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(1.) The doctrine of Spherical Trigonometry is founded on that of Plane Trigonometry; which


best be learnt from those authors, whose main design is the explanation of its theory, and of its practical uses. ANI that is intended, therefore, by the following summary, is to recapitulate, and, in as concise a manner as can well be adopted, to demonstrate its principal propositions ; with the aim of reviving, rather than of communicating, a knowledge, which is indispensibly necessary in the perusal of the latter part of this Treatise.

(2.) Of the six constituent parts of a plane triangle, namely, the three sides, and the three angles, if there be given, either any two sides, and the included angle, or the three sides, or any one side together with any two angles, it is evident, from E. 4. 1. E. 8. 1. E. 26. 1. that the triangle is thereby determined: and it might, in reality, be constructed, by means of E. 23. 1. and 3. 1. E. 22. 1. E. 23. 1. and 33. 3. according to the case.

But, if two sides and an angle opposite to one of them be given, although the position of the third side be thereby given, its length is not absolutely determined.



Thus, if BD be made equal to one of the given sides, and the angle BDC to the given angle, a circle, described from B as a center, at a distance equal to the other given side, unless it touch DC, will cut it, necessarily, in two points ; let these points be C and C': Then if B, C and B, C' be joined, it is manifest, that the data may belong to either of the triangles CBD, CBD. Since, however, a circle cannot cut a straight line in more than two points, there cannot be more than two triangles to which the data may belong.

(3.) In pure Geometry, the given quantities are supposed to be actually delineated; and the only instruments which are required, in the subsequent operations, are the rule and compass. But, in mixt mathematics, given quantities are more frequently đenoted by numerical

the same,

values. A given line, it is evident, may be expressed by the number of equal parts, of a specific magnitude, which it contains : and, since (E. 33. 6.) arches of any, circle may be made the measures of any given angles, it is plain, that given angles may be expressed by the number of equal parts, contained in the arches which measure them; the whole circumference being supposed to be divided into some definite number of such equal parts.

The Greeks supposed the whole circumference to be divided into 360 equal parts, called degrees, and marked 360°; each degree into 60 equal parts called minutes, and marked 60', each minute into 60 equal parts called seconds, and marked 60"; and so on. No geometrical construction can, indeed, be quoted, by means of which this division might actually be effected: but, it is mani. festly possible ; and it may, therefore, be supposed to be made; which is all that the reader is required to grant.

If the whole circumference had been supposed to be divided into 384°, each degree into 64', each minute into 64", and so on, such a division could have really been made, by the help of E. 15. 4. and 9. 1*. The Grecian mode of division, however, is still generally followed; although the dividing the quadrant into 100 equal parts, instead of 90, which has been partially adopted in France, greatly simplifies and facilitates the numerical calculations of Trigonometry.

* This mode of dividing the circle would have been attended with one considerable practical advantage. It would have enabled the artist to graduate, with greater ease and greater precision, those instruments, by which angular distances are actually measured: for no construction can be more simple, or admit of greater exactness, than that, by which a hexagon is inscribed in a circle, which is the first operation in this case ; and all the subdivisions are then effected, merely by the bisection of arches. Such a graduation has, accordingly, in this country been applied to a quadrant. It has been conjectured, that the Greeks adopted the division of the


If it be granted, that the circumference of a circle can be so divided, it is manifest, that a numerical value may be assigned to any given angle. But, it becomes further nécessary, in Practical Geometry, to employ two additional instruments ; a scale of equal parts, and a scale of chords ; in order first to delineate the data, and afterwards to measure the quantities, determined by the construction of the problem. The first mentioned of these instruments needs no description : and it may be constructed by means of E. 10. 6.

The Scale of Chords is a straight line, on which is marked the lengths of the chords of all arches, that contain fewer degrees than 180. If, therefore, a scale of chords be given, the radius of the circle for which it was constructed is given ; because (E. 15. 4. Cor.) that radius is equal to the chord of an arch of 60°.

It is evident, also, that, from any given scale of

circle into 360°, from having observed, that the Sun described the 360th part of his apparent course in a day: and Ptolemy assigns, as a reason for the division of the degree into 60 equal parts, the multitude of the divisors of the number 60.

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