Let S be an object whose altitude is to be taken by a fore observation, by bringing its image in contact with the apparent horizon at P; then the angle SOP will be the apparent altitude, which is evidently greater than the true altitude SOH by the arc PH, expressed by the angle of horizontal depression POH. But if the altitude of the object S is to be taken by a back observation, then, the observer's back being necessarily turned to the object, his apparent horizon will be in the direction OF, and his whole horizontal plane represented by the line DOF; in which case his back horizon OD, to which he brings the object S, will be as much elevated above the plane of the true horizon HOQ as the apparent horizon OF will be depressed below it; because, when two straight lines intersect each other, the opposite angles will be equal. (Euclid, Book I., Prop. 15.) In this case it is evident that the arc or apparent altitude S D is too little; and that it must be augmented by the arc DH = the angle of horizontal depression FOQ, in order to obtain the true altitude SH. Hence it is manifest that altitudes taken by the fore observation must be diminished by the angle of horizontal depression, and that in back observations the altitudes must be increased by the value of that angle.

The absolute value of the horizontal depression may be established in the following manner :-From where the apparent horizon OP becomes a tangent to the earth's surface at T (the point of contact where the sky and water seem to meet) let a straight line be drawn to the centre E, and it will be perpendicular to OP (Euclid, Book III., Prop. 18): hence it is obvious that the triangle ETO is right-angled at T. Now, because OT is a straight line making angles from the point O upon the same side of the straight line O E, the two angles EOT and TOH are together equal to the angle EOH (Euclid, Book I., Prop. 13); but the angle EOH is a right angle; therefore the angle of depression TOH is the complement of the angle EOT, or what the latter wants of being a right angle: but the angle TEO is also the complement of the angle EOT (Euclid, Book I., Prop. 32); therefore the angle TEO is equal to the angle of horizontal depression; for magnitudes which coincide with one another, and which exactly fill up the same space, are equal to one another. Then, in the right-angled rectilineal triangle ETO, there are given the perpendicular TE, = the earth's semidiameter, and the hypothenuse E O, = the sum of the earth's semidiameter and the height of the observer's eye, to find the angle TEO = the angle of horizontal depression TOH :-hence the proportion will be, as the hypothenuse EO is to radius, so is the perpendicular T E to the cosine of the angle TEO, which angle has been demonstrated to be equal to the angle of horizontal depression HOP. But because very small arcs cannot be strictly determined by cosines, on account of the differences being so very trivial at the beginning of the quadrant as to run several seconds without producing any sensible alteration, and there being no rule for showing

why one second should be preferred to another in a choice of so many, the following method is therefore given as the most eligible for computing the true value of the horizontal depression, and which is deduced from the 36th Prop. of the third Book of Euclid.

Because the apparent horizon OP touches the earth's surface at T, the square of the line OT is equal to the rectangle contained under the two lines CO and eO. Now as the earth's diameter is known to be 41804400 English feet, and admitting the height of the observer's eye e O to be 290 feet above the plane of the horizon; then, by the proposition, the square root of CO, 41804690 x e O, 290= the line OT, 110105.75 feet; the distance of the visible horizon from the eye of the observer independent of terrestrial refraction.

Then, in the right-angled rectilineal triangle ETO, there are given the perpendicular ET=20902200 feet, the earth's semidiameter, and the base OT 110105. 75, to find the angle TEO. Hence,

90:0.0" log. sine

As the perpendicular TE= 20902200 feet, log. arith. compt. 2. 679808
Is to the radius .
So is the base OT. .110105.75 feet, log..

10.000000

5.041810

But it has been shown that the angle TEO, thus found, is equal to the angle HOP; therefore the true value of the angle of horizontal depression HOP, is 18:7? Now, according to Dr. Maskelyne, the horizontal depression is affected by terrestrial refraction, in the proportion of about onetenth of the whole angle; wherefore, if from the angle of horizontal depression 18.7" we take away the one-tenth, viz. 1.49", the allowance for terrestrial refraction, there will remain 16:18" for the true horizontal depression, answering to 290 feet above the level of the sea. The principles being thus clearly established, it is easy to deduce many simple formulæ therefrom, for the more ready computation of the horizontal depression; of which the following will serve as an example.

To the proportional log. of the height of the eye in feet, (estimated as seconds,) add the constant log. .4236, and half the sum will be the proportional log. of an arc; which being diminished by one-tenth, for terrestrial refraction, will leave the true angle of horizontal depression.

Example.

Let the height of the eye above the level of the sea be 290 feet, required the depression of the horizon corresponding thereto?

