AXIOMS An axiom is a self-evident proposition. 1. Things which are equal to the same thing are equal to each other. 2. If equals be added to equals, the wholes will be equal. 3. If equals be taken from equals, the remainders will be equal. 4. If equals be added to unequals, the wholes will be unequal. 5. If equals be taken from unequals, the remainders will be unequal 6. Things which are double of equal things are equal to each other. 7. Things which are halves of the same thing, are equal to each other 8. The whole is greater than any of its parts. 9. The whole is equal to the sum of all its parts. SIGNS. Equality is denoted by two horizontal lines. + Addition: as 4+3=7; which signifies that 4 added to 3 equals 7. × Multiplication: as 4X3-12; which signifies that 4 multiplied by 3 equals 12. Subtraction: as 4-3-1; which signifies that 3 taken from 4 leaves 1. )(, ÷, 4, 2|4, Division: as, 2)4(2, and 4÷2—2, and 4=2, and 2|4—2. In either case it signifies that 4 divided by 2 equals 2. :::: Proportion: as, 2: 4 :: 6:12; which is read, 2 is to 4 as 6 is to 12. Vinculum: as 4+3=7; which is read, the sum of 4 and 3 equals 7, and 4—3±1, is read, the difference of 4 and 3 equals 1. ✔ Radical sign: placed before a number denotes that the square root is to be taken. 42 implies that 4 is to be raised to the second power. 43 implies that 4 is to be raised to the third power. Vimplies the third root. Reduction Descending and Ascending 121 Comparison of Numbers and Quanti- Numbers, Whole Numbers by Frac- tions, Fractions by Fractions ... ... 69 To reduce Fractions to Integers of ARITHMETIC. Art. 1.-ARITHMETIC is the science of numbers. It explains their properties, and teaches how to apply them to practical purposes. Art. 2.-The principal, or fundamental rules, are, Notation, Numeration, Addition, Subtraction, Multiplication, and Division. These are called fundamental rules, because all questions in Arithmetic are solved by one or more of them. 0 Art. 3.-Notation is the expressing of any number or quantity by figures; thus, 1 one; 2 two; 3 three; 4 four; 5 five; 6 six; 7 seven; 8 eight; 9 nine; O cipher. The first nine figures are sometimes called digits, from the Latin word digitus, which means a finger. In the early stages of society people. counted by their fingers; they were also formerly all called ciphers—hence the art of Arithmetic was called ciphering. Art. 4.—There are two methods of Notation-the Arabic, as above, and the Roman, which is expressed by the following seven letters of the alphabet: I, V, X, L, C, D, M. 1 2 3 4 5 6 7 8 9 10 20 30 40 50 I, II, III, IV, V, VI, VII, VIII, IX, X, XX, XXX, XL, L, LX, LXX, LXXX, XC, C, D, M. Art. 5.-When a letter of less, is placed before one of a greater value, it diminishes the value of the greater, by the value of itself—thus, X signifies ten, but IX is only nine. When a letter of less, is placed after one of greater value, it increases the value of the greater by the value of itself. This method is seldom used except in numbering chapters, sections, etc. QUESTIONS.-1. What is Arithmetic? 2. What are the principal, or fundamental rules? 3. Why so called? 4. What is Notation? 5. What are the first nine figures sometimes called? 6. What were they all formerly called? 7. How many methods of Notation, and what are they? 8. How many are the Arabic characters, or figures? 9. By what is the Roman method expressed? 10. How is a letter affected when one of less value is placed before it? 11. How when one of less value is placed after it? 12. For what is the Roman method of Notation principally used? NUMERATION. Art. 6.—Numeration teaches to express in words the value of any number represented by figures. Thus, 365 is read, three hundred and sixty-five. Art. 7.-Figures have a simple and relative value. When a figure stands alone its value is simply so many units, or ones; as, 2 two; 3 three; 4 four. Their relative value is derived. from the place they occupy when joined together, or from their distance from the unit's place. Thus, 2 and 3 express their own value; simply so many units; but they are made to express either 23 or 32; that is, either three units and two tens, or two units and three tens. Hence it appears that the first, or right-hand place, always expresses so many units; it is therefore called the unit's place; the second, the place of tens, expressing always as many tens as the figure contains units. The third place is hundreds; the fourth, thousands, as may be seen by the following QUESTIONS.-13. What is Numeration? 14. What is the value of a figure standing alʊne? 15. From what is their relative value derived? 16. What does the first, or right-hand figure, always express, and what is it called? 17. What are the second, third, and fourth places called? 18. What is the value of the cipher, when standing alone, or at the left hand of another figure? 19. What effect has it when placed at the right of another figure? |