new Divifion of the Quadrant, 29 
more, and therefore will not be exprefled by the fame number, 
but will have fome fmall difference in the feventh or laft figure. 
And the fame will happen in almoft all the other arcs; fo that 
generally the fines, &c. which are exact for the ares in the 
firftcolumn, will not be quite fo for thofe in the fecond, when. 
exprefled in whole feconds only, fince thefe will fometimes 
differ by the part correfponding to almoit half a fecond. How- 
ever, inthis, or any other cafe, where the difference is. exa@lly: 
known, we may profitably make ufe of the numbers in the 
old tables for conftruéting or verifying thofe of the new, by 
taking in the proportional part of the difference. . Let, there- 
fore, all the fines, &c. of every 1309. be computed from the 
old tables, and entered in the new, by adding to the fine, &c: 
of the correfponding multiple of 45’ the like multiple of the 
ig part of the proportional difference for 1.° This will give — 
about 120 fines, &e. to ferve as a verification of the computa- 
tions by the more general methods. But if the fecond column 
be exactly conftructed with all its decimal places by the conti- 
nual addition of 2:06264807, the old tables may be converted’ 
into the new, by allowing for the odd feconas and decimals. 
And for this purpofe it will, perhaps, be beft to ufe the large 
table of RuEericus, which contains the fines, tangents, and 
feconds, to ten places of figures for every 10”, and alfo the 
differences. At leaft, fuch fines, &c.. may be found in this way 
as have their feconds and decimals well adapted for the pur- 
pofe; and for fuch as would be found too troublefome in this. 
way, recourfe may be had to fome of the following methods.. 
12. Let, us now examine the expreflions for the fines,’ &cs. 
by infinite feries. 
‘The- 
