the language of figures expresses that the unit of each place increases from right to left, according to the scale of tens. This is called the decimal system of numbers, and the scale is uniform.

United States Currency.

28. United States Currency affords an example of a system of denominate units, increasing according to the scale of tens: thus,

Eagle.

Dollar.

in which ten units of any denomination make one unit of the next higher.

The dollars are denoted by $, and separated from the dimes, cents, and mills by a period (.), called the decimal point.

Varying Scales.

29. If we write the well-known signs of the English Cur "ency, and place 1 under each denomination, we shall have

27. If several figures are written by the side of each other, what does the language express? What name is given to this system of numbers ? What is the scale?-28. How do the different units compare with each other in United States Currency?

The signs, £ s. d. and far., denote the value of the unit 1 in each denomination; and they also determine the relations between the different units. For example, this simple language expresses the following ideas:

1st. That the unit of the right-hand place is 1 farthing; of the place next at the left, 1 penny; of the next place, 1 shilling; of the next place, 1 pound: and

2d. That 4 units of the lowest denomination make cze unit of the next higher; 12 of the second, one of the third; and 20 of the third, one of the fourth. Hence, 4, 12, and 20 are

the numbers which make up the scale.

30. If we take the denominate numbers of Avoirdupois weight, we have

in which the units increase in the following manner: viz., counting from the right, 16 units of the lowest denomination make 1 unit of the next higher; 16 of the second, 1 of the third; 25 of the third, 1 of the fourth; 4 of the fourth, 1 of the fifth ; 20 of the fifth, 1 of the sixth. The scale, therefore, for this class of denominate numbers, varies according to the above law

If we take any other class of denominate numbers, as the Troy weight, we shall have a different scale, and the scale will continue to vary as we pass from one class of numbers to another. But in all the formations, we shall recognize the application of the same general principles.

31. There are, therefore, two general methods of forming the different systems of integral numbers, from the unit one. The first consists in preserving a uniform law of relation between the different units. If that law of relation is expressed by 10, we have the system of decimal or common numbers.

29. Is the scale uniform or varying in the English Currency? Name the units of the scale at each change of denomination.-30. Name the units of the scale, at each step, in the Avoirdupois weight. Name them also in the Apothecaries weight?

The second method consists in the application of known, though varying laws of change in the units. These changes in the units, produce different systems of denominate numbers, each of which has its appropriate scale.

Integral Units of Arithmetic.

32. The Integral Units of Arithmetic are divided into eight classes:

1. Units of Abstract Numbers;

2. Units of Currency;

3. Units of Length, or Linear Units;
4. Units of Surface;

5. Units of Volume, or Cubic Units;

6. Units of Weight;

7. Units of Time;

8. Units of Angular Measure.

First among the units of arithmetic is the abstract unit 1. This is the primary base of all abstract numbers, and becomes the base, also, of any denominate number, by merely naming the particular thing to which it is applied.

Of the Signs.

33. The sign =, is called the sign of equality. When placed between two numbers, it denotes that they are equal; that is, that each contains the same number of units.

The sign, is called plus, which signifies more. When placed between two numbers, it denotes that they are to be added together. Thus, 3 + 2 = 5.

The sign, is called minus, a term signifying less. When placed between two numbers, it denotes that the one on the right is to be taken from the one on the left. Thus, 6-2 = 4.

31. How many general methods are there of forming numbers from the unit one? What is the first? What is the second?-32. Into how many classes are the Units of Arithmetic divided? Name them.

The sign X, is called the sign of multiplication.

When

placed between two numbers, it denotes that they are to be multiplied together. Thus, 12 x 3, denotes that 12 is to be multiplied by 3.

The parenthesis is used to indicate that the sum or difference of two or more numbers is to be regarded as a single number. Thus, (2+3+5) × 6,

shows, that the sum of 2, 3, and 5, is to be multiplied by 6. And (5 −3) × 6,

denotes that the difference between 5 and 3, is to be multiplied by 6.

The sign, is called the sign of division. When placed between two numbers, it denotes that the one on the left is to be divided by the one on the right. Thus, 45, denotes that 4 is to be divided by 5.

Properties of the 9's.

34. In any number, written with a single significant figure, as, 4, 40, 400, 4000, &c., the excess over exact the number of units in the significant figure. number may be written thus,

9's is equal to For, any such

Each of the numbers 9, 99, 999, &c., contains an exact number of 9's; hence, when multiplied by 4, the several products will contain an exact number of 9's: therefore,

33. What is the sign of Equality? What is the sign of Addition? What of Subtraction? What of Multiplication? For what is the parenthesis used? What is the sign of Division ?

34. What will be the excess over exact 9's in any number expressed by a single significant figure? How may the excess over exact 9's be found in any number whatever?

The excess over exact 9's, in each number, is 4; and the same may be shown for each of the other significant figures.

If we write any other number, as

6253,

we may read it, 6 thousands, 2 hundreds, 5 tens, and 3. Now, the excess of 9's in the 6 thousands, is 6; in 2 hundreds, it is in 5 tens, it is 5; and in 3, it is 3: hence, in them all, i is 16, which is one 9, and 7 over: therefore, 7 is the excess over exact 9's in the number 6253. In like manner,

2;

The excess over exact 9's, in any number whatever, is found by adding together the significant figures, and rejecting the exact 9's from the sum.

NOTE. It is best to reject or drop the 9, as soon as it occurs: thus, we say, 3 and 5 are 8 and 2 are 10; then, dropping the 9, we say, 1 to 6 is 7, which is the excess; and the same for all similar operations.

1. What is the excess of 9's in 48701? In 67498?

2. What is the excess of 9's in 9472021? In 2704962?

3. What is the excess of 9's in 87049612? In 4987051?

REDUCTION.

35. REDUCTION is the operation of changing a number from. one unit to another, without altering its value.

36. REDUCTION DESCENDING is the operation of changing a number from a greater unit to a less.

37. REDUCTION ASCENDING is the operation of changing a uumber from a less unit to a greater.

38. If we have 4 yards, in which the unit is 1 yard, and wish to change to feet, the units of the scale will be 3, since 3 feet make 1 yard; therefore, the number of feet will be

4 × 3 = 12 feet.

35. What is Reduction?-36. What is Reduction Descending?--37. What is Reduction Ascending?