TRIGONOMETRY 
ing through the centre, and invariably fall 
upon a diameter, drawn perpendicular to 
that right line. See Dialling, Geometry, 
and Mathematical instrmnents ; under 
which various explanations will be found, 
wheieby (he student may perceive the ne- 
cessity for such reference. 
The solution of the several cases in plane 
trigonometry depend upon four proposi- 
tions, called axioms, which cailnot be too 
perfectly understood, and ought ever to be 
adverted to. 
Axiom /. In any right-lined plane trian- 
gle, if the hypothenuse (or longest side) be 
made the radius of a circle, the other two 
sides, or legs, will be the sines of their oppo- 
site angles: but if either of the legs, includ- 
ing the right angle, be made radius, the 
other leg becomes the tangent of its oppo- 
site angle, and the hypothenuse the secant 
of the same angle. For in the triangle 
ABC, (fig. 21. Plate XV. Trigonometry) 
let A B he made the radius of a circle ; and 
with one foot of the compasses on A or B 
describe a circle: it is plain that the leg 
BC will be the sine of the angle A, and 
AC the sine of the angle B: but if AC be- 
comes radius, B C will be the tangent to 
the angle A, and B A the second thereto. 
Again, by making BC radius, AC will be 
tangent, and A B the secant of the angle B. 
Hence it is plain that the different sides 
take their names according to that side 
which is made radius. 
Remark, that to find a side, any side 
may be made radius : then say, as the name 
of the side given is to the name of the side 
required; so is the side given to the side 
required. But to find an angle, one of the 
given sides must be made radius: then as 
the side made radius is to the other side • 
so is the name of the first side (which is’ 
always radius) to the name of the second 
side; wliich fourth proportional must be 
found among the sines, or tangents, &c. to 
be determined by the side made radius : 
against it is the required angle. In a right- 
angled triangle you must always have two 
sides, or the angles and one side given to 
find tlie rest. 
To find D C. 
As the .sine angle D 101® 25'..., 
Is to the side B C 76 
So is sine angle B 44“ 42' 
11.72801 
9.99132 
To the side D C 54.53 
The foregoing is worked by logarithms, 
thus : add the logarithm of the second and 
third terms together, then deduct the loga- 
rithm of the first term, and the remainder is 
the logarithm of tue fourth term, or number 
sought. When an angle is greater than 90® 
the sine, tangent, and secant of the supple- 
ment, (i.e. of the number of degrees want- 
ing of 180») are to be used. 
Two sides, and an angle opposite to one 
of them, being given to find the other oppo- 
site angle, and the third side. fitr. o'? ” The 
B C D St® 49 given to find the angle B DC 
obtuse, and the side CD. 
To find D. 
As the sine BD 65 1.81291 
Is to the angle C 31® 49' ~9.7Z1K 
So is the side BC 106 2.02531 
11.74729 
_ 1.81291 
To sine angle D 120® 43' 
To find D C. 
As sine angle C 31“ 49' 9.72198 
Is to the sine B D 65 7.81291 
So is sine ai/gle B 217.28 9.66392 
i7!47683 
9.72198 
1.75485 
Axiom II. In all plane triangles, the 
sides are in direct proportion to the sines of 
their opposite angles. Tims, « if two angles 
and one side be given to find either of the 
other legs.’* In fig. 22. the angle B C D is 
101® 25 , the angle CBD is 44® 42', and the 
given teg B C is equal to 76 of the scale 
assumed ; to find the sides C D and B D. 
To the sine DC .56.88... 
Axwm III. In every plane triangle it 
will be as the sum of any two sides is to 
their difference; so is the tangent of half 
the sum of the angles opposite there, to the 
tangent of half their difference. Which half 
diflerence, being added to half the sum of 
theangle.s, gives, (he greater; but if sub- 
tracted, the remainder will be the lesser 
angle. ^ 
‘ Two sides, and their contained an®le 
fhTth- a a'* “"«>es, and 
the third side, fig. 24 .” The side B C 109, 
B p 76 leagues, and the angle C B D lOl® 30 ' 
being given, to find the angle BDC, or 
B CD, and the side CD, 
