page 85 note * It is perhaps worth while to enter a plea for the use of the phrase, from the very beginning, in dealing with limits. The phrases “the limit when n is ∞,” “the limit when n = ∞” can do nothing bnt harm.

page 85 note † Castle’s Manual of Practical Mathematics (Macmillan, 1903)Google Scholar; Lachlan, and Fletcher’s, Elements of Trigonometry (Arnold, 1904)Google Scholar. I refer to these as examples, merely; in many other respects both books strike me as good.

page 85 note ‡ I have found a little numerical calculation very useful in convincing the unbeliever that there is an assumption involved. Thus with n even, and , we find the product to be less than 1/10³ when n = 10; less than 1/10¹⁰ for n=30; less than 1/10²⁰ for n=50.

page 86 note * For the purpose of numerical calculation the second form is often more useful than the series.

page 87 note * The angle is taken (as usual in the Calculus) to lie between – and and π.

page 87 note † They also serve to show that the convergence to the limit is very slow, compared with that of the exponential series.

page 87 note ‡ Some of the results (perhaps all of them) are given as exercises in Prof. Gibson’s excellent Calculus; I had, however, obtained them independently before his book was published.

page 87 note § It is perhaps going too far at present to suggest that the theory of double limits and uniform convergence is essential for a professed mathematician in England; just as, for the present, it seems hopeless to expect even an elementary knowledge of the foundations of geometry from our “up-to-date” Euclids.

page 88 note * It may not be out of place to refer to the fact that the real difficulty in finding the infinite product is not to prove that x(1 – x ²/π ²), etc. are factors; but to prove that no factor of the form, e^ax is present. This remark is due to Stolz, but is not made in any of the English text-books.