It would not be possible in the short time allotted to me for this paper to discuss any scheme of Educational Reconstruction which might appear to me as ideal. Indeed it is doubtful, notwithstanding the constant use which is made of the term, whether any reconstruction of our educational system, in the full sense of the word, is probable or even possible. But many reforms, some of them fundamental and far-reaching in their effects, are both possible and probable. It is with certain of these which, directly or indirectly, will materially affect the position of mathematics, that I propose to deal as far my limited time allows.

In the Report of the Education Reform Council, published last year under the title “Education Reform,” a paragraph occurs on page 84 with reference to the Mathematics Course suitable for boys of 16 to 18 years of age attending a Secondary School. There it is suggested that such boys who take up Mathematics and Science may pursue the study of differential and integral calculus, including the easy parts of differential equations, and apply these with advantage to higher algebra, trigonometry, coordinate geometry, solid geometry and mechanics; that the course should include some introduction to infinite series; that too much time has been devoted hitherto to the binomial series; and that the thing to aim at is Taylor’s series.

If n quantities, which are known to the nearest whole number, are added together, find the probability that their sum so obtained is correct to the nearest whole number.

Let the probability that the error is between x and x + dx be p dx.

If n = 2, and x>0,

p dx=Probability that first error is between x – 1/2 and 1/2,

and that second error is within a given dx of its possible range

from – 1/2 to 1/2,

= (1– x)dx ; ∴ p = 1–x.

If x<0, p = 1+x.

I suppose that most teachers of Elementary Mathematics, when first they broach the subject of finding the circnrnference of a circle, write on their blackboards for comparison

and draw the conclusion that “as these results agree a far as the third figure, the approxinlatioll π=3 1/7 can reasonably be expected to get the first 3 figures of your answer right.” The purpose of this paper is to allalyse to some extent a few practical rules of this type.

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.”

No general agreement about a first-year syllabus in Geometry is possible unless there is some measure of agreement as to the aim we ought to have in teaching geometry at all. For to different aims there will correspond differences, not only in the manner of teaching the syllabus, but also in the actual matter taught. What, then, ought to be our aim?

In the Report of t h e Education Reform Council, published in 1917 under the title “Education Reform,” it is suggested on page 83 that for boys of 14 to 16 years of age attending a secondary school it is desirable that the mathematical course should be directed towards an elementary use of calculus methods, so that those who remain at school to a later stage may be able to use effectively the elements of infinitesimal calculus. On the next page it is further suggested that boys of 16 to 18 years of age who take up mathematics and science may pursue the study of differential and integral calculus, including the easy parts of differential equations, and apply these with advantage to higher algebra, trigonometry, co-ordinate geometry, solid geometry and mechanics; that the co-ordinate geometry requires a muchbroader treatment than that commonly adopted—a treatment in which the calculus is freely used and in which some “geometrical” conies, when elegant and convenient, is dovetailed into the analysis.

Let t be the centre of a circle, radius r, p any point in its plane, and draw the diameter through p. Then the inverse of p is a point p′ on tp such that tp. tp′ =r2. This circle is called the circle of inversion, its centre t the pole of inversion and the square of its radius the constant of inversion, and each of the points p, p′ is the inverse of the other. This relation is a metric one: a descriptive relation can be obtained as follows.

Geometry in a Mathematical Syllabus seems to me to occupy much the same position that Greek does in the Classical. The two have similar merits and similar defects, and I am doubtful whether either deserves its place except in specialist classes.

The object of the paper is to call attention to some simple means of interpolating points in a cubic graph, and of indicating gradients without actual calculation.