Examples to the Rules: with their Solutions by Sherwin's and Taylor's

5'

× 60"

Here, if we take, in the first Example, that to be the true result which is determined by the rule, it appears that the arc is not determined from its logarithmic sine by Sherwin's Tables exactly in seconds, nor by Taylor's Tables exactly in parts of seconds (see p. 96). In the second Example, the number of seconds in the result obtained by Sherwin's Tables is right: for, although the number of seconds is not necessarily right, it may, in certain cases, happen to be right: if, instead of 7.1644398, the proposed logarithmic tangent had been 7.2022871, the arc, by Sherwin, would have been 5' 30", and wrong in seconds, since the true arc is 5′ 28′′.633.

Amongst the independent methods of ascertaining the true result, there is none more simple than that which is called, technically, the differential *. If a, a', a", &c. be successive values of a quantity, a, differing by a constant interval 1, and if the 1st, 2d, 3d, &c. differences be d', d", d", &c. then any intermediate value (A) distant from a by the interval x is equal to

x-1 a + x.d' + x. 2

d" +x.

x-1 x-2
2 3

d'" + &c.

This series, by a direct application, gives the logarithmic sine and tangent of a proposed arc: thus, suppose the logarithmic tangent of 5' 1" 12" 24" to be required:

Here, stopping at the 2d difference d" and taking 48 for its mean value, and, for greater simplicity, making the series to begin from a', in which case

as it ought to be. See the former Example, p. 258.

The arc cannot by a direct process be found, by means of the preceding series, from the logarithmic sine or tangent: but, thus it may be found :

then (by the Tables) the logarithmic tangents next less and greater are 7.1641417, the log. tan. 5' 1", and, 7.1655821 the log. tan. 5' 2": hence, taking the differences as before,

and consequently the arc =5' 1".2067 as before, see Example, p. 258. This method, although it does not determine the logarithmic sines and tangents of small arcs so concisely and conveniently as Dr. Maskelyne's Rule, is of very extensive use, and especially in Astronomy. It may be considered as the foundation of the common Rule given in books of Logarithms, for finding the logarithmic sines, tangents, &c. of arcs that contain a fractional part of a minute or second: for, in mean arcs, the differences d", d", &c. are nothing or very small, and consequently

A = a + xd'.

Thus, to find by Sherwin's Tables the log, tan. 44° 29′ 30′′.

.. since x = 1, log. tan. 44° 29′ 30′′ = 9.9921670+(.0002527) = 9.9922933. See p. 96. Also Sherwin, pp. 23, 24, &c. Hutton, p. 149, &c.

Demonstration of the Formula for Computing the Approximate Reduction to the Horizon. See p. 186.

Let A be the observed angle, & the correction, that is, let A+ x be the angle required, H and h the heights,

similarly, sin. h = h
H
H2
+&c.=1-
2
2
cos. A - Hh

cos. H=1

nearly; similarly, cos. h=1

2

cos. A-Hh

2

h2

2

Hence, cos. (1+x)=(1) (1)-(+)

1

2

2

2

(cos. A-Hh) [1-(H2 + h2)]-1

•

=

or, cos. A. cos. x sin. A. sin. r=(cos. A-Hh) [1+(H2+b2)]

nearly,

but, since x is very small, cos. x=1, and sin. x=x;
sin. A.x = cos. A Hh + cos. A (H2 + h2)

... COS.

and x =

Hh- cos. A (H2 +h2)
sin. A

orx may be differently expressed; thus,

This is the expression given by M. Legendre, Mem. Acad. 1787, p. 354; also in his Trig. edit. 6, p. 413. It is nearly equal to the first term of the series given by Delambre, Connoissance des Tems, 1793, and in the first vol. of Measure of an Arc of the Meridian, p. 140; by Suanberg, p. 38; and by Dr. Maskelyne, Phil. Trans. 1797, p. 451.

then, the series deduced by the three latter Mathematicians, is

x= n sec. H. sec. h - sec. H. sec. h. cot. A. sin. 1"

n2

2

+ n3. sec.3 H. sec.3 h (

Here the first term, since sin. agrees with Legendre's expression.

+ cot.2 A). sin.2 1" + &c.
(H+h) is nearly = (H + h),

Demonstration of Legendre's Theorem, See p. 191.

Let r be radius of sphere; a, b, c, the sides of the spherical triangle; then the sides of a similar triangle on a sphere whose

site to a, and A' the angle of a rectilinear triangle, the sides of

1+cos. A = (sin. S. sin. Sa) (sin..sin.)

S

2

S3 2.3r3

r

r

r

-1

r

-1

2.3r2

2.3r2

X bc