ology to the occasion, and capable of being correctly understood by those for whom it is intended. Comparative Arithmetic, Analogy ("resemblance between things with regard to circumstances or effects,') the inductive principle, persuasive or mental Arithmetic—these terms are nearly synonymous, and may with propriety be so considered. The idea that mathematics can be taught to any extent by analogy, induction, or degrees of comparison, without rules and actual calculation, merely by the effort or operation of the mind, without the use of figures, is unquestionably absurd and inconsistent in the extreme ; and, strange it may appear, some authors and teachers advocate the principle, and call it precocity or march of intellect; the system is neither founded in common sense, nor on philosophical or mathematical principles, for it is self-evident that every effect must be produced by some corresponding or sufficient cause: hence arises the necessity of rules or data, on which to found a system of calculations, which must operate with so much certainty that they shall self-evidently prove their own theory and correctness, or which carry their own evidence with them—the eternal principle of truth; this can only be done by actual calculation-by induction never. Can the surveyor calculate the different angles of his survey by analogy, and without the use of decimal tables, and the application of rules, prepared for that express purpose ? Can the navigator take solar and lunar observations, and calculate the places, southing, and setting of the stars, on the inductive principle? Can the astronomer from analogy (induction), and without figures, trace the abstruse paths or parabolic orbits of the comets, and calculate the powers of attraction and repulsion, the distances and motions of the planetary system? Does not every mathematician know, that the sciences mentioned above require the most intense study, and the application of certain unerring rules? A variety of questions and examples have been given in square and cubical measure, mensuration, etc., which will be found highly useful and entertaining to the pupil, and in numbers sufficient for the usual occurrences of practical business life. A free use of the black-board in school is earnestly recommended, to make the ready calculator. To those who have used or recommended the preceding edition of this work, the author would embrace the present opportunity of tendering his sincere acknowledgments, for the kind and obliging manner uniformly manifested in his behalf. A Key to this work is now published. À SHORT INTRODUCTION TO AMERICAN CURRENCY When this country was subject to the crown of England, we were governed by their laws, and adopted their method of reckoning money, in pounds, shillings, and pence, which differed in value in the several colonies, as they were then called, but are now called states; neither was it of the same value of the currency of England. The pound currency of England was once equal to a pound avoirdupois, which was many times its present value. Under the colonial government, the several states issued bills of credit to supply the want of specie, and to answer a medium of trade ; but as these bills were not received by the British merchants in payment for goods at their par value, holders of the bills had to pay more. Thus, in New York they had to pay in the bills of the state at the rate of 8 shillings for 4 shillings and 6 pence sterling, and so in proportion to the depreciation of the bills of the other states. 'Taking 4 shillings 6 perce sterling as the value of a dollar, the currency of the New England states was 6 shillings to the dollar: or, 6 shillings New England currency was worth 4 shillings and 6 pence English currency; New York and North Carolina, 8 shillings to a dollar, or equal to 4 shillings and 6 pence English; Virginia, Kentucky, Tennessee, and Ohio, 6 shillings; New Jersey, Pennsylvania, Delaware, and Maryland, 7 shillings and 6 pence; South Carolina and Georgia, 4 shillings and 8 pence; (Canada and Nova Scotia, 5 shillings). This state of the cyrrency was attended with much inconvenience, and is at the present day, but is fast going into disuse, much to the relief of the community. After the war of the revolution we became a separate and independent nation, framed our own laws, coined money, fixed the value of currency, &c. Our coins are of gold, silver, and copper, and their weight and value so proportioned as to increase from the lowest to the highest by tens, or in a tenfold proportion; that is, ten of every lower denomination, or less value, make one of the next higher, and consequently one of every higher makes ten of the next lower. Thus we say, 10 mills make, or are equal to 1 cent in value ; 10 cents are equal to 1 dime; 10 dimes are equal to 1 dollar; and 10 dollars are equal to 1 eagle. There are also the half cent, half dime, quarter and hali dollar, quarter and half eagle ; but in dolls. cts. mills. 