TABLE XII. contains the refraction of the heavenly bodies, calculated by Dr. Bradley's rule, supposing the refraction to be as the tangent of the apparent zenith distance of the object decreased by three times the refraction, the horizontal refraction being supposed equal to 33'.

The rule expressed in logarithms is this:

Log. tang. (app. zen. dist.-3. refraction)-8.2438534=Log. of ref. in sec. The numbers calculated by this rule agree nearly with those published in Table I. of Maskelyne's Requisite Tables.

TABLE XIII. contains the dip of the horizon for various heights, calculated by the rule in § 197 of Vince's Astronomy, in which the terrestrial refraction is allowed for. All the numbers of this table differ a little from those published by Dr. Maskelyne, who had made a different allowance for that refraction. The rule given by Mr. Vince, expressed in logarithms, is: 1.7712711+half the log. of the height in feet-Log. dip in seconds.

TABLE XIV. contains the sun's parallax in altitude, calculated by multiplying the natural sine of the apparent zenith distance by the sun's horizontal parallax 83". The numbers in this table agree with those published by Dr. Maskelyne.

Table XV. contains the augmentation of the moon's semi-diameter= 15". 626 X sine D's altitude. This table agrees nearly with that published by Maskelyne.

TABLE XVI. contains the dip for various distances and heights, calculated by this rule.

3

7

h

d

D=d+0.56514×—

in which D represents the dip in miles or minutes, d the distance of the land in sea miles, and h the height of the eye of the observer in feet.

TABLES XVII. XVIII. and XIX. were first calculated by the author of this work, and published in the Appendix to the first edition. The correction in the first of these tables is equal to the difference between the star's refraction and 60'. The correction of Table XVIII. is equal to the difference between 60' and the correction of the sun's altitude for parallax and refraction. The correction of Table XIX. is equal to the difference between 59′ 42′′ and the correction of the moon's altitude for parallax and refraction. The logarithms in each of these tables may be found by adding together the constant log. 9.6990, the log. co-sine of the apparent altitude of the object, the proportional logarithm of the correction of the altitude of the object for parallax and refraction, and rejecting 20 from the index. The method of performing these calculations are so obvious, that it is unnecessary to enter into any farther explanation. Most of the numbers in these tables were calculated three different times.

TABLE XX. There are two columns in this table corresponding to each degree, the numbers in one column exceed those of the other by 18", the numbers in the least column express the difference b, between the base B and the hypotenuse B+b of a right angled spheric triangle, whose third side P, never exceeds 60'; the argument at the top of the table being B, and at the side 60'+P. The value of b being found by this rule by Taylor's logarithms:

Log. b in seconds-Log. co-tang. B+Log. vers. sine P-14.6855749-Diff.

log. sines of B and B+b

in which the last term may in most cases be neglected.

TABLE XX. (New Form.) corrections in seconds additive. See appendix, page 618.

TABLE XXI. for turning time into degrees, is the same as in other works of this kind.

TABLE XXII. contains the proportional logarithms for three hours. The numbers of this table may be found by subtracting the logarithm of the time in seconds from the log. of 10300"; or, which is the same thing, by the following rule:

Prop. log. T=4.0534758-log. of T in seconds, neglecting the three right hand figures of the remainder.

TABLE XXIII was first constructed by Mr. Douwes of Amsterdam, about
the year 1740, for which he received £.50 of the Commissioners of Longi-
tude in England. This table was published in the first and second editions
of the Requisite Tables; in the former of which it was carried as far as six
hours: in the latter the table of Log. Rising was extended to 9 hours: in
the present edition of this work it is extended to 12 hours. The numbers
in this table are easily deduced from the log. sines, log, co-secants, and log.
versed sines of the hour to which they correspond. Thus, if the time, op-
posite to any number of these tables turned into degrees, is H, we shail have
Log. 4 elapsed time of H=log. co-secant H-10.0000000
Log. middle time Log. sine H-4.6989700

Log. rising H=Log, versed sine H-5.0000000
=2x log. sine H-14.6939700

By means of these formulæ, the numbers of Table XXIII. were calculated by Sherwin's, Hutton's and Taylor's logarithms, and above a thousand errors were discovered in the second edition of the Requisite Tables, most of which were in the additional three hours (from six to nine hours) not published in the first edition. About two thirds of these additional numbers differ from their true values by one or two units.

TABLE XXIV. was compared with Sherwin's and Hutton's Tables, and a few errors corrected.

TABLE XXV. contains the log. sines, log. tangents, &c. corresponding to points and quarter points of the compass. This was compared with Sherwin's, Hutton's, and Taylor's logarithms.

TABLE XXVI. contains the common logarithms of numbers, which was compared with Sherwin's, Hutton's and Taylor's logarithms.

TABLE XXVII. contains the common log. sines. tangents, secants, &c. This was compared with Sherwin's, Hutton's and Taylor's tables. Two additional columns are given in this table, which are very convenient in finding the time from an altitude of the sun. The degrees are marked to 180°, which saves the trouble of subtracting the given angle from 1800 when it exceeds 30°.

