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" If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio ; the first shall have to the last of the first magnitudes, the same ratio which the first of the others has to the last. NB This is usually... "
Euclid's plane geometry, books iii.-vi., practically applied; or, Gradations ... - Seite 217
von Euclides - 1861
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Elements of Geometry: Containing the First Six Books of Euclid : with a ...

John Playfair - 1849 - 317 Seiten
...and B : A : : F : E, therefore, by this first case, since CyA, F7 D, that is, D/F. PROP. XXII. THEOR. If there be any number of magnitudes, and as many others, which, taken two ana two in order, have the same ratio ; the first will have to the last of the first magnitudes, the...
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The geometry, by T. S. Davies. Conic sections, by Stephen Fenwick

Royal Military Academy, Woolwich - 1853
...less than F. Therefore, if there be three, etc. QED c • EUCLID S ELEMENTS. PROPOSITION XXII. THEOR. If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio ; the first shall have to the last of the first magnitudes, the same ratio...
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The first six books of the Elements of Euclid, with numerous exercises

Euclides - 1853 - 147 Seiten
...therefore, d is less than f Therefore, if there be three magnitudes, &c. QED PROPOSITION XXII. — THEOREM. If there be any number of magnitudes, a/nd as ma/ny others, which, taken two and two in order, have the sa/те ratio ; the first shall have to the last of thefirst magnitudes the same ratio...
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Elements of Geometry: Containing the First Six Books of Euclid with a ...

John Playfair - 1855 - 318 Seiten
...demonstration extended to any number of magnitudes. nF(11.5.). Again, PROP. XXIII THEOR. If there le any number of magnitudes, and as many others, which,...in a cross order, have the same ratio ; the first will have to the last of the first magnitudes the same ratio which the first of the others has to the...
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The Elements of geometry; or, The first six books, with the eleventh and ...

Euclides - 1855
...the last of the others. First, let there be three magnitudes A, B and С, and other three D, E and F, which taken two and two in a cross order have the same ratio ; that is, such that A is to B, as E is to F ; and as B is to C, so is D to E. A is to С, as D is...
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The Elements of Euclid, with many additional propositions, and explanatory ...

Euclides - 1855
...3 KMHLN ABC_EFG lhen B ' C ' DF ' G ' H' therefore A : D :: E : H. PROPOSITION XXIII. THEOREM.—Jf there be any number of magnitudes, and as many others, which, taken tivo and two, in a cross order, have tlie same ratio; the first has to the last of the first magnitudes...
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Manual of Euclid: books iv., v., vi., by J.A. Galbraith and S. Haughton

Euclides - 1859
...laa rо irXqöOç, aiivSvo \apßavoptva Kai iv rif avrif \oyif Kai Stîffov iv ry avru Xóyy Éffrai. If there be any number of magnitudes, and as many others, which, taken two and two, have the same ratio; they shall be in the same ratio by equality. Statement — Let there be three...
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Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with ...

Robert Potts - 1860 - 361 Seiten
...first; that is, JD is less than JP. Therefore, if there be three, &c. QED PROPOSITION XXII. THEOREM. If there be any number of magnitudes, and as many others, which taken two and two in order, have the same ratio ; the first shall have to the last of the first magnitudes, the same ratio...
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Euclid's Elements of plane geometry [book 1-6] explicitly enunciated, by J ...

Euclides - 1860
...I> : C ; and, by inversion.(Prqp. 4), A:A + B::C:C + D. PROPOSITION XXII. THEOREM. (EUCLID tSXII.) If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio ; the first ehall have to the last of the first magnitudes the same ratio,...
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The Elements of Euclid with Many Additional Propositions and Explanatory Notes

Eucleides - 1860
...ratio taken two and two, but in a cross order, then, if the first magnitude be greater than the third. If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio. If there be any number of magnitudes, and as many others, which, taken...
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