2. Required the degrees and minutes in the angle whose tangent is 10.47464. Ans. 71° 28'.

ON GUNTER'S SCALE.

GUNTER'S Scale is an instrument by which, with a pair of dividers, the different cases in trigonometry, and many other problems may be solved.

It has on one side, a diagonal scale, and also the lines of chords, sines, tangents, and secants, with several others.

On the other side there are several logarithmic lines as follow:

The line of numbers marked Num., is numbered from the left hand of the scale towards the right, with 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 which stands in the middle of the scale; the numbers then go on 2, 3, 4, 5, 6, 7, 8, 9, 10 which stands at the right hand end of the scale. These two equal parts of the scale are similarly divided, the distances between the first 1, and the numbers 2, 3, 4, &c. being equal to the distances between the middle 1, and the numbers 2, 3, 4, &c. which follow it. The subdivisions of the two parts of this line are likewise similar, each primary division being divided into ten parts, distinguished by lines of about half the length of the primary divisions.

The primary divisions on the second part of the scale, are estimated according to the value set upon the unit on the left hand of the scale: If the first 1 be considered

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as a unit, then the first 1, 2, 3, &c. stand for 1, 2, 3, &c. the middle is 10, and the 2, 3, 4, &c. following stand for 20, 30, 40, &c. and the ten at the right hand for 100. If the first 1 stand for 10, the first 2, 3, 4, &c. must be counted 20, 30, 40, &c. the middle 1 will be 100, the second 2, 3, 4, &c. will stand for 200, 300, 400, &c. and the 10 at the right hand for 1000.

3 4

If the first 1 be considered as , of a unit, the 2, 3, 4, &c. following will be,,, &c. and the middle 1, and the 2, 3, 4, &c. following will stand for 1, 2, 3, 4, &c.

The intermediate small divisions must be estimated according to the value set upon the primary divisions.

Sines. The line of sines, marked Sin. is numbered from the left hand of the scale towards the right, 1, 2, 3, 4, &c. to 10, then 20, 30, 40, &c. to 90, where it terminates just opposite 10 on the line of numbers.

Tangents. The line of tangents, marked Tan. begins at the left hand, and is numbered 1, 2, 3, &c. to 10, then 20, 30, 40, 45, where there is a brass pin, just under 90 in the line of sines; because the sine of 90° is equal to the tangent of 45°. From 45 it is numbered towards the left hand 50, 60, 70, 80, &c. The tangent of arcs above 45° are therefore counted backward on the line, and are fɔund at the same points of the line as the tangents of their complements.

There are several other lines on this side of the scale, as Sine Rhumbs, Tangent Rhumbs, Versed Sines, &c.; but those described are sufficient for solving all the pro

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Remarks on Angles, Triangles, c.

1. If from a point D in a right line AB, one or more right® lines be drawn on the same side of it, the angles thus formed at the point D will be together equal to two right angles, or 180°; thus ADE + EDB = two right angles, or 180o: also ADC + CDF + EDB = two right angles, or 180°. Fig. 35.

2. Since the angles thus formed at the point D, on the other side of AB would also be equal to two right angles, the sum of all the angles formed about a point is equal to four right angles or 360°.

3. If two right lines cut one another, the opposite angles will be equal: thus AEC BED and AED = CEB. Fig. 36.

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4. The sum of the three angles of a plane triangle is equal to two right angles, or 180o.

5. If the sum of two angles of a triangle be subtracted from 180°, the remainder will be the third angle.

6. If one angle of a triangle be subtracted from 180°, the remainder will be the sum of the other two angles.

7. In right angled triangles, if one of the acute angles be subtracted from 90°, the remainder will be the other acute angle.

8. The angles at the base of an isosceles triangle are equal to one another.

9. If one side of a triangle be produced, the external

posite angles: thus the external angle CBD, of the triangle ABC, is equal to the sum of the internal and opposite angles A and C. Fig. 37.

10. The angle at the centre of a circle is double of the angle at the circumference, upon the same base, that is, upon the same part of the circumference: thus the angle. BEC is double of the angle BAC. Fig. 38.

11. The angles in the same segment of a circle are equal to one another: thus the angle BAD is equal to the angle BED; also the angle BCD is equal to the angle BFD. Fig. 39.

12. The angle in a semicircle is a right angle; thus the angle ECF, Fig. 45, is a right angle.

13. This mark' placed on the sides or in the angles of a triangle, indicates that they are given; and this mark ° placed in the same way, indicates that they are required.

PRACTICAL RULES FOR SOLVING ALL THE CASES OF PLANE TRIGONOMETRY.

CASE 1.

The angles and one side of any plane triangle being given, to find the other sides.

RULE.

As the sine of the angle opposite the given side,

Is to the sine of the angle opposite the required side,
So is the given side,

To the required side.*

* DEMONSTRATION. Let ABC, Fig. 40, be any plane triangle; take

Note 1.-The proportions in trigonometry are worked by logarithms; thus, from the sum of the logarithms of the second and third terms, subtract the logarithm of the first term, and the remainder will be the logarithm of the fourth term.

2. The logarithmic sine of a right angle or 90° is 10.00000, being the same as the logarithm of the radius.

EXAMPLES.

1. In the triangle ABC, there are given the angle A= 32° 15', the angle B=114° 24', and consequently the angle C = 33° 21', and the side AB = 98;* required the sides AC and BC.

By Construction, Fig. 41.

Make AB equal to 98 by a scale of equal parts, and draw AC, making the angle A = 32° 15′; also make the angle B = 114° 24' and produce BC, AC, till they meet in C, then is ABC the triangle required; and AC, measured by the same scale of equal parts, is 162, and BC is 95.

will be the sines of the angles A and B to the equal radii AC and BF. Now the triangles BDC and BEF being similar, we have CD: FE:: BC: BF:or AC; that is sin. A: sin. B::BC: AC. In like manner it is proved, that sin. A: sin. C:: BC: AB. When one of the angles is obtuse, the demonstration is the same. Hence it appears, that in any plane triangle, the sides are to one another as the sines of their opposite angles.

* This 98 may express so many feet, or yards, &c., and the other