8 → 13678 is computed to set 13678 on the board. The board and intermediate computations now look like this. The ninth row with 1231101 is the largest value smaller than the remainder, so the first digit of the fractional part of the square root is 9. The value of the ninth row is subtracted from the remainder and a few more zeros are appended to get the new remainder 11669900. The second column on the ninth row is 18 with 13678 on the board, so → 13679 → append 8 → 136798 is computed to set 136798 on the board. The steps can be continued to find as many digits needed and if the precision needed is achieved. If the remainder becomes zero, this means the exact square root was found. Rounding up Having found the desired number of digits, it is easy to determine whether or not it needs rounding up; i.e., changing the last digit. Another digit doesn't need to be found to see if it is equal to or greater than 5. 25 is appended to the root and it is compared to the remainder; if it is less than or equal to the remainder, then the next digit will be at least five and rounding up is needed. In the example above, 6839925 is less than 11669900, so the root needs to be rounded up to 6840.0. To find the square root of a number that isn't an integer, say 54782.917, everything is the same, except that the digits to the left and right of the decimal point are grouped into twos. So 54782.917 would be grouped as Then the square root can be found using the process previously mentioned. Diagonal modification During the 19th century, Napier's bones were transformed to make them easier to read. The rods were made with an angle of about 65° so that the triangles that had to be added were aligned. In this case, in each square of the rod the unit is to the right and the ten (or the zero) to the left. The rods were made such that the vertical and horizontal lines were more visible than the line where the rods touched, making the two components of each digit of the result easier to read. Thus, in the picture it is immediately clear that: Genaille–Lucas rulers In 1891, Henri Genaille invented a variant of Napier's