required the true latitude, without making use of the supposed one?

Answer. The latitude by the rule is 1°.36'.24" North, but the place being so near to the equator, it will be proper to examine arc the fourth as directed in the notes to the rule. By this method the latitude is 0°.39' South. The former is nearest the latitude by account.

(R) In all the preceding examples the two observations of the sun's altitude are supposed to have been made at the same place, and the latitude determined by the solution agrees to that place. Although the two altitudes are generally taken at the same place at land, yet at sea that is seldom the case. An allowance, therefore, ought to be made for the run of the ship during the elapsed time; thus, find the angle contained between the ship's course and the sun, if it be eight points no correction is necessary, but if less or more than eight points the correction must be applied to the first altitude, by addition or subtraction. Consider the angle contained between the ship's course and the bearing of the sun as a course, the distance made good during the elapsed time as a distance; with these find a difference of latitude and apply it as above.* The result will reduce the first altitude to what it would have been if taken at the same place where the second was taken. The latitude must be found with these altitudes, thus corrected, in the same manner as before: this will be the latitude of the place where the second altitude was taken.

"The difference of longitude during the elapsed time may likewise be taken into consideration, though in general it is of little or no consequence. The change of the sun's declination during the elapsed time might likewise be considered, but this, like the longitude, will cause no sensible error; particularly, if the declination answering to the middle time between the observations be used.

(S) PROBLEM XVII. (Plate III. Fig. 12.)

Given the apparent distance of the moon from the sun, or a star, and their apparent zenith distances, to find their true distance, as seen from the earth's centre.

Let zм be the observed zenith distance of the moon, zm the true zenith distance; Mm being the difference between the moon's refraction and her parallax in altitude. And let z represent the observed zenith distance of the sun or of a star; zs the true zenith distance, Os being the difference between the sun's refraction and his parallax, or the refraction of a star.

* Vide Mr. Cotes's De estimatione errorum in mixtâ Mathesi.

GENERAL PRINCIPLES.

I. Find the segments of the baseV and VM, (by the rule W.231.). With Oz and Ov find the angle zOv, which will be equal to the angle TOS. (N. 135.) With Os, and the angle TOS, find OT, which will be equal to SP. The triangle TOS, being indefinitely small, may be considered as a plane triangle. Again, with vм and zм find the angle zмv, with мm and the angle zмV find the base RM, considering the right-angled triangle mRM as a plane triangle.

Lastly, MO+SP-RM-Sm the true distance in all cases except where the angle at the zenith is acute, and the angle at the moon obtuse, then мO+SP+RM=SM.

II. Or, with z☹,Zм, and Oм find the vertical angle ZM, (P. 248.), and with zs, zm, and the angle OzM, find the true distance sm. (N. 245.)

The various methods which have hitherto been made use of for determining the distance between the moon and the sun, or a star, are derived from one or other of the above principles. Those methods which are derived from the latter, are generally preferable to those derived from the former, as being more correct and simple. When the observed distance is small, or the moon's parallax great, and the star's refraction considerable, two other corrections are necessary in order to render the first of these principles generally correct. We have considered the little triangles as right-angled, but the fact is NPO and NRM are each of them isosceles triangles, and, therefore, OP and Rm are not strictly perpendicular to sm and OM; these corrections, therefore, consist in determining how much SP and RM deviate from the bases of right-angled triangles. A true method of determining these four corrections may be seen in the Edinburgh Transactions, Article VII., Physical Class. (T) INVESTIGATION OF A GENERAL RULE FOR DETERMINING THE TRUE DISTANCE OF THE MOON FROM THE SUN, OR

FROM A FIXED STAR. (Plate III. Fig. 13.)

Let OM be the observed distance, and sm the true distance. Also, let zм be the observed zenith distance of the moon and zm the true zenith distance; z☺ the observed zenith distance of the sun or of a star, and zs the true zenith distance, as above. Then, by one of the formulæ F. 184, we have, in the triangle Ozm,

2 sine2 z rad. cos (z~Zм)-rad. cos M

in the triangle szm, by the same formula,

2 sine z rad. cos (zs~zm)—rad. cos sm

; and, likewise

By putting these values equal to each other, and dividing by the radius, we obtain

COS (ZO~ZM)-COS OM cos (zs~zm)-cos sm

sine ZO X sine ZM

sine zs x sine zm

Hence, cos sm-cos (ZS ~ zm)—cos OM-COS(ZC —~ZM) × sine zs x sine zm

sinez X sine zм®

Now zs~zm and likewise z~ZM will always be acute, being each less than 1°; but OM and sm, being each of the same species, may be either acute or obtuse, therefore when the observed distance is more than 90°,

CossmcoSOM +COS (ZO~ZM) ×

(zs~zm).

sine zs x sine zm

sine z × sine Zм

And when the observed distance is less than 90°,

~ COS

sine zs x sine zm

Cos sm=cos (zs ~ zm) — cosCMX

~ Cos

sine ZO sine ZM

(zs~zm.) Hence the following

GENERAL RULE.*

To the natural cosine of the difference between the apparent altitudest, add the natural cosine of the apparent distance if more than 90°, or subtract it if less than 90°, and find the common logarithm of the remainder; to which add the logarithm secants of the apparent altitudes, the logarithm cosines of the true altitudes, and reject the tens from the index. The difference between the natural number answering to this sum in the table of logarithms, and the natural cosine of the difference between the true altitudes, will give the natural cosine of the true distance.

