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An introduction to the theory and practice of plane and spherical trigonometry, and the stereographic projection of the sphere; including the theory of navigation ... - Softcover
Synopsis
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. 1810 Excerpt: ...OF RIGHT-ANGLED SPHERICAL TRIANGLES, AND THR LOGARITHMICAL SOLUTIONS OF ALL THE CASES. PROPOSITION XXI. (H) In any right-angled spherical triangle. Radius, is to the sine of any side; as the tangent of the ad-, jacent angle, is to the tangent of the opposite side., Demonstration. Let Aec be a spherical triangle, right-angled at A. rad: sine Ab:: tang /. B: tang Ac. For, let D be the centre of the sphere, and draw the radii DC, Da, and Db; also from the right-angle A, in the plane Dba, draw At perpendicular to Db, and it will be the sine of the arc Ab. At the point E, in the plane Dbc, draw Ef perpendicular to Db, then will the angle Fea be the inclination of the planes Dba and Dbc, and consequently (D. 129.) equal to the spherical angle CBA. Draw Af from the point A, a tangent to the arc AC, and produce the radius DC to F; then since the arc AC is perpendicular to the plane Dba (for by hypothesis it cuts the arc Ab at rightangles), Af will be perpendicular to Ae. Because Ac is less than a quadrant, and Da, Db, and Ah, are in the same plane Dba, and that Ef is at right-angles to DB (by construction); DC cannot likewise be at right-angles to DB. Therefore DC and Ef being in the same plane Dbc, and not parallel to each Other, must meet. The plane triangle Fae is right-angled at A; the perpendicular Af is a tangent to the spherical perpendicular Ac-, the base Ak is the sine of the spherical base Ab; and the angle Fea is equal to the spherical angle Abc. Hence, rad: Ae:: tang Fea: Af. That is, rad: sine Ab:: tang Abc: tang Ac, Q.e.d. (I) Scholium. The foregoing proposition is of considerable importance in spherical trigonometry, and therefore ought to be clearly understood.--It will perhaps be of advantage to some students, to illustrate the nature of it in a me...
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