is equal to of of an inch. From the zero point, therefore, to the second division of the vernier is .22 inch, to the third .33, and so on.

To measure any line by the scale, take the distance in the compasses, and move them along the scale until you find that they exactly extend from some division on the vernier to a division on the scale. Add the number on the scale to the number on the vernier for the distance required. Thus, suppose the compasses extended from 66 on the vernier to 110 on the scale, the length is 176.

To lay off a distance by the scale, for example 175, take 55 from 175, and 120 is left: extend the compass from 120 on the scale to 55 on the vernier. To lay off 268 = 180+ 88, extend the compasses from 180 on the scale to 88 on the vernier, as marked by the arrow heads.

The vernier scale is equally accurate with the diagonal scale, and much more readily made.

SECTION III.

TABLES OF TRIGONOMETRICAL FUNCTIONS.

129. Table of Natural Sines and Cosines. THIS table (page 87 of the Tables) contains the sines and cosines to five decimal places for every minute of the quadrant. The table is calculated to the radius 1. As the sine and cosine are always less than radius, the figures are all decimals. In the table the decimal point is omitted. If the sine and cosine is wanted to any other radius, the number taken from the table must be multiplied by that radius.

To take out the sine or cosine of an arc from this table, look for the degrees, if less than 45, at the top of the table, and for the minutes at the left; then, in the column headed properly, and opposite the minutes, will be the function required. If the degrees are 45 or upwards they will be

found at the bottom, and the minutes at the right. The name of the column is at the bottom.

Thus, the sine of 32° 17', found under 32° and opposite 17', is .53411.

The cosine of 53° 24', found over 53° and opposite 24′ in the right-hand column, is .59622.

130. The table of natural sines and cosines is of but little use in trigonometrical calculations, these being generally performed by logarithms. It is principally employed in determining the latitudes and departures of lines.

131. Table of Logarithmic Sines, Cosines, &c. This table contains the logarithms of the sines, cosines, tangents, and cotangents, to every minute of the semicircle, the radius being 10 000 000 000 and its logarithm 10. The logarithmic sine of 90°, cosine of 0°, tangent of 45°, and cotangent of 45°, is each 10.

The sine, cosine, tangent, and cotangent, of every arc being equal to the sine, cosine, tangent, and cotangent, of its supplement, and also to the cosine, sine, cotangent, and tangent, of its complement, the table is only extended to forty five pages, the degrees from 0 to 44 inclusive being found at the top, those from 45 to 135 at the bottom, and from 136 to 180 at the top. The minutes are contained in the two outer columns, and agree with the degrees at the top and bottom on the same side of the page.

The columns headed Diff. 1" contain the difference of the function for a change of 1" in the arc. These differences are calculated by dividing the differences of the successive numbers in the columns of the functions by 60. By an inspection of these columns of difference it will be seen that, except in the first few pages, they change very slowly. In these, in consequence of the rapid change of the function, the differences vary very much. The difference set down will not, therefore, be accurate, except for about the middle of the minute. The calculations for seconds, therefore, are not in these cases to be depended on. To obviate this inconvenience, and give to the first few pages a degree

of accuracy commensurate with that of the rest of the table, the sines and tangents are calculated to every 10 seconds, and these are the same as the cosines and cotangents of arcs within two degrees of 90.*

132. Use of Table. To take out any function from the table, seek the degrees, if less than 45° or more than 135°, at the top of the page, and the minutes in the column on the same side of the page as the degrees. Then, in the proper column, (the title being at the top,) and opposite the minutes, will be found the value required.

If the degrees are between 45° and 135°, seek them at the bottom of the page, the minutes being found, as before, at the same side of the page as the degrees. The titles of the columns are also at the bottom.

EXAMPLES.

Ex. 1. Required the sine of 37° 17'.

Ans. 9.782298.

Ex. 2. Required the cosine of 127° 43'. Ans. 9.786579. Ex. 3. Required the cotangent of 163° 29′.

Ex. 4. Required the tangent of 69° 11'.

Ans. 10.527932.

Ans. 10.419991.

133. If there are seconds in the arc, take out the function for the degrees and minutes as before. Multiply the number in the difference column by the number of seconds, and add the product to the number first taken out, if the function is increasing, but subtract, if it is decreasing the result will be the value required.

If the arc is less than 90° the sine and tangent are increasing, and the cosine and cotangent are decreasing; but if the arc is greater than 90° the reverse holds true.

*The rectangle of the tangent and cotangent of an arc being equal to the square of radius, their logarithms are arithmetical complements (to 20) of each other. Our column of differences serves for both these functions. It is placed

between them.

Ex. 1. What is the tangent of 37° 42′ 25′′?

Ex. 2. What is the cosine of 129° 17' 53"?

Ex. 3. What is the sine of 63° 19′ 23′′?

Ans. 9.951120.

Ex. 4. What is the cosine of 57° 28' 37"?

Ans. 9.730491.

Ex. 5. What is the tangent of 143° 52′ 16′′?

Ex. 6. What is the sine of 172° 19′ 48"?

Ans. 9.863314.

Ans. 9.125375.

If the sine or tangent of an arc less than 2° or more than 178°, or the cosine or cotangent of an arc between 88° and 92°, is required, it should be taken from the first pages of the table. Take out the function to the ten seconds next less than the given arc, multiply one tenth of the difference between the two numbers in the table by the odd seconds, and add or subtract as before.

The cotangent of an arc less than 2° may be found by taking out the tangent, and subtracting it from 20.000000; so likewise the tangent of an arc between 178° and 180° is found by taking the complement to 20.000000 of its cotangent.

Ex. 1. Required the sine of 1° 27′ 36′′.

Sine of 1° 27' 30" is

of difference

Difference 6"

8.405687

82.6

Sine of 1° 27′ 36′′

8.406183

Ex. 2. What is the cosine of 88° 18' 48"?

Ans. 8.468844.

Ex. 3. What is the sine of 179° 19′ 13"?

Ans. 8.074198.

134. To find the Arc corresponding to any Trigonometric Function.

If degrees and minutes only be required, seek, in the proper column, the number nearest that given; and if the title is at the top the degrees are found at the top, and the minutes under the degrees; but if the title is at the bottom the degrees are at the bottom, and the minutes on the same side as the degrees.

If seconds are desired, seek for the number corresponding to the minute next less than the true arc, and take the difference between that number and the given one: divide said difference by the number in the difference column, for the seconds.

Ex. 1. What is the arc whose sine is 9.427586?

9.427586

Sine of 15° 31' is

9.427354

7.58) 232.00 (31′′

The arc is, therefore, 15° 31′ 31′′.

227 4

4.60