alphabet to the mathematics through the senses-or in forming what Kant calls intuition; and which had been long adopted by the French revolutionary writers, to say nothing of Rousseau and Le Clerk, who have both recommended it; but, in carrying out this principle he fell into an error far greater, and fulfilled the truth of the adage

"Incidit in Scyllam, qui vult vitare Charybdin,"

and forgot the necessity of positive knowledge, as the material for thought, and for practical and future use. Hence the pupils of his establishment were dismissed with intellectual powers tolerably sharpened, but without the stores of knowledge important for immediate use. Well qualified for reasoning or argumentation rather, but not prepared to apply it to the business of common life. Taught, in fact, so clearly as not to see at all, and their senses so refined, as to be without common

sense.

A second leading principle was to "begin from within," and to advance outwardly-which his disciples alone understoodand then to advance gradually, step by step, from the simplest beginning to the more compound stages, without omitting a single intermediate step. Having once fixed upon a point of departure, a regular series of propositions succeeded with positive proofs for every combination; and thus geometry and drawing commenced with a point, architecture with a line, natural history with a plant, an insect, or a stone; geography with the surrounding scenery, and language with a single word. Of the soundness of this principle in a general point of view, and taken with due limitations, no one will doubt; but to judge of its propriety when applied in the rigorous manner with which Pestalozzi has adhered to it, let us look to its effects on his own pupils under his own management. The commissioners sent to examine his schools reported, that although a fair proportion of his pupils were in their sixteenth year, and had been occupied from the time of their admission ten hours per day, yet none of the branches of study were carried to any height, and some did not proceed beyond the elements. The belles-lettres were but barely entered upon, geometry extended to the mensuration of superfices, arithmetic of no practical application, geography advanced only half towards its intended limits, singing and drawing stopped at an equally low point, and religious instruc tion, paused upon the natural evidences as afforded by man's nature, as it was called, of the Supreme Being, and thus did not a rive at the first principles of Christianity.

Pestalozzi also commenced mathematical studies too early, attached too much importance to them, and devoted a portion of time to them which did not allow a reasonable attention to other studies, and which prevented a regular and harmonious cultivation of the other powers.

His method of instruction too, was defective on one important point. Simplification was carried too far, and continued too long. The mind was so accustomed to receive knowledge divided into its most simple elements and smallest portions, such infinitesimal homeopathic drops and grains, that it was not prepared to embrace complicated ideas, or to make those rapid strides in investigation and conclusion, which is one of the most important results of a sound education, and which indicates the most valuable kind of mental vigour, both for scientific purposes, and for practical life.

Analysis was also adopted by Pestalozzi where the direct contrary ought to have been employed, so as not to divide that into a multiplicity of successive operations which nature gives directly in a single lesson. The method of analysis, even as lately employed by some modern professors, subjects a child to the exercise of the understanding which is most useful when the end cannot be obtained without this labour, but which is entirely superfluous, as the same point can be equally well gained in a manner more immediate and direct. There is also a danger in separating things associated in nature; it is not less inconvenient to divide what ought to be united than to confound what ought to be kept distinct; generally speaking, it would appear that the analytical method is advantageous for the sciences and for the arts of industry when the mind is matured, but not at all suitable for teaching the first principles of things before the mind is formed, or while the understanding is without strength. Thus Pestalozzi, in aiming to be the most natural, became the most artificial of teachers.

The same principle, radically defective in most of its intellectual applications, was positively absurd and dangerous, as we have already observed, in morals He attempted to draw from the minds of children, before they had stores of knowledge, a principle carried to ridiculous excess in some of our model infant schools. He seemed to forget that the most elevated subjects were to be taught by "authority," as was the practice of the Great Teacher. He attempted to draw too much from the minds of his pupils those great truths of religion and the spiritual world which can only be acquired by Revelation; and Providence was against him, and his fundamental error was shown to be such by the unhappy experience of his own institution.

We have been thus free in animadversions at this time upon Pestalozzi and his defects, because we have remarked with regret the exertions of a clique of the semi-materio semi-spirituo empyrics busy in infusing the idea, or the method, into our public and private schools, and because we believe that such a method, although perhaps begun in arithmetic (seemingly harmless) may end in religion, and thus undermine the base of our common faith not to say anything of the danger with which our most venerable institutions may be threatened, and the confusion which may be thereby produced in education itself.

THE RELATIONS OF NUMBER.

