And, in the present enquiry, all the terms after the second may be neglected, because the 5th power of an arc of 4° divided by 120, gives a quotient not exceeding 001. Consequently, we may assume sin b = b — ¿b3, sin a = a—ža3; and thus the preceding equation will become,

b

- 6

= sin в(a — ža3) sin B- (a3. sin B

b3).

or, b = a Now, if the triangle were considered as rectilinear, we should have b = a. sin B; a theorem which manifestly gives the side b or AC too great by (a3. sin в - 63). But, neglecting quantities of the fifth order, for the reason already assigned, the last equation but one gives b3 a3. sin3 B. Therefore, by substitution, ea3. sin B(1- sin2 B): or, to have this error in seconds, take R′′ the radius expressed in seconds,

Cor. 1. If a = 42, and B = 35°16', in which case the value of sin B. cos2 B is a maximum, we shall find e =—

-41".

Cor. 2. If, with the same data, the correction be applied, to find the side c adjacent to the given angle, we should have

So that this error exists in a contrary sense to the other; the one being subtractive, the other additive.

Cor. 3. The data being the same, if we have to find the angle c, the error to be corrected will be

As to the excess of the arc over its chord, it is easy to find it correctly from the expressions in prob. 5: but for arcs that are very small, compared with the radius, a near approximation to that excess will be found in the same measures as the radius of the earth, by taking 24 of the quotient of the cube of the length of the arc divided by the square of the radius.

PROBLEM VIII.

It is required to Investigate a Theorem, by means of which, Spherical Triangles, whose Sides are Small compared with the radius, may be solved by the rules for Plane Trigonometry, without considering the Chords of the respective Arcs or Sides.

Let a, b, c, be the sides, and A, B, C, the angles of a spherical triangle, on the surface of a sphere whose radius is r;

then

L

then a similar triangle on the surface of a sphere whose radius

of brevity, we represent by a, ß, y, respectively: then, by

equa. I chap. iv, we have cos a =

cos acos.cos y
sin B. sin y

Now, r being very great with respect to the sides a, b, c, we may, as in the investigation of the last problem, omit all the terms containing higher than 4th powers, in the series for the sine and cosine of an arc, given at pa. 5 vol. ii: so shall we have, without perceptible error,

And similar expressions may be adopted for cos ß, cos y, sin y. Thus, the preceding equation will become

Multiplying both terms of this fraction by 1+(ẞ2 + y2), to simplify the denominator, and reducing, there will result, B2+22 a2 a++ B++z+ — Qa2f2 — Qa2y2 — 2ß22

COS A =

+

24B7

Here, restoring the values of a, ß, y, the second member of the equation will be entirely constituted of like combinations of the letters, and therefore the whole may be represented by

2bc

N

24bcr2

(1.)

Let, now, A' represent the angle opposite to the side a, in the rectilinear triangle whose sides are equal in length to the ares a, b, c; and we shall have

Squaring this, and substituting for cos2 A' its value 1-sin2 A',

there will result

— 4b2c2 sin2 A′ = a2 + b2 + c2 — 2a2b2 — 2a2c2 — 2b2c2 = N. So that, equa. 1, reduces to the form

Let A = Ax, then, as x is necessarily very small, its second power may be rejected, and we may assume cos a = cos a' x. sin A': whence, substituting for cos a this value of it, we

It hence appears that x is of the second order, with respect to and and of course that the result is exact to quan-.

tities within the fourth order. Therefore, because A=A'+x,

But, by prob. 2 rule 2, Mensuration of Planes, be sin a' is the area of the rectilinear triangle, whose sides are a, b, and c. Therefore A = 4' +

or A' A

area

372

area

3r2

And A'B' + c = 180° = A+B+C

Whence, since the spherical excess is a measure of the area (th. 5 ch. iv), we have this theorem: viz.

A spherical triangle being proposed, of which the sides are very small, compared with the radius of the sphere; if from each of its angles one third of the excess of the sum of its three angles above two right angles be subtracted, the angles so diminished may be taken for the angles of a rectilinear triangle, whose sides are equal in length to those of the proposed spherical triangle*.

Scholium.

We have already given, at th. 5 chap. iv, expressions for finding the spherical excess, in the two cases, where two sides and the included angle of a triangle are known, and where the three sides are known. A few additional rules may with propriety be presented here.

