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Mr. W. P. Browne, R.N., Plymouth.
Admiral Sir Isaac Coffin, bart., Titley Court, Hereford.
ranean, 6 copies.
Capt. Nevinson De Courcy, R.N., Stoketon House, Plymouth.
Admiral the Right Hon. Lord Viscount Exmouth, G.C.B.
DESCRIPTION AND USE
PRINCIPLES UPON WHICH THEY HAVE BEEN COMPUTED.
To convert Longitude, or Degrees, into Time, and conversely. THIS Table consists of six compartments, each of which is divided into two columns: the left-hand column of each compartment contains the longitude, expressed either in degrees, minutes, or seconds; and the right-hand column" the corresponding time, either in hours, minutes, seconds, or thirds. The proper signs, for degrees and time, are placed at the top and bottom of their respective columns in each compartment, with the view of simplifying the use of the Table :-hence it will appear evident that if the longitude he expressed in degrees, the corresponding time will be either in hours or minutes; if it be expressed in minutes, the corresponding time will be either in minutes or seconds; and if it be expressed in seconds, the corresponding time will be expressed either in seconds or thirds. The converse of this takes place in converting time into longitude.
The extreme simplicity of the Table dispenses with the formality of a rule in showing its use, as will obviously appear by attending to the following examples.
Example 1. Required the time corresponding to 47:47:47" of longitude ? 47 degrees, time answering to which in the Table is 3! 8" 0: 0 47 minutes, answering to which is
.0. 3. 8.0 47 seconds, answering to which is ..0.0, 3. 8
Lon. 47:47:47", the time corresponding to which is
8 hours; longitude answering to which in the Table is · 120:0:05
Time 8:52"28:, the longitude corresponding to which is . 133:07
Besides the use of this Table in the reduction of longitude into time, and the contrary, it will also be found very convenient in problems relating to the Moon, where it becomes necessary to turn the right ascension of that object into time.
Exampie. The right ascension of the Moon is 355:44:48"; required the corresponding time?
355 degrees, time answering to which
23.40" 0: 0 44 minutes, answering to which is 0. 2. 56. 0
48 secs., answering to which is 0. 0.3.12
Right ascension 355:44:48", the time corresponding to
Since the Earth makes one complete revolution on its axis in the space of 24 hours, it is evident that every part of the equator will describe a great circle of 360 degrees in that time, and, consequently, pass the plane of any given meridian once in every 24 hours; whence it is manifest that any given number of degrees of the equator will bear the same proportion to the great circle of 360 degrees that the corresponding time does to 24 hours; and that any given portion of time will be in the same ratio to 24 hours that its corresponding number of degrees is to 360.
Now since 24 hours are correspondent or equal to 360 degrees, 1 hour must, therefore, be equal to 15 degrees; 1 minute of time equal to 15 minutes of a degree; 1 second of time to 15 seconds of a degree, and so
And as 1 minute of time is thus evidently equal to 15 minutes or one fourth of a degree, it is very clear that,4 minutes of time are exactly equal to 1 degree ; wherefore since degrees and time are similarly divided, we have the following general rule for converting longitude into time, vice versa.
Multiply the given degrees by 4, and the product will be the corresponding time :-observing that seconds multiplied by 4 produce thirds ; minutes, so multiplied, produce seconds, and degrees minutes ; which, divided by 60, will give hours. The converse of this is evident :-thus,
reduce the hours to minutes; then these minutes, divided by 4, will give degrees; the seconds, so divided, will give minutes, and the thirds, if any, seconds. Hence the principles upon which the Table has been computed. The following examples are given for the purpose of illustrating the above rule. Example 1.
Example 2. Required the time corresponding Required the degrees correspondto 36:44:32"?
ing to 3.45"48:20: ? Given degrees = 36:44:32". Given time=345"48:20: Multiplied by 4
Corresponding time 2:26:58:8:
Divide by 4) 225.48.20
Corresponding degs. 56:27.5"
Depression of the Horizon. The depression or dip of the horizon is the angle contained between a horizontal line passing through the eye of an observer, and a line joining his eye and the visible horizon.
This Table contains the measure of that angle, which is a correction expressed in minutes and seconds answering to the height of the observer's eye above the horizon; and which being subtracted from the observed central altitude of a celestial object, when the fore observation is used, or added thereto in the back observation, will show its apparent central altitude. The corrections in this Table were deduced from the following considerations, and agreeably to the principles established in the annexed diagram.
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 SO H 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 O F 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 SD 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 O P 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 E TO 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 E OT and TO H 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 To H 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 T E O 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 E TO, there are given the perpendicular TE,
= the earth's semidiameter, and the hypothenuse EO, = 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