Unquestionably, much more will be attained by instruction in the mathematics and the natural sciences than now is, when more natural methods are adopted. One point of importance here is that young students should not be spoiled by premature abstraction, but should be made acquainted with their material from living pictures of it before they are made to work with it by purely ratiocinative methods. A good stock of geometrical experience could be obtained, for example, from geometrical drawing and from the practical construction of models. In the place of the unfruitful method of Euclid, which is only fit for special, restricted uses, a broader and more conscious method must be adopted, as Hankel has pointed out.[124] Then, if, on reviewing geometry, and after it presents no substantial difficulties, the more general points of view, the principles of scientific method are placed in relief and brought to consciousness, as Von Nagel,[125] J. K. Becker,[126] Mann,[127] and others have well done, fruitful results will be surely attained. In the same way, the subject-matter of the natural sciences should be made familiar by pictures and experiment before a profounder and reasoned grasp of these subjects is attempted. Here the emphasis of the more general points of view is to be postponed. Before my present audience it would be superfluous for me to contend further that mathematics and natural science are justified constituents of a sound education,--a claim that even philologists, after some resistance, have conceded. Here I may count upon assent when I say that mathematics and the natural sciences pursued alone as means of instruction yield a richer education in matter and form, a more general education, an education better adapted to the needs and spirit of the time,--than the philological branches pursued alone would yield. But how shall this idea be realised in the curricula of our intermediate educational institutions? It is unquestionable in my mind that the German _Realschulen_ and _Realgymnasien_, where the exclusive classical course is for the most part replaced by mathematics, science, and modern languages, give the _average_ man a more timely education than the gymnasium proper, although they are not yet regarded as fit preparatory schools for future theologians and professional philologists. The German gymnasiums are too one-sided. With these the first changes are to be made; of these alone we shall speak here. Possibly a _single_ preparatory school, suitably planned, might serve all purposes.