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" If the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth... "
Pantologia. A new (cabinet) cyclopędia, by J.M. Good, O. Gregory, and N ...
by John Mason Good - 1813
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The First Six, and the Eleventh and Twelfth Books of Euclid's Elements: With ...

Euclid, James Thomson - Geometry - 1845 - 352 pages
...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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A system of practical mathematics; being no.xvi. of a new series of school-books

Scottish school-book assoc - 1845
...multiple of the third is also equal to that of the fourth; or if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth. NOTE. A multiple of a quantity is the result of repeating that quantity...
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Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's ...

Dennis M'Curdy - Geometry - 1846 - 138 pages
...5 Wherefore, if the first be the same multiple, &c. QED 4 Th. If the first of four magnitudes have the same ratio to the second which the third has to the fourth; then any equimultiples of the antecedents shall have the same ratio as any equimultiples of the consequents....
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The Elements of Euclid, the parts read in the University of Cambridge [book ...

Euclides - 1846
...has a greater ratio to the second than the fifth has to the sixth. PROP. 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 equal,...
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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
...respectively. If, therefore, the first, &c. QED PROP. IV. THEOREM. If the first of f oar magnitudes has the same ratio to the second which the third has to the fourth ; any equimultiples whatever of the first and third have the same ratio to any equimultiples of the...
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The Elements of Euclid, with many additional propositions, and explanatory ...

Euclides - 1855
...any whatever of G, H ; therefore as E is to G so is F to H (c). COROLLARY. Likewise, if the first has the same ratio to the second, which the third has to the fourth, then also any equimultiples whatever of the first and third shall have the same ratio to the KEA second...
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