In the ancient story of Barlaam and Ioasaph, which edified and delighted our medieval forefathers through many centuries, there is a charming apologue of a fowler and a nightingale whom the former had snared and was about to slay. “Free me from my fetters,” said the nightingale, “and I will give thee three precepts by the keeping of which thou shalt be greatly benefited all thy life long.” The astonished fowler accepted the terms and set the bird free, receiving as his reward the precepts : “Never try to attain to the unattainable; never regret the thing past and gone; and never believe the word that passes belief.” But as the bird flew aloft she cried to her captor, “Shame, sir, on thy recklessness! What a treasure thou hast lost to-day! For I have inside me a pearl larger than an ostrich egg.” When the fowler heard this he was distraught and would fain have taken her again. “Come hither,” he said, “into my house; I will make thee right welcome, and send thee forth with honour.” But the nightingale replied, “Now I know thee to be a mighty fool. Though thou didst receive my words readily and gladly, thou hast gained no profit thereby. I bade thee never regret the thing past and gone; and behold thou art distraught with grief because I am escaped out of thy hands. I charged thee not to try to attain to the unattainable; and thou triest to catch me again. Besides which I bade thee never believe a word past belief; and behold thou hast believed that I had in me a pearl exceeding the size of my whole body.”

The object of the “Advanced Section” being to study and find out the best methods of teaching Scholarship work, it seems advisable briefly to pass in review the higher algebra syllabus which is at present being taught to pupils in the departments of advanced mathematics in our secondary schools.

There is nothing more extraordinary in the educational world at the present day than the change of attitude among teachers towards the subject they have to teach. Teachers of bygone ages were to a large extent content to regard their pupils as so many buckets into which they were required to pour a prescribed amount of intellectual material. In most cases the teacher did not know himself the sources whence came his stock-in-trade. He had so many hard facts at his disposal which he had either got out of a printed text-book or from lecture notes, and it was his business according to his genius and ability to breathe upon these dry bones that they should live.

The algebraic solution of any problem formulated in mathematical symbols takes one of two forms: it is either given in the explicit form x=f(a, b, c,...), where → is some functional form of a, b, c, ... , which are supposed known in any particular case, or the solution required is to be found from an equation F(x, a, b, c, ...)=0, where F is a known functional form of the unknown x and of the quantities a, b, c, ... . The mathematician is generally concerned on the one hand with reducing f(a, b, c, ...) into its simplest, often its “prettiest” form, and on the other hand with investigating what method, if any, exists for “solving” the equation F(x, a, b, c, ...) = 0, i.e. finding x in terms of a, b, c, ... . The course included in any text-book of algebra consists principally of the former, since the latter is distinctly more difficult.

In spite of the achievements of mathematics, and of a very general recognition of the fact that mathematics is essential to the progress of knowledge in most of its departments, there exists a certain amount of contempt for the symbols, the formulae, and the abstractions of the subject. For a generation we have struggled against this feeling, and have endeavoured to correct the defects to which the feeling is due, but I do not think that many are satisfied with the success of our efforts. Those who have attended to the widespread demands for a more generous and thorough provision for education, as representing the only sure foundation for national efficiency and progress, have remarked with dismay the little interest that is shown in mathematical education. Surely this state of things demands, at the present time, our most urgent attention.