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THE

ELEMENTS OF GEOMETRY.

BOOK V.

DEFINITIONS.

1. A less magnitude is said to be an aliquot part or sub-,

multiple of a greater, when the less measures the greater.

Note. One magnitude is said to measure another when it is contained in it a certain number of times exactly; for example, 3 is an aliquot part of 15; for it measures, or is contained in it exactly 5 times; but 4 is not a submultiple of 15, for it is contained in it more than three times, and less than four times.

2. A greater magnitude is said to be a multiple of a less,

when the less measures it.

3. Ratio is the mutual relation of two magnitudes of the

same kind, with respect to quantity.

Note. It is necessary that the magnitudes should be of the same species, as two lines, two surfaces, or two numbers; for a ratio could not subsist between a line and a surface, or between a surface and a number.

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4. Magnitudes are said to have a ratio to one another,

when they are such that the less can be multiplied so as to exceed the greater.

Note. All commensurable * magnitudes have a ratio to one another, the magnitudes may be =, and then the ratio is called a ratio of equality ; or they may be unequal in various degrees of inequality, and then the ratio is called a ratio of greater or less inequality; a ratio of greater inequality, when the first magnitude is greater than the second, and a ratio of less inequality, when the first is less than the second.

The first magnitude is called the antecedent, and the second the consequent.

It is to be observed, that ratio of equality, and equality of ratio are not to be confounded; for they are by no means synonymous terms, since two or more ratios may be though the quantities that are compared are unequal ; thus the ratio of 4:12 is equal to the ratio of 8: 24 though the numbers are all unequal.

But magnitudes which are incommensurablet have no ratio to one another that can be determined, ex. gr. a finite line has no determinate ratio to an infinite, an angle of contact has no determinate ratio to a rectilineal angle, a side of a o’ has no determinate ratio to its diagonal, for the value of one is unity and of the other the ✓2 5. Magnitudes are said to be in the same ratio, the first

to the second as the third to the fourth, when as often as any submultiple whatever of the

first is contained in the second, so often is an equisubmultiple of the third con

tained in the fourth. 6. Magnitudes which have the same ratio to one another

are called proportionals. Note. Thus the numbers 3 and 27 have to one another the same ratio that two lines, one an inch, and the other nine inches long have, for 3 is the same submultiple of 27 that I

Commensurable magnitudes are such as bave some one magnitude that measures them both, this is called their common measure ; thus 19 and 16 have for their common measure 4, &c.

+ Incommensurable magnitudes are such as have no common measure.

inch is of 9; or if any submultiple of A is contained in B as often as an equisubmultiple of C is contained in D, then A has the same ratio to B that C has to D. This being the case, we may conclude that those quantities are proportionals, i. e. 3:27:: 1 inch : 9 inches, and A:B :: C: D. But if there be any submultiple of the first that measures the second, while an equisubmultiple of the third does not measure, or is not contained an equal number of times in the fourth, as if there be two ratios 8: 13 and 16: 27, although the submultiples 2, 4, 8 of the antecedent 16 are contained in its consequent 27 as often as the equisubmultiples 1, 2, 4, of the antecedent 8 are contained in its consequent; yet if we take the equisubmultiples 1 and of the antecedents 16 and 8, we will find that 1 is contained in 27 oftener than is contained in 13, and therefore that the ratios of 16:27, and of 8:13 are not =, but the first antecedent has a less ratio to its consequent than the second antecedent has to its consequent; we may then conclude that those four magnitudes 8:13 and 16:27 are not proportional.

