page no 258 note * See H. G. Forder, Foundations of Euclidean geometry (1927).

page no 258 note † See G. H. Hardy, Pure Mathematics, p. 139, or Hardy, Littlewood and Pól;ya, Inequalities, Th. 41, p. 39. These are referred to as P.M. and H.L.P. respectively.

page no 259 note * The obvious extension has “≤” in place of “<”, but this is enough for our purpose. For the extension with “<” see H.L.P., p. 41, or use methods suggested on p. 18 of H.L.P.

page no 259 note † We have used the rule

page no 260 note * In P.M. (5th ed., p. 387) these are based on a definition of The method here avoids the infinities of tan x and enables us to use Euler’s beautiful proof of the addition theorem.

page no 260 note † C(x) and finite because, if 0≤x≤1–2⁻ⁿ ,

page no 261 note * The verification of the derivatives is almost trivial except at multiples of ½Π, and here is much simpler than if our basis were tan⁻¹ x, as we do not have to deal with the infinities of tan x.

page no 262 note * We have defined and use only the principal value of log z.

page no 262 note † The error term is the sum of a finite number of terms each of the form

O(|x| ^r |y| ^s ) where n + 1 ≤ r + s ≤ 2(n + 1).