466 PHILOSOPHICAL TRANSACTIONS. [aNNO 16Q2. 



And first, infinite length, or a line infinitely long, is to be considered either 

 as beginning at a point, and so infinitely extended one way, or else both ways 

 from the same point; in which case the one, which is a beginning infinity, is 

 the one half of the whole, which is the sum of the beginning and ceasing infi- 

 nity, or as I may say of infinity a parte ante and a parte post, which is analogous 

 to eternity in time or duration, in which there is always as much to follow, as is 

 past, from any point or moment of time, nor does the addition or subduction of 

 finite length, or space of time, alter the case, either in infinity or eternity, since 

 neither the one nor the other can be any part of the whole. 



As to infinite surface or area, any right line, infinitely extended both ways on 

 an infinite plane, divides that infinite plane into equal parts, the one to the right 

 and the other to the left of the said line ; but if from any point, in such a 

 plane, two right lines be infinitely extended, so as to make an angle, the infi- 

 nite area, intercepted between those infinite right lines, is to the whole infinite 

 plane, as the arch of a circle, on the point of concourse of those lines as a 

 centre, intercepted between the said lines, is to the circumference of the circle; 

 or as the degrees of the angle to the 36o degrees of a circle. For example, 

 two right lines meeting at a right angle, include, on an infinite plane, a 

 quarter part of the whole infinite area of such a plane. And if two parallel in- 

 finite lines be supposed drawn on such an infinite plane, the area intercepted 

 between them will be likewise infinite ; but at the same time it will be infinitely 

 less than that space which is intercepted between two infinite lines that are in- 

 clined, though with never so small an angle, since in the one case the given 

 finite distance of the parallel lines diminishes the infinity in one degree of dimen- 

 sion ; whereas in a sector there is infinity in both dimensions : and consequently 

 the one quantity is infinitely greater than the other, and there is no proportion 

 between them. 



From the same consideration arise the three several species of infinite space 

 or solidity ; for a parallelopipid or a cylinder infinitely long, is greater than 

 «ny finite magnitude, how great soever ; and all such solids supposed to be 

 formed on given bases, are in the same proportion to one another as those bases. 

 But if two of these three dimensions are wanting, as in the space intercepted 

 between two parallel planes infinitely extended, and at a finite distance ; or with 

 infinite length and breadth, with a finite thickness ; all such solids shall be as 

 the given finite distances to one another ; but these quantities, though infinitely 

 greater than the other, are yet infinitely less than any of those wherein all the 

 three dimensions are infinite. Such are the spaces intercepted between two 

 inclined planes infinitely extended ; the space intercepted by the surface of a 

 cone, or the sides of a pyramid, likewise infinitely continued, &c. of all which 



