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or coversed of complement of an angle = versed of angle; and similarly sucovers. (90-1) = suvers. A.

We have spoken of the problem of the longitude. The determination of the latitude, if we could always observe the sun's meridian altitude, would be an easy problem : but clouds very frequently prevent this observation. The mariner, therefore, requires other methods of determining the latitude ; and the one which is best adapted, and has been most improved, is that which gives the latitude from two altitudes of the sun, and from the interval of time elapsed between the observations. M. Mendoza's rule and tables are founded on the direct solu. tion of this problem ; in which solution it is necessary, or rather it is customary, to find four quantities by four independent processes.

The analytic art has not bitherto afforded a solution by which the latitude is exhibited under one formula. In the first of these processes, is found a side (S) dependent on the complement of the declination and on the interval of time : this side is found arithmetically, that is, in specific in stances, by table 21, and is easily obtained by that table. In the second process is found an angle (a) dependent also on the declination and interval of time, but expressed mathematically by means of the abovementioned side and interval. This angle (or, if we mistake not, in M. Mendoza's rule, the complement of this angle,) is found by table 22d : the pages of which, for the sake of convenience, are printed opposite to the corresponding pages of table 21. In the third process, it is necessary to find an angle (B), which is easily had from this form; (a, á' meaning greatest and least altitudes ;) cos. (Statá).

sir
Sta ta'

a )cosec.S

. sec. a' This versed sine is easily had by nieans of table 17, already mentioned. It is no small recommendation of that table, that it enables us to perform Logarithmic Trigonometrical Operations by addition solely.--By the remaining part of the process, the latitude is found : thus if a be the latitude, and d the declinaa tion, (1 + sin. 2) = suver. (140)

suvers. (B+C)) cos. a'. co s.

į suvers. (d--a)

suvers. ( B-+) cos. a' cos. d versin. o or, if we put

suvers. (d-a) 1 + sin. = suvers. (d–o') {

} = suvers.(da) {

and

v. sin.B

2

2

2

{ 1- (suvers. (B+C))

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2

versin.

2

{su

Suvers.

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2

2

and.. log. covers. 1 = log. surers. (d.-a') + log. suvers. On these forms, as it appears to us, the rule is founded which is given by the author in page 31, problem 8. The rule, however, for finding the latitude, cannot under every circumstance be applied; and the author, in his remarks, lays down certain limitations, and promises to speak farther on this subject in a future work,

More instances should, in our opinion, have been adduced to shew the use and application of the tables 21 and 22.

The rule for finding the latitude (such is the necessity of the case,) is rather long and complicated : at first sight, it appears very formidable : but, as in other instances, a little acquainte ance with it causes its terrors to disappear, and after a short time we were surprised to find how expeditiously we could resolve cases. It is proper, however, to mention that in those cases which we have resolved by the present tables, the latitude was not obtained within those limits of exactness which the author thinks are attainable by his tables. T'inding that the latitude did not come out exactly right from several of our own observations, we distrusted them, and applied the tablcs to instances given in the Requisite Tables, Nautical Almanac, &c. In the example of the Requisite Tables, p. 17, the lati tude we found to be 39° 28' which agrees with the latitude determined by the common rule of approximation.-In the cxample, p. 10. of Nautical Almanac for 1800, where the greater altitude is 50° 1', the less altitude is 41° 33', and the interval between the observations is 1" 30", we determine the latitude by the present tables to be 52° 6' 15": whereas Mr. Brinkley makes the latitude 52° 10'.-In an instance taken from the Philosophical Transactions for 1799, p. 115, where, the two altitudes are 20° 12' and 20° i and the interval 20 4, we determine the latitude to be 52° 11' 10'; whereas it is 520 12' 42".

We consider it as our duty thus fairly to state our doubts concerning the first determination of the latitude with no greater error than that of one minute ; and we also give the instances in which the operations were performed. Yet, before we dismiss this part, we must add, without any affectation of modesty, that we do not pledge ourselves to have committed 20 oversight; since some trifling misconception of the rules, or some slight mistake in the use and application of the tables, may have occasioned that deviation from the right results which has already been mentioned.

The problem of the determination of the latitude from two observed altitudes and the time between the observations has

Kod

engaged

engaged the attention of several eminent mathematicians; and it is more difficult than the determination of the true distance of the moon from a star. M. Mendoza's rule is founded on the direct solution : but mariners have bitherto adopted an approximate method ; employing, as an element of calculation, the latitude by account. An approximate method called Douwe's method, is given in the Requisite Tables : but it is sometimes tedious, requiring a repetition of the operation, and, when repeated, not always leading to right conclusions. this account, Mr. Brinkley, an astronomer at Dublin, ioverted a supplementary process of computation; by which the latitude, once found by Douwe's method, might be conveniently corrected; and this process, with its demonstration and tables, was published in the year 1793, when the Nautical Almanac for 1797 appeared. Mr. Brinkley, we apprehend, was the author of this method ; for, although a memoir on the same subject, containing the same formula from which the rules are deduced, was published as early as the year 1791 in the Connoissance des Temps for 1793, (page 289) by M. Mendoza, yet, from a passage in that memoir, the priority of invention seems in some degree awarded to the Dublin Astronomer *. At this distance of time, however, the circumstance is involved in some perplexity. If Mr. Brinkley merely calculated

t

COS.