Though incommensurable magnitudes have not a determined ratio to one another, yet they have a certain ratio, and two other magnitudes might have just the same ratio. The fifth definition, when applied to magnitudes even of this kind is adequate, for it says that the ratio of two magnitudes is the same with the ratio of two others, provided any equisubmultiples whatever of the antecedents are contained an equal number of times in their respective consequents; it is not necessary that those submultiples should measure the consequents, or be contained in them a certain number of times without a remainder, ex. gr. if the 10th part of A be contained in B 15 times.(though with a remainder, suppose x) then the 10th part of C is contained in D 15 times and not 10 times, and this is supposed the case not only for the 10th parts of A and C, but also if we take any parts of them ever so small; for if we take 1,000,000th part of A, it must be contained in B as often as the 1,000,000th part of C is contained in D, so that in this case the remainder must be less than any assigned. And this being so, those four magnitudes A:B::C:D are proportional.

7. When a submultiple of the first is contained in the se

cond oftener than an equisubmultiple of the third is contained in the fourth, then the first is said to have to the second a less ratio than the third has to the fourth; and on the other hand, the third is said to have to the fourth a greater ratio than the first has to the second,

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Note. For example, 4 has a less ratio to 12 than 10 has to 20, for any submultiple of 4, suppose 2, is contained in 12 oftener than an equisubmultiple of 10, i. e: 5 is contained in 20. If then there be any two magnitudes, A and B, and two others of the same kind, C and D, and if the half of A be contained in B six times, and the half of C be contained in D only four times, then A has a less ratio to B than C has to D; for A:3 A::C: 3C (or A:A::C: C); but 3 C is greater than D..C has a greater ratio to D than C has to 3 C, i.e. than A has to 3 A, which is a quantity not greater than B.'. &c.

8. Proportion is the similitude of ratios.

9. Proportion consists in three terms at least.

Note. When proportion exists only between three terms, they are said to be in continued proportion, the first consequent being the second antecedent. When this is the case the middle term is a mean proportional between the extremes.

10. When three magnitudes are proportional (A: B::B:

C) then the first is said to have to the third ( A: C) a duplicate ratio of that which it has to the second, (i. e. of A : B).

11. When there are four magnitudes in continued propor

tion (A: B::B: Cand B:C::C:D), then the first is said to have to the fourth (A :D) a triplicate ratio of

that which it has to the second (i. e. of the ratio of A:B). 12. If there be any number of magnitudes of the same kind (A, D, C, F) the first is said to have to the last (A:F) a ratio compounded of the ratios which the first has to the second, the second to the third, the third to the fourth (A: D and D: C and C: F) and so on to the last.

Note. By the duplicate ratio of A to B is meant the ratio of A to a third proportional to A and B, if C be the third proportional, the ratio of A to C is the one that is meant, thus the duplicate ratio of 1 to 3 is the ratio of 1 to 9, and the triplicate ratio of 1 to 3 is the ratio of 1 to 27.

There are other kinds of ratios sometimes mentioned by mathematicians, namely subduplicate, subtriplicate, &c. sesquialterate or sesquiplicate.

If several magnitudes be in continued proportion, the ratio of the first to the second is said to be subduplicate of the ratio of the first to the third, and so on.

If there be three magnitudes proportional (2 : 16 : 128) and four others also proportional (1:4:16: 64), and if the first be to the last in the first series (2 : 128) as the first to the last in the second (as I : 64), the ratio which the first has to the second in the first series (2 : 16) is said to be sesquialterate or sesquiplicate of the ratio which the first has to the second in the last series (1 : 4).

It is not necessary that the magnitudes mentioned in the twelfth def. should be proportionals, for if there be any numbers 2, 9, 5, 15, then 2 will be to 15 in a ratio compounded of the ratios of 2:9, 9 : 5 and 5:15; but a ratio may be compounded of several ratios, whose terms are not taken in continuation, as the ratio compounded of A: B and C:D would not be the ratio of A : D, but of A to another quantity, suppose X, which is to B as D: C, for if B: X ::C: D, then the ratio of A : X is compounded of the ratios of A : B and B :X;i. e. of the ratios of A: B and C: D.

13. In proportionals the antecedents are said to be homo

logous to the antecedents, and the consequents to the consequents.

Thus if A:B::C:D, then A is homologous to C, and B to D.

Geometers make use of the following terms, to express

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