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" 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 whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple... "
The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ... - Page 105
by Euclides - 1816 - 528 pages
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Elementary Algebra

James Hamblin Smith - Algebra - 1882 - 398 pages
...representing the ratios must be equal. Euclid's test is given in Book v. Def. 5, where it stands thus : " The first of four magnitudes is said to have the same...the fourth, when any equimultiples whatsoever of the f1rst and third being taken and any equimultiples whatsoever of the second and fourth : " If the multiple...
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Elements of Plane and Solid Geometry

George Albert Wentworth - Geometry - 1882 - 398 pages
...±Pa : b ± ?-b : : a : b. Ч Ч QED 272. DEF. Euclid's test of a proportion is as follows : — " The first of four magnitudes is said to have the same...second which the third has to the fourth, when any equimultiplos whatsoever of the first and third being taken, and any equimultiples whatsoever of the...
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The Elements of Euclid for the Use of Schools and Colleges: Comprising the ...

Euclid, Isaac Todhunter - Euclid's Elements - 1883 - 400 pages
...said to have a ratio to one another, when the less can be multiplied so as to exceed the other. 5. The first of four magnitudes is said to have the same ratio to the second, that the third has to the fourth, when any equimultiples whatever of the first and the third being...
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The first six books of the Elements of Euclid, and propositions i.-xxi. of ...

Euclides - 1885
...ratio and its reciprocal is unity." v. The first of four magnitudes has to the second the same ratio which the third has to the fourth, when any equimultiples...equimultiples whatsoever of the second and fourth, if, according as the multiple of the first is greater than, equal to, or less than the multiple of the...
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The First Six Books of the Elements of Euclid: And Propositions I-XXI of ...

Euclid, John Casey - Euclid's Elements - 1885 - 312 pages
...ratio and its reciprocal is unity." v. The first of four magnitudes has to the second the same ratio which the third has to the fourth, when any equimultiples...equimultiples whatsoever of the second and fourth, if, according as the multiple of the first is greater than, equal to, or less than the multiple of the...
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Notes on algebra

Howard Candler - 1885
...fourth, that is, д, l>, c, d are said to be proportionals if — = b а The geometrical definition is 2. The first of four magnitudes is said to have the same...the third has to the fourth, when, any equimultiples whatever being taken of the ist and 3rd, and any equimultiples whatever of the 2nd and 4th, if the...
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A Text Book of Algebra

William Steadman Aldis - Algebra - 1887 - 588 pages
...f3,ft This condition is evidently equivalent to the definition 5 of the Fifth Book of Euclid, viz.— The first of four magnitudes is said to have the same ratio to the second, that the third has to the fourth, when any equimultiples whatever of the first and third being taken,...
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The Harpur Euclid: An Edition of Euclid's Elements

Edward Mann Langley, W. Seys Phillips - 1890 - 515 pages
...number of metres may make, a greater length can always be found containing an exact number of yards. 5. The first of four magnitudes is said to have the ' same ratio ' to the second that the third has to the fourth, when, any equimultiples whatever of the first and third being taken,...
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THE CONTENTS OF THE FIFTH AND SIXTH BOOKS OF EUCLID

M. J. M. HILL - 1900
...following manner : — The first of four magnitudes is said to have the same ratio to the second, as the third has to the fourth, when any equimultiples...equimultiples whatsoever of the second and fourth being taken; if the multiple of the first be less than that of the second, the multiple of the third...
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The Contents of the Fifth and Sixth Books of Euclid

Euclid, Micaiah John Muller Hill - Euclid's Elements - 1900 - 143 pages
...Art. 67. Def. 13. EUCLID'S TEST FOR EQUAL RATIOS. Euclid states this Test in the following manner: — The first of four magnitudes is said to have the same ratio to the second, as the third has to the fourth, when any equimultiples whatsoever of the first and third being taken,...
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