8. The following application of Dedekind’s principle is the most important one in practice.

If s1, s2, s3, sn … be a sequence of positive rational nucmbers which steadily increase and never become greater than a given positive number K, then the given sequence approximates to a definite limit, K being rational.

Isaac newton S investigations will always rank among the chief achieve¬ments of our race- It is intended in this sketch to give an outline of his life and what he did.

Newton was born prematurely on the morning of Christmas Day, 1642, in the manor house of AYoolsthorpe, a tiny hamlet a few miles south of Grantham. His mother had been married less than a year, and was already a widow; her means were narrow, but when lie was three years old she espoused a neighbouring parson, it being apparently a condition of the contract that Isaac should not accompany her to her new home, but should he given by the bridegroom a field to be added to the Wools-thorpe estate His grandmother, Mrs. Ayscough, then came to live at the manor, and took charge of him : he continued with her till lie was twelve years old, attending the village schools, where lie learnt reading, writing, and arithmetic.

Mr. D.K. Picken has done well to direct attention to the unsatisfactory nature of the elementary discussion of the Simple Pendulum, current in the text-books.

What is usually passed off as a proof is a downright wicked deceit, and deserves the serious attention of the advocate of rigour, when he can spare time to look elsewhere, beyond a fundamental axiom or a convergency test.

I am not acquainted with any investigation of the problem of dissecting two given rectilineal figures of equal area, by straight lines, so that the parts of either will fit on the other, and the results given here are new to me. The problem is one which suggests a variety of interesting geometrical constructions. It is natural to consider first how, and in how many different ways, a dissection can be made so as to give the smallest possible number of parts ; but types of dissection which do not give the smallest possible number of parts may also be worth attention. I exclude cases of dissections depending on some accidental or unspecified coincidence of dimensions in the given figures.

THE first Mathematical Colloquium held in Edinburgh met during the first week of August, and proved a great success. It was organised by the office-bearers of the Edinburgh Mathematical Society, A. G. Burgess, M.A., F.R.S.E., and Peter Comrie, M.A., B.Sc, F.R.S.E., respectively the President and Secretary of that Society, being also President and Secretary of the Colloquium. The idea of holding such a colloquium was an outcome of Professor Whittaker’s announcement that he purposed organising, as part of the Mathematical Honours curriculum in the University of Edinburgh, a mathematical laboratory for systematic numerical discussion of functions and methods of calculation. Several correspondents had expressed the hope that vacation courses in this line of study might be established; and it was decided to make a first experiment. It was resolved, however, not to limit the colloquium to a discussion of one branch of mathematics, but to enlarge its scope by the inclusion of two other domains of mathematical thought. The broad features of the programme we owe to Professor Whittaker; and its variety was such as to appeal to all types of mathematical mind.

It is sometimes very difficult to recognise prime numbers at sight, or to discover them without laborious opertions, and mathematicians from the remotest times have endeavoured with varying success to discover suitable methods for the purpose.

In the present note I will assume that the junior student is familiar with the Sieve of Eratosthenes, and that he is required to make a complete list of primes and composite numbers between two limits such as 1601 to 1639.

In his Foundations of Geometry, Hilbert has shown “the impossibility of demonstrating Desargues’ Theorem for plane geometry without the help of the axioms of congruence.” But it must be remembered that his demonstration depends entirely on the form which he has given to the other axioms, or definitions as I should prefer to call them, and in particular on the way he defines “straight line,” namely “two distinct points always determine completely a straight line,” and “any two distinct points of a straight line completely determine that line.” What he has shown then is that in plane geometry this is an inadequate definition of “straight line.” He has not shown that that definition cannot be amended otherwise than by the introduction of the conception of congruence, or by that of three-dimensional space.

One of the most interesting results of the theory of Order as defined by Boundaries is that it explains this apparent paradox. I propose to add to the papers already published in the Gazette on this theory a brief note on this point. To aid the imagination of my readers I shall not attempt to state the problem in general terms, but shall merely state it in homely geometrical language, and for the sake of brevity I shall not pretend to give formal demonstrations of each successive point in the argument, but shall leave obvious deductions to the reader.

The algebraical transformation

should be introduced into dynamical instruction at an early stage, and interpreted as the resolution or rearrangement of the left-hand side, the kinetic energy of two bodies, say railway trucks, of weight W1 and W2 tons, moving on the same line with velocity U1 and U2 feet per second, into (A) and (B) in the right-hand side.