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

John Playfair - Euclid's Elements - 1835 - 316 pages
...H. In the same manner is the demonstration extended to any number of magnitudes. 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 will have to the last of the first magnitudes the same... ## The Teacher's Assistant in the "Course of Mathematics Adapted to the Method ...

Mathematics - 1836 - 472 pages
...last of the first magnitudes, the same ratio which the first of the others has to the last. XXIII. 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 will have to the last of the first magnitudes the same... ## The Element of Geometry

John Playfair - Geometry - 1836 - 114 pages
...therefore ^=^ , or A : B : : E : F. JJ r D r PROP. XVI. THEOR. If there be any number of quantities, and as many others, which taken two and two in order have the same ratio, the first shall have to the laist of the first quantities the same ratio which the first of the others... ## Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - Euclid's Elements - 1836 - 488 pages
...: E, therefore, by the first case, since C^A, F^^D, that is, Therefore, &c. QED PROP. XXII. THEOR. If there be any number of magnitudes, and as many others, which, taken two and tico in order, have the same ratio ; the first will have to the last of tlie jirst magnitudes, the... ## Elements of Plane Geometry According to Euclid

Andrew Bell - Euclid's Elements - 1837 - 240 pages
...same manner is the demonstration 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... ## The First Six and the Eleventh and Twelfth Books of Euclid's Elements: With ...

Euclid - Geometry - 1837 - 410 pages
...the number of magnitudes. Therefore, if there be any number of 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... ## Elements of Geometry: Containing the First Six Books of Euclid : with a ...

John Playfair - Geometry - 1837 - 332 pages
...: A : : F : E, therefore, by this first case, since C 7 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 Elements of Euclid

Euclid - Geometry - 1838 - 478 pages
...first ; and therefore D is less than F. Therefore, if there be three, &c. QED PROP. XXII. THEOR. Ir 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... ## The Elements of Euclid: The Errors, by which Theon, Or Others, Have Long Ago ...

Robert Simson - Geometry - 1838 - 434 pages
...whatever be the number of magnitudes. Therefore, if there be any number, &c. QED 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 shall have to the last of the first magnitudes the same... 