...fourth D. If, therefore, the first, &c. QED PROPOSITION IV. THEOREM. If the first of four magnitudes hat the same ratio to the second which the third has to the fourth ; then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples...

...Sivripov Kai rfrápruv, кaQ' ùiroiovovv iro\\air\aaia.afièv, rbv àvrbv ejfi Xóyo»< If the first has the same ratio to the second which the third has to the fourth; any equimultiples whatever of the first and third shall have the same ratio to any equimultiples of...

...representing the ratios must be equal. Euclid's test is given in Book v. Def. 5, where it stands thus: " The first of four magnitudes is said to have the same...equimultiples whatsoever of the first and third being takeu and any equimultiples whatsoever of the second and fourth : " If the multiple of the first be...

...representing the ratios must be equal. Euclid-s test is given in Book v. Def. 5, where it stands thus: " The first of four magnitudes is said to have the same...whatsoever of the first and third being taken and auy equimultiples whatsoever of the second and fourth : " If the multiple of the first be less than...

...only one hitherto devised that applies equally to commensurable and incommensurable quantities ; " the first of four magnitudes is said to have the same ratio to the second, that the third has to the fourth, when any equi-multiples whatsoever of the first and third being taken,...

...be proportionals, and any equimultiples be taken of the first and third, and also any equimultiples of the second and fourth, if the multiple of the first be greater than that of the second, the multiple of the third must be greater than that of the fourth...

...third is also greater than that of the fourth. The Algebraical Definition answering to this would be ' The first of four magnitudes is said to have the same...the second which the third has to the fourth, when the first is the same multiple, part, or fraction of the second which the third is of the fourth '...

...third is also greater than that of the fourth. The Algebraical Definition answering to this would be ' The first of four magnitudes is said to have the same...the second which the third has to the fourth, when the first is the same multiple, part, or fraction of the second which the third is of the fourth '...

...be compared with one which differs from the latter by less than any assignable quantity. DEFINITION. The first of four magnitudes is said to have the same...the second which the third has to the fourth when, if the first be divided into any number whatever of equal parts and the third be divided into the same...

...ratio to the second, than the fifth has to the sixth. PROPOSITION 14.— Theorem. If the first has the same ratio to the second which the third has to the fourth ; then, if the first be greater than the third, the second shall be greater than the fourth ; ana if...