I have been honoured with your favour of the 16th. inclosing a method of finding the longitude without a time piece by Mr. Moore; on which, as well as on other methods for the same purpose, suggested by yourself, & sometime ago shewn me by Mr. Freeman, you are pleased to ask my opinion.

It is scarcely necessary to remark that all the methods here proposed are strictly true in theory; and therefore the only enquiry will be with respect to their practicability, and how far they will be affected by the unavoidable inaccuracies to which all instrumental measurements are liable—

On the 1st proposed by yourself, viz. “See what is the moon’s Right ascension when she passes the meridian of Greenwich at noon or Midnight:—Watch the exact moment she attains that ascension where you are—You know she is at that moment on the meridian of Greenwich. Measure her distance from the meridian of your place—That gives you the difference of longitude.

“The prerequisites to this operation are—

“1st. To have a meridian mark

“2nd. To have calculated at what distance the moon will be from a given star when she has attained the R.A. desired; so that you will need only to watch for her gaining a given distance from a star.” I would remark—

1. The distance at which the moon will be from the sun or any given star at the moment she attains a given R.A. may indeed be readily calculated by the revolution of a spheric triangle in which there will be given two sides & the angle included, viz. the ⊙ or * ’s polar distance, the ☽’s polar distance, &d their difference of R.A to find the third side assuming however the declinations & R.A of both the bodies as known at the time their distance is to be observed. The accuracy of this assumption will depend on that of the supposed longitude of the place and, in part on the quickness or slowness in the change of the ☽’s declination at the time—Let it however be remarked that the distance thus computed is the true distance, unaffected either by parallax or refraction, and not the apparent, which alone can be observed by an instrument. this apparent distance therefore must be found from the true by a calculation similar to that used in computing the true from the apparent in the common method of finding the Longitude by the lunar distance. But for this purpose the altitudes of the two bodies must be previously known and if these be taken at the time of measuring their distance the hour of the day & thence the longitud may be readily computed, either with or without the latitude of the place—being given,. as will hereafter be shown.

2. The distance of the ☽ from the meridian of the place at the moment she is on the meridian of Greenwich is indeed the longitud from Greenwich But how is this distance or ☽’s hour-angle from the meridian to be measured? unless by an equatorial instrument? which I suppose—is not contemplated. the Equatorial was precisely the instrument contemplated. By a sextant all you can do is to measure the ☽’s altitude from which & her declination together with the latitud of the place you may by spherüs calculate her hour-angle without any meridian mark & therefore I apprehend that this mark will be unnecessary

3 In finding the true angular distance: of the ☽ from the ⊙ or a * great accuracy in taking the altitudes of these bodies is not necessary since they are only employed in ascertaining their parallax & refraction But when the distance or R.A. is to be calculated from their altitudes & polar distance the utmost accuracy will be necessary since any error in the distance or R.A. will be increased about thirty fold in the longitude deduced therefrom.

On method 2d viz. “In the moment of the ☽’s passing your meridian ascertain her Right ascension—Find from the Tables the time at Greenwich when she has that RA—This time is her then distance from the meridian of Greenwich & consequently the distance of your meridian on which she then is.” I would remark

1. That having your meridian mark accurately placed, you may, with a Transit-instrument find the time within a few seconds when the center of the ☽ is on your meridian: but without some instrument of this kind the necessary degree of accuracy is scarcely attainable since every second, of an error in the time of the ☽’s transit will produce an error of about 8 minutes of a degree in the longitude

2. The ☽’s R.A. at the moment of her transit can only be found, directly, by an equatorial instrument measuring the hour-angle of the ⊙ or of some known star which will be difference of R.A. between this body & that of the ☽ then on the Meridian. But when this is to be done by the sextant the altitude of the sun or star must be taken at the moment of the ☽’s transit and from this with the declination & latitude of the place the hour-angle may be computed The degree of accuracy however necessary in this case will scarce be practically possible

3. The time at Greenwich when the ☽ is on your Meridian is said to be “her then distance from the meridian of Greenwich & consequently the distance—of your meridian on which she then is” But by “the time at Greenwich” must here be understood: the lunar time or the ☽’s hour-angle from the meridian of Greenwich which indeed may be found from the solar time by simple proportion using the correction for 2d. difference.

Upon the above two methods it may be remarked in general that they afford no opportunity of taking a set of successive observations, a matter of the greatest importance to secure an accurate result & which the common method by lunar distance possesses in full perfection.

On Method 3d. viz. “Observe the ☽’s distance from your meridian at any moment & take her Right ascension by observation at that moment—Find from the Tables her distance from the Meridian of Greenwich at the moment she had that R.A.—If the ☽ was between the two meridians add the two distances. If she was E. or W. of both take their difference & the result in either case will be the distance of the two meridians.”

