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4 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
more striking than the gain in speed. There could be very little 
doubt, therefore, that a fair degree of skill in arithmetic with a 
binary notation could be acquired by many to whom it is impossible 
under the present system. , 
The only practicable division of arcs and angles, and the most 
natural division of all things, is by continued bisections. This is 
shown by the ratio of value in our coins, weights, and capacity 
measures ; by any table of prices; and by the prevalent subdivision 
of lowest nominal units, as of the carpenter’s inch into eighths and 
sixteenths, and of percentages into quarters, etc., in stock quotations, 
where convenience of calculation by our present arithmetic seems 
almost gratuitously sacrificed. The American coinage is inconye- 
nient in practice, because of the awkward fractional ratio 22, which 
it introduces between successive pieces; and there would be the 
same difficulty in a decimal system of weights or of measures, should 
it be imposed upon us. We have thus another powerful reason for 
endeavoring to introduce a binary arithmetic. 
In the remarks which followed, Mr. E. B. E:uiorr expressed the 
hope that Congress would adopt the metric system of weights and 
measures for international purposes. It would be better to secure 
what advantage could be gained from uniformity and consistency, 
even though the basis of consistency was an arithmetic not ideally 
the best attainable. Such a course would not prevent, but might 
pave the way for a better arithmetic. 
Mr. W. B. Taytor said the world was losing so much by the 
employment of the denary arithmetic that he thought even a single 
generation might find economy in substituting the octonary. The 
introduction of decimal measures, while it would aid the computer, 
would injure the remainder of the community. The paper of Mr. 
Farquhar had an especial value, in that it proved the ability of 
binary systems to compete with the established system in rapidity 
of computation. 
Other remarks were made by Messrs. Harkness, Mussry, Pow- 
ELL, and GILBERT. 
The next communication was by Mr. 8. M. Burnert on 
REFRACTION IN THE PRINCIPAL MERIDIANS,OF A TRIAXIAL ELLIP- 
SOID; REGULAR ASTIGMATISM AND CYLINDRICAL LENSES ; 
and he was followed by Mr. W. Harkness on 
