4th. With half the elapsed time enter the tables, and from the column of half elapsed time take out the logarithm answering thereto, and set it down under the logarithm ratio.

5th. Add these three logarithms together, and with their sum enter the tables in the column of middle time, where having found the logarithm nearest thereto, take out the time corresponding to it, and put it down under half the elapsed time.

6th. Subtract the less from the greater, and the difference will be the time from noon, when the greatest altitude was taken.

7th. With this time enter the tables, and from the column of rising, take out the logarithm corresponding to it; from this logarithm subtract the logarithm ratio, the remainder will be the logarithm of a natural number, which being found in a common table of logarithms, and added to the natural sine of the greatest altitude, will give the natural sine of the sun's meridian altitude.

Having the meridian altitude of the sun at noon, the latitude is found by the usual method.

N. B. If the latitude, found by the above process, should differ widely from the latitude by account, it will be proper to repeat the operation, using the latitude last found instead of the latitude by account, till the result gives a latitude nearly agreeing with the latitude used in the computation.

EXAMPLE I.

Being at sea in latitude 46° 50′ north by account, when the sun's declination was 11° 17' N. at 10 h. 2 m. in the forenoon, the sun's true altitude was 46° 55', and at 11 h. 27 m. in the forenoon, the second altitude was 54° 9'. Required the true latitude, and true time of the day, when the greatest altitude was taken?

H. M. S.

Nat. Sines Lat. 46° 50' Sec. 0.16487

11 27 0 Gr. Alt. 54° 9'-81055 Dec. 11° 17' Sec. 0.00848

10 2 0 Lea. Alt.46° 55'-73036 Log. ratio

0.17335

Ela. T. 1 25 0

8019 Com log.

3.90412

Ela. T. 42 30 In the column of elapsed time

0.73429

1 15 30 In the column of middle time

4.81176

T.f.noon 33 0 In the column of log. rising
From which subtract the log. ratio

The natural number in the logarithms 694

to which the nat. sine of the greatest alt. 81055 gives the nat. sine of the sun's mer. alt. = 81749 Sun's zenith distance

3.01488

0.17335

2.84153

90 0

51 50

35 10

33 As the time agrees with the observation, the watch is right.

EXAMPLE II.

Being at sea in lat. 47° 19' N. by account, when the sun's declination was 12° 16' N. at 10 h. 24 m. A. M. per watch, the true alt. of sun's centre was 49° 9'; at 1h. 14 m. P. M. his alt, was 51° 59′. Required the latitude?

Here the latitude found by computation may be relied on, as it iffers but one mile from that used in the operation.

EXAMPLE III.

Being at sea in lat. 50° 40' North per account, when the sun's eclination was 20° 0' South, at 10 h. 17 m. A. M. per watch, the in's alt. was found 17° 13', at 11h. 17 m. A. M. per watch, it was bund 19° 41'. Required the latitude?

Its log. from col. half elap. time is 0.88430

Its col. of mid. time corresponding to 4.72032

From noon, its log. from col. of rising 2.96067 log. ratio sub.

544 N. num. of

0.22504

2.73563

33682 N. S. greatest alt. +

But as this latitude differs 41 miles from that by account, it will be proper to repeat the operation, using the lat. last found instead of the lat. by account.

H. M. S.

Elapsed time 0 SO O

Lat.
Decl.

40° 59' 0.19178 20 0

0.02701

517 Nat. num, of

33682 Nat. S. gr. alt.

Nat. S. sun's mer. alt. 34199=20° 0′

0.21879

2.71344

The latitude last found, differing only one mile from that used in the operation, may be depended on as the true latitude. Hence t is plain, that the operation is repeated with very little additional ouble, few alterations being necessary.

EXAMPLE IV.

Being at sea in latitude 60° 0' north by account, when the sun was on the equator, and consequently had no declination, at 1 H. OM. P. M. per watch, his altitude was 28° 53', and at 3 H. 0 M. P. M. per watch, it was 20° 42'. Required the true latitude?

The latitude by computation, coming the same with the latitude y account, shows that the latitude by account was right. From e foregoing examples it is plain, that the operation is the same, hether the sun hath north or south declination. And it will be e same whether the ship is in a north or south latitude. It is also car, that when the sun has no declination, the secant, rejecting e index of the latitude, is the log. ratio.

EXAMPLE V.

Wanting to go through the N. Channel among the Maldives, and account being in latitude 7° 40' N. the declination being then °47' N. at 7 H. 25 M. 40 S. A. M. the true altitude of the sun's tre was 22o 30', and at 10H. 31 M. 48. S. A. M. it was found 40'. Required the ship's true latitude.

Lat. by ac. 7° 40′ 0,00390
Declin.

mes

H. M. S. Alt. 10 31 48 63° 40′

Nat. S.

89623

22 47 0,03528

p. T. 3 6 8

. T. 1 33 04 Its. log. in col. of elap. time is

Its log. in col. of rising is

Log. ratio 0,03918

Its log.

4,71058

0,40368

H. M. S.

3

1 30

5,15344

3,86709

Mer. alt. 74 29

96351 N. S. sun's mer. alt. 74° 29

N. B. As the Tables are only calculated to 10 seconds, the log. for any intermediate second is found by taking the difference between the log. next greater and next less; and saying, As 10 seconds is to that difference, so is the given seconds to the difference of the logarithms; or, if it be any even part, take such a part of the difference, and apply it to the next less logarithm; but in these operations a few seconds are not regarded.

N. S. Sun's m. alt. The latitude last found agreeing with that used in the operation, it may be taken as the true latitude; and the operation is repeated with very little additional trouble, few alterations being necessary. Hence it is plain, that if you are mistaken in the latitude by account, yet by repeating the work two or three times, making use of the latitude last found in the next operation, it will at last discover itself to be true, by being equal to the last supposition, which evidently shows the excellency of these Tables.

9635874 29. Hence the lat. in is 7° 16' N.

In the former examples we have considered both altitudes taken at the same place or station; but as that is seldom the case at sea, the necessary correction for any alteration of station may be readily made as follows:

Suppose the first altitude in the forenoon, at

The second altitude in the afternoon, at 2 h. 43 m.
Difference of longitude made is 30 miles W. equal to

Subtracted is the elapsed time

H. M.

10 26

14 43

0

2

14 41

10 26

4 15

If a ship has been sailing to the Eastward, the above two minutes must be added; but unless the difference of longitude be considerable, it is not worth notice, as it will make a very inconsiderable error in the latitude.