SEXAGESIMAL, a mode of arithmetical calculation introduced by the ancient Greek astronomers, especially by Ptolemy (q.v.), into astronomical and geometrical reckoning. It was founded upon the division of the circle into 360 parts, and, the radius being nearly of the circumference, was considered to contain GO of these parts or degrees. Con tinuing the same mode of subdivision, each degree (`) on the radius was divided into 60 minutes ('), each minute into 60 secoLls ("), and thirds CI fourths (""), etc., followed in the same relation to each other. Addition and subtraction are not altered in this method, but multiplication, division, and the extraction of roots are so to a considerable extent. Multiplication, the most used of these three operations, was carried on in the descending scale, as in the following example. where Ace° d' 4" is to be multiplied by sifr rd", or (substituting Arabic numerals) al' 4' 27' by 29° 18' 54': Here, each of the three numbers, 31, 4, 27, is multiplied by 29; the same three by 18, and the results placed in the line below. one step to the right; and again by 54, and the results placed auothet- step to the right. This arrangement proceeds on the principle
flit the product of degrees by minutes gives minutes; • of minutes by minutes, seconds; of minutes by seconds, thirds; and, in general. the denomination of a product is indi cated by the sum of the marks superposed ou the two factors: The columns are added and rearranged by reduction (q.v.). This system, though clumsy and intricate. was a great improvement, as regards facility and accuracy, on the former Greek method; and so much was it admired that succeeding geometers founded on it a complete system of general calculation, and a work on sexagesimal computation was written by Nubian], who died in 1348. It is almost unnecessary to state that the terms minutes, seconds, thirds, etc., here employed only denote sixtieths, sixtieths of sixtieths, etc., and have no other signification; further, that the degrees, minutes, and seconds in flue multiplier are, for the time being, merely abstract units and parts of units. The operation of modern arithmetic known as duodecimal multiplication is effected in the same way, the subdisions being twelfths in place of sixtieths.