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*** Book V. is now very rarely read. The subject matter,
so far as it is introductory to Book VI., is dealt with in a simpler manner at page 317, in the chapter called ' Elementary Principles of Proportion. The student is advised to proceed at once to that chapter, leaving Book V. in its stricter form to be studied at a later stage, if it is thought desirable,
Book V. treats of Ratio and Proportion, and the method adopted is such as to place these subjects on a basis independent of arith. metical principles.
The following notation will be employed throughout this section.
Capital letters, A, B, C, ... will be used to denote the magnitudes themselves, not any numerical or algebraical measures of them, and small letters, m, n, p, ... will be used to denote whole numbers. Also it will be assumed that multiplication, in the sense of repeated addition, can be applied to any magnitude, so that m. A or mA will denote the magnitude A taken m times.
The symbol > will be used for the words greater than, and < for less than.
Definition 1. One magnitude is said to be a multiple of another, when the first contains the second an exact number of times.
Definition 2. One magnitude is said to be a submultiple of another, when the first is contained an exact number of times in the second.
The following properties of multiples will be assumed as selfevident. (1) mA >, =, or <mB according as A >, =, or < B ; and
Definition 3. The Ratio of one magnitude to another of the same kind is the relation which the first bears to the second in respect of quantuplicity.
The ratio of A to B is denoted thus, A:B; and A is called the antecedent, B the consequent of the ratio.
The term quantuplicity denotes the capacity of the first magnitude to contain the second with or without remainder.
If the magnitudes are commensurable, their quantuplicity may be expressed numerically by observing what multiples of the two magnitudes are equal to one another.
Thus if A=ma, and B=na, it follows that nA =mB. In this case A=B, and the quantuplicity of A with respect to B is the arithmetical fraction
But if the magnitudes are incommensurable, no multiple of the first can be equal to any multiple of the second, and therefore the quantuplicity of one with respect to the other cannot exactly be expressed numerically: in this case it is determined by examining how the multiples of one magnitude are distributed among the multiples of the other.
Thus, let all the multiples of A be formed, the scale extending ad infinitum ; also let all the multiples of B be formed and placed in their proper order of magnitude among the multiples of A. This forms the relative scale of the two magnitudes, and the quantuplicity of A with respect to B is estimated by examining how the multiples of A are distributed among those of B in their relative scale.
In other words, the ratio of A to B is known, if for all integral values of m we know the multiples nB and (n+1) B between which mA lies.
In the case of two given magnitudes A and B, the relative scale of multiples is definite, and is different from that of A to C, if C differs from B by any magnitude however small.
For let D be the difference between B and C; then however small D may be, it will be possible to find a number m such that mD>A. In this case, mB and mC would differ by a magnitude greater than A, and therefore could not lie between the same two multiples of A ; so that after a certain point the relative scale of A and B would differ from that of A and Č.
Definition 4. Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other,
Definition 5. The ratio of one magnitude to another is equal to that of a third magnitude to a fourth, when if any equimultiples whatever of the antecedents of the ratios are taken, and also any equinultiples whatever of the consequents, the multiple of one antecedent is greater than, equal to, or less than that of its consequent, according as the multiple of the other antecedent is greater than, equal to, or less than that of its consequent.
Thus the ratio A to B is equal to that of C to D when mC>, =, or <n D according as mĂ>, =, or <nB, whatever whole numbers m and n may be.
Again, let m be any whole number whatever, and n another whole nunber deterinined in such a way that either mA is equal to nB, or m A lies between aB and (n+1)B; then the definition asserts that the ratio of A to B is equal to that of C to D if mC=nD when mA=nB; or if mC lies between nD and (n+1)D when mA lies between nB and (n+1) B.
In other words, the ratio of A to B is equal to that of C to D when the multiples of A are distributed ainong those of B in the same manner as the multiples of C are distributed among those of D.
When the ratio of A to B is equal to that of C to D the four magnitudes are called proportionals. This is expressed by saying ' A is to B as C is to D,” and the proportion is written
A:B::C: D, or A:B = C:D. A and D are called the extremes, B and C the means ; also D is said to be a fourth proportional to A, B, and C.
Definition 6. Two terms in a proportion are said to be homologous when they are both antecedents, or both consequents of the ratios.
Definition 7. The ratio of one magnitude to another is greater than that of a third magnitude to a fourth, when it is possible to find equimultiples of the antecedents and equimultiples of the consequents such that while the multiple of the antecedent of the first ratio is greater than, or equal to, that of its consequent, the multiple of the antecedent of the second is not greater, or is less, than that of its consequent.
This definition asserts that if whole numbers. m and n can be found such that while mA is greater than nB, mC is not greater than n D, or while m A =nB, mŐ is less than n D, then the ratio of A to B is greater than that of C to D.
If A is equal to B, the ratio of A to B is called a ratio of equality.
If A is greater than B, the ratio of A to B is called a ratio of greater inequality.
If A is less than B, the ratio of A to B is called a ratio of less inequality.
Definition 8. Two ratios are said to be reciprocal when the antecedent and consequent of one are the consequent and antecedent of the other respectively ; thus B : A is the reciprocal of A : B.
Definition 9. Three magnitudes of the same kind are said to be proportionals, when the ratio of the first to the second is equal to that of the second to the third. Thus A, B, C are proportionals if
A: B::B:C. B is called a mean proportional to A and C, and C is called a third proportional to A and B.
Definition 10. Three or more magnitudes are said to be in continued proportion when the ratio of the first to the second is equal to that of the second to the third, and the ratio of the second to the third is equal to that of the third to the fourth, and so on.
Definition 11. When there are any number of magnitudes of the same kind, the first is said to have to the last the ratio compounded of the ratios of the first to the second, of the second to the third, and so on up to the ratio of the last but one to the last magnitude.
For example, if A, B, C, D, E be magnitudes of the same kind, A : E is the ratio compounded of the ratios A : B, B:C, C:D, and D: E. This is sometimes expressed by the following notation :
on 12. If there are any number of ratios, and a set of magnitudes is taken such that the ratio of the first to the second is equal to the first ratio, and the ratio of the second to the third is equal to the second ratio, and so on, then the first of the set of magnitudes is said to have to the last the ratio compounded of the given ratios.
Thus, if A : B, C :D, E: F be given ratios, and if P, Q, R, S be magnitudes taken so that
P:Q :: A:B,
Definition 13. When three magnitudes are proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second. Thus if
A: B::B:C, then A is said to have to C the duplicate ratio of that which it has to B. Since
A : C=
it is clear that the ratio compounded of two equal ratios is the duplicate ratio of either of them.
Definition 14. When four magnitudes are in continued proportion, the first is said to have to the fourth the triplicate ratio of that which it has to the second.
It may be shewn as above that the ratio compounded of three equal ratios is the triplicate ratio of any one of them.
Obs. Of the propositions of Book V., which, it may be noticed are all theorems, we here give only the more important.
PROPOSITION 1. Ratios which are equal to the same ratio are equal to one another. Let A :B::P:Q, and also C:D::P:Q; then shall A:B::C:D.
For it is evident that two scales or arrangements of multiples which agree in every respect with a third scale, will agree with one another.
If two ratios are equal, the antecedent of the second is greater than, equal to, or less than its consequent according as the antecedent of the first is greater than, equal to, or less than its consequent. Let
A :B::C:D, then
or <D, according as
A>, =, or < B. This follows at once from Def. 5, by taking m and n each equal to unity.
C>, = ,