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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.
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
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.
This is founded on the following methods of applying a circle to a triangle.
Fig. 40. PROPOSITIONI. In every right
C angled triangle, as ABC, Fig. 40, it is plain from Fig. 7, compared with the Geometrical definitions to which that Figure refers, that if the hypothenuse A AC be made radius, and with it an arc of a circle be described from each end, BC will be the sine of the angle at A, and AB the sine of the angle at C; that is, the legs will be sines of their opposite angles.
Fig. 41. PROPOSITION II. If one leg, AB,
C Fig. 41, be made radius, and with it on the point A an arc be described, then BC, the other leg, will be the tangent, and AC the secant of the angle at A; and if BC be made radius, and an arc be described with it on the point C, then
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 ;
To ITS NAME, (which will be either sine, tangent or ser cant.
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 SUBTRACT THE LOGARITHM OF THE FIRST TERM : THE REMAINDER WILL BE THE LOGARITHM OF THE FOURTH TERM, WHICH SÉEK 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 radiás is always 10, which is
the logarithmic sine of 90°, and the logarithmic tangent of 45.
The preceeding PROPOSITIONS and RULES being duly attended to, the solution of the following cases of Rectanger lar Trigonometry will be easy.*
The angles and hypothenuse given to find the legs. Fig. 42.
In the triangle ABC, given the hypothenuse AC 25 rods or chains; the angle at A 35° 30': and consequently the angle at C 54° 30': (See Note Geom. Def. 39.) to find the legs. Making the hypothenuse radius, the proportions will be: To find the leg AB.
To find the leg BC. As radius 10.000000 As radius
10.000000 : hyp. AC, 25 1.397940 : hyp. AC, 25
1.397940 ::: sine ACB, 54°30/ 9.9106861: :sine CAB, 350 3049.763954
:leg AB, 20.35 nearly 1.308626 : leg BC 14.52 nearly 1.161894
NOTE. When the first term is radius, it may
be subraeted-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 proportions which can be made, until he is able to do it. with fa-cility.]
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.
Leg BC 14.52
Leg AB 20.35 NOTE. The third decimal figure in the first produét be
ing 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.
The angles and one leg given to find the hypothenuse and the other leg. Fig. 43.