VOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 4()7 



therefore i : *- :: | X — — i - X ±~ :: 1 + t 2 : 1 + ? :: the square of the 



T t 2 T o* t x 



secant of the azimuth : the square of the secant of the hour-angle, which may be 

 considered as a ratio of equality, when the angles are very small. The fluxions 

 therefore of the tangents are as the tangents themselves; and consequently they 

 must always preserve the same ratio towards each other. Let us now suppose that 

 an altitude of the sun is taken at the distance ae from the meridian, but that, in 

 consequence of an error in the assumed latitude, the calculated time is ag ; and 

 that, with a lat. differing from the former by 1 minute, we compute again, and 

 the time is found equal to af \ then will the area gc be to gb as V to the whole 

 error in latitude. Let another altitude be taken at the distance ae from noon, and 

 let the times computed with the two different latitudes that were employed before 

 be ag and af ; then, in this case likewise, the area gc will be to the area gb, as l' 

 to the error in latitude. Now the latter curve is the " figura tangentium," whose 

 quadrature is given by Cotes, in his Harmonia Mensurarum, and the expression 

 for which is extremely simple. For the fluxion of the area is = rr-si and the 

 area itself = log. (l -j- *-)£ = log. secant of the angle at the pole. The difference 

 of the log. secants, or log. cosines, will of course be equal to the area intercepted 

 between the tangents which correspond to them. 



Hence a table might easily be constructed with a double argument, — the distance 

 from noon, and the variation in time arising from the different suppositions of lati- 

 tude, — which might immediately exhibit the logarithm of the area corresponding 

 to any particular base eg supposed to be given. A 2d table might have for its ar- 

 gument the difference of the logarithms of the area gb and the area gc, which is 

 also conceived to be known, and discover at once, in degrees, minutes, and 

 seconds, the correction to be made in the assumed latitude. This correction, as 

 it appears from a comparison of the signs of i, and z in the equation L = xtz, 

 must be added or subtracted, according as the distance from noon obtained by 

 computation is too great or too little, when the azimuth is less than 90 degrees; 

 but the contrary, when the azimuth exceeds a right angle. Tables of the above 

 description shall be constructed, if this method be received with approbation ; and, 

 in the mean time, it is proposed to subjoin a short specimen which is already com- 

 pleted. 



I have presumed that we are able to determine eg, the error of time arising from 

 an error in the assumed latitude, at either of the observations; and hence it 

 becomes necessary, before we can avail ourselves of the principles which have been 

 laid down, to point out the manner in which this may be accomplished. The 

 clock gives us the whole interval between the observations, supposed to be made on 

 different sides of the meridian, equal to ae -+- ae, and by computation we obtain 

 ag -+- ag, and then we deduce eg + eg the whole error in time. Now the area gb 

 is equal to gb; and therefore, if we make a rough division of the whole error, 



3o 2 



