Take BAD from BAC,

ACD;

find one of the other sides AD. the remainder, DAC is the angle included between two known sides AD, AC; from which the angles ADC and ACD may be found, by chap. iii. case 2. The angle CAP = 180° — (APC + ACD). Also, BCP = BCA and PBC = ABC + PBA = ABC + sup. ADC. Hence, the three required distances are found by these proportions. As sin APC: AC:: sin PAC: PC, and :: sin PCA: PA; and lastly, as sin BPC: BC :: sin BCP: BP. The results of the computation are, PA = 709.33, pc = 1042.66, PB 934 yards.

***The computation of problems of this kind, however, may be a little shortened by means of the following

General Investigation.

Put AC = a, BC = b, APC = p, bpc = p′, acb = c, and let there be taken for unknown quantities PAC = x, PBC=y. The triangles PAC and PBC give

sin APC: sin CAP:: AC: CP

and sin BPC: sin CBP :: BC: CP;

y = R x, and consequently,

a sin p' sin x - b sin P (sin R cos x cos R sin x) = 0. Dividing by sin x there results,

This expression being separated into two parts, we have

Hence, a being thus determined, we get y from the equation y = R x, and cr from either of the expressions above given.

Note. It will be a useful exercise for the student, to work out the computation by both these methods. The comparison of the results will serve to give him confidence in the deductions from the analytical investigations.

EXAMPLE XVIII.

It is required to find the distances from Edystone light-house to Plymouth, Start Point, and the Lizard, respectively, from the following data:

The distance from

Plymouth to Lizard.... 607
Lizard to Start Point.. 70 miles.
Start Point to Plymouth 20)

bears from Edystone rock

Ans. From Edystone to Lizard..... 53.04

North.

W.S.W.

E. by N. miles

to Plymouth.. 14-333 ditto.

to Start Point.. 17.36 ditto.

Remark. The general problem of which the last two examples are particular cases, was originally proposed by Richard Townley, Esq. and solved by Mr. John Collins, in the Philosophical Transactions, N° 69, A. D. 1671. See New Abridgement, vol. i. p. 563.

The six cases there considered are, 1. When the station is out of the triangle made by the objects, but in one of its sides produced. 2. When the station is in one of the sides of the triangle. 3. When the three objects lie in a right line. 4. When the station is not within the triangle formed by the objects. 5. When the station is within that triangle. 6. When, by reason of the rela tion of the sides and angles, the points c and d (see the preceding diagram) fall so near together as to make the continuation to P of doubtful accuracy. To these, later writers on trigonometry have added another case, viz. 7. When the point c falls between the line AB and the station P.

It will be advantageous to the student, to modify the construction and computation to suit all these cases.

CHAPTER VI.

Spherical Trigonometry.

SECTION 1.

Fundamental Principles, General Properties, and

Formula.

1. THREE planes, AOC, AOB, BOC, all of which pass

through the centre o of a sphere, intersect the surface of that sphere in portions of great circles which form a spherical triangle ABC. Thus also is constituted the spherical pyramid or tetraedron which has for its base the triangle ABC, and for its vertex the centre o of the sphere. The angle a

C

a

B

of the triangle, is the same as the diedral angle, between the two planes BAO, CAO; it is, also, the angle formed by the tangents to the two arcs AC, AB. The like may be said of the other angles. The sides are manifestly the measures of dependent plane angles, viz. a the measure of the angle coв, b the measure of Coa, c the measure of AOB.

2. A right angled spherical triangle has one right angle; the sides about the right angle are called legs; the side opposite to the right angle is called the hypothenuse.

3. A quadrantal spherical triangle has one side equal to 90°, or is a quarter of a great circle.

4. An isosceles, or an equilateral spherical triangle, has respectively two sides or three sides equal.

5. When the sides of a triangle are each 90°, it is not only an equilateral, but a quadrantal, and a right angled triangle. All its angles as well as its sides are equal; and these sides may any of them be regarded as an hypothenuse, any of them as legs. Such is the case with the triangle that would be formed on a celestial or terrestrial globe, by the horizon, the brazen meridian, and a quadrant of altitude, fixed at the zenith, and passing through the east or west point.

6. Two arcs or angles, when compared together, are said to be alike, or of the same affection, when both are less, or both greater than 90°. They are said to be unlike, or of different affections, when one is greater and the other less than 90°.

7. Every spherical triangle has three sides and three angles; of which, if any three be given, the remaining three may be found.

8. In plane trigonometry, the knowledge of the three angles is not sufficient for ascertaining the sides (chap. i. 6): but, in spherical trigonometry, the sides may always be determined when the angles are known. In plane triangles, again, two angles always determine the third; in spherical triangles they never do. So, farther,

the surface of a plane triangle cannot be determined from its angles merely; that of a spherical triangle always can.

9. A line perpendicular to the plane of a great circle, passing through the centre of the sphere, and terminated by two points diametrically opposite, at its surface, is called the axis of such circle; and the extremities of the axis, or the points where it meets the surface, are the poles of that circle.

If we conceive any number of less circles, each parallel to the said great circle, this axis will be perpendicular to them likewise; and the poles of the great circle will be their poles.

10. Hence, each pole of a great circle is 90° distant from every point in its circumference; and all the arcs drawn from either pole of a little circle to its circumference, are equal to each other.

11. It likewise follows that all the arcs of great circles, drawn through the poles of another great circle, are perpendicular to it; for, since they are great circles by the supposition, they all pass through the centre of the sphere, and consequently through the axis of the said circle. The same thing may be affirmed in reference to small circles.

12. Hence, in order to find the poles of any circle, it is merely necessary to describe, upon the surface of the sphere, two great circles perpendicular to the plane of the former, the points where these circles intersect each other will be the poles.required.

13. All great circles bisect each other. For, as they have a common centre, their common section will be a diameter; and that manifestly bisects them.

14. The small circles of the sphere do not fall under consideration in spherical trigonometry; but such only as have the same centre with the sphere itself. Hence appears the reason why spherical trigonometry is of such great use in practical astronomy, the apparent heavens being regarded as in the shape of a concave