10. Two spaces are said to be CONTIGUOUS, when there is no other space betwixt them. But two times, betwixt which there is no other time, are called immediate, as A B, B C.|A B C| And any two spaces, as well as times, are said to be CONTINUAL, when they have one common part, as A C, B D, where the part B C is common; |A B C D| and more spaces and times are continual, when every two which are next one another are continual.
Beginning, end, way, finite, infinite.
11. That part which is between two other parts, is called a MEAN; and that which is not between two other parts, an EXTREME. And of extremes, that which is first reckoned is the BEGINNING, and that which last, the END; and all the means together taken are the WAY. Also, extreme parts and limits are the same thing. And from hence it is manifest, that beginning and end depend upon the order in which we number them; and that to terminate or limit space and time, is the same thing with imagining their beginning and end; as also that every thing is FINITE or INFINITE, according as we imagine or not imagine it limited or terminated every way; and that the limits of any number are unities, and of these, that which is the first in our numbering is the beginning, and that which we number last, is the end. When we say number is infinite, we mean only that no number is expressed; for when we speak of the numbers two, three, a thousand, &c. they are always finite. But when no more is said but this, number is infinite, it is to be understood as if it were said, this name number is an indefinite name.
What is infinite in power. Nothing infinite can be truly said to be either whole or one; nor infinite spaces or times, many.
12. Space or time is said to finite in power, or terminable, when there may be assigned a number of finite spaces or times, as of paces or hours, than which there can be no greater number of the same measure in that space or time; and infinite in power is that space or time, in which a greater number of the said paces or hours may be assigned, than any number that can be given. But we must note, that, although in that space or time which is infinite in power, there may be numbered more paces or hours than any number that can be assigned, yet their number will always be finite; for every number is finite. And therefore his ratiocination was not good, that undertaking to prove the world to be finite, reasoned thus; If the world be infinite, then there may be taken in it some part which is distant from us an infinite number of paces: but no such part can be taken; wherefore the world is not infinite; because that consequence of the major proposition is false; for in an infinite space, whatsoever we take or design in our mind, the distance of the same from us is a finite space; for in the very designing of the place thereof, we put an end to that space, of which we ourselves are the beginning; and whatsoever any man with his mind cuts off both ways from infinite, he determines the same, that is, he makes it finite.
Of infinite space or time, it cannot be said that it is a whole or one: not a whole, because not compounded of parts; for seeing parts, how many soever they be, are severally finite, they will also, when they are all put together, make a whole finite: nor one, because nothing can be said to be one, except there be another to compare it with; but it cannot be conceived that there are two spaces, or two times, infinite. Lastly, when we make question whether the world be finite or infinite, we have nothing in our mind answering to the name world; for whatsoever we imagine, is therefore finite, though our computation reach the fixed stars, or the ninth or tenth, nay, the thousandth sphere. The meaning of the question is this only, whether God has actually made so great an addition of body to body, as we are able to make of space to space.
Division proceeds not to the least.
13. And, therefore, that which is commonly said, that space and time may be divided infinitely, is not to be so understood, as if there might be any infinite or eternal division; but rather to be taken in this sense, whatsoever is divided, is divided into such parts as may again be divided; or thus, the least divisible thing is not to be given; or, as geometricians have it, no quantity is so small, but a less may be taken; which may easily be demonstrated in this manner. Let any space or time, that which was thought to be the least divisible, be divided into two equal parts, A and B. I say either of them, as A, may be divided again. For suppose the part A to be contiguous to the part B of one side, and of the other side to some other space equal to B. This whole space, therefore, being greater than the space given, is divisible. Wherefore, if it be divided into two equal parts, the part in the middle, which is A, will be also divided into two equal parts; and therefore A was divisible.