NOTE. The moon's correction, which is the difference between the refraction and the parallax in altitudes, must always be added to the apparent altitude to obtain the true altitude;

* Several other rules, differing in the form of expression, may be deduced from the foregoing demonstration, but this has been preferred on account of its shortness, and the ease with which it may be applied: requiring no other tables in its application than those which are common and well known.

A collection of short rules, without demonstration, may be seen in Nicholson's Philosophical Journal, for November 1806, vol. XV. page 254.

Dr. Mackay's 1st Method (page 150, vol. I. 3d edition) of his valuable treatise on the longitude, is the simplest I have ever met with, when his tables are used. Mendoza's method is likewise very short and easy, but his tables are large and expensive.

The apparent altitudes, are the observed altitudes corrected for dip and semidiameter, and the apparent distance, is the observed distance corrected for semidiameters.

but the sun or star's correction must always be subtracted. The sun's correction is the difference between the refraction (Table IV.), and the parallax in altitude (Table VI.), and a star's correction is the refraction (Table IV.).

EXAMPLE I.

The apparent distance of the moon's centre from the star Regulus was 63°.35′.13", when the apparent altitude of the moon's centre was 24°.29'.44", the apparent altitude of the star 45°.9.12"; and the moon's horizontal parallax 55'.2"; required the true distance?

D's horizontal parallax=55'.2", the parallax in altitude= 50'.5" (T.96.), refraction 2.4" (Table IV.) hence the D's correction (50′.5′′-2′.4′′) 48′.1", and the 's correction=57′′. (Table IV.)

Diff.app. altitudes 20°.32′.28′′ Nat. cos. 93570
Apparent distance 63°.35'.13′′ Nat. cos. 44485

Moon's apparent altitude ( 24°.29.44"log. sec. 10.04097 Star's apparent altitude* 45°. 9.12′′log. sec. 10.15169 Moon's corrected altitude 25°.17.45"log. cos. 9.95622 Star's corrected altitude 45°. 8'.15" log. cos. 9.84843)

Natural number

48782 4.68826

Diff. true altitudes 19°.50'.30" Nat. cos. 94063

+

* Instead of the logarithmical secants of the apparent altitudes, you may take the logarithmical cosines, add them together and take the sum from 20, the remainder will be the same as the sum of the logarithmical secants without the indices.

+ The sum of these four logarithms, rejecting the tens, is 9.99731. This number may be found at once from the XLIId Table of Dr. Mackay's treatise on the Longitude, 3d edition, using the) 's horizontal parallax 55.2", and the apparent altitude 24°.29'.44". The operation would be rendered shorter by such a table, but it occupies 45 pages of Royal Octavo, and, therefore, could not be inserted in this work. The IXth of the Requisite Tables is not so extensive, but then it proportionably imperfect. The author of the article Longitude in Dr. Rees's new Cyclopedia has been pleased to remark, that the rule given by me may be simplified by using the IXth of the Requisite Tables, or Dr. Mackay's XLIId Table; this I was fully aware of: my object was not to give a short rule for solving the problem by auxiliary tables not in common use, but to render the subject plain and easy to a learner without the help of any other tables than those at the end of this treatise.

EXAMPLE II.

The apparent distance of the moon's centre from the sun's was 106.46'.44", when the apparent altitude of the sun's centre was 45°.32.30", and the moon's 19°.43'.22"; the moon's correction 50'.3", and the sun's 50"; required the true distance of their centres?

Diff. of app. alts. - 25°.49'. 8′′ Nat. cos. 90018
Apparent distance 106°.46'.44" Nat. cos. 28868

Diff. of true alts. 24°.58'.15" Nat. cos. 90652

True distance 106°. 2.19" Nat. cos.

27629

EXAMPLE III.

The apparent distance of the sun and moon's centres was 68°.42′.11", when the apparent altitude of the sun's centre was 32°.0'.1", and that of the moon's 24°.0'.10"; the sun's correction was 1'.23", and the moon's 51'.1"; required the true distance of their centres?

Answer. 68°.19′.46".

EXAMPLE IV.

The apparent distance of the moon's centre and a star was 20.20', when the apparent altitude of the star's centre was 11°.14, and that of the moon's 9°.39'; the moon's correction was 51'.30", and the star's 4'.40"; required the true distance of their centres?

Answer. 1°.49'.

EXAMPLE V.

The apparent distance of the moon from the star Spica was observed to be 31°.17'.53", when the apparent altitude of the moon's centre was 18°.56′.45′′, and the apparent altitude of the star 20°.10'.56"; the moon's correction was 50'.46", and the star's 2'.35"; required the true distance of their centres? Answer. The true distance is 31°.11'.44".