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" This will reduce the whole to the triangle MGD, which is equal to the original figure. The area of the triangle may then be found by multiplying its base into half its height ; and this will be the contents of the field. In practice, it will not be necessary... "
A Course of Mathematics: Containing the Principles of Plane Trigonometry ... - Page 69
by Jeremiah Day - 1851
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

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

Jeremiah Day - Measurement - 1815 - 388 pages
...original figure. The area of the trianfle may then be found by multiplying its base into half its eight ; and this will be the contents of the field. In practice,...the computed area of the field, to the true area. Ex. If the area of a field measured by a chain 66.4 feet Jong, be computed to be 32.6036 acres ; what...
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The complete measurer: or, The whole art of measuring, containing the ...

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

Adrien Marie Legendre - Geometry - 1822 - 394 pages
...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 - 440 pages
...intersection G. Then the triangles DBC and DBG *re equal. (Eue. 37. 1.) Taking from each the common part OBH, there remains BGH equal to DCH. If then the triangle...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 that by...
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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 - 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...
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Elements of Geometry: Being Chiefly a Selection from Playfair's Geometry

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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A Treatise of Plane Trigonometry: To which is Prefixed a Summary View of the ...

Jeremiah Day - Logarithms - 1831 - 418 pages
...such a manner as, at each step, to make the number of sides one less, till they are reduced to three.S Let the side AB (Fig. 35.) be extended indefinitely...the computed area of the field, to the true area. Ex. If the area of a field measured by a chain 66.4 feet long, be computed to be 32.6036 acres ; what...
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The Principles of Plane Trigonometry, Mensuration, Navigation and Surveying ...

Jeremiah Day - Measurement - 1831 - 394 pages
...intersect AB continued. Draw also a line from D to the point of intersection G. Then the triangles DEC and DBG are equal. (Euc. 37. 1.) Taking from each...the computed area of the field, to the true area. Ex. If the area of a field measured by a chain 66.4 feet long, be computed to be 32.6036 acres ; what...
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