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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 O P 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 e O. Now as the earth's diameter is known to be 41804400 English feet, and admitting the height of the observer's eye e 0 to be 290 feet above the plane of the horizon; then, by the proposition, the square root of CO, 41804690 x e0, 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,
As the perpendicular TE = 20902200 feet, log. arith. compt.= 2.679808 Is to the radius. 90:0:0:
10.000000 So is the base OT= . . 110105.75 feet, log..
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 187". 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 Constant log.
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".
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.= 6.277366 Is to radius
Logarithmic Sine. 10.000000 So is height of the eye 25 feet, Logarithm
16'. 17"= Log. Tang. = 7.675306 Deduct one-tenth for terrestrial refraction
True horizontal dip
14:40", or15: 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 elose 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
and by the latter, taken at the same instant, 39:21!; required the dip of the shore horizon corresponding to each altitude ?
to be 39:37,
Note.-When the dip answering to an obstructed horizon is thus earefully 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,
As horizontal dip = 5:20"? Log. tang. ar. compt.=2.809275
1.380211 So is radius.
90: Logarithmic sine 10.000000
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.
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 de pends upon the following principles,
Let the circle ABCD represent the earth; A E its semidiameter, and M the moon in the horizon. Let A represent the place of an observer on the earth's surface; BDM his rational horizon, and HAO, drawn parallel thereto, his sensible horizon extended to the moon's orbit; join AM, then AME is the angle under which the earth's semidiameter A E is seen from the moon M, which is equal to the angle MA O, the moon's horizontal parallax ; because the straight line AM which falls upon the two parallel straight lines E M and AO makes the alternate angles equal to one another. (Euclid, Book I. Prop. 29.) Let the moon's horizontal parallax be assumed at 57:30", which is about the parallax she has at her mean distance from the earth; then in the right angled triangle A E M, there are given the angle A ME=57:30", the moon's horizontal parallax, and the side A E=3958. 75 miles, the earth's semidiameter ; to find the hypothenuse A M=the moon's distance from the observer at A: hence by trigonometry,
As the angle at the moon, A ME=57.30”. Log. sine ar, comp. 1.776626 Is to the earth's semidiameter=A E=3958.75 miles, Log. 3.597558 So is radius
Log. sine 10.000000
To moon's horizontal distance A M=236692.35 miles, Log. . 5.374184
Now, because the moon is nearer to the observer at A, by a complete semidiameter of the earth when in the zenith Z, than she is when in the horizon M, as appears very evident by the projection; and, because the earth’s semidiameter A E thus bears a sensible ratio to the moon's distance ; it hence follows that the moon's semidiameter will be apparently increased when in the zenith, by a small quantity called its augmentation ; and which may be very clearly illustrated as follows, viz.
Let the arc ZOM represent a quarter of the moon's orbit ; Z her place in the zenith, and Z S her semidiameter : join EZ, AS, and ES; then the angles ZE S and Z A S will represent the angles under which the moon's semidiameter is seen from the centre and surface of the earth; their diffe