Both terms in the title of this paper are probably suspect. The mathematician, versed in a subject with hundreds of years of noble tradition, tends to be suspicious of a paper on psychology; the teacher of mathematics, immersed daily in psychological relationships with ordinary children, tends to be suspicious of a paper on mathematical ability. The discussion promises to be unsatisfactory, to the one because psychology lacks mathematical precision, to the other because mathematical ability seems to present no urgent psychological problems. The teacher-mathematician may then be pardoned if he reacts sceptically to our title with the petition, “Psychology, if there be a science of psychology, help me to cope with mathematical disability.” But these suspicions are not well founded. Not only is psychology, as we shall see, becoming increasingly mathematical, but also, if we interpret our terms as mathematicians should, disability is seen to be merely the negative aspect of ability. As such, it will not be excluded from our discussion.

Some twenty-five years ago I was in the unpleasant position of “professing” Geometry, with (roughly) six weeks start of the class; it is no disparagement to the abilities of the Galway Honour Students to confess (at this late hour) that—however hard put to it—I never had reason to admit greater ignorance than that of the class. Amongst other things I was obliged to learn that there is a certain projective relation between the 27 lines on a cubic surface and the 28 bitangents to a (general) plane quartic; this fact impressed me so much that it remains (also a general idea of where a proof can be found) even now, when my memory has discarded many things which are (perhaps) more vital.

In the Mathematical Gazette of October 1928, page 234, there appears a quotation from a letter of Chief Baron Pollock to his son, when an undergraduate at Trinity, in which, speaking of Euclid, he says: “I consider (I. 4) to be much more of a ‘pons asinorum’ than the 5th. Indeed I regard it as the test of a geometric spirit. If it be fairly conquered, it proves the existence of that logical accuracy which is the soul of mathematics, and to elicit and cultivate which is the great benefit which such studies confer as a branch of education.”

To our modern minds it seems obvious that scientific progress must be based upon alternate induction and deduction. The first inductive step is, indeed, not always evident. It was only in the nineteenth century that the emergence of non-Euclidean Geometry made it clear that even a purely deductive science like Geometry must be’ based upon a preliminary induction—often subconscious, and the result of generations of tacit acceptance of intuition. Modern classical dynamics commences with the Galilean induction in the form of Newton’s Laws of Motion, thus discarding the old Greek view of motion and its causes. In some branches of human knowledge, e.g. the sociological, the first induction is hardly yet accomplished.

The ancient Greeks recognised two branches of arithmetic. One of them, the philosophical study, expounded by Euclid in Books VII-IX, developed as a logical science the properties of whole numbers. It was not so much concerned with addition, multiplication and the manipulation of fractions, in which the notation developed in the middle ages has greatly improved the methods of the ancients, as with prime, odd and even numbers, with digressions to triangular, polygonal and perfect numbers. The Greeks were familiar with the facts (1) that when a prime number is a factor of the product of two whole numbers ab it must be a factor of either a or 6, and (2) that every whole number can be expressed, in only one way, as a product of prime factors. A late offshoot of this branch of the subject, destined to develop into one division of the modern Theory of Numbers, was considered by Diophantos of Alexandria (English edition by Sir T. L. Heath), who was mainly concerned with finding solutions in rational numbers of indeterminate equations such as

yz + a=u2, zx + a = v2, xy + a=w2.

The second branch of arithmetic known to the Greeks (Aoyto-riK?/) was concerned with its applications to measurements, to monetary calculations and to various problems in which the science was used in practical life. No exposition of this is known to have survived the ages.

Geometrical Optics as a part of the mathematical curriculum in schools, universities, and technical colleges is now a dead subject, yet the generalised study of ray-tracing in optics started by Sir William Rowan Hamilton in 1824 is well worth attention on all sides. He writes: “Those who have meditated on the beauty and utility in theoretical mechanics of the general method of Lagrange, who have felt the power and dignity of that central dynamical theorem which he deduced in the Mécanique Analytique must feel that mathematical optics can only then attain a coordinate rank with mathematical mechanics when it shall possess an appropriate method and become the unfolding of a central idea.” With this end in view, Sir Wm. Hamilton submits his three papers on the Theory of Systems of Rays to the Royal Irish Academy, which, as Lord Rayleigh said, bear the mark of genius throughout. The enormous task that Sir Wm. Hamilton set himself will be evident when we discuss the wanderings of rays through optical systems in an elementary way to-night leading up to Hamilton’s general method of the Characteristic Function and its development.

