UC-NRLF V $B 53E nb LIBRARY OF THK University of California. ^yc^,^-^^?yC Received Accession No. (d 7^ 9 ^ i8g '^ Class No. y. lo\ 1= ?— \R\ JAgG^^^THO IV^PSOK. f^' This Question-Book, not being an exclusive text-book, but a means of additional illustration, does not necessarily require to be formally adopted by Boards of Education, but can be introduced and used at the option of Teachers. For hints on the object and proper use of these Questions, Teachers are referred to the *' Preface," and "Advantages," see pages 2, 3, 4. STANDARD SERIES OF ARITHMETICS: By JAMES S. EATON, Principal of the English Department of Thillips Academy, Andover, Mass. I. Primary Arithmetic, 100 pp. Beautifully illustrated. II. Intellectual Arithmetic, 176 pp. On a progressive plan. III. Common School Arithmetic, 312 pp. A complete practical text-book. IV. High School Arithmetic, 356 pp. A thorough and exhaustive treatise. ALSO Eaton's Grammar School Arithmetic, 336 pp. This is the ** Common School " with 24 additional pages,taken from the " High School.** Designed for schools so graded as to require but one book on Written Arithmetic. Eaton's Elements of Written Arithmetic, about 200 pp. In Press. Designed for Elementary Classes. This is the best consecutive series by the same author, on the latest im- proved plan, and adapted to the wants of Primary, Intermediate, Grammar, and High Schools, Academies, and Normal Schools. The popularity of Eaton's Books, and the largely increasing demand, may well sustain their claim to be called a NEW NATIONAL STANDARD SERIES. *** Teachers experiencing the disadvantages of using text-books by dif- ferent authors, or old, revamped and re-revised books, will avoid this use of a multiplicity of different editions by the same author, constantly appearing, by adopting the clear, scientific, concise, and practical system prepared by Mr. Eaton. These Arithmetics are Exclusively used in the Public Schools of Boston, and very extensively in Massachusetts and New England. They are the authorized and exclusive text-books in the State of California. They are also used in many hundred cities and towns, where the best schools are maintained, in all parts of the United States. Very low terms are offered for their first introduction, or they may be in- troduced gradually, at the regular prices, as new classes are formed. Send for Descriptive Catalogue. Specimen copies of the Arithmetics fur- nished for examination, with reference to introduction, on receipt of Postage. Primary 5 cents, Intellectual 10 cents. Common School and High School 20 cents each. Address, TAGGARD & THOMPSON, 29 Cornhill, Boston. QUESTIONS PEINCIPLES OF AEITHMETIC, TO INDICATE AN OUTLINE OF STUDY; TO INCITE AMONG PUPILS A SPIRIT OP INDEPENDENT INQUIRY; AND ESPECIALLY FITTED TO FACILITATE A THOROUGH SYSTEM OF REVIEWS. ADAPTED TO ANT TEXT-BOOKS AND TO ALL GRADES OP LEAENEfiS. By JAMES S. EATON, M. A. AUTHOR OP A SERIES OP ARITHMETICS, ETC. *'It should be the chief aim, in teacliing' Arithmetic, to lead the learner to a clear understand- ing of the Peinciples of the Science." — IIox. Johx D. Puilbkick, Sup't Boston Schools. BOSTON: TAGGAE.D AND THOMPSON. 18 66. TO TEACHERS. THE ADVANTAGES OF USING THESE QUESTIONS. 1. They are separate from any text-book, and equally well adapted to all text-books. On this account they present all the benefits of the Question Method^ and none of its defects. 2. They indicate a definite outline of study, and thus afford a substantial and useful guide to the pupil in the preparation of his lesson. 3. They incite the pupil to inquiry, awakening that thirst for knowledge which is the best motive to its acquirement. 4. They open up the several subjects by such short and sugges- tive steps, one question following upon another in the chain, that the pupil is thus led to follow out and develop the subject for himself. 5. By inciting the pupil to inquiry, and by guiding him in de- veloping the subject for himself, they subserve the highest and only true style of teaching, namely, to draw out and develop iJie facul- ties, and thus lead the pupil, instead of dictating to him and driv- ing him. 6. They afford the best means for frequent reviews and exam- inations, since it is the Principles of Arithmetic that should be reviewed, and not the mechanical operations. ' 7. The use of these Questions will not fail to ground the prin- ciples of Arithmetic in the mind of the pupil, and thus give him the Key which will command all practical operations. Entered, according to act of Congress, in the year 1805, BY TAGGARD AND THOMPSON, In the Clerk's Oflace of the District Court of the District of Massachusettsu PEEFACE. These Questions are offered to the intelligent Teachers of this country with the hope that they may serve as an essential help in teach- ing the important branch of Arithmetic. The plan of this book is in some respects new,- and it is thought that the use of it will tend to promote a more thorough and successful method than would perhaps otherwise be attained. It is generally agreed that the subject of Arithmetic is apt to be taught too mechanically, too much by mere rote, — by ** ciphering " from formulas rather than by an intelligent discussion of principles. Instead of directing the pupil **to cipher according to rule," he should rather be taught to perform examples by analysis, according to principles which he has mastered, and whose wide application he has been led fully to understand. If one be well grounded in the principles of a science, he has con- stantly at his command the Key to all operations pertaining to it. He is then Master of the Situation. So in Arithmetic, it is more important that the pupil should know, and be able to tell, upon what principle any given operation depends, than that he should be able to solve, according to a set formula, which he does not under- stand, any number of" similar examples whose answers are all before hi?n. That he should spend montlis and years of his schooling- m adding, multiplying, and dividing, simplij as an exercise, is no less absurd than that" a mechanic should exercise with dumb-bells and