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wholly artificial and accidental way in which our system of weights and measures has originated should, when the proper time comes, be explained, and a comparison be made with some other system, especially the French Système Métrique. Generally it may be said that when you find yourself confronted with any arithmetical devices or terminology which are arbitrary in their character, you will do well to shew their arbitrariness by comparing

them with some others which are equally possible. Illustra- The first occasion comes when you explain the decition of the mal character of our common arithmetic, the device of method of distinguishing the meaning of the various multiples of notation.

ten and of the powers of ten, by their places and nearness to the unit; and the use of the cipher or nought (.). Here an appeal to some visible or tangible illustration will help you much. I take from an ingenious French book' an example of such an appeal.

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Here you observe small balls or marbles are used to represent units, bags containing ten of them to represent tens, boxes containing ten such bags to represent hundreds, and baskets containing ten boxes each represent thousands. When this has been shewn, you may further illustrate the nature of our notation by an addition sum: thus,

1 L'Arithmétique du grand-papa; histoire de deux petits marchands de pommes, par Jean Macé. Paris, Collection Hetzel.

Illustrations of the decimal notation.


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You require in succession that the numeration of each line should be explained

orally, you call special attenGAG000

tion to the need and special 4 5 3 6

use of the o in the second 56 008

line. It is seen that the first 000

column makes 33, and that 8 7 0

of them 30 may be included in 3 bags, and 3 remain.

The addition of the next 8 9

line gives 30, and shews the

need of a device for mark2 4 8 ing the vacant place, and

shewing that there are no 000

odd tens. The 26 hundreds ] 3 7

are then shewn to consist of 2 baskets full contain

ing 10 boxes each, and of 6 4 2 3

boxes or hundreds remaining. These two baskets added to the four baskets represent six thousands.

Thus the fundamental parts of our system of nota

tion--the device of place, 6 6 0 3

the counting by tens, the use

of the cipher, and the need of carrying, are all made clear to the eye and to the understanding of your pupil.

Many other forms of visible illustration have been devised, but it will be far better for you to exercise your own ingenuity in inventing them. Only bear in mind the rule of action already urged upon you. When

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Scalcs of notation other than decimal.

your box of cubes, your abacus, your number pictures, your diagrams representing collections of tens, have succeeded in making the subject intelligible; have the courage to cast them aside. Arithmetic is an abstract science, and the sooner scholars can see its truths in a pure and abstract form, the better. It is not an uncommon fault among Pestalozzian teachers to employ what are sometimes called intuitional methods, long after they have served their purpose, and when the pupil is quite ready to deal intelligently with abstract rules.

One very effective way of making the decimal notation clear is to assume some other number than ten as the possible base of a system of notation, and to invite the scholars to consider with you how numbers would have been represented on that system. It may be shewn that as a system founded on tens requires 9 digits and a cipher, so a quaternary system would have required three digits only, an undenary would have required one more digit than we use, say x; and that a binary scheme of notation applicable to the highest numbers would have been possible with one digit and a cipher only, since all large numbers would then have been gathered into twos and powers of two, instead of into tens and powers of ten.

By questions and suggestions you and your scholars come to frame on the black board some such table as this:

Decimal scale. Scale of two, Scale of six, Scale of eleven.

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13 14 15 16




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17 18 19 20



on the

13 23


15 25

16 30

17 31

13 32

19 A few easy sums to be worked out in numbers arranged on these scales, and afterwards verified by conversion into ordinary numbers will do much to clear the mind of the pupil as to the wholly artificial character of the decimal notation.

When you come to Weights and Measures and before Lesson requiring tables to be learned by heart, it is well as I

Système have said to give a short historical lesson shewing how our métrique. system grew up. The fact that we want fixed units of length, of weight, and of capacity to serve as the basis of all calculation; and the curious fact that nature does not supply by any single object a determinate and unalterable unit of any one of them, will partly account for the queer and irregular way in which we have from time to time based our calculations on grains of barley, on the vibrations of the pendulum, or the length of Henry I.'s arm. With a good diagram, such as is in use in all the French schools, it may then be shewn how the unit of length, the Mètre, which forms the base of the metric system is obtained from the measurement of a definite part of the earth's meridian ; how this unit squared gives the unit of surface, the Are, how the same

unit cubed gives the units both of magnitude and of capacity, the Litre and the Stere; how a given bulk so measured of distilled water gives the unit of weight, the Gramme; how a certain weight of silver gives the unit of value, the Franc; and how all these units by a simple nomenclature, are subject to decimal multiplication and sub-division. It is only when a simple and scientific system like this is seen in all its details—and the whole of it may easily be explained and learned in one halfhour's lesson that the real nature of the confusion and anomalies of our own system of compound arithmetic

comes into clear light. All rules Every rule you teach should be first of all made the should be

subject of an oral lesson and demonstration. The demon. strated method of experiment and induction, will often enable are learned you to arrive at the rule, and shew its necessity. One or prac.

of the first rules in which the difference between a tised.

skilled teacher, and a mere slave of routine becomes apparent, is the early rule of Subtraction. You want for example to take 479 from 853, and the method of socalled explanation is apt to be like this:

“9 from 3 I cannot; Borrow 10. 9 from 13 853

leaves 4. Set down 4. 479

Carry 1 to the 7. 7 and i are 8; 8 from 374

5, I cannot; borrow 10, 8 from 15 leaves 7. Set down 7.

“ Carry 1 to the 4. 4 and I are 5. 5 from 8 leaves 3. Set down 3." ow, of course, if the object is to get the right answer that object is fulfilled, for 374 is undoubtedly correct. But as an exercise in intelligence I hope you see that this is utterly worthless. The word "borrow' has been put into the children's mouths, but whence the ten is

Subtrace tion.

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