page no 111 note * Mr. Broadbent has kindly drawn my attention to Whittaker and Robinson’s Calculus of Observations, p. 79—The Principle of Iteration.

This is absolutely the process that I am now describing but with this important difference. The authors of the book and presumably those previous writers to whose work they refer, never appreciated the importance of discarding decimals. When more than a year ago I used the process, which I had independently hit upon for finding the approximate square root of an algebraical quantity, also upon plain numbers and naturally used decimals, I saw at once that the process was nearly as exhausting as the method universally taught in the schools and was quite unsuitable where a root was required with a very high degree of accuracy; but that if I kept to vulgar fractions I obtained the advantages herein described. Perhaps eminent mathematicians would not naturally use vulgar fractions, and it needed someone much more prosaic to descend so low as to prefer them vulgar or even improper. However, the change over has converted a relatively useless process into by far the best that I have seen, and whereas in either, a mistake does not alter the result but only delays its attainment, the vulgar fraction method has the further advantage that the difference rule for the numerators shows if a mistake has been made and where, and Barlow’s Tables provide much of the arithmetic ready done.