So far as mathematics is concerned the teaching in the Universities must, in view of the position of the subject in the schools, have been of the most elementary character. The educational ideas that dominated the Universities till the middle of the 16th century and that were by no means completely transformed till a much later date were mediaeval; the curricula were drawn up in the interests of the church rather than for the promotion of learning and science. Again, the funds at the disposal of the Universities were quite inadequate to meet the expense of courses such as the charters implied and it must be admitted, I fear, that within the teaching and administrative bodies of the Universities themselves there was frequently a want of harmony that was fatal to educational interests. Many, probably most, of the Regents (or, as we should now call them, Professors) were learned men as judged by the standard of the time, but their training seems to have developed a tendency to concentrate on words rather than facts, on language rather than ideas. The disciplinary code was extremely detailed and the limits prescribed for teachers and students alike were totally inconsistent with that atmosphere of freedom which is essential to all higher education. While it was certainly better for Scotland that it should have institutions devoted to the higher learning I confess that I have derived from my study of the conditions of our Universities for the first two centuries of their existence the conviction that they failed to realise the aspirations of their founders and that Scottish students still required to go abroad - as many did - if they were bent on getting access to the best scholarship of the day.
It is perhaps worth noting that for many years (till the beginning of the 18th century, Grant, I., 147) the system of Regents was in force; the individual subjects were not assigned to different teachers, but to each teacher, or Regent as he was called, was assigned a class. The students of that class - which often meant all the students of a particular year - received their whole instruction during their University course from the same Regent who was thus kept in close touch with them and came to know their individual aptitudes. In the infancy of the Universities the system, I think, had some advantages but it lasted far too long; at King's College in Aberdeen it was not superseded till the end of the 18th century. But one can readily understand that the predilections of the Regents would often lead to difference of emphasis on different subjects and there is some evidence that mathematics was one of the subjects that at times received less than the normal attention.
Up to the Reformation the curriculum in Arts seems to have been practically identical in the three Universities. The Statutes of the Faculty of Arts in Glasgow University outline the course for graduation and I think we may assume that it represents the courses of the other two for all practical purposes. The preponderating subjects are drawn from Aristotelian philosophy; the mathematical subjects are:- The Sphere (of Sacrobosco, I suppose), perspective, algorism and the elements of geometry. The physics, the de caelo and the meteorology of Aristotle were also included in the course. Of the mathematical textbooks, other than the Sphere of Sacrobosco which is rather astronomical than mathematical, I can find no mention. The mathematical programme was thus very meagre and probably did not go beyond arithmetic of the scholastic type and the very elements of geometry; so far as I am aware there is no name of any special note in the history of mathematics that is associated with the Universities in these years.
The Reformation had a profound influence on the general life of Scotland but the educational proposals put forward were, as is well-known, very imperfectly realised in practice. Still a change for the better had set in as James Melville's account of his studies at St Andrews shows. Under the date 1572 he writes that in his "the second yeir of my course ... the Primarius [Mr James Wilkie] a guid, peaceable, sweit auld man, wha luiffed me weill, teached the four speaces of the Arithmetik and sum thing of the Sphere" ... "In the thrid yeir of our course we hard the fyve buikis of the Ethiks with the aught buiks of the Physiks - [and de Ortu et Interitu]." ... "The fourt and last yeir of our course ... we lerned the buikis de Caelo and Mateors, also the Spher, more exactlie teachit be our awin Regent" (Diary, pp. 27, 28). When Andrew Melville was made Principal of Glasgow University James accompanied him and was appointed a Regent. In the Diary, p. 49, he gives a sketch of the courses laid down by his uncle and I quote the passage that deals with mathematics and physics:- "He teatched the Elements of Euclid, the Arithmetic and Geometrie of Ramus, the Geographie of Dyonisius, the Tables of Hunter, the Astrology of Aratus. ... From that to the Naturall Philosophie; he teatched the buiks of the Physics, De Ortu, De Caelo, etc., also of Plato and Fernelius. With this he joined the Historie, with the twa lights thairof, Chronologie, Chorography, out of Sleidan, Menarthes and Melanethon." He concludes his sketch with the words, "Finalie, I dar say thar was na place in Europe comparable to Glasgow for guid letters, during these yeires, for a plentifull and guid chepe mercat of all kynd of langages, artes and sciences." It is, perhaps, worth adding that James says (p. 54), "the second yeir of my regenting I teatchit the elements of Arithmetic and Geometrie out of Psellus for shortness."
Again one of the Old Laws of King's College, Aberdeen, promulgated anew in 1641, gives a syllabus of the subjects taught in each of the four classes or years. [Fasti Aberd. p. 231]. No mathematics appear in the first year, but in the second Alsted's Compendium of Arithmetic and Geometry is prescribed, while in the fourth year the De Caelo, de Ortu et Interitu and, as occasion permits, the Elements of Astronomy, Geography, Optics and Music from Alsted's Admiranda Mathematica are among the subjects of study. The textbooks of Alsted are of interest as indicating the nature of the courses, but as none of Alsted's books were published before 1610 it is obvious that they could have formed no element in the instruction till long after the Reformation. The extracts given by Mr Anderson [The Arts Curriculum. Aberdeen 1892] from the Report of the Commissioners of 1647-48 do not add anything material to what has been already stated.
There can be no question, I think, that the latter half of the sixteenth century witnessed a decided improvement in University teaching but, after all, the improvement was rather an indication of a more liberal outlook than a solid advance in higher learning; some years had still to elapse before the Universities produced men of genuine mathematical distinction.
So far nothing has been said of the curricula in Edinburgh University. A detailed account, such as we do not possess for any of the other Universities, is given by Sir Alexander Grant in his Story of the University of Edinburgh; it will be sufficient for my purpose to state the demands in mathematics at the foundation of the CoIlege. In the Bajan or first year, no mathematics; in the Semi-Bajan or second year, towards the close of the session a compendium of Arithmetic was given to the students; in the Bachelor or third year there is no mention of mathematics while in the Magistrand or fourth year "the De Caelo of Aristotle and the Sphere of Johannes de Sacrobosco were read and demonstrations of Practical Astronomy were given. Then the students read the De Ortu, the Meteorologica and the De Anima and also Hunteri Cosmographia (a work on Geography)."
From this syllabus it is plain that mathematics was not placed in any better position than in the older Universities; in fact, the course is not so good as that which Melville set up in Glasgow - so hard is it to break away from long established tradition.
I shall not do more in describing general conditions in the Universities, though their detailed study is of great interest and full of lessons that modern educationalists might study with profit. Throughout the 17th century the political and ecclesiastical conditions in Scotland were very unfavourable to higher learning but, though harassed by many troubles, the Universities carried on, and towards the end of the century the place of mathematics became more assured and worthy exponents of it began to appear.