A Practical Application of the Principles of Geometry to the Mensuration of Superficies and Solids

make the transverse distance from 5 to 5 equal to the given line; and the distance from 3 to 3 will be of it.

In working the proportions in trigonometry on the sector, the lengths of the sides of triangles are taken from the line of lines, and the degrees and minutes from the lines of sines, tangents, or secants. Thus in art. 135, ex. 1,

35:R:: 26: Sin 48°.

To find the fourth term of this proportion by the sector, make the lateral distance 35 on the line of lines, a transverse distance from 90 to 90 on the lines of sines; then the lateral distance 26 on the line of lines, will be the transverse distance from 48 to 48 on the lines of sines.

For a more particular account of the construction and uses of the Sector, see Stone's edition of Bion on Mathematical Instruments, Hutton's Dictionary, and Robertson's Treatise on Mathematical Instruments.

Multiplying both the numerator and denominator by cot a cot b, dividing by R2, and proceeding in the same manner, for cot(a—b) we have,

NOTE L. p. 109.

The errour in supposing that arcs less than 1 minute are proportional to their sines, can not affect the first ten places of decimals. Let AB and AB' (Fig. 41.) each equal I minute. The tangents of these arcs BT and B'T are equal, as are also the sines BS and B'S. The arc BAB' is greater than BS+B'S, but less than BT+B'T. Therefore BA is greater than BS, but less than BT: that is, the difference between the sine and the arc is less than the difference between the sine and the tangent.

Now the sine of 1 minute is

0.000290888216

And the tangent of 1 minute is 0.000290888204

The difference is

0.000000000012

The difference between the sine and the arc of 1 minute is less than this; and the errour in supposing that the sines of 1', and of 0' 52′ 44′′" 3""" 45"""" are proportional to their Eres, as in art. 223, is still less.

NOTE M. p. 110.

There are various ways in which sines and cosines may be more expeditiously calculated, than by the method which is given here. But as we are already supplied with accurate trigonometrical tables, the computation of the canon is, to the great body of our students, a subject of speculation, rather than of practical utility. Those who wish to enter into a minute examination of it, will of course consult the treatises in which it is particularly considered.

There are also numerous formulæ of verification, which are used to detect the errours with which any part of the calculation is liable to be affected. For these, see Legendre's and Woodhouse's Trigonometry, Lacroix's Differential Calculus, and particularly Euler's Analysis of Infinites.

A TABLE OF

NATURAL SINES AND TANGENTS;

TO EVERY TEN MINUTES OF A DEGREE.

IF the given angle is less than 45°, look for the title of the column, at the top of the page; and for the degrees and minutes, on the left. But if the angle is between 45° and 90°, look for the title of the column, at the bottom; and for the degrees and minutes, on the right.

D. M. Sine Tangent [Cotangent

Cosine D. M.

0° 0' 0.0000000 0.0000000 Infinite

1.0000000 90° 0' 0029089 0029089 343.77371 0.9999958 50 20 0058177 0058178 171.88540| 9999831 40 30 0087265 0087269 114.58865 9999619 40 0116353 0116361 85.939791 0° 50' 0145439 0145454 68.750037

9999323 20

9998942 89° 10'

1° 0' 0.0174524 0,0174551 57.289962 0.9998477 89° 0