compaffes, transfer thefe divifions to the chord line A D, which mark with the correfponding numbers, and it will become the line of chords, which may be transferred to the ruler.

4. For the line of thumbs, divide the quadrant B D into eight equal parts: then with the compaffes, from the centre B, transfer the divifions to the line B D, which will be the line of rhumbs.

5. For the line of fines, through each of the divifions of the arc A D draw right lines parallel to the radius DC, which will divide the radius A C into the fines, or verfed fines; numbering it from C to A for the fines, and from A to C for the verfed fines.

6. For the line of tangents, lay a ruler on C, and the feveral divifions of the arc A D; and it will interfect the line E A, which will become a line of tangents, transferring the numbers from the arc A D to that line.

7. For the line of fecants transfer the divifions from the tangent line to the line F D with the compaffes, and from C as the centre, marking the divifions with the correfponding numbers on the tangent line.

8. For the line of femi-tangents, lay a ruler on B and the feveral divifions of the arc A D, which will interfect the radius C D in the feveral divifions of the femi-tangents, which are to be numbered according to the arc A D.

9. For the line of longitude, divide the radius A C into 60 equal parts, through each of thefe draw lines parallel to the radius CD; the points where thefe lines interfect the arc AD are to be transferred with the compaffes from A as a centre to the chord A D, and numbered thereon, which will give the line of longitude.

10. For the line of latitude, the femicircle A D B must be completed to a circle, then a ruler laid on the point D, and on the feveral divifions on the line of fines, A C will interfect the next quadrant of the circle, in as many points; when from the opposite part of the circle to D, as the centre,

the interfections of the arc are to be transferred to its chord, and numbered according to the numbers on the line of fines.

The chief uses of the lines of fines, tangents, fecants, and femi-tangents, are to find the poles, and centres of the several circles, reprefented in a projection of the sphere.

I have been more particular in defcribing the construction of this fcale, as it is an inftrument in most general use in mathematics and by the foregoing directions the learner may. conftruct any lines on the scale himself, where there happens not to be a mathematical inftrument maker nigh at hand, and place them on a rule, as feen fig. 3.

SECT. I.

OF PLANE TRIGONOMETRY.

THE three methods of refolving triangles, or cafes in trigonometry, are:-1. By geometrical conftruction. 2. Arithmetical computation. And, 3. Inftrumental operation. In the first method, the triangle is conftructed, by drawing, and laying down the feveral parts, viz.-the fides from a fcale of equal parts, and the angles from a fcale of chords, or other inftrument: then the unknown parts are measured by the fame scales; and thus they become known.

In the fecond method, the terms of the proportion are ftated according to rule; which terms confift partly of the numbers of the given fides, and partly of the fines, &c. of angles taken from the tables; the proportion is then refolved like all other proportions, in which a fourth term is to be found from three given terms, viz. by the Rule of Three.

In the third method of refolving the triangle, by inftrumental operation, recourfe must be had to the logarithmic lines, on one fide of the two foot fcales; extending the compaffes from the first term to the second or third, which happens to be of the fame kind with it; then that extent will

reach

reach from the other term to the fourth term. In this operation for the fides of triangles, is used the line of numbers, and for the angles the line of fines or tangents, according as the proportion refpects fines or tangents.

In every cafe in plane trigonometry, there must be given' three parts, one of which, at least, must be a fide. And every triangle that can be proposed, will fall under one of the three following cafes:

CASE I.

When two of the three given Parts are a Side, and its oppofite Angle.

CASE II.

When there are given two Sides, and their contained

Angle.

CASE HI.

When the three Sides are given.

RULE. For the firft cafe, viz.--That the fides are proportional to the fines of their opposite angles: that is, as the one fide given, is to the fine of its oppofite angle, so is another fide given to the fine of its oppofite angle. Or, as the fine of a given angle is to its oppofite fide, fo is the fine of another given angle to its oppofite fide. Thus, to find an angle, we must begin the proportion with a given fide, that is oppofite to a given angle; and to find a fide, we must begin with an angle oppofite to a given fide.

EXAMPLE. In the triangle, B D C, (fig. 4.) having the fide B D equal to 106, the fide B C equal to 65, and the angle BDC, 31 degrees, 49 minutes, to find the angle B C D, and the fide CD.

1. By geometrical Construction.

Draw a line B D equal to 106; at D, make an angle of 31° 49' by drawing DC; take 65 in the compaffes, and with

one

one foot at B, extend the other foot to C, in the line DC, then draw the line B C, and it is done: for the angle C will be 120° 43′; the angle D 31° 49'; and the angle B, 27° 28′; and the fide D C, 56.

Or, it may be wrought as follows:

180° o' The fum of three angles
59°17′ Supplement of angle C
Angle C

120 43′

31°49' Angle D

152°32' Their fum

180° o'

152°32'

27°28′ Angle B.

Here it must be noted, that when the given angle is obtufe, the angle fought will be acute; but when the given angle is acute, and oppofite to a lefs given fide, then the required angle is doubtful, whether acute or obtufe; it ought therefore to be determined before the operation be performed.

For

For the above proportion gives 59° 17′ for the required angle; but as it is obtuse, its fupplement to 180° muit be taken, viz. 120° 43'.

3. By Gunter's Line, or inftrumental Operation.

RULE. Extend the compaffes from 65 to 106 on the lines of numbers, and that extent will reach from 31° 49′ to 59° 17′ on the line of fines.

Secondly. The extent from 31° 49′ to 27° 28′ on the line of fines, will reach from 65 to 56.88 on the line of numbers.

CASE II.

When the three given Parts are two Sides and their contained Angle.

RULE. As the fum of the two given fides is to the difference of the fides, fo is the tangent of half the fum of the two oppofite angles or cotangent of half the given angle to the tangent of half the difference of those angles.

Then the half difference added to the half fum gives the greater of the two unknown angles, and fubtracted, leaves the less of the two angles,

Thus, having all the angles, the remaining third fide win be found by the former cafe.

EXAMPLE. Having the fide B C, equal to 109, B D equal to 76, (fig. 5,) and the angle C B D, 101° 30', to find the angle B DC, or B C D, and the fide C D.

1. By geometrical Conftruction.

109. and B D, fo as to make and make B D equal to 76;

Draw the line B C equal to an angle with B C, of 101° 30′, join BC and BD with a right line, and it is done; for the angle D being measured, is found to be equal to 47° 32′, the angle C 30° 58', and the fide DC 144,8.

2. Arith