a+b+c+&c a+bb::c+d a+c+e+&c a=bk A=ma A=mX b+d+f+&c B=mb B=nX Book c=dk c=md C=rX clearing of fractions commensurable consequents Convertendo cross order deduce conclusion deduce equation thence divide equal equation thence deduce equimultiples Euclid excess Express enunciation fifth four magnitudes four proportionals greater ratio greatest identical Inequalities inferred instance kind least less lesser magnitude taken magnitudes be proportionals meaning method multiple multiply corresponding sides number of magnitudes portionals process will prove proof proportionals proportionals when taken prove Q. E. D. PROP remainder required for conclusion right-hand column second is greater second rank Show Simplify terms single magnitude sixth substituting subtract taken Taking given propor thence deduce con thence deduce equa third third to fourth tion deduce equation tion required tiple true whole σα
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 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.