466 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1799. 



separated from it by more gentle means? What power exists here, to protect the 

 inflammable particles, which afterwards turn to coal, so effectually against a degree 

 of heat which nothing else can resist? Of what nature is the salt obtained in con- 

 junction with the coal? These are all questions which excite great interest, but 

 which are not easily answered. How far I have been successful in resolving them, 

 some subsequent Essays will show ; which I shall have the honour to lay before the 

 r. s., as soon as I shall have sufficiently repeated the experiments I have already made. 



VII. A Method of Finding the Latitude of a place, by means of Two Altitudes of 



the Sun, and the Time Elapsed between the Observations. By the Rev. W. Lax, 



A.M.) Lowndes's Prof, of Astronomy, Cambridge, p. 74. 



I hope the following method of determining the latitude, by means of 2 alti- 

 tudes of the sun and the time elapsed between the observations, will be found not 

 less convenient for nautical purposes than the rules which are commonly employed. 

 But I would rather recommend it in those cases where rigid accuracy is required, 

 and the astronomer is provided with no better instrument for taking the sun's alti- 

 tude, than a Hadley's sextant of the most improved construction. The process 

 will be neither difficult nor tedious; and, if the observations are made with proper 

 exactness, I conceive the latitude will generally be obtained within a few seconds of 

 the truth. 



In the spherical triangle, whose sides are the complements of the latitude, de- 

 clination, and altitude, let z represent the angle at the pole, and t its tangent; z 

 the azimuth, and t its tangent; l the latitude, and x its cosine, radius being 

 unity; then, if the altitude and declination remain constant, we shall have L = 

 xiz, and consequently 1 will vary as tz, when the increment of x, compared with 

 x itself, is inconsiderable. Hence, if the abscisse of the Fig. ] . 



curve abcd, fig. 1, be always proportional to z, and its 

 ordinate to t, the area gb, intercepted between any 2 of 

 these ordinates may represent the increment of the latitude 

 corresponding to the increment of the time eg. Let abed, 

 fig. 2, be another curve, whose abscisse ae is always equal 

 to ae in the preceding figure, but whose ordinate eb is pro- 

 portional to /, the tangent of the hour-angle; then will 

 the area gb vary as gb, at small distances from the meri- 

 dian, and of course may represent the increment of the A 

 latitude. Now, to prove this, we have only to show that t 

 and t, when both are small, bear to each other a given ratio. 

 Let s and £ be the sine and cosine of the azimuth ; s and <r 

 the sine and cosine of the angle at the pole; then will -= z . 



— - — , and - = z . — - — ; z = - , and z = — . But since the 



complements of the declination and altitude remain constant 

 while the latitude is made to vary s will be to j as s to s; and 



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