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(21) 1/3 is to the quotient. But the divisor 1/2 is but half of 1, & therefore the dividend must be but half the quotient: consequently the quotient must be double the dividend, that is double of 1/3; but this doubled is 2/3, which is the quotient according to the rule. The same reasoning will hold good in all other cases.

80. A ratio, or fraction, may be encreased two ways; 10. either by encreasing the antecedant or numerator; or 20. by diminishing the consequent or denominator. Thus if the ratio 4 to 7, or the fraction 4/7 were proposed it is evident that by adding any number, 1 for instance to the numerator 4, the fraction or the ratio of which it is the exponent will be encreased. For 4+1/7 = 5/7 > 4/7. But the same may be done by diminishing the denominator. For 4/7-1 = 4/6 > 4/7.