PHYSICAL ARITHMETIC.*

THERE is a story told of the late Professor de Morgan that, after examining a school in algebra, and even trigonometry, he asked the head class what they understood by the number seven, to which he got no reply, and that thereupon he recommended -them to begin their mathematical studies over again. It is cer- tainly not too much to say, that no children who had received their first notions of number by the method used by Messrs. Sonnen- schein and Nesbitt would have any difficulty in replying to such a question most satisfactorily.

This book, at present, consists of two parts, to be used by the teacher, the exercises of each part being printed separately for the use of the pupils. The four little volumes are accompanied by three most toylike-looking boxes, which contain little cubes,— staves (just like the longer pieces in a child's box of bricks), squares, and framework, with which the teacher is enabled to give a child a physical idea of any number from one to a million.

Before describing the manner of using these cubes, &c., we 'must compare the system, most favourably, with another of some- what the same kind, in which materials are used to represent the tens and hundreds, &c., but which materials have no physically numerical connection with one another ; and by it one is rather reminded of Major Jackman's teaching of Jemmy Lirriper, where the child, after adding up three saucepans, an Italian iron, a hand- bell, a toasting-fork, a nutmeg-grater, four pot-lids, a spice-box, two egg-cups, and a chopping-board, answers " fifteen, put down the five, and carry the chopping-board." In Messrs. Sonnenschein and Nesbitt's method, their physical ten is really ten of their physical units ; the hundred, a hundred units; and even the million, really a million units. One of the boxes contains one hundred small cubes of wood—two of their sides being coloured black—and with these cubes the child is taught the names of numbers, from one to a hundred. He is taught to arrange them in rectangles, and when possible in squares, thereby learning the different combina- tions of numbers.

It seems a pity that the term prime " should not be introduced even at this early stage, as nothing can explain the term better to a child's mind than the simple fact that he cannot place a prime number of cubes into a rectangle. The word " square " is introduced and explained, but not soon enough ; and even children might be taught the possibility of piling up twenty-seven of the cubes into a solid of the same shape as the little cube itself, and so get the notion of a cube number. The other side of this box contains ten " staves," as they are called, which are long pieces of wood, containing exactly ten cubes ; and one square plate, con- taining ten staves ; these are coloured black and white alternately, for the purpose of easily seeing that they are physically exact in their representation. These are introduced as soon as the pupil bas got the notion of ten cubes and the meaning of the arbitrary names one, two, to ten. But though all the numbers up to 100 are represented and discussed with these staves for the tens and cubes for the ones, yet they are dealt with also with the cubes alone, with- out using the staves, as it is manifest it would be quite impossible with three indivisible tens and six ones to show that thirty-six was a square number, or that it was four nines. The second box con- tains ten of these hundred-plates, and is, therefore, a perfect cube ; but it certainly seems a great pity that the apparatus does not contain a solid cube, with its sides marked, to show the smaller cubes. If the apparatus did contain such a piece, there would be all the requisites for teaching cube-root, as almost universally taught in America, viz., one large cube, one small one, three plates, and three staves, as these can be built up into a perfect cube, each side of which contains eleven of the small cubes. Many teachers, feeling the ex- treme value of scales by which to teach thoroughly the principles * The A, B, C of Arithmetic. By Sonnenschein and Nesbitt. London: Whittaker and Co. of the common, or denary scale, would be sadly tempted to imitate Mr. Sonnenschein in his idea, and have staves and plates of other numbers manufactured. The fact that there are two kinds of division is strongly brought out in the book, and a new sym- bol is introduced, which means " is contained in." In ordinary arithmetic-, 643 =2 may mean, if 6 concrete quantities be divided into 3 equal parts, there will be two such concrete quantities in each part ; or it may mean that 6 concrete quantities contain 3 of such concrete quantities twice. But in the method under consideration, 64.3=2 can only mean the former operation, and the latter would be represented by 64.3=2. As some mathema- ticians use this symbol for "unequal to," it seems a mistake not to have invented an utterly unused symbol.

Subtraction, of abstract numbers, is taught through the sub- traction of £ a. d., and the tens in the numbers of shillings are called half-sovereigns, and the tens, hundreds, &c., in the pounds are called ten, hundred, &c.,—pound-notes ; but the difficulty of borrowing one, or rather of carrying one, is overcome by marking the number from which the number has been borrowed, to remind the worker that he must read that number one less, thereby losing the opportunity of teaching and impressing one of the most charming little pieces of rigid reasoning that can come into a child's early education. When, instead of taking away one from the number, whence we borrowed one, we add on one to the number to be subtracted, an operation practically performed by every one who has learnt arithmetic a year, we really do that which, to thoroughly explain, requires the use of two or three axioms and as many distinct steps of proof.

Casting out nines is taught very early, and is insisted upon as a proof for both multiplication and division. A long method of casting out elevens is also referred to and half explained in the appendix of the second part. If learners knew how to cast out both nines and elevens, they would have a perfect test by which to examine their work in multiplication and division, as it is im- possible for a result to be wrong by nine and eleven at the same time.

