The Secretary of State having early in the present month, reported to Congress on the subject of a first meridian for the United States, in which he has recommended the establishment of an Observatory as of essential utility to ascertain the position of the Capitol in this City with due precision, permit me to send some rules by which the parallaxes in longitude and latitude may be found with great accuracy, and for some of which we are indebted to Mr Seth Pease, an assistant postmaster general of the United States.—

The usual method for the parallax in longitude is—to add together the log. sine of the Moon’s horizontal parallax, reduced, (or, in case of a Solar eclipse, the hor. par. ☽ à ⊙,) the Sine of the altitude of the nonagesimal, and the arith. comp. cosine of the Moon’s true latitude, the sum of these three log’s, rejecting radius, will be = Sine of an arch which we will call (a)

To (a) add the log. sine of the Moon’s true distance from the nonagesimal, for the sine of 1st approxn of parallax;—which is to be added to the Moon’s true distance from the nonagesimal: this process ought to be repeated until the correct parallax in longitude, and consequently, the Moon’s apparent distance from the nonagesimal, be found.—

But Mr Pease, in order to save the trouble of four or five approximations, has given the following rules, which are found on trial, to produce an accurate result.

Log. sine ☽’s hor. par: + sine altitude nonag: = log. (x)

Log. (x) + cosine ☽’s true dist. à nonag. = log. (y)

Find the corresponding natural number of log. (y.)

Natural cosine ☽’s true lat. – nat. number (y) = natural number of (z) to which find the corresponding log.

Arith. comp. log. (z) + log. (x) + log. sine ☽’s true dist. à nonag. = log. tangent parallax in longitude.

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Rule 2.

Log. cosine ☽’s true lat. + cosine ☽’s true dist. à nonag. = log. A, to which find its corresponding natural number.

Log. Sine ☽’s hor. par. + log. sine altitude of the nonag. = log. B; find its corresponding natural number, and take the difference of it and the natural number A, which will be the natural number C; find the log. corresponding to C; then

arith: comp. log. C, + log. cosine ☽’s true lat. + log. sine ☽’s true dist. à nonag. = log. tangent ☽’s apparent distance à nonag.—the Difference between which and the true dist. à nonag. is = parallax in longitude.

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Mr Pease has also furnished a rule to find the Moon’s apparent latitude, as follows.—

Log. sine ☽’s hor. parallax, + log. cosine alt. nonag. = log. D; to which find its corresponding natural number.

The natural sine of ☽’s true latitude is E.

E, ∓ D, = F; find the corresponding log. of F.

Log. F, + log. cosecant Moon’s true dist. à nonag. + log. sine ☽’s apparent dist. à nonag. + log. secant ☽’s true lat. = log. tangent ☽’s Apparent latitude, – the difference between which and the Moon’s true latitude, is = parallax in latitude.

☞ If the latitude of the place, and the Moon’s true lat. be both north, the difference of E and D, will be = F; but if the Moon’s lat. be south, their sum; and if D be greater than E, the Moon’s apparent lat. will be South, when the true lat. is north, which sometimes happens in the case of solar eclipses.

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Mr Pease has given the following rule for ascertaining the Moon’s augmented semidiameter arising from her apparent altitude—

Arith. comp. log. F, (found as before directed) + log. sine ☽’s horizontal Semidiameter, + log. sine ☽’s apparent latitude, = log. sine ☽’s augmented Semidiameter, from which the inflexion of the Moon’s light being subtracted, will give the Moon’s corrected Semidr

The rule used byMessrs Garnett and Ferrer, (supposed to be taken from the works of M. de la Lande) to find the Moon’s parallax in latitude, is as follows.—

Log. sine hor. parallax, + cosine alt. nonag. + ar. comp. sine ☽’s true dist. à nonag. + sine ☽’s apparent distance à nonag. = sine first of parallax in lat.

Log. sine Moon’s true latitude, + ar. comp. sine ☽’s true dist. à nonag. + Sine parallax in longitude, + cosine ☽’s true dist. à nonag. + par. in longitude⁄2, = Sine of second part of the parallax.

The difference of these parts, if the latitude of the place and the Moon’s true latitude be both north, or their sum if the Moon’s true latitude be south, will be the parallax in latitude, nearly approximated, which call (p.) Find therefrom, the Moon’s apparent lat. approximated, which call (l) then,

Log. sine (p) + cosine (l) = sine parallax in latitude, correct.

By Dr Maskelyne’s rule, to give a correct result, the Moon’s apparent latitude must be obtained by several approximations, using the Moon’s true lat. in the first instance.—

Log. sine hor. parallax, + cosine altitude of the nonag: + cosine Moon’s true latitude, = Sine (a)

Find the Moon’s approxd apparent latitude, by means of the true lat. and arch (a) then,

Log. sine hor. parallax, + sine altitude nonag. + Sine Moon’s approxd apparent latitude, + cosine Moon’s true dist. à nonag. + par. in lat. long.⁄2 = Sine b)

The Sum or difference of (a) and (b), as before recited, will be an approximate parallax in latitude; but the process must be repeated, to have the parallx sufficiently correct.

