tion, Multiplication, and Division of COMPOUND NUMBERS-Notation, Reduction, Addition, Subtraction, Multiplication, and Division of FRACTIONS, Vulgar and Decimal - ALGEBRA ARITHMETICAL AND GEOMETRICAL PROGRESSION-PROPORTION, as founded on Geometrical Progression - PRACTICE-Involution and Evolution of Numbers. In Notation it is explained how figures change their value by a change in their position, and thus the reason for carrying and borrowing in Addition and Subtraction is made easily perceptible to the student, when he enters on these rules. These rules are treated of conjointly, and made subservient in proving each other; and here the reason for carrying is explained: Multiplication and Division are also made to prove each other, and are shown to be repeated Additions and Subtractions. The principle of carrying, and the relatative value of figures, is further explained. Measure is explained, and the principles of the rules for finding the greatest common measure, and the least common mul. tiple, are made plain. The relationship of multiple and sub-multiple is fully considered. The Notation of simple and compound numbers is compared. The principle of carrying in compound numbers is explained. The origin and nature of Fractions are considered, and their connexion with Proportion and Algebraic Equations is shown. The powers and roots of numbers are treated of scientifically, and illustrated by Arithmetical and Geometrical Progression. Proportion is also treated of with relation to Geometrical Progression. Under the head of General Proportion are included, the Rule of Three, Double Rule of Three, The Chain Rule, Interest, Exchange, Barter, Loss and Gain, Fellowship, Discount, and some others; also, the method is shown of working the same questions by Equations, and the two modes are compared. In the course of the work are exhibited many concise modes of calculation, adapted to matters of business, and tending to facilitate mercantile calculations. The method of calculating Interest, Commission, Discount &c. is very brief, but founded on principles of unerring certainty, which principles are fully explained. In all the hitherto published Arithmetics, the answers are given to the different questions. Of this plan the Author disapproves. When a boy at school has a question given him to work, the first thing he does is to look and see what the answer is; he then exercises his ingenuity, not to discover the principles which he should apply, and which would infallibly lead to the right result, but how he may make his solution conform with the given answer. Principles are thus entirely lost sight of, the consequence of which is, that when he leaves school and goes to business, where he has to work questions promiscuously, as they arise, and where no answers are given, he is totally at a loss how to proceed. How common it is to hear school boys, when a question is put to them, ask, What is the answer? What rule is it in ? Their knowledge of the leading principles of the different rules is so imperfect that they can seldom discover to which of those rules the examples belong. These observations do not equally apply to all scholars. Exceptions there certainly are. Some will find out the right course in spite of the obstacles thrown in their way, but they apply to the greater part of them; and I believe there are few that are thoroughly versed in the principles of arithmetic who have not obtained a knowledge of these principles for themselves, after they have left school and gone to business, when they find that a mere acquaintance with abstract rules, without the knowledge of principles, affords them but little assistance in the solution of practical examples. These are the Author's reasons for declining to give answers to the questions, that the student may depend on his knowledge of principles for an accurate solution of the questions submitted. For the convenience of Teachers, however, and of such students as may pursue their studies without the assistance of a Teacher, a KEY will shortly be published. As a school book, it is hoped that this work will be found useful, by relieving the teacher in a great measure from a laborious and irksome part of his duties, by enabling the pupil to acquire a knowledge of the principles of this useful science with little other explanation than what this work supplies. In using this Work as a basis of instruction, different methods may be followed. That which the author would take the liberty to recommend, would be, to cause the pupils in each particular rule, to read deliberately in class what is said on each rule, and request them to reflect on what they have read. The questions for examination given with the rule may then be asked, to see how far they understand the subject; and when they can answer these questions pretty correctly, the Teacher may then catechise them more minutely, by putting questions of his own. Technical and other difficult terms are explained either in the text or in a note. If this course, or something similar, be pursued, and care be taken that the pupil clearly understands not only every sentence, but every word that he reads, and if these exercises be associated with the mechanical or practical part of the rule, there is no boy of common parts who may not be made an expert arithmetician. The great object of the Teacher should be to get his pupil to reflect; this attained, the Teacher's labour will be trifling. Throughout the Work, it has been the object of the Author to render Arithmetic easy, by rendering it intelligible - not only to lay down rules, but to explain the reasons upon which every rule is founded; and thus to exhibit Arithmetic, not merely as a mechanical art. but as a pleasing science. The paragraphs are numbered for the purpose of reference. Marlborough Crescent Academy, 30th of 8 Mo., 1836. ARITHMETICAL CHARACTERS. =(equal to) denotes equality; thus 20s. = £1. is the sign of Proportion; as 2:4:3:6; that is, as 2 is to 4, so is 3 to 6. .. signifies therefore. =3. the Radical Sign, denotes the Square Root; thus 9 denotes the Cube Root; * denotes the Biquadrate Root, &c. The root of a number may also be expressed by a fractional exponent: that iɛ, a small fraction at the upper corner, on the right hand of the number. Thus, 4* denotes the square root of 4, and 16 denotes the cube root of 16. When a number is multiplied a number of times into itself, the number of factors employed in such multiplication may be denoted by a small figure written at the upper corner of the right hand of the number. Thus 2' denotes 2 X 2 = 4, that is, the square or second power of 2; 23 denotes 2 × 2 × 2 8, that is, the cube or third power of 2, &c. Both roots and powers of numbers may be expressed by fractional exponents. Thus 83 denotes the cube root of the square of 8, or it denotes the square of the cube root of 8, :, 83 = 3√/82 = 18 Division is frequently denoted by writing the dividend and divisor in the form of a fraction, thus denotes that 12 is to be divided by 4. abc denotes a × 6 × c, that is, the continued product of a, b, and c. 4 a denotes 4 X a. (2 + 5) × 3, denotes being multiplied by 3. aaaa Xa Xaa3. a that 2 is to be added to 5, previous to When several quantities are enclosed in brackets; thus (2 + 4 − 3), or have à vinculum over them, thus, 243, they are to be considered as one quantity in all the operations performed upon them. Multiplication is frequently denoted by a dot, thus 4·2 — 4X2=8 ERRATA. Page 14. First word, for en, read ten. 31. Fifth line from the bottom of par. 82, for right of the multiplier, read right of the multiplicand. 100. Seventh example, for average quantity, read average quality. 106. Formula, ninth line from the top, instead of or 6 = 3 of of 6 = 11⁄2 &c., read of 6 = 3, 4 of 6 = 14, &c. 108. Last line but one, for we und, read we find. 149 First line at the beginning, for of these decimals, read of the fractions. 168. Example, divide 68437658 by 990, for 9128 — 938 in the quotient, read 69128 - 938. 174. Examples, divide by 999, and by 990, the top line of each example, are advanced a space too much to the left, which must be taken into account in adding up. 177. Last line of paragraph 454 for (64) read (64)5 178. Last line, but one of paragraph 455, at the beginning, for 4×, read 4 + 1. 180. Fifth line of paragraph 463, at the end, for a, read a3 and at the seventh line of the same paragraph, at the end, for a3, read a}. |