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At a given point A in a given right line AB, to make an angle of any proposed number of degrees, suppose 38 degrees. Fig. 33.
With the centre A, and a radius equal to 60 degrees, taken from a scale of chords, describe an arc, cutting AB in m; from the same scale of chords, take 38 de. grees and apply it to the arc from m to n, and from A through n draw the line AC, then will the angle A contain 38 degrees.
Note.-Angles of more than 90 degrees are usually taken off at twice.
To measure a given angle A. Fig. 34. Describe the arc mn with the chord of 60 degrees, as in the last problem. Take the arc mn between the dividers, and that extent applied to the scale of chords, will show the degrees in the given angle.
Note. When the distance mn exceeds 90 degrees, it must be taken off at twice, as before.
OF THE TABLE
LOGARITHMIC OR ARTIFICIAL SINES, TANGENTS, &e.
This table contains the logarithms of the sine, tangent, &c. to every degree and minute of the quadrant, the radius being 10000000000, and consequently its logarithm 10.
Let the radius CB, Fig. 32, be supposed to consist of 10000000000 equal parts as above, and let the quadrant DB be divided into 5400 equal arcs, each of these will therefore contain l'; and if from the several points of division in the quadrant, right lines be drawn perpendicular to CB, the sine of every minute of the quadrant, to the radius CB will be exhibited. The lengths of these lines being computed and arranged in a table, constitute what is usually termed a table of natural sines. The logarithms of those numbers taken from a table of logarithms and properly arranged form the table of loga. rithmic or artificial sines. In like manner the logarithmic tangents and secants are to be understood.
The method by which the sines are computed is too abstruse to be explained in this work, but a familiar exposition of this subject as well as the construction of lo. garithms may be seen in Simpson's Trigonometry.
To find, by the table, the sine, tangent, &c. of an arc or
angle. If the degrees in the given angle be less than 45, look for them at the top of the table, and for the minutes, in the
left hand column; then in the column marked at the top of the table, sine, tangent, &c: and against the minutes, is the sine, tangent, &c. required. If the degrees are more than 45, look for them at the bottom of the table, and for the minutes, in the right hand column; then in the column marked at the bottom of the table, sine, tangent, &c. and against the minutes, is the sine, tangent, &c. re. quired.
Note. The sine of an angle and of its' supplement being the same, if the given number of degrees be above 90. subtract them from 180°, and find the sine of the remainder.
EXAMPLES. 1. Required the sine of 32° 27' 2. Required the tangent of 57° 39' 3. What is the secant of 890 31 4. What is the sine of 157° 43'
To find the degrees and minutes, corresponding to a given
sine, tangent, &c. Find, in the table, the nearest logarithm to the given one, and the degrees answering to it will be found at the
top of the table if the name be there, and the minutes on the left hand; but if the name be at the bottom of the table, the degrees must be found at the bottom, and the minutes at the right hand.
1. Required the degrees and minutes in the angle 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 seale; 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 subdivi. sions 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 I be considered
as a unit, then the first 1, 2, 3, &c. stand for 1, 2, 3, &c. the middle 1 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.
If the first 1 be considered as is, of a unit, the 2, 3, 4, &c. following will be jo, io, io, &c. and the middle 1, and
4 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 found 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