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.

It cannot be said too often that each portion of a mathematical curriculum must be chosen with some definite purposes in mind, and that these must be the controlling influence in the scheme of development. Such purposes may be roughly divided into two classes : the first includes all desires to illustrate or elucidate general ideas and principles, and the second includes all desires to relate one part of the subject to another, or to some other domain of thought. It is not here implied that possible fulfilment of some such purposes enforces the inclusion of any particular branch of mathematics ; its difficulty may be too great for the students in question, and in any case only a limited amount can be included in a given time-table. But it is very definitely intended that no branch shall be included merely because it happens to be sufficiently easy or to tickle the fancy. There is ample real and human interest to be found without sacrifice of purpose ; and while undue difficulty is to be deprecated, it must always be remembered that definite achievement resulting from continued effort is of the very essence of the learning of mathematics.

If we wish to develop a race of geometers, I think we should begin by encouraging the habit of speculation. This we do nowadays by using squared paper for plotting graphs. Here, however, we place ourselves very much in the hands of the printer : and although ideas of geometrical shape are enlarged and systematised, the right angle plays too great a part in whatever knowledge of curves is acquired ; indeed a knowledge of curves is not the professed object in plotting graphs so much as a visualised representation of corresponding changes in related physical quantities.

In teaching elementary geometry there lingers an aversion from the use of dividers in pricking out equal distances : for the transference of distances is not recognised in the postulates of Euclid, and Euclid's influence remains.

Later on a study of the conics, carried on for a prolonged period, and the application of the theorems of Pascal and Brianchon, must exercise their fascination. Ideas about infinity become clarified : the use of parallels is seen to be necessary, to exercise the right accorded by Euclid's first postulate to join any given point to any other, even if one be at infinity. But the mind inevitably tends to become obsessed with the second-degree idea. Mere straight-edge methods are found to be so potent in the construction of conics that metrical properties are neglected. Primary notions about equal distances interfere with the whole scheme of a projective geometry confined to the conics ; and the bisection of a given finite straight line becomes a search for the harmonic conjugate of the point at infinity in its direction, with regard to its two ends.

When, with the other members, I received a request from the Secretary, it seemed only fair to him that I should express my willingness to do whatever might be in my power towards the provision of a subject for discussion ; and therefore, with an earnest wish that he would if possible ignore my suggestions, I proposed the rather well worn subject of this paper. To my dismay he did not ignore the suggestions, and in consequence you will have to listen to the trite observations of one who, sufficiently remote from great centres of education, and in great measure precluded from intimate contact with modern methods, is nevertheless sufficiently interested in them to form theories, or rather opinions, of his own. Those to whom the element of novelty may be absent in the points that I proceed to consider will, I trust, extend a charitable reception on the great moral principle of the Western States that the pianist who does his best is not to be shot.

In the ordinary course of lecturing to students who come stamped with the approval of some of the entrance examination Boards, and their name is legion, I find that, especially when they have had a more or less practical course, very great difficulties arise on account of their weak grasp of mathematical reasoning.

John Napier's Logarithmorum Canonis Mirifici Descriptio was published in 1614 ; and it is proposed to celebrate the tercentenary of this great event in the history of mathematics by a Congress, to be held in Edinburgh on Friday, 24th July, 1914, and following days.

The Celebration is being held under the auspices of the Royal Society of Edinburgh, on whose invitation a General Committee has been formed, representing the Royal Society of London, the Royal Astronomical Society, the Town Council of Edinburgh, the Faculty of Actuaries, the Royal Philosophical Society of Glasgow, the Universities of St. Andrews, Glasgow, Aberdeen, and Edinburgh, the University College of Dundee, and many other bodies and institutions of educational importance.

The President and Council of the Royal Society of Edinburgh have now the honour of giving a general invitation to mathematicians and others interested in this coming Celebration.