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Elements of plane (solid) geometry (Higher geometry) and trigonometry (and mensuration), being the first (-fourth) part of a series on elementary and higher geometry, trigonometry, and mensuration - Softcover
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Inhaltsangabe
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1845 Excerpt: ... will be equal in their polar triangles P and Q, (Prop. X.:) but since the triangles P and Q are mutually equilateral, they must also (Prop. XIV.) be mutually equiangular; and, lastly, the angles being equal in the triangles P and Q, it follows (Prop. X.) that the sides are equal in their polar triangles A and B. Hence the mutually equiangular triangles A and B are at the same time mutually equilateral. Scholium. This proposition is not applicable to rectilineal triangles; in which equality among the angles indicates only proportionality among the sides. Nor is it difficult to account for the difference observable, in this respect, between spherical and rectilineal triangles. In the proposition now before us, as well as in Propositions XII, XIII, XIV, which treat of the comparison of triangles, it is expressly required that the arcs be traced on the same sphere, or on equal spheres. Now similar arcs are to each other as their radii; hence, on equal spheres, two triangles cannot be similar without being equal. Therefore it is not strange that equality among the angles should produce equality among the sides. The case would be different, if the triangles were drawn upon unequal spheres; there, the angles being equal, the triangles would be similar, and the homologous sides would be to each other as the radii of their spheres. PROPOSITION XVIII. THEOREM. The sum of all the angles in any spherical triangle is less than six right angles, and greater than two. For, in the first place, every angle of a spherical triangle is less than two right angles (see the following Scholium): hence the sum of all the three is less than six right angles. Secondly, the measure of each angle of a spherical triangle (Prop. X.) is equal to the semicircumference minus the correspond...
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Reseña del editor
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1845 Excerpt: ... will be equal in their polar triangles P and Q, (Prop. X.:) but since the triangles P and Q are mutually equilateral, they must also (Prop. XIV.) be mutually equiangular; and, lastly, the angles being equal in the triangles P and Q, it follows (Prop. X.) that the sides are equal in their polar triangles A and B. Hence the mutually equiangular triangles A and B are at the same time mutually equilateral. Scholium. This proposition is not applicable to rectilineal triangles; in which equality among the angles indicates only proportionality among the sides. Nor is it difficult to account for the difference observable, in this respect, between spherical and rectilineal triangles. In the proposition now before us, as well as in Propositions XII, XIII, XIV, which treat of the comparison of triangles, it is expressly required that the arcs be traced on the same sphere, or on equal spheres. Now similar arcs are to each other as their radii; hence, on equal spheres, two triangles cannot be similar without being equal. Therefore it is not strange that equality among the angles should produce equality among the sides. The case would be different, if the triangles were drawn upon unequal spheres; there, the angles being equal, the triangles would be similar, and the homologous sides would be to each other as the radii of their spheres. PROPOSITION XVIII. THEOREM. The sum of all the angles in any spherical triangle is less than six right angles, and greater than two. For, in the first place, every angle of a spherical triangle is less than two right angles (see the following Scholium): hence the sum of all the three is less than six right angles. Secondly, the measure of each angle of a spherical triangle (Prop. X.) is equal to the semicircumference minus the correspond...
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