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Higher periods than those in the Table, may be easily formed by following the above analogy.

4. The foregoing law, which assigns superior values to these ten characters, according to the order or place which they occupy and the use of so many derivative and compound words in forming the names of numbers, saves an inconceivable amount of time and labor in learning Notation and Numeration, as well as in their application.

40. To read numbers which are expressed by figures.

Point them off into periods of three figures each ; then, beginning at the left hand, read the figures of each period in the same manner as those of the right hand period are read, and at the end of each period, pronounce its name.

OBs. 1. The learner must be careful, in pointing off figures, always to begin at the right hand; and in reading them, to begin at the left hand.

2. Since the figures in the first or right hand period always denote units, its name is not pronounced. Hence, in reading figures, when no period is mentioned, it is always understood to be the right hand, or units' period.

EXERCISES IN NUMERATION.

Note.-In numerating large numbers, it is advisable for the pupil first to apply to each figure the name of the order which it occupies. Thus, beginning at the right hand, he should say, " Units, tens, hundreds,” &c., and point at the same time to the figures standing in the order which he mentions.

Read the following numbers : Ex. 1.

3506 11. 706305 2.

6034 12. 1640030 3.

5060 13. 830006 4. 90621 14. 70900038 5. 73040 15. 3067300 6. 450302 16. 12604321 7. 603260 17. 70003000 8. 130070

18.

161010602 9. 2021305 19. 80367830 10. 4506580 20. 400031256

21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

967058713

32100040 106320000 780507031

4063107 29038450 1046347025 20380720000

8503467039 450670412463

QUEST.-40. How do you read numbers expressed by figures ? Obs. Where begin to point them off? Where to read them? Do you pronounce the name of the right hand period ? When no period is named, what is understood ?

37.

38.

31. 430812000641 36.

120340078910356 32. 5200240301000

43601000345000 33. 98760000216

506302870045380 34.

82600381000000 39. 42008120537062035 35.

403070003462000 40. 653107843604893048 41. 210 256 031 402 385 290 845 381 467. 42. 361 438 201 219 763 281 572 829 318 278.

41. The method of dividing numbers into periods of three figures, was invented by the French, and is therefore called the French Numeration. The English divide numbers into periods of six figureri

, in the following manner:

Ifo Hundreds of Thousands of Billions.
No Tens of Thousands of Billions.
co Thousands of Billions.
or Hundreds of Billions.
o Tens of Billions.

Billions.

No Hundreds of Thousands of Millions.
co Tens of Thousands of Millions.
A Thousands of Millions.
0 Hundreds of Millions.
N Tens of Millions.
O Millions.

A Hundreds of Thousands.
W Tens of Thousands.
♡ Thousands.
co Hundreds.

Tens.
er Units,

Period III.
Period II.

Period I. According to this method, the preceding figures are read thus : 423561 billions, 234826 millions, and 479365.

Obs. 1. It will be perceived that the two methods agree as far as hundreds of millions; the former then begins a new period, while the latter continues on through thousands of millions, &c.

2. The French method is generally used throughout the continent of Europe, as well as in America, and has been recently adopted by some English authors. It is very generally admitted to be more simple and convenient than the Eng. lish method.

Qoyst.-41. What is the French method of numeration? What the English method 3 Ons which is the more simple and convenieni?

EXERCISES IN NOTATION.

42. To express numbers by figures.

Begin at the left hand, and write in each order the figure which denotes the given number in that order.

If any intervening orders are omitted in the proposed number, write ciphers in their places. (Art. 38.) Write the following numbers in figures : 1. Two thousand, one hundred and nine. 2. Twenty thousand and fifty-seven. 3. Fifty-five thousand and three. 4. One hundred and five thousand, and ten. 5. Seven hundred and ten thousand, three hundred and one 6. Two millions, sixty-three thousand, and eight. 7. Fourteen millions, and fifty-six. 8. Four hundred and forty millions, and seventy-two. 9. Six billions, six millions, six thousand, and six. 10. Forty-five billions, three hundred and forty thousand, and seventy-six. 11. Five hundred and fifty-six millions, three thousand, two

hundred and sixty-four. 12. Eight hundred and ten billions, ten millions, and seventy

five thousand. 13. Ninety-six trillions, seven hundred billions, and fifty-four. 14. Three hundred and forty-nine quadrillions, five trillions,

seven billions, four millions, and twenty. 15. Nineteen quintillions. 16. Six hundred and thirty sextillions. 17. Two hundred and ninety-eight septillions. 18. Seventy-four octillions. 19. Four hundred and ten decillions. 20. Eight hundred and sixty-three duodecillions. 21. Nine hundred and thirty-five tredecillions.

