Cases 1, 2. Making AB the middle part.

As rad. sine AB :: tang. A: tang. BC (Theor. 4.); whence, by inversion and alternation, sine AB : tang. BC :: rad. : tang. A :: co-tang. A consequently r. sine AB

radius (Cor. 5. Prop. 1.); co-tang. A x tang. BC.

Again, r. sine C: sine AC: sine AB (Theor. 1.); whence r. x sine AB = sine C × sine AC.

In the same manner the theorems may be proved when BC is made middle part.

Cases 3, 4. Making the complement of A middle part.

:

As rad. co-sine A:: tang. AC: tang. AB (Theor. 1.); and, by inversion and alternation, co-sine A: tang. AB::

rad. : tang. AC :: co-tang. AC: rad. (Cor. 5. Prop. 1.); whence rad. x co-sine A co-tang. AC x tang. AB.

Again, rad. sine C

:

co-sine BC: co-sine A (Theor.

3.); whence rad. x co-sine A = sine C x co-sine BC.

In like manner the theorems may be proved when the complement of C is middle part.

Case 5. Complement of AC middle part.

:

As rad. co-sine AC:: tang. A: co-tang. C (Theor. 5.); whence, by inversion and alternation, co-sine AC co-tang. C rad. tang. A co-tang. A consequently rad. x co-sine AC

rad. (Cor. 5. Prop. 1.);

co-tang. A x co-tang. C.

Again, rad. co-sine AB: co-sine BC: co-sine AC (Theor. 2.), and therefore rad. x co-sine AC co-sine AB X co-sine BC.

=

When any two of these circular parts are given to find a third, we must assume such one middle part as will make the other two either both adjacent or both opposite extremes.

The solution of the cases of oblique spherical triangles.

Two sides AC, The included Upon AB produced (if need be) let fall angle ACB the perpendicular CD; then (by Theor.

BC, and an an

gle A opposite 2 to one of them)

5.) rad. co-sine AC tang. A co. tang. ACD; but (by Cor. 2. to Theor. 1.) as tang BC: tang. AC:: co-sine ACD : co-sine BCD. Whence ACB ACDBCD is known.

As rad. : co-sine A:: tang, AC tang. AD (by Theor. 1.); and (by Cor. to Theor. 2.) as co-sine AC co-sine BC: cosine AD: co-sine BD. Note. This and the last case are both ambiguous when the first is so.

As rad. : co-sine A: tang AC tang. AD (by Theor. 1.); whence BD is also known; then (by Cor. to Theor. 2.) as co-sine AD: co-sine BD; co-sine AC co-sine BC.

As rad. : co-sine A: tang. AC: tang. AD (by Thror. 1.); whence BD is known; then (by Cor, to Theor. 4) as sine BD sine AD: tang. A tang.

B.

As rad. : co sine AC : : tang. A : có tang. ACD (by Theor. 5.); whence BCD is also known; then (by Cor. to Theor. 3.) as sine ACD: sine BCD: : co-sine A: co.sine B.

As rad. co sine AC: tang. A cotang. ACD (by Theor.5.); whence BCD is also known; then as co sine of BCD : co sine ACD:: tang, AC: tang. BC |(by Cor. 2. to Theor. 1.)

Case.

Given

Sought

Solution.

9

:

Two angles A, The side BC As sine B: sine AC sine A: sine
B, and a side opposite the BC (by Cor. 1. to Theor. 1.)

Two angles A, The side ABAs rad. : co-sine A: :: tang, AC tang.
B, and a side betwixt them AD (by Theor. 1.); and as tang. B:
AC opposite to
tang. A sine AD: sine BD (by Cor.
to Theor. 4.); whence AB is also known.

one of them

Two angles A,

The other

As rad.: co-sine AC

tang. A co

B, and a side angle ACB tang. ACD (by Theor. 5.); and as co

10 AC opposite to

one of them

sine A co-sine B:: sine ACD: sine BCD (by Cor. to Theor. 3.); whence ACB is also known.

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

Of the Nature and Construction of Logarithms, with their application to the doctrine of Triangles.

AS the business of trigonometry is wonderfully facilitated by the application of logarithms; which are a set of artificial numbers, so proportioned among themselves and adapted to the natural numbers 2, 3, 4, 5, &c. as to perform the same things by addition and subtraction, only, as these do by multiplication and division: I shall here, for the sake of the young beginner (for whom this small tract is chiefly intended), add a few pages upon this subject. But, first of all, it will be necessary to premise something, in general, with regard to the indices of a geometrical progression, whereof logarithms are a particular species.

Let, therefore, 1, a, a2, à3, aa, a3, ao, a", &c. be a geometrical progression, whose first term is unity, and common ratio any given quantity a. Then it is manifest,

1. That the sum of the indices of any two terms of the progression is equal to the index of the product of those terms. Thus 2+3 (5) is the index of a2 × a3, or a5; and 3 + 4 (= 7) is the index of a3 × a1, or a". This is universally demonstrated in p. 19 of my book of Algebra.

2. That the difference of the indices of any two terms of the progression is equal to the index of the quotient of one of them divided by the other. Thus 5 3 is the index of