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The phrases in their context!


I know there is no mathematician, who will not refuse to be judged by the diagrams he describes upon paper, these being loose draughts, as he will tell us, and serving only to convey with greater facility certain ideas, which are the true foundation of all our reasoning.
This I am satisfyed with, and am willing to rest the controversy merely upon these ideas.
I desire therefore our mathematician to form, as accurately as possible, the ideas of a circle and a right line; and I then ask, if upon the conception of their contact he can conceive them as touching in a mathematical point, or if he must necessarily imagine them to concur for some space.
Whichever side he chuses, he runs himself into equal difficulties.
If he affirms, that in tracing these figures in his imagination, he can imagine them to touch only in a point, he allows the possibility of that idea, and consequently of the thing.
If he says, that in his conception of the contact of those lines he must make them concur, he thereby acknowledges the fallacy of geometrical demonstrations, when carryed beyond a certain degree of minuteness; since it is certain he has such demonstrations against the concurrence of a circle and a right line; that is, in other words, be can prove an idea, viz.
that of concurrence, to be INCOMPATIBLE with two other ideas, those of a circle and right line; though at the same time he acknowledges these ideas to be inseparable.
If the second part of my system be true, that the idea of space or extension is nothing but the idea of visible or tangible points distributed in a certain order; it follows, that we can form no idea of a vacuum, or space, where there is nothing visible or tangible.
This gives rise to three objections, which I shall examine together, because the answer I shall give to one is a consequence of that which I shall make use of for the others.
First, It may be said, that men have disputed for many ages concerning a vacuum and a plenum, without being able to bring the affair to a final decision; and philosophers, even at this day, think themselves at liberty to take part on either side, as their fancy leads them.
But whatever foundation there may be for a controversy concerning the things themselves, it may be pretended, that the very dispute is decisive concerning the idea, and that it is impossible men coued so long reason about a vacuum, and either refute or defend it, without having a notion of what they refuted or defended.
Secondly, If this argument should be contested, the reality or at least the possibility of the idea of a vacuum may be proved by the following reasoning.
Every idea is possible, which is a necessary and infallible consequence of such as are possible.
Now though we allow the world to be at present a plenum, we may easily conceive it to be deprived of motion; and this idea will certainly be allowed possible.
It must also be allowed possible, to conceive the annihilation of any part of matter by the omnipotence of the deity, while the other parts remain at rest.
For as every idea, that is distinguishable, is separable by the imagination; and as every idea, that is separable by the imagination, may be conceived to be separately existent; it is evident, that the existence of one particle of matter, no more implies the existence of another, than a square figure in one body implies a square figure in every one.
This being granted, I now demand what results from the concurrence of these two possible ideas of rest and annihilation, and what must we conceive to follow upon the annihilation of all the air and subtile matter in the chamber, supposing the walls to remain the same, without any motion or alteration? There are some metaphysicians, who answer, that since matter and extension are the same, the annihilation of one necessarily implies that of the other; and there being now no distance betwixt the walls of the chamber, they touch each other; in the same manner as my hand touches the paper, which is immediately before me.
But though this answer be very common, I defy these metaphysicians to conceive the matter according to their hypothesis, or imagine the floor and roof, with all the opposite sides of the chamber, to touch each other, while they continue in rest, and preserve the same position.
For how can the two walls, that run from south to north, touch each other, while they touch the opposite ends of two walls, that run from east to west? And how can the floor and.
roof ever meet, while they are separated by the four walls, that lie in a contrary position? If you change their position, you suppose a motion.