| John Playfair - Euclid's Elements - 1806 - 311 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 - 304 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 - 367 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
...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 - 129 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 - 158 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 - 252 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 - 186 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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