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PRINCIPLES.

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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. Afixing 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.

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EXERCISES.

1. Read the following combinations :
1. XV.
10. XLV.

19. DCCXC.
2. IV.
11. XCIX.

20. MXXIX. 3. XIV. 12. LXV.

21. VDLV. 4. XXIV. 13. CIX.

22. DLDC.
5. XIX.
14. CXI.

23. COXDVI. 6. XXXIX. 15. XCI.

24. VIII. 7. XXXIII. 16. DCXC.

25. CCXC. 8. XXIX. 17. CCCXXXIX. 26. CXLIX. 9. XLIX. 18. DCCXXXIV. 27. MMD.

28. LXXDCCCXCIX. 29. MDXCVDCCCLXIV.

66. 86.

2. Write in Roman characters the following:
1. 15.
11. 18. 21.

27.
2. 36.
12. 42.

22. 81.
3. 87.
13.

23. 95.
4. 56.
14.

24. 40.
5. 49.

15. 63. 25. 45. 6. 99.

16. 100. 26. 534. 7. 1050.

17. 3600. 27. 5000. 8. 5010. 18. 587. 28. 436. 9. 789.

19. 207. 29. 999. 10. 1898. 20. 8004. 30. 76,959. 3. Which of these are correct expressions and which incorrect? VV. DD.

VXX.
LV.
CCC.

VLC.
XCC.

VDC.
LXIX. MMXL.

CIXVIIX.
XLX.
XLIX.

LXXXVIII.
XCIX. CIXXVII. DMCC.

• XIL

REVIEW.

1. Define the following terms :

1. Unit.
2. Number.
3. Arithmetic.
4. Principle.
5. Concrete number.
6. Abstract number.
7. Analysis.
8. Synthesis.
9. Rule.
10. Problem.

11. Solution.
12. Notation.
13. Numeration.
14. Word method.
15. Arabic method.
16. Roman method.
17. Simple value.
18. Local value.
19. Zero.
20. Period

21. Decimal point.

25. Dime. 22. United States Money. 26. Dollar. 23. Mill.

27. Eagle. 24. Cent.

28. Significant figures. 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.

Explanation. 7 acres

Since one field contains 7 acres and the other 8 acres, the 8 acres

two fields contain 7 acres and 8 acres, which are 15 acres. 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.

Explanation. 120 ANALYSIS.—There are three numbers to be added, each con331

taining units, tens, and hundreds.

PRINCIPLE.-Only like orders can be added. 478

SYNTH ESIS.—Hence we write the numbers with the units' 929 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.

2. Find the sum of 25, 206, and 9837.

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Process.

Explanation.
25

ANALYSIS.
206
25

2 tens + 5 units. 9,837 206

2 hundreds + 0 tens + 6 units. 9,837 9 thousands + 8 hundreds + 3 tens + 7 units. 10,068

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.

EXERCISES.

1. Find the sum of:

(1.) 234 365

(2.) 134 542

(3.)
712
314

473
321

(5.)
535
213

(6.) $2.14 $3.31 $1.50

(7.)
$6.10
$2.11
$1.34

(8.)
$31.12
$41.23
$20.44

(9.)
$231.25
$512.30
$210.44

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