DEFINITION VII When of the equimultiples of four magnitudes (taken as in the fifth definition), the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth; then the first is said to have to the second a greater ratio than the third magnitude has to the fourth and, on the contrary, the third is said to have to the fourth a less ratio than the first has to the second. If, among the equimultiples of four magnitudes, compared as in the fifth definition, we should find 0000, but — or ▲▲▲▲, or if we should find any particular multiple M' of the first and third, and a particular multiple m' of the second and fourth, such, that M' times the first is m' times the second, but M' times the third is not m' times the fourth, i.e.,orm' times the fourth; then, the first is said to have to the second a greater ratio than the third has to the fourth; or the third has to the fourth, under such circumstances, a less ratio than the first has to the second: although several other equimultiples may tend to show that the four magnitudes are proportionals. This definition will in future be expressed thus : : In the above general expression, M' and m' are to be considered particular multiples, not like the multiples M and m introduced in the fifth definition, which are in that definition considered to be every pair of multiples that can be taken. It must also be here observed, that⇓⇓,, and the like symbols, are to be considered merely the representatives of geometrical magnitudes. 1 2 3 In a partial arithmetical way, this may be set forth as fol lows: Let us take the four numbers, 8, 7, 10, and 9. that is, twice the first is greater than twice the second, and twice the third is greater than twice the fourth; and 16 21 and 2027; that is, twice the first is less than three times the second, and twice the third is less than three times the fourth; and among the same multiples we can find 72 56 and 90 72: that is, 9 times the first is greater than 8 times the second, and 9 times the third is greater than 8 times the fourth. Many other equimultiples might be selected, which would tend to show that the numbers 8, 7, 10, 9, were proportionals, but they are not, for we can find a multiple of the first a multiple of the second, but the same multiple of the third that has been taken of the first the same multiple of the fourth which has been taken of the second; for instance, 9 times the first is 10 times the second, but 9 times the third is not 10 times the fourth, that is, 7270, but 90 not 90, or 10 times the first we find 11 times the second, but 10 times the third not greater than 11 times the fourth. When any such multiples as these can be found, the first (8) is said to have to the second (7) a greater ratio than the third (10) has to the fourth (9). not E PROP. VIII. THEO. Of unequal magnitudes the greater has a greater ratio to the same than the less has and the same magnitude has a greater ratio to the less than it has to the greater. Δ Let and be two unequal magnitudes, and any other. We shall first prove that unequal magnitudes, has a greater ratio to than than the which is the greater of the two as m' is the first multiple which first becomes is not M' than (m1) or m' M' m' +0 :. m2 O−O + O must be M' ... + M' ▲ ; Δ M' ; m'; but it has been shown above that m', therefore, by the seventh definition, M'A and M' will be each, and m' the least multiple of, which first becomes greater than M' m' ○ − 0 + ○ is M' + M'A; The contrivance employed in this proposition for finding, among the multiples taken, as in the fifth definition, a multiple of the first greater than the multiple of the second, but the same multiple of the third which has been taken of the first, not greater than the same multiple of the fourth which has been taken of the second, may be illustrated numerically as follows: The number 9 has a greater ratio to 7 than 8 has to 7; that is, 978: 7; or, 8+1:78:7. The multiple of 1, which first becomes greater than 7, is 8 times, therefore we may multiply the first and third by 8, 9, 10, or any other greater number; in this case, let us multiply the first and third by 8, and we have 64 + 8 and 64: again, the first multiple of 7 which becomes greater than 64 is 10 times; then, by multiplying the second and fourth by 10, we shall have 70 and 70; then, arranging these multiples, we have― Consequently 64 + 8, or 72, is greater than 70, but 64 is not greater than 70, .. by the seventh definition, 9 has a greater ratio to 7 than 8 has to 7. The above is merely illustrative of the foregoing demonstration, for this property could be shown of these or other numbers very readily in the following manner; because, if an antecedent contains its consequent a greater number of times than another antecedent contains its consequent, or when a fraction is formed of an antecedent for the numerator, and its consequent for the denominator be greater than another fraction which is formed of another antecedent for the numerator and its consequent for the denominator, the ratio of the first antecedent to its consequent is greater than the ratio of the last antecedent to its consequent. Thus, the number 9 has a greater ratio to 7, than 8 has to 7, for is greater than 2. 7 8 7 Again, 17: 19 is a greater ratio than 13: 15, because |