"What then is man, that thou art mindful of him, or the son of man, that thou visitest him?

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Thy power is circumscribed by no bounds, both great and small are alike unto thee.

"From the sun in the firmament of heaven, to the sand on the sea shore, all is the operation of thy hand.

"From the cherubim and seraphim which stand before thee, to the worm in the bowels of the earth, all living creatures receive of thee what is good and expedient for them."*

Praise then the Lord, O my soul, praise his name for ever and ever.

*See "Hymns, to the Supreme Being, in imitation of the Eastern Songs," London, 1780,

ESSAY IV.

AN

INTRODUCTION

ΤΟ

PRACTICAL ASTRONOMY.

THERE is no part of mathematical science more truly calculated to interest and surprize mankind, than the measurement of the relative positions and distances of inaccessible objects.

To determine the distance of a ship seen on a remote spot of the unvaried face of the ocean, to ascertain the height of the clouds and meteors which float in the invisible fluid above our heads, or to shew with certainty the dimensions of the sun, and other bodies, in the heavens, are among the numerous problems which, to the vulgar, appear far beyond the reach of human art, but which are nevertheless truly resolved by the incontrovertible principles of the mathematics.

These principles, simple in themselves, and easy to be understood, are applied to the construction of a variety of instruments; and the following pages

contain an account of their use in the quadrant and the equatorial.

The position of any object, with regard to a spectator, can be considered in no more than two ways; namely, as to its distance, or the length of a line supposed to be drawn from the eye to the object; and as to its direction, or the situation of that line with respect to any other lines of direction; or, in other words, whether it lies to the right or left, above or below those lines. The first of these two modes bears relation to a line absolutely considered, and the second to an angle. It is evident, that the distance can be directly come at by no other means than by mea suring it, or successively applying some known measure along the line in question; and therefore, that in many cases the distance cannot be directly found; but the position of the line, or the angle it forms, with some other assumed line, may be readily ascer tained, provided this last line do likewise terminate in the eye of the spectator. Now the whole artifice in measuring inaccessible distances consists in finding their lengths, from the consideration of angles, observed about some other line, whose length can be submitted to actual mensuration. How this is done I shall proceed to shew.

Every one knows the form of a common pair of compasses. If the legs of this instrument were mathematical lines, they would form an angle greater or less, in proportion to the space the points would have passed through in their opening. Suppose an are of a circle to be placed in such a manner, as

to be passed over by these points, then the angles will be in proportion to the parts of the arc passed over; and if the whole circle be divided into any number of equal parts, as, for example, 360, the number of these comprehended between the points of the compasses will, denote the magnitude of the angle. This is sufficiently clear; but there is another circumstance which beginners are not always sufficiently aware of, and which, therefore, requires to be well attended to: it is, that the angle will be neither enlarged nor diminished by any change in the length of the legs, provided their position remains unaltered; because it is the inclination of the legs, (and not their length) or the space between them, which constitutes the angle. So that if a pair of compasses, with very long legs, were opened to the same angle as another smaller pair, the intervals between their re spective points would be very different; but the number of degrees on the circles, supposed to be applied to each, would be equal, because the degrees themselves on the smaller circle would be exactly proportioned to the shortness of the legs. This property renders the admeasurement of angles very easy, because the diameter of the measuring circle may be varied at pleasure, as convenience requires.

In practice, however, the magnitude of instruments is limited on each side. If they are made very large, they are difficult to manage; and their weight, bearing a high proportion to their strength, renders them liable to change their figure, by bending when their position is altered: but, on the contrary, if

they are very small, the errors of construction and graduation amount to more considerable parts of the divisions on the limbs of the instrument.

GENERAL PRINCIPLES OF CALCULATION.

Before we proceed any farther, I shall slightly notice the general principles of the calculations we are going to use.

Plane Trigonometry is the art of measuring and computing the sides of plane triangles, or of such whose sides are right lines.

In most cases of practice, it is required to find lines or angles whose actual admeasurement is difficult or impracticable. These mathematicians teach us to discover by the relation they bear to other given lines or angles, and proper methods of calculation.

Finding the comparison of one right-line with another right-line more easy than the comparison of a right-line with a curve, they measure the quantities of the angles not by the arc itself, which is described on the angular point, but by certain lines described about that point.

If any three parts of a triangle are known, the remaining unknown parts may be found either by construction or by calculation.

If two angles of a triangle are known in degrees and minutes, the third is found by subtracting their sum from 180 degrees; but if the triangle be right