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.

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.

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!

It is well known that the simpler regular solids can be assembled into compounds which have, as a whole, the symmetry of a more complicated solid : two tetrahedra with a symmetry of a cube, five cubes with that of a dodecahedron, and so forth ([1] p. 134, [2] p. 134).

In Fig. 1 ABC is any triangle on the sides of which similar isosceles triangles are erected. For the present these are shown external to the triangle, but later on the possibility of triangles erected internally is included.

In a recent paper [1], Victor Bryant shows how the number of axioms required to define a vector space can be reduced to seven (in addition to closure requirements). The main result of his article is that commutativity of addition can be deduced from the other axioms. In the present article we show how to reduce this number to six. For certain underlying fields one or more of these axioms can be deduced from the others. However, the six axioms are in general independent; we invite interested readers to show this by constructing their own counter-examples, which the editor of the Gazette will be pleased to receive.