 | John Playfair - Mathematics - 1806 - 320 pages
...is the same with the duplicate ratio of their sides ; and hence, also, any two similar rectilineal figures are to each other as the squares of their homologous sides. PROP. XXI. THEOR. Book VI. RECTILINEAL figures, which are similar to the same rectilineal figure, are... | |
 | Jeremiah Day - Measurement - 1815 - 388 pages
...chain is found to be too long or too short ; the true contents may be found, upon the principle that similar •figures are to each other as the squares of their homologous sides. (Euc. 20. 6.) The proportion may be stated thus ; As the square of the true chain, to the square of... | |
 | Thomas Keith - 1817 - 306 pages
...I = 7- X VD — d 3/ \D — d 3/ a — d 3 the solidity of the frustum ABD c. Now all similar plane figures are to each other as the squares of their homologous sides. Therefore A 1 a : : D* : d* or — = — , put each off these equal to m ; then A — * D* and a —... | |
 | Adrien Marie Legendre - Geometry - 1822 - 394 pages
...second polygon, we shall have S' =p'2? : (A, B, A' B', &c.) Hence S : S': :p2 :p's ; hence the surfaces of similar figures are to each other as the squares of their homologous sides. Let us now proceed to polyedrons. We may take it for granted, that a face is determined by means of... | |
 | Jeremiah Day - Geometry - 1824 - 440 pages
...chain is found to be too long or too short; the true contents may be found, upon the principle that similar figures are to each other as the squares of their homologous sides. (Euc. 20.6.) The proportion may be stated thus ; As the square of the true chain, to the square of... | |
 | Euclid, Dionysius Lardner - Euclid's Elements - 1828 - 542 pages
...polygon, we shall have S' = ^/i0 : (A, B, A', B', &c.) Hence S : S' : : p2 : p'1 ; hence the surfaces of similar figures are to each other as the squares of their homologous sides. Let us now proceed to polyedrons. We may take it for granted, that a face is determined by means of... | |
 | Timothy Walker - Geometry - 1829 - 138 pages
...them, than those just explained. We shall demonstrate the following general proposition. — Any two similar figures are to each other as the squares of their homologous sides — . We begin with two similar triangles. Let these be ABC and DEF (fig. 77), and let AG be the altitude of... | |
 | Timothy Walker - Geometry - 1829 - 156 pages
...ABCDE:FGHIK::ABC:FGH. But ABC : FGH : : AB* : p G*. Therefore ABCDE : FGHIK : : AB* : FG*. In other words, similar figures are to each other as the squares of their homologous sides. 117. THEOREM. — Circles are to each other as the squares of their radii. No diagram is necessary... | |
 | John Bonnycastle - Geometry - 1829 - 256 pages
...such as have the same number of sides, and the angles contained by those sides respectively equal. 8. The areas of similar figures are to each other as the squares of their like sides. MENSURATION OP SUPERFICIES, . THE area of any figure is the measure of its surface,... | |
 | John Playfair - Geometry - 1829 - 210 pages
...other is the same with the duplicate ratio of their sides; and hence, also, any two similar rectilineal figures are to each other as the squares of their homologous sides. CoR. 3. Two similar triangles, or two similar polygons, are to each other as any rectilineal figure... | |
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... similar figures are to each other as the squares of their homologous sides. 
