TABLE

Of Chords corresponding to every 100 feet on curve from 200 to 1000 feet, calculated to every 15 minutes' rate of curvature, from 15 minutes to 8 degrees, radius of 1° being 5730 feet.

Rate of

curvature.

200 feet. 300 feet. 400 feet. 500 feet. 600 feet. 700 feet. 800 feet. 900 feet. 1000 ft.

15'

'30'

45'

1°

199.99

299-97 399-92

499.85

599:73

200.00 300.00 400.00 499.99 599.98 699.97 799.96 899-94 999.92
200:00 299.99 399.98 499.96 599.93 699.89 799.84 899.77
200 299.98 399.95 499.91 599.84 699.76 799.64 899.49
699.57

999.69

999.30

799-36 899.09

998-75

TABLES OF NATURAL AND LOGARITHMIC VERSED SINES, AND EXTERNAL SECANTS.

On the Construction of the Tables of Versed Sines and External

Secants.

In the above figure it is required to find the value of versed sine FBCG, of arc B CAB angle a, and of external secant CD in terms of sine CF and tangent B D.

The chord BC = 2 sine † B C, and angle FCB is measured by arc A Barc B C. AB

Therefore making chord B C radius, B F will be the sine of angle FCB, and we have:

Versed sine BF 2 x sine FCB

2 × (sine a)2.

That is, twice the square of sine of half given arc versed sine. Making CF radius. BF becomes tangent, and we have, versed sine BFCF x tangent FCB, or sine a x tangenta. Now by similar triangles v. s. a: ex. sec. a :: cos, a: radius; v. s. a ex. sec. a :: sine a tangent a;

and

or, ex. sec. a = v. s. a x radius
ax

cosine a

tan. a x tangenta.

Then log. v. s. a = log. sine a + log. tan. † a (10=log. of R.), and log. ex. sec. a= log. v. s. a 10-log. cos. a.; or, log. ex. sec. a = log. tan. a + log. tan. 1 a 10.