5 25 5 reckoning money, it is customary to use dollars, cents, and mills, omitting the other denominations. In writing numbers in dollars, cents, and mills, we leave a small space between them, or separate them with a comma (,) : five dollars twenty-five cents and five mills ; dollars 5, 25, 5; or dollars 5, 2, 5, 5-five dollars two dimes five cents and five mills. The rule for adding several sums together in dollars, cents and mills, is this :-add all the mills, and carry one for every ten, and add this one to the next figure in the place of cents ; but if it is less than ten, set it down directly under the column of mills, then add the cents in the first column, and carry one for every ten to the next column of cents, which add, and carry one for every ten, as before, to the place of dollars ; add the dollars and set down the full amount ; for even tens write a cipher (0). If A owes you dollars 5, 25, 5, and B dollars 8, 82, 9, how much do both owe you? A 5, 25, 5 B 8, 82, 9 14, 08, 4 In this example, we say 9 and 5 are 14, set down 4, and carry 1 to the next figure 2, which will make 3, and 5 are 8; set down the 8, and nothing to carry, because it is less than 10; then 8 and 2 are 10, set down a cipher, because it is even (10), and carry 1 to the next figure 8, will make 9, and 5 are 14; set down all the last column, so that the two sums owing by A and B, when added together, will make dollars 14, 08, 4—fourteen dollars, eight cents, and four mills. Note.- No scholar should be permitted the use of an arithmetic until he has been exercised in the first four rules, both mentally and on the board; every school should have several boards, one for each class, and not let a day pass without their use, by having the teacher give examples first, then the pupils, beginning with the first, and continue through the class, without the book; prove their questions, and give their reasons: this will exercise the mind, and bring forward all the faculties; in this way, there will be more knowledge acquired in one month than there can be in ten in conning over a book, which is a blank to them, and but little better to their teachers. It will attract attention and excite emulation, more than any other course, which is the mainspring in the acquisition of knowledge -STRICT ATTENTION. TABLE OF CONTENTS. 101 114 to 135 136 140 159 162 166 168 172 177 179 183 185 187 188 194 195 195 202 208 212 219 220 221 225 23C 235 235 to 240 246 249 258 263 Signs. Significations. = Equal; as, 10 mills=1 cent, 10 cents=1 dime, 10 dimes=1 dollar, 10 dollars=1 eagle; and when placed between two numbers it denotes that they are equal to each other. + More, or addition; as, 6+2=8, 3+2=5; when the num bers are small, we can say 6 and 2 are 8, 3 and 2 are 5. This sign is sometimes placed at the right of the quotient, or answer, denoting that there is a sinall remainder. It is also called plus, meaning more. Less (minus), or subtraction; when placed between two numbers, it denotes that the one on the right is to be taken from the one on the left; thus, 5--3=2 denotes that 3 is to be taken from 5, will =2, or 2 over, which is the remainder; 5 is the minuend, and 3 the subtrahend; when the numbers are sinall, you can say 3 from 5 leaves 2. It is sometimes used in division, signifying a remainder, as 2 in 5, 2–1 remain der; 2 in 5 twice and 1 over. x Into, or multiplied by, sign of multiplication ; thus, 4 x 4=16, 5x5=25, denotes when placed between two numbers that they are to be multiplied together; 6 x 6=36, that is, 6 times 6 are 36. X is sometimes used. • Division, or divided by; as, 25=5=5, or 8)64(8, or $4 = 8, meaning that 25 divided by 5, the answer will be 5, : 64 divided by 8=8 answer: again, 4 denotes that 64 is divided, 8: and 8)** signifies that 64 is divided by 8. E. eagle, D. or $, dollar, d. dime, c. or cts. cents, m. mills. Period, or decimal point, separatrix, used to distinguish decimals from whole numbers, as those on the right are decimals, and on the left (if any) are whole numbers ; thus, .25 ; 46 .24: it is used to separate D. c. m.; the comma (,) is also used for this purpose. :::: Proportion; as, 2:4 :: 8:16, that is, as 2 is to 4, so is 8 to 16; or the same proportion that 4 is to 2, so is 16 to 8. 12--3+5=4, a vinculum; the line over the 3 and 5 connectés all the numbers over which it is drawn as simple numbers, as 3 and 5 are 8, from 12=4. ✓ or ) Square Root; as, ?/6438: denotes that the square root of 64 is 8. Cube Root; as, /64=4:1 Biquadrate Root, as V64=2. Dollars, dimes, and mills increase, and are calculated the same as whole numbers: |