Books Books 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 ## The Elements of Euclid; viz. the first six books,together with the eleventh ...

Euclides - 1841 - 378 pages
...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 - Euclid's Elements - 1841 - 144 pages
...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 - 320 pages
...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 - Euclid's Elements - 1842 - 332 pages
...: 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... ## Euclid in Paragraphs: The Elements of Euclid: Containing the First Six Books ...

Euclid - Geometry - 1845 - 218 pages
...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... ## The First Six, and the Eleventh and Twelfth Books of Euclid's Elements: With ...

Euclid, James Thomson - Geometry - 1845 - 380 pages
...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's Elements of geometry [book 1-6, 11,12] with explanatory notes ...

Euclides - 1845 - 544 pages
...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 - 292 pages
...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 - Euclid's Elements - 1846 - 332 pages
...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... 