Numeration Subtraction Multiplication INDEX. Division Tables of Money, Weights, and Measures Compound Subtraction Reduction Compound Reduction Reduction of the old New England currency Multiplication of Federal Money Bills in Federal Money Compound Multiplication Bills in English Money Compound Division Vulgar Fractions Decimal Fractions Mental Operations in Fractions Exchange of Currencies Circulating Decimals Simple Interest Interest with Partial Payments Miscellaneous Problems in Interest Page 9 13 17 21 25 34 40 44 49 55 57 58 61 65 68 71 75 78 109 120 122 133 140 146 148 150 151 152 153 154 155 158 161 163 169 Extraction of the Square Root Extraction of the Cube Root Rule for extracting the Roots of all Powers Arithmetical Progression Geometrical Series Infinite Series Discount by Compound Interest 179 182 184 186 188 190 192 195 196 197 Annuities at Compound Interest . 217 Alligation Medial 220 Alligation Alternate 221 Permutations and Combinations 224 CHARACTERS USED IN THIS WORK. $ Contraction, for U. S. United States' currency, and is prefixed to dollars and cents. = Sign of equality; as 12 inches1 foot, signifies, that 12 inches are equal to one foot. : Sign of addition; as 8+6=14, signifies, that 8 added to 6 is equal to 14. Sign of subtraction; 8—6—2, that is, 8 less 6 is equal to 2. × Sign of multiplication; as 7×6=42, that is, 7 multiplied by 6, is equal to 42. Sign of division; as 42÷6-7, that is, 42 divided by 6 is equal to 7. Numbers placed in this manner imply, that the upper line is to be divided by the lower line. :: : Signs of proportion; thus, 2:4::6:12, that is, 2 has the same ratio to 4, that 6 has to 12; and such numbers are called proportionals. 12—3+4=13. Numbers placed in this manner show, that 3 9 83 is to be taken from 12, and 4 added to the remainder. The line at the top is called a vinculum, and connects all the numbers over which it is drawn. Implies, that 9 is to be raised to the second power; that is, multiplied by itself. 8 Implies, that 8 is to be multiplied into its square. This sign prefixed to any number shows, that the square root is to be extracted. This sign prefixed to a number, shows, that the cube root is to be extracted. ARITHMETIC. SECTION I. ARITHMETIC is the art of computing by numbers. Its five principal rules are Numeration, Addition, Subtraction, Multi plication, and Division NUMERATION. Numeration teaches to express the value of numbers either by words or characters. The numbers in Arithmetic are expressed by the following ten characters, or Arabic numeral figures, which the Moors introduced into Europe about nine hundred years ago; viz. 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, O cipher, or nothing. The first nine are called significant figures, as distinguished from the cipher, which is of itself insignificant. Besides this value of those figures, they have also another, which depends on the place in which they stand, when connected together; as in the following table. Here any figure in the first place, reckoning from right to left, denotes only its simple value; but that in the second place, denotes ten times its simple value; and that in the third place a hundred times its simple value; and so on; the value of any successive place being always ten times its former value. Thus in the number 1834, the 4 in the first place denotes only four units, or simply 4; 3 in the second place signifies three tens, or thirty; 8 in the third place signifies eighty tens or eight hundred; and the 1, in the fourth place, one thousand; so that the whole number is read thus,—one thousand eight hundred and thirty-four. As to the cipher, 0, though it signify nothing of itself, yet being joined to the right hand of other figures, it increases their value in a tenfold proportion; thus 5 signifies only five, but 50 denotes 5 tens or fifty; 500 is five hundred ; and so on. NOTE. The idea of number is the latest and most difficult to form. Before the mind can arrive at such an abstract conception, it must be familiar with that process of classification, by which we successively remount from individuals to species, from species to genera, from genera to orders. The savage is lost in his attempts at numeration, and significantly expresses his inability to proceed, by holding up his expanded fingers or pointing to the hair of his head. See Lacroix. |