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" ... we suppose the error A to be of any magnitude whatever. 286. Definition. Similar sectors and similar segments are such as correspond to similar arcs. 287. Theorem. Similar sectors are to each other as the squares of their radii. Proof. The similar... "
An Elementary Treatise on Plane and Solid Geometry - Page 83
by Benjamin Peirce - 1847 - 150 pages
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Elements of Geometry: With Notes

John Radford Young - Euclid's Elements - 1827 - 246 pages
...the one arc to the other as the radius of the former to that of the latter. Cor. 3. Therefore also similar sectors are to each other as the squares of their radii, for each sector is to the circle as the arc to the circumference (Prop. XXIII. Cor. 2. B. VI) ; consequently...
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An Elementary Treatise on Plane and Solid Geometry

Benjamin Peirce - Geometry - 1837 - 216 pages
...greater than the aggregate of the quantities neglected, and yet we cannot escape this absurdity as long we suppose the error A to be of any magnitude whatever....sectors are to each other as the squares of their radii. JDemonstration. The similar sectors AOB, A'O'B' (fig. 136) are, by the same reasoning as in art. 97,...
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An Elementary Treatise on Plane and Solid Geometry

Benjamin Peirce - Geometry - 1841 - 186 pages
...represented by A. Since the aggregate of the quanties neglected is infinitely small, that is, as tmall as tec choose ; we can choose it to be less than the error...squares of their radii. Proof. The similar sectors AOB, A'QfB' (fig. 136) are, by the same reasoning as in § 97, the same parts of their respective circles,...
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Elements of Geometry: With Practical Applications ...

George Roberts Perkins - Geometry - 1847 - 326 pages
...the one arc to the other as the radius of the former to that of the latter.. Cor.. 3. Therefore also similar sectors are to each other as the squares of their radii ; for each sector is to the ciflle as the arc to the circumference : consequently the one sector is...
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Elements of Geometry: With, Practical Applications

George Roberts Perkins - Geometry - 1850 - 332 pages
...the one arc to the other as the radius of the former to that of the latter. Cor. 3. Therefore also similar sectors are to each other as the squares of their radii; for each sector is to the circle as the arc to the circumference : consequently the one sector is to...
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Elements of Geometry and Trigonometry

Adrien Marie Legendre - Geometry - 1852 - 436 pages
...of their diameters. 151 PBOPOSITION XIV. THEOEEM. Similar arcs are to each other as their radii: and similar sectors are to each other as the squares of their radii. Let AB, DE, be similar arcs, and ACB, DOE, similar sectors: then will and AB : ACB DE :: DOE CA For,...
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Elements of Geometry and Trigonometry from the Works of A.M. Legendre ...

Charles Davies - Geometry - 1854 - 436 pages
...diameters. BOOK V. 151 PROPOSITION XIV. THEOREM. Similar arcs are to each other as their radii: and similar sectors are to each other as the squares of their radii. Let AB, DE, be similar arcs, and ACB, DOE, similar sectors : then AB : DE :: CA : OD • and A CB DOE...
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An Elementary Treatise on Plane and Solid Geometry

Benjamin Peirce - Geometry - 1855 - 184 pages
...infinitely small, that is, as small as we choose ; we can choose it to be less than the error JI ; a manifest absurdity, for the error cannot be greater...squares of their radii. Proof. The similar sectors AQB, A'Q'B' (fig. 136) are, by the same reasoning as in ^ 97, the same parts of their respective circles,...
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Elements of Geometry and Trigonometry from the Works of A. M. Legendre ...

Adrien Marie Legendre, Charles Davies - Geometry - 1857 - 442 pages
...squares oj their diameters. PROPOSITION XIV. THEOREM. Similar arcs are to each other as tf1eir radii: and similar sectors are to each other as the squares of their radii. Let AB, DE, be similar arcs, and ACB, DOE, similar sectors : then AB : DE :: CA : OD; and ACB DOE CA...
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Annual Report on Public Schools in Rhode Island, Volumes 16-19

1861 - 420 pages
...equal to the radius. 3. Book V, Proposition XIV. Similar arcs are to each other as their radii; and similar sectors are to each other as the squares of their radii. 4. Book VI, Proposition IV. If a straight line be perpendicular to two straight lines at their point...
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