VOL. LXXXVII.] PHILOSOPHICAL TRANSACTIONS. Q5 



zenith, the result must be true if the angles are accurately taken; and it is only 

 necessary to observe, that when the sum of the distances is less than 1 80°, the half 

 difference must be added to the distances, as by the last reduction. There is a dif- 

 ficulty in making this observation when the sun passes the meridian very near the 

 zenith, as the change in azimuth from east to west is too quick to allow sufficient 

 time; nor can it be obtained by the sextant when the sun passes the meridian more 

 than 30 degrees from the zenith: for I never could adjust the back observation of 

 the Hadley's quadrant with sufficient accuracy to be depended on. 



III. Researches on the Chief Problems of Nautical Astronomy. By Joseph de 

 Mendoza Rios, Esq., F. R. S. From the French, p. 43. 



In these researches the author proposes to consider the chief problems of nau 

 tical astronomy in a general manner, and to give formulas for all the cases. He 

 divides the subject into 2 parts. In the first part he comprises what relates to the 

 determination of the latitude by 2 solar altitudes; also the calculation of the hour 

 angle by the observed height of a star, and reversely the height by the hour angle. 

 The subject of the 2d part is the reduction of the observed distances of the moon 

 from the sun or a star, for determining the longitude. 



In this paper Mr. Mendoza has collected together a great number of rules, or 

 analytical formulas, for the purposes above specified, expressed chiefly by means of 

 the versed sines of arcs; for which reason he first, by way of lemmas, adopted certain 

 analytical expressions for the versed sines, coversed sines, and supversed sines of 

 arcs, in terms of the sines and cosines of those arcs. He then enters on the sub- 

 ject of finding the latitude by observation. This, it is well known, is easily done 

 by one observation only, at noon day. But as the weather and other convenient 

 circumstances are not always or often suitable for this purpose, other methods have 

 been devised for cases when the sun is out of the meridian of the place, and parti- 

 cularly that of observing 2 altitudes and interval of time between them, or the azi- 

 muth. Among the methods for these cases, Mr. M. more especially notices those 

 of Peter Nunnez and Cornelius Douwes ; which he investigates and transforms in 

 many different ways. 



Having noticed the formulas for the latitude, Mr. M. then, in the 2d part of the 

 paper, in like manner, collects the various rules for the longitude of the place, accord- 

 ing to what is called the lunar method, or that by the observed distance between the 

 moon and the sun by day, or the stars by night. And here their chief excellencies 

 consist in the reduction of the observed to the true distance of these luminaries, by 

 a proper reduction on account of parallax and refraction ; for which purpose various 

 modes are given. But as the uses of these formulas are best shown by means of 

 logarithmic and other tables, adapted expressly for the purpose, we cannot do better, 

 for this purpose, than refer to the ingenious author's very elaborate collection of 

 tables and rules, published in the year 1805, under the title, " A Complete Collec- 

 tion of Tables for Navigation and Nautical Astronomy," where the above formulas 



