18 PHILOSOPHICAL TRANSACTIONS. [aNNO IGQS. 



space whose fluxion is y \^ y- •\- ■:^ a a ; but this space is no other than the 

 exterior equilateral hyperbola a b g e (fig. 5), whose semi-axis is a b ^ 4- a, 

 abscisse a e ^ ?/, and ordinate e g =: x. 



For measuring the superficies produced by the rotation of a curve about its 

 axis, there must be assumed for its fiuxion, the cylindrical superficies, whose 

 altitude is the fluxion of the curve, and its distance from the axis is the ordinate 

 belonging to this fluxion. For example, let a c (fig. 1) be a circular arc, 

 which, by revolving about the axis A b, generates a spherical superficies, pro- 

 posed to be measured. The fluxion c d of the arc is already found to be 



. — ; hence, if we multiply this by the circumference to the radius 



V 2 r X — X X 



B c, that is by W 1 r x — a; a: (supposing the ratio of the circumference to 



the radius to be - ), we shall have the fluxion of the spherical surface = c.r, 



and therefore the surface itself is c x. 



As to what respects the centre of gravity : having found the fluxion of the 

 superficies or solid, and drawn this into its distance from the vertex, we must 

 then resort back to the fluent ; which being divided by the superficies or solid 

 itself, will give the distance of the centre of gravity from the vertex. Thus, 

 let it be required to find the centre of gravity of all paraboloids : the fluxion of 



!2+i 



these is generally expressed by a;" i- ; this multiplied by x gives a?" i, the 



fluent of which is - ■ ^ x" , which being divided by the area of th6 para- 



boloid, — — — a;" , gives — x for the distance of the centre of gravity 



from the vertex. 



Much in the same way is found the centre of gravity in the portion of a 

 sphere. For its fluxion An.dx i — x x x, drawn into x, is 4 ?i . d x'^ x — x^ .f, 

 the fluent of which 4 7i . ^ d x^ — -^ x*, divided by 4 jz . ±.dx- — ^ x^, the solidity 



of the portion, produces ^ ."T^ ^, or ^~/ x, for the distance of the cen- 



v"— T* Otf — 4* 



tre of gravity from the vertex. 



A most compendious and facile Method for constructing the Logarithms, ex- 

 emplified and demonstrated from the Nature of Numbers, without any regard 

 to the Hyperbola ; with a speedy Method for finding the Number from the 

 Logarithm given. By E. Halley. N° 21 6, p. 58. 

 The invention of the logarithms is justly esteemed one of the most useful 



discoveries in the art of numbers, and accordingly has had an universal recep- 



