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

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

40. The Sign of Involution is a number written above, and a little to the right, of another number. It indicates the power to which the latter is to be raised. Thus, 123 indicates that 12 is to be taken 3 times as a factor; the expression is equivalent to 12 x 12 x 12. The number expressing the sign of involution is called the Index or Exponent.

41. The Sign of Evolution, ✔, is a modification of the letter r. It indicates that some root of the number after it is to be extracted. Thus, 25 indicates that the square root of 25 is to be extracted; 364 indicates that the cube root of 64 is to be extracted.

AXIOMS.

42. An Axiom is a self-evident truth. The principal axioms required in arithmetical investigations are the following:

1. If the same quantity or equal quantities be added to equal quantities, the sums will be equal.

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

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

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

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

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

8. Like powers or like roots of equal quantities are equal.

9 The whole of any quantity is greater than any of its parts. 10. The whole of any quantity is equal to the sum of all its parts.

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.

THE ARABIC NOTATION

48. Employs ten characters or figures to express numbers.

Thus,
Figures,

0 1 2 3 4 5 6 7 8 9
two, three, four, five,
six, seven, eight, nine.

Names and values.

naught
cipher.

one,

or

49. The cipher, or first character, is called naught, because it has no value of its own. It is otherwise termed nothing, and zero. The other nine characters are called significant figures, because each has a value of its own. They are also called digits, a word derived from the Latin term digitus, which signifies finger.

50. The ten Arabic characters are the Alphabet of Arithmetic. Used independently, they can express only the nine numbers that correspond to the names of the nine digits. But when combined according to certain principles, they serve to express all numbers.

51. The notation of all numbers by the ten figures is accomplished by the formation of a series of units of different values, to which the digits may be successively applied. First, ten simple units are considered together, and treated as a single superior unit; then, a collection of ten of these new units is taken as a still higher unit; and so on, indefinitely. A regular series of units, in ascending orders, is thus formed, as shown in the following

52. The various orders of units, when expressed by figures, are distinguished from each other by their location, or the place they occupy in a horizontal row of figures. Units of the first order are written at the right hand; units of the second order occupy the second place; units of the third order the third place; and so on, counting from right to left, as shown on the following page :

53. In this notation we observe

1st. That a figure written in the place of any order, expresses as many units of that order as is denoted by the name of the figure used. Thus, 436 expresses 4 units of the 3d order, 3 units of the 2d order, and 6 units of the 1st order.

2d. The cipher, having no value of its own, is used to fill the places of vacant orders, and thus preserve the relative positions of the significant figures. Thus, in 50, the cipher shows the absence of simple units, and at the same time gives to the figure 5 the local value of the second order of units.

54. Since the number expressed by any figure depends upon the place it occupies, it follows that figures have two values, Simple and Local.

55. The Simple Value of a figure is its value when taken alone; thus, 4, 7, 2.

56. The Local Value of a figure is its value when used with another figure or figures in the same number. Thus, in 325, the local value of the 3 is 300, of the 2 is 20, and of the 5 is 5 units. NOTE.-When a figure occupies units' place, its simple and local values are the same.

57. The leading principles upon which the Arabic notation is founded are embraced in the following

GENERAL LAWS.

I. All numbers are expressed by applying the ten figures to dif ferent orders of units.

II. The different orders of units increase from right to left, and decrease from left to right, in a tenfold ratio.

III. Every removal of a figure one place to the left, increases its local value tenfold; and every removal of a figure one place to the right, diminishes its local value tenfold.