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 Hence, if from the vertical angle в we demit

bc. sin A the perpendicular BD upon the base ac, dividing it into the two segments a, ß, we shall have 6 = a+ß,

R

and thence E=

"

2,2

c(x+ẞ) sin a =

R"

R Bc. sin A. But the two right

272

sin A +
angled triangles ABD, CBD,being nearly rec-
tilinear, give a=a. cos c, and ß=c. cos ▲;

whence we have

2,2

R 2r2

B

sin A. COS A.

B

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 A sin 2A, and sin c. cos c = sin 2c

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 6 and height h, of the triangle are given, then we have evidently E = 4bh 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 Level; to find its Measure when Reduced to the Level of the Sea.

that

Let 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 i being determined by the rule for the measurement of such altitudes by the barometer and thermometer, (in this volume); let в be the length of the base measured at the elvation h, and 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

rhir Bb

Hence, also B—b =

ding Bh by r+h, we

:

TB

rth

Bh

B

b

+hrth; or, by actually divi

shall have

B-6 = BX (~

Bh

h h2

h3

+3

Which is an accurate expression for the excess of в above 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 7600000000; and may therefore safely be neglected so that for all practical purposes we may assume Bb = . 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

pass through it are acted on as if they passed successively through media of increasing density, and are therefore inflected more and more towards the earth as the density augments. In consequence of this it is, that rays from objects, whether celestial or terrestrial, proceed in curves which are concave towards the earth; and thus it happens, since the eye always refers the place of objects to the direction in which the rays reach the eye, that is, to the direction of the tangent to the curve at that point, that the apparent, or observed elevations of objects, are always greater than the true ones. The differ

ence of these elevations, which is, in fact, the effect of refraction, is, for the sake of brevity, called refraction: and it is distinguished into two kinds, horizontal or terrestrial refraction, being that which affects the altitudes of hills, towers, and other objects on the earth's surface; and astronomical refraction, or that which is observed with regard to the altitudes of heavenly bodies. Refraction is found to vary with the state of the atmosphere, in regard to heat or cold, humidity or dryness, &c so that, determinations obtained for one state of the atmosphere, will not answer correctly for another, without modification. Tables commonly exhibit the refraction at different altitudes, for some assumed mean state.

:

2. With regard to the horizontal refraction the following method of determining it has been successfully practised in the English Trigonometrical Survey.

Let A, A', be two elevated stations on the surface of the earth, BD the intercepted arc of the earth's surface, c the earth's centre, AH', AH, the horizontal lines at A, A', produced to meet the opposite vertical lines CH', CH. Let a, a, represent the apparent places of the objects A, A', then is a'AA' the refraction observ

A

B

ed at A, and an A the refraction observed at A'; and half the sum of those angles will be the horizontal refraction, if we assume it equal at each station.

Now, an instrument being placed at each of the stations A, A', the reciprocal observations are made at the same instant of time, which is determined by means of signals or watches previously regulated for that purpose; that is, the observer at a takes the apparent depression of A', at the same moment that the other observer takes the apparent depression of A.

In the quadrilateral ACA'1, the two angles A, A', are right angles, and therefore the angles I and c are together equal to two right angles but the three angles of the triangle IAA'

are

are together equal to two right angles; and consequently the angles A and A' are together equal to the angle c, which is measured by the arc BD. If therefore the sum of the two depressions Ha'a, H'aa', be taken from the sum of the angles HA AH'A A' or, which is equivalent, from the angle c (which, it known, because its measure BD is known; the remainder is the sum of both refractions, or angles aa ́a, a'AA'. Hence this rule, take the sum of the two depressions from the measure of the intercepted terrestrial arc, half the remainder is the refraction

3. If, by reason of the minuteness of the contained arc BD, one of the objects, instead of being depressed, appears elevated, as suppose A' to a": then the sum of the angles a" AA and an A will be greater than the sum IAA+IA'A, or than c, by the angle of elevation a" AA'; but if from the former sum there be taken the depression HAA, there will remain the sum of the two refractions. So that in this case the rule becomes as follows: take the depression from the sum of the contained arc and elevation, half the remainder is the refraction.

5. The quantity of this terrestrial refraction is estimated by Dr. Maskelyne at one-tenth of the distance of the object observed, expressed in degrees of a great circle. So, if the · distance be 10000 fathoms, its 10th part 1000 fathoms, is the 60th part of a degree of a great circle on the earth, or 1',which therefore is the refraction in the altitude of the object at that distance.

But M. Legendre is induced, he says, by several experiments, to allow onlyth part of the distance for the refraction in altitude. So that, on the distance of 10000 fathoms, the 14th part of which is 714 fathoms, he allows only 44" of terrestrial refraction, so many being contained in the 714 fathoms. See his Memoir concerning the Trigonometrical operations, &c.

Again, M Delambre, an ingenious French astronomer, makes the quantity of the terrestrial refraction to be the 11th part of the arch of distance. But the English measurers, especially Col. Mudge, from a multitude of exact observations, determine the quantity of the medium refraction, to bethe 12th part of the said distance.

The quantity of this refraction, however, is found to vary considerably, with the different states of the weather and atmosphere, from the 4th to theth of the contained arc. See Trigonometrical Survey, vol. 1 pa. 160, 355.

Scholium

Scholium.

Having given the mean results of observations on the terrestrial refraction, it may not be amiss, though we cannot enter at large into the investigation, to present here a correct table of mean astronomical refractions. The table which has been most commonly given in books of astronomy is Dr. Bradley's, computed from the rule r = 57'' x cot (a + 3r), where a is the altitude, r the refraction, and r = 2′35′′ when a = 20o. But it has been found by numerous observations, that the refractions thus computed are rather too small.Laplace, in his Mecanique Celeste (tome iv pa. 27) deduces a formula which is strictly similar to Bradley's; for it is r=mx tan (z-nr), where z is the zenith distance, and m and n are two constant quantities to be determined from observation. The only advantage of the formula given by the French philosopher, over that given by the English astronomer, is that Laplace and his colleagues have found more correct coefficients than Bradley had.

R

Now, if R = 579-2957795, the arc equal to the radius, if we make m= (where k is a constant coefficient which,as well as n, is an abstract number,) the preceding equation will

72

become- =k× tan (z-nr). Here, as the refraction r is

R

always very small, as well as the correction nr, the trigono

metrical tangent of the arc nr may be substituted for

we shall have tan nr = k . tan (z—nr).

But nr = z—(z—nr)....z—nr = i̟z + (4z—nr) ;

R

-; thus

n

This formula is easy to use, when the coefficients 7 and

1

1+

are known and it has been ascertained, by a mean of many observations, that these are 4 and 99765175 respectively. Thus Laplace's equation becomes

sin (z-8r) =-99765175 sin z: and from this the following table has been computed. Besides the refractions, the differences of refraction, for every 10 minutes of altitude, are given; an addition which will render the table more extensively useful in all cases where great accuracy is required.

Table