The propositions of the fifth book of Euclid proved algebraically

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Page 12 - 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 whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth...
Page 62 - 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,
Page 58 - IF there be any number of magnitudes, and as many others, which, taken two and two, in a cross 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
Page 54 - Ij the first be to the second as the third to the fourth, and if the first be a multiple, or a part of the second ; the third is the same multiple, or the same part of the fourth.
Page 83 - This Grammar is in general use at Oxford, Cambridge, Dublin, and Durham; at Eton, King's College, London, and most other public schools. MADVIG'S LATIN GRAMMAR. A Latin Grammar for the Use of Schools. By Professor MADVIG, with additions by the Author. Translated by the Rev. G. WOODS, MA Uniform with JELF'S "Greek Grammar.
Page 38 - IF one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other ; the remainder shall be the same multiple of the remainder, that the whole is of the whole.
Page 90 - ANNALS OF ENGLAND. An Epitome of English History. From Cotemporary Writers, the Rolls of Parliament, and other Public Records. 3 vols. Fcap. 8vo., with Illustrations, cloth, 15s. Recommended by the Examiners in the School of Modern History at Oxford.
Page 25 - 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.
Page 57 - IF there be three magnitudes, and other three, which, taken two and two, have the same ratio ; if the first be greater than the third, the fourth shall be greater than the sixth ; and if equal, equal ; and if less, less...
Page 31 - ... that they are proportionals when taken two and two of each rank, and it is inferred, that the first is to the last of the first rank of magnitudes, as the first is to the last of the others: ' Of this there are the two follow' ing kinds, which arise from the different order in ' which the magnitudes are taken two and two.

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