ELEMENTS OF GEOMETRY. BOOK V. OF PROPORTION. THE preceding Books treat of magnitude as concrete, or having mere extension; and the simpler properties of lines, of angles, and of surfaces, were deduced, by a continuous process of reasoning, grounded on the principle of superposition. But this mode of investigation, how satisfactory soever to the mind, is by its nature very limited and laborious. By introducing the idea of Number into geometry, a new scene is opened, and a far wider prospect rises into view. Magnitude, being considered as discrete, or composed of integrant parts, becomes assimilated to multitude; and under this aspect, it presents a vast system of relations, which may be traced out with the utmost facility. Numbers were at first employed, to denote the aggregation of separate, though kindred, objects; but the subdivision of extent, whether actually effected or only conceived to exist, bestowing on each portion a sort of individuality, they came afterwards to acquire a more comprehensive application. In comparing together two quantities of the same kind, the one may contain the other, or be contained by it; that is, the one may result from the repeated addition of the other, or it may in its turn produce this other by a process of successive composition. The one quantity is, therefore, equal, either to so many times the other, or to a certain aliquot part of it. Such seems to be the simplest of the numerical relations. It is very confined, however, in its application, and is evidently, in this shape, insufficient altogether for the purpose of general comparison. But that object is attained, by adopting some intermediate term of reference. Though a quantity should neither contain another exactly, nor be contained by it; there may yet exist a third and smaller quantity, which is at once capable of measuring them both. This measure corresponds to the arithmetical unit; and as number denotes the collection of units, so quantity may be viewed as the aggregate of its component measures. But mathematical quantities are not all suscep may tible of such perfect mensuration. Two quantities be conceived to be so constituted, as not to admit of any other quantity that will measure them completely, or be contained in both of them without leaving a remainder. Yet this apparent imperfection, which proceeds entirely from the infinite variety ascribed to possible magnitude, creates no real obstacle to the progress of accurate science. The measure or primary element, being assumed in succession still smaller and smaller, its corresponding remainder must be perpetually diminished. This continued exhaustion will hence approach nearer than any assignable difference to its absolute term. Quantities in general can, therefore, either exactly or to any required degree of precision, be represented abstractly by numbers; and thus the science of Geometry is at last brought under the dominion of Arithmetic. It is obvious, that quantities of any kind must have the same composition, when each contains its measure the same number of times. But quantities, viewed in pairs, may be considered as having a similar composition, if the corresponding terms of each pair contain its measure equally. Two pairs of quantities of a similar composition, being thus formed by the same distinct aggrega tions of their elementary parts, constitute a Proportion. DEFINITIONS. 1. Quantities are homogeneous, which can be added together. 2. One quantity is said to contain another, when the subtraction of the smaller-continued if necessary-leaves no remainder. 3. A quantity which is contained in another, is said to measure it. 4. The quantity which is measured by another, is called its multiple; and that which measures the other, its submultiple. 5. Like multiples and submultiples are those which contain their measures equally, or which equally measure their corresponding compounds. 6. Quantities are commensurable, which have a finite common measure; they are incommensurable, if they will admit of no such measure. 7. That relation which one quantity is conceived to bear to another in regard to their composition, is named a ratio. 8. When both terms of comparison are equal, it is called a ratio of equality; if the first of these be greater than |