5. In the oblique-angled spherical triangle ABC, given the angle B=52° 20', the angle C=63° 40, and the side AB=830 25'; required the rest? Answer, AC=61° 19' 53", BC=118° 21' 25, LA=127° 26° 47'. 6. In the oblique-angled spherical triangle ABC, given the angle C=48° 20', the angle B=125° 20', and the side AC=114° 30° ; required the rest ? Answer, AB=56° 40', BC=83° 11', A-62° 54'. 7. In the oblique-angled spherical triangle ABC, given the angles B–53° 30°, C=60° 15', and the side BC=115° 40'; required the rest ? Answer, AB=75° 56' 31", AC=67° 2' 39", A=126° 13' 35". 8. In the oblique-angled spherical triangle ABC, suppose the angle A=125° 20', the angle C=48° 30', and the side AC=83° 12', required the rest? Answer, BC=114° 30', AB=65° 30', _ B=62° 54'. 9. In the oblique-angled spherical triangle ABC, given AB=81° 10, AC=600 20, BC=112° 25'; required the angles ? Answer, L A=122. 11' 6", B=520 42' 11", LC=64° 46' 36". 10. In the oblique-angled spherical triangle ABC, given AB=56° 40', AC=83° 13', and BC=114° 30°; required the angle A opposite the side BC? Answer, LA=1250 18' 56'. 11. In the oblique-angled spherical triangle ABC, given the angle A=125° 20', the angle B=48° 30', and the angle C=68° 54'; required the sides ? Answer, AB=83° 12' 4", BC=114° 29' 56", AC=560 39' 29". 12. In the oblique-angled spherical triangle ABC, given the angles A=129° 30°, B--540 36', and C=63° 5'; required the sides ? Answer, AB--82° 19', BC=120° 57'4", AC=64° 55' 36". NAPER'S RULES OF THE CIRCULAR PARTS. The proportions upon which the solution of the various cases of right-angled spherical triangles depend, are simple, and perfectly adapted to logarithmic computation ; but they are not easily remembered. All these cases may be solved by Naper's Rules of the Circular Parts, which supply an artificial memory to the computist; and in the whole compass of the mathematical science it will not be easy to find rules equally ingenious and conducive to facility and brevity of computation. The nature and application of these rules will readily be understood from the following explanation: 3. To find the angle B. Here the angle B is the middle part, and the leg BC, and the hypothenuse AB are adjacent parts; therefore rad * cos B=tang BC X cot AB whence B is found=65° 45' 57". By the above rule of Naper's we are enabled to solve all the cases of right-angled spherical triangles; and also those cases of oblique-angled spherical triangles in which we have directed a perpendicular to be drawn from an angle to the opposite side, provided that two of the given parts remain in one of the two triangles thus formed. Trigonometry is a branch of mathematical science which is indispensible on the calculations of remote and inaccessible objects; and hence its use in geography, in ascertaining the various distances and position of places on the earth ; in navigation, in directing the course, the latitude and longitude of a ship. Spherical Trigonometry is particularly applied to the sublime science of astronomy, in discovering the positions, magnitudes, and distances of the heavenly bodies. It may also be applied to the useful arts, as in architecture the theorems of Naper will be found useful in ascertaining the angles which two adjacent planes of a roof at a hip make with each other, the inclination of the planes being given to the horizon. In short, a catalogue of its applications would be too formidable to be inserted in this place. CONIC SECTIONS. 1. By Conic Sections are understood the sections produced from a cone by cutting it with a plane. For example, the circle is a conic section, because if we cut a right cone by a plane, parallel to its base, the section will be circle. The triangle is also a conic section, because if we cut a right cone by a plane perpendicular to its base, and passing through its vertex, the section will be a triangle. But the name of conic sections is more particularly applied to three other sections of a cone, of which we shall explain the origin and properties, after having made known the manner of treating them analytically Descartes first conceived the idea of applying Algebra to Geometry. The utility of this application soon became apparent, and the geometers who succeeded him, availed themselves of this discovery to such an extent, that it will for ever be celebrated for its great fecundity. The principal uses of this doctrine are perceived in its application to the theory of curves, the study of which is indispensable to those who desire to obtain a thorough knowledge of the Physico Mathematical sciences. 2. The object of this theory is to express, by equations, the laws, according to which we suppose any given curves have been described ; and reciprocally, to direct the Analyst, either in the description of the curves of which he knows the equations, or in the investigation of the properties of those curves. For this purpose, each point of the curve that we desire to trace, is referred to two right lines, of which one is called the line, or axis of the abscissæ ; the other, the line, or axis of the ordinates. We then determine the ratio which obtains between the abscissæ and ordinates, and the analytical expression of this ratio gives the equation of the curve. |