280 NUMBER IN VISIBLE OBJECTS. elements of addition, multiplication, and division, and become conversant with their nature by the repeated representation of the relations which they express, in visible objects, subtraction is to be exercised upon the same plan, as follows: The ten squares being put up together, the teacher takes away one of them, and asks: “If I take one from ten, how many remain? The child counts, finds nine, and answers: 'If you take one from ten, there remain nine.' The teacher then takes away a second square, and asks: 'One less than nine, how many? The child counts again, finds eight, and answers: •One less than nine are eight;' and so on to the end. “This exemplification of arithmetic is to be continued in successive exercises, and in the manner before described. For instance: “As soon as the addition of one series is gone through, the subtraction is to be made at the same rate, thus: having counted together one and two make three, and two make five, and two make seven, and so on up to twenty-one squares, the subtraction is made by taking away two squares at a time, and asking; two from twenty-one, how many are there left?' and so on. “The child has thus learned to ascertain the increase and diminution of number, when represented in real and moveable objects; the next step is to place the same successions before him in arithmetical tables, on which the numbers are represented by strokes or dots.” These tables have since fallen into disuse, and various sorts of apparatus have been substituted in their place; such, for instance, as the ball frame, well known in this country as part of the outfit of infant schools. For the sake of illustration, however, we add, on the other side, a representation of the original table* which, executed on a sufficiently large scale, might still form a useful implement on the wall of the schoolroom, if it was for no other purpose than to induce the habitual intuition of the first hundred numbers, in decimal series. • For specimens of the exercises to be performed by means of this numerical table, sce the “ Christian Monitor and Family Friend,” Part IV. p. 18. “These tables,” continues Pestalozzi, “ on which the child is still calculating in real numbers, answer the same purpose as my spelling-book for the exemplification of sound in words exhibited on the slate. When the child has been exercised in this intuitive method of calculation as far as these tables go, he will have acquired so complete a knowledge of the real properties and proportions of number, as will enable him to enter with the utmost facility upon the common abridged modes of calculating by ciphers. His mind is above confusion and trifling guesswork; his arithmetic is a rational process, not mere memory work, or mechanical routine; it is the result of a distinct and intuitive apprehension of number, and the source of perfectly clear ideas in the farther pursuit of that science. “ But the increase and diminution of things is not confined to the number of units; it includes the division of units into parts. This forms a new species of arithmetic, in which we find every unit capable of division and subdivision into an indefinite number of parts. “ In the course before described, a stroke representing the unit, was made the intuitive basis of instruction; and it is now necessary, for the new species of calculation just mentioned, to find out a figure which shall be divisible to an indefinite extent, and yet preserve its character in all its parts, so that every one of them may be considered as an independent unit, analogous to the whole; and that the child may have its fractional relation to the whole as clearly before his eyes, as the relation of three to one, by three distinct strokes. “ The only figure adapted to this purpose is the square. By means of it the diminution of each single part, and the proportionate increase of the number of parts by the continued division and subdivision of the unit may be made as intuitively evident as the ascending scale of numbers by the addition or multiplication of units. A fraction table has been prepared, counting eleven lines, with ten squares in each line. The squares of the first line are undivided, those of the second divided in halves, those of the third in thirds, &c. “This table is to be followed up by others, in which farther subdivisions are introduced in the following order. The squares which, in the first table, were divided into halves, are now divided into 2, 4, 6, 8, 10, 12, 14, 16, 18 20, parts; those in the third line into 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30. “Now as the alphabet of forms is chiefly founded upon the division of the square into its parts, and the above fractional tables serve to illustrate the same division in a variety of manners, the alphabet of forms, and that of fractions, prove in the end the same; and the child is thus naturally led to connect in his mind the elements of form with those of number, both explaining and supporting each other. “My method of arithmetic is therefore essentially founded upon the alphabet of forms, which was originally intended only for the purposes of measuring and drawing. "THE FUNDAMENTAL FORMULA” EXAMINED. 283 “By means of these fractional squares, the child acquires such an intuitive knowledge of the real proportions of the different fractions, that it is a very easy task, afterwards, to introduce him to the use of ciphers for fractional calculation. Experience has proved, that by my method they arrive at this part of arithmetic from three to four years earlier than by the usual mode of proceeding. And it may be said of this, as of the former course, that it sets the child above confusion and trifling guesswork, his knowledge of fractions being founded upon intuitive and clear ideas, which give him both a desire for truth and the power of discovering and realizing it in his mind." We have purposely abstained from interrupting the above extracts by stating our objections to that view of number upon which Pestalozzi's reasoning is founded. While we perfectly coincide with him, as regards the necessity of substituting an intuitive instruction in number, for the dull drudgery of “doing sums,” we entirely dissent from him, and we trust not without good reason, on those points which have reference to the nature of number, and the mode of presenting to the mind its gradual increase. Pestalozzi considers number only seriatim, and therefore considers all arithmetic as a mere enlargement or abridgment of the formula“one and one are two,"overlooking altogether the important fact, that this formula, which expresses the juxta-position of two objects, presupposes in the mind the idea of two; and in the same manner its enlargement in “one and one and one are three," presupposes the idea of three; for this simple reason, that it is impossible to conceive the operation of putting together, without having an idea of that which is put together, no more than it is possible to conceive the operation of building, without any idea of building-materials. The origin of number must not be sought in the repetition of units; because without the previous idea of number, the idea of repetition could not exist. And for the same reason, as we have shewn elsewhere, “ the usual way of teaching number," seriatim, by the repetition of units, “is a mere self-deception, inasmuch as it presupposes the knowledge which it pretends to impart. Thus, for instance, to teach the child what eight is, by counting one and one, and one and one, and one and 284 THE GENERIC POWER OF NUMBER. one, and one and one, supposes in the child the capacity of comprehending in one view these eight successive ones, a mental operation which is impossible without a previous knowledge of eight.” The question then arises: if the repetition of units be not the source from which the knowledge of numbers is to be derived, whence shall we obtain it? If “ one and one are two,” be not the fundamental formula, what is it? The answer to this question is given in what might appropriately be termed the generic power of number, or the power of every number,* to produce out of itself an indefinite series of numbers, in such order that the number attained by every operation is superior to those, the knowledge of which is requisite for performing the operation. The idea of two, for instance, suffices for the operation of taking two twos; the result of which is the number four. The same idea of two again suffices, strictly speaking, for the operation of taking two two-twos; and even a quibbler cannot trace in it the idea of a higher number than three; a number far inferior to the result of the operation, which is eight. Here it is quite evident, that from the idea of two, the mind is led to that of eight, whereas by the eightfold repetition of units the mind is, in fact, only led from the idea of eight, to the idea of eight, that is to say, left exactly where it was before. Instead of the formula “one and one are two,” or in other words, “two ones are two,” or “two is two;" we have the formula axa=b. The next question which presents itself, is: But whence do we get the a, which by its generic power produces the b; where does the idea two come from? Without entering upon “the metaphysics of number,” we have a right to assume the fact, that the mind has a primitive idea of two; since that fact is clearly established by the very possibility of • The term number with us excludes the idea of one, which is the opposite of all number; whilst none is the negation both of unit and number. |