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" ... similar figures are to each other as the squares of their homologous sides. "
Plane and Solid Geometry: To which is Added Plane and Spherical Trigonometry ... - Page 107
by George Roberts Perkins - 1860 - 443 pages
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

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...
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A Practical Application of the Principles of Geometry to the Mensuration of ...

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...
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The complete measurer: or, The whole art of measuring, containing the ...

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 —...
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Elements of Geometry and Trigonometry: With Notes

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...
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The Second [-fifth and Sixth] Part of A Course of Mathematics: Adapted to ...

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...
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The First Six Books of the Elements of Euclid, with a Commentary and ...

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...
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Elements of Geometry: With Practical Applications, for the Use of Schools

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...
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Elements of Geometry: With Practical Applications, for the Use of Schools

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...
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An Introduction to Mensuration and Practical Geometry

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,...
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Elements of Geometry: Being Chiefly a Selection from Playfair's Geometry

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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