22. Six hundred and seventy-three quintillions, seventeen quadrillions, and forty-five.

23. Twenty trillions, six hundred and forty-eight billions, and twenty-five thousand.

OBs. The great facility with which large numbers may be expressed both mn language and by figures, is calculated to give an imperfect idea of their real magnitude. It may assist the learner in forming a just conception of a million, a billion, a trillion, &c., to reflect, that to count a million, at the rate of a hundred a minute, would require nearly seventeen days of ten hours each; to count a billion, at the same rate, would require more than forty-five years; and to count a trillion, more than 45,662 years.

43. From the preceding illustrations, the learner will perceive that a variety of other systems of notation may be formed upon the same principle, having different numbers for their radices. Thus, if we wished to form a quinary system; that is, a system in which the numbers should increase in a five-fold ratio, or has five for its radix, it would require four significant figures and a cipher. Let the figures 1, 2, 3, 4, and 0, be the characters employed; then five would be expressed by 1 and 0, and would be written thus 10; six by 1 and 1, thus 11 ; seven by 1 and 2, thus 12 ; eight by 1 and 3, thus 13; nine by 1 and 4, thus 14; ten by 2 and 0, thus 20; eleven by 2 and 1, thus 21, &c.

44. In the binary or diadic system of notation developed by Leibnitz, there are two characters employed, 1 and 0. The cipher when placed at the right hand of a number, in this system, multiplies it by two. Thus the number one is expressed by 1; two by 10; three by 11; four by 100; five by 101; six by 110; seven by 111; eight by 1000; nine by 1001; ten by 1010; oleven by 1011, &c.

OBS. 1. In like manner other systems of notation may be formed, having three, four, six, eight, twelve, or any given number for their radix.

When the radix is two, the system is called binary or diadic; when three it is called ternary; when four, quaternary; when five, quinary; when six, senary; when seven, septenary; when eight, octary; when nine, nonary, &c.

2. It should be observed that every system of notation, formed upon the foregoing principles, will require as many distinct characters, as there are units in the radix, and that one of them must be a cipher, and another a unit.

For the method of changing numbers from the decimal to other scales of notation, and the converse, see Arts. 162, 163.

QUEST.-43. Is the decimal notation the only system that can be formed on the same principles? How would you form a quinary system of notation ?

Write six in the qui

Obs. How many nary scale on the black-board. Write seven, nine, ten, eleven, twelve. tharacters will any system formed upon this principle require ?

45. About the commencement of the second century, Ptolenıy introduced the sexagesimal notation, which has sixty for its radix.

Obs. 1. It is said that the Chinese and some other eastern nations now employ this system in measuring time, using periods of sixties, instead of centuries. Relics of the sexagesimal notation may also be seen in our division of the circle, and of time, where the degree and hour are each divided into 60 minutes, the minute into 60 seconds, &c.

2. The Roman notation seems to have been commenced with V or five for its radix, which was afterwards changed to X or ten. It may therefore be regarded as a kind of combination of the quinary and decimal systems.

46. Since the number eight may be divided and sub-divided so many times without a remainder, some contend that a system of notation having eight for its radix, would be preferable to the decimal system.

Others claim that the duodecimal notation; that is, a system with twelve for its radix, would be more convenient than either.* However this may be, the decimal system is so firmly rooted, it were hopeless to attempt a change.

Obs. It may be doubted whether any other ratio of increase would, on the whole, be more convenient, than that of the present system. If the ratio were less, it would require more places of figures to express large numbers; if the ratio were larger, it would not indeed require so many figures, but the operations would manifestly be more difficult than at present, on account of the numbers in each order being larger. Besides, the decimal system is sufficiently comprehensive to express with all desirable facility, every conceivable number, the largest as well as the smallest; and yet it is so simple, that a child may understand and apply it. In a word, it is every way adapted to the practical operations of business, as well as the most abstruse mathematical investigations. In whatever light, therefore, it is viewed, the decimal notation must be regarded as one of the most striking monuments of human ingenuity, and its beneficial influence on the progress of science and the arts, on commerce and civilization, must win for its unknown author the everlasting admiration and gratitude of mankind.

* Barlow's Theory of Numbers, Leslie's Philosophy of Arithmetic, Edinburgh Ency. clopedia.

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