Elements of Plane and Spherical Trigonometry: Written Originally in Russian, and Translated Into English by the Authors, Basil Nikitin, M.A. Inspector, and Prochor Souvoroff, M.A. Vice-Inspector, of the Imperial Academy of Cadets at Cronstadt
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Page 1 - The Circumference of every Circle is fuppofed to be divided into 360 equal Parts...
Page 59 - AB be either of the fides, the fine of the fidcAB will be to the radius, as the tangent of the other fide AC to the tangent of the angle ABC, oppofite to AC. Let D be the centre of the fphere ; join AD, BD, CD, and let AE be drawn perpendicular to BD, which therefore will be the fine of the arch AB, and from the point E, let there be drawn in the plane BDC the tIraight line EF at right angles to BD, meeting DC in F, and let AF be joined.
Page 2 - A straight line CD drawn through C, one of the extremities of the arch AC, perpendicular upon the diameter passing through the other extremity A, is called the Sine of the arch AC, or of the angle ABC, of which it is the measure.
Page 48 - ... 36th Prop, may be demonstrated without the 35th. PROP. XXXVI. B. XI. TACQUET in his Euclid demonstrates this proposition without the help of the 35th ; but it is plain that the solids mentioned in the Greek text in the enunciation of the proposition as equiangular, are such that their solid angles are contained by three plane angles equal to one another each to each ; as is evident from the construction. Now Tacquet does not demonstrate, but assumes these solid angles to be equal to one another;...
Page 66 - BFC, (as the fquare of AD is to the fquare of DG, that is) as the fquare of the radius to the fquare of the tangent of the angle DAG, that is, the half of BAC : But HA is half the perimeter of the triangle ABC, and AD is the...
Page 6 - ... complement of any angle is called the Cosine, Cotangent, or Cosecant of that angle. Thus, let CL or DB, which is equal to CL, be the sine of the angle CBH ; HK the tangent, and BK the secant of the same angle : CL or BD is the cosine, HK the cotangent, and BK the cosecant of the angle ABC. COR. 1. The radius is a mean proportional between the tangent and the cotangent of any angle ABC ; that is, tan. ABC X cot.
Page 19 - IN a plain triangle, the fum of any two fides is to their difference, as the tangent of half the fum of the angles at the bafe, to the tangent of half their difference.
Page 47 - NY two ficles of a fpherical triangle are greater than the third. • Let ABC be a fpherical triangle, any two fides AB, BC will be greater than the other fide AC. , Let D be the centre of the fphere ; join DA, DB, DC. The folid angle at D is contained by three plane angles ADB, ADC, BDC; and by 20, n.