clubs to perfect the muscles of his arm, before he shall -touch the blacksmith^s hammer or the carpenter's chisel. Those few who advocate so much practice with abstract numbers in the fundamental rules, in order to perfect the pupil in addition, &c., before proceeding further 4 PREFACE. (since no subsequent page of the Arithmetic is without practice in one or all of the fundamental rules), remind one of the person who advised the news-carrier, whose business required him to use his legs all 'day, to take a walk of a few miles for exercise before beginning his day^s work. Practical examples sliould illustrate priy^ciples, but should not hurij tliem out of sight* Facility in combining numbers with celerity and accuracy attends only upon long-continued and uninterrupted prac- tice. When the occasion for such practice comes, as with the ac- countant and book-keeper, this facility also comes. The use of these Questions will, it is believed, awaken curiosity and stimulate a spirit of inquiry among pupils, and thus interest them in the subject, a thing of paramount importance. The only way in which the principles of any science can be well and thoroughly established in the mind, is by constant reviewing^ and to this end, it is believed that no method can surpass the use of theso Questions. They are adapted to any text-book, and each scholar being pro- vided with a copy of the Question-book, can seek wherever his choice may dictate for the best, and most appropriate answers. It is proper to state that though these Questions are sufficiently based upon material of Mr. Eaton's to entitle him to be called the author, the sudden decease of this distinguished teacher has devolved upon another the labor of preparing them for the press. The Editor takes pleasure in acknowledging his obligations to the several eminent teachers who examined the proof-sheets of these Questions, to whom he is indebted for valuable suggestions. It is hoped that the generous favor so universally shown towards Mr. Eaton's Series of Arithmetics will be extended to the ** Ques- tions," and that this endeavor to aid his fellow-teachers will meet with their cordial approbation. The Editor. Boston, INfovEMBER 20, 1865. QUESTIO^^S. SECTION I. NOTATION AND NmyiERATION. 1. What is Arithmetic ? 2. Define the word science* 3. Define the word aH^ 4. What is a number ? Give an example. 6. What is a unit ? Give an example. 6. What is a concrete number? Name one. 7. What is an abstract number? Name one. 8. How many fundamental operations in Arithmetic ? Define the word fundamental. What is Notation ? ^ What is Numeration ? What is the exact meaning of the word notation ? How many methods of notation are employed ? Which is most convenient in Arithmetic ? 15. How many and what are the figures of the Arabic notation ? 16. Whence the name of this method ? AVhat are the Arabic figures sometiines called ? Whence the name digit 7 What is a significant figure ? What is the largest number a single figure can express ? JHow is any number from ten to one hundred expressed? 22. How is a larger number expressed ? t23. What is understood by the place a figure occupies ? 24. Where is the &rst placCy and what is it called? \* 6 ARITHINIETIC. 25. The second place, and what called? third? fourth? &c. 26. What Is the simple value of a figure ? 27. What is the local value of a figure ? 28. What is the effect of removing a figure one place to the left? 29. Removing It two places to the left? Three? four? &c. 30. What principle is thus made evident? 31. Define the word principle, 32. What is the use of the zero (0) ? 33. How many and what are the methods of numerating? 34. How many figures comprise a period in each method ? 35. Which is mostly used in this country ? '36. Name the first six periods in the French method." 37. What is the first step In numerating ? 38. What is the second step ? 39. Explain how any given number may be written. 40. Explain the process of reading any number. 41. Illustrate the two preceding questions by examples. 42. How does the English method differ from the French? 43. How many and what characters are employed In Roman notation ? 44. Upon how many principles are the letters combined in Roman notation? 45. Name and Illustrate each of them. 46. For what is Roman notation chiefly used ? SECTION n. 1. What Is Addition ? Give an example. 2. What is meant by the sum or amounts 3. What Is the sign of addition, and how used? 4. What is the sign Indicating dollars ? 5. What is the sign of equality, and how used? 6. How are numbers written for addition ? 7. Which column is first added ? next added ? 8. Where Is the sum of each column placed ? 9. What is done if the amount of any column exceeds ten^ 10. Upon what principle does this last process depend? 11. Explain the process of addition by an example. ARITHMETIC. 12. Must numbers to be added be of the same kind ? Why ? 13. What are the methods oi proof in addition? Illustrate; 14. Define the word proof. SECTio:Nr III. SUBTrwlCTION. 1. What is Subtraction ? Give an example. 2. AYhat is the greater number called ? 3. What is the less number called ? 4. What is the result obtained by this process called ? 5. What is the sign of subtraction, and how used? 6. How are numbers written for subtraction ? 7. With which figure do we begin to subtract ? 8. Where is the difference or remainder written ? 9. If any figure in the upper number is less than the one under- neath it, what is to be done ? 10. How then do we proceed with the next column ? 11. Upon what principle does this operation depend? 12. Illustrate the principle last named. 13. Describe the complete process of eubtraction, and give an example. 14. What is the method oi proof ^ The reason for it. 15. If the difference be added to the subtrahend^ what is ob- tained ? 16. If the difference be subtracted from the minuend, what is ob- tained? 11, Of what operation is Subtraction the opposite ? SECTioN^ rv. MULTIPLICATION. 1. What is Multiplication ? Give an example. 2. W^hat process does it resemble, and how does it differ from it ? 3. What is the name of the number to be multiplied ? 