PRINCIPLES.

1. Repeating I., X., C., or M. repeats its value.

XX. denotes 20; CC. denotes 200. V., L., and D. cannot be thus repeated.

2. Prefixing I., X., or C. to a letter of greater value diminishes that value by I., X., or C.

3. Affixing I., V., X., L., C., or D. to a letter of greater value increases that value by I., V., X., L., C., or D.

4. Inserting I., X., or C. between two letters,* each of greater value, diminishes the united value of the two by I., X., or C.

The first of the two must not be of less value than the second. XIV., not VIX., denotes 14; XIX. denotes 19.

5. A bar placed over a letter, except I., increases its value a thousand-fold.

C. denotes 100,000.

6. IIII. is sometimes used instead of IV., as on the dials of clocks and watches. 400 may be expressed by CCCC. or by CD.

3. Which of these are correct expressions and which in

2. Repeat the four principles of notation.

3. Name the periods from the 1st to the 22d.
4. Repeat the five principles of the Roman notation.
5. Repeat the rule for numeration.

ADDITION.

INDUCTIVE STEPS.

1. How many units are 5 units and 3 units? 2 tens and 7 tens? 4 thousands and 6 thousands?

2. A certain field has 7 acres, and an adjoining field 8 How many acres in both fields?

acres.

Process.

7 acres 8 acres 15 acres

Explanation.

Since one field contains 7 acres and the other 8 acres, the two fields contain 7 acres and 8 acres, which are 15 acres.

3. If on one shelf there are 9 books and on another shelf 5 books, how many books are on

both shelves?

Write and explain the process.

4. How many pounds are 8 pounds and 6 pounds ?

Write and explain.

5. The process of thus uniting quantities in a single quantity is called adding.

6. What is the unit of 8 pounds? Of 6 pounds? Of 14 pounds?

7. Can quantities having like units be added?

8. Add 6 pounds and 5 dollars. Can you show a process? If you can, is your result 11 pounds or 11 dollars?

9. What, then, does addition require as to the units to be added?

10. What does addition require as to the unit of the result or sum?

11. Numbers having like units are called Like Numbers.

DEFINITIONS.

1. Addition is the process of finding the sum of two or more like numbers. The sum is, therefore, the result of addition. 2. A Sign indicates some process or condition. The sign of addition is an upright cross, +. It is read "plus."

3. The Sign of Equality is two short horizontal lines, =. It is read "equals," or "is equal to." 3+2 = 5, is read "3 plus 2 equals 5." 3+2=5, being an expression of equality, is called an Equation.

PRINCIPLES.

1. Only like numbers and orders can be added.

2. The numbers added and their sum are like numbers.

EXERCISES.

1. Find the sum of 120, 331, and 478.

Process.

120

331

478

929

Explanation.

ANALYSIS.-There are three numbers to be added, each containing units, tens, and hundreds.

PRINCIPLE.-Only like orders can be added.

SYNTHESIS.-Hence we write the numbers with the units'

figures (0, 1, 8) in the first column on the right, the tens' figures (2, 3, 7) in the second column, and the hundreds' figures (1, 3, 4) in the third column.

The sum of the first column is 9 units; the sum of the second column is 12 tens 1 hundred + 2 tens. We write the 2 tens, and add the 1

hundred to the hundreds' column, making 9 hundreds. Hence the sum required is 929.

206

9,837

10,068

206 9,837

=

Explanation.

2 tens 5 units.

2 hundreds + 0 tens + 6 units.

= 9 thousands +8 hundreds + 3 tens +7 units.

PRINCIPLE.-Only like orders can be added.

SYNTHESIS.-We, therefore, write the units' figures (5, 6, 7) in the first column; the tens' figures (2, 0, 3) in the second column; the hundreds' figures (2 and 8) in the third column; and the 9 thousands alone in the fourth place. Adding the first column we have 18 units 1 ten and 8 units. We add the 1 ten to the tens' column and have 6 tens. Adding the third column we have 10 hundreds 1 thousand and 0 hundreds. We say finally 1 thousand + 9 thousand

10 thousand.

RULE FOR ADDITION.

1. See that the numbers to be added are like numbers. 2. Write units of the same order in the same column. 3. Begin at units' column, and find the sum of each column separately.

4. Write the units of a sum, but add the tens with the next column.

5. Write the entire sum of the last column.