The equations governing the motion of a viscous fluid were first obtained by Navier more than 100 years ago (“ Memoire sur les Lois du Mouvement des Fluides,” Mem. de l’Acad. des Sciences, vi. 389, 1822), and in spite of their close study by Stokes, Helmholtz, Kelvin, Rayleigh, Lamb and numerous other mathematicians of great eminence, no complete unrestricted solution for any case of practical importance has yet been discovered. As they stand, the mathematical difficulties presented by the equations have so far been found to be too formidable, and whatever progress has been achieved has been by imposing restrictions on the form of the equations, and therefore serious limitations on the nature of the fluid motion studied. It may be that the mathematical symbolism in the formulation of the problem of viscous fluid flow as usually presented is not that best adapted for its purpose, that it is not as natural a medium of expression as for example Tensor Analysis is for Relativity ; that in fact the essential factors that govern the eddying, for instance, in the wake of a moving body are not presented as governing the structure of the equations.

The idea of a function of an infinite number of variables, like so many other mathematical ideas, has its origin in the study of problems of applied mathematics. Such problems, as is usual, preceded by some considerable time the explicit formulation of the idea. As an introduction to this discussion it will be sufficient to consider briefly two of them.

The Problem of the Brachistochrone.—Given two points P, Q in a vertical plane, what is the shape of a smooth curve through P, Q for which the time required to descend from P to Q under gravity is a mimimum.

It is hardly necessary for me to express here my indebtedness to the patience and learning displayed by Sir Thomas Heath in his edition of the Works of Archimedes (Cambridge, 1897); but for his statements and suggestions I should never have even known that the greatest master of Greek geometry had left so fascinating a problem for the speculations of later mathematicians. In what follows I can only claim to have tried to frame my conjectures on the hypothesis that so consummate a geometer was at least on the same intellectual plane as such men as Newton or Gauss, so far as a humble admirer can compare those who have towered as giants in the subject.

The solution of the problem of drawing a circle to touch a given line BC at a given point X and to touch also a given circle v is very familiar; it is obvious that this solution must lead to an elementary verification of Eeuerbach’s theorem, but the extreme simplicity of this verification is commonly overlooked.

While major mathematical problems were discussed already in the seventeenth century in such serial publications as the Philosophical Transactions (1665) and Acta Eruditorum (1682), it was not till the early part of the eighteenth century that serials containing elementary problems of wide appeal commenced to appear. The first of these seems to have been the Lady’s Diary, started in London in 1704, and “designed for the sole use of the female sex”. It had an immediate success, and continued to appear in various forms for 168 consecutive years. Of the second, Delights for the Ingenious, there were only eight numbers, in 1711. The third publication of the kind was possibly Kunstfrüchte 1. Sammlung (1723), a publication (I have not seen it) of the Hamburg mathematical society, founded thirty-three years before. The Jahrbriefe of this same society were published irregularly during the 140 years 1732-1871. For the fifth and sixth publications we come back to England, Miscellanece Curiosce (York, 1734-35) and Gentleman’s Diary, started in 1741 and continued for a century before its union with the Ladies’ Diary. Then followed five other English serials before Holland’s Mathematische Liefhebberye (Purmerende, 1754), issued annually for seventeen years. During the next 175 years the number of these minor publications became very large. It is my purpose to bring together brief notes on minor English serials and their editors of the past two and a quarter centuries, and to indicate where more information regarding them may be found. It will not be possible, within the limits of this article, to indicate more than very occasionally anything of the contents (often rich and varied) of such serials, many results appearing in their pages for the first time. Problems and their solutions usually occupied the greater part of the space, and most of the prominent English mathematicians of the time were contributors. While some may incline to frown upon such mathematical occupations, it may be recalled that “Sätze und Aufgaben” were to be found in such an exalted source as Crelle’s Journal, so recently as 1858.

The whole of ordinary algebra, dealing with real or complex numbers a, b, c, is based on three fundamental laws which are usually called:

1. I. The Associative Law,

2. II. The Distributive Law,

3. III. The Commutative Law.