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]. |
From inside the book
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Page 19
... take the Propositions in the follow- ing order . § 1. 1 , 5 , 3 , 2 ... magnitudes be equimultiples of as many : whatever multiple any one of them is of its part , the same multiple are all the first magnitudes of all ... magnitude be the C 2.
... take the Propositions in the follow- ing order . § 1. 1 , 5 , 3 , 2 ... magnitudes be equimultiples of as many : whatever multiple any one of them is of its part , the same multiple are all the first magnitudes of all ... magnitude be the C 2.
Page 38
... taken separately , be propor- tionals : they are also proportionals when taken jointly . That is - If the first be ... magnitude be to another as a magnitude taken from the first is to a magnitude taken from the other : the remainder is ...
... taken separately , be propor- tionals : they are also proportionals when taken jointly . That is - If the first be ... magnitude be to another as a magnitude taken from the first is to a magnitude taken from the other : the remainder is ...
Page 39
... magnitude ( c ) taken from the first is to a magnitude ( d ) taken from the other : the re- mainder ( a – c ) is to the remainder ( b - d ) as the whole ( a ) is to the whole ( 6 ) . ] Express enunciation . Taking given propor- tion ...
... magnitude ( c ) taken from the first is to a magnitude ( d ) taken from the other : the re- mainder ( a – c ) is to the remainder ( b - d ) as the whole ( a ) is to the whole ( 6 ) . ] Express enunciation . Taking given propor- tion ...
Page 47
... magnitude the same ratio as each of the second rank has to another single ... Taking given propor- tions • deduce equations . thence deduce equa- tion ... magnitudes of the same PROPOSITION XXIV . 47.
... magnitude the same ratio as each of the second rank has to another single ... Taking given propor- tions • deduce equations . thence deduce equa- tion ... magnitudes of the same PROPOSITION XXIV . 47.
Page 48
... Taking that ratio first which contains the greatest magnitude , and inverting if necessary , we obtain a proportion in which the greatest stands first . Let magnitudes , so arranged , be named a , b , c , d , of which a is greatest ...
... Taking that ratio first which contains the greatest magnitude , and inverting if necessary , we obtain a proportion in which the greatest stands first . Let magnitudes , so arranged , be named a , b , c , d , of which a is greatest ...
Common terms and phrases
A+B+C+ a+b+c+&c a+bb::c+d a+c+e+&c a=bk A=mX abcd b+d+f+&c B=mb B=nX c=dk C=mc c=md C=rX clearing of fractions clusion Let common measure conclusion thence deduce Convertendo cross order deduce equation thence divide e=nb equal equation required equation thence deduce equimultiples EUCLID excess Express enunciation Express first enuncia Express second enun fifth four magnitudes greater ratio hypothesis Taking given inequality less Let A=ma Let a>b Let ab::c:d magnitude taken magnitudes be proportionals multiple multiply by ab multiply corresponding sides number denotes number of magnitudes Preliminary Algebra process will prove proportionals when taken Q. E. D. COROLLARY Q. E. D. PROP remainder required for conclusion right-hand column second a greater second rank Simplify terms single magnitude Taking given inequa Taking given propor thence deduce con thence deduce equa thence prove three magnitudes tion deduce equation tion required tiple unequal ratios Universal Proposition vinculum σα
Popular passages
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 30 - 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 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.