| 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, it is 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. |
| It is 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 coued any thing exist without length, without breadth, or without depth? |
| Two different answers, I find, have been made to 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; I 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 anything 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. |
| It is in vain to search for a contradiction in any thing that is distinctly conceived by the mind. |
| Did it imply any contradiction, it is impossible it coued ever be conceived. |
| There is therefore no medium betwixt allowing at least the possibility of indivisible points, and denying their idea; and it is on this latter principle, that the second answer to the foregoing argument is founded. |
| It has been pretended [L'Art de penser.], 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 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. |
| A surface terminates a solid; a line terminates a surface; a point terminates a line; but I assert, that if the ideas of a point, line or surface were not indivisible, it is impossible we should ever conceive these terminations: For let these ideas be supposed infinitely divisible; and then let the fancy endeavour to fix itself on the idea of the last surface, line or point; it immediately finds this idea to break into parts; and upon its seizing the last of these parts, it loses its hold by a new division, and so on in infinitum, without any possibility of its arriving at a concluding idea. |
| The number of fractions bring it no nearer the last division, than the first idea it formed. |
| Every particle eludes the grasp by a new fraction; like quicksilver, when we endeavour to seize it. |