1. The spherical excess E, may be found in seconds, by the expression E = R; where s is the surface of the triangle:

R'S

bc, sin a = ab. sin c = ac. sin B = 풀어?.

sin B. sin c
sin (B+c)

r is

the radius of the earth, in the same measures as a, b, and c, and B" 206264" 8, the seconds in an arc equal in length to the radius.

log

R"

If this formula be applied logarithmically; then log R′′ =

1

arc 1"

5.3144251,

This curious theorem was first announced by M. Legendre, in the Memoirs of the Paris Academy, for 1787. Legendre's investigation is nearly the same as the above: a shorter investigation is given by Swanberg, at p. 40, of his "Exposition des Opérations faites en Lapponie ;" but it is defective in point of perspicuity,

2. From

2. From the logarithm of the area of the triangle, taken as a plane one, in feet, subtract the constant log 9.3267737, then the remainder is the logarithm of the excess above 180°, in seconds nearly*.

3. Since s bc. sin A, we shall manifestly have E = R" bc. sin A. Hence, if from the vertical angle B we demit

2,2

the perpendicular BD upon the base Ac, dividing it into the two segments a, ß, we shall have b

R 2r2

= a+ ß,

and thence E = c(a + B) sin a =

sin A+

212

Bc. sin a.

R

"

Bc. sin A. But the two right

angled triangles ABD, CBD, being nearly rectilinear, give a=a.cos c, and B = c. cosa; whence we have

A

272

B

a

In like manner, the triangle ABC, which itself is so small as to differ but little from a plane triangle, gives c. sin A = a. sin c. Also, sin A COS Asin 2A, and sin c . cos csin 2c (equa. xv ch. iii). Therefore, finally,

From this theorem a table may be formed, from which the spherical excess may be found; entering the table with each of the sides above the base and its adjacent angle, as argu

ments.

R"

72

4. If the base b and height h, of the triangle are given, then we have evidently E = bh. Hence results the following simple logarithmic rule: Add the logarithm of the base of the triangle, taken in feet, to the logarithm of the perpendicular, taken in the same measure; deduct from the sum the logarithm 9.6278037; the remainder will be the common logarithm of the spherical excess in seconds and decimals.

5. Lastly, when the three sides of the triangle are given in feet; add to the logarithm of half their sum, the logs. of the three differences of those sides and that half sum, divide the total of these 4 logs. by 2, and from the quotient subtract the log. 9.3267737; the remainder will be the logarithm of the spherical excess in seconds &c, as before.

One or other of these rules will apply to all cases in which the spherical excess will be required.

*This is General Roy's rule given in the Philosophical Transactions, for 1790, pa. 171.

PROBLEM

PROBLEM IX.

Given the Measure of a Base on any Elevated evel; to find its Measure when Reduced to the Level of the Sea.

Let r represent the radius of the earth, or the distance from its centre to the surface of the sea, r+h the radius referred to the level of the base measured, the altitude h being determined by the rule for the measurement of such altitudes by the barometer and thermometer, (p. 255 vol. ii, 6th edition); let в be the length of the base measured at the elevation h, and b that of the base referred to the level of the sea. Then because the measured base is all along reduced to the horizontal plane, the two, B and b, will be concentric and similar arcs, to the respective radii r+h and r. Therefore, since similar arcs, whether of spheres or spheroids, are as their radii of curvature, we have

B

Thor, by actually di

h3

13

-

Which is an accurate expression for the excess of в above b.

Bh

B

But the mean radius of the earth being more than 21 million feet, if h the difference of level were 50 feet, the second and all succeeding terms of the series could never exceed the fraction 176050503000; and may therefore safely be neglected: so that for all practical purposes we may assume B-b 113 Or, in logarithms, add the logarithm of the measured base in feet, to the logarithm of its height above the level of the sea, subtract from the sum the logarithm 7-3223947, the remainder will be the logarithm of a number, which taken from the measured base will leave the reduced base required.

PROBLEM X.

To determine the Horizontal Refraction.

1. Particles of light, in passing from any object through the atmosphere, or part of it, to the eye, do not proceed in a right line; but the atmosphere being composed of an infinitude of strata (if we may so call them) whose density increases as they are posited nearer the earth, the luminous rays which

pass