The use of the slips for the purpose of multiplication is now evident; thus to multiply 2085 by 736 we take out in this manner the multiples corresponding to 6, 3, 7, and set down the digits as they are obtained, from right to left, shifting them back one place and adding up the columns as in ordinary multiplication, viz.
Each of the four faces of each rod contains multiples of one of the nine digits, and is similar to one of the slips just described, the first rod containing the multiples of o, I, 9, 8, the second of o, 2, 9, 7, the third of o, 3, 9, 6, the fourth of 0, 4, 9, 5, the fifth of I, 2, 8, 7, the sixth of I, 3, 8, 6, the seventh of I, 4, 8, 5, the eighth of 2, 3, 7, 6, the ninth of 2, 4, 7, 5, and the tenth of 3, 4, 6, 5.
Each rod therefore contains on two of its faces multiples of digits which are complementary to those on the other two faces; and the multiples of a digit and of its complement are reversed in position.
Then a binary n", equated to zero, represents n straight lines through the origin, and the x, y of any line through the origin are given constant multiples of the sines of the angles which that line makes with two fixed lines, the axes of co-ordinates.
As new axes of co-ordinates we may take any other pair of lines through the origin, and for the X, Y corresponding to x, y any new constant multiples of the sines of the angles which the line makes with the new axes.