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The application of the foregoing rules will be shewn in the following example.

Let the Moon’s horiz. parallax à ☉, be 53.′ 56.″ 7, the altitude of the nonagesimal 55.° 0′, the Moon’s true latitude, north, 32.′ 53.″ 4, and her true distance from the nonagesimal (East) 10.° 59.′ 32.″—required the parallaxes in longitude and latitude?

 

For the parallax in longitude.

		° ′ ″
Sine hor. par. ☽ à ☉,		0.53.56. 7		8.1956595.
Sine alt. nonag.		55.0		9.9133645.
ar. co. cosine ☽’s true lat.		0.32.53. 4		0.0000199.
		° ′ ″	(a)	8.1090439.
Sine ☽’s true dist. à nonag.		10.59.32.	(b)	9.2802955
Sine 1st approxn	a + b,	8.25.55	(c)	7.3893394
Sine	b + c,	11. 7.57.55.	(d)	9.2857398
Sine 2d approx.	a + d,	8.31.93	(e)	7.3947837
Sine	b + e,	11. 8. 3.93	(f)	9.2858081.
Sine	a + f	8.32.01	(g)	7.3948520

 

If the process be continued, the parallax in longitude will be found 8.′ 32.″ 01, which may be considered as sufficiently correct; hence the apparent distance from the nonagesimal is 11.° 8.′ 4.″ 01. dec.

 

By Mr Pease’s Rule 1.

	′ ″
Sine hor. par. ☽ à ☉,	53.56.7		8.1956595
Sine altitude nonag	55. 0.0		9.9133645
		log. ( x )	8.1090240.
Cosine ☽’s true dist. à non. 10.° 59.′ 32.″			9.9919580.
Corresponding nat. no	0126178	log. (y)	8.1009820
Nat. cosine ☽’s true lat.	9999542
Nat. number (z)	9873364.	corresponding log.	9.9944695
		ar. comp.	0.0055305
		log. (x)	8.1090240.
Sine ☽’s true dist. à non.	10.° 59.′ 32.″		9.2802955
parallax in longitude,	8.′ 32.″ 0	tang.	7.3948500

Rule 2.

Log. cosine Moon’s true lat. 0.° 32.′ 53.″ 4		9.9999801.
〃 cosine ☽’s true dist. à nonag. 10.° 59.′ 32.″		9.9919580.
Corresponding nat. number, 9816080.	(A) log.	9.9919381
Log. sine hor. par. ☽ à ☉, 53.′ 56.″ 7		8.1956595.
〃 Sine alt. nonag  55.° 0		9.9133645
Natural number 0128536	(B) log.	8.1090240.

Nat. number	(A)	9816080
〃	(B)	0128536
	(C)	9687544.  log	9.9862137
		ar. comp.	0.0137863.
Log. cosine ☽’s true lat. 0.° 32.′ 53.″ 4		° ′ ″	9.9999801.
〃  Sine ☽’s true dist. à nonag.		10.59.32	9.2802955
〃  tang. ☽’s apparent dist:		11.  8.  4	9.2940619
diff: = par: in longitude		0.  8.32.

For the parallax in latitude.

By M. de Lande’s rule.

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	′ ″
Log. sine hor. par. ☽ à ☉,	53.56. 7	8.1956595.
〃  cosine alt. nonag.	55. 0. 0	9.7585913.
	°
ar. co. sine ☽’s true dist à nonag.	10.59.32	0.7197045.
Sine apparent dist.	11. 8. 4.	9.2858090.
Sine 1st part par. in lat	31.′ 20.″ 16.	7.9597643.

	° ′ ″
Log. sine Moon’s true lat.	0.32.53.4		7.9807833.
ar. co. sine ☽’s true dist. à non.	10.59.32		0.7197045.
Sine parallax in longitude	0. 8.32.01		7.3948520.
cosine true dist. par. in long.⁄2 }	11. 3.48 –		9.9918530.
	′ ″ dec.
Sine 2nd part par.	0.25.21		6.0871928.

	′ ″
First part	31.20.16
Second part	– 0.25.21
(p)	30.54.95.	Sine	7.9539012
Moon’s true lat. North	32.53.40
〃  apparent lat. approxd  (l)	1.58.45.	cosine	9.9999999.
par: in lat. correct	30.54.95.	sine	7.9539011.

Dr Maskelyne’s rule

To avoid several approximations, the above result will be tested by this rule, using the Moon’s apparent latitude already found.

