468 PHILOSOPHICAL TRANSACTIONS. [ANNO 17QQ. 



without any regard to accuracy, in the inverse ratio of the hour-angles at the two 

 observations, and, entering the first table with these times, mark the area cor- 

 responding to each other at their respective distances from noon, and increase the 

 one and diminish the other equally, till we get the areas of the same magnitude, 

 this, we may conclude, is the proper value of each. 



If the table were constructed to every second of time, we might ascertain these 

 logarithmic areas merely from inspection; but, as it will be advisable to confine it 

 within narrower limits, we shall sometimes find it necessary, as in other tables, to 

 deduce their ultimate value by the rule-of-three. When we have increased one 

 portion of time and diminished the other, till the difference of their corresponding 

 areas becomes a minimum, we must divide this difference between them in the 

 proportion of their respective increments in the next interval of time, and subtract 

 or add the part assigned to each, according as it is greater or less than the 

 other. The table however might easily be carried to such an extent, that exact- 

 ness in this division could never be required; but, on the contrary, it would be 

 quite sufficient, when the hour-angles were nearly equal, to add the arms together, 

 and take half the sum for the value of each. 



From these principles may be deduced the following practical rule for determining 

 the latitude of a place. When the sun comes within 15 degrees of the meridian, 

 in the morning, let his altitude be taken, and the time of the observation be ac- 

 curately marked; and let another altitude be taken after he has passed the meridian, 

 while his distance from it is less than 15°; and let the time of this observation 

 likewise be noted. Then, with the supposed latitude of the place, compute the 

 times corresponding to each of the altitudes in terms of the log. cosine of the 

 hour-angle, and take the difference of the intervals, as shown by the clock, and 

 determined by calculation, and divide it between the observations in the manner 

 explained above. Compute the log. cosine of the hour-angle a 2d time, with the 

 greatest altitude and the latitude increased or diminished by a minute, according 

 as it appears, from a comparison of the intervals, to have been too little or too 

 great; and take the difference between this log. cosine and that which resulted 

 from the first operation, when the same altitude was employed. Having thus ob- 

 tained the two areas gb and gc, we must subtract their logarithms from each 

 other, and with their difference entering the 2d table we shall find the degrees, 

 minutes, and seconds, by which the assumed latitude is to be increased or di- 

 minished. 



It will be needless perhaps to suggest that, in the higher latitudes we may ex- 

 tend the limits above specified a few degrees farther from the meridian, without 

 offering any material violence to the theory, as it has hitherto been explained; and 

 that, on the other hand, when the declination and latitude are nearly equal, and 

 of the same denomination, it will be expedient to confine our observations within a 

 much shorter distance from noon. But it will afterwards be demonstrated that, 



