2d. By Calculation.

1. Making AC the radius, the required sides are found by these propositions, as in plate 4, case 1.

If from the sum of the second and third logs. that of the first be taken, the number will be the log. of the fourth; the number answering to which will be the thing required; but when the first log. is radius, or 10.000000, reject the first figure of the sum of the other two logs. (which is the same thing as to subtract 10.000000), and that will be the log. of the thing required. 2. Making AB the radius.

* For finding the logarithmic sine, co-sine, &c. of any number of degrees and minutes in the table, also the degrees, minutes, &c. of any logarithmic sine, co-sine, &c., the reader is referred to table 2, at the end of this treatise.

Or, having found one side, the other may be obtained by cor. 2, theo. 14, sect. 4.

3d. By Gunter's Scale.

The first and third terms in the foregoing proportions being of a like nature, and those of the second and fourth being also like to each other; and the proportions being direct ones; it follows, that if the third term be greater or less than the first, the fourth term will be also greater or less than the second: therefore the extent in your compasses from the first to the third term will reach from the second to the fourth.

Thus, to extend the first of the foregoing proportions;

1. Extend from 90° to 46° 30', on the line of sines; that distance will reach from 250, on the line of numbers, to 181, for BC.

2. Extend from 90° to 43° 30', on the line of sines; that distance will reach from 250, on the line of numbers, to 172, for AB.

If the first extent be from a greater to a less number; when you apply one point of the compasses to the second term, the other must be turned to a less; and the contrary.

By def. 20, sect. 4. The sine of 90° is equal to the radius; and the tangent of 45° is also equal to the radius; because if one angle of a right-angled triangle be 45°, the other will be also 45°; and thence (by the lemma preceding theo. 7, sect. 4) the tangent of 45° is equal to the radius: for this reason the line of numbers of 10.000000, the sine of 90°, and tangent of 45°, being all equal, terminate at the same end of the scale.

The first two statings of this case answer the question without a secant; the like will be also made evident in all the following cases.

4th. Solution by Natural Sines.

From the foregoing analogies, or statements, it is obvious that if the hypothenuse be multiplied by the natural sine of either of the acute angles, the product will be the length of the side opposite to that angle; and multiplied by the natural cosine of the same angle, the product will be the length of the other side, or that which is contiguous to the angle. Thus : The given angle=47° 30′

The base and angles given, to find the perpendicular and hypothenuse.

PL. 5. fig. 5.

In the triangle ABC, there is the angle A 42° 20′, and of course the angle C 47° 40′ (by cor. 2, theo. 5), and the side AB 190 given; to find BC and AC.

1st. By Construction.

Make the angle CAB (by prop. 16, sect. 4) in blank lines, as before. From a scale of equal parts lay 190 from A to B, on the point B erect a perpendicular BC (by prob. 5, sect. 4), the point where this cuts the other blank line of the angle will be C; so is the triangle ABC constructed: let AC and BC be measured from the same scale of equal parts that AB was taken from, and the answers are found.

Or, having found one of the required sides, the other may obtained by one or the other of the cors. to theo. 14, sect. 4. 3d. By Gunter's Scale.

1. When AC is made the radius.

be

Extend from 47° 40′ to 90° on the line of sines; that distance will reach from 190 to 257, on the line of numbers, for AC.

2. When AB is made the radius, the first stating is thus performed:

Extend from 45° on the tangents (for the tangent of 45° is equal to the radius, or to the sine of 90° as before) to 42° 20'; that extent will reach from 190, on the line of numbers, to 173, for BC.

3. When BC is made the radius, the second stating is thus performed:

Extend from 47° 40', on the line of tangents, to 45°, or radius; that extent will reach from 190 to 173, on the line of numbers, for BC; for the tangent of 47° 40' is more than the radius, therefore the fourth number must be less than the second, as before.

The first two statings of this case answer the question without a secant.

Thus, .739239)190.000000(257.02, &c.=AC.

147 8478

4215220

3696195

5190250

5174673

1557700

1478478

and, .673443=Nat. S of A.

190 side AB.

60609870

673443

127.954170