INFINITE quantities. The very idea of magnitudes infinitely great, or such as ex ceed any assignable quantities, does in clude a negation of limits: yet, if we nearly examine this notion, we shall find that such magnitudes are not equal among themselves, but that there are really, be sides infinite length and infinite area, three several sorts of infinite solidity ; all of which are quantitates eel generis, and that those of each species are in given proportions.
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 infinity ; or, as may be said of infinity, a parte ante and a parte post, which is analogous to eternity in time and duration, in which there is always as much to follow as is past, from any point or mo ment of time : nor doth the addition or subduction of finite length, or space of time, alter the case either in infinity or eternity, since both the one or the other cannot be any part of the whole. As to infinite surface, or area, any right line, in. finitely extended both ways on an infinite plane, does divide 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 infinite 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 three hundred and sixty degrees of a circle : for example, right lines meeting at a right angle do include, on an infinite plane, a quarter part of the whole infinite area of such a plane.
But if two parallel infinite lines be sup posed drawn on such an infinite plane, the area intercepted between them will be likewise infinite ; but at the same time will be infinitely less than that space, which is intercepted between two infinite lines that are inclined, though with never so small an angle ; for that, in the one case, the given finite distance of the pa rallel lines diminishes the infinity in one degree of dimension ; whereas in a sec tor, there is infinity in both dimensions : and consequently the quantities are the one 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 parallelopiped, or a cylin der, infinitely long, is greater than any finite magnitude, how great soever ; and all such solids, supposed to be formed on given bases, are as those bases in propor tion to one another. 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 one to another ; but these quantities, though infinitely greater than the other, are yet infinitely less than any of those wherein all the three dimensions are in finite. Such are the spaces intercepted between two inclined planes infinitely ex tended ; the space intercepted by the sur face of a cone, or the sides of a pyramid, likewise infinitely continued, &c. of all which, notwithstanding the proportions one to another, and to the To 24Y, or vast abyss of infinite space (wherein is the lo cus of all things that are or can be ; or to the solid of infinite length, breadth and thickness, taken all manner of ways) are easily assignable ; for the space between two planes is to the whole as the angle of those planes to the three hundred and sixty degrees of the circle. As for cones and pyramids, they are as the spherical surface intercepted by them is to the sur fhce of the sphere, and therefore cones are as the versed sines of half their angles to the diameter of the circle : these three sorts of infinite quantity are analogous to a line, surface, and solid ; and, after the same manner, cannot be compared, or have no proportion the one to the other.