4. The name of the number which we multiply by ? 8 ARITHMETIC. 5. Wliat is the result obtained by the process called ? « 6. What are the multiplier and multiplicand, taken together, called ? 7. Define the word factor, 8. AVhat is the sign of multiplication, and how used ? 9. How are the numbers usually written in multiplication? 10. Can the multiplier be a concrete number ? 11. What is the first step towards multiplying by a single figure? 12. AVhere are the units of the product written? 13. What is done with the tens, if any are obtained? 14. Upon what principle does this " carrying" depend? 15. Describe the full process when the multiplier consists of a single figure, 16. Illustrate the above process by an example. 17. In. multiplying by more than one figure, where is the first figure in each partial product written ? 18. Give the reason for so writing it. . 19. What is meant by ** partial product? " 20. How is the complete product obtained? 21. Explain the full process of multipljing by any number of fig- ures? 22. Illustrate the above process by an example. 23. May the multiplier and the multiplicand exchange pbces ? 24. What is the method of proof in multiplication ? 25. What is a composite number ? ' 26. What are factors ? 27. How may you multiply by a composite number ? 28. Illustrate the above process, and give the reason for it. 29. How may a number be multiplied by a unit with any number of ciphers at its right ? 30. How, when there are ciphers at the right of both multiplier and multiplicand? 31. How, when there are ciphers between the significant figures of the multiplier ? . 32. Illustrate and give the reason for the last three processes. 33. Name any other methods of contraction in multiplication which you know. 34. Define the word contraction. ARITHMETIC. SECTION V. DIVISION. 1. What is Division? Give an example. 2. What process does it resemble, and how does it differ from it ? 3. What is the number to be divided called ? 4. What is the number by which we divide called ? 5. AVhat is the result called ? 6. AYhen the number cannot be exactly divided, what is that part of the dividend left called ? 7. What is the most common sign of division, and its use ? 8. What other signs of division are there ? 9. Write all the signs of division, and tell their use. 10. How many ways of performing division, and what are they usually called ? 11. How are the numbers usually written for division ? 12. What is the first step when the divisor consists of but one fig- ure ? 13. What is the next step, and so on ? 14. Of what order is any quotient figure ? 15. Describe the complete process of Short Division ? ** 16. Illustrate the above process by performing an example. 17. When is the division said to be complete ? 18. When, is the dividend said to be indivisible by a number? 19. AVhat is. an exact divisor? 20. When the divisor is so large as to require all the process to be written out, what kind of .divlson is it ? 21. What Is the first step In the process of Long Division ? The second? Tli« third? The fourth? 22. When the remainder (In partial division) and the next figure of the dividend brought down, will not contain the divisor, what is to be done? 23. If the product of the divisor into the quotient figure exceeds any partial dividend, what Is to be done ? 24. What if the last remainder equals or exceeds the divisor? 25. Of what is division the opposite ? 26. How, then, would you prove division ? 27. Upon what principle does this proof depend ? 10 AEITHMETIC. 28. How is division by a composite numbf^r porformed?* 29. Illustrate the above process by an example. 30. If there arc several remainders how is the true remainder found ? 31. Explain the process of (inding the true remainder, by an example. 32-. Give the reasons for the steps taken in the last process ? 33. How do you divide by a unit with any number of ciphers at its right ? 34. Upon what principle does this give the correct result? 35. What is the reason for this process ? 36. How may we proceed when there is one or more ciphers at the right of the divisor ? 37. How is the true remainder found in this case ? 38. Can you name any other contractions in division.^ 39. The divisor and quotient are factors of what? 40. How is the dividend found from the divisor and quotient? 41. How is the divisor found from the dividend and quotient? SECTION VI. GENERAL PRINCIPLES OF DIVISION. 1. Will a large dividend and large divisor always produce a large quotient? 2. Will a small dividend and a small divisor always produce a small quotient? 3. Does, then, the size of the quotient in division depend upon the absolute size of the divisor and dividend ? ^ 4. Define the word absolute. "* 5. Define the word relative. 6. Upon what does the size or value of the quotient depend ? 7. If the divisor remains unaltered, how does multiplying the dividend affect the quotient? Give an example.* 8. Tf'tlic divibor remains unaltered, how does dividing the divi- dend affect the quotient ? ~ Give an example. * In the illustrative examples, let tlie dividend be written above tlie line, and the divisor below the line, as ] 2 4 ARITHMETIC. 11 9. If the dividend remains unaltered, how does multiplying the divisor affect the quotient? Give an example. 10. If the dividend remains unaltered, how does dividing the divi- sor affect the quotient? Give an example. 11. How does increasing or decreasing the dividend, in either case, affect the quotient ? * 12. How does increasing or decreasing the divisor, in either case, affect the quotient ? 13. Multiplying or dividing both dividend and divisor by the same number, how in either case affects the quotient ? 14. AYhat three general principles, in regard to the quotient, may be here laid down ? 15. Give examples illustrating each. SECTIO:^T YII. REDUCTION. 1. What is a Compound Number? Give an example. 2. What is a Simple Kumber? Give an example. 3. Are abstract numbers simple or compound? 4. What is a Denominate Number ? _ 5. Is the expression 3 weeks, 2 pints, and 6 yards, a compound number? 6. Give a compound number. 7. How does it differ from the numbers in question 5 ? 