ELEMENTS

OP

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 originally on superposition. But this mode of investigation, however satisfactory to the mind, is, from 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 that aspect, it

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presents a vast system of relations, which may be traced out with the utmost facility.

Numbers were first employed, to denote the collection of distinct, though kindred, objects; but the subdivision of extent, whether actually effected or only conceived, bestowing 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 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 numerical relations. It is very confined, however, in its application, and is evidently, in that shape, insufficient altogether for the purpose of general comparison. But this object is attained, by adopting some intermediate reference. Though a quantity 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 susceptible of such perfect mensuration. Two quantities may be conceived to be so constituted, as not to admit another which will measure them completely, or be contained in both 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 still smaller and smaller, its corresponding remainder must be perpetually diminished. This continued exhaustion will hence approach its absolute term, nearer than any assignable difference.

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 aggregations 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 this,-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 the second, it is a ratio of majority; and if the first be less than the second, it is a ratio of minority.

9. The identity of ratios constitutes a proportion or analogy.

10. Four quantities are said to be proportional, when a submultiple of the first is contained in the second as often as a like submultiple of the third is contained in the fourth.

11. Of proportional quantities, the first of each pair is named the antecedent, and the second the consequent.

12. The antecedents are homologous terms; and so are the consequents.

13. One antecedent is said to be to its consequent, as another antecedent to its consequent.

14. The first and last terms of a proportion are called the extremes, and the intermediate ones, the means.

15. A ratio is direct, if it follows the order of the terms compared; it is inverse or reciprocal, when it holds a reversed order.

Thus, if the ratio of A to B be direct, that of B to A is the inverse or reciprocal ratio.

16. Quantities form a continued proportion, when the intervening terms stand in the double relation of consequents and antecedents.