In his Foundations of Geometry, Hilbert has shown “the impossibility of demonstrating Desargues’ Theorem for plane geometry without the help of the axioms of congruence.” But it must be remembered that his demonstration depends entirely on the form which he has given to the other axioms, or definitions as I should prefer to call them, and in particular on the way he defines “straight line,” namely “two distinct points always determine completely a straight line,” and “any two distinct points of a straight line completely determine that line.” What he has shown then is that in plane geometry this is an inadequate definition of “straight line.” He has not shown that that definition cannot be amended otherwise than by the introduction of the conception of congruence, or by that of three-dimensional space.

One of the most interesting results of the theory of Order as defined by Boundaries is that it explains this apparent paradox. I propose to add to the papers already published in the Gazette on this theory a brief note on this point. To aid the imagination of my readers I shall not attempt to state the problem in general terms, but shall merely state it in homely geometrical language, and for the sake of brevity I shall not pretend to give formal demonstrations of each successive point in the argument, but shall leave obvious deductions to the reader.

Passing very briefly here, but by no means very briefly in the classroom, over this treatment of velocity, a few words may be given to the problem of the tangent.

Another useful application is that the parabola, in the form y = –k.x2, represents the path of a projectile. The boys at this stage have already dealt with equations of the form y= ±(x–a)(x–b), and are able to find the points where the curve is cut by the x or the y-axis.

§ 1. The Anharmonic Ratio expressed in terms of Squared Distances. In Cremona’s notation, the symbol (AB, CD) stands for the quantity

where ABCD are four collinear points, A, B and C, D are Conjugate pairs. It is easy to show that

There is, of course, nothing really new to be said about the elementary theory of the Elliptic Functions sn, cn, dn; but it may be not too late in the day for a didactic note on the existence and one-formedness of these functions to have some usefulness. One reason, and perhaps the chief one, why the older analysis has fallen into disfavour in the treatment of elliptic functions is the obscurity on the subject of one-formedness which prevails in so many presentations of the early parts of the theory. Halphen’s Fonctions Elliptiques is exceptionally valuable in that it neither avoids nor postpones the difficulty; but it is common for analysts who follow the older methods either to pass the matter over without proper emphasis, or to confess weakness by appealing to the ideas of the general theory of functions, thus encouraging a belief that the newer analysis is absolutely necessary for soundness.