Height of the eye 290 feet, esteemed as secs. 4:50", propor.log.=1.5710

.4236

method.

True horizontal depression 16:18", the same as by the direct

In using the Table, it may not be unnecessary to remark that it is to be entered with the height of the eye above the level of the sea, in the column marked Height, &c.; opposite to which, in the following column, stands the corresponding correction; which is to be subtracted from the observed altitude of a celestial object when taken by the fore observation; but to be added thereto when the back observation is used, as before stated. Thus the dip, answering to 20 feet above the level of the sea, is 4:17"

TABLE III.

Dip of the Horizon at different Distances from the Observer.

If a ship be nearer to the land than to the visible horizon when unconfined, and that an observer on board brings the image of a celestial object in contact with the line of separation betwixt the sea and land, the dip of the horizon will then be considerably greater than that given in the preceding Table, and will increase as the distance of the ship from the land diminishes in this case the ship's distance from the land is to be estimated, with which and the height of the eye above the level of the sea, the angle of depression is to be taken from the present Table. Thus, let the distance of a ship from the land be 1 mile, and the height of the eye above the sea 30 feet; with these elements enter the Table, and in the angle of meeting under the latter and opposite to the former will be fonnd 17. which, therefore, is the correction to be applied by subtraction to the observed altitude of a celestial object when the fore observation is used, and vice versa.

The corrections in this Table were computed after the following manner; viz.,

Let the estimated distance of the ship from the land represent the base of a right-angled triangle, and the height of the eye above the level of the sea its perpendicular; then the dip of the horizon will be expressed

by the measure of the angle opposite to the perpendicular : hence, since the base and perpendicular of that triangle are known, we have the following general

Rule. As the base or ship's distance from the land, is to the radius, so is the perpendicular, or height of the eye above the level of the sea to the tangent of its opposite angle, which being diminished by one-tenth, on account of terrestrial refraction, will leave the correct horizontal dip, as in the subjoined example.

Let the distance of a ship from the land be 1 mile, and the height of the eye above the level of the sea 25 feet, required the corresponding horizontal dip

As distance 1 mile, or 5280 feet, Logarithm Ar. Comp.

90, Logarithmic Sine.

Deduct one-tenth for terrestrial

refraction

True horizontal dip =

6. 277366 10.000000

16. 17" Log. Tang. = 7.675306

1.37

14:40%, or 15 nearly as in the Table.

Remark. Although a skilful mariner can always estimate the distance of a ship from the shore horizon to a sufficient degree of accuracy for taking out the horizontal dip from the Table, yet since some may be desirous of obtaining the value of that dip independently of the ship's distance from the land, and consequently of the Table, the following rule is given for their guidance in such cases :

Let two observers, the one being as near the mast head as possible, and the other on deck immediately under, take the sun's altitude at the same instant. Then to the arithmetical complement of the logarithm of the difference of the heights, add the logarithm of their sum, and the logarithmic sine of the difference of the observed altitudes; the sum, rejecting 10 from the index, will be the log. sine of an arch; half the sum of which and the difference of the observed altitudes will be the horizontal dip corresponding to the greatest altitude, and half their difference will be that corresponding to the least altitude.

Example.

Admit the height of an observer's eye at the main-topmast head of a ship close in with the land, to be 96 feet, that of another (immediately under) on deck 24 feet; the altitude of the sun's lower limb found by the former to be 39:37', and by the latter, taken at the same instant, 39:21; required the dip of the shore horizon corresponding to each altitude?

Note. When the dip answering to an obstructed horizon is thus carefully determined, the ship's distance from the land may be ascertained to the greatest degree of accuracy by the following rule: viz. As the Log. tangent of the horizontal dip of the shore horizon is to the logarithm of the height of the eye at which that dip was determined, so is radius to the true distance.

Thus, in the above example where the horizontal dip has been determined to the corresponding height of the eye and difference of altitudes,

The same result will be obtained by using the greatest dip and its corresponding height; and since the operation is so very simple, it cannot fail of being extremely useful in determining a ship's true distance from the shore.

TABLE IV.

Augmentation of the Moon's Semidiameter.

Since it is the property of an object to increase its apparent diameter in proportion to the rate in which its distance from the eye of an observer is diminished; and, since the moon is nearer to an observer, on the earth, when she is in the zenith than when in the horizon, by the earth's semidiameter; she must, therefore, increase her semidiameter by a certain. quantity as she increases her altitude from the horizon to the zenith. This increase is called the augmentation of the moon's semidiameter, and depends upon the following principles,