I. Let ABO be a triangle in which = 30°, B = 60°, C = 90°.

Produce AC to D so that AD = AB. Join BD.

The problem of the greatest term of a binomial expansion is a favourite one in elementary text-books, and its solution is often difficult to a beginner. The difficulty, at least in the case where the index is negative or fractional, seems to be caused by the fact that a “formula” is provided which gives a value for r, such that the (r + 1)th term is the greatest. Moreover, this formula is not always the same. Sometimes it is sometimes ; and unless the student has a very good memory he is sure sometimes to make mistakes. Elementary mathematics ought not to be a memory exercise. It is a platitude to say that the educational value of the teaching of mathematics lies in its training of the powers of reasoning. This element is eliminated when processes of reasoning are reduced to a rule of thumb. As well might one use “Molesworth” as a text-book of the principles of mechanics.

Let ABCD be a horizontal section of a rectangular slab of glass. A pin is set up vertically at P, close to the face AB, or at a short distance from it. It is possible to see the pin P through the glass if we look through the face CD; but the pin is invisible if we look through the face AD. If, however, we look through the face CD, into the face AD, we shall see an image of the pin in AD, which acts like a mirror. The experiment illustrates total reflection of light; the explanation is easy. Let PO be a ray of light which after passing into the glass will be incident on AD at O′. Let NON′ be the normal to AB at O,

and KO′K′ be normal to AD at O′. The critical angle for glass being 41° 45′ approximately,

∴ the ray OO′ will not emerge through AD, but will be reflected in the direction O′Q, and will emerge at Q in the direction QS.

The Theorem that an – a is exactly divisible by n, if n be a prime number, may be established as follows from the Binomial Theorem.

is an integer, if x is an integer, and n + 1 is a prime integer which is not a factor of x.

This is not a neat proof of Fermat's theorem, but, as far as I know, it is a new proof, and it may have some little interest, as it seems very possible that the theorem may have been suggested to Fermat in this way. Fermat published the theorem without any demonstration and without indicating what had suggested it, and any proofs, that I have seen, give no indication why one should look for such a theorem, and would very possibly never have been given if the theorem had not been already known.