Euclid, Book V. Proved Algebraically, So Far as it Relates to Commensurable Magnitudes: To which is Prefixed a Summary of All the Necessary Algebraical Operations, Arranged in Order of Difficulty

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Side 44 - If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio ; the first shall have to the last of the first magnitudes, the same ratio which the first of the others has to the last. NB This is usually cited by the words "ex sequali,
Side 54 - The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples...
Side 13 - A LESS magnitude is said to be a part of a greater magnitude, when the less measures the greater; that is, when the less is contained a certain number of times exactly in the greater.
Side 44 - Next, let there be four magnitudes A, B, C, D, and other four E, F, G, H, which, taken two and two in a cross order, have the same ratio, viz.: A...
Side 41 - If there be three magnitudes, and other three, which, taken two and two, have the same ratio; then if the first be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less.
Side 16 - In a direct proportion, the first term has the same ratio to the second, as the third has to the fourth.
Side 54 - Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other.
Side 40 - THEOB.—If four magnitudes be proportionals, they are also proportionals by conversion; that is, the first is to its excess above the second, as the third to its excess above the fourth. Let AB be to BE, as CD to DF: then BA shall be to AE, as DC to CF.
Side 57 - Dividendo, by division ; when there are four proportionals, and it is inferred, that the excess of the first above the second, is to the second, as the excess of the third above the fourth, is to the fourth.
Side 56 - ... compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and so on unto the last magnitude.

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