in hot-houses, but as something vigorous enough to stand the common air and be a delight and ornament to common life. The fact seems almost beyond question, and the vital issue which the public schools have to determine is in what way they shall deal with a problem with which they are directly face to face. Possibly they may still continue to ignore it and, either through indolent neglect or the preoccupation of sordid cares, allow a study which is their oldest heritage slowly to decline and disappear. But if they do so, if they forget alike their traditions and responsibilities, they will incur the charge not merely of being false to their own honour, but of having betrayed the true interests of liberal education. For assuredly no form of education can justly be called "liberal" in which the study of science and preparation for active life are not associated as their necessary complement with that study of polite letters to which classical learning is certainly the best, and possibly the indispensable foundation. II. MATHEMATICS. BY T. J. GARSTANG. MATHEMATICS in England were at a low ebb when Babbage, Peacock, Herschel, and other devoted students founded the Analytical Society at Cambridge in 1813. Its purpose, as Babbage wittily described it, was "to inculcate the principles of pure d-ism as opposed to the dot-age of the University"; or, in other words, to substitute the Continental notation of the calculus for the original dots of Newton. The objects of the society were entirely achieved, and the foundation laid for that growth and persistent activity of the Cambridge School which has long been the admiration of the country. The value of initiative and co-operative effort is well illustrated by the part played by the Analytical Society in the reform of University mathematics. For over a quarter of a century after 1860 the agitation for the reform of school teaching was centred, firstly, in the British Association for the Advancement of Science; secondly, in the Association for the Improvement of Geometrical Teaching. The former is too well known to need further comment; but a few words about the latter may not be superfluous. The Association for the Improvement of Geometrical Teaching was founded in 1871, with the late Dr. T. A. Hirst, F.R.S., as first president. In 1881 it widened its basis so as to include all branches of elementary mathematics; and eventually, in 1894, adopted the name of the Mathematical Association. Its original members were either actively engaged or much interested in the teaching of mathematics in schools; and this characteristic of its members still holds good. From the beginning the public schools have had distinguished representatives; and it is impossible to describe the present position of mathematics in the great English secondary schools, and the part played by the body of teachers engaged therein, without referring in more or less detail to the work of this association. Since its foundation it has enlisted the support and been honoured with the devoted service of the most able and far-seeing mathematical teachers; moreover, from its now somewhat lengthy line of presidents, it has had, through a series of annual addresses, advice and encouragement from many who fill so successfully the various professorships in the ancient Universities. The teaching of mathematics in schools has developed in a way, common to so many English institutions, by a process of growth from a somewhat obscure origin; it exhibits the defects, if not the virtues, of a lack of system. No one of recognised authority has fully stated the claim of mathematics to its just place in general education; nor has the public been warned in clear statements of the danger to growing youth of a too exclusive application to the perplexing but often fascinating mysteries of its lines and symbols. Both these issues present points of difficulty, if not of danger, in these democratic days, when the control of education lies in the hands of a majority gathered in the turmoil of a general election. But though little has been done towards the solution of such general questions, there have been changes in the last few years which have excited more than passing interest. That the modern schoolboy learns no Euclid, that the name is fast becoming buried in a forgotten past, appears strange to the last generation, to whom Euclid was one of those bitter experiences, the pain of which memory so kindly refuses to recall. The old verse : If there should be another flood, Were the old world to be submerged, This book would still be dry. has now lost its raison d'être; and modern youth has not yet had time to fashion a successor. But the final passing away of Euclid's Elements of Geometry, a book which has held sway for nearly 2,200 years, is an event rare enough to deserve a more serious treatment. a England was the last great country of the world to discard Euclid in favour of some more suitable text-book. Though all that was said in praise of Euclid could be admitted-that it had been the encouragement and guide of scientific thought throughout so many centuries, that it had been a reference book of thoroughly reliable knowledge, and standard of form which scientific writing might justly emulate but seldom excel-the fact remained that the greatness of Euclid had proved no blessing; the text was treated with idolatry, and the slightest deviation of phrase visited on the head of the unhappy schoolboy as a crime of sacrilege. No atmosphere could have been more foreign to the proper requirements of true education. The outcry against the slavish adherence to a single book was heard almost without intermission for more than forty years. The Geometrical Association made repeated but ineffectual attempts to persuade University authorities to permit departure from from Euclid's sequence. Strong opposition was encountered in unexpected quarters. De Morgan in a trenchant article showed up the deficiencies of some early suggestions; Lewis Carroll brought the satire of his unrivalled pen to discredit the growing movement for reform; while Cayley openly avowed his preference for Euclid without the slightest change. The British Association, however, through its committee specially appointed "to "to consider consider the possibility of improving the methods of instruction in elementary geometry," as a whole favoured the proposed reforms; but, conscious of its inability to enforce any decision against public opinion, recommended no immediate policy. The growth of opinion amongst the educated public was doubtedly stimulated by the presidential addresses to Section A, of which those of Sylvester, Smith, Henrici, and Chrystal may be specially mentioned. But, despite the most unstinted efforts, the results were trifling; the movement flagged; and Euclid appeared for the moment to be enshrined more firmly than before. un But new forces were discovered in unexpected quarters. The development of engineering generally, and particularly of the electrical branches, necessitated the training of a body of students, as versatile in mind, as quick in action and decision. A knowledge of mathematics beyond the range of Euclid was only one of the conditions of the required intelligence; fertility of invention was a sine qua non. The effect of Euclid was disastrous when tried with such students; and newer methods, involving trigonometry, analytical geometry, and the calculus, became the leading feature of the mathematical work in the most efficient engineering schools. At length the failure of scientific foresight at the time of the Boer war, both in the Army abroad and in the nation at home, reacted on public emotion, and seemed to provide that opportunity for success without which counsels of perfection are usually given in vain. This later feeling was ably |