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 |
Common terms and phrases
a+b a-b a+b+c+ a+b+c+&c a+bb::c+d a+c+e+&c a=bk a=mb A=mX Algebraical Definition b-d b-d b+d+f+&c B=mb B=nX c=dk C=mc c=md C=rX called proportionals clearing of fractions clusion Let common measure Convertendo cross order deduce equation thence divide e=nb equation thence deduce equimultiples Euclid excess Express enunciation Express first enuncia Express second enun f=nd four magnitudes four proportionals greater magnitude greater ratio instance magnitude taken magnitudes be proportionals multiple multiply corresponding sides nb nd number of magnitudes Preliminary Algebra process will prove proportionals when taken Q. E. D. COROLLARY Q. E. D. PROP QUÆSITA ratio of third remainder required for conclusion right-hand column second is greater second rank Simplify terms single magnitude subtract Taking given inequa Taking given propor thence deduce con thence deduce equa thence prove third to fourth three magnitudes tion deduce equation tion required tiple unequal ratios Universal Proposition σα
Popular passages
Page 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,
Page 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...
Page 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.
Page 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...
Page 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.
Page 16 - In a direct proportion, the first term has the same ratio to the second, as the third has to the fourth.
Page 54 - Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other.
Page 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.
Page 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.
Page 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.