8. Is dvery compound number a denominate number ? 9. Is every denominate number a compound number ? 10. What is Reduction ? 11. How many kinds of Reduction, and what are they? 12. Why are they so named ? 13. Which of the fundamental operations is employed in Reduction Descending? 14. Give an example of the above. 15. Explain the full process of Reduction Descending, by an example. I 16. Which of the fundamental operations is employed in Reduction |A.scending_? ^ 17. Give an example. 12 ARITHMETIC. 18. Explain tlie full process of Reduction Ascending, by an example. 19. How Is each of these processes proved? 20. What is English Money? Repeat the table. Give the abbreviations indicating the denominations. 21. What is meant by a scale in Reduction ? 22. Give an example of an ascending and descending scale. 23. What is the scale for English Money ^ 24. For what is Troy Weight used ? 25. Repeat the table for Troy Weight. Give the abbreviations which mark the denominations. 26. Give its descending scale. Its ascending scale. 27. For what is Apothecaries' Weight used? Give the table. Give the signs which mark the denominations. 28. By what weight aj-e medicines bought and sold ? 29. What denomination in the above weights are alike ? 30. For what is Avoirdupois Weight used? 31. Repeat the table. Give the abbreviations which mark the denominations. Give the scale. 32. How many pounds now make a ton ? How many formerly ? 33. What is a long or gross ton ? A short ton ? 34. Where is the gross ton now used ? 35. One pound Avoirdupois equals how many grains of Troy or Apothecaries' Weight? 36. How many grains heavier is the Avoirdupois pound than the Troy? 37. In exchanging a quantity of gold dust for cotton, by what weight would each be weighed ? 38. For what is Cloth Measure used ? 39. Repeat the table of Cloth Measure. Give the abbreviations which mark the denominations. 40. For what is Long Measure used? 41. Repeat the table of Long Measure. Give the abbreviations which mark its denominations. 42. Give the scale of Long Measure. 43. How long is a degree upon a circle of the earth ? 44. For what is Chain Measure used ? 45. Repeat the table of Chain Measure. Give the abbreviations which mark its denominations. 46. What is the difference between Long Measure and Chain Measure ? ^ ARITHMETIC. • 13 47. For what is Square Measure used? 48. Repeat the table of Square Measure. Give the abbreviations, of the denominations. 49. Give the scale of Square Measure. 50. Define the word area, 51. Define the word angle, 52. What is a rectangle ? 53. What is each angle of a rectangle called ? 54. How is the area of a rectangle obtained ? 55. How is the breadth of a rectangle found, when the area and length are given ? 56. How is the length found, when the area and width are given ? 57: What is Solid or Cubic Measure ? 58. Repeat the table. Give the abbreviations. 59. What is a prism ? A rectangular prism ? 60. Wiiat is a cube ? • 61. How are the solid contents of a rectangiilar prism, or a cube found? 62. How is the depth, length, or breadth of a rectangular prism or cube found, if the solid contents and the area of one face is known? 63. For what is Liquid Measure used ? ^ 64. Repeat the table. Give the abbreviations. Qb, What is the standard unit of Liquid Measure ? Qi^. How majny cubic inches does this unit contain ? 67. How many cubic inches does a gallon contain? 68. What are the different names for casks containing from 50 to 150 gallons or more ? 69. For what is Dry Measure used? ^ 70. Repeat the table of Dry Measure. Give the abbreviations of its denominations. 71. How many solid inches does a bushel contain? 72. How many gallons does a bushel contain ? 73. What are the dimensions of the common bushel measure? 74. For what is Time used ? 75. What are the natural divisions of Time ? 76. What are the artificial divisions of Time? 77. Repeat the table for Time Measure. Give the abbreviations. 78. Give the scale of Time Measure ? 79. What are the names of the calendar months P 14 ARITHMETIC. 80. How many days in each ? 81. "What is meant by a solar year, and what is its length? 82. What is meant by a lunar month ? 83. What is meant by a leap year? 84. For what is Circular Measure used ? 85. Repeat the table of Circular Measure. Give the signs which mark its denominations. 86. What is a circle? . / 87. What is the circumference of a circle ? 88. What is an arc of a circle ? A quadrant of a circle ? Its diameter ? Its radius ? 89. Can you repeat a Miscellaneous .Table, by which different sorts of merchandise are weighed or measured ? 90. By what measure is land measured ? 91. By what measure is distance reckoned ? 92. By what measure is lumber surveyed ? 93. How is wood measured ? V 94. How is coal estimated ? 95. How is railroad freight often reckoned ? 96. How is a ship's freight reckoned ? SECTioi^ vm. GENERAL ARITHMETICAL PRINCIPLES. * 1. What is an even number ? 2. What is an odd number ? 3. What is a prime number ? 4. Define the \fOvdi prime. 5. What is a composite number ? 6. What is the only even prime number ? 7. When are numbers said to be mutually prime? 8. Define the word mutually, 9. What is a Power? 10. What is a Eoot ? 11. How is the power of a number expressed ? 12. How is the root of a number expressed ? 13. A number is what power of itself? 14. A number is what root of itself? ARITHMETIC^ 15 15. What is the square root of a number? The cube root? ' 16. Is there any similarity between the relations of a composite number and its factors, and a number and its rooty or a number and its power?- 17. What are the factors of a number ? 18. Is a number a factor of itself ? 19. What are the prime factors of a number ? 20. What is meant by factoring a number ? 21. What numbers are divisible by 2 ? 22. What numbers are divisible by 3 ? 4? 5? 6? 23. ' What numbers are divisible by 9 ? 10 ? 11 ? 12 ? 24. What general principle is there in regard to the divisibility of numbers ? 25. Define the word problem, 26. Define the word 50^1^^1071. 27. What is the meaning of each of these words in Arithmetic? 28. What is meant by solving a problem ? 29. What are ^n'me factors ? 