It may be remarked—

1. That it has the advantage of the two former methods, in that the observations may be repeated at pleasure, and a mean of the results taken—But here also the ☽’s distance from the meridian & her R.A. can be found only by an equatorial instrument or by measuring altitudes & making calculations as before observed and in this case a meridian mark will be of no service The utmost degree of accuracy however attainable in this way would fall far short of what might be attained & with much less difficulty in common method by lunar distance.

Method 4th. without a Meridian—“Knowing the latitude of your place you can calculate the sun’s rising or setting for the day which is his distance from your meridian: If the moon is visible at the moment of sunrise or sunset observe her R.A. at that moment & seeing by the Nautical Almanac at what time she had that R.A. at Greenwich you have the difference of longitud The irregularity of the refraction (it is observed) is one objection to this Method.”

The same remarks that have been made—on the former methods so far as relates to hour-angles & Right ascension with the unavoidable inaccuracy of the results are equally applicable to this method.—Upon the whole it may be observed that the method of finding the longitude by lunar distance is preferable to those in which her RA. is used evin supposing what is far from being true that both could be measured with equal ease and accuracy for in the Nautical Almanac we have these distances computed for every 3 hours and therefore the time corresponding to any intermediate distance may be found with sufficient accuracy by simple proportion. But the ☽’s R.A. being computed only for noon & midnight the time corresponding to any intermediate R.A. or the contrary ought to be found by the more difficult method of second differences. Mr. Moore in his method asserts that “nothing but sextants are required” but how the time can be found by this instrument otherwise than by taking an altitude & calculating by spherüs I confess I cannot well comprehend.

In this method there is certainly much more calculation reqd. than in the common method from the lunar distance directly & yet I cannot perceive that it possesses any peculiar advantages for though as Mr. M. observes after the trouble of the beginning calculation observations may be multiplied at pleasure—I presume that a single calculation made on the mean of a set, or number of observations taken in succession would be equally accurate & certainly much more simple—Besides this method in common with some of the foregoing where you are to wait for a certain moment or distance at which you are to make your observation is liable to this objection, that when the given moment arrives a cloud or some other obstacle may prevent the observation & then all your previous labour is lost. And indeed the arm & eye cannot fail to be fatigued with so long an exertion as in this case is necessary—

I am indeed, Sir, fully sensible of the great importance of some practical method of finding the longitude without the precarious aid of a time-piece or any other instrument than a good sextant and that accuracy is much more to be regarded than simplicity or ease in calculation And as I hinted above this can be done with as much accuracy & ease, as can reasonably be wished and that even without the aid of any assistant.

From the observed angular distance of the ☽ from the ⊙ or from any of the stars from which her distance is computed in the Nautical almanac together with their respective altitudes: to find the latitude and longitude of the place with the correct apparent time & if required the azimuth of either or both of the bodies & thence the variation of the magnetic needle—is one of the problems in nautical astronomy that has been taught in this seminary for near twenty years. A copy of one of the Formulae which I use for this purpose I gave Mr. Briggs the last time (I think) he was in this city. Since that time I have met with a similar problem in the 2d. edition of “Mackay’s Treatise on the theory & practice of finding the longitude at sea” published in 1801, and for which it is said he recd the thanks of the Board of Longitude—and lest you should not be furnished with a copy of this treatise, I shall subjoin his method with an example expressing however his precepts in a kind of Algebraic Formula in which every stop of the process is intelligibly pointed out in the margin and shall then work the same example by the method I generally use There is in fact little difference in the two methods, unless that Mr. M. makes use of nal. versed sines (which however not being found in the common book of requisite tables I have changed into nal. cosines) while I make use only of the log. functions.

Example—Being in south latitude Sunday 24th. 1792 in the afternoon the distance between the nearest limbs of the ⊙ & ☽ was from the mean of a set of observations 55°.48’.34" the alt. of ☽ lower limb 43°.23’, that of the ⊙’s 17°.40’. and height of eye 12 feet Reqd the lat. & long. of the place of observation?—

The estimated Greenwich time being about 4¼ hours P. M Correcting the alts. from the N.A. taking out the declinations ½ diameters &c & then computing the true distance of center & true Gr. time by any of the common methods for that purpose we will have the following eliments viz.

