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I. The first of these objections which I shall take notice of, is more proper to prove this connexion and dependence of the one part upon the other than to destroy either of them. It has often been maintained in the schools, that extension must be divisible, in infinitum, because the system of mathematical points is absurd; and that system is absurd, because a mathematical point is a nonentity, and consequently can never, by its conjunction with others, form a real existence. This would be perfectly decisive, were there no medium betwixt the infinite divisibility of matter, and the nonentity of mathematical points. But there is evidently a medium, viz. the bestowing a colour or solidity on these points; and the absurdity of both the extremes is a demonstration of the truth and reality of this medium. The system of physical points, which is another medium, is too absurd to need a refutation. A real extension, such as a physical point is supposed to be, can never exist without parts different from each other; and wherever objects are different, they are distinguishable and separable by the imagination.
II. The second objection is derived from the necessity there would be of penetration, if extension consisted of mathematical points. A simple and indivisible atom that touches another must necessarily penetrate it; for 'tis impossible it can touch it by its external parts, from the very supposition of its perfect simplicity, which excludes all parts. It must therefore touch it intimately, and in its whole essence, secundum se, tota, et totaliter; which is the very definition of penetration. But penetration is impossible: mathematical points are of consequence equally impossible.
I answer this objection by substituting a juster idea of penetration. Suppose two bodies, containing no[Pg 64] void within their circumference, to approach each other, and to unite in such a manner that the body, which results from their union, is no more extended than either of them; 'tis this we must mean when we talk of penetration. But 'tis evident this penetration is nothing but the annihilation of one of these bodies, and the preservation of the other, without our being able to distinguish particularly which is preserved and which annihilated. Before the approach we have the idea of two bodies; after it we have the idea only of one. 'Tis impossible for the mind to preserve any notion of difference betwixt two bodies of the same nature existing in the same place at the same time.
Taking then penetration in this sense, for the annihilation of one body upon its approach to another, I ask any one if he sees a necessity that a coloured or tangible point should be annihilated upon the approach of another coloured or tangible point? On the contrary, does he not evidently perceive, that, from the union of these points, there results an object which is compounded and divisible, and may be distinguished into two parts, of which each preserves its existence, distinct and separate, notwithstanding its contiguity to the other? Let him aid his fancy by conceiving these points to be of different colours, the better to prevent their coalition and confusion. A blue and a red point may surely lie contiguous without any penetration or annihilation. For if they cannot, what possibly can become of them? Whether shall the red or the blue be annihilated? Or if these colours unite into one, what new colour will they produce by their union?
What chiefly gives rise to these objections, and at the same time renders it so difficult to give a satisfactory answer to them, is the natural infirmity and unsteadiness[Pg 65] both of our imagination and senses when employed on such minute objects. Put a spot of ink upon paper, and retire to such a distance that the spot becomes altogether invisible, you will find, that, upon your return and nearer approach, the spot first becomes visible by short intervals, and afterwards becomes always visible; and afterwards acquires only a new force in its colouring, without augmenting its bulk; and afterwards, when it has increased to such a degree as to be really extended, 'tis still difficult for the imagination to break it into its component parts, because of the uneasiness it finds in the conception of such a minute object as a single point. This infirmity affects most of our reasonings on the present subject, and makes it almost impossible to answer in an intelligible manner, and in proper expressions, many questions which may arise concerning it.
III. There have been many objections drawn from the mathematics against the indivisibility of the parts of extension, though at first sight that science seems rather favourable to the present doctrine; and if it be contrary in its demonstrations,'tis perfectly conformable in its definitions. My present business then must be, to defend the definitions and refute the demonstrations.
A surface is defined to be length and breadth without depth; a line to be length without breadth or depth; a point to be what has neither length, breadth, nor depth. 'Tis evident that all this is perfectly unintelligible upon any other supposition than that of the composition of extension by indivisible points or atoms. How else could any thing exist without length, without breadth, or without depth?
Two different answers, I find, have been made to[Pg 66] this argument, neither of which is, in my opinion, satisfactory. The first is, that the objects of geometry, those surfaces, lines, and points, whose proportions and positions it examines, are mere ideas in the mind; and not only never did, but never can exist in nature. They never did exist; for no one will pretend to draw a line or make a surface entirely conformable to the definition: they never can exist; for we may produce demonstrations from these very ideas to prove that they are impossible.
But can any thing be imagined more absurd and contradictory than this reasoning? Whatever can be conceived by a clear and distinct idea, necessarily implies the possibility of existence; and he who pretends to prove the impossibility of its existence by any argument derived from the clear idea, in reality asserts that we have no clear idea of it, because we have a clear idea. 'Tis in vain to search for a contradiction in any thing that is distinctly conceived by the mind. Did it imply any contradiction, 'tis impossible it could ever be conceived.
There is therefore no medium betwixt allowing at least the possibility of indivisible points, and denying their idea; and 'tis on this latter principle that the second answer to the foregoing argument is founded. It has been pretended,[5] that though it be impossible to conceive a length without any breadth, yet by an abstraction without a separation we can consider the one without regarding the other; in the same manner as we may think of the length of the way betwixt two towns and overlook its breadth. The length is inseparable from the breadth both in nature and in our[Pg 67] minds; but this excludes not a partial consideration, and a distinction of reason, after the manner above explained.
In refuting this answer I shall not insist on the argument, which I have already sufficiently explained, that if it be impossible for the mind to arrive at a minimum in its ideas, its capacity must be infinite in order to comprehend the infinite number of parts, of which its idea of any extension would be composed. I shall here endeavour to find some new absurdities in this reasoning.
