Fig. 38.
A

PROBLEM XVIII. To describe a circle which shall pass through any three given points, not lying in a right line, as A, B, D. Fig. 39.

B

*

Draw lines from A to B, and from B to D; bisect those lines by PROBLEM II, and the point where the bisecting lines intersect each other, as at C, will be the centre of the circle.

PROBLEM XIX. To find the centre of a circle.

By the last PROBLEM it is plain, that if three points be any where taken in the given circle's periphery, the centre of the circle may be found as there taught.

Directions for constructing irregular figures of four or more sides, may be found in the following treatise on SUR

VEYING.

TRIGONOMETRY.

TRIGONOMETRY is that part of practical GEOMETRY, by which the sides and angles of triangles are measured whereby three things being given, either all sides, or sides and angles, a fourth may be found; either by measuring with a scale and dividers, according to the PROBLEMS IN GEOMETRY, or more accurately by calculation with logarithms, or with natural sines.

TRIGONOMETRY is divided into two parts, rectangular and oblique-angular.

PART I.

RECTANGULAR TRIGONOMETRY.

This is founded on the following methods of applying a circle to a triangle.

PROPOSITION I. In every right angled triangle, as A B C, Fig. 40, it is plain from Fig. 7, compared with the Geometrical definitions to which that Figure refers, that if the hypothe-A nuse AC be made radius, and with it an arc of a circle be described from each end, B C will be the sine of the angle at A, and A B the sine of the angle at C; that is, the legs will be sines of their opposite angles.

PROPOSITION II. If one leg, A B, Fig. 41, be made radius, and with it on the point A an arc be described, then B C, the other leg, will be the tangent, and A C the secant of the angle at A; and if B C be made radius, and an arc be described with it on the point C, then A B will be the tangent, and AC the se

Fig. 40.

C

Fig. 41.

B

leg be made radius, the other leg will be a tangent of its opposite angle, and the hypothenuse a secant of the same angle.

Thus, as different sides are made radius, the other sides acquire different names, which are either sines, tangents, or

secants.

As the sides and angles of triangles bear a certain proportion to each other, two sides and one angle, or one side and two angles being given, the other sides or angles may be found by instituting proportions, according to the following rules.

RULE I. To find a side, either of the sides may be made radius, then institute the following proportion:

AS THE NAME OF THE SIDE GIVEN, (which will be either radius, sine, tangent, or secant ;)

IS TO THE LENGTH OF THE SIDE GIVEN ;

SO IS THE NAME OF THE SIDE REQUIRED, (which also will be either radius, sine, tangent, or secant ;)

To THE LENGTH OF THE SIDE REQUIRED.

RULE II. To find an angle, one of the given sides must be made radius, then institute the following proportion : AS THE LENGTH OF THE GIVEN SIDE MADE RADIUS; IS TO ITS NAME, (that is, radius ;)

SO IS THE LENGTH OF THE OTHER GIVEN SIDE;

TO ITS NAME, (which will be either sine, tangent, or secant.)

Having instituted the proportion, look for the corresponding logarithms, in the logarithms of numbers for the length of the sides; and in the table of artificial sines and tangents, for the logarithmic sine, tangent, or secant.

ADD TOGETHER THE LOGARITHMS OF THE SECOND AND THIRD TERMS, AND FROM THEIR SUM SUBSTRACT THE LOGARITHM OF THE FIRST TERM THE REMAINDER WILL BE THE LOGARITHM OF THE FOURTH TERM, WHICH SEEK IN THE TABLES, AND FIND ITS CORRESPONDING NUMBER, OR DEGREES AND MINUTES.

See the introduction to the table of logarithms; which should be attentively studied by the learner, before he proceeds any further.

NOTE. The logarithm for radius is always 10, which is the logarithmic sine of 90°, and the logarithmic tangent of 45.

The preceding PROPOSITIONS and RULES being duly at tended to, the solution of the following CASES of Rectangular Trigonometry will be easy.*

In the triangle AB C, given the hypothenuse A C 25 rods or chains; the angle at A 35° 30'; and consequently the angle at C 54° 30': (See Note, GEOм. Def. 38.) to find the legs.

Making the hypothenuse radius, the proportions will be:
To find the leg A B.
To find the leg B C.

As radius

: hyp. AC, 25

10.000000 As radius

10.000000

1.397940: hyp. AC, 25 1.397940

:: sine ACB, 54° 30′ 9.910686: : sineCAB, 35°30′ 9.763954

11.308626

10.000000

11.161894 10.000000

leg AB, 20.35 nearly 1.308626 : legBC,14.52 nearly 1.161894

NOTE. When the first term is radius, it may be substracted by cancelling the first figure of the sum of the other

two terms.

As secants are excluded from the tables, such cases in Trigonometry as are solved by secants are also excluded.

[The learner should exercise himself, in this and the following rules in TRIGONOMETRY, in stating all the propor tions which can be made, until he is able to do it with facility.]

BY NATURAL SINES.

This CASE may be solved by natural sines,* according to the following proportions:

AS UNITY, OR 1, IS TO THE LENGTH OF THE HYPOTHENUSE, SO IS THE NATURAL SINE OF THE SMALLEST ANGLE, TO THE LENGTH OF THE SHORTEST LEG. OR, SO IS THE NATURAL

SINE OF THE LARGEST ANGLE, TO THE LENGTH OF THE LONGEST LEG.

Or, which is the same thing, MULTIPLY THE NATURAL SINES OF THE TWO ANGLES BY THE HYPOTHENUSE; THE PRODUCTS WILL BE THE LENGTH OF THE TWO LEGS.

NOTE. The third decimal figure in the first product being 7, the preceding figure may be called one more than it is, viz. 2. And whenever in any product, &c. there are more places of decimals than you wish to work with, if the one at the right hand of the last which you wish to retain is more than 5, add a unit to the last, because a greater decimal number than 5 is more than half.

CASE II.

The angles and one leg given, to find the hypothenuse and the other leg. Fig. 43.