a tangent to the surface at N, then will this line represent the visual ray along which the spectator at M will see the visible horizon; and as this tangent sweeps round M, and comes successively into the positions MO o, MP p, M Q q, the point of contact N will mark out on the surface the circle NOPQ. The area of the spherical surface compre

hended within this circle is the portion of the earth's surface visible to a spectator at M, and the angle N M Q included between the two extreme visual rays is the measure of its apparent angular diameter. Leaving, at present, out of consideration the effect of refraction in the air below M, of which more hereafter, and which always tends, in some degree, to increase that angle, or render it more obtuse, this is the angle measured by the dip sector. Now, it is evident, 1st, that as the point M is more elevated above m, the point immediately below it on the sphere, the visible area, i. e. the spherical segment or slice N O P Q, increases; 2dly, that the distance of the visible horizon* or boundary of our view from the eye,

* "Opigw, to terminate.

viz. the line MN, increases; and, 3dly, that the angle NMQ becomes less obtuse, or, in other words, the apparent angular diameter of the earth diminishes, being nowhere so great as 180°, or two right angles, but falling short of it by some sensible quantity, and that more and more the higher we ascend. The figure exhibits three states or stages of elevation, with the horizon, &c. corresponding to each, a glance at which will explain our meaning; or, limiting ourselves to the larger and more distinct, MNOPQ, let the reader imagine n N M, MQ7 to be the two legs of a ruler jointed at M, and kept extended by the globe N m Q between them. It is clear, that as the joint M is urged home towards the surface, the legs will open, and the ruler will become more nearly straight, but will not attain perfect straightness till M is brought fairly up to contact with the surface at m, in which case its whole length will become a tangent to the sphere at m, as is the line xy.

(23.) This explains what is meant by the dip of the horizon. Mm, which is perpendicular to the general surface of the sphere at m, is also the direction in which a plumb-line* would hang; for it is an observed fact, that in all situations, in every part of the earth, the direction of a plumb-line is exactly perpendicular to the surface of still water; and, moreover, that it is also exactly perpendicular to a line or surface truly adjusted by a spirit-level. Suppose, then, that at our station M we were to adjust a line (a wooden ruler for instance) by a spirit-level, with perfect exactness; then, if we suppose the direction of this line indefinitely prolonged both ways, as XMY, the line so drawn will be at right angles to Mm, and therefore parallel to a my, the tangent to the sphere at A spectator placed at M will therefore see not only all the vault of the sky above this line, as X ZY, but also that portion or zone of it which lies between X N and Y Q; in other words, his sky will be more than a hemisphere by the zone YQXN. It is the angular breadth of this redundant zone-the angle YMQ, by which the visible horizon appears depressed below the direction of a spirit-level- that is called See these instruments described in Chap. III.

m.

the dip of the horizon. It is a correction of constant use in nautical astronomy.

(24.) From the foregoing explanations it appears, 1st, That the general figure of the earth (so far as it can be gathered from this kind of observation) is that of a sphere or globe. In this we also include that of the sea, which, wherever it extends, covers and fills in those inequalities and local irregularities which exist on land, but which can of course only be regarded as trifling deviations from the general outline of the whole mass, as we consider an orange not the less round for the roughness on its rind. 2dly, That the appearance of a visible horizon, or sea-offing, is a consequence of the curvature of the surface, and does not arise from the inability of the eye to follow objects to a greater distance, or from atmospheric indistinctness. It will be worth while to pursue the general notion thus acquired into some of its consequences, by which its consistency with observations of a different kind, and on a larger scale, will be put to the test, and a clear conception be formed of the manner in which the parts of the earth are related to each other, and held together as a whole.

(25.) In the first place, then, every one who has passed a little while at the sea side is aware that objects may be seen perfectly well beyond the offing or visible horizon—but not the whole of them. We only see their upper parts. Their bases where they rest on, or rise out of the water, are hid from view by the spherical surface of the sea, which protrudes between them and ourselves. Suppose a ship, for instance, to sail directly away from our station; at first, when the distance of the ship is small, a spectator, S, situated at some certain height above the sea, sees the whole of the ship, even to the water line where it rests on the sea, as at A. As it recedes it diminishes, it is true, in apparent size, but still the whole is seen down to the water line, till it reaches the visible horizon at B. But as soon as it has passed this distance, not only does the visible portion still continue to diminish in apparent size, but the hull begins to disappear bodily, as if sunk below the surface. When it has reached a certain

T

distance, as at C, its hull has entirely vanished, but the masts and sails remain, presenting the appearance c. But if, in this state of things, the spectator quickly ascends to a higher station, T, whose visible horizon is at D, the hull comes again in sight; and, when he descends again, he loses

D

E

it. The ship still receding, the lower sails seem to sink below the water, as at d, and at length the whole disappears: while yet the distinctness with which the last portion of the sail d is seen is such as to satisfy us that were it not for the interposed segment of the sea, ABCDE, the distance TE is not so great as to have prevented an equally perfect view of the whole.

(26.) The history of aëronautic adventure affords a curious illustration of the same principle. The late Mr. Sadler, the celebrated aëronaut, ascended on one occasion in a balloon from Dublin, and was wafted across the Irish Channel, when, on his approach to the Welsh coast, the balloon descended nearly to the surface of the sea. By this time the sun was set, and the shades of evening began to close in. He threw out nearly all his ballast, and suddenly sprang upwards to a great height, and by so doing brought his horizon to dip below the sun, producing the whole phenomenon of a western sunrise. Subsequently descending in Wales, he of course witnessed a second sunset on the same evening.

(27.) If we could measure the heights and exact distance of two stations which could barely be discerned from each other over the edge of the horizon, we could ascertain the actual size of the earth itself: and, in fact, were it not for the effect of refraction, by which we are enabled to see in some small degree round the interposed segment (as will be hereafter explained), this would be a tolerably good method of ascertaining it. Suppose A and B to be two eminences, whose perpendicular heights A a and Bỏ (which

for simplicity, we will suppose to be exactly equal) are known, as well as their exact horizontal interval a Db, by measurement; then it is clear that D, the visible horizon of both, will lie just half-way between them, and if we suppose a Db to be the sphere of the earth, and C its centre in the figure C Db B, we know Db, the length of the arch of the circle between D and b, - viz. half the measured interval, and b B, the excess of its secant above its radius which is the height of B,- data which, by the solution of an easy geometrical problem, enable us to find the length of the radius D C. If, as is really the case, we suppose both the heights and distance of the stations inconsiderable in comparison with the size of the earth, the solution alluded to is contained in the following proposition:

The earth's diameter bears the same proportion to the distance of the visible horizon from the eye as that distance does to the height of the eye above the sea level.

When the stations are unequal in height, the problem is a little more complicated.

(28.) Although, as we have observed, the effect of refraction prevents this from being an exact method of ascertaining the dimensions of the earth, yet it will suffice to afford such an approximation to it as shall be of use in the present stage of the reader's knowledge, and help him to many just conceptions, on which account we shall exemplify its application in numbers. Now, it appears by observation, that two points, each ten feet above the surface, cease to be visible from each other over still water, and in average atmospheric circumstances, at a distance of about 8 miles. But 10 feet is the 528th part of a mile, so that half their distance, or 4 miles, is to the height of each as 4 × 528 or 2112:1, and therefore in the same proportion to 4 miles is the length of the earth's diameter. It must, therefore, be equal to