a2 Dr. Hurton’s Projeé for a 
above proportion will become 100000 : 200000 ~ ‘cooor, or 
1120000000001 :: fin.a : fin.m — 1+ fin.n+13 confe- 
quently fin. n—-t + fin.n+t is = 2 fin. 2.—ococegdc001 
fin.m, and the fines are in arithmetical-progreflion except 
only for the fmall difference of -oocogo000r1fin.z, hence 
fin.z +1 is = 2—‘0000000001 x fin.z—fin.n —1; and there- 
fore taking # fucceffively equal to+1, 2, 3, 4, &c.-the feries 
of fines will be as follows: 
fin. I = I —'000000000023 
fin. 2=2—‘oo000e0001 x fin. 1; 
fin. 3 = 2—*oo00000001 » fin. 2=fin. 15 
fin. 4=2— ‘000000050! x fin. 3 — fin. 2; 
fin. 5 = 2 ~*0000000001 x fin. 4— fin. 3 
&&c. 
And by this’ theorem, vzz. fin. a+ 1=2-— 0000000001 x fin. 
n—{fin.m—1, may be eafily filled up the intervals between 
thofe primary numbers mentioned in former articles. 
16. In like manner, as radius : 2 cof. a:: cof. 2a@: cof. 2—1 Ya 
+ cof.m +1.a3; and hence this theorem, cof. 2+1= 
2 ‘0000000001 x caf. m—cof.z—1, by which the cofines 
will.be all eafily filled up. And thefe two theorems for the 
fines and cofines are fo eafy and accurate, that we need not 
have recourfe to any other, but only to check and verify thefe at 
certain intervals, as at every 1ooth number, by a proportion from 
RHETICUs’s canon, as mentioned at art. 11. or by any other way. 
17. The-fines and cofines being compleated, the difference 
between the radius and cofine will be the verfed fine; the dif- 
ference between radius and fine will be the co-verfed fine; and 
thefum of the radius and cofine will be the fup.verfed fine. 
13. From 
