is by no means to be regretted-it is in conformity with the spirit of the age. In this country, Mr. Robert Lowe, himself a great classical professor, spoke last year rather more in favour of the cultivation of modern languages. Hence, the great importance of the cultivation of music and painting as an essential element in the culture of a nation. These branches should not be studied as a profession, but should be taught to children of both sexes along with their "Step by Step." This cultivation would accompany them and put its impress upon all their doings, and all their productions would be marked by the same love and appreciation of the beautiful. It was to be hoped, at any rate as far as Glasgow was concerned, that this discussion would lead to some practical results. The second city was most favourably situated for the initiation of a technical school, which might not only be made a model to the whole country, but might rival the German system under consideration. They in this city possessed a plant such as no other city can boast of. They had the power, if they only appreciated the interests at stake and put forth an effort to collect the means scattered among the tiny foundations or mortifications, to establish a technical college which would reflect renown upon this city, and be the means of initiating throughout the country a system of technical education which will maintain the prosperity of the empire.

Mr. DAVID SANDEMAN said--The great obstacle in the way of carrying out a course of thorough technical education, such as that advocated in Mr. Dixon's admirable paper, is the unwillingness of parents to allow their children to remain at school the necessary length of time. It is possible that their ideas may undergo a change in this respect, but until then, I think that we should do our best with the means and material at our command.

The Anderson's College, the Mechanics' Institution, Allan Glen's Institution, and the School of Design, when properly endowed or supplemented, would be in a position to undertake the scientific instruction of the youth of Glasgow for many years to come, the practical application of which could be acquired in the school or workshop, as might be considered most suitable, by those engaged in the various branches of industry.

The Weaving School in Well Street appears to be supplying a a want which had been felt, the number of students in the

and I may mention, as a gratifying fact and as a proof that trades' unions are not unfriendly to technical education, that two tenters are among this year's students. I may further state, that the success attending this school has led its Trustees to consider the desirability of adding to it a dye-house as soon as adequate funds shall be forthcoming.

Mr. THOMAS MUIR, Vice-President, said that, rather than give any opinions of his own, as the President had suggested, he would prefer to draw attention to one or two unsettled points in the hope of eliciting still more information from the gentlemen of practical experience who were present. In the first place, it seemed to him that there had been made manifest a considerable diversity of opinion not only as to when technical education was to be begun in a pupil's course and where it was to be carried on, but also as to what it really was. In these circumstances it would be a very valuable contribution to the discussion if Mr. Dixon would sketch in its full extent such a curriculum as he would approve of for a pupil who was intended by his parents for a technical pursuit.

Again, reference had been made by more than one speaker to the harmful practice of withdrawing boys from school at too early an age, and it had been well said that in the great majority of cases a year at school, after the age of thirteen, was worth more than any two before it. But how often, on the other hand, did one hear it asserted that soundly-educated lads of seventeen or eighteen were not wanted in business-that the boy of thirteen or fourteen was more serviceable at first, and greatly superior afterwards? Now, where lay the truth in this matter? The question, it must be borne in mind, was not which of the two at eighteen was the more useful to his employer, but which was the more useful at twenty, the more enterprising and prosperous at twenty-five or thirty, and the more valuable citizen generally for the remainder of his life.

Mr. WM. CALDWELL gave it as his experience of more trades than one for many years, that the demand for higher skill on the part of the workman was by no means so great as many advocates of technical education appeared to suppose. He said it was by no means uncommon to find an inferior workman with comparatively little knowledge of his trade preferred to a more experienced man. This occurred principally in shops where articles of an inferior quality were made to meet the popular demand for cheapness. He believed that nothing would contribute so much to the raising of

the skill of our artisans as an increased demand on the part of the public for good substantial work.

Mr. DAVID THOMSON, I.A., said that the increasing demand of the public for cheapness tended powerfully to repress the skill of our artisans, as it led to the manufacture of inferior articles by other than highly skilled and conscientious workmen. The keen competition between the masters in the matter of price prevented them from combining to put an end to the many malpractices which now prevail in all classes of manufactures, and the only hope of remedy appeared to him to be in getting the trades' unions to take up the subject and make it imperative that all apprentices be thoroughly trained to their craft, and that they be duly proved before receiving certificates of craftsmanship; and, further, that they, the craftsmen, should take combined action, and refuse to execute work of a shoddy, artificial, and fraudulent character. Some reformation of this sort would, he thought, have to be made before technical schools could be properly developed, and do the good they are so well calculated to perform.

