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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 ... - Σελίδα 83
των Dennis M'Curdy - 1846 - 138 σελίδες
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The Elements of Euclid; viz. the first six books,together with the eleventh ...

Euclides - 1841 - 351 σελίδες
...greater than D, that is, D is less than F. Therefore, if there be three, &c. QED PROP. 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 fi'..*t has to th...

The Doctrine of Proportion Clearly Developed: On a Comprehensive, Original ...

Oliver Byrne - 1841 - 98 σελίδες
...d' e' f a' V U * c' 1/ 7' > a ' 7 * ? ' d' a' ( r У Of Ь edefg Ь' tí ОГ y g PROP. XXIII. THEO. 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...

Elements of geometry: consisting of the first four,and the sixth, books of ...

Euclides - 1842
...proportion. The proposition is usually cited by the words, ex cequali (sc. distantia), or ex &quo. PROP. XI. 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 of the one set of magnitudes will have to the last,...

Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - 1842 - 317 σελίδες
...: A : : F : E, therefore, by this first case, since Cy A, F "7 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...

The First Six, and the Eleventh and Twelfth Books of Euclid's Elements: With ...

Euclid, James Thomson - 1845 - 352 σελίδες
...the number of magnitudes. Therefore, if there be any numberof magnitudes, &c. PROP. XXIII. THEOR. — 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 has to the last of the first magnitudes the same ratio...

Euclid in Paragraphs: The Elements of Euclid: Containing the First Six Books ...

Euclid - 1845 - 199 σελίδες
...number of magnitudes. Therefore, if there be any number, &c. QED ABCD EFGH PROPOSITION XXIII. THEOR. — 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...

Euclid's Elements of geometry [book 1-6, 11,12] with explanatory notes ...

Euclides - 1845
...be the number of magnitudes. Therefore, if there be any number, &c. 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 Jirsl shall have to the last of the first magnitudes the same...

The Elements of Euclid, the parts read in the University of Cambridge [book ...

Euclides - 1846
...greater than D, that is, D is less than F. Wherefore, If there be three %c. QED PROP. XXII. THEOS. If there be any number of magnitudes, and as many...two in order, have the same ratio, then the first shall have to the last of the first magnitudes the same ratio which the first of the others has to...

Elements of Geometry: Containing the First Six Books of Euclid, with a ...

Euclid, John Playfair - 1846 - 317 σελίδες
...the same manner is the demonstration extended to any number of magnitudes. PROP. XXIII. THEOR. iff there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio ; thefirst will have to the last of the first magnitudes the same...

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

Euclides - 1848
...others. • NB This is usually cited by the words "ex tequali," or "ex aguo." PROP. 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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