Binary number 1197. Note that the first Prior Value of 0 is simply an initial decimal value. This method is an application of the Horner scheme. The fractional parts of a number are converted with similar methods. They are again based on the equivalence of shifting with doubling or halving. In a fractional binary number such as 0.11010110101, the first digit is formula_7, the second formula_8, etc. So if there is a 1 in the first place after the decimal, then the number is at least formula_7, and vice versa. Double that number is at least 1. This suggests the algorithm: Repeatedly
Binary number as by Bells, by Trumpets, by Lights and Torches, by the report of Muskets, and any instruments of like nature". (See Bacon's cipher.) John Napier in 1617 described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters. Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results; they were found later among his papers. Possibly the first publication of the system in Europe was by Juan Caramuel y Lobkowitz, in 1700. Leibniz studied binary numbering in 1679; his work appears in his article "Explication de l'Arithmétique Binaire" (published