fluxion of gw, equal to cc the fluxion of the curve ac. Consequently 



GW = AC. Q. E. D. 



21. It may be noted that this equitangential curve gives the quadrature of a 

 figure of tangents standing perpendicular on their radius. In fig. 8, let AyT be 

 a curve whose ordinates, as gy, gF, are equal the tangents of their respective 

 intercepted arcs Ak, ah. Let Tg be produced to touch the curve Ac in c : then is 

 the area aFg equal to the rectangle contained by radius ar and gc the produced 

 part of the ordinate; or aFg = ar X gc. The demonstration of which, and 

 df the following section, I for brevity omit. 



22. If we suppose the figure acb &c kr, fig. 8, infinitely continued^ to 

 be turned about its asymptote rk as an axe, the solid so generated will be equal 

 to a rectangled cone whose altitude is equal to ar. And its curve surface will 

 be equal to half the surface of a globe, whose radius is ar. So that if the 

 curve be continued both ways infinitely, as its nature requires, the whole sur- 

 face will be equal to that of a globe of the same radius ar. 



The description of the ruler and wheel, fig. Q, is sufficient for the demon- 

 stration of the properties of the curve; but in order to an actual construction 

 for use, I have added fig. 11, where ab is a brass ruler: wh the little wheel, 

 which must be made to move freely and tight upon its axe, like a watch wheel, 

 the axe being exactly perpendicular to the edge of the ruler ; s represents a 

 little screw-pin, to set at several distances for different radii, and its under end 

 is to slide by the edge of the other fixed ruler; p is a stud for conveniently 

 holding the ruler in its motion. 



Note. — Most of these properties of this curve by the name of la tractrice, 

 are to be found in a Memoir of M. Bomie, among those of the Royal Academy 

 of Sciences for the year 1712, but not published till 1715; whereas this paper 

 of Mr. Perks was produced before the Royal Society in May 1714, as appears 

 by their Journal. 



An Account of a Book j entitled Methodus Incrementorum. Auctore Brook Taylor ^ 

 LL.D. et R, S. S. By the Author. N° 345, p. 339. 



When I applied myself to consider thoroughly the nature of the method of 

 fluxions, which has been the occasion of so much honour to its great inventor 

 Sir Isaac Newton, I fell by degrees into the method of increments, which I 



