PREFACE. THE following pages are principally deductions from the Author's manuscripts, which he has, from time to time, composed, for the use of his own Pupils; and which he now offers to the Public, under a conviction, from long experience, that the different Rules and Examples are practically adapted to the present improved state of the science. The chapter on Mental Calculations, by which the value of any number of articles from 1 to 10,000, as well as the interest of money, may be found with as little trouble, and in most cases less, than by the use of a Ready Reckoner, merits a careful perusal. That much neglected rule, "Profit and Loss," is reduced to actual practice. It is divided into four Cases, with practical examples to each. The questions under this head are, for the most part, taken from commercial transactions. The Table at page 155, contains an easy and concise method for finding the selling price of any commodity, in order to gain or lose a given rate per cent. Questions similar to 9, 10, 11, and 12, in Multiplication of Vulgar Fractions, page 106, are frequently given in books on arithmetic. They may have their use; but are here only introduced to show the fallacy of their nature, As to question 9, to multiply 2s. 6d. by 3s. 6d., correctly speaking, this is to multiply 2s. 6d. by 34, and not 3s. 6d. In question 10, the 17s. 6d. is multiplied by, and not 13s,4d. Question 11, the only intelligible meaning, as De Morgan justly observes, is, if a stock of money is to be increased at the rate of £99 19s. 11 d. for every £. in it; how much would that be, when the stock itself is £99 19s. 11 d.? Similar observations will hold good for Question 12, as also Question 14 in Division. Were we to incorporate the idea of Duodecimals, we should readily conceive, that to multiply shillings by shillings, the product would be square shillings; as also, to multiply cwt. by cwt., the product would be square cwt., terms with which we are totally unacquainted. In the Cube Root, a rule is given for finding nearly one-half of the root by Simple Division; and this portion, comprising the latter part of the figures in the root, will, in long calculations, be found particularly advantageous. The rule for finding the number of gallons that a pump raises per stroke, will be found of use to the plumber, &c. The work is concluded with an Appendix, containing some additional questions and illustrations on the principles of Arithmetic; and Demonstrations to the principal Rules which will form an agreeable exercise to the more advanced student. Leeds, January 2nd, 1843. EXPLANATION OF THE CHARACTERS USED IN THIS WORK, WITH WHICH THE PUPIL CANNOT BE TOO EARLY ACQUAINTED. +(Plus) The sign of addition signifies that the numbers between which it stands are to be added together; thus 48, denotes that 8 is to be added to the 4. -(Minus.) The sign of subtraction; thus, 8-4,denotes that 4 is to be subtracted from the 8. (Multiple.) The sign of multiplication; thus, 3×4 denotes that 3 is to be multiplied by the 4. ÷ (Division.) The sign of division; when placed between two quantities, signifies that the former is to be divided by the latter; thus, 8 4 denotes that the 8 is to be divided by 4. : = (Equals.) The sign of equality generally placed between two equal quantities; thus 5+2 = 7, i. e. 5 added to 2 is equal to 7. ::: (Proportion.) When four quantities are connected by these signs, it is called a proportion; thus, 2 : 4 :: 6 : 12 denotes an equality of ratio, and is read, as 2 is to 4, so is 6 to 12. { } (Brackets.) They are employed to show that all the quantities within them are to be treated as though forming but one quantity. .. This sign stands for therefore. ... This sign is used for because or since. ✔or () denotes that the square root of the quantity which 플 3 it includes is to be extracted, as 25 or (25)± = 5. 3√or ( ) } The cube root. ( or () The 4th root. I -or () The nth root, and so on. 2 as Denotes that the quantity inclosed is to be squared, [25]2 or(25)2 is equal to 25 × 25 = 625. ()3 Denotes that the quantity enclosed is to be cubed. ()" Denotes the nth power, and so on. ERRATA. Line 4, page 7, for as the, read as if the. Page 23, in the price of gold, for 101d. read 10žd. Quest. 10, page 45. Ans. 4 yrs. 9m. 22 days, 3 hrs. 10, page 94, Ans. 40 30 and 128 809 809 80⚫ Page 119, in the note, for add infinitum, read ad-infinitum. Quest. 16, page 121, for decimal of a £. read decimal of £5. |