with the remaining counter on the bottom row: and read off 26, the final result. Some properties of the grid Unlike addition and subtraction, the entire grid is used to multiply, divide, and extract square roots. The grid has some useful properties utilized in these operations. First, all the squares on any diagonal going from the bottom left to the top right have the same value. Since a diagonal move can be broken down into a move to the right (which halves the value) followed by a move up (which doubles the value), the value of the square stays the same. In conjunction with that diagonal property, there is a quick way to divide the numbers on the bottom and right edges of the grid. Locate the dividend 32 along the right side and the divisor 8 on the bottom edge of the grid. Extend a diagonal from the dividend and locate the square where it intersects a vertical line from the divisor. The quotient lies at the right end of the grid from this square, which for our example is 4. Why does this work? Moving along the diagonal does not change the value; the value of the square on the intersection is still the dividend. But we also know it is the product of the squares along the bottom and right edge. Since the square on the bottom edge is the divisor, the square on the right edge is the quotient. Napier extends this idea to divide two arbitrary numbers, as shown below. Multiplication To multiply a pair of binary numbers, first mark the two numbers on the bottom and the right side of the grid. Say we want to multiply 22 (= 10110) by 9 (= 1001). Now place counters at every "intersection" of vertical and horizontal rows of the 1s in each number. Notice that each row of counters on the grid is just 22 multiplied by some power of two. In fact, the total value of the counters is the sum of two rows 22*8 + 22*1 = 22*(8+1) = 22*9 So the counters on the board actually represent the product of the two numbers, except it is not possible to "read off" the answer just yet. Recall that moving counters diagonally does not change the value, so move all the counters on inner squares diagonally until they hit either the bottom row or the