	′ ″
Log. sine hor. par. ☽ à ☉,	53.56. 7		8.1956595.
〃  cosine alt. nonag.	55. 0. 0		9.7585913.
〃  cosine ☽’s apparent lat.	0. 1.58.45		9.9999999
Sine 1st part parallax in lat.	30.56.44		7.9542507

	′ ″
Log. sine hor. par. ☽ à ☉,	53.56. 7		8.1956595
〃  Sine alt: nonag.	55. 0. 0		9.9133645
〃  Sine ☽’s apparent lat. north.	1.58.45		6.7591100.
〃  cosine true dist. + par. in long.⁄2	11. 3.48		9.9918530
	″
Sine 2nd part par.	0. 1.49 Sine		4.8599870.

	′ ″
First part parallax	30.56.44.
Second part do	– 0. 1.49.
parallax in latitude, correct,	30.54.95	the same as above.

Mr Pease’s rule.

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	° ′ ″
Log. sine hor: par. ☽ à ☉,	0.53.56.7			8.1956595.
〃  cosine alt. nonag.	55. 0. 0			9.7585913.
Correspondg nat. number,	0090001.7	(D).	log.	7.9542508
Nat. sine ☽’s true lat. ,	0095671.7	(E)
Nat. number,	0005670.0	(F)	log.	6.7535831.
Log. cosecant ☽’s true dist. à nonag.				10.7197045.
〃  sine ☽’s apparent dist.				9.2858090.
〃  Secant ☽’s true lat. (north)	′ ″			10.0000199
Moon’s apparent lat. north	1.58.45.		tang.	6.7591165
〃  true lat. north	32.53.40.
diff. = par: in latitude	30.54.95

By the parallactic angle.

 

Here are two sides of an oblique spherical triangle.—the distance from the zenith to the pole of the ecliptic = 55.° = altitude of the nonagesimal, and the Moon’s distance from the said pole, = 89.° 27.′ 6.″ 60 dec, and the angle between them, 10.° 59.′ 32.″ = Moon’s true distance from the nonagesimal, to find the angle opposite to the first side. (i.e.) the parallactic angle.

	° ′ ″
1st Side	55. 0. 0		tangent		10.1547732.
Angle	10.59.32.		Cosine		9.9919580
arch 1 =	54.29.59.82		tangent		10.1467312
2nd side	89.27. 6.60
diff. = 2nd arch	34.57. 6.78.		cosecant		10.2419299
angle	10.59.32		tangent,		9.2883374.
arch 1st	54.29.59.82.		Sine		9.9106857.
parallactic angle,	15.25.51.57		tang.		9.4409530.
parallactic angle,	15.25.51.57		cosine		9.9840554.
arch II	34.57. 6.78		cotang.		10.1555049.
Moon’s true altitude,	54. 3. 5.44.		tangent,		10.1395603.

The complement of which = 35.° 56.′ 54.″ 56. dec, is the third side of the triangle, or the Moon’s true distance from the Zenith.

 

With the hor. parallax ☽ à ☉, = 53.′ 56.″ 7, and the Moon’s true altitude, 54.° 3.′ 5.″ 44, find the Moon’s apparent altitude, (Exclusive of refraction) = 53.° 31.′ 1.″ 0 dec; then,

	′ ″
Log. sine hor. parallax ☽ à ☉,	°  53.56. 7		8.1956595.
〃  cosine ☽’s apparent alt.	53.31. 1. 0		9.7743845
〃  cosine parallactic angle,	15.25.51.57		9.9840554.
〃  Sine parallax in lat, nearly,	0.30.55.78		7.9540994.
Moon’s true lat.	0.32.53.40
Moon’s apparent lat. nearly	–1.57.62	. north

	° ′ ″
Log. sine hor. parallax ☽ à ☉,	0.53.56. 7		8.1956595.
〃 cosine ☽’s app: alt.	53.31. 1. 0		9.7743845.
〃 Sine parallactic angle	15.25.51.57		9.4250083.
〃 ar. comp. cosine ☽’s app. lat.	0. 1.57.62		0.0000001
〃 Sine parallax in longitude	8.32.26		7.3950524.

This result differs about ¼ of a Second in the parallax in longitude, and ^⅚ of a Second in the parallax in latitude, from the former; the other rules are entitled to a preference in nice calculations of the parallaxes; and in the case of Mr Pease’s, the result would always be more accurate, if the natural numbers, sines, cosines &c. were extended to eight places of figures, instead of seven.

The most correct determination of the longitude of any place is known to be from Solar eclipses and occultations of fixed Stars by the Moon; and the best rules for ascertaining the parallaxes in longitude and latitude, may not be unworthy of your attention, and that of the American Philosophical Society at Philadelphia.