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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.
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.
For example, let the curve AMBM be a circle, AB its diameter, C the center, and AD an indefinite line, drawn from A perpendicular to AB. Suppose now any point M of the circum- A
PC ference to be referred to the two lines AB, AD, by the lines MP, ME parallel to AD, AB. Then will MP be an ordinate ; and ME, or its equal AP, an abscissa to the point M. The lines AD, AB, are the axes of the ordinates and abscissæ.
Call the diameter AB, 2a ; AP, x; PM, y. Then by (Euclid 8.6) PM’=AP X PB, or in symbols y*= (2-x)=2 ax-xx. As this relation is the same for any point M whatever, this is the equation of the circle.
3. Any expression in which a quantity enters, is for the sake of conciseness, called a function of that quantity. Thus, for example, we say, that the equation of the circle expresses the constant equality of a function of each ordinate (its square), to a function of the corresponding abscissa (its product by the remainder of the diameter).
The abscissæ, and corresponding ordinates of a curve, are called the co-ordinates; and as the length of these lines varies at each instant, they are called variable, or indeterminate quantities, in opposition to the constant or determinate quantities.
The point from which we begin to compute the abscissæ, is called the origin of the abscisse. We may fix this point at pleasure before we commence investigating the equation of the co-ordinates ; but this position once determined, it must always remain the same during the details of the same calculation.
Usually the origin of the abscissæ is placed at either the summit or center of the curve.
And as at setting out from their origin, we may take the coordinates on two opposite sides, it is customary to designate those on one side by the sign +, and those on the other by the sign —; so that an abscissa is considered as positive, when it is on that part of the axis which is considered as positive. The choice of this part is altogether arbitrary; but when once made, it must be adhered to.
4. The ordinates may be either perpendicular or oblique to the line of the abscissæ, provided that they are parallel to one another. Generally they are supposed to be perpendicular, and are distinguished as positive or negative, according as they are on the one or the other side of the axis of the abscissæ. Sometimes they proceed from a fixed point, as we shall hereafter explain.
These things premised, let us describe the curve whose equation is y'=2 az-ix. We already know (2) that it is the circumference of a circle whose diameter is 2 a; but if we were ignorant of this, the construction of this equation would soon inform us of it.