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" MAGNITUDES which have the same ratio to the same magnitude are equal to one another ; and those to which the same magnitude has the same ratio are equal to one another. "
The Solutions of the Geometrical Problems: Consisting Chiefly of Examples in ... - Page 16
by Thomas Gaskin - 1847 - 263 pages
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The Elements of Euclid for the Use of Schools and Colleges: Comprising the ...

Euclid, Isaac Todhunter - Euclid's Elements - 1883 - 428 pages
...it has to AB. [V. Dtfnitien 7. Wherefore, of unequal magnitudes &c. QED LKD PROPOSITION 9. THEOREM. Magnitudes which have the same ratio to the same magnitude,...magnitude has the same ratio, are equal to one another. First, let A and B have the same ratio to C: A shall be equal to B. For, if A is not equal to B, one...
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Elements of geometry, containing books i. to vi.and portions of books xi ...

Euclides, James Hamblin Smith - 1883 - 376 pages
...mqA, .'. D has to B a greater ratio than D has to A. V. Def. 7. QED PROPOSITION VIII. (Eucl. v. 9.) Magnitudes, which have the same ratio to the same...another ; and those, to which the same magnitude has the so/me ratio, are equal to one another. Let A and B have the same ratio to C. Then must A = B. For if...
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The Elements of Euclid, books i. to vi., with deductions, appendices and ...

Euclides - 1884 - 434 pages
...A by m, but does not exceed the multiple of A + B by m ; .-. C: A is greater than C : A + BV Def. 9 Magnitudes which have the same ratio to the same magnitude...magnitude has the same ratio are equal to one another. First let A : C = B : C : it is required to prove A = B. For if A be greater than B, then A : C is...
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The Harpur Euclid: An Edition of Euclid's Elements

Edward Mann Langley, W. Seys Phillips - 1890 - 538 pages
...nC>mB, while nC<mh, which is contrary to the hypothesis. Similarly A cannot be less than B. Hence : — (1) Magnitudes which have the same ratio to the same magnitude are equal to one another. (2) Magnitudes to which the same magnitude has the same ratio are equal to one another. PROPOSITION...
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Euclid Revised: Containing the Essentials of the Elements of Plane Geometry ...

Euclid - Geometry - 1890 - 442 pages
...when in (A + M) > (n + i) B, mA< (n + i) B; .-. A + M : B > A : B, .-. also B : A + M < B : A. 2">, magnitudes which have the same ratio to the same magnitude are equal. Let A, B, C be three magnitudes of the same kind, such that A : C = B : C. Then, by i°, if A > B,...
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The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ...

Euclid - Geometry - 1892 - 460 pages
...magnitude has the same ratio to equal magnitudes. For if A = B, then C : A = C : B. PROPOSITION 6. (i) Magnitudes which have the same ratio to the same magnitude are equal to one another. That IB, if A : C = B : C. then A = B. (ii) Those magnitudes to which the same magnitude has the same...
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Uniplanar Algebra: Being Part I of a Propædeutic to the Higher Mathematical ...

Irving Stringham - Algebra - 1893 - 164 pages
...because nC > mR while nC is either < m A, or at most = m A, .-. C : B > C : A. (Def. 7.) PROPOSITION 6. "Magnitudes which have the same ratio to the same...magnitude has the same ratio are equal to one another. ' ' That is, A, B, C being three magnitudes of the same kind ; if A : C : : B : C, then A = B. and...
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Ellen: Or, Whisperings of an Old Pine, Volume 1

Joseph Battell - Force and energy - 1903 - 722 pages
...— will hold a greater number of pints than quarts. No demonstration necessary. PROPOSITION IX. ' Magnitudes which have the same ratio to the same magnitude...magnitude has the same ratio are equal to one another.' '• Still another proposition that needs not demonstration. PROPOSITION X. "Ratio being the number...
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A Text-book of Euclid's Elements for the Use of Schools, Book 1

Euclid - Euclid's Elements - 1904 - 488 pages
...taking m and n as before, nC > mB, while nC is not > mA ; .: C:B>C:A. Def. 1. PROPOSITION 6. Mar/nitndes which have the same ratio to the same magnitude are...one, another ; and those to which the same magnitude hax the same ratio are equal to one another. First, let A : C : : B : C ; • then shall A = B. For...
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Syllabus of Plane Geometry: (corresponding to Euclid, Book I-VI) ; Prepared ...

Cora Lenore Williams - Geometry - 1905 - 122 pages
...same magnitude has a greater ratio to the less of two magnitudes than it has to the greater. Prop. 26. Magnitudes which have the same ratio to the same magnitude...magnitude has the same ratio are equal to one another. Prop. 27. That magnitude which has a greater ratio than another has to the same magnitude is the greater...
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