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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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The definitions, postulates, axioms, and enunciations of the propositions of ...

Euclides - 1848
...of the second, and the other of the fourth. PROP. IV. THEOREM. 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 Elements of Euclid. Books I.-VI.; XI. 1-21 XII. 1-2. A New Text ..., Book 1

Euclides - Geometry - 1853
...is no necessity for all four to be of the same kind. OBS. 3. When the first of four magnitudes has the same ratio to the second which the third has to the fourth, the third clearly has the same ratio to the fourth which the first has to the second. Such will appear...
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The geometry, by T. S. Davies. Conic sections, by Stephen Fenwick

Royal Military Academy, Woolwich - Mathematics - 1853
...fourth D. If, therefore, the first, etc. QED PROPOSITION IV. THEOB. 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 first six books of the Elements of Euclid, with numerous exercises

Euclides - Geometry - 1853 - 147 pages
...If, therefore, the first, &c. QED PROPOSITION IV. — THEOREM. If the first of four magnitudes has the same ratio to the second which the third has to the fowrth ; then any equimultiples whatever of tlie first and third shall have the same ratio to any equimultiples...
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The Elements of geometry; or, The first six books, with the eleventh and ...

Euclides - 1855
...magnitude, they cannot be said to be of the same Hud, and so cannot have any ratio to each other. V. The first of four magnitudes is said to have the same ratio to the «cond, which the third has to the fourth, when any equimultiples whatsoever of the first and third...
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The Elements of Euclid, with many additional propositions, and explanatory ...

Euclides - 1855
...b = 6, which is the first case in Euclid. PROPOSITION A. THEOREM.—If the first of four magnitudes have 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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The principles and practice of arithmetic

John Hind - 1856
...is stated in the fifth Book of Euclid's Elements, that "Proportion is the Similitude of Ratios ; and the first of four magnitudes is said to have the same...the third has to the fourth, when any equimultiples whatever of the first and third being taken, and any equimultiples whatever of the second said fourth...
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Elements of Geometry: Containing the First Six Books of Euclid with a ...

John Playfair - Euclid's Elements - 1856 - 318 pages
...therefore A=mnC /(, tj',;t<f'£ PROP. IV. THEOR. /, / 2. // //>5 If the first of four magnitudes has thr, 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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Practical Arithmetic: Uniting the Inductive with the Synthetic Mode of ...

James Bates Thomson - Arithmetic - 1858 - 384 pages
...answer is greater than the third term, arises from the fact, that theßrst türm of a proportion has the same ratio to the second, which the third has to the /он г £Л or answer ; consequently, if the answer is greater than the third term, the second term...
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Manual of Euclid: books iv., v., vi., by J.A. Galbraith and S. Haughton

Euclides - 1859
...àiroiovovv ToXXaя-Xaíriaff/iov, rev áurоv i'Ctí Xóyov Xq^öivra raráXXqXa. If the first has the same ratio to the second which the third has to the fourth; any equimultiples whatever of the first and third shall have the same ratio to any equimultiples of...
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