10. Figures, then, have a certain fixed value in themselves, distinct from their relative value.. 2 is always two; but whether it be two units, two tens, two hundreds, or two thousands, depends on its position with regard to other figures with which it may be in connexion. Let the pupil bear in mind, that the last figure of a series of figures, is always so many unitsthe last but one, so many tens-the last but two, so many hundreds-the last but three, so many thousands, and so on. 11. The following table will render this plain :—

2,

~ Units.

2,

The number expressed by these seven twos is read thus: Two millions, two hundred and twenty-two thousands, two hundred and twenty-two.

12. The student will perceive that there is a great difference in the value between the two at the beginning and the two at the end of the above table. In both cases the figure is the same, but how different the value! one two is a million times the value of the other, and this increase in value is acquired merely by a change in its place, by advancing it six paces to the left hand. The student should reflect on this.

13. The cypher, though it expresses no value in itself, performs the important office of giving value to figures with which it may be in connexion. Without its assistance we should be unable to express any even number of tens, hundreds, or thousands. Suppose we should want to express the number ten, for example. How is this to be done? It cannot be done by any one of the significant figures, or by any combination of these figures. But let us call to our aid a cypher, and the matter is done at once. Ten, is expressed by the figure 1 and a cypher, thus, 10; the cypher being in the place of units, advances the to the place of tens. Twenty is expressed thus, 20-thirty thus, 30-one hundred thus, 100-one hundred and fifty thus, 150-one hundred and five thus, 105. In the last example, observe the 1 is in the place of hundreds, the 0 in the place of tens, and the 5 in the place of units.

*Series means a succession of figures, numbers, or of things in regular order.

Arithmetic; and, secondly, of the calculation of numbers, written or conceived. The first of these divisions will form the subject of the following chapter.

CHAPTER II.

NOTATION AND NUMERATION.

6. NOTATION* is the art of expressing numbers in figures. 7. NUMERATIONt is the art of reading or enumerating numbers when expressed in figures.

8. For the purpose of expressing number, we have ten signs or characters. By these we are enabled to express all numbers, however large, or however small.

9. These characters are as follow:

1 One 3 Three 5 Five | 7 Seven | 9 Nine

8 Eight 0 Cypher

Of these characters, nine are called significant§ figures, to distinguish them from nought, or cypher. These significant figures have a certain fixed value in themselves, besides a relative value, which they acquire from position. Any of these figures, when placed alone, is of a certain fixed value, representing only the number which is assigned to it in the above table. The figure 6, for example, when so placed, means only six, and is called a simple number; but join to it another figure, 2, for instance, and its value becomes conditional, depending on its relative position with regard to the annexed figure. If the 2 be placed on the left of it, thus, 26, the value of the 6 remains unchanged; it expresses only six, and the two figures taken together express twenty-six. But change their position; place the 6 on the left of the 2, thus, 62, and the 6 becomes increased in value ten-fold; it now expresses sixty, and the 2, two, and the two figures taken together express sixty-two, and form a compound|| number. Annex another

figure, thus, 620, and the 6 now expresses six hundred, and the 2, twenty. Both figures have acquired a still further augmentation of value-they are increased again ten times, and would continue to increase in the same ratio,** by every addition of a figure to the right hand.

*Notation, from the Latin not-a, a mark.

+ Numeration, from the Latio numer-us, a number.

Character, from a Greek word which signifies a mark or stamp. Significant; expressive, important, from the Latin sign-um, a mark, a sign, a seal.

Compound, formed by the union of two or more; mingled; from com, with or together, and pon-o, to put, to place.

¶ dugmentation, the act of increasing.

