Would you indicate 3 + 3 + 3+3 as 4x3 or as 3x4† Most colleagues and boys that I have asked plump for the former. 4x3 means to them “four threes” or “four times three”, which is emphatically not the meaning “four multiplied by three”. Clearly “four times three” is the same as “three multiplied by four”; (I do not mean that they are equal, I mean that the expressions are two ways of saying the same thing). If we indicate this common meaning by the symbols 4x3, we are in fact doing the opposite of what we do in the case of the other three arithmetical processes. 4 + 3 means (to most people) “take four and add to it three”, or “four increased by three”; 4-3 means “take four and subtract three from it” , or “four diminished by three”; so 4 + 3 means “take four and divide it by three”. Obviously, then, 4x3 must mean “take four, and multiply it by three”; i.e. “four taken three times” or three fours.

In the extension from the use of natural numbers with zero to integers, i.e. the introduction of directed numbers including negative numbers, much experimental work is now being done with the use of mathematical structure. Some attempts are being made to give this as the initial treatment at about the age of 12, and some teachers are getting their children very interested and consider it a correct method. For other children the formal treatment will come later, when they are more mature and begin to query the rules they are using. This may be when they are starting to work ahead of their O level syllabus, the beginning of a more advanced course with work on groups. It could give work for those doing some mathematics when they are beginning to specialise in some other subjects and want to try to understand more about the mathematics they have learnt.

My attention was first called to this attractive curve by a pupil, M. J. Burleigh; so, lacking any other name, we called it “Burleigh’s Oval”. It is mentioned in Salmon’s Higher Plane Curves (p. 107), where a cartesian equation is given; and in Hilton’s Plane Algebraic Curves (p. 64), where it is described, with a diagram, as the reciprocal of the Limaçon. It also occurs in a very interesting general treatment of negative pedals by A. Ameseder, (Archiv der Mathematik und Physic, LXIV, 1879).

In Note 2566 (Gazette XXXIX, 305) Professor Arscott quoted some results on divisibility, generalised to cover the cases of odd numbers of the form N = 10M + z, where z = 1, 3, 7 or 9. He stated that the proofs are “based upon simple congruences . . .” and the proof for divisor 53 is given.

This article develops the four cases from one central proof, and then indicates how these may be applied as tests for all odd divisors. If the need is to resolve a given number into its prime factors, the appropriate powers of 2 and 5 may easily be removed; if, however, all that is required is to establish whether (or not) a given odd number is a factor, this preliminary operation is unnecessary.