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Ex Æquali. If there are two sets of magnitudes, such that the first is to the second of the first set as the first to the second of the other set, and the second to the third of the first set as the second to the third of the other, and so on to the last magnitude : then the first is to the last of the first set as the first to the last of the other.
First, let there be three magnitudes A, B, C of one set, and three, P, Q, R, of another set,
and let A:B::P:Q,
and B:C::Q :R;
V. 8, Cor. and because B:C::Q:R,
.. mB : nC :: MQ : nR, .., invertendo,
nC : mB :: nR : MQ. Now, if
V. 5. .. mP : mQ>nR : mQ,
and :: mP>nR. Similarly mP= or <nR according as mA= or <nC. A:C :: P: R.
V. 9. V. 3.
Secondly, let there be any number of magnitudes, A, B, C, ... L, M, of one set, and the same number P, Q, R, ... Y, 2, of another set, such that
A :B :: P:Q,
L:M::Y : Z; then shall A:M:: P: 2.
For A:0::P :R,
and C:D :: R:S; :. by the first case A :D :: P:S, and so on, until finally A : M P: 2.
If A:B ::P:Q, and B:C:: R:P: then A : 0 :: R:Q.
If A :B:: X: Y,
and C:B:: 2:Y; then shall A +C:B:: X+2: Y.
For since C :B:: 2 :Y, .., invertendo,
B:C:: Y: 2.
Also A:B :: X : Y, .., ex æquali,
A :C:: X : 2, .., componendo, A+C:C:: X+Z: Z.
Again, C:B::Z:Y, .., ex æquali,
A+C: B:: X + Z: Y.
V. 14. v. 13.
If two ratios are equal, their duplicate ratios are equal.
Let A :B::C:D; then shall the duplicate ratio of A to B be equal to that of C to D.
Let X be a third proportional to A and B, and Y a third proportional to C and D,
so that A :B::B:X, and C :D::D:Y;
:: B:X ::D:Y; .., ex æquali,
A :X :: 0 :Y.
Def. 13. :. the duplicate ratio of A : B=that of C :D.
NOTE. The converse of this theorem may be readily proved ; namely,
If the duplicates of two ratios are equal, the ratios themselves are equal.
ELEMENTARY PRINCIPLES OF PROPORTION.
INTRODUCTION TO BOOK VI.
1. The first four books of Euclid deal with the absolute equality or inequality of geometrical magnitudes. In Book VI. such magnitudes are compared by considering their ratio or relative greatness.
2. The meaning of the words ratio and proportion in their simplest arithmetical sense may be given as follows:
(i) The ratio of one number to another is the multiple or fraction which the first is of the second.
Four numbers are in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth
3. These definitions are however not strictly applicable to the purposes of Pure Geometry, for the following
(i) Pure Geometry deals only with magnitudes as represented by diugrams, without measuring them in terms of a common unit: in other words, it makes no use of number for the purpose of comparing magnitudes.
(ii) It commonly happens that Geometrical magnitudes of the same kind are incommensurable, that is, they are such that it is impossible to express them exactly in terms of some common unit. Nevertheless it is always possible to express the arithmetical ratio of two such magnitudes within any required degree of accuracy. [See Note, p. 131: also Hall and Knight's Elementary Algebra, Art. 289.]
4. Accordingly, the object of Euclid's Fifth Book is to establish the Theory of Proportion on a basis independent of number. But as Book V. is now very rarely read, we propose here merely to illustrate algebraically such principles of proportion as are required before proceeding to Book VI. The strict treatment of the subject given in Book V. may be studied at a later stage, if it is thought desirable.
Obs. In what follows the symbol > will be used for the words greater than, and < for less than.
5. The following definitions are selected from Book V.
Definition 1. One magnitude is said to be a multiple of another, when the first contains the second an exact number of times.
Thus ma is a multiple of a, if m is any whole number.
Definition 2. One magnitude is said to be a submultiple of another, when the first is contained in the second an exact number of times. is a submultiple of if m is
Definition 3. The ratio of one magnitude to another of the same kind is the relation which the first bears to the second in regard to quantity; this is measured by the fraction which the first is of the second.
Thus if two such magnitudes contain a and b units respectively, the ratio of the first to the second is expressed by the fraction
The ratio of a to b is generally denoted thus, a : b; and a is called the antecedent and b the consequent of the ratio.
The two magnitudes compared in a ratio must be of the same kind; for example, both must be lines, or both angles, or both
It is clearly impossible to compare the length of a straight line with a magnitude of a different kind, such as the area of a triangle.
INTRODUCTION TO BOOK VI.
Definition 5. Four quantities are in proportion, when the ratio of the first to the second is equal to the ratio of the third to the fourth.
When the ratio of a to b is equal to that of x to y, the four magnitudes are called proportionals. This is expressed by saying “a is to b as x is to y,” and the proportion is written
a:b:: X: Y ;
or a :b = x : y. Here a and y are called the extremes, and b and x the means.
(i) Algebraical Test of Proportion. The ratios a : b and 2: y may be expressed algebraically by the fractions b
and ; thus the four magnitudes a, b, a, y are in proportion if
(ii) Geometrical Test of Proportion. 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 equimultiples 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 of a to b is equal to that of x to y, that
is to say,
a, b, x, y are in proportion, if
or < ny, according as ma >, or < nb, whatever whole numbers m and n may be.
Note. The Algebraical and Geometrical Tests of Proportion, though differing widely in method, really determine the property; for each may be deduced from the other. This is fully explained on the following page.