B

The folution of the cafes of right-angled Spherical

triangles.

Given Sought

4

Solution.

The hyp. The oppo-As radius: fine hyp. AC ::

AC and one

fite leg
BC

fine A fine BC (by the for

mer part of Theor. I.)

As radius: co-fine of A: tang. AC: tang. AB (by the latter part of Theor. 1.)

The hyp.
3 AC and one
angle A

The other

As radius

co-fine of AC

anglè C

tang. A

co-tang. C (by)

Theor. 5.)

The hyp.

The other

AC and one leg BC

As co-fine AB radius :: co-fine AC: co-fine BC (by Theor. 2.)

The hyp. The oppo-As fine AC radius:: fine AC and one fite angle AB fine C (by the former

leg AB

As tang. AC: tang. AB:
: co-fine A (b
radius

A

Theor. I.)

The other [As radius: fine AB :: tan

leg BC gent A tangent BC (by Theor. 4.)

Cafe,

Cafe.

8

Given

Sought

Solution.

One leg The oppo-As radius fine A coAB and the fite angle fine of AB: co-fine of C (by

adjacent
angle A

C

The hyp.
AC

Theor. 3.)

As co line of A; radius :: tang. AB tang. AC (by Theor. I.)

The other

BC and the

leg AB

As tang. A tang. BC:: radius: fine AB (by Theor. 4.)

10

oppofite

Both legs
AB and BC

Both angles 15 A and C

Both angles 16 A and C

co-fine of A: fine C (by, Theor. 3.)

As fin. A: fin. BC:: radius fine AC (by Theor. 1.)

As radius co-fine AB : co-fine BC: co-fine AC (by Theor. 2.)

:

An angle, As fine AB radius: tang. fuppofe ABC: tang. A (by Theor. 4•)

ra

A leg, As fin. A co fine C::
fuppofe dius co-fine AB (by Theor.
AB
13.)

The hyp.
AC

As tang. A co-tang. C:: radius co-fine AC (by Thear 5.).

Note, The 10th, 11th, and 12th cafes are ambiguous; fince it cannot be determined, by the data, whether AB, C, and AC,

be

greater or less than 90 degrees each.

D 2

The

44

The folution of the cafes of oblique Spherical tri

angles.

D

Either of the
other angles,
suppose B

Solution.

As fine BC: fine A: : fine AC : fine B (by Cor. 1. to Theor. 1.) Note, This cafe is ambiguous when BC is lefs than AC; fince it cannot be determined from the data whether B be acute or obtufe.

Upon AB produced (if need be) let
fall the perpendicular CD; then (by
Theor. 5.) rad. co-fine AC : :
tang. A co-tang. ACD; but (by
Cor. 2. to Theor. 1.) as tang. BC:
tang. AC: co-fine ACD: co-fine
BCD. Whence ACB ACD +
BCD is known.

As rad. : co-fine A tang. AC:
tang. AD (by Theor. 1.) and (by Cor.
to Theor. 2.) as co-fine AC: co-fine
BC :: co-fine AD: co-fine BD.
Note. This and the last cafe are
both ambiguous when the first is fo.
As rad. co-fin. A tang. AC:
tang. AD (by Theor. 1.) whence BD
is alfo known; then (by Carol. to
Theor. 2.) as co fine AD: co-fine
BD: co-fine AC: co fine BC.
As rad. : co-fine A: tang. AC:
tang. AD (by Theor. 1.) whence BD
is known; then (by Cor. to Theor. 4.)
as fine BD fine AD :: tang. A

tang. B.

Cafe.

Two angles A, ACB and the 6 fide AC betwixt them.

Sought

The other
angle B

Two angles A, Either of the ACB and the other fides, 7 fide AC be- fuppofe BC twixt them.

Solution.

As rad. co-fine AC :: tang A:
co-tang. ACD (by Theor. 5.) whence
BCD is alfo known; then (by Cor.
to Theor. 3.) as fine ACD fine
BCD:: co-fine A: co-fine B.
As rad. : co-fine AC:: tang. A:
co-tang. ACD (by Theor. 5.) whence
BCD is alfo known; then as co-fine
of BCD: co-fine ACD :: tan. AC
: tang. BC (by Cor. 2. to Theor. 1.)

Two angles A, The fide BC As fine B: fine AC :: fine A: fine
B and a fide oppofite the BC (by Cor. 1. to Theor. 1.)

AC oppofite to

Jone of them.

other

Two angles A, The fide AB As rad.: co-fine A:: tang. AC
Band a fide AC betwixt them.jtang. AD (by Theor. 1.) and as tan.

B: tang. A :: fine AD: fine BD (by Cor. to Theor. 4.) whence AB is also known.

As rad.: co-fine AC :: tang. A: co-tang. ACD (by Theor. 5.) and as co-fine A: co-fine B:: fine ACD : fine BCD (by Cor. to Theor. 3.) whence ACB is alfo know!).

As tang. AB: tang. AC+ BC

2

: tang. DE, the

pofe AC

: tan.

included by the perpendicular and a line bifecting the vertical angle; whence ACD is also known; then (by Theor. 5.) tang. A co-tang. ACD:: rad.: co-fine AC.

Note, In letting fall your perpendicular, let it always be from the end of a given fide and oppofite to a given angle.

:

Of the nature and conftruction of Logarithms, with their application to the doctrine of Triangles.

A

S the bufinefs of trigonometry is wonderfully facilitated by the application of logarithms; which are a fet of artificial numbers, fo proportioned among themfelves and adapted to the natural numbers 2, 3, 4, 5, &c. as to perform the fame things by addition and fubtraction, only, as thefe do by multiplication and divifion: I fhall here, for the fake of the young beginner (for whom this fall tract is chiefly intended) add a few pages upon this fubject. But, firft of all, it will be necefiary to premife fomething, in general, with regard to the indices of a geometrical progreffion, whereof logarithms are a particular fpe

cies.

Let, therefore, 1, a, a2, a3, aa, a3, a3‚ àa1‚, &c. be a geometrical progreffion whofe firft term is unity, and common ratio any given quantity a. Then it is manifeft,

1. That, the fum of the indices of any two terms of the progreffion is equal to the index of the product of thofe terms. Thus 2+3 (5) is the index of a'a', or a; and 3+ 4 (7) is the index of a xat, or a". This is univerfally demonftrated P. 19. of my book of Algebra.

in

2. That, the difference of the indices of any twa terms of the prografion is equal to the index of the quotient of one of them divided by the other. Thus

5-3 is the index of or a. Which is only

the converfe of the preceding article.

.