30. How may a number be factored, so that all its factors will be prime ? 31. What are composite factors ? 32. What is the difference between the above kinds of factors ? 33. How are the composite factors obtained from the prime? 34. What is a common divisor ? 35. What factors must a common divisoi*contain ? 36. What is the greatest common divisor? 37. What factors must a greatest common divisor contain ? 38. Are there other names for the greatest common divisor? 39. How is the greatest common divisor of two or more numbers found ? 40. Give an example illustrating this. 41. When the numbers can not readily be resolved into their prime factors , how may the greatest common divisor be found ? If there are more than two numbers ? 42. Explain the above process, and give an example. 43. State the principles upon which this operation is based. 44. What is a multiple of a number ? 45. What factors must a multiple contain? 46. What is a common multiple of two or more numbers P^^ 47. What factors must a common multiple contain ? '^-^'^ 16 ARITHMETIC. 48. What IS the least common multiple of two or more mimbers ? 49. What factors must the least common multiple contaia? ^50. How is the least common multiple found ? 51. Upon what principle does this process depend? 52. Illustrate this process by an example. 53. How can several numbers be readily resolved into their prime factors ? 54. How in the last, case is the least common multiple found .^ Give an example. 65. When any of the given numbers are measures of one another, how may the process 'be shortened? 56. Name other methods of shortening the process, if any. 57. What is the least common multiple of prime or mutually prime numbers ? 58. What is the least common fliultiple of two numbers ? SECTION IX. FRACTIONS. 1. What is a Fraction ? Give an example. 2. What is a Common or Vulgar Fraction ? How expressed ? 3. What does the number below the line indicate ? 4. Define the word denominator. - 5. Why is the number below the line so called ? 6. AVliat is the number above the line called, and why ? 7. What are the numbers above and below the line, £aken together, ' called ? 8. How does the expression for a Common Fraction resemble the expression for Simple Division ? 9. What may all. such fractions be termed m respect to Division ? 10. To what term in Division does the value of a Common Frac- tion correspond ? 11. Which fundamental operatipn, then, do all fractions arise from ? 12. Give the fractional expression for Division. 13. Which number is the divisor, and what is the name of it in the fractional expression ? 14. Which number is the dividend, and what is the name of it in the fractional expression ? , ARITHMETIC. 17 15. Is, then, tlie divisor always larger than the dividend ? 16. Name the general principles o? Division which indicate how the multiplication or divisio-n of either the dividend or divisor, or both, affect the value of the quotient? 17. How does multiplying the numerator [dividend] affect the value of the fraction [quotient] ? 18. How does dividing the numerator [dividend] affect the value of the fraction [quotient] ? ' 19. How does multiplying the denominator [divisor] affect the value of the fraction [quotient] ? 20. How does dividing the denominator [divisor] affect the value of the fraction [quotient] ? 21. How does multiplying or dividing both numerator [dividend] and denominator [divisor] by the same number, affect the value of the fraction [quotient] ? - 22. Always, then, an increase or decrease of the numerator [divi- dend] how in either case affects the value of the fraction [quotient] ? 23. Always, then, an increase or decrease of the denominator [divisor] how in either case affects the value of the fraction [quo- tient] ? I 21. What is a Proper Fraction ? Give an example. I 25. What is an Improper Fraction ? Give an example. 26. What is a Simple Fraction? Give an example. 27. What is a Compound Fraction ? Give an example. 28. What is a Mixed Number ? Give an example. 29. What is a Complex Fraction? Give an example. 30. Define the word complex. 31. What is the reciprocal of a number? . 32. Define the word reciprocal. 33. How is a mixed number reduced to an improper fraction? Give an example. 34. How is an improper fraction reduced to a whole or mixed Dumber? Give an example. 35.' Is the value of the expression changed in either of these reductions ? 36. What is an integer ? 37. How is an integer reduced to the form of a fraction ? Is the mlue altered ? •^)8. How is a fraction reduced to its lowest terms ? cJ9. Meaning of lowest terms of a fraction ? 18 ARITHIMETIC. 40. Is the value of the fraction changed by this reduction ? 41. Upon what principle does this reduction to lowest terms depend ? 42. By what two methods is a fraction multiplied by a whole number ? Give examples. 43. What general principles explain the reason for these pro- cesses ? 44. What determines which method should be adopted ? 45. What is the product of a fraction multiplied by its denomi- nator? 46. By what two methods is a mixed number multiplied by a whole number? Give examples. 47. By what two methods may a fraction be divided by a whole number? Give examples. 48. 'What determines which method is to be preferred ? 49. What general principles explain this process ? 60. By what two methods may a mixed number be divided by a whole number? Give examples. . 51. How is one fraction multiplied by another ? Give an example. 52. How many times smaller is a fraction than the integer which^ the numerator represents ? 53. Having, then, multiplied any fraction by the numerator of another given fraction, is the product too large or too small for the correct result of multiplying by the given fraction ? 54. If too large or too small, what determines how many times so ? 55. What, then, is the next step to complete the multiplication of a fraction by a fraction ? bQ, Give an example and illustrate the complete process. 57. What is meant by cancelling^ in Arithmetic ? 58. Upon what general principle, already treated of, does cancel- ling depend ? 59. AYhen may cancelling be used to advantage ? Give an example. 