** Ratio meaus in the same rate, proportion, or by the same difference.

10. Figures, then, have a certain fixed value in themselves, distinct from their relative value.. 2 is always two; but whether it be two units, two tens, two hundreds, or two thousands, depends on its position with regard to other figures with which it may be in connexion. Let the pupil bear in mind, that the last figure of a series of figures, is always so many unitsthe last but one, so many tens-the last but two, so many hundreds-the last but three, so many thousands, and so on. 11. The following table will render this plain

The number expressed by these seven twos is read thus: Two millions, two hundred and twenty-two thousands, two hundred and twenty-two.

12. The student will perceive that there is a great difference in the value between the two at the beginning and the two at the end of the above table. In both cases the figure is the same, but how different the value! one two is a million times the value of the other, and this increase in value is acquired merely by a change in its place, by advancing it six paces to the left hand. The student should reflect on this.

13. The cypher, though it expresses no value in itself, performs the important office of giving value to figures with which it may be in connexion. Without its assistance we should be unable to express any even number of tens, hundreds, or thousands. Suppose we should want to express the number ten, for example. How is this to be done? It cannot be done by any one of the significant figures, or by any combination of these figures. But let us call to our aid a cypher, and the matter is done at once. Ten, is expressed by the figure 1 and a cypher, thus, 10; the cypher being in the place of units, advances the to the place of tens. Twenty is expressed thus, 20-thirty thus, 30-one hundred thus, 100-one hundred and fifty thus, 150-one hundred and five thus, 105. In the last example, observe the 1 is in the place of hundreds, the 0 in the place of tens, and the 5 in the place of units.

* Series means a succession of figures, numbers, or of things in regular order.

14. The following table will show how figures become increased in value, by every addition of a cypher to the right hand :

15. The student should carefully read the foregoing paragraphs six times over at least, and think as he reads, for without this the mere reading will do little. He should also, by a careful observation of the foregoing tables, inform himself how figures increase in value as they advance towards the left hand; and for this purpose, he should copy on his slate, in a neat manner, and with its explanations, the table in paragraph 11. Then beginning at the right hand 2, that is, at the units, he should erase* that figure and substitute some other in its place he should do the same with the tens figure, and with the others. He should do so repeatedly, and after every such alteration he should read the figures. By this means he will soon become acquainted with figures, and their use in expressing number, especially if, at the same time, he be taught the connexion of signs and objects.

16. Suppose he had a number of matches, and should count, and tie them up in bunches, ten in each bunch-then take these bunches of ten, and tie ten of them together, he would then have a bunch of a hundred. Ten bunches of a hundred would make a bunch of a thousand, and so on. Suppose the number of matches should be 1,214, and he should proceed to count and tie them up in bunches of ten in a bunch, of such bunches he would have 121 and four matches over;-if he should then take these bunches of ten and tie every ten of them together, he would have twelve bunches of a hundred, and one bunch of ten over;—if he should, lastly, tie ten of

*Erase; destroy, rub out, efface.

+ Especially; particularly, chiefly, principally.

these bunches of a hundred together, he would then have one bunch of a thousand, two bunches of a hundred each, one bunch of ten, and four single matches.

17. By this process he would acquire a perfect idea of the number 1,214,-an idea which would be fixed permanently in the mind by the operation of the senses, and not be forgotten so soon as the lesson is over, which is too frequently the case in the old mode of teaching. A lesson performed in this manner, showing the connexion of objects with their signs, I have no hesitation in pronouncing to be worth a hundred performances in mere abstract numbers. Let the lesson be repeated and varied, until the student clearly understands his subjectand each time he counts his matches, let him express their number in figures.

18. When a number consists of many figures, it is best to divide it into places of three figures each, by placing a comma at each division. This greatly facilitates the reading of large numbers. See the following table :—

QUESTIONS FOR EXAMINATION.

What is Notation? What is Numeration? What are their uses ?

How many characters have we for the purpose of expressing number? Describe them. Which are the significant characters, and why are they so called?

Describe how figures change their value by changing their situation.

What is the office of the cypher? What influence has it on other figures?

* Facilitate; to make clear or easy, from the Latin facil-is, easy.