2

tan, I'

tan. d the values of

- 1, he ought not to have

COS. M published the method as his own in the Nautical Almanac. On the other hand, if a copy of the method was in M. Men. doza's possession before the publication of the Connoissance des

Temps, then is Mr. B. in fact the author of the method, and M. Mendoza has not acted fairly in rendering him so slight, and, which is worse, so ambiguous an homage.

The two methods for correcting Douwe's rule are in fact the same.

If L be the true and I the computed latitude, then, according to Mr. Brinkley to

* The passage is this : " L'expression ci dessus est assez commode; mais pour en faciliter l'usage, on pourra substituer deux tables qui donnent

cost les valeurs de

I et de tan. d cotan. l. M. Brinkley, AstroCOS. M de l'Université de Dublin, les ayant calculées, a eu la bonté de m'en donner une copie, et c'est avec plaisir que je lui rends cet hommage."

+ We have observed two errors in the Nautical Almanac ; p. 14. merid. Altitude is put for Zenith distance, and p. 15. 1. 9. cos. L, is put for sin. L.

nome

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L

tan. Msin. T- vers. T tan.d cot. T being the time nearest to noon and M the middle time.

According to M. Mendoza, (p. 294, Connoissance des Temps, psur l'Année 1793.)

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The two espressions, then, are precisely the same ; since, in this latter case, dL answers to l' in the former, and dl to L

Nir. Brinkley and M. Mendoza have both resclved the problem of Douwe by means of the orthographic projection of the circle: but this is by no means necessary; and in fact, all things considered, the resolution is not so simple as by the ordinary and obvious processes of spherical trigonometry, The simplification of problems, by the adventitious aid of new principles and new branches of science, is generally delusive.

Ní Kraft, who, as we have already mentioned, solved the problem of clearing the moon's distance from the effects of parallax and refraction, has also given in the Petersburgh Acts for 179, p. 353. a short memoir on the determination of the Latitude from two altitudes and the elapsed time: but his solution of the problem appears to us to have no advantages over the common and direct solution.

In our account of this volume, we have hitherto dwelt only on the two principal problems of Nautical Astronomy; and indeed the other problems which are here solved by rule, and for which tables are constructed, are of inferior concern. In page 26, problem 3, which proposes to determine the Azimuth, it appears to us that versed sine is printed instead of suversed : for if a be the Azimuth, a the polar distance, , the latitude, and a the altitude, then

sin. a sin. a cos. O

cos. cos, a cos.(a tal+cos. ata-t- atata ,,itcos, 6 = cos. À cos. a

COS, à cos, a

COS.

COS

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2

2

or

(atat1 – 7) sec. a. sec. a

cos.

COS.

(

2

2

suve TS.

atata or

sec. a sec, a 2 Table 31. is constructed for the purpose of finding the amplitude: but we doubt whether much time is gained by the formation of this table: because in using it we are obliged to make two multiplications, and in finding the amplitude by the common logarithmic tables, we have only to add the logarithmic sine of the declination to the logarithmic secant of the latitude, and to take out the angle corresponding to the L sine of the sum.-An error also appears to us to be made in the example given for the illustration of this table, p. 14, where the sun's declination is put 6° 17': from which the amplitude does not result from the common method, 90 18: the true declination ought, it seems to us, to be 6° 47': the error is nothing in itself, but it tends to raise suspicion in the mind of the calculator who makes use of the tables.

Table 43. of proportional parts, is very useful and commodious; and the author has fully explained its use.

We believe that the present volume contains all the tables which are requisite in Nautical Astronomy: that is, the tables here given suffice for the solution of all nautical problems: but the analytical solution of such problems must be accommodated to the tables. Solutions, for instance, that involve natural sines or cosines, must be rejected, or altered ; and solutions also, which are expressed in terms of tangents or cotangents, are not adapted to the present tables, which contain no logarithmic tangents. We rather wish that the tangents had been inserted; for the present publication is much more sizeable than Taylor's quarto, and it is desirable to use it in cases for which the author has not given solutions, which solutions may be expressed by means of tangents. For instance, we had occasion to compute, from the lunar tables, the distance of the moon from a certain fixed star; and we found ourselves obliged, when we had gone half way in the computation, to put aside the present tables, and to seek in those of Taylor for a logarithmic tangent which we wanted : thus, if D and d be the declinations of the moon and a star, and r the difference of their right ascensions, then D and d being opposite declinations,

vers. (dist.) = vers. (D+d). (sec. 6)? where & is such that

cos d cos. D. vers. I (tan. 6)?

vers. (D+d). This is the most obvious solution : but it is true that another may be given, in which M. Mendoza's tables will be very commodious ; thus

suvers.

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