Questions have been inserted at the bottom of each page, designed to direct the attention of teachers and pupils to the most important principles of the science, and fix them in the mind; it is not intended, however, nor is it desirable, that the teacher should servilely confine himself to these questions, but vary their form, and extend them at pleasure, and invariably require the pupil thoroughly to understand the subject, and give the reasons for the various steps in the operation by which he arrives at any result in the solution of a question. The object of studying mathematics is not only to acquire a knowledge of the subject, but also to secure mental discipline, to induce a habit of close and patient thought, and of persevering and thorough investigation. For the attainment of this object, the examples for the exercise of the pupil are numerous, and variously diversified, and so constructed as necessarily to require careful thought and reflection for the right application of principles. The author would respectfully suggest to teachers, who may use this book, to require their pupils to become familiar with each rule before they proceed to a new one; and, for this purpose, a frequent review of rules and principles will be of service, and will greatly facilitate their progress. If the pupil has not a clear idea of the principles involved in the solution of questions, he will find but little pleasure in the study of the science; for no scholar can be pleased with what he does not understand. The article on weights, measures, and money, will be found, it is believed, to contain valuable information, and such as no similar work places within the reach of pupils. This addition, it is hoped, will be found interesting to teachers and scholars. BENJAMIN GREENLEAF. Bradford Teachers' Seminary, Nov. 15th, 1848. NOTICE. CONTENTS. Page 8 Apothecaries' Weight, Table, 88 Exercises in French Numeration, 12 Long Measure, Table. Exercises in French Notation and Nu- Surveyors' Measure, Table, 14 Cubic or Solid Measure, Table, 97 Exercises in English Numeration, . 15 Wine Measure, Table, Exercises in English Notation and Ale and Beer Measure, Table, 101 Circular Measure or Motion, Table, 105 16 Miscellaneous Exercises in Reduc. ADDITION OF COMPOUND NUMBERS, — SUBTRACTION — Mental Exercises, 26 English Money, 27 Examples for Practice in the different MULTIPLICATION — Mental Exercises, 35 SUBTRACTION OF COMPOUND NUMBERS, MISCELLANEOUS EXERCISES IN ADDI. CONTRACTIONS IN MULTIPLICATION AND Contractions in Multiplication, 59 MULTIPLICATION OF COMPOUND NUM- MISCELLANEOUS EXAMPLES INVOLVING 63 DIVISION OF COMPOUND NUMBERS, 125 MISCELLANEOUS EXAMPLES IN MULTI- Reduction of United States Money, Addition of United States Money, . 69 PLICATION AND DIVISION OF COM- Subtraction of United States Money, 71 Multiplication of U. States Money, 72 Division of United States Money, 73 CANCELLATION, Practical Questions by Analysis, 74 PROPERTIES AND RELATIONS OF NUM- . . . The Greatest Common Divisor, 136 ASSESSMENT OF Taxes, 229 SECTION XIX. 232 FRACTIONS -- VULGAR FRACTIONS, . 140 Reduction of Vulgar Fractions, 141 Subtraction of Vulgar Fractions, Multiplication of Vulgar Fractions, 155 PROPORTION, 233 Miscellaneous Exercises in Vulgar Reduction of Fractions of Compound PARTNERSHIP, OR COMPANY BUSINESS, 243 Addition of Fractions of Compound Subtraction of Fractions of Com- Miscellaneous Examples in Profit and Questions to be performed by Analy. Miscellaneous Questions by Analy- 179 Addition and Subtraction of Duodeci. SECTION XX. Multiplication of Duodecimals, 259 Numeration of Decimal Fractions, 182 Notation of Decimal Fractions, 188 Extraction of the Square Root, 264 190 Application of the Square Root, 263 Miscellaneous Exercises in Decimals, 192 Extraction of the Cube Root, 273 277 Application of the Cube Root, ARITHMETICAL PROGRESSION, 279 Annuities at Simple Interest by Arith- SECTION XXIII. SECTION XLI. Miscellaneous Exercises in Interest, 206 Annuities at Compound Interest by Geometrical Progression, Table, SECTION XXV. SECTION XLIII. SECTION XXVI. SECTION XLIV. SECTION XXVII. SECTION XL COMMISSION AND BROKERAGE, 222 MENSURATION OF SOLIDS, · SECTION XXVIII. SECTION XLVI. Stocks, 224 MENSURATION OF LUMBER AND TIM- BER, 310 SECTION XXIX. • 226 SECTION XLVII. MISCELLANEOUS QUESTIONS, 311 SECTION XXX. DUTIES, 227 | WEIGHTS, MEASURES AND MONEY, , 318 . 1 ARITHMETIC. ARTICLE 1. ARITHMETIC is the science of numbers and the art of computing by them. A number is a unit or an assemblage of units. A unit or unity is the number one, and signifies an individual thing or quantity. The introductory and principal rules of arithmetic are Notation, Numeration, Addition, Subtraction, Multiplication, and Division. The last four are called the principal or fundamental rules, because all arithmetical operations depend upon them. § I. NOTATION AND NUMERATION. Art. 2. Notation is the art of expressing numbers by figures or other symbols. There are two methods of notation in common use; the Roman, and the Arabic or Indian.* ART. 3. The Roman notation employs seven capital letters, viz. : I, for one; V, for five ; X, for ten ; L, for fifty; C, for one hundred; D, for five hundred; M, for one thousand. The intermediate numbers and the numbers greater than one thousand are expressed by the use of these letters in various combinations; thus, II expresses two; IV, four; VI, six; IX, nine; XV, fifteen; &c. * For the origin of our present numeral characters, see the History of Arithmetic in the larger work of the author. QUESTIONS. - Art. 1. What is arithmetic? What is number? What is a unit or unity ? Which are the principal or fundamental rules of arithmetic ? Why are they called the principal rules ? - Art. 2. What is notation How many and what methods of notation are in common use ? — Art. 3. What are used to express numbers in the Roman notation ? What are their names ? When two or more equal numbers are united, or a less number follows a greater, the sum of the two represents their value ; as, XX, twenty; VI, six. But when a less number is placed before a greater, the difference of the two represents their value; as, IV, four ; IX, nine. one. seven, TABLE OF ROMAN LETTERS. 1 LX sixty. II two. LXX seventy. III three. LXXX eighty. IV four. XC ninety. V five. C one hundred. VI six. CC two hundred. VII CCC three hundred. VIII eight. CCCC four hundred. IX nine. D, or 15 five hundred. X ten. six hundred. XX twenty. DCC seven hundred. XXX thirty. DCCC eight hundred. XL forty. DCCCC nine hundred. L fifty. one thousand. Any number between unity and two thousand may be expressed by the letters in the preceding table, By first writing down the largest part of the required number, found in the table, and then annexing to this the next less, that will not make a number greater than the one required, and thus proceeding until the number is complete. EXERCISES IN ROMAN NOTATION. The learner may write the following numbers in letters : 1. Ninety-six. Ans. XCVI. 2. Eighty-seven. 3. One hundred and ten. 4. One hundred and sixty-nine. 5. Two hundred and seventy-five. 6. Five hundred and forty-two. 7. One thousand three hundred and nineteen. 8. One thousand eight hundred and forty-eight. QUESTIONS. When is the sum of two letters taken for their value ? When the difference? Repeat the Table of Roman Letters. What direc. tion is given for writing numbers in the Roman notation ? |