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Obs. 1. This method of expressing numbers was invented by the Romans, and is therefore called the Roman Notation. It is now seldom used, except to lenote chapters, sections, and other divisions of books and discourses.

2. The letters C and M, are the initials of the Latin words centum, and mille, the former of which signifies a hundred, and the latter a thousand; for this reason it is supposed they were adopted to represent these numbers.

31. a. It will be perceived from the Table above, that every time a letter is repeated, its value is repeated. Thus I, standing alone, denotes one ; II, two ones, or two, &c. So X denotes ten ; XX, twenty, &c.

When a letter of a less value is placed before a letter of a greater value, the less takes away its own value from the greater; but when placed after, it adds its own value to the greater.

32. A line or bar -) placed over a letter, increases its value a thousand times. Thus, V denotes five, V denotes five thousand; X, ten ; X, ten thousand, &c.

Obs. 1. In the early periods of this notation, four was written IIII, instead of IV; nine was written VIII, instead of IX; forty was written XXXX, instead of XL, &c.

The former method is more convenient in performing arithmetical operations in addition and subtraction; while the latter is shorter and better adapted to ordinary purposes.

2. A thousand was originally written CI?, which, in later times, was changed into M; five hundred was written 1Ɔ instead of D. Annexing to Iincreased its value ten times. Thus, 15 denoted five thousand; 1553, filly thousand, &c.

3. Prefixing C and annexing to the expression CIƆ, makes its value ten times greater: thus, CCIdenotes ten thousand; CCC1005, a hundred thousand. According to Pliny, the Romans carried this mode of notation no further. When they had occasion to express a larger number, they did it by repetition. Thus, CCCIO,CCCI°°°, expressed two hundred thousand, &c.

33. The common method of expressing numbers is by the Arabic Notation. The Arabic method employs the following ten characters or figures, viz:

1 2 3 4 5 6 7 8 9 0 one, two, three, four, five, six, seven, eight, nine, zero.

QUEST.--Obs. Why is this method called Roman? 31. a. What is the effect of repeating a letter? If a letter of less value is placed before another of greater value, what is the effect? If placed after, what? 32. When a line or bar is placed over a letter, how does it affect its value? 33. What is the common way of expressing numbers? How many characters does this method employ?

The first nine are called significant figures, because each one always has a value, or denotes some number. They are also called digits, from the Latin word digitus, which signifies a finger.

The last one is called a cipher, or naught, because when standing alone it has no value, or signifies nothing.

OBS. 1. It must not be inferred, however, that the cipher is useless ; for when placed on the right of any of the significant figures, it increases their value. It may

therefore be regarded as an auxiliary digit, whose office, it will be seen hereafter, is as important as that of any other figure in the system.

2. Formerly all the Arabic characters were indiscriminately called ciphers; hence the process of calculating by them was called ciphering ; on the same principle that calculating by figures is called figuring.

34. It will be seen that nine is the greatest number that can be expressed by any single figure in the Arabic system of Notation.

All numbers larger than nine are expressed by combining together two or more of these ten figures, and assigning different values to them, according as they occupy different places. For example, ten is expressed by combining the 1 and 0, thus 10; eleven by two ls, thus 11; twelve by 1 and 2, thus 12; twenty, thus 20; thirty, thus 30; &c. A hundred is expressed by combining the 1 and two Os, thus 100; two hundred, thus 200; a thousand by combining the 1 and three Os, thus 1000; two thousand, thus 2000; ten thousand, thus 10,000; a hundred thousand, thus 100,000; a million, thus 1,000,000; ten millions, thus 10,000,000; &c. Hence,

35. The digits 1, 2, 3, &c., standing alone, or in the right hand place, respectively denote units or ones, and are called units of the first order.

When they stand in the second place, they express tens, or ten ones ; that is, their value is ten times as much as when standing

QUEST.-What are the first nine called ? Why? What else are they called? What is the last one called ? Why? Obs. Is the cipher useless? What may it be regarded ? What is the origin of the term ciphering? 34. What is the greatest number that can bo expressed by one figure? How are larger numbers expressed ? 35. What do the digits, 1, 2, 3, &c., denote, when standing alone, or in the right hand place? What are they "hen called? What do they denote when standing in the second place ?

in the first or right hand place, and they are called units of the second order.

When occupying the third place, they express hundreds; that is, their value is ten times as much as when standing in the second place, and they are called units of the third order.

When occupying the fourth place, they express thousands ; that is, their value is ten times as much as when standing in the third place, and they are called units of the fourth order, &c. Thus, it will be seen that,

Ten units make one ten, ten tens make one hundred, and ten hundreds make one thousand ; that is, ten in an inferior order are equal to one in the next superior order. Hence, universally,

36. Numbers increase from right to left in a tenfold ratio; consequently each removal of a figure one place towards the left, increases its value ten times.

Note.-1. The number which forms the basis, or which expresses the ratio of increase in a system of Notation, is called the Radix of that system. Thus, the radix of the Arabic notation is ten.

2. The reason that numbers increase from right to left, instead of left to right, is probably owing to the ancient practice of writing from the right hand to the left.

37. The different values which the same figures have, are called simple and local values.

The simple value of a figure is the value which it expresses when it stands alone, or in the right hand place. Hence the simple value of a figure is the number which its name denotes.

The local value of a figure is the increased value which it expresses by having other figures placed on its right. Hence the local value of a figure depends on its locality, or the place which

QUEST.-What is their value then ? What are they called? What is a figure called when it occupies the third place? What is its value then? What is it called when in the fourth place? What is its value? How many units are required to make one ten ? How many tens make a hundred? How many hundreds make a thousand ? How many of an inferior order are required to make one of the next superior order ? 36. What is the general law by which numbers increase? What effect has it upon the value of a figure to remove it one place towards the left ? Note. What is the number called which forms the basis or the ratio of increase in a system of notation? What is the radix of the Arabic notation? Why do numbers increase from right to left? 37. What are the different values of the same figure called? What is the simple value of a figure? What the local ?

it occupies in relation to other numbers with which it is connected (Art. 35.)

Obs. 1. This system of notation is called Arabic, because it is supposed to have been invented by the Arabs.

2. It is also called the decimal system, because numbers increase in a tenfold ratio. The term decimal is derived from the Latin word decem, which signifies ten.

3. The early history of the Arabic notation is veiled in obscurity. It is the opinion of some whose judgment is entitled to respect, that it was invented by the philosophers of India. It was introduced into Europe from Arabia about the eighth century, and about the eleventh century it came into general use, both in England and on the continent. The application of the term digit to the significant figures, affords strong presumptive evidence that the system had its origin in the ancient mode of counting and reckoning by means of the fingers; and that the idea of employing ten characters, instead of twelve or any other number, was suggested by the number of fingers and thumbs on both hands. (Art. 33.)

NUMERATION. 38. The art of reading numbers when expressed by figures, is called NUMERATION.

The pupil will easily learn to read the largest numbers from the following scheme, called the

NUMERATION TABLE.

{ Tredecillions.

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# Nonillions.

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e Thousands

Units.

685, 876, 389, 764, 391, 827, 218, 649, 853, 123, 234, 579, 793, 465, 623.

XV. XIV. XIII. XII.

XI.

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II.

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39. The different orders of numbers are divided into periods of three figures each, beginning at the right hand. The first period, which is occupied by units, tens, hundreds, is called units'

Quest.-Upon what does the local value of a figure depend? Obs. Why is this system of notation called Arabic? What else is it sometimes called? Why? What do you say of its early history? When was it introduced into Europe? What is the probable origin of the system? Why were ten characters, rather than any other number, adopted ? 38. What is Numeration ? 39. How are the orders of numbers divided? What is the first period called ? By what is it occupied ?

period; the second is occupied by thousands, tens of thousands, hundreds of thousands, and is called thousands' period; the third is occupied by millions, tens of millions, hundreds of millions, and is called millions' period; the fourth is occupied by billions, tens of billions, hundreds of billions, and is called billions' period; and so on, the orders of each successive period being units, tens, and hundreds.

The figures in the table are read thus: 685 tredecillions, 876 duodecillions, 389 undecillions, 764 decillions, 391 nonillions, 827 octillions, 218 septillions, 649 sextillions, 853 quintillions, 123 quadrillions, 234 trillions, 579 billions, 793 millions, 465 thousand, 6 hundred and twenty-three.

Note.-1. The terms thirteen, fourteen, fifteen, &c., are obviously derived from three and ten, four and ten, five and ten, which by contraction become thirteen, fourteen, fifteen, and are therefore significant of the numbers which they denote. The terms eleven and twelve, are generally regarded as primitive words; at all events, there is no perceptible analogy between them and the numbers which they represent. Had the terms oneteen and twoteen been adopted in their stead, the names would then have been significant of the numbers one and ten, two and ten; and their etymology would have been similar to that of the succeeding terms.

The terms twenty, thirty, forly, &Q., were formed from two tens, three tens, four tens, which were contracted into twenty, thirty, forty, &c.

The terms twenty-one, twenty-two, twenty-three, &c., are compounded of twenty and one, twenty and two, &c. All the other numbers as far as ninetynine, are formed in a similar manner.

2. The terms hundred, thousand and million are primitive words, and bear no analogy to the numbers which they denote. The numbers between a hundred and a thousand are expressed by a repetition of the numbers below a bundred. Thus we say one hundred and one, one hundred and two, one hundred and three, &c.

3. The terms billion, trillion, quadrillion, &c., are formed from million and the Latin numerals bis, tres, quatuor, &c. Thus, prefixing bis to million, by a slight contraction for the sake of euphony, it becomes billion ; prefixing tres to million, it is easily contracted into trillion, &c. The Latin word bis signifies two; tres, three; quatuor, four; quinque, five; sex, six; septem, seven; ocio, eight; norem, nine; decem, ten; undecim, eleven; duodecim, twelve ; tredecin, thirteen.

QUEST.-What is the second period called ? By what occupied ? What is the third called ? By what occupied ? What is the fourth called ? By what occupied ? What is the fifth called ? By what occupied ? Repeat the Numeration Table, beginning at the right hand.

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