The simple approximation for the rth root of a number which is nearly an exact rth power, given by Mr. Whipple last month, is found as a rule in Bonnycastle’s Arithmetic. Perhaps the most elegant form is this:

The following system of algebraic inequalities is of great value and of wide application:

for all positive proper fractional values of a,

for positive proper fractional values of α1, α2, αn, and value of k Such that 2k<n; ∑r denoting the sum of the r-products of the α’s.

In order to give an adequate account of the progress of Geometry in the century that has just come to a close, it is important to cast a rapid glance over the condition of Mathematical Science at the beginning of the nineteenth century.

The hopeless complexity, from the elementary didactic point of view, of the hitherto accepted accurate trigonometrical proofs of the power series for sin x, cos x has been so fully insisted upon in these pages (this volume, pp. 85, 86), that a more elementary method seems urgently called for.

The measurement of the acceleration due to gravity by means of the compound pendulum is a very common laboratory experiment; but there are certain simple principles involved in the theory of it which are not, as it seems to me, sufficiently emphasized by teachers, and which are not stated (at all events explicitly) in any text-book known to me. It may be of use to some to enunciate them here, and to shew their importance both to the instrument-maker and the student.

We may now turn our attention to the questions which arise specially in connection with a reversible pendulum, such as Kater’s. Of course the object in view is so to place two knife-edges on opposite sides of the C.G. that the period shall be the same from either, the two positions being selected according to the rule at the end of § 4. But the usual procedure will be to decide on the distance between the knife-edges at the start, say one metre; then to clamp them to the more or less uniform bar of the pendulum, using a standard distance-piece to secure the proper interval between them, and finally to adjust a sliding weight in such a position as to make the period the same for both. We have seen in § 2 how the period, or rather the length of the equivalent S.P., varies with the position of the sliding weight, but it will be convenient now to change the notation. Let I1, I2, ... denote the moments of inertia of the various masses making up the pendulum, each about its own C.G., M1, M2.. the masses; x1, x2... the co-ordinates of the centres of gravity, referred to one of the knife-edges as origin; let the co-ordinate of the other knife-edge be d, and let the letters I, M, x without suffixes refer to a moveable mass which is being adjusted. We shall at once see how much simpler it is to discuss the graph whose ordinate is the length of the S.P., than the one where the period is used, for the position which the moveable mass should occupy of course corresponds to the intersection of the two graphs relating respectively to the two knife-edges: and it is surely better to have to deal with the intersections of conics than of cubics.

In the following articles the method of Averages, used in the Oct. 1903 issue of the Gazette to effect the expansion of the simple Algebraical functions, is employed for the Trigonometrical Expansions and for Taylor’s Theorem, and the results are applied to the question of convergence. The method was adopted as the simplest I could devise for even the Binomial, subject to the condition that no infinite series should be used unless the limits of error after a finite number of terms were explicitly stated. I would suggest that this condition might well be laid down in a school course, as the uncertainty that comes over our Algebra as soon as we leave Geometrical Progressions ‘to infinity’ delays a clear grasp of Mathematics more than a fuller consideration of the simpler infinite series would. The method is, of course, essentially integral calculus, but I have found—and I hope my subjects have not been specially favourable—that the proofs can be readily followed by students to whom the ideas and notation of the Differential and Integral will not be familiar for a considerable time, and in whom perhaps they will never inspire a sense of absolute security.

The Mean Rate of Increase which was investigated by Prof. Bryan in No. 48 of the Mathematical Gazette may evidently be thought of as the inclination of a straight line to an axis. The object of the following note is to point out an example in which this inclination naturally occurs.

It is very far from my purpose to construct an antithesis between a purely scientific and a technical conception of the Problem of Mechanics. The conflict which undoubtedly existed some years ago, and at times was waged with bitterness, has come to a satisfactory conclusion; and a generous appreciation of the two different directions of investigation has taken the place of unseemly strife. From my personal knowledge of the Technical High School in Aachen, I at least can lay stress on the fact that I have been encouraged in the most ready manner by my technical colleagues of all departments.

The Association of Public School Science Masters fixed their annual meeting for Jan. 16, and accepted a proposal by Mr. R. H. Thwaites to read a paper for discussion at that meeting on the subject of the possible fusion of the teaching of mathematics and science in public schools. I wrote to Mr. Pendlebury, pointing out the interest of the subject to us, and suggesting that our meetings ought not to clash. In reply, he requested me to open a similar discussion at our meeting to-day. I hope I have made it clear that the suggestion that the subject is ripe for consideration is originally due to Mr. Thwaites.

Were it not for Euclid’s Propositions iii. 26 (proved by superposition) and vi. 33, the statement that he nowhere defines the circumference of a circle or any curved line would surely be a truism. His definition of a circle is substantially only a locus definition. It may well be argued that neither does he define a straight line (or an angle) ; but in the case of a straight line he introduces sufficient ideas and axioms to make his subsequent propositions convincing, whereas he gives us nothing to bridge the interval between a straight line and a curved one.

And so, wise after the event, we may say that all the thought spent in old days over ‘squaring the circle’ had no adequate foundation, and was foredoomed to fail.

As to iii. 26 and vi. 33, they are only further examples of the truth with which we are familiar in the case of angles, that it is sometimes easier to prove that things are equal etc., than to say exactly what they are.

During the last few years the methods of teaching Mathematics in schools have been subjected to much adverse criticism: this has induced teachers to apply themselves to the task of setting their house in order. It is now generally deemed advisable to give greater prominence to practical applications, to make more use of accurate drawing as a method of solution of problems and for purposes of verification, to curtail numerical calculations by dispensing with figures of doubtful or unnecessary accuracy, and to lay more stress on the employment of graphical methods.

It appears from the recently published life of Archbishop Temple that he is supposed to have found a proof of the proposition that any map can be coloured with four colours, no two adjacent segments of it being coloured alike. The writer had heard of this before the publication of the Life, but had not succeeded in finding what the proof was, and was under the impression that it had never been published, especially as it was not mentioned in Mr. Rouse Ball’s book. As stated in the Life the supposed solution was published in the Journal of Education, June 1, 1889. It turns out to be what one has learned from experience (and the writer owns that he has spent much time on the subject) to expect from any theory which looks like a short solution of this extremely elusive problem.