...said to have a ratio to one another, when the less can be multiplied so as to exceed the other. V. The first of four magnitudes is said to have the same ratio to the scoond, which the third has to the fourth, when any equimultiples whatsoever of the first and third...

...be equal to that of the second, the multiple of the third will be equal to that of the fourth : and if the multiple of the first be less than that of the second, the multiple of the third will be less than that J of the fourth. It is this proposition which is deduced as a necessary consequence...

...having the same ratio to two others В and D, of the same kind with the former two respectively. Def. 7. The first of four magnitudes is said to have the same ratio to the se* By the word " corresponding" here used, it U R«TÍy iat«*nded to point out the fact of there...

...ought to be defined, is still a subject of controversy among geometers. Euclid defines them thus : The first of four magnitudes is said to have the same...equimultiples whatsoever of the second and fourth being taken, the equi-multiples of the first and third either together exceed, are equal to, or are...

...said to have a ratio to one another, when the less can be multiplied so as to exceed the other. V. The first of four magnitudes is said to have the same...taken, and any equimultiples whatsoever of the second arid rt fourth, if the multiple of the first be less than that of li T.* • * the second, the multiple...

...less, that is, ' when the ' greater contains the less a certain number of times ex' actly.' III. ' Ratio is a mutual relation of two magnitudes of the...magnitudes is said to have the same ratio to the second, that the third has to the fourth, when any equimultiples whatsoever of the first and third being taken,...

...less, that is, ' when the ' greater contains the less a certain number of times ex' actly.' III. ' Ratio is a mutual relation of two magnitudes of the...magnitudes is said to have the same ratio to the second, that the third has to the fourth, when any equimultiples whatsoever of the first and third being taken,...

...equal to, or less than the corresponding magnitude of the other ; and conversely (ax. 1, 2, 3, 4).* • "The first of four magnitudes is said to have the same ratio tu the second, which the third has to the fourth, when any equimultiples whatsoever of the first and...

...oft as there are units in the product of these two numbers. IV. If the first of four magnitudes has the same ratio to the second which the third has to the fourth, and if any equimultiples whatever be tcriken of the first and third, and any whatever of the second...

...GEOMETRY. ELEMENTS or GEOMETRY. EUCLID, Books *i, *II, *III, IV. Book v. *Definition of Proportion. The first of four magnitudes is said to have the same...second which the third has to the fourth, when^ — any eq'wi-multiples whatsoever of the Jirst and third being taken, and any equi-multiples whatsoever of...