292 MR EDWARD SANG ON THE TABULATION OF ALL FRACTIONS 
we get 169 new fractions between these limits, not one of which is decompos- 
able into prime factors below 240; we get, however— 

Fraction. Year. Annual Error. 
618355 - 5.19.23.283 | 
620 048  16.11.13.271 242 175 +011 
879138 2.3.3.13.13.17.17 ; 
881545 ~ 57.89.9083 ses all soils 






The latter of these has the inconveniently large prime factor 283, but gives 
an approximation perhaps within the limits of error in the determination in the 
length of the equinoctial year. 
This mode of interpolation enables us to suit the order of proceeding to the 
circumstances of the case. We might, for example, have at first taken very 
narrow limits of error, pushed the interpolation therewithin to the utmost 
extent of our table of divisions, and only enlarged the limits of error when com- 
pelled by the non-discovery of useful results. 
On representing the ratio 365°242 217 :366:242 217 by a chain-fraction we 
get the quotients 1, 365, 4, 7, 1, 3, 1, 1, 3, 7, which give the approximations— 
10592 12053 46751 58804 105555 375469 2733838 & 
10 621’ 12086’ 46879’ 58965’ 105844’ 376497’ 2741 323? 

. 
alternately on the one and on the other side of the true value. Arranging these 
in the order of their magnitude, and writing the corresponding length of the 
year opposite each, we get 



10 592 46 751 105 555 2 733 838 ; 
Tosa ~*! 389; ae B79 879 aes yy ee ee, 2 TAL 323 Se 
375 469 58 804 12 053 
a76 497 242218; Feogs 242286; yo qgg 242 424. 
Now here we observe that the values -242 214 and ‘242 218 are within the 
range of the error of astronomical determination; wherefore any fraction between 
105555 4 375 469 
105 884 8° 9376 g07 
Between these limits we have already inserted and examined all those fractions 
whose denominators are below one million, and have now to treat those expressed 
by numbers between 1 000 000 and 3 036 000, the limit of BurckHarpr’s table. 
In this way, by a process exceedingly simple in its principles and mode of 
application, but very laborious in the actual work, we can discover all those 
combinations of prime factors which express, with sufficient precision, any pro- 
posed ratio. The great labour is in seeking the divisions of large numbers ; and 
the limits 


may be regarded as expressing the true ratio. 