60. In cancelling, when should the quotient, if a unit, be retained ? 61. How is a compound^ fraction reduced to a simple one? 62. What process already given explains this operation ? 63. How is a whole number multiplied by a fraction ? Give an example. 64. How is a mixed number multiplied by a fraction or by another mixed number? Give an example. ARITHMETIC. 19 65. How IS a fraction divided by a fraction ? 66. Having divided, a fraction by the numerator of any given fraction, is the quotient thus obtained too large or too small for a correct result of dividing by the given fraction?. 67. How many times smaller is the fraction representing the divisor than the integer repreapnted by its numerator ? 68. Having, then, divided by the numerator of tiie fraction, what determines how many times too small or too large the quotient ob- tained is for a correct result of dividing by the given fraction ? 69. What, then, is the next step to complete the division of one fraction by another ? 70. Why is the divisor inverted in dividing one fraction by another? 71. Upon what ** principles of division " are the processes of mul- tiplying and dividing fractions by one another explained ? 72. What is the reciprocal of a fraction, and how is it obtained? 73. How Is division of fractions performed when the denominators are alike ? 74. How may the division of fractions be performed when the numerator and denominator of the divisor are factors of the corre- sponding terms of the dividend? 75; How is a whole number divided by a fraction ? How is a inixed number divided by a mixed number? Give examples. 76. How is a complex fraction reduced to a simple one ? Give an example. 77. What is a common denominator? 78. How is a common denominator for several fractions obtained ? Give an example. 79. Upon what ** principle of division " already explained, does this process depend ? 80. AVhat is the least common denominator? 81. How is a least common denominator for several fractions ob- tained ? Give an example. 82. What principle gives the reason for this process ? 83. Is It always necessary to reduce fractions to their lowest terms, before proceeding to find the least common denominator? 84 How may numerators effractions be made alike ? 20 ARITHMETIC. SECTIOI^ X. KEDUCTION, ADDITION, AND SUBTRACTION OF FRACTIONS. 1. 'How is a fraction of a higher deijomlnation reduced to a frac- tion of a lower denomination ? 2. Give an example of the above, and explain the process of mul- tiplying a fraction by a whole number. 3. How is a fraction of a lower denomination reduced to a fraction of a higher denomination ? 4. Give an example of the above, and explain the process_of divid- ing a fraction by a w^ole number. 5. How is a fraction of a higher denomination reduced to whole number^ of a lower denomination ? 6. Give an example of the above, and explain the process of re- ducing improper fractions to whole or mixed numbers; 7. When one quantity is a part of another, how is it expressed fractionally? Giv^ an example. 8. Must quantities thus compared be of the same denomination ? Why.P 9. How are whole numbers of a lower denomination reduced to a fraction of a higher denomination ? 10. What two methods of this reduction ? 11. Explain the process of each. 12. In what cases is each preferable? 13. Is what is called the part in this connection, ever equal to or greater than the unit with which it is compared ? 14. What kind of a fraction results when the part is the greater ? 15. Can numbers of different denominations be added directly? 16. When numbers are added, of what kind or denomination is the amount ? 17. If 2 fifths and 3 fifths are added, what is the result? 18. Can 2 sevenths and 3 fifths be added like the fifths above ? 19. When different fractions are to be adde4j what is the first step to be taken ? Why ? 20. What is the next? Explain the full process. 21. How can the process of adding fractions which have a common numerator be performed ? 22. To what is this last process equivalent? ARITHMETIC. 21 23. !Must numbers to be subtracted from one anotlief, be of tlie same denomination ? 24:. How is one fraction subtracted from another? The first step? the second ? Give an example. 25. How can the process of subtracting fractions which have like numerators be performed ? 26. To what is this contraction equivalent ? 27. How is a whole or mixed number- subtracted from a whole gr mixed number ? 28. Define the word analysis. 29. How is an example in Arithmetic analyzed ? Give an exam- ple. 80. Can all examples in Arithmetic be performed by analysis ? 31. Why are other methods, according to set rules, often adopted ? SECTION" XI. DECIMAL FRACTIONS. 1. What is a Decimal Fraction? 2. Define the word dscimal, axid give its derivation. 3. When we speak of *' Decimals '' what is usually meant ? 4. What may be the denominator of a Common Fraction ? 5. Of what class of numbers, must the denominator of a Decimal Fraction always be ? 6. Are all the principles of Common Fractions equally applicable to Decimals ? 7. Is the denominator of a decimal fraction expressed ? How is it known ? 8. How are decimals distinguished from whole numbers ? 9. What is the decimal j^oint ? 10. What is the first place at the right of the decimal point named ? Second? Third? 11. Do whole numbers and decimals increase in the same ratio towards the left ? 12.. Write a decimal often places, and numerate it? 13. What does a mixed number in Decimal Fractions consist of ? Give an example. 14. In which direction is the integral part numerated, and in which direction the decimal ? 22 AEITIOIETIC. 15. "\Ybat determines the name and value of a figure in a Decimal Fraction ? In whole numbers ? , 16. Moving the decimal point, in a number towards the right, has what effect ? Towards the left, what effect ? 17. In what two ways may a decimal be read ? Give examples. 18. How many numerations are required in order to read a decimal ? Explain. 19. What is the effect upon the value of a (decimal of annexing one or more ciphers ? 20. What principle explains the above result ? 