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BC = a.



sin. a sin. B

sin. (B-a)
sin.c. cos B

sin. (6-a) Let the Z CBD be observed = y

DC = BC. tan. y

sin. a . cos.ß. tan. y

sin. (1-7) CP and CD being thus determined, DP is known

3. Suppose the plane on which the observer is stationed not to be horizontal. The inclination (8) of AB to the horizon may be determined by a level. Let the angles EAP, FBP be observed, = Q and ß as before. CP=g, CB= .

sin. CBP sin. PCB sin.(B-0) ),


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y = PB

= PB

cos. 5



sin. PAB
also PB = AB.

sin. A PB
sin. (a–0);
sin. (B-a)

sin. («-8) sin. (B-8)
:: y = a

ccs. sin. (B-a)
log: y = log. a + log, sin. (6-0) + log: sin. (B-d)

log, cos. c-logsin. (-a);

sin. CPB
Also x = BP

sin. PCB
sin. (ac)

cos. ß
log, r = log, a + log, sin. («-0) + log, cos. B

- log, sin. (B – a) log, cos. d.

cos. à


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AD= y cot.a; BD = y cot. B; DC=y cot. y. But

A B + B D - A D cos. ABD =


B C + BD - DC cos. DBC=

and cos. A B D =

cos. DBC
A B2+ BD— A D2 DC2 - BCR - BD2

Substituting the values of these quantities as determined above, we find

ab (a + b) =

a (cot.) - (a + b) (cot. 3) + 6 (cot. a) 5. If the side of a hill form an inclined plane whose inclination to the horizon is = a, to find the direction in which a rail-road must run along the side of the hill, in order that it may have an ascent of 1 foot in every 100 feet.

P Let A P represent the rail-road, AN the foot of the hill, PN a line on the side of the hill perpendicular to AN; PN vertical, and NM horizontal and perpendicular to AN. AM will be the projection of AP on the horizontal plane, and the angle MAN will determine the direction of the rail-road. Let this angle 0; also PNM will be the inclination of the hill Now if PAM : B

MP 1 sin. B E

.01 AP 100

MN also sin. O =


MN cot. a


MP tan. B


cot. a

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sin. o

tan. 8
sin. O tan.ß. cot. O

(sin. 3)* (.01)
(tan. B)* =

1-(sin. B) 1 - 1.01) : tan. B =.01 {1. .0001}+

=.01 {1+.00005} nearly

.. sin. 03.0100005 cota

log: sin. 0 = logt .0100005 + log: cot. a. 6. To find the distance of two objects which cannot be approached, situated in the same plane with the observer.

Let PQ be the two objects; A B a line = a, which the observer can measure, lying in the same plane with PQ. At A let the angle PAQ PAB be observed respectively = a and B; and at 8 let the angles P AB, A BQ, be observed respectively equal y and d.

PA sin. PBA
PB sin. APB



AQ = a.

= a.

Ρ Α Ξ α

.. log, PA = log: a + log, sin, y
sin. (B+r)
sin. ABQ

+A.C. log, sin. B+y-10 sin. A QB

sin. (8 + Q AB)

sin. 8

sin. (B - at 8) :: log: AQ = log, a + log, sin. 8 + A.C.log: sin. (B-a + d) – 10.

Also P Q = AP+AQ - 2 AP.AQ cos. PAQ.

From the expression PQ must be determined exactly as in Arts. (56, 57, or 58,) substituting the values AP and A Q as found above.

The distances AP, BQ, BP, are easily found from the triangles A BP, A B Q, and then the positions of P and Q, as well as their distances, are determined. If there be any other object, R situated in the same plane, its position may be determined in the same manner, either by observing it from A and B (if visible from thence), or from any other stations, whose positions have been previously found. Hence it will easily be seen how a tract of country may be mapped, provided it be not too extensive; but if the extent be considerable, we cannot, on account of the spherical form of the earth, consider objects situated on its surface ås lying in one plane, with due regard to the accuracy of our results. Consequently, extensive and accurate trigonometrical surveys can only be conducted by means of spherical trigonometry. A complete account of the method of doing this, and of the use and construction of the instruments for this purpose, will be given in a subsequent part of this work.

We shall here conclude the first part of our treatise on Trigonometry. A knowledge of the theorems contained in it, with that of their appli

cation to Plane Trigonometry, is essential to enable the student to pass on to the differential calculus and its simpler applications. The remainder of the subject, as we have already observed in the Introduction, may be better studied in more immediate connexion with some of the higher branches of analysis ; and on this account we conceive that it ought not, in a well-arranged course of mathematical reading, to precede, but to follow the calculus. This first part may be considered as forming one complete division of the subject; and therefore the reason above stated is sufficient for the postponement of the higher branches of it till after our treatise on the Differential Calculus shall have made its appearance.

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