9. Twenty undecillion six hun- 10. Seventy duodecillion, nine dred nonillion ninety-four septillion octillion five hundred and thirtythree hundred and one billion fifty-two sextillion four hundred trileight thousand three hundred and lion eight million one hundred four. and ten.

59. Orders. Since we may have 2 tens, 3 hundreds, etc., the same as 2 apples, 3 books, etc., these differen* groups may be regarded as units of different orders; thus,

60. From this it is seen that ten units of a lower order make one unit of the next higher order; the system of notation is therefore called the Decimal System, from the Latin decem, ten.

NOTE. The pupil should notice carefully the distinction between periods and orders of units. The first period, called units period, consists of units of the 1st, 2d and 3d order; the second period, called thousands period, consists of units of the 4th, 5th and 6th orders, etc. Periods increase by thousands; orders by tens.

EXAMPLES FOR PRACTICE.

Write and read the following:

1. Two units of the 2d order, and four of the 1st.

5. Eight units of the 8th order, six of the 6th, three of the 3d, and

2. Nine units of the 4th order, one of the 1st. and three of the 1st.

3. Five units of the 7th order, four of the 4th, and eight of the 2d.

4. Three units of the 9th order, five of the 5th, two of the 2d, and four of the 1st.

6. One unit of the 11th order, four of the 10th, nine of the 7th, two of the 6th, and seven of the 3d.

7. Five units of the 10th order, two of the 6th, and three of the 1st. 8. Six units of the 13th order, and four of the 5th.

THE DECIMAL SCALE.

61. In Numeration and Notation we have two classes of units, Simple and Collective.

62. A Simple Unit is a single thing, or one; a collective unit denotes a group or collection, regarded as a whole

ΝΟΤΑΤΙΟΝ.

19 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

tonths 1 hundredthe and 5 thousandthe OP 1Q and 275

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 $8.463.

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.)

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:

1. One million.

2. One billion.

3. One trillion.

4. One quadrillion.

5. One quintillion.

6. One sextillion.

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

1. 468756054.

2. 8630685025.

3. 70685973284.

4. 5637240250167.

5. 76557004032854.

6. 3205056702436057.

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

1. Five million six thousand and

one.

2. Six billion five thousand and three million seven hundred and nine.

3. Nine thousand trillion five hundred thousand billion seventeen thousand and three.

4. How many times one trillion French is one trillion English? 5. How many times one decillion French is one decillion English?

6. How many times one quintil. lion French is one quintillion English?

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

82. To express other numbers these characters are com bined 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 value 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 L xher.

83. These principles arexhibited in the following table: