| And even this correction is susceptible of a new correction, and of different degrees of exactness, according to the nature of the instrument, by which we measure the bodies, and the care which we employ in the comparison. |
| When therefore the mind is accustomed to these judgments and their corrections, and finds that the same proportion which makes two figures have in the eye that appearance, which we call equality, makes them also correspond to each other, and to any common measure, with which they are compared, we form a mixed notion of equality derived both from the looser and stricter methods of comparison. |
| But we are not content with this. |
| For as sound reason convinces us that there are bodies vastly more minute than those, which appear to the senses; and as a false reason would perswade us, that there are bodies infinitely more minute; we clearly perceive, that we are not possessed of any instrument or art of measuring, which can secure us from ill error and uncertainty. |
| We are sensible, that the addition or removal of one of these minute parts, is not discernible either in the appearance or measuring; and as we imagine, that two figures, which were equal before, cannot be equal after this removal or addition, we therefore suppose some imaginary standard of equality, by which the appearances and measuring are exactly corrected, and the figures reduced entirely to that proportion. |
| This standard is plainly imaginary. |
| For as the very idea of equality is that of such a particular appearance corrected by juxtaposition or a common measure. |
| the notion of any correction beyond what we have instruments and art to make, is a mere fiction of the mind, and useless as well as incomprehensible. |
| But though this standard be only imaginary, the fiction however is very natural; nor is anything more usual, than for the mind to proceed after this manner with any action, even after the reason has ceased, which first determined it to begin. |
| This appears very conspicuously with regard to time; where though it is evident we have no exact method of determining the proportions of parts, not even so exact as in extension, yet the various corrections of our measures, and their different degrees of exactness, have given as an obscure and implicit notion of a perfect and entire equality. |
| The case is the same in many other subjects. |
| A musician finding his ear becoming every day more delicate, and correcting himself by reflection and attention, proceeds with the same act of the mind, even when the subject fails him, and entertains a notion of a compleat TIERCE or OCTAVE, without being able to tell whence he derives his standard. |
| A painter forms the same fiction with regard to colours. |
| A mechanic with regard to motion. |
| To the one light and shade; to the other swift and slow are imagined to be capable of an exact comparison and equality beyond the judgments of the senses. |
| We may apply the same reasoning to CURVE and RIGHT lines. |
| Nothing is more apparent to the senses, than the distinction betwixt a curve and a right line; nor are there any ideas we more easily form than the ideas of these objects. |
| But however easily we may form these ideas, it is impossible to produce any definition of them, which will fix the precise boundaries betwixt them. |
| When we draw lines upon paper, or any continued surface, there is a certain order, by which the lines run along from one point to another, that they may produce the entire impression of a curve or right line; but this order is perfectly unknown, and nothing is observed but the united appearance. |
| Thus even upon the system of indivisible points, we can only form a distant notion of some unknown standard to these objects. |
| Upon that of infinite divisibility we cannot go even this length; but are reduced meerly to the general appearance, as the rule by which we determine lines to be either curve or right ones. |