The use of the logarithmical lines on Gunter's Scale. By these lines and a pair of compasses, all the problems of Trigonometry, &c. may be solved.

These problems are all solved by proportion; Now in natural numbers, the quotient of the first terin by the second is equal to the quotient of the third by the fourth: therefore, logarithmically speaking, the difference between the first and second term is equal to the difference between the third and fourth, consequently on the lines on the scale, the distance between the first and second term will 'be equal to the distance between the third and fourth. And for a similar reason, because four proportional quantities are alternately proportional, the distance between the first and third terms, will be equal to the distance between the second and fourth. Hence the following

General Rule.

The extent of the compasses from the first term to the second, will reach, in this same direction, from the third to the fourth term. Or, the extent of the compasses from the first term to the third, will reach, in the same direction, from the second to the fourth.

By the same direction in the foregoing rule, is meant that if the second term lie on the right hand of the first, the fourth will lie on the right hand of the third, and the contrary. This is true, except the two first or two last terms of the proportion are on the line of tangents, and neither of them under 45°; in this case the extent on the tangents is to be made in a contrary direction: For had the tangents above 45° been laid down in their proper direction, they would have extended beyond the length of the scale towards the right hand; they are therefore as it were folded

back upon the tangents below 45°, and conse quently lie in a direction contrary to their proper and natural order.

If the two last terms of a proportion be on the line of tangents, and one of them greater and the other less than 45°; the extent from the first term to the second will reach from the third beyond the scale. To remedy this inconvenience, apply the extent between the two first terms from 45° backward upon the line of tangents, and keep the left hand point of the 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 trian-. gle; 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.

It

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. is usually divided into Right angled and Oblique angled 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, with the same

angles, the sides may be greater or less in any proportion.

RIGHT ANGLED PLANE TRIGONOMETRY.

PL. 5. fig. 1.

1. In every right-angled plane triangle ABC, if the hypothenuse AŬ 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 C; 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 302, Simpson's Algebra) &c. for its sine, and its co-sine by 1

a3

2.3

2.3.4.5

am +

a5

a7 2.3.4.5.6.7

2

+

and

a3

2.3.4.5.6

as.

2.3.4.5.6.7.8'

In which series, the value of a is found thus, as the number of degrees or mi.. nutes 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 value of a in this case, 000000000004102, &c. Consequently, .000290888208665 .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

2.3 000000000004102

Consequently .08715574 the sine of 5 degrees. Also, .00380771,

a4 24

and 00000241, hence 1- .00380771.00000241.9961947 sine of 5 degrees.

After the same manner, the sine and co-sine of any other are 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 (Plat 1. fig. 8.)

Fig. 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 de scribed 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;

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 Cl the cosecant, 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 shove.