to express different kinds of quantity. Thus, 36 miles, and 15 days; 5 hours 36 minutes, and 7 bushels 3 pecks.

15. A Power is the product arising from multiplying a number by itself, or repeating it any number of times as a factor.

16. A Root is a factor repeated to produce a power.

17. A Scale is the order of progression on which any system of notation is founded. Scales are uniform and varying.

18. A Uniform Scale is one in which the order of progression is the same throughout the entire succession of units.

19. A Varying Scale is one in which the order of progression is not the same throughout the entire succession of units.

20. A Decimal Scale is one in which the order of progression is uniformly ten.

21. Mathematics is the science of quantity.

The two fundamental branches of Mathematics are Geometry and Arithmetic. Geometry considers quantity with reference to positions, form, and extension. Arithmetic considers quantity as an assemblage of definite portions, and treats only of those conditions and attributes which may be investigated and expressed by numbers. Hence,

22. Arithmetic is the Science of numbers, and the Art of computation.

NOTE.-When Arithmetic treats of operations on abstract numbers it is a science, and is then called Pure Arithmetic. When it treats of operations on concrete numbers it is an art, and is then called Applied Arithmetic. Pure and Applied Arithmetic are also called Theoretical and Practical Arithmetic.

23. A Demonstration is a process of reasoning by which a truth or principle is established.

24. An Operation is a process in which figures are employed to make a computation, or obtain some arithmetical result.

25. A Problem is a question requiring an operation.

26. A Rule is a prescribed method of performing an operation. 27. Analysis, in arithmetic, is the process of investigating principles, and solving problems, independently of set rules.

28. The Five Fundamental Operations of Arithmetic are, Notation and Numeration, Addition, Subtraction, Multiplication, and Division.

SIGNS.

29. A Sign is a character indicating the relation of numbers, or an operation to be performed.

30. The Sign of Numeration is the comma (,). It indicates that the figures set off by it express units of the same general name, and are to be read together, as thousands, millions, billions, etc.

31. The Decimal Sign is the period (.). It indicates that the number after it is a decimal.

32. The Sign of Addition is the perpendicular cross,+, called plus. It indicates that the numbers connected by it are to be added; as 3+ 5 +7, read 3 plus 5 plus 7.

33. The Sign of Subtraction is a short horizontal line, called minus. It indicates that the number after it is to be subtracted from the number before it; as 12-7, read 12 minus 7.

34. The Sign of Multiplication is the oblique cross, X. indicates that the numbers connected by it are to be multiplied together; as 5 × 3 × 9, read 5 multiplied by 3 multiplied by 9.

35. The Sign of Division is a short horizontal line, with a point above and one below, ÷ It indicates that the number before it is to be divided by the number after it; as 18÷6, read 18 divided by 6.

Division is also expressed by writing the dividend above, and the divisor below, a short horizontal line. Thus, 18, read 18 divided by 6.

36. The Sign of Equality is two short, parallel, horizontal lines,=. It indicates that the numbers, or combinations of numbers, connected by it are equal; as 4+ 8 = 15-3, read 4 plus 8 is equal to 15 minus 3. Expressions connected by the sign of equality are called equations.

37. The Sign of Aggregation is a parenthesis, (). It indicates that the numbers included within it are to be considered together, and subjected to the same operation. Thus, (8 + 4) × 5 indicates that both 8 and 4, or their sum, is to be multiplied by 5. A vinculum or bar,, has the same signification. Thus, 7x9÷3=21.

38. The Sign of Ratio is two points,. Thus, 7:4 the ratio of 7 to 4.

39. The Sign of Proportion is four points, :: 36 48, is read, 3 is to 6 as 4 is to 8.

40. The Sign of Involution is a number written abo little to the right, of another number. It indicates the which the latter is to be raised. Thus, 123 indicates th to be taken 3 times as a factor; the expression is equi 12 x 12 x 12. The number expressing the sign of inv called the Index or Exponent.

41. The Sign of Evolution, ✔, is a modification of th It indicates that some root of the number after it is to be e Thus, ✔25 indicates that the square root of 25 is to be e 64 indicates that the cube root of 64 is to be extracte

AXIOMS.

42. An Axiom is a self-evident truth. The princip required in arithmetical investigations are the following: 1. If the same quantity or equal quantities be added quantities, the sums will be equal.

2. If the same quantity or equal quantities be subtra equal quantities, the remainders will be equal.

3. If equal quantities be multiplied by the same nu products will be equal.

4. If equal quantities be divided by the same number tients will be equal.

5. If the same number be added to a quantity and s from the sum, the remainder will be that quantity.

6. If a quantity be multiplied by a number and th divided by the same number, the quotient will be that q 7. Quantities which are respectively equal to any othe are equal to each other.

8. Like powers or like roots of equal quantities are e 9 The whole of any quantity is greater than any of

mi

NOTATION AND NUMERATION.

43. Notation is a system of writing or expressing numbers by characters; and,

44. Numeration is a method of reading numbers expressed by characters.

45. Two systems of notation are in general use the Roman and the Arabic.

`NOTE. — The Roman Notation is supposed to have been first used by the Romans: hence its name. The Arabic Notation was first introduced into Europe by the Moors or Arabs, who conquered and held possession of Spain during the 11th century. It received the attention of scientific men in Italy at the beginning of the 13th century, and was soon afterward adopted in most European countries. Formerly it was supposed to be an invention of the Arabs; but investigations have shown that the Arabs adopted it from the Hindoos, among whom it has been in use more than 2000 years. From this undoubted origin it is sometimes called the Indian Notation.

THE ROMAN NOTATION.

46. Employs seven capital letters to express numbers. Thus,

47. The Roman notation is founded upon five principles, as

follows:

1st. Repeating a letter repeats its value. Thus, II represents two, XX twenty, CCC three hundred.

2d. If a letter of any value be placed after one of greater value, its value is to be united to that of the greater. Thus, XI represents eleven, LX sixty, DC six hundred.

3d. If a letter of any value be placed before one of greater value, its value is to be taken from that of the greater. Thus, IX represents nine, XL forty, CD four hundred.

4th. If a letter of any value be placed between two letters, each of greater value, its value is to be taken from the united value of the other two. Thus, XIV represents fourteen, XXIX twentynine, XCIV ninety-four.

5th. A bar or dash placed over a letter increases its value one thousand fold. Thus, V signifies five, and V five thousand; L fifty, and L fifty thousand.

NOTES.-1. Though the letters used in the above table have been employed the Roman numerals for many centuries, the marks or characters used origi nally in this notation are as follows:

2. The system of Roman Notation is not well adapted to the purposes of numerical calculation; it is principally confined to the numbering of chapters and sections of books, public documents, etc.

11. Five hundred fifty-five.

12. Seven hundred ninety-eight.

13. One thousand three.

14. Twenty thousand eight hundred forty-five.