that kind of notation in which unity or 1 and 0 are only used. This was the invention of M. Leibnitz, who shows it to be very expeditious in discovering the properties of numbers, and in constructing tables; and M. Dangecourt, in the history of the Royal Academy of Sciences, gives a specimen of it concerning arithmetical progressionals, where he shows, that because in binary Binary arithmetic only two characters are used, therefore the laws of progression may be more easily discovered by it than by common arithmetic. All the characters used in binary arithmetic are 0 and 1; and the cipher multiplies every thing by two, as in the common arithmetic by 10. Thus 1 is one; 10, two; 11, three; 100, four; 101, five; 110, six; 111, seven; 1000, eight; 1001, nine; 1010, ten; which is founded on the same principles with common arithmetic. Hence appears the reason of the celebrated property of the duplicate geometrical proportion in whole numbers, namely, that one number of each degree being taken, we may thence compose all the other whole numbers above the double of the highest degree. For example, 111 being the sum of 4, 2, and 1, or, analytically, of 100 = 4, 10 = 2, and 1 = 1, or 7 in all, this property may serve assayers to weigh all kinds of masses with a little weight; and may be used in coins, to give several values with small pieces. The binary method of expressing numbers once established, all the operations will be easy; in multiplication, particularly, there will be no need for a table, or getting any thing by heart. But the inventor does not recommend it for common use, because of the great number of figures required to express a number; adding, that if the common progression were from 12 to 12, or from 16 to 16, it would be still more expeditious; and that its use consists chiefly in discovering the properties of numbers, in constructing tables, and the like. What makes the binary arithmetic the more remarkable is, that it appears to have been the same with that used four thousand years ago among the Chinese, and left as an enigma by Fohi, the founder of their empire as well as of their sciences.