compasses where it falls; bring the right-hand point from 45° to the third term of the proportion; this extent now in the compasses applied from 45° backward will reach to the fourth term, or the tangent required. For, had the line of tangents been continued forward beyond 45°, the divisions would have fallen above 45° forward, in the same manner as they fall under 45° backward.

SECTION V.

TRIGONOMETRY.

The word Trigonometry signifies the measuring of triangles. But under this name is generally comprehended the art of determining the positions and dimensions of the several unknown parts of extension, by means of some parts which are already known. If we conceive the different points which may be represented in any space to be joined together by right lines, there are three things offered for our consideration; 1, the length of these lines; 2, the angles which they form with one another; 3, the angles formed by the planes in which these lines are drawn, or are supposed to be traced. On the comparison of these three objects depends the solution of all questions that can be proposed concerning the measure of extension and its parts; and the art of determining all these things from the knowledge of some of them is reduced to the solution of these two general questions.

1. Knowing three of the six parts, the sides and angles, which constitute a rectilineal triangle, to find the other three.

2. Knowing three of the six parts which compose a spherical triangle, that is, a triangle formed on the surface of a sphere by three arches of circles which have their centre in the centre of the same sphere, to find the other three.

The first question is the object of what is called Plane Trigonometry, because the six parts considered here are in the same plane it is also denominated Rectilineal Trigonometry. The second question belongs to Spherical Trigonometry, wherein the six parts are considered in different planes. But the only object here is to explain the solutions of the former question, viz.

PLANE TRIGONOMETRY.

Plane Trigonometry is that branch of Geometry which

teaches how to determine or calculate three of the six parts of a rectilineal triangle, by having the other three parts given or known. It is usually divided into Right-angled and Obliqueangled Trigonometry, according as it is applied to the mensuration of right or oblique-angled triangles.

In every triangle or case in trigonometry three of the parts must be given, and one of these parts at least must be a side; because, if the three angles only were given, it is obvious that all similar triangles would answer the question.

RIGHT-ANGLED PLANE TRIGONOMETRY.
PL. 5. fig. 1.

1. In every right-angled plane triangle ABC, if the hypothenuse AC be made the radius, and with it a circle or an arc of one be described from each end, it is plain (from def. 20), that BC is the sine of the angle A, and AB is the sine of the angle that is, the legs are the sines of their opposite angles.*

*The sine and co-sine of any number of degrees and minutes is found by the series (which is given and illustrated in page 49, Ryan's Differential and Integral Calculus)

a

a3
+

αδ

a7

2.3 2.3.4.5 2.3.4.5.6.7

+2.3.4

ασ
a8
+
2.3.4.5.6 2.3.4.5.6.7.8"

In which series the value of a is found thus: as the number of degrees or minutes in the whole semicircle is to the degrees or minutes in the arc proposed, so is 3.14159, &c. to the length of the said arc, which is the value of a. For example, let it be required to find the sine of one minute; then, as 10800 (the minutes in 180 degrees): 1 :: 3.14159, &c. : .000290888208665 the length of an arc of one minute, which is the a3 a3 value of a in this case, and

2.3

6

-.000000000004102, &c. Conse

quently, .000290888208665- .000000000004102.000290888204563=

the required sine of one minute.

Again, let it be required to find the sine and co-sine of five degrees, each true to seven places of decimals. Here, .0002908882, the length of an arc of one minute (found above), being multiplied by 300, the number of minutes in 5 degrees, the product .08726646 is the length of an arc of 5 degrees; therefore in this case we have

a =

a3

.08726646

and 4.00000241; hence 1-.00380771+.00000241.9961947-the co-sine of 5 degrees.

Fig. 2.

2. If one leg AB be made the radius, and with it on the point A an arc be described, then BC is the tangent and AC is the secant of the angle A, by def. 22 and 25.

Fig. 3.

3. If BC be made the radius, and an arc be described with it on the point C, then is AB the tangent and AC is the secant of the angle C, as before.

Because the sine, tangent, or secant of any given arc in one circle is to the sine, tangent, or secant of a like arc (or to one of the like number of degrees) in another circle, as the radius of the one is to the radius of the other; therefore the sine, tangent, or secant of any arc is proportional to the sine, tangent, or secant of a like arc, as the radius of the given arc is to 10.000000, the radius from whence the logarithmic sines, tangents, and secants in most tables are calculated; that is,

If AC be made the radius, the sines of the angles A and C, described by the radius AC, will be proportional to the sines of the like arcs or angles in the circle that the tables now mentioned were calculated for. So if BC was required, having the angles and AB given, it will be,

After the same manner the sine and co-sine of any other arc may be derived; but the greater the arc is the slower the series will converge, and therefore a greater number of terms must be taken to bring out the conclusion to the same degree of exactness.

