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19. If to nine hundreds one hundred more is added, we shall have ten hundred, which by analogy may be said to form a new unit, called a thousand : and although it would be easy to continue the course we have pointed out still further, it is more simple to view the thousand as forming a new species of units, in all respects similar to the primary units of which we have been speaking. Thus we count by a thousand from one to nine hundred and ninety-nine, as we have already counted by units; and again, in the same manner, and on the same principle, ten hundred of thousands will form an unit of a new kind called millions, and the families, so to speak, of billions, trillions, &c., are of course formed in the same manner as we have formed thousands and millions.

20. It will be readily conceived, that to all these collections of families, of whatever denomination, will be joined necessarily those of inferior grade, and that in this manner the complete series of whole numbers, from one to infinity, is obtained. It will also be seen, that every number differs by a unit only from that which precedes and follows it, this having been already pointed out when we first spoke of whole numbers, from 1 to 9; again 9 + 1 = 10, which differs from 9 by 1 only ; and as to 10 were then added successively all the simple numbers from 1 to 9, the series was completed as far as 19. After 19 came the number 20, which differs from 19 by one only; for 19 + 1 = 10 + 9 + 1, or is equal to twice ten, or two tens, or 20. From 20 we then proceed on the same principle to 30 and 40, and as far as 99, by adding continually 1 to that number, which we had already formed or reckoned. Again, 99 + 1 - 90 + 9 + 1 = 90 + 10, or is equal to nine tens and one ten, or to ten tens, forming the number 100, which also only differs from 99 by one unit. By a series of numbers continued in the same way, we then reckon from 100 to 999; but 999 + 1 = 900 + 90 + 9 + 1 = 900 + 90 + 10 = 900 + 100, or 10 hundreds, which we have called a thousand. Thus, too, we proceed from 1 thousand to 999 thousands; and, so on to 999 thousands + one thousand a new family, which is called a million. It is evident, that this process might easily be continued to any extent that the teacher might deem requisite. " 21. The thousands, the millions, the billions may be considered as units, and all these families may consist of units, of tens, and of hundreds, which will occupy, as already pointed out, the first, the second, and the third rank. Each family, therefore, will give a set of 3 figures, for which

may

be substituted, either in whole or in part, noughts or zeros. The rank or place which each family should occupy is readily found by analogy. Thus the family of units, in going from right to left, occupies the

1st rank, that of thousands the

2d rank, that of millions the

3rd rank, and so on to any extent. This will be sufficiently evident from the accompanying numeration table, where the different values of each family are represented by zeros, and in which each family (or set of three figures) is separated from that preceding and succeeding it by commas.

When once the pupil comprehends this table, it will be very easy for him both to understand and remember the two following rules :

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22. To express in words, or to read any number of figures, we should first divide the number by commas, into families or sets of 3 figures, beginning on the right hand. Afterwards each set or family should be read separately (beginning at the left hand), as if it consisted of units only, taking care at the same time to give to each the name which belongs to it. Thus, if we have to read the numbers 45260604297, we should divide it by commas, thus, 45,260,604,297, and commencing from the left, we have 45 billions, 260 millions, 604 thousands, 297 units, or expressed in full (in words at length), it will be forty-five billions, two hundred and sixty millions, six hundred and four thousand, two hundred and ninety-seven, the word "units" being suppressed as unnecessary. It will be perceived, that this number contains neither units of millions nor tens of thousands.

23. To express in written or printed figures any given number expressed in words, we may in the first instance write down separately each set or family as if it consisted of units only.

It will be easy afterwards to write them down in the proper order, taking care to place three zeros or noughts where any set or family is altogether wanting, and one nought for each separate value which may be wanting in each family. Thus, let me be required to write down in figures five trillions, two hundred and eight millions, six hundred and forty thousand, and eight. In the first instance I write 5 trillions,

second

208 millions, third

640 thousand, fourth

8 units; and then, by placing each family in the rank to which it properly belongs I shall have 5,000,208,640,008.

After the 5 trillions I placed 3 noughts, because the family of billions is altogether wanting; and for the same reason, because units of thousands and the hundreds and tens of units are wanting, I have substituted zeros in the place of them. It was not, however, necessary to mark the place of the tens or hundreds of trillions which are wanting, as there are no units of a superior grade to trillions. With a little practice it will be easy for the pupil to commence writing down the num

bers stated, as soon as he hears the name of the first or highest num

ber. *

24. Examples.—Read or write down in words the numbers signified by the following figures. In 1828, the rule of England was acknowledged in Asia by upwards of 123,020,400 human beings.

The independent states in that quarter of the globe are reckoned at 134005604 persons.

In Asia there are six hundred and thirty-four million, nine hundred and eight thousand inhabitants.

In Europe there are two hundred and thirteen million, eight hundred and twenty-two thousand, five hundred and seventy-four souls. Africa is supposed to contain above seventy millions of people. The population of America is about thirty-four million, three hundred and seventeen thousand, seven hundred in number.

Europe contains three million, three hundred and eighty-seven thousand, and nineteen square miles.

Asia contains twenty million, four hundred and thirty-two thousand, two hundred and five square miles.

Africa contains thirteen million, four hundred and fifty thousand

square miles.

America contains fifteen millions of square

miles.

* There is a ready and striking way of palpably representing numeration, and the principles on which it depends, by means of cubes ; for unity itself, in its primary signification, being represented by a cube, every other unit of each family whatever, thousands, millions, billions, will take the shape of a cube (though of larger dimensions) likewise; the tens also being represented by rows or lines of ten cubes, and hundreds by squares of ten rows of ten cubes.

