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tions, 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 eatent, whether actually effected or omly 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 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 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

tible of such perfect mensuration. Two quantities may be conceived to be so constituted, as not to admit of any other quantity that 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 successively 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 aggregations of their elementary parts, constitute a Proportion.

DEFINITIONS.

1. Quantities are homogeneous, which can be added tâgether.

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.

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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. w

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. A proportion or analogy consists in the identity of ratios.

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 eatremes, 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. s

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

17. When a proportion consists of three terms, the middle one is said to be a mean proportional between the tWO extremes.

18. The ratio which one quantity has to another may be considered as compounded of all the connecting ratios among any interposed quantities.

Thus, the ratio of A to D is viewed as compounded of that of A to B, that of B to C, and that of C to D.

19. Of quantities in a continued proportion, the first is said to have to the third, the duplicate ratio of what it has to the second; to have to the fourth, a triplicate ratio; to the fifth, a quadruplicate ratio; and so forth, according to the number of ratios introduced between the extreme terms.

20. If quantities be continually proportional, the ratio of the first to the second is called the subduplicate of the ratio of the first to the third, the subtriplicate of the ratio of the first to the fourth, &c.

To facilitate the language of demonstration relative to numbers or abstract quantities, it is expedient to adopt a clear and concise mode of notation.

1. The sign = expresses equality, to majority, and 2. minority; Thus A= B denotes that A is equal to B,

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