INTRODUCTION

ΤΟ

PART II.

ON THE PRINCIPLES OF PLANE TRIGONOMETRY.

(1.) THE doctrine of Spherical Trigonometry is founded on that of Plane Trigonometry; which may best be learnt from those authors, whose main design is the explanation of its theory, and of its practical uses. All 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.

B

K

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, C'BD. 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 denoted by numerical

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, the same, 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 manifestly 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

* 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

circle

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.

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.

chords might be constructed, another scale, adapted to a circle of any other given radius, by means of E. 10. 6.

The uses to which such an instrument may be applied are very obvious.

If, from the center of any proposed circular arch, or the summit of any proposed plane rectilineal angle, at a distance equal to the chord of 60°, a circle be described, and the scale of chords be applied to the two points in which its circumference is cut, by two straight lines, produced if necessary, which contain the proposed angle, or which are drawn, from the center, through the extremities of the proposed arch, the number of degrees will be indicated, which measure the proposed arch or angle.

And, conversely, on a given plane, an arch may be described, or an angle may be made, by the help of a scale of chords, which shall have for its measure any given number of degrees.

If, likewise, a scale of chords be constructed for a great circle, in a given sphere, any proposed arch of such a circle may thereby, very readily, be measured. For, the distance between the extremities of the arch may easily be taken, by the opening of a spherical compass, and may thus be transferred to the scale of chords. And, by a converse operation, there may be cut off, from the circumference of a given great circle of the sphere, an arch of any proposed number of degrees.