—The writer when introducing his pupils to the use of Logarithmic Tables is accustomed to give them exercise with the number 1·00023. This number, he tells them, without explaining how he obtains it, is such that 1·00023 × 1·00023 × … 10,000 times would give a, number which read to the 3rd decimal place is equal to 10. He makes this the basis of practical exercises in approximations, and builds on it lessons in Indices, and in the use of Logarithmic Tables, with particular explanation of the Mean Difference Columns.

The Derived Function: In what follows by f(x) is meant a rational integral function of x of degree n with the coefficient of xn unity. By the “roots of f(x)” is meant the roots of f(x) = 0; by f′(x) is meant df(x)/dx.

The subjoined proof is not given in the current text books, but is handed on by oral tradition.

The usual proof requires in general the assumption of an infinite series of operations (with the consequent limit theorems involved), as all the n given numbers tend to equality. Let us take an arithmetical example, and let us tabulate the sequences involved in the way suggested in my paper on “The Teaching of Limits and Convergence to Scholarship Candidates” in the May (1911) issue of the Mathematical Gazette.

On pp. 95 and 96 of No. 8 (October 1911) of Mathematical Notes is given a numerical method of finding rational right-angled triangles.

It has been known for centuries that

are the sides of a rational right-angled triangle whatever be the values of p and q; for