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COMPARISON BETWEEN THE ALGEBRAICAL AND GEOMETRICAL
TESTS OF PROPORTION.
(i) If a, b, x, y satisfy the Algebraical test of proportion, to shew that they also satisfy the geometrical test. By hypothesis
ma numbers, we obtain
ny thus these fractions are both improper, or both proper, or both equal to unity; hence mx >, =, or <ny, according as ma >, =, or < nb, which is the Geometrical test of proportion.
(ii) If a, b, x, y satisfy the Geometrical test of proportion, to shew that they also satisfy the Algebraical test.
By hypothesis mx>, =, or <ny, according as ma>, =, or <nb, it is required to prove that
y' Suppose >2.; then it will be possible to find some fraction
y which lies between them, n and m being positive integers.
.(2) y From (1),
ma>nb; from (2),
mx <ny ; and these contradict the hypothesis.
are not unequal; that is y
y Definition 6. Two terms in a proportion are said to be homologous, when they are both antecedents or both consequents of the ratios. Thus if
a:b:: 3 : y, a and x are homologous ; also b and y are homologous,
INTRODUCTION TO BOOK VI.
Definition 8. Two ratios are said to be reciprocal, when the antecedent and consequent of one are respectively the consequent and antecedent of the other.
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. Here b is called a mean proportional to a and c; and c is called a third proportional to a and b. When four magnitudes are in proportion, namely when
a :b::c:d, then d is called a fourth proportional to a, b, and c.
Definition 10. A series of magnitudes of the same kind are said to be in continued proportion, when the ratios of the first to the second, of the second to the third, of the third to the fourth, and so on, are all equal. Thus a, b, c, d, e are in continued proportion, if
a:b = - b:c = c:d
а ъ d that is, if
Definition 11. When there are any number of magnitudes of the same kind, the first 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.
Thus if a, b, c, d, e are magnitudes of the same kind, then a :e is the ratio compounded of the ratios
a:b, b:c, cid, d:e. NOTE. Algebra defines the ratio compounded of given ratios
that formed by multiplying together the fractions which represent the given ratios. In the above illustration it will
b d be seen that on multiplying together the ratios
to ê þ obtain the ratio
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 :c is said to be the duplicate of the ratio a : b.
NOTE. In Algebra the duplicate of the ratio a :b is defined as the ratio of a' to 6%.
It is easy to show that the two definitions are identical.
a b Now
b'c bb that is,
a:c:: a : 12.
6. The following theorems from Book V. are here proved algebraically. Reference is made to them in Book VI. under certain technical names.
THEOREM 1. By Equal Ratios. Ratios which are equal to the same ratio are equal to one another.
That is, if = x : y, and c:d=X : Y; then shall
THEOREM 3. Invertendo, or Inversely. If four magnitudes are proportionals, they are also proportionals taken inversely. That is, if
X : Y, then shall
b:a=y:. Since, by hypothesis, it follows that ;
THEOREM 11. Alternando, or Alternately. If four magnitudes of the same kind are proportionals, they are also proportionals when taken alternately. That is, if
a:b=X : Y, then shall
a : x=b:y.
a For, by hypothesis,
у Multiplying both sides by
NOTE. In this theorem the hypothesis requires that a and b shall be of the same kind, also that w and y shall be of the same kind ; while the conclusion requires that a and a shall be of the same kind, and also b and y of the same kind.
THEOREM 12. Addendo. In a series of equal ratios (the magnitudes being all of the same kind), as any antecedent is to its consequent so is the sum of the antecedents to the sum of the consequents.
That is, if a : x=b:y=C:2= ; then shall a : x=a +b+c+... :X +y++....
a b c Let each of the equal ratios
be equal to k.
x y z
Then a=kx, b=ky, C=kz, .. ; .., by addition,
a+b+c+... =k(x+y++ ...);
2 + y +2+...
a + 6
6 +1 = + 1, or
THEOREM 13. Componendo. If four magnitudes are proportionals, the sum of the first and second is to the second as the sum of the third and fourth is to the fourth. That is, if
a: b=X:Y; then shall
a+b:b=+y: y. For, by hypothesis,
y that is,
a+b:b=x+y: y. Dividendo. Similarly it may be shewn that a-b: b=x – Y:y.
THEOREM 14. Ex Æquali. If there are three magnitudes a, b, c of one set, and three magnitudes x, y, z of another set ; and if these are so related that
a:b=X : Y, and
b:cry: %) then shall
b For, by hypothesis, and
X Y .., by multiplication,
THEOREM 15. If two proportions have the same consequents, that is, if
y then shall
=x+2: Y. For, by hypothesis, and
to y ū
atc X + 2 .., by addition,