.............. 225 227 Evolution .... ..... Compound Subtraction .............. 133 Involution..... To reduce Longitude to Time........ 136 Compound Addition and Subtraction. 137 Extraction of the Square Root........ 228 Supplement to Compound Numbers.. 142 Extraction of Roots in General ....... ..... 252 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. 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; 0 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 illed 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 10 20 30 40 50 I, II, III, IV, V, VI, VII, VIII, IX, X, XX, XXX, XL, L, 60 70 80 90 100 500 1000. 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 alone? 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? Art. 8. The cipher, when standing alone, or at the left hand of another figure, signifies nothing, as 05, 005, is five in either case, because it still occupies the unit's place. But when placed at the right hand of another figure, it increases its value in a tenfold ratio, by removing the figure farther from the unit's place. This may be seen by the following— TABLE II. 0 Nothing. 20 Twenty. 200 Two hundred. 2,000 Two thousand. 20,000 Twenty thousand. 200,000 Two hundred thousand. 2,000,000 Two millions. Art. 9.-To know the value of any number of figures. RULE.—1. Numerate from the right hand to the left, by saying units, tens, hundreds, &c., as in the Table. 2. To the simple value of each figure join the name of its place, reading from the left hand to the right. |783|694 542 987 562 714 923 610 782 184 542 365 987 963| HHHHHHM The first division of the foregoing Table is according to the French method, into periods of three figures each: the name of the period is superadded. The second division is according to the English method, into periods of six figures each. The name of each period is subjoined. The two divisions of the QUESTIONS.-20. How may the value of any number be found? 21. What are the two methods of numeration in the third table? 22. In what respect do they differ? |