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" IF there be any number of magnitudes, and as many others, which, taken two and two, in a cross 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... "
Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's ... - Page 83
by Dennis M'Curdy - 1846 - 138 pages
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Elements of Geometry: Containing the First Six Books of Euclid : with a ...

John Playfair - Euclid's Elements - 1849 - 332 pages
...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 - Mathematics - 1853 - 400 pages
...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...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 tke others...
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The first six books of the Elements of Euclid, with numerous exercises

Euclides - 1853 - 176 pages
...number of magnitudes. Therefore, if there be any number, <fec. QED PROPOSITION XXIII. — THEOREM. If there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio ; the, first sliall have to the last of the first magnitudes the...
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Elements of Geometry: Containing the First Six Books of Euclid with a ...

John Playfair - Euclid's Elements - 1855 - 340 pages
...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, taken two and two, in a cross order, have the same ratio ; the first will have to the last of the first magnitudes the same...
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The Elements of geometry; or, The first six books, with the eleventh and ...

Euclides - 1855 - 262 pages
...Therefore, if there be three, &c. QED PROP. XXII. THEOREM. If there be any number of magnitudes, andas many others, which taken two and two in order, have the same ratio ; (he first shall have to the last of the first magnitudes, the same ratio which the first has to the...
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The Elements of Euclid, with many additional propositions, and explanatory ...

Euclides - 1855 - 230 pages
...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 - 136 pages
...laa rо irXqO, aiivSvo \apavoptva Kai iv rif avrif \oyif Kai Stffov iv ry avru Xyy ffrai. 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 - Geometry, Plane - 1860 - 380 pages
...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...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 has to the...
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The Elements of Euclid with Many Additional Propositions and Explanatory Notes

Eucleides - 1860 - 396 pages
...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 two and two, in a cross order, have the same ratio. If magnitudes are equal . . If two magnitudes are unequal. If a...
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Euclid's Elements of plane geometry [book 1-6] explicitly enunciated, by J ...

Euclides - 1860 - 288 pages
...same manner the demonstration is extended to any number of magnitudes. PROPOSITION XXIII. THEOREM. If there be any number of magnitudes, and as many others, which, taken two and two in a cross order, have the same ratio ; the first shall have to the last of the first magnitudes the same...
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