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For example, the leading dates in chronology might be embodied in a variety of questions. A few sums and differences derived from the reigns of the English sovereigns, would be a collateral aid in stamping these on the memory; and might be the more effectual that it is not given as the essential stress of the exercise. Such simple examples in subtraction as how many years have elapsed since the Conquest, since the death of Charles I., since the Union of England and Scotland, the dates being either given in the question, or assumed to have been otherwise given--would help to impress these on the memory.

In a similar way, important Geographical numbers could be stamped on the recollection by being manipulated in a variety of questions. The dimensions, area, and population of the three kingdoms, the proportion of cultivated and uncultivated land, the population of the largest cities, the productions, trade, taxation of the country, all which become the subject of reference and the groundwork of reasonings in politics,-could receive an increased hold on the mind by their iteration in the Arithmetical sums.

The common weights and measures should be familiar to everyone; and these might be so wrapped up in exercises, that the pupil could not avoid taking note of them. The mere act of writing them a number of times on the slate, with a view to solving questions, would render it almost impossible to escape being struck by them. A most valuable datum in the ordinary contingencies of life is the relation of weight to bulk, given through the medium of water. A cubic foot of water weighs 62 lbs, and a gallon weighs 10 lbs.; these are data that no mind

should be without. If a few leading specific gravities cork, wood (of some of the commoner kinds), building stone, iron, lead, gold-were added, there would be the means of readily arriving at many interesting facts.

Frequent reference might be made to foreign moneys and scales of weights and measures, as of almost universal interest; and especially to the decimal system of foreign countries. All this could be done in questions.

Next, I might cite the scales of the thermometer. For want of knowing these, the statements of temperature are, in the majority of cases, unintelligible: the Centigrade and Reaumur being now more in use than Fahrenheit.

The comparative strength in alcohol of spirits, malt liquors, and wines might be incidentally remembered by being involved in exercises of computation.

We might range over the various sciences for interesting data. Thus, in Astronomy, such leading numbers as the sun's distance, and magnitude, the moon's distance, the distances of the greater planets from the sun, the periods of revolution, and rotation-could be chosen, as not unlikely to make an impression through their incidental use in questions. Such is the so-called perversity of human nature, that the mind would often take a delight in dwelling upon these casual figures, because to remember them was not a part of the task. And further, by a general law of the mind, if a question for some reason or other has engaged the attention in an unusual degree, the memory will receive the indelible stamp of all its parts and accompaniments.

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The Higher Mathematics.

The Methods in Geometry, Algebra, and the Higher Mathematics, are the methods for impressing abstract and symbolical notions and principles. The understanding must now accompany the work throughout; the stage of routine manipulation, worked up to automatic dexterity, is left behind. To a certain extent, the mechanical processes may enter into Algebra; the pupil may receive certain instructions, and, without understanding the reasons, perform the simpler operations of adding subtracting, multiplying, as in Arithmetic, but in the resolution of equations, the principles must be understood.

To advance at a moderate, steady, pace, to see each step well familiarized, before entering on the next,-are the rules of all difficult acquisition, from the beginning of time till the end. The earlier parts of such subjects as Geometry and Algebra need the longest iteration: the progress should be at an accelerating rate. The higher Mathematics should not be commenced with immature or incapable minds.

The fundamental axioms of Mathematics, including Arithmetic, are brought forward exclusively in connection with Geometry, which has always been the purest type of a demonstrative science. This makes Geometry now, what it was to Plato, the portal of the sciences. The scheme of formal demonstration, proceeding from Definitions, Axioms, and Postulates, is first unfolded in Euclid; hardly anything corresponding is found in the usual modes of commencing Arithmetic

and Algebra. No one knows Geometry, in the proper scientific way, without comprehending the precise drift of all these preparatory elements, as well as the nature of consecutive demonstration. But there is a concrete handling of Geometry precisely analogous to the Pestalozzi system for Arithmetic, and having the same effect. It familiarizes the mind with the figures or diagrams, enables sides and angles to be understood, and gives a mode of experimental proof of some of the leading theorems that is really conclusive in itself, although not the sort of proof that belongs to the science. That the sum of the three angles of a triangle is two right angles, can be proved in the concrete; just as we can prove that six times four is twenty-four. It would be a mistake, however, to suppose that the experimental proof of propositions by cutting and folding cards is either Geometry, or a preparation for entering on the march of Euclid, or of any other system of Geometry conceived in the scientific form. When we come to the real business of Geometry, we have quite another sort of work before us; we are refused appeals to the senses or the concrete, and must establish each property as a consequence of some previous property, starting first of all from the definitions and axioms, which are to be conceived as purely representative abstractions. The serious work of the teacher lies in following this plan, and in using his concrete instances only in aid of the abstractions as they are given in the definitions. 'A line is length without breadth.' Examples of lines in the concrete may be given with this definition, but what the pupil must learn to understand (with no small difficulty) is, that every concrete line is false to the definition; and that

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the mental operation to be performed is thinking of the length, and neglecting, or leaving out of account, the breadth. Next the straight line is taken up in a fashion that leaves the concrete far behind. No doubt, a little concrete illustration is useful as a help to the definition—'lines cannot coincide in two points without coinciding altogether;' but the notion must thenceforth be grasped as an abstraction, and conjoined with other abstractions in chains of demonstration. So with the other definitions. So also with the axioms: a few concrete examples are provided at the outset, and their support is thenceforth withdrawn; the mind must hold by the abstract conceptions, as embodied partly in diagrams, and partly in general language; and must be ready to draw inferences from clusters of propositions given in this naked form. The concrete preparation soon exhausts its efficacy; and the pupil has to depend upon the power to retain and to accumulate abstractions for the purposes of the work in hand.

The aid to be afforded by the teacher in mastering the demonstrations of Geometry, consists chiefly in making the essential steps prominent among the long-winded repetitions of subordinate matters. The propositions as given in Euclid could be simplified by giving a larger type to the main statements; and the living voice of the teacher can still further contribute to put the stress of attention where it is most required, and withdraw it from the tedious repetitions.

Algebra is better learnt after Geometry, inasmuch as it works in part by demonstration or deduction from principles, for which by far the best commencement is Geometry. It has its own speciality, which consists in