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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... "
Elements of Geometry, Geometrical Analysis, and Plane Trigonometry: With an ... - Page 458
by Sir John Leslie - 1809 - 493 pages
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Euclid's Elements of plane geometry [book 1-6] with explanatory appendix ...

Euclides - 1840 - 192 pages
...multiples or submultiples which are equal, those pairs of numbers are proportional, or the first has the same ratio to the second which the third has to the fourth. But it must be remembered that there are incommensurable magnitudes, the relative values of which,...
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The Doctrine of Proportion Clearly Developed: On a Comprehensive, Original ...

Oliver Byrne - 1841 - 144 pages
...understand this definition before proceeding further. с 2 PROP. IV. THEO. If the first of four magnitudes have the same ratio to the second, which the third has to the fourth, then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples...
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The Elements of Euclid; viz. the first six books,together with the eleventh ...

Euclides - 1841 - 378 pages
...the fourth. If, therefore, the first, &c. QED PROP. IV. THEOR. If the first of four magnitudes has the same ratio to the second which the third has to the fourth; then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples...
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The Doctrine of Proportion Clearly Developed: On a Comprehensive, Original ...

Oliver Byrne - 1841 - 140 pages
...a, artd (m — n) b с = b, when m — n = 1. PROP. A. THEO. If the first of the four magnitudes has the same ratio to the second, which the third has to the fourth, then, if the first be greater than the second, the third is also greater than the fourth ; and if equal,...
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Elements of geometry: consisting of the first four,and the sixth, books of ...

Euclides - 1842 - 316 pages
...the ratio of the third to the fourth. Definition of proportion according to Euclid, (Def. V., Book " The first of four magnitudes is said to have the same...being " taken, and any equimultiples whatsoever of the second and " fourth ; if the multiple of the first be equal to, greater " than, or less than the multiple...
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The principles of arithmetic. [Followed by] The principles of algebra

Wales Christopher Hotson - 1842 - 306 pages
...:: a + c + e ... : 6 + d +/... 149. Geometrical Definition of Proportion. (Euclid, book v. def. 5). The first of four magnitudes is said to have the same...first and third being taken, and any equimultiples whatsover of the second and fourth ; if the multiple of the first, be less than that of the second,...
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - 1842 - 332 pages
...mB=mnC, and by hypothesis A=mB, therefore A=wmC PROP. IV. THEOR. If the first of four magnitudes has the same ratio to the second which the third has to the fourth, and if any equimultiples whatever be taken of the first and third, and any whatever of the second and...
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A system of practical mathematics; being no.xvi. of a new series of school-books

Scottish school-book assoc - 1845 - 444 pages
...geometrical magnitudes, and therefore it is necessary to substitute another, which is as follows: — Def. 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 First Six, and the Eleventh and Twelfth Books of Euclid's Elements: With ...

Euclides, James Thomson - 1845 - 382 pages
...whatever of G, H: therefore (V. def. 5) as E : G : : F : H. Therefore, &c. Cor. Likewise, if the first have the same ratio to the second, which the third has to the fourth, then also any like multiples whatever of the first and third have the same ratio to the second and...
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Euclid in Paragraphs: The Elements of Euclid: Containing the First Six Books ...

Euclid - 1845 - 218 pages
...ratio to the second, than the fifth has to the sixth. PROPOSITION XIV. THEOR. — If the first has the same ratio to the second which the third has to the fourth; then, if the first be greater than the third, the second shall be greater than the fourth ; and if...
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