Mr. DIXON said in reply that, with the exception of a remark by Mr. Jacoby upon the meaning of the term "technical school," there had been throughout the discussion no direct criticism of his paper. Whether the Handels-schule of Germany were or were not usually included under the term "technical," he concurred in the opinion of Mr. Jacoby regarding its value, and the desirability of having similar schools in at least the large commercial centres of this country.

The discussion had been at anyrate suggested by his paper, which had therefore supplied a certain amount of recent and, as he believed, reliable information concerning the technical schools of Germany, and had also called forth the opinions of many who were interested in the instituting of similar schools in this country. He had been much gratified to witness the great amount of interest now taken in technical education, although he must admit that, as yet, opinion on the subject is somewhat chaotic. He had noticed, however, that a few fundamental points had received clear recognition from several speakers. He referred to such points as the necessity for scientific instruction as a basis for all thorough technical training, to the superiority of instruction in day-classes over instruction in evening-classes, and to the necessity of school attendance till about the sixteenth year of the pupil's age. However opinions might differ regarding the best or most convenient

ways of carrying out these principles in practice, it seemed to him that little divergence of opinion existed regarding the principles themselves. The whole discussion had, indeed, struck him as being decidedly in advance of previous discussions on the same subject to which he had listened, and he fancied that it had shown. in particular a decreasing liability to dread the "theoretical" as the negative of the "practical."

Mr. Muir had proposed to him the task of sketching out a full curriculum of study for a boy intended for a technical pursuit. This was more than he could undertake to do, as the curriculum must be adapted to the trade or profession the boy is intended to follow, and to the amount of time that he can devote to the learning of it. It was for these reasons that in Germany the organization of the technical high schools, the organization of the technical middle schools, and the organization of the Baugewerkschulen are all different from each other; and a corresponding difference would show itself in this country when the organization of our technical schools came to be taken up in earnest. Mr. Smith had advised all who were interested in the organization of technical schools to visit Allan Glen's Institution, and he (Mr. Dixon) would be happy to show them everything connected with it, as perhaps the best means he had of meeting Mr. Muir's request. He might say that they would there find a school organized to carry on the education of a boy from his sixth to about his sixteenth year, and to supply him, particularly during the last four years of his course, with instruction in those branches of pure and applied science that chiefly underlie the industrial arts. They would find that great importance is attached to the study of English, and, after that, to drawing, modern languages, mathematics, physics, and chemistry. They would find that as pupils advanced towards the highest class in the school, their attention was more and more directed to mathematics, freehand and mechanical drawing, machine construction, theoretical and applied mechanics, and chemistry. They would find generally a liberal supply of scientific apparatus, and, in particular, they would find a well-equipped laboratory, in which pupils were, in the first place, trained to repeat the chemical experiments they witnessed upon the lecture-table, and were afterwards taught a systematic course of chemical analysis. They would also find provision made, in the shape of a workshop, for teaching the advanced pupils the use of the ordinary tools for the working of

wood and metal, and for showing them the practical application of much of the theoretical instruction given them under the heads of Theoretical and Applied Mechanics, and of Machine Construction. They would find all this done with the object of enabling boys to proceed to the learning of their trade elsewhere with intelligence and ease. They would certainly not find any belief in the possibility of teaching any trade in the school.

Referring to this latter point, he hoped that from the paper that he had read no one would gather that any considerable educational authority in Germany was at present advocating the teaching of trades in connection with schools. It was expressly stated in the paper that it appeared to him to be as good as universally accepted in Germany "that a youth can only learn his trade properly when he is surrounded by all the tools and appliances actually used in his trade, and with all its processes continually under his view." There was, however, something more to be done than to protest against the attempt to teach lads their trade in school. They had to recognize the fact that in this country there is almost no such thing as teaching a trade anywhere. He could have liked to hear in the course of the discussion a protest against the neglect of teaching trades in the right place, no less emphatic than the protest they had heard against the attempt to teach trades in a wrong place.

VI.-On a Substitute for Euclid's Third Postulate.

By Mr. R. F. MUIRHEAD, B.Sc.

[Read before the Society, January 7th, 1880.]

EVERY logical system of geometry must presuppose as its foundation a certain number of principles expressed in the form of geometrical theorems. In Euclid's "Elements" the theorems that are assumed without proof are mostly of a very self-evident character, and they are called by Euclid Axioms. But the "Elements" not only form a system of geometry, but also give the solutions of a number of problems. In order to accomplish the latter result it is necessary