It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not always define, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid’s earlier proofs fail before this test.

So far as I know the properties of a tetrahedron are not given in any mathematical text-book. In the late Dr. Wolstenholme’s Examples for Practice in the use of Seven-figure Logarithms, the formulae which are necessary to determine the plane and dihedral angles and the volume of the tetrahedron, having given the six edges, are stated, but nothing beyond this. Dr. Wolstenholme worked at the subject for some years, and several of his results were sent as problems to the Educational Times, where they appeared for the most part after his death. In one set of questions he remarks that they are “the equations I have been looking for for years,” and so I think we may fairly infer that these questions represent the more advanced results of his investigations. The solutions of these questions, so far as I have been able to find them, are practically all given in this paper. “With regard to the rest of the matter, I hardly dare claim any originality. Wolstenholme must have known all or nearly all the results, and any one seriously attacking the subject must come upon them. All I can say is that I have never seen them in print, and that under any circumstances the bringing of them together must be helpful to students in many branches of mathematics.

In plane coordinate geometry, with a system of rectangular axes, I wish to press the claim of points whose coordinates are given by complex numbers to be represented on the plane in just the same way as all other points in the plane.

If A, B, C, G are four given points in a plane, it is possible to place masses x, y, z at A, B, C respectively, so that the centroid of the system is at G. This, it is to be observed, can only be done in one way as x, y, z must be proportional to the areal (or “barycentric”) co-ordinates of G with respect to the triangle ABC.

We have seen the connection between the general theory of verniers and the indeterminate equation of the form pn − qm = r; the equation pn + qm = r might be illustrated in much the same way. But, taking the case originally considered, we observe that it is the same question as before, from a different point of view, if instead of starting with an interval AB between divisions of two contiguous scales, and asking how far we must go to reach a coincidence, we suppose the zeros of two scales to coincide and ask how far we have to go to reach a certain interval, between divisions of the two. The question only differs in so far as it suggests the simultaneous consideration of more than one such interval. But further if, instead of repeated divisions of each kind, we take a circle whose entire circumference = one of the larger divisions, a say, and from a fixed point A on the circumference measure off continually arcs = to the smaller division 6, (figs. 5, 6), the question how far we have to go to reach a certain approximation to the starting point is the same thing in another form, although the arrangement brings to light many new relations, which in their turn throw light on the result. We have the subdivisions of the different a-intervals superposed as it were upon one a-interval, and the circle is thus continually divided into more and more parts until (if a and b are commensurable) exact coincidence with the starting point is reached.

While all teachers of Mathematics should be interested in the paper read last September by Prof. Perry and in the debate which followed it, members of our own Association, founded as it was for the Improvement of Geometrical Teaching, should give it special attention and should take advantage of the upheaval at Glasgow to press some of their immediate aims. Although Prof. Perry presented his case with a certain amount of picturesque exaggeration, I feel sure that his contentions were in the main true, and that the mathematical training of the schools, however well fitted for those who are being prepared for a University course, turns out the ordinary boy with a modicum of Mathematics which he finds of little use for any professional purpose. I do not however think that we need look upon Prof. Perry as an opponent but rather as a valuable ally. I am convinced that much might be done, at any rate in the earlier stages of mathematical teaching, which besides going far to meet Prof. Perry’s views and to remove the reproach (in my opinion deserved) of the Engineer, would be a considerable gain from our own point of view. Our cause may be advanced considerably if we make a proper use of the present opportunity.

The British Association this year has inaugurated a new Section, that of Education; and at the recent Glasgow meeting this new Section made a good start, under the presidency of Sir John Gorst, by holding a series of joint discussions with several of the older Sections, in addition to the usual reading of sectional papers. Thus it came that on Saturday the 14th of September a three hours’ discussion took place on the “Teaching of Mathematics.”

Let x0, y0, z0 be the areal coordinates of a particular point, x, y, z the areal coordinates of any other point. Put ξ =x - x0, etc. Then Σξ = 0, and ξ,η,ζ(or any two of them) serve to determine the position of a point, and a relation between them gives a locus.

The advantage of Areal “over Cartesian coordinates is in the greater symmetry of the analytical forms, and in the more convenient treatment of the line infinity, and of points at infinity.”—Cayley, Art. “Geometry,” Encyc. Britt., 9th Edn.

So many questions either directly on Areals, or easily and elegantly treated by means of that system of coordinates, have appeared in recent years in the various Entrance Scholarship Examinations that no apology is necessary for introducing such questions into the following paper, which contains solutions of all those set at Cambridge from 1894-1900.

The practical use and working of a vernier is very simple. It is applicable to all measurements where minute accuracy is required, whether the quantity to be measured is itself a length, as the reading of a barometer, or is determined by the distance passed over by a moving point relatively to a uniformly divided scale straight or curved, as in the angular readings of an astronomical instrument. In its simplest form too, the principle on which it depends is very obvious. Supposing, for instance, we require to read a scale to the one hundredth of an inch. To graduate such a scale directly would require a large number of minute subdivisions, which would be troublesome to make with the requisite accuracy and very confusing to the eye in reading.

The aim of the following articles is to meet certain difficulties and defects in the accepted book-work of the “Higher Algebra” of school work, especially in connection with infinite series.