TRIGONOMETRY. 
621 
TRIGONOMETRY is that part of geometry 
which teaches howto measure the sides and an- 
j gles of triangles. Trigonometry is either plane 
[ or spherical, of each of which we shall treat in 
! order. 
Plane Trigonometry is the science which 
treats of the analogies "of plane triangles, and of 
! the methods of determining their sides and an- 
| gles. For this purpose, it is not only requisite 
I that the peripheries of circles, but also that cer- 
tain right lines in and about a circle, are supposed 
I to be divided into some assigned number of 
[ equal parts. These lines are denominated sines, 
I tangents, secants, &c. The sides of plane tri- 
angles may be estimated in feet, yards, chains, or 
1 by any other definite measures, or by abstract 
numbers : but the angles are measured by the 
arcs of a circle, contained between the two legs, 
having the angular point for its centre. 
Every circle is supposed to be divided into 
fl(iO equal parts, called degrees ; each degree 
into 60 equal parts, called minutes; each minute 
into 60 equal parts, called seconds. An angle 
is said to be of as many degrees, minutes, se- 
, conds, See. as are contained in the arc, or part 
of the circumference, by which it is measured. 
A right angle is measured by the fourth part 
of the circumference, or 90° ; an obtuse angle 
is greater than 90°, and an acute angle is less 
than 90°. Degrees, minutes. See. are marked at 
the top of the figures by which the arc is de- 
noted. Thus we say 34° 28' 50", thirty-four 
degrees, twenty-eight minutes, and fifty seconds. 
The difference of an arc from 90°, or a quad- 
rant, is called its complement-, and its difference 
from 180°, its supplement-, thus in Plate Miscel. 
fig. 246, the arc AB is the complement of HB : 
but AB is the supplement of BD. 
A chord, or subtense, is a right line drawn 
from one extremity of an arc to the other : 
thus BE is the chord or subtense of the arc 
BAE, or BDE. 
The sine, or, as it is sometimes called, the right 
sine, of an arc, is a right line drawn from one 
extremity of the arc, perpendicular to the dia- 
meter passing through the other extremity : 
thus BF is the sine of the arc AB, or BD. 
The versed sine of an arc is the part of the dia- 
meter, intercepted between the arc and its sine : 
AF is the versed sine of AB, and DF of the arc 
DB. 
The co-sine of all arc is the part of the diame- 
ter intercepted between the center and the sine, 
and is equal to the sine of the complement of 
that arc. Thus CF is the co-sine of the arc AB, 
and is equal to BI, the sine of its complement 
HB. 
The tangent of an arc, is a right line touching 
the circle in one extremity of that arc, con- 
tinued from thence to meet a line drawn from 
the center through the other extremity ; which 
line is called the secant of the same arc : thus 
AG is the tangent, and CG the secant of the 
arc AB. 
The co-tangent and co-secant of an arc, are the 
tangent and secant of the complement of that 
arc : thus HK and CK are the co-tangent and 
co-secant of the arc AB. 
The lines here described, belong equally to 
an angle as to the arc by which it is measured ; 
and, except the chords and versed sines^they 
are all common to two arcs or angles which are 
the supplements of each other. 
So that if the sine, tangent, &c. of any arc or 
angle above 90° are required, it is the same 
thing as to find the sine, tangent, &c. of its sup- 
plement, or what it wants of 180°. 
They are also called the natural sines, tan- 
gents, &c. of the arc6 or angles to which they 
belong; and the logarithms of the numbers 
by which they are represented, are the loga- 
rithmic sines, tangents, &e. 
And as one or other of these lines make a part 
of every trigonometrical operation, they have 
been calculated to a given radius, for every de- 
gree, minute, &,c. of the quadrant, and ranged 
in tables for use. 
Whence, by the help of such a table, the sine, 
tangent, &c. of any arc or angle, may be found 
by inspection ; and, vice versa, the arc, or an- 
gle, to which any sine, tangent, &c. belongs. 
Upon this table also, and the doctrine of si- 
milar triangles, depends the solution of the se- 
veral cases of plane trigonometry, which may be 
performed either by the natural or logarithmic 
sines, tangents, &c., as occasion requires. 
But the logarithmic sines, tangents, See., are 
those most commonly used : as the calculations, 
in this case, are all performed by adding and 
subtracting only, instead of multiplying and di- 
vidingj as is required by the natural sines, &c. 
The sine, tangent, Sec. of any arc or angle 
being of the same magnitude as the sine, tan- 
gent, Sec. of its supplement, it is plain that a 
table of these lines made for every degree, mi- 
nute, 8e c. of the quadrant, or 90°, will serve for 
the whole circle. 
It is also to be observed that, in every such 
table, the natural sines, tangents, Sec. are usually 
calculated to radius 1 ; but in order that the lo- 
garithmic sines, tangents, Sec. may be all posi- 
tive, they are calculated to radius 1.0000000000, 
or 1 with 10 cyphers, the logarithm of which is 
10, so that the latter are the logarithms of the 
former, with 10 added to the index. 
