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The first table contains the equatorial semidiameter of the Moon in time; the arguments of which are, the daily variation of the Moon's passing the meridian at the top, and the apparent semidiameter in the side column. The other table contains the increase of the seinidiameter, answering to the Moon's declination and equatorial semidiameter in time.

EXAMPLE.

Required the time the Moon's semidiameter takes to pass the meridian, its semidiameter being 16' 10", diurnal retardation 52', and declination 21°?

To semidiam. 16' 10", and diur. retard. 52', equat. semid. is 67.0 To equat. semidia. 67". O, and declinat. 21°, the increase is 4.7

Time of Moon's semidiameter passing the meridian

71.7

TABLE XLI.

Natural Versed Sines*,

As the solution of the problem for correcting the observed distance between the Moon and the Sun or a fixed star, is rendered

very short, when the operation is performed by means of a table of natural versed sines ; upon this account, therefore, the above table is inserted; and in order to make the operation as simple as possible, this table is extended to every tenth second of the semicircle, or 180 degrees, the proportional part to each intermediate second is given at the bottom of the page. Hence the natural versed sine of any given arch may

be taken out at sight. The natural co versed sine of a given degree and minute is found on the same line with the next less minute, in the column marked 60° at the bottom of the page; and that answering to an arch, expressed in degrees, minutes, and seconds, is obtained, by deducting the proportional part, answering to the intermediate seconds, from the co versed sine of the given degree, minute, and next less tenth second.

When the nat. versed sine of the supplement of an arch above 90' is wanted, take that answering to the natural co-versed sine of the excess of that arch above 90. Thus the natural versed sine of the supplement of 114° is the natural co-versed sine of 24o.

This table was first given in this work published in 1793 ; M. Van Swinden copied it in his Verandeling, printed at Amsterdam in 1796, saying that to his other Tables he had added the very beautiful Table of Natural Versed Sines by Mackay.' This table has since been copied by others, without acknowledging from whence they took it.

The logarithmic versed sine of an arch is found by taking out the logarithm answering to the natural versed sine ; and since the radius of this table is 1,000000, the index will be 9, if the natural versed sine consists of six figures; 8, if it consists of five figures ; 7, if of four, &c.

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EXAMPLES.

I.

Required the logarithmic versed sine of 23° 28' ? The nat. versed sine of 23° 28' is .082708, the log. of which is 8.917548, the log. versed sine of 29° 28'.

II.

Required the logarithmic versed sine of 146° 8' 40" ?

The natural versed sine of 146° 38' 40" is 1.835275, the logarithm of which is 0.263701, the logarithmic versed sine of 146° 38' 40".

Beside the application of this table to the solution of the above problem, it may be used very successfully in many other calculations; of which the following may serve as a specimen :

PROBLEM:

Given the Course and Difference between the Distance and Difference,

of Latitude, to find each separately, and also the Departure.

RULE.

To the given difference, annex six cyphers, and divide this sum by the natural versed sine of the course, and the quotient will be the distance, from which the diff. being subtracted, the remainder will be the diff. of latitude: and the distance multiplied by the versed sine of the complement of the course, will give its excess above the departure. If logarithms be used, the operation, in most cases, will become more simple.

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Let the course be S. W. by S. the difference between the dist ance and diff. of lat. 40 miles. Required the distance, diff. of latitude, and departure ? Course 3 pts.N. V.S,16853log.9.22668 Natural co-V.S. 44443 Diff. 40. log. 1.60205 log.

9.64780

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The natural sine of any given arch may be found, by subtracting the natural co-versed sine of that arch from the radius, or 1.000000, and the natural co-sine is obtained by subtracting the natural versed sine from the radius.

EXAMPLE.

Required the natural sine, and natural co-sine of 340 17, 2017 Radius 1 000000

1.000000 Nat co-ver, sine of 34° 17' 21" .436634 Nat, ver. sine .173792

Natural sine of

346 17' 20.563366

Nat. co sine

.826208

REMARK,

M. Van Swinden's table of auxiliary angles, which angle he calls p, for reducing the apparent to the true distance, according to the method proposed by M. Krafft, is easily deduced from the above, by subtracting the constant log. 0.301030, from the logarithmic difference answering to the Moon's apparent altitude and horizontal parallax, and the remainder will be the logarithmic co-sine of the auxiliary angle.

Thus, in the preceding example, the logarithmic diff. is 9,994729 Constant logarithm

0,301030

Auxiliary angle 60° 23' 54."

Co-sine

9,693699

This method is given in page 157 of the first vol. and illustrated by only one example; it is, therefore, thought proper to subjoin the following

EXAMPLE.

Let the observed distance between the nearest limbs of the sun and moon be 110° 49' 24" the observed altitude of the Sun's lower limb 1.5° 46, that of the upper limb of the Moon 43° 34', Sun's semidiameter 16' 10", Moon's semidiameter 15' 41", and horizontal parallax 57' 32". Required the true central distance ?

Observed distance between the Sun and Moon's nearest limbs

110° 42' 24" Sun's semidiameter

16 10 Moon's semidiameter

+

15 4) Augmentation

11

Apparent central distance

lll 14 56

Ob. alt. Sun's lower limb
Sun's semidiameter

15° 46' Ob. alt. Moon's ref.log. 3° 341 to 16

Moons semidiameter 16

Apparent alt. Sun's center 16 2 App. alt. )'s center

App. alt. O's center

4319 16' 2

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TABLES

XLII. XLIII. AND XLIV.

Logarithmic Difference. These tables are also intended to facilitate the method of reducing the apparent to the true distance. The first is to be entered with the Moon's apparent altitude at the top, and its horizontal parallax in the side column. The proportional part answering to the excess of the given altitude above the next less altitude, at the top of the table, is found in the right hand column; and that corresponding to the excess of the given parallax above the next less tabular argument, is found at the bottom of the table. The other two tables, which serve to correct the former, according as the Moon is compared with the Sun or a fixed star, are to be entered with the apparent altitude of the Sun or observed star respectively.

EXAMPLE.

Let the Sun's apparent altitude be 27° 43', that of the Moon 46 18', and horizontal parallax 59' 33". Required the logarithmic difference ?

Log. diff. to ap. alt. 46' 10' and hor. par 59' 30" 9.994755 Proportional part to 8' of altitude

12 3" of parallax

5 Number from Table XLIII. answering to the Sun's alt.

9

to

Logarithmic difference required

9.994729

TABLE XLV.

Logarithms of Numbers. Logarithms are a series of numbers invented by Lord Napier* Baron of Merchiston, for the purpose of facilitating the arithmetical computations in plane and spherical trigonometry. His treatise, entitled “ Mirifici Logarithmorum Canonis Descriptio" was published at Edinburgh in the year 1614.

Or, Nepes, as in the title of the book.

VOL. II.

D

Logarithms

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