5. If any number of straight lines meet in a common point (V) they form a number of angles occupying all the angular space around that point. The whole of that angular space is supposed likewise to consist of 360 equal parts, called also Degrees, which are divided into Minutes, Seconds, &c., like the degrees of a circle.

Degrees are marked by a small circle over them; minutes by one accent; seconds, by two accents, &c. : thus 6° 4' 14" 37"" means 6 degrees, 4 minutes, 14 seconds, 37 thirds.*

6. An arc or angle is thus measured by the number of degrees, minutes, &c., which it contains; and, in Trigonometry, when we speak of an angle, we commonly mean the number of degrees, &c., in the angle.

COROLLARY. If any number of lines meet in a point V, and a circle be described about that point as a centre, with any radius, then any angle, as BVC, is the same part of all the angles at V, that the arc BC is of the whole circumference: that is, each is the same part of 360°; or the angle BVC contains the same number of degrees as the arc BC. Hence, in the same cir

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cle, or in equal circles, angles at the centre are said to be measured by the arcs on which they stand.

7. The Complement of an arc or of an angle is what it wants of 90°. The Supplement is what it wants of 180°.

Thus the arc BG is the complement, and BA the supplement, of the arc BC; and the angle BOG is the complement, and BOA the supplement, of the angle BOC.

G

B

* Minutes are frequently divided decimally, and sometimes the degrees themselves are so divided, thus dispensing, in the former case, with the use of seconds, thirds, &c., and, in the latter case, with that of minutes also.

8. Let BC be an arc of a circle and O its centre. OB, OC, be radii drawn to its extremities from B and C let BD and CF be drawn perpendicular to OC and both terminated by the lines* of the two radii. Then BD is called the Sine of the arc BC; DC is its Versed Sine; CF, its Tangent; and OF its Secant.

EXERCISE. Let the scholar draw a diagram like the first on this page, and exhibit, upon it, the sine, the versed

B

Let

F

sine, the tangent, and the secant, first of the arc SP, then, successively, of the arcs NS, RN, and MR.

9. Returning to the arc BC, let BG be its complement. Then BH, the sine of the complement, is called the Cosine of the arc BC: GH, the versedsine of the complement, is called the Co-versed-sine of BC: and, in like manner, GI is called the Cotangent, and OI the Cosecant of BC.

EXERCISE. In the annexed diagram let the scholar point out the sine, versed-sine, tangent, and secant of the arc BC; then the sine, &c., of the arc BG; then the cosine, co-vefsed-sine, cotangent, and cosecant, of the arc BC; then the cosine, &c., of the arc BG.

G

I

H

B

G

I

H

B

C

C

D

By the expression lines of the radii, is meant the radii produced as far as may

be necessary either way.

11. The sine, tangent, &c., of the arc BC, are often called the sine, tangent, &c., of the angle BOC, to the radius OC.

12. Sometimes also they are called the sine, tangent, &c., of the number of degrees contained in the arc BC or in the angle BOC, the radius being at the same time named or understood.

Thus, if the arc BC, or the angle BOC, is of 50°, and the radius OC is 10 of any equal parts, then BD is called the sine of 50°, CF the tangent of 50°, &c., to the radius 10.

EXERCISE 1. If the radius is 10, what is the sine ot 90°; what is the chord of 60°; what, the sine of 30°; what, the cosine of 60°; and what, the versed-sine of 60°?

2. If the radius is 1, what is the tangent of 45°; what, the cotangent of 45°; what, the secant of 60°; what, the cosine of 0°; and what, the tangent of 90° ?

3. The radius being 1, compute the sine and tangent of 60°, and the secant of 45°.*

13. When the arc BC is greater I than a quadrant, the sine BD, the versed-sine DC, the tangent CF, the secant OF, the cosine BH or OD, the co-versed-sine HG, the cotangent GI, and the cosecant OI, occupy the places shown in the annexed diagram.

B

D

G

H

C

COR. The sine of an arc is equal to the sine of its supplement.

EXERCISE. What is the sine of 30°; what, of 150°; what, of 60°; and what, of 120°; the radius being 100?

14. For use in calculation, the sines, tangents, secants, and versed-sines, of arcs, for every minute in the quadrant, to the radius 1, have been computed and arranged in tables; and, on the discovery of logarithmic arithmetic, the logarithms of the sines, tangents, secants, and versed

The scholar is supposed to be acquainted with the first principles of Practical Geometry and of Mensuration, and nothing more is required for the above Exercises. The answers, when involving decimal fractions, may carried out as far as the teacher pleases.

be

sines, for every minute of the quadrant, to the radius 10,000;000,000, were arranged in similar tables. The latter are called Logarithmic Sines, Tangents, &c.: they may be found in Table III of this volume. In contradistinction to these, the former are called Natural Sines, Tangents, &c.; but such is the facility which logarithms afford, that the natural sines, &c., have now been almost superseded by the logarithmic, and are seldom used.

15. The words sine, tangent, secant, versed-sine, cosine, cotangent, cosecant, and co-versed-sine, are often abbreviated thus: sin, tan, sec, versin, cos, cot, còsec, and coversin.

CHAPTER VI.

ON THE USE OF THE TABLES OF SINES, TANGENTS, AND SECANTS.*

PROBLEM I.

To find, from Table III, the Logarithmic Sine, Tangent, Secant, Cosine, Cotangent, or Cosecant of an acute Angle † of any given number of Degrees and Minutes.

RULE. If the given number of degrees is less than 45, look for the degrees in the top line of the table, turning over the leaves till the proper page is found. In that page, look in the second line for the name of the column wanted; and on the left margin of the page for the given number of minutes.

But, if the given number of degrees is not less than 45, look for the degrees in the bottom line of the page; for the name of the column, in the second line from the bottom; and for the minutes, on the right margin.

In either case, having found the minutes, then, in the same line, in its proper column, will be found the logarithm wanted.

*These, of course, also include cosines, cotangents, and cosecants, which are merely sines, tangents, and secants of the complemental arcs or angles.

Or of an arc less than a quadrant. But it is properly angles and not arcs on which Trigonometry is employed. Some given radius is understood, which is, of course, the radius of the table used.

EXERCISE 1. What is the logarithmic sine of 9° 10′?

2. What is the log. tangent of 37° 26'?

3. Find the log. secant of 64° 19′.

4. Find the log. cosine of 88° 44'.

Ans. 9.202234.

Ans. 9.883934. Ans. 10.363114. Ans. 8.344504.

PROBLEM II.

To find, from Table III, the Logarithmic Sine, Tangent, or Secant of an Acute Angle of any given number of Degrees, Minutes, and Parts of a Minute.

RULE. Neglect the parts of a minute; but, if they amount to more than half a minute, increase the number of minutes by a unit. Then proceed as in the last problem.

EXAMPLE 1. What is the logarithmic sine of 22° 32' 48" ?

Log sine of 22° 33'............9′583754, Ans.

EXAMPLE 2. What is the log tangent of 28° 47'?

Log tan 28° 48'............9-740169, Ans.

EXAMPLE 3. Find the log. secant of 56° 22·38′.

Log sec 56° 22'....

..10.256587, Ans.

COR. The logarithmic cosine, cotangent, and cosecant may be found in the same manner as the sine, tangent, and secant.

EXAMPLE 4. Find the logarithmic cosine of 26° 13·55'.

Log cos 26° 14'.

.9.952793, Ans.

EXERCISE. Find the logarithmic tangent of 33° 17′ 57′′; the log. sine of 68° 153'; and the log. secant of 49° 28.84'. Answers: 9.817484; 9.967927; and 10·187308.