15. Let X1, Y1, Z1 be the second points in which AH, BH, CH meet the circle GH

Then GX1 ┴ HX, and hence ║ BC,

X1X =⅓AX;

tan PX1X= 3 tan PAX = 3 tan θ,

i.e. -PX1 X= ø.

But OX1H = OGH= ø; 0X1P is a straight line.

Area Of a Rectangle.—Consider a rectangle A of height h and base b (Fig. 18). By moving the rectangle so that its base slides along the x-axis through a distance b, it takes up the second position A. Thus, on a length 26 of the x-axis, commencing at O, we have a rectangle of height h and area 2A, and so on.

The Note on Momentum and Kinetic Energy—432. [6. a. β.]—by Mr. W M. Roberts, is interesting in showing a return in the Royal Military Academy to the gravitation unit of force in Dynamics, as more tangible for application to practical gunnery and engineering than the absolute system, introduced into theoretical instruction about 40 years, and described in Maxwell's Matter and Motion.

The use of Mathematics we understand as not merely utilitarian applications, such as those by which astronomy is employed in navigation, “whereby our nautical logbooks can be better kept, and water transport of all kinds has grown more commodious”; flight too can be made in the air, but also the cultivation at the same time of the intellect, to think correctly, and arrive at the right conclusion from data which are certain ; and the enlargement generally of the interest in this life.

And so our subject received its name from MA⊖HMATA, meaning disciplined exact knowledge, and formed an important branch in the old University ideal, to teach how to live a life, not merely to make a living. To keep the soul alive, body and soul together, and to cure intellectual timidity, the worst form of cowardice.

An account of singular solutions of differential equations of the first order is given in every text-book on differential equations. The theory of singular solutions of differential equations of higher order is not usually given in elementry treatises. It is discussed by Goursat (American Journal of Mathematics, xi., pp. 329-372), who bases his discussion on the theory of singular solutions of simultaneous equations ot the first order. From the didactic point of view a more elementary treatment may be acceptable, and an attempt is here made to indicate how singular solutions of a given equation of the second order may be found by simple means.

Major Dixon has given in the Gazette for May an interesting representation, suggested, perhaps, by Hilbert's example of a non-desarguesian geometry, of a plane geometry, in which straight lines are represented by closed curves on a closed convex surface devoid of singularitles. Employing his theory of order and “collation,” he constructs this geometry so as to be desarguesian.

AT the beginning of this century we find that there was a lively movement on foot to insist on the application of mathematics to actual problems occurring in daily life—a reaction from that purely formal mathematical teaching which for half a century had treated the subject-matter as of secondary importance so long as the logical training was achieved. This reaction became apparent amongst schoolmasters, especially in the meetings of the Forderungsverein, and it was even asserted by the extremists that Mathematics was nothing more than the handmaid of Science.

Now that there is a tendency to cut out a good deal of useless matter in arithmetic and algebra, and to spend less time over formal geometry, it is possible, and I think profitable, to introduce boys quite early to a course of numerical trigonometry. For the last fifteen years, I have seen some such course as is here described taught successfully to boys of thirteen and fourteen years of age. The inclusion of certain types of question in arithmetic and algebra courses—such, for instance, as those wonderful creations that you will find in chapters on vulgar fractions—is, I suppose, justified on the ground that they call for patience, orderly arrangement of work, and accuracy. But scope can be found for the exercise of all these qualities in numerical problems which arise in connection with other branches of their work—geometry, mechanics, physics, etc.—and the range of such problems is greatly increased if the boy has an elementary knowledge of trigonometry Moreover, the work involved in solving triangles is often fairly complicated and must be done carefully and arranged in an orderly manner, otherwise it is liable to end in disaster. There is a certain amount of drudgery involved, but it is drudgery with some interest attached to it.

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.