TABLE XXVIII. was calculated by proportioning the daily variation of the time of the moon's passing the meridian.

TABLE XXIX. contains the correction of the moon's altitude for parallax and refraction, corresponding to the parallax 57' 30".

TABLES XXX. and XXXI. are tables of proportional parts, taken from the Requisite Tables, with a few corrections.

TABLE XXXII. contains the variation of the altitude of any heavenly body for one minute of time from noon, for various degrees of latitude and declination. The following method was used in constructing the ta ble:-A and B were calculated for each degree of declination by these formulæ.

Log. A=Log. 1′′.96349+2 log. cos. declination-20.00000.
Log. B Log. A.+log. tang. declination-10.00000.

and then the correction of the table corresponding to the zenith distance Z
(=Lat. +Dec.) was found by this formula. AX co-tang. Z+B. To fa-

S

cilitate the computation of these numbers, a table of the products of A by the whole numbers from 1 to 9 was calculated.

TABLE XXXIII. contains the squares of the minutes and parts of a minute corresponding to every second from 0 to 12′ 59′′. This requires no explanation.

TABLE XXXIV. contains the error of an observed angle arising from a deviation of 1' in the parallelism of the surfaces of the central mirror, those surfaces being supposed to be perpendicular to the plane of the instrument. The correction in the fifth column of this table corresponding to any angle

A in the first column may be found nearly by Hutton's logarithmns, as follows: to the constant logarithm 0.07345 add the log. secant of A, find thús in the column of log-tangents and take out the corresponding natural secant B, then the correction will be 2' (B-1,55.) The numbers in the second column are nearly equal to those in the fifth corresponding to the angle A+ 200, decreased by 1". 68. The numbers in the third column are equal to the difference between 1". 68, and the numbers in the fifth corresponding to A 200. The numbers in the fourth column are equal to the half difference of the numbers on the same horizontal line in columns second and third, when it exceeds 400, otherwise their half sum.

TABLE XXXV. contains the correction to be applied to an observation taken in a direction inclined to the plane of the instrument. The following rule was used in calculating this table: Find an arch A such that

Log. sine A Log. sine observed angle+log, co-sine of error of inclina tion. Then the difference between 2 A and the observed angle will be the tabular correction.

TABLE XXXVI. contains the variation of the mean refraction (given in Table XII.) for various temperatures and densities of the air. The correction given in this Table is nearly the same as that deduced from Dr. Bradley's rule, which is as follows:-As the mean height of the barometer 29,6 inches is to the true height, so is the mean refraction to the corrected refraction: and as 350 increased by the height of Fahrenheit's thermometer is to 490, so is the corrected to the true refraction.

TABLE XXXVII. contains the latitudes and longitudes of the fixed stare of the 1st. 2d. and 3d. magnitudes. The nine stars, from which the distances are marked in the Nautical Almanac, are given from the table published in the Nautical Almanac for 1820. The rest were deduced from the table published in the second edition of Doctor Mackay's treatise on longitude, supposing the annual precession 50". 35 and the secular equation as in his table. In the third edition of his work, the precession was allowed on the latitude as well as the longitude, which causes an error of about 8' in the latitudes of the stars in that edition.

TABLE XXXVIII. was calculated by this rule. Suppose L to be the latitude, R the reduction of latitude, then log. co-tang. (L-R) =0.0629001 + log.co-tang. L. The reduction of parallax corresponding to 53', 57′ and 61', was found by these formulas respectively 5′′.3—5", 3 cos. 2 L; 5′′.7—5′′. 7 cos. 2 L; 6". 1-6′′.1 còs. 2 L.

TABLE XXXIX. was calculated by the rule in vol. I. page 334 of Vince's Astronomy, supposing S to be the place of the sun, P that of the planet, and T that of the earth.

SP

Aberration=-20.cos. STP-20 ✓ ST.cos. SPT. Making use of the distances, &c, given by La Place in vol. III. of his Mecanique Celeste. A small alteration was made in the rule in calculating the aberration of Mercury.

TABLE XL. was calculated by 17′′.9. sine long. D's node.

• TABLE XLI. was calculated by - 20". cos. argument.

TABLE XLII. Part I. =-19". 173 cos. arg. Part II. =0".827 cos. arg. =-3'.9814 cos. arg.

Part III.

TABLE XLIII. Part I.

Part III.

TABLE XLIV. Part I.

8".33 cos. arg. Part II. =-1". 22 cos. arg. -16".382 sine arg.

8".1845 sine arg. Part II. =(arg. in seconds)

960"

Part III.=960"X sine D's par. in lat. X tang. D's true lat.

-960. versed sine par. in lat.

If we suppose the sum of these three parts to be S seconds, and the moon's horizontal semi-diameter to be D minutes. Part IV. corresponding to S and D will be $x (D+16)(D-16)

256

B