And, as the natural sines, tangents, Sec. of any 
angles or arcs of different circles, are propor- 
tional to the radii of those circles, their values 
may be readily found, or made to correspond 
to any radius whatever. 
In every plane triangle, three things must be 
given to find the rest ; and of these three one at 
least must be a side, because the same angles are 
common to an infinite number of triangles. 
It is also to be observed, that all the varieties 
that can possibly happen in the solution of plane 
triangles, are comprised under the three follow- 
ing cases : viz. 
1. When two of the three given things are a 
side and its opposite angle. 
2. When two sides and their included angle 
are given. 
3. When the three sides are given. 
Each of which cases may be resolved, either 
by geometrical construction, by arithmetical 
computation, or instrumentally. 
In the first of these methods, the triangle is 
constructed, by laying down the sides from a 
scale of equal parts, and the angles from a scale 
of chords, or a protractor, and then measuring 
the unknown parts by the same scale or instru- 
ment from which the others were taken. 
In the second method, having stated the pro- 
portion, according to the proper rule, multiply 
the second and third terms together, and divide 
the product by the first, and the quotient will be 
the fourth term required, for the natural num- 
bers. Or, in working by logarithms, add the 
logarithms of the second and third terms toge- 
ther, and from the sum take the logarithm of the 
first, and the number answering to the remain- 
der will be the fourth term sought. 
In the third method, or instrumentally,- as 
i suppose by the logarithmic lines on one side of 
the common two-foot scales, extend the com- 
passes from the first term to the second or third 
i as they happen to be of the same kind ; and that 
I extent will reach from the other term to the 
fourth term required, taking both extents to- 
wards the same end of the scale. 
The second of these methods, however; or 
j that in which the operation is performed by lo- 
garithms, is the one generally employed ; the 
other two being chiefly of use as checks on the 
calculations, or, in certain simple cases, where a 
near approximate value of the quantities to be 
determined is thought sufficient. 
It may here also be further remarked, that 
when one or more logarithms are to be sub- 
tracted, in any operation, it will be better to 
write down their complements, or what each of 
them wants of 10.0000000 instead of the loga- 
rithms themselves, and then add them together, 
abating as many tens in the index of the sum as 
there were logarithms to be subtracted. 
Thus, if the logarithm to be subtracted is 
3.4932758, it will be the same thing- as to add 
its complement 6.5067242 ; and if it is 
9.07432600, its complement, or the number to 
be added, will be 0.92567400 ; which numbers 
are readily found by beginning at the left hand 
and subtracting each figure of the logarithm 
from 9, except the last significant figure on the 
right, which must be subtracted from 10. 
If the index of the logarithm, whose comple- 
ment is to be taken, is greater than 10, write 
down what the index wants of 19, and the rest 
of the figures as before ; and, after the addition, 
subtract 20 from the index of the sum. And if 
the logarithm of a decimal is to be subtracted, 
add 10 to the index, and then take the comple- 
ment of the resulting number, and the rest of 
the figures, as before. 
Thus the complement of the logarithm : 
12.4907327 is 7.5092673 ; and the complement 
of the logarithm of 3.5972648 is 2.4027332. 
PROPERTIES OF PLANE THIANGI.E3, REQUIRED 
IN THE PRACTICAL PART OF THIS SCIENCE. 
The sum of all the three angles of any plane 
triangle is equal to two right angles, or 180°. 
The greater side is opposite to the greater an- 
gle ; and the less side to the less angle. 
The sum of any two sides is greater than the 
third ; and th * difference of any two sides is less 
than the third. 
The triangle is equilateral, isosceles; or sca- 
lene, according as its three angles are all equal, 
or only two of them equal, or all three unequal. 
The angles opposite to the two least sides are 
acute ; and if there is an obtuse angle, it is op- 
posite to the greatest side. 
A perpendicular drawn from the opposite 
angle to the longest side will fall within the tri- 
angle ; and the greater and less segment will be 
next the greater and less side. 
In an isosceles triangle, a perpendicular drawn 
from the vertex will bisect both the base and 
the vertical angle.- 
In a right-angled triangle the hypothenuse is 
equal to the square- root of the sum of the 
squares of the other two sides; and either of 
the sides is equal to the square root of the (dif- 
ference of the squares of the 'hypothenuse and 
the other side. 
Note, also, that if the half difference of any 
two quantities is added to their half sum, it 
will give the greater of those quantities and, 
if it is subtracted from the half sum, it will give 
the. less. 
Case I. When two of the three given things 
are a side and its opposite angle, to find the rest. 
Rule. The sides of any plane triangle are to 
each other as the sines of their opposite angles, 
and vice versa That is, 
As any side is to the sine of its opposite an- 
gle, so is any other side to the sine- of its oppo- 
site angle. 
Or, As the sine of any angle is to its opposite 
side, so is the sine of any Other angle to its op- 
posite side. 
Hence, to find an angle) begin the proportion 
with a side opposite to a given angle ; and to 