The third box contains three pieces of framework, which frame- work, constructed of wood of the same breadth and thickness as the small cube, really shows the space that would be occupied by 10,000, 100,000, and 1,000,000 of the small cubes, that is, of the units. The two latter frameworks are jointed with hinges for the purpose of packing, and the million, though of some size when extended, can be put away into a box much smaller than that used for croquet.

The ten plates which represent a thousand, being contained in a cubical box, are called by the technical name of "a box "; but in the second part, where the numbers up to a million are con- sidered, they, as a whole, have a double name, viz., " box" as to the small cube, and CUBE as to the higher numbers, thereby impressing most thoroughly the important system of reading numbers by periods of three figures,—that is, that whereas a cube is the name of the right-hand figure of the three used to notate hundreds, tens, and units, CUBE is the name of the right-hand figure of the three used to notate the units of thousands or boxes. And thus CUBE and box imply the same quantity. According to the same system of notation, a million is called a BOX.

By making the Box our unit, it is evident that the PLATE, CUBE, &c., represent tenths, hundreds, &c., and that the original unit (the cube) is now a millionth ; and by this process, the teaching of terminating decimals is hinted at in the appendix. There is also something said about a recurring decimal whose period was less than seven figures, but we doubt considerably whether any one who did not previously understand decimals would be able to follow such an explanation. To give an example of the method suggested, let us take one in what is called short division. The question would be worked something like this :-

CUBES.

Bence. PLATES. STAVES. Box. Plates. Staves. Cubes.

2) 3 4 2 5 2 5 9 1 7 1 2 6 2 9

And one cube over.

and the reading with regard to the first one carried would simply involve such language as 1 box = 10 plates, as the pupil by means of the apparatus can see with his eyes, instead of saying 1 million = 10 hundreds of thousands, which he has to take on faith at the mouth of his teacher.

The amount of theory taught, almost imperceptibly, by this method is considerable ; but still that which is simply useful is aimed at, as has been already tried to be shown in the matter of subtraction. It will be an evil day for England, or any other country, when instruction is substituted for education, and in no science is this so true as it is in arithmetic, since arithmetic is the one single science requiring logical reasoning that is taught to all,

young and old, rich and poor, who pretend to any education. No one can doubt that it is very possible to make arithmetic a merely mechanical operation, and we are very much afraid, from testimony that we get from different directions, that to do other- wise is the exception. We cannot but hope that the adoption of Messrs. Sonnenschein and Nesbitt's method may, to some extent at least, oblige learners to understand what they are doing, on the one hand, and to enable them to talk about the principles and rules, on the other.

Before concluding, we must notice a cabalistic-looking penta- gon, enclosed in a circle, which appears on every book, box, &c., connected with the system, and which is, as we suppose, used as a trade-mark. The figure, however, appears in the book itself, and its use is explained. Every good teacher of arithmetic has made his younger pupils count by twos, threes, &c., up to nines, or even elevens and twelves. This pentagon is constructed for the purpoie of teaching boys to count by some constant quantity quickly, so as to ensure accuracy of addition. Its invention is certainly very ingenious. To describe it :—A pentagon inscribed in a circle ; within the angular points of the pentagon, reading in the direction of the hands of a, clock, are the numbers 1, 7, 3, 9, 5, and outside the middle points of its sides are the numbers, 4 (between the 1 and 7), 0, 6, 2, 8. This clock or counting-machine is used in the following way :—To count by threes, begin with any number, as 25, and read the numbers one after the other (going - as the hands of the clock), and (omitting the tens) the units will be suggested by the numbers on the circle. To count by sevens, we must go the other way. To count by sixes or fours, begin with any number, and read off every alternate figure clockwise or in the opposite direction respectively. To count by nines or ones (that is, ordinary counting), read off every third figure from where you begin, clockwise or non-clockwise respectively ; and for twos and eights, every fourth figure. Pro- vided this circle were marked by figures which did not differ successively by an even number or five, it is evident that any system of marking would produce exactly the same result, but practically there are only two such methods of marking the circle; as marking it by three in one direction is the. same as marking it by seven in the opposite direction. So, also, marking it by nine in the one direction would be marking it by one in the other direction. One can easily perceive that it is far better to use the former• method rather than the latter, as being less intelligible to the uninitiated and more useful, since successive additions of the lower numbers (except one and two) are performed by using the figures in closer proximity to one another.

. When we consider the price of those parts of the book in- tended to be circulated amongst the scholars (4d. a copy), one cannot help hoping that a large circulation may attend it, and that by its use teachers, no less than students, may profit, and learn something which they had not learned, or at any rate, realised before, of the first principles of our notation. And we cannot help trusting that the introduction of this and similar books may do something to raise the standard of our arithmetical teaching, which at present, we are bound to believe, according to the returns of examiners of all sorts and kinds, is far below what it ought to be and what it might be.