		°	′	″			°	′	″
	True distance	56.	26.	21		☉’s polar dist	113.	24.	42
	True Gr. time	4h.	14.	22		☽’s polar dist.	99.	13	—
	☉’s Zen. dist—	72.	10.	18.		☽’s being nearest the meridian
	☽’s Zen. dist—	45.	46.	19

Formula from Mackay’s Rules

	True dist.	A	56.26.21				Cosec	a	10.07920
	Zen. dist. of body farst fr. mer.	B	72.10.18				Cosec	b	10.02138
	A—B	C	15.45.57	n. cos	c	96254
	Zen. dist of body nst mer.	D	45.46.19	n. cos	d	69752
	c—d		d m s	numb	—	26502	Log	e	4.42328
	a+b+e–20	F	3.12.59.				L. rising		4.52386
	Pol. dist. of body B	G	113.24.42.				cosec	g	10.03731
	A—G	H	56.58.21	n. cos	h	54504
	Pol. dist. of body D	I	99.13.	n. cos	i	16087
1	h ± i		d m s	numb	__	70521	Log.	k	4.84832
	a+g+k–20	L	5.42.9				L. rising		4.96483
	F—L	M	2.29.10				L. rising	m	4.31056
	B—G	N	41.14 24	n. cos	n	75196
	20+m–b– g.			numb	o	17859	Log.		4.25187
	m.o. Lat of place	P	34.59.7	n. sin		57337	cos.	p	9.91344
	F—L	Q	2.29.10				H.E.T.	q	0.21762
	b+p+q–10 hr. ∠ of Body B	R	2.58.59				H.E.T.		0.15244
	Gr. time	S	4.14.22
	R—S = Long in time	T	1.15.23	= 18.° 50¾′ No

1 Add when of arches H & I one is greater and the other less than 90°

 

Formula from P

	True dist		A	56.26	cosec	a	10.07923
	☉ pol. dist		B	113.25	cosec	b	10.03733
	☽ pol. dist		C	99.13
	A+B+C		D	269. 4
	½ D		E	134.32	sin	e	9.85299
	E—C		F	35.19	sin	f	9.76200
	a+b+e+f–20					g	19.73155
	½ g		H	42.46	cos	__	9.86577
	☉ Zen. dist		I	72.10	cosec	i	10.02139
	☽ zen. dist		K	45.46
	A+I+K		L	174.22
	½ L		M	87.11	sin	m	9.99947
	M—K		N	41.25	sin	n	9.82055
	a+i+m+n–20					o	19.92064
	½ o		P	24. 7	cos.	__	9.96032
	B+J		Q	185.35
	½ Q		R	92.47½	cosec	r	10.00051	sec	r	11.31243
	R—B		S	20.37½	sin	s	9.54680	cos	s	9.97123
1	H∓P		T	18.39	Cot.	t	10.47171	cot	t	10.47171
2	r+s+t–10		U	91.00.				tan	_	11.75837
	r+s+t–10		V	46.15½	tan	_	10.02902
	U—V, ☉i hr. ∠ = 2.d 58.m 58s		W	44.44½	sin	w	9.84752
	2 T.		X	37.18	cosec	x	10.25754
	i+w+x–10, Lat. of place		Y	34.° 58′			10 08645
	Green time	4.14.22.
	Z—W= Long-W.	1 15.24	=	18.° 51′ W.__

1 Add when of plane passing thro’ the bodies lies between the zenith & elevated pole; otherwise sub.

2 Take the supp. of the ∠ found in the table when Q exceeds 180°—

U+V= 137.°15½′ is the sun’s az. from midnight merid.

I shall close with a few general remarks, relative to the manner of conducting the lunar observations

1. When an observation for the purpose of finding the lat & long. is to be made by one person alone which may be frequently better than using assistants, let him take his set of observations in the following order viz.

Dist. of limbs	}	All at small intervals of time & as nearly equal as possible. And let the calculation be made on a mean f[. . . .] thus respective alts. & distances which will then correspond to the time when the middle distance was taken—
⊙ or * ’s alt.
☽’s . alt.—
Dist
☽’s Alt.—
⊙’s Alt.—
Dist.—

The alts. on land will be best taken by reflection from the surface of water or some other fluid, defended from the motion of the air by means of a piece of talc or the like

2 As the long. of the place will generally be known previous to the observation within a few degrees, the ☽’s hor. par semi diam. &c taken from the Naut. Alm. for the estimated Gr. time will give the true dist. with sufficient accuracy without repeating the calculation or using any correction.

3 When the lat. of the place is previously known which will be generally the case then it will be sufficient to measure the alt. of one of the bodies only—and let this be the sun or star, if not too near the meridian.

The calculations are then to proceed thus. (1) From the lat. of the place with the dec. & alt. of the body find the hour of the day. (2) From the lat. of the place with the hour of the day & dec. of the other body find its alt. (3) From the app. dist. & app. alts of the bodies find the true dist. and thence the long. by some of the common methods—

4 If the observer be furnished with a’watch, he may by taking the ⊙’s alt. in the forenoon, & again in the afternoon repeatedly find the error & rate of going of his watch and then at any other time not far dist. from the time of such an observn. he may by making the proper allowance know the hour of the day with suff accy—and when this is known the distance only need be measured as the alt. may thus be made [. . . .].

I am. Sir, with sentiments of the most profound respect & esteem your most obedient servant