^{page 195 note *} Gazette, May 1922, Dec. 1923, etc.; Report on the Teaching of Geometry in Schools (referred to hereafter as Report, pp. 35-40.

^{page 195 note †} Gazette, Dec. 1922; Proc. of London Math. Soc. ii. 23, part i. p. 45.

^{page 195 note ‡} Articles on “Ratio and Proportion,” by the writer, in the Gazette, Jan. and May 1920, may be referred to here—in order to save the space that would be occupied by restating what is set out at length there. On p. 10 of that volume (vol. x.) the relation between the concepts of Ratio and of the Real Numbers is shown. This I have elaborated in a little book on Number recently published.

^{page 195 note §} Observe that the statement covers the other case, of the concave angles PAX′, PBX′, … which are in decreasing order of absolute magnitude—since ∠XAP+∠PAX′ = st. ∠.

^{page 196 note *} See Gazette, vol. xi. p. 188 (Dec. 1922).

^{page 196 note †} See Carslaw’s Non-Euc. Geom. and Trig. (Longmans), pp. 6-8, on the basic constructions of Plane Geometry.

^{page 196 note ‡} Cf. Report, p. 48. Symmetry has great value in concise expression.

^{page 196 note §} Afterwards called x, and so in Fig. 1.

^{page 196 note ‖} See Gazette, vol. xi. p. 189.

^{page 196 note **} It is to be clearly understood that all references to “infinity” are simply abbreviated “limit” statements, which require interpretation in terms of the more elementary ideas of the context.

^{page 197 note *} In the article in the Gazette, vol. xi. (p. 190), the theory of Parallels was taken up at this point.

^{page 197 note †} Here we come expressly upon the “axioms of order,” which have generally been left (unformulated) to the learner’s intuitions in Elementary Geometry. For these see Whitehead’s Axioms of Descriptive Geometry (Cambridge Tracts, No. 5), ch. i., especially p. 8.

^{page 198 note *} “Sameness of shape”—or “similarity”—being defined in this connection, first for an integral “scale.”

^{page 198 note †} Nor are the elementary ideas of Ratio satisfactorily taught. See Omette, vol. x. p. 9.

^{page 199 note *} The mode of transition to the general case is indicated in the Gazette, vol. x. p. 10.

^{page 199 note †} The aim should be to depart as little as possible from standard elementary formulations of axioms (in so far as these are sound) while taking the utmost pains to keep in line with modern ideas—as presented in Whitehead’s tracts.

^{page 199 note ‡} The relation of (c) to the facts of § 3, above, should be noted.

^{page 199 note §} See footnote to § 4 (i), (2), p. 197.

^{page 199 note ‖} To take this fact as the axiom—which is implicit in Euclid, explicit in Hilbert—appears to me obiectionable: the more so when it is stated in the form “at least one other.”

^{page 200 note *} The difference in mode of proof from Bue. XI. 9 is of crucial importance to the treatment. See next sub-section, (vi).

^{page 200 note †} If l, l₂′ were to intersect, at a point A, both would be parallel through A to l ₁.

^{page 200 note ‡} § 4, (ii).

^{page 200 note §} It is convenient to keep the term “perpendicular” for intersection at right angles.

^{page 200 note ‖} Note the slight difference from Bue. XI. 4, 5, in the formulation of (1) and (2).

^{page 201 note *} See footnote at the end.

^{page 201 note †} The difference of direction is specified by angle quantities, and parallelism is bound up with angles of inclination.