The task you have set me, of surveying in the space of 50 minutes the contribution of the Scots to mathematics, is a well-nigh impossible one. Such a survey must begin with the revival of learning in western Europe, and may appropriately end shortly before the middle of the nineteenth century, when the concept of Scottish mathematics begins to lose definition; an exact date might be fixed, rather arbitrarily, by the first appointment of an Englishman to a Scottish mathematical chair in 1838, when Philip Kelland succeeded William Wallace at Edinburgh. These limits will exclude many distinguished mathematicians of Scottish birth active during the last 100 years, such as Sir Thomas Muir and Maclagan Wedderburn, not to speak of the applied mathematicians Kelvin, Tait and Clerk Maxwell.

It is a commonplace that pictures leave a deeper impress on the human mind than abstractions and, in spite of its abstract nature, that is also true of many part? of mathematics. A proof of a theorem or the construction of a counter-example is often suggested by pictorial considerations, and that is true of algebra and analysis as well as of obviously geometrical subjects like topology. (For entertaining examples see Professor J. E. Littlewood’s book [1].) There is obviously a connection between a developed geometrical intuition of that kind and the kind of spatial experience which ought to be furnished by a school mathematics course. No one would deny its importance and the writers of textbooks on ‘modern’ mathematics are well aware of the need to stimulate geometrical intuition. What has been called in question is the relation between that kind of geometrical intuition and the formal treatment of the euclidean plane based, albeit insecurely, on Euclid’s Elements. Indeed, the traditional treatment has been abandoned with indecent haste and without serious question. In Part I of this paper we make a plea for a re-consideration of the case for some kind of formal geometry as an indispensable part of the education of the more mathematically able children.

The standard procedure for turning right at a cross-road is first to move as far over to the right as possible, while remaining on one’s side of the road, which leaves other traffic free to go straight on or to turn left, provided the road is wide enough.

At the end of the nineteenth century David Hilbert made the decisive step in the foundations of geometry (for the historical development see [1]): a road was opened to a theory of geometric structures. In spite of the fact that one spoke (and speaks still today!) strangely enough of ‘implicitly’ defining the geometrical concepts through the axioms, it became clear in the sequel that the axiom system of Hilbert constitutes an ‘explicit’ definition: not of course for ‘point’, ‘line’, etc. but of the concept ‘euclidean geometry’ (or ‘euclidean space’).

A number of introductory statistics books published during recent years have presented the basic theory of point estimation (see, for example, [1]–[4]). The time now seems ripe for a few cautionary tales to be told, or retold, for there is more to the estimation of a parameter than searching for an unbiased estimator or maximising the likelihood of a random sample. Having said that, I need at once to emphasise the distinction between methods of estimation on the one hand, and properties of estimators on the other. For a particular problem, the method of maximum likelihood, fcr example, provides a possible set of estimators for the parameters involved. Other methods such as least squares or minimum-χ2 might provide alternatives; but no estimator should ever be put to use until its properties have been investigated in some depth. And here lies the rub; for, as I intend to show in the following paragraphs, the choice of criteria by which one estimator is judged superior to another rests, ultimately, with the individual statistician or researcher responsible for the estimate produced. In other words, no completely objective solution to the estimation problem is tenable.

It is sometimes assumed that the solution of a problem can be found to the same number of significant digits as the data. That this is not always so can be illustrated by the following example: the solution of the set of equations

123x ― 124y = 61, 248x―250y = 123

is x = 1, y = 0.5, while the solution of the perturbed set

123·01x ― 124y = 61, 248x ― 250y = 123

is x = ― 4, y = ―4·46!