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
Results 1-5 of 7
Page 19
... Corollary ) , 6 , Corollary of 2 . § 2. C , D , 15 . § 3. 11 , 13 . § 4. 7 , 9 , 8 , 10 . § 5. 16 , B , 18 , 17 , 17 * , E , 12 , 19 , 24 , 22 , 23 . § 6. 4 . § 7. A , 14 , 20 , 21 , 25. ] PROP . I. If any number of magnitudes be ...
... Corollary ) , 6 , Corollary of 2 . § 2. C , D , 15 . § 3. 11 , 13 . § 4. 7 , 9 , 8 , 10 . § 5. 16 , B , 18 , 17 , 17 * , E , 12 , 19 , 24 , 22 , 23 . § 6. 4 . § 7. A , 14 , 20 , 21 , 25. ] PROP . I. If any number of magnitudes be ...
Page 20
... COROLLARY . The Proposition holds true of two ranks of magni- tudes , of which each of the first rank is the same multiple of a single magnitude as each of the second rank is of another single magnitude . If [ This Corollary is usually ...
... COROLLARY . The Proposition holds true of two ranks of magni- tudes , of which each of the first rank is the same multiple of a single magnitude as each of the second rank is of another single magnitude . If [ This Corollary is usually ...
Page 22
... - tion required for conclusion .. , xng by m ma n nb = тс nd ; C A · . .. , substituting , = B D .. A : B :: C : D. thence deduce con- clusion Q. E. D. COROLLARY 1 . With the same data : the multiple 22 PROPOSITION IV .
... - tion required for conclusion .. , xng by m ma n nb = тс nd ; C A · . .. , substituting , = B D .. A : B :: C : D. thence deduce con- clusion Q. E. D. COROLLARY 1 . With the same data : the multiple 22 PROPOSITION IV .
Page 39
... COROLLARY . With the same data : the remainder is to the re- mainder as the magnitude taken from the first is to the magnitude taken from the other . [ With the same data : the remainder ( a – c ) is to the remainder ( b - d ) as the ...
... COROLLARY . With the same data : the remainder is to the re- mainder as the magnitude taken from the first is to the magnitude taken from the other . [ With the same data : the remainder ( a – c ) is to the remainder ( b - d ) as the ...
Page 45
... ƒ : d , с e f ; = / ; 7 = 2 , and 2 = ; a b adding corresponding sides , ate . b c + f . = - ; d thence deduce con- clusion .. a + ebc + f : d . Q. E. D. COROLLARY 1 . With the same data ; the excess PROPOSITION XXIV . 45.
... ƒ : d , с e f ; = / ; 7 = 2 , and 2 = ; a b adding corresponding sides , ate . b c + f . = - ; d thence deduce con- clusion .. a + ebc + f : d . Q. E. D. COROLLARY 1 . With the same data ; the excess PROPOSITION XXIV . 45.
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.