Or, having found the sine, the co-sine will be found from it (by theo. 14), the co-sine CL (plate I, fig. 8)=√ CH2, HL3, or c =

$2.

For other methods of constructing the canon of sines and co-sines, the reader is referred to Hutton's Mathematics, Simpson's Algebra, &c. The sines and co-sines being known or found by the foregoing method, the tangents and secants will be easily found from the principle of similar triangles, in the following manner:

In plate 1, fig. 8, where of the arc BH, HL is the sine, CL or FH the co-sine, BK the tangent, CK the secant, DI the co-tangent, and CI the co-secant, the radius being CH, or CB, or CD, the three similar triangles CLH, CBK, CDI give the following proportion (by theo. 14): 1. CL: LH:: CB: BK; whence the tangent is known, being a fourth proportional to the co-sine, sine, and radius.

2. CL: CH:: CB: CK; whence the secant is known, being a third proportional to the co-sine and radius.

3. HL: LC:: CD: DI; whence the co-tangent is known, being a fourth proportional to the sine, co-sine, and radius.

4. HL: HC:: CD: CI; whence the co-secant is known, being a third proportional to the sine and radius.

As for the logarithms, sines, tangents, and secants in the tables, they are only the logarithms of the natural sines, tangents, and secants calculated as above.

Fig. 1.

As S.C: AB:: S.A: BC.

That is, as the sine of the angle C in the tables is to the length of AB (or sine of the angle C in a circle whose radius is AC), so is the sine of the angle A in the tables to the length of BC (or sine of the same angle in the circle whose radius is AC).

In like manner the 'tangents and secants represented by making either leg the radius will be proportional to the tangents and secants of a like arc, as the radius of the given arc is to 10.000000, the radius of the tables aforesaid.

Hence it is plain, that if the name of each side of the triangle be placed thereon, a proportion will arise to answer the same end as before thus, if AC be made the radius, let-the word radius be written thereon; and as BC and AB are the sines of their opposite angles, upon the first let S.A, or sine of the angle A, and on the other let S.C, or sine of the angle C, be written. Then,

When a side is required, it may be obtained by this proportion, viz.

As the name of the side given

is to the side given,

So is the name of the side required

to the side required.

Thus, if the angles A and C and the hypothenuse AC were given, to find the sides; the proportion will be

1. R: AC:: S.A: BC.

Fig. 1.

That is, as radius is to AC, so is the sine of the angle A to BC. And,

2. R: AC:: S.C: AB.

That is, as radius is to AC, so is the sine of the angle C to AB. When an angle is required we use this proportion, viz.

As the side that is made the radius

is to radius,

So is the other given side

to its name.

Thus, if the legs were given, to find the angle A, and if AB be made the radius, it will be

AB: R:: BC: T.A.

Fig. 2.

That is, as AB is to radius, so is BC to the tangent of the angle 4.

After the same manner, the sides or angles of all right angled plane triangles may be found, from their proper data.

We here, in plate 4, give all the proportion requisite for the solution of the six cases in right-angled trigonometry; making every side possible the radius.

In the following triangles this mark in an angle denotes it to be known, or the quantity of degrees it contains to be given; and this mark' on a side denotes its length to be given in feet, yards, perches, or miles, &c. and this mark°, either in an angle or on a side, denotes the angle or side to be required.

From these propositions it may be observed, that to find a side, when the angles and one side are given, any side may be made the radius; and to find an angle, one of the given sides must be made the radius. So that in the 1st, 2d, and 3d cases any side, as well required as given, may be made the radius, and in the first statings of the 4th, 5th, and 6th cases, a given side only is made the radius.

RIGHT-ANGLED TRIANGLES.

CASE I.

The angles and hypothenuse given, to find the base and perpendicular.

PL. 5. fig. 4.

In the right-angled triangle ABC, suppose the angle A= 46° 30'; and consequently the angle C=43° 30′ (by cor. 2, theo. 5); and AC 250 parts (as feet, yards, miles, &c.); required the sides AB and BC.

1st. By Construction.

Make an angle of 46° 30' in blank lines (by prob. 16, geom.), as CAB; lay 250, which is the given hypothenuse, from a scale of equal parts, from A to C; from C let fall the perpendicular BC (by prob. 7, geom.), and that will constitute the triangle ABC. Measure the lines BC and AB from the same scale of equal parts that AC was taken from, and you have the answer.

* It is proper to observe, that constructions, though perfectly correct in theory, would give only a moderate approximation in practice, on account of the imperfection of the instruments required in constructing them; they are called graphic methods. Trigonometrical methods, on the contrary, being independent of all mechanical operation, give solutions with the utmost accuracy: they are founded upon the properties of lines called sines, co-sines, tangents, &c., which furnish a very simple mode of expressing the relations that subsist between the sides and angles of triangles.