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Thus, let a be the cube which represents unity, and a b being a row of ten of such cubes, will represent ten units. The square a b c d, formed of ten rows of ten cubes, will then be one hundred, and ten of these hundreds, forming the cube b, e, will represent a thousand, in the same way that a represented unity in the first instance. Now, if we called this large cube a'; a'b', which would be formed of a row of ten of such cubes, would represent tens of thousands; and again, a' B'd done hundred thousand, whilst ten of these hundredths would form a new cube b'e', which would represent millions. It would be easy, by a repetition of the process, to represent billions, and even trillions.

3. Corollaries or Deductions from Decimal Numeration, as explained

in the preceding Chapter.

When we examine with attention the principles upon which the system of decimal numeration is framed, we may deduce from it several important consequences (or inferences) which will not only serve to explain it more fully, but which will be found most useful to us as we proceed.

25. By whatever rank of figures we take unity to be represented, we must bear in mind that its value is ten times greater than that of unity taken in the rank immediately to the right of it, and consequently ten times less than unity taken in the rank immediately to the left of it. Thus one ten is equivalent to ten units, and ten tens are only equal to one hundred. Again one hundred is equivalent to ten tens, and ten hundreds are required to form one thousand.

26. It follows, that unity taken in any rank, is in value one hundred times greater than unity, as represented in the second rank to the right of it, and one hundred times less than unity in the second rank to the left of it. Thus one thousand is equivalent to 100 tens, which is one hundred times less than one hundred thousand :or 1 thousand

This the pupil should demonstrate 10 hundreds, and 1 hundred

10 tens,

by the diagram in the note to the

23 paragraph. Hence one thousand is equal to as many times 10 tens as it contains hundreds; that is, it is equal to ten times 10 tens, or to one hundred tens: and conversely, one ten is one hundred times less than the thousand, which in the same manner may be shown to be the hundredth part of 100 thousand ; or, expressed generally, by whatever rank unity is represented, it is always 10 times, 100 times, or 1000 times, &c. greater than unity which is in the 1st, 2nd, or 3rd ranks to its right; and for the same reason, it is the tenth, hundredth, or thousandth part of that which is in the 1st, 2nd, or 3rd rank to the left.

27. Each of the 9 figures (or digits, as they are sometimes called) which we employ in numeration, have two values perfectly distinguishable one from the other, and it is by means of this distinction that we are enabled to represent any number whatever with ten signs only. The first value of any figure is that which depends upon its form, which form indicates invariably one particular or constant number of units. Thus, the figure 5 indicates always a collection of fives, whether that collection consists of units properly so called, or of tens, or of hundreds, &c. In this respect its value never varies, for which reason this is called its absolute or unconditional value. The other value of any figure depends on its position, or the rank which it occupies, and which determines to what kind or family we must refer it. Thus, according to the rank in which we find the figure 5, it represents either a collection of primary units, or of tens, or of hundreds, &c.; and each 5 will be 10 times, 100 times, or 1000 times greater than any other 5 in the 1st, 2nd, or 3rd rank to the right; and conversely, will be 10

times, or 100 times, or 1000 times less than any 5 in the 1st, 2nd, or 3rd rank to the left. This, therefore, is called its relative value.

28. To know how often any given number contains either units, or tens, or hundredt, &c., we must read the numbers as if unity was represented by each family successively; thus the numbers 36475 contain 3647 times ten, 364 times one hundred, 36 times one thousand, and 3 tens of thousands; that is, the tens are joined to the hundreds, to the thousands, &c., in the same way that the units are joined to the tens, the hundreds, &c. As, then, we have 3647 tens, so in like manner we should have 3647 units, if the 7 represented primary units; or, again, we have 364 hundreds, as we should have 364 units, if the 4 had represented 4 units. Or, taking the number 3647 tens, we might say a ten of thousands is equivalent to 1000 tens; and therefore 3 tens of thousands are three times as much, or 3 times 1000 .

or 3000 tens, and I thousand 100 tens, therefore 6 thou.. sand 6 times 100

or 600 tens, 1 hundred 10 tens, therefore 4 hundred

40 tens, and besides, there are 7 tens to be added, or

7 tens,

measur

3647 tens. It would of course be easy to show, that the same thing holds good in all cases.

29. All quantities, of whatever kind, inasmuch as they may be ed, may be represented by numbers. In the following part of the work, we will examine the several combinations of numbers, by means of which answers may be returned to many useful and interesting arithmetical questions. We shall find, that these combinations naturally divide themselves into four kinds, and that to all of them numeration is the proper basis or foundation.

All the operations in arithmetic are in general termed calculations ; and it is because all calculations depend in the first instance on numeration, that we consider it so necessary for the pupil, before he proceeds to combinations of a higher grade, fully to understand the principles of numeration.

F. Curie. (To be continued.)

Notices of Books.

Religious Instruction, in a graduated Series of Lessons for Young Children,

by the Author of " Lessons on Objects,&c. foolscap 8vo. pp. 226.

(Seeley.) It is sadly to be feared that a considerable portion of the religious instruction of the day, even when sound in doctrine, loses more than half its force by the vague and desultory way in which it is given; and that this error especially prevails, where it ought to be most guarded ag ainst-namely, in our infant and Sunday schools. As far as the

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