CHAPTER X.

In the same way that abstract terms of qualities are derived from a contemplation of real objects, are ideas of number also gained. A blind man has no idea of colour, having never seen any object of which it is a property, and his mind cannot receive such an impression until it be first enstamped upon the retina of his eye, and thence conveyed to the sensorium by the optic nerve. So must a child be mentally in the dark regarding number as a property of objects, unless his understanding has in some way or other been impressed by the fact in numbering them. There can be no reflections without something to be reflected from, namely, ideas of sensation, and there can be no sensations unless proceeding directly or indirectly from sensible objects. Ideas of sense are indeed almost material emanations themselves, but from the myriad hues they reflect by falling under the prismatic influence of the mind, the imagination can realise out of them scenes brighter than any presented by nature, and the judgment construct an artificial world more complicated, but no less real than the natural. Yet the foundation of these thoughts and fancies must all rest on the materialism of nature, else like the "baseless fabric of a vision" they will vanish at the test of reason. The law of gravitation would never have been discovered nor become the source of so many other discoveries and calculations, had not its in

fluence been seen in operation in some such familiar instances as the cohesion of dew-drops, or the falling of an apple. Such qualities or properties are only known by their effects on material substances; and if the latter had not been seen, the former would never have been dreamed of. In like manner the student of nature should not take for granted the existence of any such principle until he has abstracted it by his own observation. His faith will rest most securely upon the evidence of sense, and he will penetrate much farther into the unseen world of abstractions by standing upon the eminence of nature. The mind must lay the foundation of its thoughts deep in sense before it can raise a tower of observation high enough to obtain glimpses of things spiritual.

Thus, then, for a child to be set to count up fifty or a hundred, to add, subtract, multiply, and divide so many sounds and figures, without having first associated these with realities, is an attempt to climb without a ladder, or fly without wings. Unless found to be the names of the numbers of his fingers, balls, marbles, or other familiar objects, the shapes and sounds of the nine digits will be shapes and sounds alone, and their combinations on a slate, or even in mind, as aimless as a French puzzle. The arithmetic of tangible and visible objects should, therefore, be among the earliest and most frequent studies of a child from his entrance into school, as clear notions of number and quantity throw so much light upon other branches, and are so well calculated to train and methodize the mental faculties themselves. Its first steps,

however, should be of the most gradual and easy kind, and each new idea worked into the very constitution of the mind by repeated examples. The elements of number consist of but a very few leading ideas, which it is imperative should be clearly understood at the outset, that all subsequent combinations may be free from that

perplexity which is the necessary consequence of dealing with principles not arrived at by an inductive process of the mind itself. As the largest volume contains but the twenty-six letters of the alphabet, so the most complicated calculations have only the nine digits and a zero, and the most involved process of arithmetical reasoning is merely a ramification of units. The system of decimal progression, upon which so many of our calculations are based, most probably has been suggested by the number of fingers on the two hands, and the natural tendency to employ them in counting. The student, must, therefore, begin at the root of the science, thus, one finger, two fingers, three fingers; two fingers and three fingers are five fingers, and so on; these must be his first concrete ideas of arithmetic. The objects must be seen in combination, and ought to be given before the arbitrary sounds, one, two, three, &c. Marbles, buttons, apples, oranges, everything that can be seen in combination, may form media for impressing these elementary principles.

The numerical value of money should also be first proved by tangible evidence, and farthings, halfpence, pence, sixpences, and shillings, seen and handled while added and subtracted. Two halfpence in one hand placed beside three in the other, make the sum twopence halfpenny, and a halfpenny taken out of it leaves twopence. The superior value of silver to copper, and of gold to silver, should also be shown by a practical exchange. The sum must thus be seen to increase and diminish before an idea of addition and subtraction can be formed, and the idea must be obtained before there is any necessity for giving such names.

A very appropriate instrument for facilitating visible calculations of this nature, has been brought into use by Mr. Wilderspin, named the Arithmeticon. It is simply

à frame, with a certain number of wooden balls, painted alternately black and white, which move horizontally along the wires on which they are strung. The master combines and arranges these by moving them with his finger, while the children observe and name the different arrangements. Such an exercise gives a reality to the ideas of number which thus attach to palpable and distinct objects. While the eye rests upon the balls, the mind easily calculates the number of them, whereas, in absence of such a point d'appui, it cannot grasp the same amount of combination. The intellect has then nothing tangible to compute, and, strictly speaking, computes nothing; or, if forced to make the attempt, it must first suggest ideas of objects, and calculate these, which is a more complicated process. This principle of object calculation is also akin to that which necessity prompts à rude community to adopt. Hence we hear of one people using knotted strings as a calculating apparatus, another notched sticks, shells, and pebbles-calculi, from whence the term calculation is derived; and even among the Roman and Greek mathematicians there was in use an instrument very similar to the arithmeticon itself, called the Abacus, on which they cast up certain accounts. This instrument, of which Mr. Wilderspin seems to claim the merit of invention-or at least the Abacus-was in familiar use, even in Europe, until a recent period. The modern system of notation by the nine digits and a cipher, is believed to have been derived from the Indians. Through them it descended to the Arabians, and was introduced into Europe by the Moors about the beginning of the eleventh century.

The Roman Abacus was a board with parallel grooves placed perpendicularly, along which the balls or counters were moved. The simple value of each ball was one, but

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it had also a positional value, as our common digits have. In an Abacus with a certain number of upright grooves, the one to the right expressed units, that next to it tens, the next hundreds, the next thousands, and so on. For example, the number 31452 would be expressed by two balls on the groove nearest the right hand, five balls on the one next it, four on the next, one on the next, and three on the left hand groove, and any other conceivable number might thus be noted according to the different arrangement of the counters and number of the grooves. It is on the same principle as modern notation, by which different degrees of unity are expressed by different marks, and their values changed by position, and it might approach still nearer to this were differently sized or coloured balls employed to denote the different quantities of 1, 2, 3, 4, &c.

The Abacus was used by both Greeks and Romans, the latter having borrowed it from the former, and it was employed less frequently by the Greeks only because their system of notation by the alphabet was more perfect than the figurative notation of the Romans. The science was indeed but little cultivated among the latter people compared with what it was among the Greeks. Their minds were less adapted to abstract calculations, and hence the necessity of this more palpable mode of computation. This is, therefore, the same reason that renders such means better adapted to the capacities of children than to those more advanced in the science. appeals to the eye before the mind, and exhibits something calculable before they are required to calculate. The chief difference in point of form between the Arithmeticon and the Abacus is, that the balls move vertically in the latter, and in the former horizontally, and in this respect it more resembles an instrument

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