21.N How is the value of a decimal affected by prefixing one or more cii:>hers? 22. What principle gives the reason for the above result ? 23. What sort of a fraction results from placing a common fraction at the right of a decimal ? Give an example. 24 How are decimal fractious written, and what particular care is necessary? ^ 25. How can any ambiguity of meaning be avoided in reading a whole number and a decimal, as 200.003? 26. How are addition, subtraction, multiplication, and division of decimals performed ? 27. How are decimals wHtten for addition, and what special care is necessary ? 28. How for subtraction ? • 29. Where does the decimal point fall in the result, in e^ch case ? 30. If there are more decimal places in the subtrahend than in the minuend, what must be done ? What is the effect upon the value of the decimal ? 31. In the multiplication of decimals, how is the place of the point in the product, determined ? 32. If the number of places in the, product is less than the number of decimal places required, what is to be done ? 33. Illustrate by multiplying by decimals, with the denominators written out, the principle on which the pointing off in decimal multi- plication depends. 34. How can a decimal be multiplied by 10, 100, 1000, &c. ? How multiplied by any number with one or more ciphers at the right ? 35. How is the place of the point in the quotient determined in the division of decimals? ARITHMETIC. 23 36. If the number of decimal places in the quotient is too small for the requi^rem5tract number ? Explain the process by an example. 52. What is 1 inch in Luiear Measure? In Square Measure? In Cubic Measure ? * - 53. To what is V equivalent ? 1" ? V" ? Explain by multiplying 1 inch jDy 1 inch expressed as the fraction (y^^^) of a foot. 28 ARITHIVIETIC. / 54. How are the denominations of the product ascertained In the multiplication of Duodecimals ? Explain find Illustrate by an example, in which the inches and seconds are expressed as the fractions of a foot. 55. Give the complete process of multiplication of Duodecimals, illustrated by an example. 56. How is the division of Duodecimals performed? Give an example. 57. How is the division performed when the dividend and divisor are both compound numbers ? . 58. Can compound numbers be reduced to whole numbers and decimals, and the various operations of adding, subtracting, mul- tiplying and dividing, performed, and then the product reduced back to compound numbers ? 59. Is the latter method often simpler and easier ? ,60. In what country are all the Measures on the scale of 10 ? 61. Is such a scale an advantage ? Why ? 62. Could ITumbers themselves be founded upon a scale of 12, or any other number than 10 .^^ ' SECTION XIY. PERCENTAGE AND INTEREST. 1. What is the meaning of the expression per cent. 7 Its derivation ? . 2. What sign is used to indicate it ? 3. What is meant by rate per cent. ? Give an example. What is meant by per annum ? 4. What is meant by Percentage ? Give an example. 5. What is called the Base of Percentage? Give^n example. 6. In what ways may the rate per cent, be expressed? 7. If expressed by a decimal of more than two places, how must the places after the second decimal place be regarded ? 8. How is the Percentage found when the Base and Rate are given ? Give an example. 9. How is the rate per cent, found when the Base and Percentage are given ? Give an example. 10. Do the last two operations prove each other ? . ^ AEITHMETIC. . 29 11. How is the Base found when the Percentage and the Rate per cent, are given ? Give an example. 12. What is Interest .5^ 13. ^Yhat is the Principal ? 14. What is the Amount ? 15. How are Percentage and Interest related .5* 16. In Interest what corresponds to the Base? The Percentage ? the Rate ? 17. How is the rate of interest usually fixed ? 18. What is a higher rate than legal interest called ? 19. What is the legal rate of interest in New England and in many of the United States .^^ What in New York? What in various other States ? 20. ^Yhat is the rate of the United States' Courts ? Of England ? Of France ? 21. How is the interest of $ 1 at 6 per cent, found for any given time? 22. How is the interest of any sum at 6 per cent, found for any given time ? Give an example, and analyze the process. 23: Do you know or can you reason out any other methods of finding the interest of any sum at 6 per cent, for any given time, and what are they? Illustrate by analyzing examples. 24. How is interest computed on pounds, shillings, and pence ? 25. How can interest be computed on any foreign money ? 26. How is interest found at any other rate than 6 per cent., first finding it at 6 per cent. ? 27. Can you reason out any other methods of finding the interest of any sum for any given tune at other rates than 6 per cent. ? Illustrate by an example. 28. How is the amount of any sum at any rate for any time ascer- tained? 29. What is a Promissory note ? 80. What is the person who signs a promissory note called ? 81. What is the person to whom, or to whose order, it is to be paid called? 82. What is meant by indorsing a note ? Who is the indorser ? 83. What is meant by the face of a note ? 84. Is a note ever signed by more than one person ? What is such note called ? 35. What is a negotiable note? 8* 80 ARITHMETIC. 36. What are partial payments ? 37. How is the interest computed on promissory notes, when no partial payments have been made ? 38. How is such interest computed when partial payments have been made ? 39. What is to be dohe, if any payment does not equal or exceed the interest due on the note at the time such payment is made ? 40. Why is this provision made ? 41. Explain the full process of computing the interest on notes, on which partial payments have been made? 42. When a note is settled within a year after the interest com- menced, and partial payments have been made, how is it customary ta compute the interest ? 43. What other method of computing interest on promissory notes ? 44. Considering that the interest of $ 1 at 6 per cent, is 1 mill for 6 days, andl cent for 60 days, how may the interest on any number of dollars, for anytime, be computed by a short process? 