I have been asked to write a short account of the present-day work in the algebra of invariants for the benefit of those who have not made a special study of the subject. A good summary account can be found in the various continental works of reference, and particularly in the Encyklopädie der Mathematischen Wissenschaften, Band III. Teil iv. 7, written by Dr. R. Weitzenböck, and brought up to about tho date 1920 with tolerable completeness, both in subject-matter and in references. Nothing has, however, been published in the last thirty years quite so good and readable as Meyer’s Bericht, which reviewed the first fifty years of progress from the time when Boole inaugurated the theory in the Cambridge Mathematical Journal, iii. pp. 1-20, of the year 1841. As things have turned out, and especially because of the impossibility of buying separate parts of the Encyklopädie without embarking on the whole section (Geometry in this particular case), it seems a pity that we have no machinery in England for the systematic issue of corresponding reviews, perhaps in the form of this Bericht, to cover by degrees the field of present-day mathematics.

Analysis Situs, or Topology, is the investigation of those properties of surfaces and spaces which are unaltered by deformation, that is, by bending and stretching without tearing. For example, a circular disc with a hole in it can be stretched so that its boundary circuits have any of a great variety of forms, but the number of circuits (two) remains unaltered—it is a topological invariant. The fundamental problem of topology (at present unsolved) is to determine necessary and sufficient conditions that a given surface or space A should be deformable into a given surface or space B.

In the preface to his Principles of Geometry, Professor Baker states that the “Foundations” volume “assumes only those relations of position, for points, lines and planes, which, furnished with a pencil, some rods and some string, a student may learn by drawing diagrams and making models.” When the beginner attempts to make these diagrams, it is not unusual for one or more points to go off the page. To illustrate how, in an important set of cases, it is possible to get all of the diagram on the page at the first trial is the purpose of this note.

While Professor Nunn’s treatment of Asymptotes (Math. Gazette, May 1926) gives the elementary student a method for finding the equations of linear asymptotes, the present writer feels that there is some weakness in the method, as the geometrical consequences of the various types of asymptote on the general form of a curve are not emphasised. There is, moreover, a serious difficulty for the advanced student. Professor Nunn regards two parallel asymptotes as a degenerate case of the asymptotic parabola, thus throwing over entirely the ideas of multiple points at “infinity”. This must lead to considerable modifications in the values and properties of Pliicker’s numbers. In fact, the change would affect the whole theory of the deficiency of curves.

(1) The institution of Local Education Authorities in 1902. These Authorities have been able to survey over a wide range of Secondary Schools the working of the examination system and to estimate to some extent its effects on Secondary Education as a whole. Consequent on this, but not wholly the result of it, came some general reforms, e.g. the formation of the Secondary Schools Examination Council. But L.E.A.’s are also involved in the examination system as examiners, since they test candidates for scholarships and free places in Secondary Schools. Their examination view-point is therefore a wide one, embracing that of both teacher and examiner. The full effect of this dual point of view has not yet matured, but some innovations or improvements in junior scholarship examinations have already resulted from it; for instance, the development of the oral examination and the use of the teacher’s assessment of a pupil. In some areas, too, teachers participate in marking scripts. Examinations of School Certificate standard are so far little affected, but it would be in accord with the trend of events if the wide powers of University Examining Authorities were further modified and the school record of the candidate made to play a more important part in the examination than at present.

“General Intelligence” is a will-o’-the-wisp, and Psychologists have been chasing it with Tests for a considerable time. Whether it has yet been brought to heel and can be exactly measured is still open to question, but the modern Intelligence Tests certainly discover some kind of innate ability. Is this ability purely “general,” or do some types of ability show up better than others in the Tests?