| It's first principles are still drawn from the general appearance of the objects; and that appearance can never afford us any security, when we examine, the prodigious minuteness of which nature is susceptible. |
| Our ideas seem to give a perfect assurance, that no two right lines can have a common segment; but if we consider these ideas, we shall find, that they always suppose a sensible inclination of the two lines, and that where the angle they form is extremely small, we have no standard of a I @ right line so precise as to assure us of the truth of this proposition. |
| It is the same case with most of the primary decisions of the mathematics. |
| There remain, therefore, algebra and arithmetic as the only sciences, in which we can carry on a chain of reasoning to any degree of intricacy, and yet preserve a perfect exactness and certainty. |
| We are possest of a precise standard, by which we can judge of the equality and proportion of numbers; and according as they correspond or not to that standard, we determine their relations, without any possibility of error. |
| When two numbers are so combined, as that the one has always an unite answering to every unite of the other, we pronounce them equal; and it is for want of such a standard of equality in extension, that geometry can scarce be esteemed a perfect and infallible science. |
| But here it may not be amiss to obviate a difficulty, which may arise from my asserting, that though geometry falls short of that perfect precision and certainty, which are peculiar to arithmetic and algebra, yet it excels the imperfect judgments of our senses and imagination. |
| The reason why I impute any defect to geometry, is, because its original and fundamental principles are derived merely from appearances; and it may perhaps be imagined, that this defect must always attend it, and keep it from ever reaching a greater exactness in the comparison of objects or ideas, than what our eye or imagination alone is able to attain. |
| I own that this defect so far attends it, as to keep it from ever aspiring to a full certainty: But since these fundamental principles depend on the easiest and least deceitful appearances, they bestow on their consequences a degree of exactness, of which these consequences are singly incapable. |
| It is impossible for the eye to determine the angles of a chiliagon to be equal to 1996 right angles, or make any conjecture, that approaches this proportion; but when it determines, that right lines cannot concur; that we cannot draw more than one right line between two given points; it's mistakes can never be of any consequence. |
| And this is the nature and use of geometry, to run us up to such appearances, as, by reason of their simplicity, cannot lead us into any considerable error. |
| I shall here take occasion to propose a second observation concerning our demonstrative reasonings, which is suggested by the same subject of the mathematics. |
| It is usual with mathematicians, to pretend, that those ideas, which are their objects, are of so refined and spiritual a nature, that they fall not under the conception of the fancy, but must be comprehended by a pure and intellectual view, of which the superior faculties of the soul are alone capable. |
| The same notion runs through most parts of philosophy, and is principally made use of to explain oar abstract ideas, and to shew how we can form an idea of a triangle, for instance, which shall neither be an isoceles nor scalenum, nor be confined to any particular length and proportion of sides. |
| It is easy to see, why philosophers are so fond of this notion of some spiritual and refined perceptions; since by that means they cover many of their absurdities, and may refuse to submit to the decisions of clear ideas, by appealing to such as are obscure and uncertain. |
| But to destroy this artifice, we need but reflect on that principle so oft insisted on, that all our ideas are copyed from our impressions. |
| For from thence we may immediately conclude, that since all impressions are clear and precise, the ideas, which are copyed from them, must be of the same nature, and can never, but from our fault, contain any thing so dark and intricate. |
| An idea is by its very nature weaker and fainter than an impression; but being in every other respect the same, cannot imply any very great mystery. |
| If its weakness render it obscure, it is our business to remedy that defect, as much as possible, by keeping the idea steady and precise; and till we have done so, it is in vain to pretend to reasoning and philosophy. |
| SECT. II. OF PROBABILITY, AND OF THE IDEA OF CAUSE AND EFFECT. |
| This is all I think necessary to observe concerning those four relations, which are the foundation of science; but as to the other three, which depend not upon the idea, and may be absent or present even while that remains the same, it will be proper to explain them more particularly. |