Plate V.

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 10.00000, reject the first figure of the sum of the other two logs. (which is the same thing as to subtract 10.00000; and that will be the log. of the thing required.

2. Making AB the radius.

Secant A: AC::R: AB.
Secant A: AC:: T.A: BC.

Or, having found one leg, the other may be obtained by cor. 2. theo. 14. sect. 1.

M

By Gunter's Scale.

On this scale there are lines of numbers, sines, and tangents, as well as lines of sine and tangent rumbs, versed sines, meridional parts, and equal parts; but the three first lines are sufficient for our present purpose.

The divisions on these respective lines, are the logarithms of numbers, sines, and tangents, taken from a scale of equal parts, and applied on the lines of the 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. 22. sect. 1. 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. 1.) the tangent of 45° is equal to the radius : for this reason the line of numbers of 10.00000, the sine of 90°, and tangent of 45° being all equal, terminate at the same end of the scale; where there are small brass centres, usually placed to preserve the scale.

It was said before, that the tangents ended at 45°; but because the logarithms of tangents more than 45°, must pass off the scale; such distances therefore, as exceed 45°, are set backwards from 45, and numbered 50, 60, 70, &c.

There is no line of secants on the scale; for every thing requisite can be performed without

them.

Thus the two first statings of this case, answers the question without a secant: the like will be also made evident in all the following cases.

CASE II.

The base and angle given; to find the perpendicular and hypothenuse.

Plate V. 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 leg AB 190, given; to find BC and AC.

Geometrically.

Make the angle CAB (by prob. 16. sect. 1.) 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. 1.) 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 you have the answer.