It is becoming more widely recognised each year, among leaders of educational thought, that in the field of educational science, where mere opinion holds sway, this opinion should be replaced by conclusions scientifically obtained from carefully collected data.

In the teaching of mathematics we have such questions as:

(i) What is the most economical length of drill periods in first-year algebra ?

(ii) What is the best method of procedure for acquiring skill in the mechanical work in algebra ?

(iii) To what extent does mechanical skill in algebra help the pupil intelligently to use algebraic processes, such as the equation, the graph and the formula ?

(iv) What proportion of pupils benefits by a certain type of course and up to a given standard ?

(v) The results of (iv) would be a factor in deciding whether all secondary school pupils should take mathematics throughout their course.

This quotation from Einstein is merely one of the many disturbing statements that a teacher of geometry may find in recent books on physics or even in the newspapers. It will not be long before some of our pupils will ask if it is worth while to continue the study of a subject of which the principal results are apparently contradicted by the latest discoveries. The object of the present article is to examine these apparent contradictions. To do this we shall trace the evolution of ideas concerning the nature of geometry and consider their relation to school work. The conclusion will be reached that certain changes in our teaching are desirable. Fortunately these changes will actually make school geometry easier as well as more scientific.

The subject of my address has recently been “ventilated” in more than one organ of educational opinion; and it is stated that a committee will shortly be assembled to explore the possibility of an escape from the present chaos to the sweet simplicity of an agreed and authoritative sequence.

I know nothing about the constitution of the committee nor about the proposals that are likely to be brought before it; nor had it been announced or foreshadowed when I suggested the subject for discussion to-night. My intention in suggesting that subject was merely to revive and develop further certain proposals brought forward in an address given to the Association at a time when many of its members were engaged in a vastly more serious discussion elsewhere. I venture, however, to hope that the circumstances I have referred to may add materially to the usefulness of our debate.

Statistical method is coming to be so extensively applied, that those who use it must from time to time feel the need of assuring themselves as to the security of its foundations. For this assurance they will naturally turn to books on the subject. But do these give much help?

OA,OB = x1, x2, being distances, carrying signs, along an axis, the “Bearing” of B with respect to A (called a vector, or step, or displacement) is given by AB = OB – OA = x2 – x1 in all cases, and is independent of the position of O.

If we wish to add two such steps, a construction is necessary, viz. they are placed “ end on ” in either order; and the sum is dependent on the position of O ; if M be the mid-point of AB, the sum = 20M.

This great work, generally attributed to Eudoxus, has attracted the attention of mathematicians of all time, but is now neglected both in the Schools and in the Universities.

The first person to discover a flaw in the logic of any of the propositions, as given in the text that has come down to us, was (I believe) Saccheri, who, in his Euclides ab omni naevo vindicatus, pointed out that in the demonstration of the 18th proposition it was assumed that if there be three magnitudes of which the first and second are of the same kind, then a fourth magnitude of the same kind as the third must exist such that the ratio of the first magnitude to the second was the same as that of the third to the fourth. This proposition he endeavoured unsuccessfully to prove, but it is now known that its validity depends on the Axiom of Continuity.

It is well known that the St. Andrew’s cross (X) occurs as the symbol for multiplication in W. Oughtred’s Clavis Mathematicae, 1631, and also (in the form of the letter X) in an anonymous Appendix which appeared in E. Wright’s 1618 edition of John Napier’s Descriptio. This Appendix is very probably from the pen of Oughtred. The question has arisen, is this the earliest use of X to designate multiplication? It has been answered in the negative—incorrectly so, we think, as we shall endeavour to show.