PUBLISHERS' NOTICE.

It is more than sixteen years since the first Arithmetical work of Prof. STODDARD was introduced to the Teachers of our coun try. Its production was occasioned by an inquiry into the causes which have made the study of Mental Arithmetic the most irksome to the pupil, and laborious to the Teacher, of all the studies pursued in our Schools. His success in removing the difficulties which have obstructed the pupil's path, has given him an enviable reputation. No work of a similar character has ever before gained so wide spread a popularity in so short a time, as "STODDARD'S AMERICAN INTELLECTUAL ARITHMETIC." It is now the text-book in many of the Public and Private Schools and Academies of New York, Penn. sylvania, Ohio, Michigan, and Missouri, and is used extensively in the Schools of New England, where it has recently been introduced. It has received from many of the most distinguished Mathematicians of our country the unequivocal encomium: "By far the best system of Mental Arithmetic ever published."

In introducing to those who uphold our educational interests, this new work, by Prof. STODDARD, on PRACTICAL or WRITTEN ARITH METIC; the Publishers solicit for it a thorough and candid examination. They think it unnecessary to say more than that they believe it exhibits the same striking peculiarities that characterize his former work, viz.: a perfect adaptation to the gradually expanding faculties of the pupil. The cheerfulness with which classes pursue the study of the Intellectual Arithmetic, shows clearly that Prof. STODDARD has apprehended and remedied the defects of former works.

With the deepest convictions of the value of this work, it is presented to practical instructors. They are asked to judge how far its author has succeeded in making a work, which clearly states, and thoroughly explains the principles of the science; and which, at the same time, can be mastered without discouraging toil, and the knowl. edge it imparts retained without laborious effort. From their d cision there can be no appeal.

PRACTICAL ARITHMETIC.

DEFINITIONS.

1. Quantity is anything that can be increased or diminished, and made a subject of measurement and computation; as space, duration, matter, force, motion, and number.

2. A Number is a definite expression of quantity.

3. Figures are characters used to represent numbers; as, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The 0 expresses no number.

4. Like Numbers express the same kind of quantity; as 2 and 4, 2 pounds and 4 pounds, 8 yards and 9 yards.

5. Unlike Numbers express different kinds of quan tity; as, 2 yards and 4 pounds, 2 feet and 4 dollars.

6. A Unit is one. The Unit of a Number is one of that number. Thus, 1 is the unit of 6, 1 apple is the unit of 6 apples.

7. Numbers are, in expression.

1. Simple,

Abstract; as 4, 7, 9, 5.

Denominate, or Concrete; as 4 rods, 7 books. 2. Compound Denominate, or Concrete; as 4 feet 7 inches. 3. Integral; as 4, 7 miles, 8 dollars.

4. Fractional; as, of a foot, of a dollar.

5. Mixed; as 4, 5 dollars, 7 yards.

8. A Simple Number is an expression of like units, either abstract or denominate; as 3, 5 bushels, 7 miles. 9. An Abstract Number is a number that ex. presses no particular kind of quantity; as 4, 7, 8.

10. A Denominate Number is a number that expresses some particular kind of quantity; as 4 yards, 9 dollars. 7 miles.

11. A Compound Number is a number expressed by more than one denomination considered as a whole; as 4 feet 7 inches, 4 pounds 9 ounces.

12. An Integer, or an Integral Number, is any whole number.

13. A Fraction, or a Fractional Number, is one or more of the equal parts of a quantity considered as a whole; as, of a dollar, of 2 bushels which equals one bushel.

of

14. A Mixed Number is a number expressed by an integer and a fraction; as 41, 73 miles, 8 yards.

15. Arithmetic is the science of numbers and the art of computation.

As a science, Arithmetic classifies our knowledge of numbers and of computation. As an art, it applies this knowledge. By the Art of Arithmetic we perform operations in it. By the Science of Numbers, we know the principles of this Art.

16. A Problem is a question proposed for solution. 17. A Rule, in Arithmetic, is a method for the solu tion of a problem.

18. Analysis, in Arithmetic, is a statement in rega lar order of the different steps required in the solution of ✩ problem.

19. A Demonstration is a process of reasoning which establishes a truth.

20. A Sign, in Arithmetic, is a character used to indi cate an operation or a relation.

21. The Fundamental Rules, or Operations of Arithmetic are, Notation and Numeration, Addition, Subtraction, Multiplication, and Division.

NOTE. It is not necessary that these definitions be committed to memory at present. For Axioms and Scales see Stoddard's Complete Arithmetic, page 304.

NOTATION AND NUMERATION.

22. Notation is the method of expressing numbers by figures or other characters.

23. Numeration is the method of reading numberg

There are three methods of expressing numbers :

1. By words; as one, two, three, four, &c.

2. By letters; as I, V, X, &c., called the Roman Method. 3. By figures; as 1, 2, 3, 4, &c., called the Arabic Method. NOTE.-1. The Roman Notation had its origin among the Romans. 2. The Arabic Notation originated in India, and was introduced into Europe by the Arabians in the eleventh century. This method is some times called the Indian Notation.

ROMAN NOTATION.

24. The Roman Notation expresses numbers by the following seven letters:

I V X L Values, one, five, ten,

C

D

M

one

five

one

fifty, hundred, hundred, thousand.

25. All other numbers may be expressed by combinations of these letters, in accordance with the following principles :

1. Repeating a letter repeats its value. Thus, I denotes one, II two, III three, X ten, XX twenty.

2. If a letter be placed at the left of one of greater value, the difference of their values is the number represented. Thus, IV denotes four, IX nine, XL forty.

3. If a letter be placed ai the right of one of greater value, the sum of their values is the number represented. Thus, Ví denotes six, XI eleven, XV fifteen.

NOTE.-A number expressed by principle 2d, instead of a single letter, may be placed on the right of one denoting a greater value. Thus, XIV denotes fourteen, CIX one hundred and nine, XXIV twenty-four, XCIX ninety-nine.

4. A dash,—, drawn over a letter multiplies its value by ona thousand. Thus, V denotes five thousand.

ROMAN TABLE.

NOTE.-The Roman method of Notation, at present, is used, principally, for numbering lessons, chapters, sections of books, and public documents.

11. Thirty-five thousand and ninety-two.

12. Seven thousand five hundred and ten.

13. Sixteen hundred and eighty-nine.

14. Fifty-four thousand and fifty-eight.

15. One thousand eight hundred and sixty-three.