The term Geometrography is new to mathematical science, and it may be defined, in the words of its inventor, as “the art of geometrical constructions.”

1. In Fig. 8, take

and denote DE, EF, FD by x, y, z respectively;

From these equations

The following proof of the well-known result, n being any number, a, b, c the different prime factors that singly, or multiply, compose it

seems worthy of notice.

1. If x, y, z are the distances of the point P from the sides BC, CA, AB of a triangle, to find P so that the product xyz may be a maximum.

§ 1. When it is necessary to find the factors of such an expression as x2 – 7x – 120 the difficulty for beginners lies in finding the pair of factors of 120 which satisfy the middle term: in this case so that their difference is 7. They may write down the pairs in order (1 × 120; 2 × 60; etc.) and then choose the suitable pair, but this method is often long; or they may guess until they chance to find the pair required. This is often done in so haphazard a fashion that much time is wasted, especially if the given expression have no rational factors.

If we put θ = π/2 in the well known equation

we get eiπ/2 = i; and raising both sides to the power i,

1. My only justification for presenting this paper to the Society lies in the fact that, so far as I am aware, the uniform convergence of the Fourier Series is nowhere alluded to, and far less discussed, in any English textbook; while the precautions that are necessary in differentiating the series are hardly ever mentioned even in treatises which give a very thorough treatment of its convergence. I have confined myself almost exclusively to what may be called ordinary functions, as a complete discussion of what has been done in recent years for functions that lie outside the category of “ordinary” would make the paper much too long. For information as to the original authorities, I would refer to the paper which I communicated to the Society last session On the History of the Fourier Series. It is sufficient to say here that the proof I now give is simply an adaptation of that of Heine (Kugelfunctionen, Bd. I. 57–64, Bd. II. 346–353) and of that of Neumann (Über die nach Kreis … Functionen fortsch. Entwickelungen, 26–52).

In continuation of the results given in Vol. XI. of the Proceedings I note that the equations to AP, BQ, CR are respectively

and to AP′, BQ′, CR′ are

Cette note a pour objet de familiariser les élèves avec l'emploi des coordonnées tangentielles, en appliquant ces coordonnées, concurremment avec les coordonnées ponctuelles, à la résolution d'un certain nombre de questions, d'ordre très général, concernant les surfaces du second ordre.

1. Let one of the series of circles, which can be drawn so as to touch the sides AB, AC of a triangle, touch those sides in K, L; and let AK = AL = δ.

Then the points K, L are

and the lines BL, CK are givn by

Hence eliminating δ we get for the locus of P the hyperbola

The allied hyperbolas are

The rational treatment of Geometry has this important disadvantage, that for want of suitable demonstrations it seems impossible to preserve the natural grouping of the facts developed. The study of Rational Geometry, in fact, should always be supplemented by a systematic attempt to array the facts demonstrated according to their subject-matter; for it will hardly be denied that a direct and systematic knowledge of the Properties of Geometrical Figures has an intrinsic value apart from the knowledge of their demonstrations. In pursuing such a retrospective scheme as this in connection with the Second Book of Euclid, I have found that a very comprehensive view of the subject-matter is obtained by adding a Third Mode of Section of a straight line to the two which are already recognised. This third mode of section, for which I have not been able to find a more suitable name than “Circuitous Section,” along with the other two known as Internal and External Section respectively, exhausts the possible modes of section of a line—for three-dimensional space at any rate. From this point of view, the elementary treatment of the subject may be arranged as follows. It will be observed that several important properties of triangles and polygons acquire a new-interpretation as cases of circuitous section.

The area of the pedal triangle of a given triangle is easily shown by trilinear co-ordinates to bear to that of the original triangle the ratio R2 – S2: 4R2 where S is the distance of the point from the circumcentre of the triangle. A proof, by purely geometrical methods, of this theorem was read before the Society (Proceedings, Vol. III., pp. 78–79) by Mr Alison.

The notation employed in the following pages is that recommended in a paper of mine on “The Triangle and its Six Scribed Circles”* printed in the first volume of the Proceedings of the Edinburgh Mathematical Society. It may be convenient to repeat all that is necessary for the present purpose.

The method employed in this paper is first to ascertain in how many ways a cube can be cut into tetrahedra without making new corners, and then, taking each of these divisions of the cube as the type of a genus of divisions of the general parallelepiped, determining the number of species in each genus.

These equations are solved by the ordinary methods applicable to linear equations with constant coefficients.