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. [2 other copies, in unbound sheets].
a+b a-b a+b+c+ a+b+c+&c a+c+e+&c a=bk A=ma b+d+f+&c B=mb BOOK c=dk c=md clearing of fractions commensurable conclusion thence deduce consequents covering cross order deduce conclusion deduce equation thence directions divide divide by x equal equation thence deduce equimultiples Euclid excess Express enunciation fifth follows four magnitudes greater ratio greatest identical Inequalities inferred instance kind least left-hand column less magnitude taken meaning method multiple multiply corresponding sides number of magnitudes Preliminary Algebra process will prove proof proportionals Proposition Q. E. D. PROP reference remainder required for conclusion right-hand column second rank Show Simplify terms single magnitude sixth substituting subtract supply taken Taking given propor thence deduce con thence deduce equa third third to fourth tion deduce equation tion required true unequal vinculum whole σα
Page 30 - 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 27 - 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 2 - In a direct proportion, the first term has the same ratio to the second, as the third has to the fourth.
Page 40 - Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other.
Page 26 - 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 43 - 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 42 - ... 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.
Page 45 - Those magnitudes of- which the same, or equal magnitudes, are equimultiples, are equal to one another. 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude.