63. The Orders of Units are the units represented by the figures in the different places of a numerical expression.

64. Simple Units are called units of the first order; tens are called units of the second order; hundreds, units of the third order, etc.

65. The Scale of a system of notation is the law of relation between its successive orders of units.

66. The Radix of the scale is the number which expresses the relation of the successive orders.

67. The Decimal Scale of notation is that in which the radix is ten. The system of numeration and notation explained is therefore called the decimal system.

68. Since figures in different parts of the scale express different units, figures may be regarded as having two values, a Simple and Local value.

69. The Simple Value of a figure is the number of units it expresses when it stands alone, or in units place. 70. The Local Value of a figure is the number it expresses when in any other than units place.

71. The Decimal System of numeration had its origin in the practice, common to all nations, of counting by groups of tens.

72. The Arabic System of notation is based on the simple but ingenious device of place. The system would be the same in principle, whatever the radix of the scale.

73. If we fix the place of units by a point (.), we may extend the scale to the right of units place, and have the scale descending as well as ascending.

74. The first place on the right of the point will be onetenth of units or tenths, the second place one-tenth of tenths, or hundredths, the third place, thousandths, etc.

75. Such terms are called decimals, and the point is called the decimal point. Thus, 48.375 is read 48 and 3 tenths 7 hundredths and 5 thousandths, or 48 and 375 thousandths.

76. The Currency of the United States is expressed by the decimal system in integers and decimals. The dollar is the unit and is indicated by the symbol $. The first place at the right of the decimal point is called dimes; the second place, cents; and the third place, mills.

77. Dimes and Cents, in practice, are read as a number of cents. Thus, $4.65 is read 4 dollars and 65 cents; and $72.485 is read 72 dollars 48 cents and 5 mills. Mills are often expressed as a fractional part of cents; thus $8.465 is written $8.461.

NOTES.-1. Pupils will notice the difference between the Arabic system of notation and the decimal system of numeration. The Roman method of notation bears the same relation to the decimal system as the Arabic. 2. Any number could have been taken as the basis of the scale; hence the Decimal System is not essential, but merely accidental or conventional. 3. The decimal scale originated from the custom, among primitive races, of reckoning by counting the fingers, the number on both hands, including the thumbs, being ten.

4. The Arabic notation is named from the Arabs, who introduced it into Europe by their conquest of Spain during the 11th century. The Arabs obtained it from the Hindoos, by whom it was probably invented more than 2000 years ago.

5. The first nine of the Arabic characters are called significant figures, because they always denote a definite number of units. They are also called digits, from the Latin digitus, a finger, because they were employed as a substitute for the fingers, with which the ancients used to reckon.

6. The character 0 is called naught, because it indicates no value. It is also called zero, which is an Italian word, signifying nothing. It is also called cipher, which is derived from the Arabic sifr or sifrum, meaning empty, vacant. The term was subsequently applied to all the Arabic characters, and the use of them was called ciphering.

7. There are three theories for the origin of the Arabic characters: 1st, that they are modifications of characters formed by the combination of straight lines; 2d, that they are modifications of characters formed by the combination of angles; and 3d, that they are derived from the initial letters of the Hindoo words for numbers. The last theory is given by Prinseps and indorsed by Max Müller, and is probably the true one. (See Brooks's Philosophy of Arithmetic.)

ENGLISH METHOD OF NUMERATION.

78. The method of numeration by dividing numbers into periods of three figures each, is called the French Method. There is also another method called the English Method.

79. The English Method uses periods of six figures each, calling the first period units, the second millions, the third billions, the fourth trillions, etc.

80. The places in each period are units, tens, hundreds, thousands, tens of thousands, hundreds of thousands. The method is represented in the following table:

Trillions Period. Billions Period. Millions Period. Units Period.

The remaining periods have the same names as in the French method.

EXAMPLES FOR PRACTICE.

1. Write the following numbers by both the French and English methods and show their difference:

2. Read the following by both the French and English methods:

3. Write the following by either method, and read the results by the other method:

ROMAN NOTATION.

81. The Roman Method of Notation employs seven letters of the Roman alphabet. Thus, I represents one; V, five; X, ten; L, fifty; C, one hundred; D, five hundred; M, one thousand.

82. To express other numbers these characters are combined according to the following principles:

1. Every time a letter is repeated its value is repeated. 2. When a letter is placed before one of a greater value, the DIFFERENCE of their values is the number represented. 3. When a letter is placed after one of a greater value, the SUM of their values is the number represented.

4. A dash placed over an expression increases its value a thousand fold. Thus VII denotes seven thousand.

NOTE. In applying these principles write the different orders of units in succession, beginning with the higher.

83. These principles are exhibited in the following table:

NOTE.-The Roman method is named from the Romans, who invented and used it. It is now employed only to denote the chapters and sections of books, pages of preface and introduction, and in other places for prominence and distinction.

EXAMPLES FOR PRACTICE.

1. Write the following numbers by the Roman Method: 1. Nine hundred and thirty-six. 2. One thousand five hundred and sixteen. 3. Four thousand two hundred and four. 4. Seven thousand and sixty-eight. 5. Thirty thousand and thirteen.

2. Read the following numbers:

LXXXVIII; VDLIX; MDCCCLXXV; MMDXC; LCCCXXX; XDCLVI; LIXCCCCXLIV; MMMMXC; cliv; xcvii; clxix.

SECTION II.

FUNDAMENTAL OPERATIONS.

ADDITION.

84. Addition is the process of finding the sum of two or more numbers.

85. The Sum of several numbers is a number which contains as many units as the numbers added.

86. The Sign of Addition is +, and is read plus. It denotes that the numbers between which it is placed are to be added.

87. The Sign of Equality is, and is read equals. It denotes that the numbers between which it is placed are equal.

NOTES.-1. The Sign of Addition consists of two short lines bisecting each other, the one in, and the other perpendicular to, the line of writing. 2. The symbol was introduced by Stifelius, a German mathematician, in a work published in 1544.

3. The symbol was introduced by Robert Recorde, an English mathematician, in his "Whetstone of Wit," a work on algebra published in 1557.

PRINCIPLES.

1. The numbers added must be similar.

2. Units of the same order only can be added directly. 2. The sum is a number similar to the numbers added. 4. The sum is the same in whatever order the numbers are added.

PROBLEM.

88. To find the sum of two or more numbers.

OPERATION.

571

395

683 1649 Ans.

1. What is the sum of 571, 395, and 683? SOLUTION. We write the numbers so that terms of the same order stand in the same column, and begin at the right to add. 3 and 5 are 8 and 1 are 9, units; we write the 9 units under the column of units: 8 and 9 are 17 and 7 are 24, tens, or 2 hundreds and 4 tens; we write the 4 tens under the column of tens, and add the 2 hundreds to the column of hundreds: 2 and 6 are 8 and 3 are 11 and 5 are 16, hundreds, or 1 thousand and 6 hundreds; we write the 6 hundreds under the column of hundreds, and place the 1 at the left in the place of thousands. Hence the sum of 571, 395, and 683 is