four last figures of the logarithmic difference that are given: hence, in taking out the numbers from these columns, they are always to be prefixed by the characteristic, and the two leading figures in the first column. The logarithmic difference is to be taken out in the following manner.

Enter the Table with the moon's apparent altitude in the left-hand column of the page, or the altitude next less if there be any odd minutes, opposite to which, and under the minutes of the moon's horizontal parallax, at top, will be found a number, which call the approximate logarithmic difference.

Enter the compartment of the " Proportional parts to seconds of parallax," abreast of the approximate logarithmic difference, with the tenths of seconds of the moon's horizontal parallax in the vertical column, and the units at the top, and take out the corresponding correction. Enter the right-hand compartment of the page,* abreast of where the approximate logarithmic difference was found, or nearly so, with the odd minutes of altitude, and take out the corresponding correction, which place under the former. Enter Table XXV or XXVI., with the sun's, star's, or planet's apparent altitude, and take out the corresponding correction, which also place under the former. Now, the sum of these three corrections being taken from the approximate logarithmic difference, will leave the correct logarithmic difference.

Example 1.

Let the moon's apparent altitude be 19:25, her horizontal parallax 60′38′′, and the sun's apparent altitude 33 degrees; required the logarithmic difference?

Log. difference to app. alt. 19:20, and hor. par. 60 is 9.997669
Propor. part to 38 seconds of parallax is 28
Propor. part to 5 minutes of altitude is
Cor, from Tab. XXV. ans. to sun's apparent alt. is 10

Logarithmic difference, as required

11

sum

49

9.997620

* In taking out the correction corresponding to the odd minutes of altitude in this compartment, attention is to be paid to the moon's horizontal parallax: thus, if the parallax be between 53′ and 56', the correction is to be taken out of the first column, or that adjoining the minutes of altitude; if it be between 56′ and 59′, the correction is to be taken out of the second, or middle column; and if it be between 59′ and 62'; the correction is to be taken out of the third, or last column.

Example 2.

Let the moon's apparent altitude be 63:37, her horizontal parallax 58:43, the apparent altitude of a planet 35:10, and its horizontal parallax 23"; required the logarithmic difference?

Log. difference to appar. alt. 63:30, and hor. par. 58., is 9.993622 Propor. part to 43% of parallax is

Propor. part to 7 of altitude is

Cor. from Tab. XXVI. ans. to planet's appar. alt.

83

7 sum-118

28

Logarithmic difference, as required

9.993504

Remark. The logarithmic difference was computed by the following

Rule.

To the logarithmic secant of the moon's apparent altitude, add the ⚫ logarithmic cosine of her true altitude, and the constant log..000120;* the sum of these three logs., abating 10 in the index, will be the logarithmic difference.

Example.

Let the moon's apparent altitude be 19:20, and her horizontal parallax 60 minutes; required the logarithmic difference?

The difference between the log, cosines of the true and apparent altitude of a star

betwixt 30 and 90 degrees.

TABLE XXV.
xxv.

Correction of the Logarithmic Difference.

This Table is divided into two parts: the first, or left-hand part, contains the correction of the logarithmic difference when the moon's distance from the sun is observed; and the second, or right-hand part, the correction of that log. when the moon's distance from a star is observed. Thus, if the sun's apparent altitude be 35 degrees, the corresponding correction will be 11; if a star's apparent altitude be 20 degrees, the corresponding correction will be 1; and so on. These corrections are always to be applied by subtraction to the logarithmic difference deduced from the preceding Table.

The corrections contained in this Table were obtained in the following manner, viz.

To the log. secant of the apparent altitude, add the log. cosine of the true altitude; and the sum, rejecting 10 from the index, will be a log. ; which being subtracted from the constant log. .000120,* will leave the tabular correction.

Example 1.

Let the sun's apparent altitude be 35 degrees; required the tabular correction?

Let the apparent altitude of a star be 10 degrees; required the tabular

correction?

See Note, page 50.

Star's apparent altitude 10: 0 0 Log. secant 10.006649

Correction of the Logarithmic Difference when the Moon's Distance from a Planet is observed.

The arguments of this Table are, the apparent altitude of a planet in the left or right-hand marginal column, and its horizontal parallax at top; in the angle of meeting stands the corresponding correction, which is to be applied by subtraction to the logarithmic difference deduced from Table XXIV., when the moon's distance from a planet is observed. Hence, if the apparent altitude of a planet be 20 degrees, and its horizontal parallax 21 seconds, the corresponding correction will be 16 subtractive, and so on. This Table was computed by the rule in page 51, under which the correction corresponding to the sun's apparent altitude in Table XXV. was obtained, as thus:

Let the apparent altitude of a planet be 23 degrees, and its horizontal parallax 21 seconds; required the correction of the logarithmic difference?

TABLE XXVII.

Natural Versed Sines, and Natural Sines:

Since the methods of computing the true altitudes of the heavenly bodies, the apparent time at ship or place, and the true central distance between the moon and sun, or a fixed star, are considerably facilitated by the application of natural versed sines, or natural sines, this Table is given; which, with the view of rendering it generally useful and convenient, is extended to every tenth second of the semicircle, with proportional parts corresponding to the intermediate seconds; so that either the natural versed sine, natural versed sine supplement, natural co-versed sine, natural sine or natural cosine of any arch, may be readily taken out at sight. The numbers expressed in this Table may be obtained in the following

manner :

Let ABC represent a quadrant, or the fourth part of a circle; and let the radius CB =unity or 1, be divided into an indefinite number of decimal parts: as thus, 1.0000000000, &c. Make BD the radius CB; and since the radius of a circle is equal to the chord of 60 degrees, the arc BD is equal to 60 degrees: draw DM, the sine of the arc BD, and, at right-angles thereto, the cosine DE: bisect the arc BD in F, and draw FN and FG at rightangles to each other; then will the former represent the sine, and the latter the cosine of the arc BF 30 degrees: bisect BF in H; then HO will express the sine, and HI the cosine of the arc BH = 15 degrees.

Proceeding in this manner, after 12 bisections, we come to an arc of 0:0:52"44"3":"45"", the cosine of which approximates so very closely to the radius C B, that they may be considered as being of equal value. Now, the absolute measure of this arc may be obtained by numerical calculation, as follows, viz.