TABLE XIV.

Equations to equal Altitudes.-PART SECOND.

In this Table, the interval between the observations is marked at top or bottom, and the sun's declination in the left or right-hand margin; under or over the former, and opposite to the latter, stands the corresponding equation, expressed in seconds and thirds: thus, the equation answering to 6 hours 40 minutes, and declination 18:30, is 2 seconds and 48 thirds. The equations contained in this Table were computed as follows, viz.:To the log. co-tangent of the declination, add the log. tang. of half the interval in degrees; the proportional log. of the whole interval in time (esteemed as minutes and seconds), and the constant log. 8.8239;* the sum of these four logs., rejecting 29 from the index, will be the proportional log. of the corresponding equation in minutes and seconds, which are to be considered as seconds and thirds.

Example.、

Let the sun's declination be 18:30, and the interval between the observed equal altitudes of the sun 4 hours; required the corresponding equation?

Sun's declination

18:30:

Log. co-tang. 10. 4755

Half interval 2 ho. in degs. 30. 0.

Whole interval 4 ho. esteemed as 4 min. Prop. log.

Log. tang.

9.7614

1.6532

To find the Equation of Equal Altitudes by Tables XIII. and XIV.

Rule.

Enter Table XIII., with the latitude in the side column and the interval between the observations at top; and find the corresponding equation, to which prefix the sign + if the sun be receding from the elevated pole, but the sign - if it be advancing towards that pole.

Enter Table XIV., with the declination in the side column, and the interval between the observations at top, and take out the corresponding equation, to which prefix the sign + when the sun's declination is increasing, but the sign. when it is decreasing.

* The arithmetical complement of 12 hours considered as minutes.

Now, if those two equations are of the same signs; that is, both affirmative or both negative, let their sum be taken; but if of contrary signs, namely, one affirmative and the other negative, their difference is to be taken then,

J

To the proportional log. of this sum or difference, considered as minutes and seconds, add the proportional log. of the daily variation of the sun's declination; and the sum, rejecting I from the index, will be the proportional log. of the true equation of equal altitudes in minutes and seconds, which are to be esteemed as seconds and thirds, and which will be always of the same name with the greater equation.

Example 1.

In latitude 49: south, the interval between equal altitudes of the sun was 7:20; the sun's declination 18: north, increasing, and the variation of declination 15:12; required the true equation of equal altitudes? Opposite lat. 49° under 7:20" Tab. XIII. stands +15"27" Opposite dec. 18: under 7'20" Tab. XIV. stands + 2.30

Sum

17"57" Pro. log. 1.0012 Variation of declination . 15:12 Pro. log. 1.0734

True equation, as required + 15"10" Pro. log. 1.0746

Example 2.

In latitude 50: north, the interval between equal altitudes of the sun was 5:20; the sun's declination 18:30: north, increasing, and the daily variation of declination 14:34"; required the true equation of equal altitudes?

Op. lat. 50°

under 520 Tab. XIII.stands-14"50" Op.dec.18:30:under 5 .20 Tab. XIV. stands + 3.11

True equation, as required-9:26" Pro. log. = 1.2808

Remark.-In north latitude the sun recedes from the elevated pole from the summer to the winter solstice; that is, from the 21st June to the 21st December; but advances towards that pole from the winter to the summer solstice: viz., from the 21st December to the 21st June. The converse of this takes place in south latitude: thus, from the 21st June to the 21st December, the sun advances towards the south elevated pole; but recedes from that pole the rest of the year, viz., from the 21st December to the 21st June.

Here it may be necessary to observe, that in taking out the equations from Tables XIII. and XIV. allowance is to be made for the excess of the given, above the next less tabular arguments, as in the following examples :

Example 1.

Required the equation from Table XIII., answering to latitude 50:48", and interval between the observations 5 hours 10 minutes ?

Equation to latitude 50%, and interval 4:40" =

Tabular diff. to 1:of lat.=+31"; now,

31" x 48"

14:33"

+ 0.241

Tab. diff. to 40% of inter.=+17"; now,

60. 17" × 30!

+ 0.124

40:

Equation, as required

15%10"

Example 2.

Required the equation from Table XIV., answering to sun's declination 20:47, and interval between the observations 5 hours 10 minutes?

Equation to declination 20:30 and interval 4:40"= 3:44"

Note. Should the latitude exceed the limits of Table XIII., which is only extended so far as to comprehend the ordinary bounds of navigation, viz., to 60 degrees, the first part of the equation, in this case, must be determined by the rule under which that Table was computed, as in page 21.

TABLE XV.

To reduce the Sun's Longitude, Right Ascension, and Declination; and also the Equation of Time, as given in the Nautical Almanac, to any given Meridian, and to any given Time under that Meridian.

This Table is so arranged, that the proportional part corresponding to any given time, or longitude, and to any variation of the sun's right ascension, declination, &c. &c., may be taken out to the greatest degree of accuracy.

Precepts.

In the general use of this Table it will be advisable to abide by the solar day; and hence, to estimate the time from noon to noon, or from 0 to 24 hours, after the manner of astronomers, without paying any attention to either the nautical or the civil division of time at midnight. And to guard against falling into error, in applying the tabular proportional part to the sun's right ascension, declination, &c. &c., it will be best to reduce the mean time at ship or place, to Greenwich time; as thus:

Turn the longitude into time (by Table I.), and add it to the giyen time at ship or place, if it be west; but subtract it if east; and the sum, or difference, will be the corresponding time at Greenwich.

From page II. or III. of the month in the Nautical Almanac, take out the sun's right ascension, declination, longitude, &c., for the noons immediately preceding and following the Greenwich time, and find their difference, which will express the variation of those elements in 24 hours; then,

Enter the Table with the variation, thus found, at top, and the Greenwich time in the left-hand column; under the former and opposite the latter will be found the corresponding equation, or proportional part. And, since the Greenwich time may be estimated in hours, minutes, or seconds, and the variation of right ascension, &c. &c. &c., either in minutes or seconds: the sum of the several proportional parts making up the whole of such time and variation will, therefore, express the required proportional part. The proportional part, so obtained, is always to be applied by addition to the sun's longitude and right ascension at the preceding noon; but it is to be applied by addition, or subtraction, to the sun's declination and the equation of time at that noon, according as they are increasing or decreasing.

Example.

Required the sun's right ascension and declination, and also the equation of time, May 6th, 1824, at 5:10", in longitude 64:45 west of the meridian of Greenwich?

To find the Sun's Declination :

Sun's declination at noon, May 6th, 1824, per Nautical

Pro. part

0.8 0. 0. 9.40

to 9:29 and 16'38" is 6.34. 20.55 + 6:34"

Equation of time at noon, May 6th, 1824, per Nautical

Remark. Should the proportional part corresponding to the daily variation of the sun's longitude and any given time be required, it may be taken from the first page of the Table, by esteeming the seconds of variation, in that page, as minutes, and then raising the signs of the corresponding proportional parts one grade higher than what are marked at the top of the said page: the seconds of variation are of course to be taken out in the usual manner.

Note. The present Table was computed agreeably to the established principles of the rule of proportion; viz. As one day, or 24 hours, is to the variation of the sun's right ascension, declination, &c. &c., in that time, so is any other portion of time to the corresponding proportional part of such variation.