I have been honoured with your favour of the 2d. and thank you for your confidence, which I will never abuse—I am preparing a set of astronomical formulæ for Mr. L. and will, with the greatest pleasure, render him every assistance in my power—I take the liberty of subjoining the formula which I commonly use for computing the longitude from the common lunar observation, illustrated by an example—The other formulæ for computing the time, alts. &c are all expressed in the same manner, viz. by the common algebraic signs; which renders the process extremely easy even to boys or common sailors of but moderate capacities.

Example

Suppose the apparent angular distance of the sun & moon’s nearest limbs (by taking the mean of a set of observations) to be 110°.2′.30″ the app. Alt of ☉’s lower limb measuring 20°.40′ and that of ☽’s lower limb 35°.24′ height of the eye 18 feet, estimated Greenwich time Sept. 18th. 1798 about 6 hours p.m. time at place of observation, allowing for error of watch, or computed from the sun’s alt. & lat. of place 4 h. 20 m 30 s p.m. apparent time. Reqd. the longitude of the place of observation, from the merid. of Greenwich.

Solution

From the app. alts. of the lower limbs of ☉ & ☽ find the app. alts. of their centers by subtracting the dip corresponding to the height of the eye, and adding the app. semidiameters: Also from the app. dist. of limbs find the app. dist. of centers by adding the semidiameters. The longitude may then be computed by the following formula; in which the capital letters represent the corresponding arches in the adjoining column; & the small letters, the logarithmic functions of these arches. Where the small letter is omitted, the arch is found from the log. funct. The logs. need not be taken out to more than 4 decimal places, and to the nearest minute only of their corresponding arches except in the case of proportional logs. Where an ambiguous sign [occurs]1 as ± or ⨦ (expressing the sum or difference) the one or the other is to be used as directed in the explanatory note to which the number in the margin refers

		☉		☽
° ′ ″		° ′ ″		° ′ ″
110. 2. 30		20. 40. 00		35. 24. 00
15. 59		3. 18		3. 18	dip
15. 20		20 36. 42		35. 20. 42
110. 33. 49	app. dist. Cents	15. 59		15. 20	semi. diam
		20. 52. 41		35  36   2	App. Alt of Cents

Explanatory notes

1.	Add when C is greater than B otherwise subtract
2	Subtract when C is greater than B otherwise add
3	Subtract when either H or I exceeds 90°, or when H is greater than I, otherwise add.
4	Add when either H or I exceeds 90°, or when H is less than I, otherwise subt.
5	In Tab. 13 (reg. tab.) under the nearest degree to Q at top find two numbers, one opp. the nearest min. to ☽’s corr. of alt. found in tab. 8, and the other opp. the nearest min. to 1st. corr. (N) and the diff. of these two numbers will be the 3d corr. This corr. may generally be omitted
6	Add when Q is less than 90°. otherwise sub.
7	These are to be found in N.A. from p. 8h. to p. 11th. of the month, and the sun or star from which the moons dist. was obsd taking out the two distances which are next greater, & next less than the true dist. (S) calling that the preceding dist. which comes first in the order of time, and the other the folling2 dist.
8.	The Gr. time and time at place of ob. must both be reckoned from the same n[. . .]3
9.	When Y is greater than Z the long. is W. otherwise it is E—and when the long. comes out more than 12 hours or 180°. subt. it from 24h or 360° & change it name

Formula

	App. dist. of cents.	A	110.33.49
	☽’s app. alt	B	35.36	cosec	b	10.2350
	☉ or ✶’s app. alt	C	20.53
	½ B + C	D	28.14	tan	d	9.7299
	C ~ D	E	7.21	cot	e	10.8894
	½ A	F	55.17	tan	f	10.1593
	d + e + f – 20	G	80.33	tan	–	10.7786
1	F ⨦ G	H	25.16
2	F ⨤ G	I	135.50	tan	i	9.9874
	☽’s hor. par. N.A. page 7.	K	55.42	pr lg	k	.5994
	b + i + k – 20	L	27. 8	pr lg	–	.8218
	Refr. of I (considd as an Alt)  Tab. I. (reg. tables)	M	59
	L – M = 1st. corr.	N	26. 9
3	A ± N	O	110. 7.40
	Refr. of H for ✶,  refr-par for ☉ = 2d Corr	P	1.54
4	O ± P	Q	110. 9.34
5	Corr. from Tab. 13th = 3d Corr	R	4
6	Q ± R = true dist. of cents	S	110. 9.30
7	Preceding dist. in N.A.	T	110. 2.21
7	Following dist. in N.A	U	111.27.47
	T ~ S	V	7.9	pr lg	v	1.4010
	T ~ U	W	1.25.26	pr lg	w	3236
	v - w	X	15. 4	pr lg	–	1.0774
8	Hour above T, in N.A, + X  = true Green. time	Y	6.15. 4
8	Time at pl. of obs.	Z	4.20.30
9	Y ~ Z = Long. in time	A	1.54.34
9	A ÷ 4 = Long. in degrees &c	B	28°.38½′	West.

Note, the logarithmetrical part of the operation may, with sufficient accuracy be wrought on Gunters scale thus

1	Extend the compasses from Tang E to Tang D, and that extt. will reach from Tang. F to Tang. G.
2	Extend from Tang. I to sine B and that extent will reach (on the line of Numbers) from K to L
3	Extend (on the line of Numbers) from W to 180m and that extent will reach from V to X

I am, Sir, with the most perfect respect & esteem, your obedient Servant