The powers of numbers, and their relation to each other, have been in a variety of ways demonstrated; but rarely indeed with any important practical application: we have ingenious theories of the wondrous powers of the number 9, and a variety of arithmetical legerdemain is abroad, which appears to the curious very singular and astonishing. Napier's bones or rods afford some good illustrations of the multiplying powers; but there appears to have been no instance of the successful application of the "occult powers of numbers" till the invention of the "Arithmetical Frames," by Mr. Martin, which are, without question, applied to a use the most important and extensive. But when we come to make an examination of these, we are unable to ascertain, except in one or two cases, the principles upon which they are constructed. In these frames we have embodied a system of arrangement which carries out, ad infinitum, practical exhibitions of all the elementary rules, not singly only, but also in every variety of combination which the ten digits will make, affording demonstrable proofs of the correctness or incorrectness of every figure; at the same time none but the teacher who has been previously informed of the mode of detecting error, can by any possibility be informed of it. A dozen exercises of fifteen or sixteen figures each, may be worked in one rule only, or through the whole four rules, and be checked by the master at a mere glance, while those exercises may be varied to the extent of many thousands millions of times, and be proved by the same mode and with the same facility. It has often occurred to mathematicians, that a series of numbers might by some possibility be arranged, so as to produce uniform and known results in an almost infinite series; but this suspected power of the arrangement of numbers has never been shown, excepting in a few cases of particular numbers; and even these have not been applied to any practical purpose, excepting by Patrick Whytock, to whom we shall hereafter refer. But this arrangement which is founded on the peculiar properties of certain decimal fractions is defective, as it only refers to the simple rules, whereas the arithmetical frames or tablets constructed by Mr. Martin, comprise also the compound rules; and this appears most extraordinary; for there cannot well be worse decimal relations, supposing they are constructed on this principle, than those of the numbers 4, 12, and 20, which form the integral parts of our common currency; but Mr. Martin has arranged and can apply, if necessary, his principle through all the weights and measures, affording an infinite variety of examples, whose solutions bear such a relation to their propositions, that their correctness or incorrectness is immediately discoverable by him who has learnt the mode of discovery; and which may be acquired, by any one conversant with addition and multiplication,

in a few minutes. Nor is this all, for the frames are so arranged that the smallest as well as the largest examples may be given; while the working of the examples of one rule gives examples in another, and the working again of these, examples in a third, and so on-proving the correctness of each, even to the pupils themselves, and pointing out error; at the same time that the master has a counter check, which he can apply in a moment to a whole morning's set of exercises. Such a plan, where a large number of boys are to be taught, as in national schools, must be of incalculable advantage; and even in private schools must afford great assistance to teachers, from the variety of examples presented, and the ease with which their answers may be ascertained.

Of the kind of calculation, or of the application of the powers of numbers to the construction of these frames it is difficult to speak; but we may now refer to a system of numbers invented by Patrick Whytock. About the year 1820, a problem was proposed for solution in the Mathematical Repository, requiring 6 digits a, b, c, d, e, f, such that their products by 1. 2. 3. 4. 5. and 6, shall contain among them the following arrangements -viz.:

In this problem was contained the leading principle of Mr. Whytock's arrangement; but which, however, was quite unknown to him at that time. The construction of his diagrams. are, therefore, founded on the peculiar properties of certain decimal fractions, to which he gives the name of perfect quantities. In reducing vulgar fractions to decimals, every decimal fraction is comprised in one of these two general forms:

1st. It is finite or terminate, the decimal expression concluding in one or more digits, leaving no remainder.

2nd. It is infinite or interminate, because in continuing the division by the denominator of the vulgar fraction to any given length, a remainder will still be found; it is only with this last genus of fractions that we have concern, and it will be found to consist of the two following species:

1st. Repeaters, which either commence or come to issue in the perpetual recurrence of the same remainder and same quotient figure.

2nd. Circulators, which either at first, or after some steps of the division, issue in the constant rotation of a plurality of figures in regular order.

Circulating decimals consist of two principal varieties, upon the latter of which Mr. Whytock's diagrams are founded.

The first variety consists of all the circulators, the extent of whose rotation of figures consists of fewer steps than half the number of units contained in the denomination of the equivalent vulgar fraction. Thus the decimal of the 13ths is expressed by a rotation of six places, being one half 13 minus 1. The rotation of 11ths consists of only two places or figures, being one-fiftof 11 minus 1, and so of many others.

The second variety of circulating decimals arises from certain denominators, which, whatever numerator we assume, always give out the same rotation of places or figures in the decimal expression. The number of rotatory figures is uniformly 1, minus the units contained in the denominator of the equivalent vulgar fraction, this being the maximum of places, which for reasons that will appear, it can never exceed. To these decimal expressions are confined the term Perfect Quantities.

Though there are many denominators that give out decimal expressions assuming the form of perfect quantities, yet the greater part are so inconveniently large, that the rotation of their figures, taking so many places, becomes quite overpowering to the memory, and they are, therefore, rejected. The four smallest, and therefore the most eligible, are the denominators .7 .17 .19 and 23, which therefore give out decimal expressions of 6. 16 18 and 22 places respectively.

To place this in the clearest light, let us trace the steps by which 1-7th is reduced to a decimal expression

Quotient or

On inspecting this operation, 7.)1.0/1.42857 perfect quantity. it will, in the first place, be seen 132647 Remainders or that we started with the numeRadix numbers. rator or remainder 1, and after 6 steps of continuous multiplying by 10 and dividing by 7, we arrived again at the remainder 1, with which we commenced. The rotation is, therefore, complete ; for, were we to proceed further, we should obtain a repetition of the same rotation; that is to say, the same quotient figures, and the same remainder in regular order.

3.0

28

2.0

14

6.0

56

40

35

5.0

49

1