In his interesting article on Newton in the July Gazette Dr. Rouse Ball had space only for a slight reference to Newton’s views on what we should now call the Foundations of the Differential Calculus. The subject “ involves,” as Dr. Rouse Ball has remarked, “ some awkward questions of philosophy which before Weierstrass’s researches were usually slurred over.” Some of the logical difficulties incident to Newton’s method of approach were clearly seen and pointed out by a contemporary, the celebrated Bishop Berkeley, in a tract which drew much attention at the time, and was followed by a considerable controversy.

If we form the sum S(n) of the digits of a number n, and then the sum of the digits of S(n), and so on, in a finite number of steps we reach a number of one digit, R(n), which we call the reduce of n.

Starting with a number prime to 9, say 2, consider the reduces of powers of 2; we obtain the sequence 2, 4, 8, 7, 5, 1, 2, 4, 8, ..., which recurs after the sixth term. Consider next a five-digit number, say 16427; from this we derive 15794 formed by replacing each digit of 16427 by the reduce of the sum of the remaining digits. Repeating the transformation, we obtain in turn 73184, 72461, 49751, 48137, 16427, the number being restored to its original value after only six transformations. It will be found that a three-digit number repeats after eighteen transformations and yet for a twenty-digit number only six transformations are needed.

The method usually employed for the extraction of square roots involves a series of successive approximations, one for each significant decimal figure of the root. A binomial series may be found which converges quicker than Σ10-n, but this involves a number of separate divisions. Iteration by dividing an approximate value of the root into its square and taking the mean nearly doubles the number of significant figures at each operation. The methods here given converge rather faster than this type of iteration, and enables a root to be expressed as a simple infinite series. They arose from a question put to me by Mr. K. A. Kermack, as to how an ancient Roman could have obtained a good approximation to a square root as a rational fraction. I believe them to be novel, at least in part. It will clearly be sufficient to give methods for finding the square root of a positive integer k.

The sequence 1, 3, 7, 13, 21, … arises in many ways, for example from the integers placed in a spiral on a square grid, the terms arising where the spiral intersects a diagonal on the grid, and as the central axis of a triangular array of numbers: