to obtain the altitude a.b. from 2. stations c. & d. each in the plane of the axis of a mountain a.b. and themselves on an inclined plane.

a. the apex, or summit of the mountain.

a.b. it’s Axis. b. it’s base in the level of a river.

c. the plane of the instrument at it’s 1st station.

d. it’s plane at it’s 2d station.

c.d. the inclined plane measured.

de. the same continued to the axis at e.

df. the horizontal plane in the level of d.

cg. the horizontal plane in the level of c.

ch. the height of the instrument above the surface of the river in b.h.

di. the height of the instrument at d. above the level of c.g.

dk. continuation of ad. to the plane of c.g.

at c.	observe	the vertical ∠ acg. and rhumb of c.g.
		and the height of the instrument above the water, to wit, ch.
		the vertical ∠ icd1
	measure	c.d.
at d.	observe	the ∠ of altitude a.d.f.

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In the △ cdi.	to reduce cd. to ci. to find di. and c.i.
	given cd and all the angles
in the △ dik.	to find i.k. in order to reduce c.i. to ck. (given di. & all the ∠s)
in the △ ack.	to find ak. (given ck. and all the ∠s)
in the △ akg.	to find ag. and gk. (given ak. and all the ∠s)
to ag. add ch. or gb. for the whole altitude ab.

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To ascertain the height and distance of an inaccessible object (e.g. the summit of the sharp, or South peak of Otter) by a measured base between two stations on an inclined plane, nearly parallel with the mountain.

let l. be the apex, or summit of the mountain.

q. a point in it’s axis in the level of it’s base q.r.s & of the surface of Otter river

m. the plane in which the instrument is placed at it’s 2d and highet2 station.

n. it’s plane at the 1st3 or lowest station.

m.n. the inclined plane measured from station to station.

m.o.u. the horizontal plane of m.

n.t.p. the horizontal plane of n.

m.t. the difference of level between these two planes

m.r. the height of m. above the base, or water of the river.

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l.o.p.q.		are vertically coincident. then
m.t.r
u.n.s
m.u.o		are 3. horizontal △s vertically coincident, & consequently similar & equal.
t.n.p.
r.s.q.

in this figure the line p.n. is assumed longer than o.m. that it may project in the profile, & shew in perspective the parallelograms m.u.s.r. & m.u.n.t. the diagonal m.n. & △ l.m.n.

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at m.	observe	the vertical ∠s l.m.o. & u.m.n.
		the horizontal ∠ o.m.u. & rhumbs of m.o. & m.u.
		the height of m.r.
		[the altitude, horizontal ∠ & rhumb of the summit of the North peak L. for another object]
	measure the inclined plane m.n.
at n.	observe	the vertical ∠ l.n.p.
		the horizontl ∠ t.n.p. & rhumb of n.p.
		[the altitude, horizontal ∠ & rhumb of the summit of the North peak L.]

with these data proceed as follows.

to reduce n.m. to n.t. = u.m. & to find m.t. in the △ m.n.t. given m.n. & all the ∠s

to find t:p = m.o. and n.p.	in the horizontl △ n.p.t. given n.t. & all the ∠s
to find l.o.	in the vertical △ l.m.o. given m.o. & all the ∠s.
to find l.p.	in the vertical △ l.n.p. given n.p. & all the ∠s.

add l.o. to o.q. and l.p. to p.q. and take the mean for the true height l.q.

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The observations of the North peak from m. and n. may be used to obtain the distance between the two peaks, according to the diagram & method of Fig. 1. page 1. by reducing m.n. to t.n. which will then correspond with a.b. there, as c.d. there will correspond with l. and L. in these observations.