WITHOUT EQUAL ALTITUDES. 499 
I send you a type of the calculation complete, in order that 
any one who wishes to pursue it, may easily be enabled to put 
it in practice. As the altitudes are all successive, the inter- 
vals ought to be nearly equal ; by which means, merely casting 
the eye over the results, you readily discover any inconsisten- 
ey, if it should exist. Although there may appear a number 
of figures in the work, it is extremely simple, as I have only to 
combine the error of the instrument with the refraction, paral- 
Jax, and semi-diameter of the sun, for the first and last obser- 
vation, the tenth part of which difference must be equally dis- 
tributed throughout the series, in order to obtain each altitude ; 
and the same system applies to the rest of the work. As the al- 
titudes increase or diminish by a uniform quantity, their natural 
sines are all taken out at the same opening of the book, and the 
proportional parts for seconds, applied as appears in the mode 
I have adopted, and have herewith transmitted. The 
Table entitled Logarith. Rising, in the Requisite Tables, al- 
though perfectly equal to all nautical purposes, I did not consi- 
der as sufficiently extended to give the tenths of seconds, which 
modern instruments and Logarithmic Tables afford the means 
of arriving at. I therefore computed a Table entirely a-new, 
from the formula 
sin. (co. lat. + dec. ©) — sin. (alt. ©) , 
cos. lat. x cos. dec. © i 
2 sin.* (4 a) = 
but taking the logarithms from tables to ten places of figures, 
although only retaining eight for those in the computation, 
as being sufficient to give me the last figure correct. These 
Tables begin at 2h. from Noon, and are carried on to 6 h., and 
to every 10’, with the differences corresponding. The re- 
sults were proved by the first and second differences, and 
lastly, by comparing the first five figures with those given in the 
Requisite 