45. What is the most simple method of proof in interest? 46. In reckoning interest, how many days are usually considered a month ? Is this exactly correct ? 47. In what cases must it be reckoned on the exact number of days? ' 48. If the interest of $1 is 5 mills for one month, what is the inter- est for one fifth of a month or 6 days ? ' How can interest, then, easily be reckoned for 6 days multiplied by 10,100, &c. that is, for 2 months, 20 months, 200 months ? 49. What is Annual Interest, and how is it computed ? 50. What four quantities, or elements, do we deal with in In- terest ? 51. How many are given and how many required in each problem in Interest ? 52. "When the Principal, Interest, and Time are given, how is the Rate found ? Give an example, and analyze it. 53. When the Principal, Interest, and Rate are given, how is the Time found ? Give an example, and analyze it. 54. How is the Time found at which any principal will double itself at any rate per cent. ? 55. When the Interest, Time, and Rate are given, how is the Prin- cipal found? Give an example, and analyze it. I ARITHMETIC. 31 56. When the Amount, Rate, and Time are given, how istMiP¥nf=^ cipal found ? Give an example, and analyze it. 57. When the Principal, Rate, and, Time are given, how is the Interest found ? * 58. What is Compound Interest? 59. How often may the interest b.e compounded? 60. Can compound interest be legally collected ? Is it usury ? 61. How is compound interest computed ? 62. Is compound interest at 3 per cent /iaZ/* as much as at 6 per cent? Why? SECTION XV. DISCOUNT AND BANKING. 1. What is Discount? 2. What is meant by present worth ? 3. Is it correct to say the present worth of a sum of money ? Why ? 4. What is 2ideU'} 5. What term in interest does the debt in discount correspond to ? What the present worth ? What the discount ? 6. How, then, is the present worth of a debt found? Give an example. 7. How is discount found ? Give an example. 8. To what is the interest on the present worth equal ? 9. What is a Bank? 10. Meaning of the word incorporated^ ? 11. What is a cAar^er ? 12. What is the Capital Stock of a Bank? . 13. How many kinds of banks are there ? Describe them. 14. What threefold purpose do banks in this country usually serve ? 15. Who control the affairs of a bank? 16. What are the officers of a bank? 17. What are Bank Bills ? 18. What is a Bank Check? *Let the teacher take the equation,I.=P.xE.XT. and illustrate each of tho Problems In Interest. 32 ARITHMETIC. 19. What is the face of a note ? ' , . 20. What is meant by the maturity of a note ? , 21. When is a note said to be given on time ? - 22. When is such a note legally due, that is, when does it mature? 23. What are days of grace ? ^ 24. When a note becomes due on Sunday, or any legal holiday, when must it be paid ? 25. When is a note due that is giveu for a certain number of days ? When given for a certain number of months, when due ? 26. When is the interest paid on money borrowed of a bank? What is this interest called ? 27. What is the sum of money received by the borrower from the bank called ? 28. What is said to be done with the note thus used at a bank? 29. How is bank interest, that is, bank discount, reckoned ? 30. When a note hearing interest is discounted, what is made the base for discounting ? • 31. How is the sum found for which a note must be written, that the proceeds may be a specified sum? Give an example. 32. The difference between true Discount and Bank Disc(5unt ? Illustrate. SECTION XYI. INSURANCE, STOCKS, COMMISSION, &C. 1.. What is Insurance ? 2. What is the Premium, and how is it computed? 3. Does the per cent, for the premium-vary ? 4. The more hazardous the risk, the higher or lower the'per cent, premium? 5. Is any property so hazardous that it is not easy to effect an insurance? 6. By whom is insuring in this country usually carried on ? Ever by individuals ? 7. What is meant by an underwriter ? 8. Can you name any of the benefits of insurance, especially to a conmaercial community? ^ , ARITHMETIC. 83 9. AVliat is an Insurance Policy ? What does it specify ? 10. Is property usually insured for its full value ? Why ? 11. May it be insured at more than one office, that is, by more than one company ? When ? 12. How is the premium computed ? 13. Are there charges for insurance besides the premium? 14. What is Capital Stock of a?2?/ Company ? 15. Name some of the different kinds of incorporated companies. 16. How is the Capital usually divided ? . ' 17. What other securities are called stocks ? 18. Meaning of securities in the above connection ? 19. What is thenar value of stock? ' ' 20. What is the market value of stock, and is it, or not, usually the same as the par value ? 21. Give an example of stock at par. Above par. Below par. 22. What are dividends ? 23. What are assessments ? 24. When the dividends are large, is the stock likely tb be above or below par ? Why ? 25. When the assessments are large, and dividends small, is the stock likely to be above or below par ? 26. How are dividends and assessments computed ? 27. How is the market value of stock found, when the par value, and premium or discount, is known ? 28. How is the number of shares found that may be bought for a certain sum, when the discount or premium is known ? 29. What is Commission or Brokerage ? 30. What is an Agent ? By what other name is he sometimes designated? 31. How is Commission or Brokerage computed? Give an ex- ample. 32. How is the commission or brokerage found, the rate per cent commission being given, when the agent is to take his pay from a certain sum and invest the balance? Give an example, and explain it. 33. What is a Tax? 34. By whom are taxes assessed ? ' 35. To what uses are taxes applied? 36. How is a tax on property assessed? 37. How is a tax on persons assessed? 38. What is meant by a poll ? 84 ARITHMETIC. 39. Wliat are the two cMef divisions of property? 40. What is Real Estate ? 41. What is Personal Property?: 42. AVhat is an Inventory P 43. Are the details of taxation the same in all the States? What peculiarity in Vermont ? What in Connecticut ? 44. What is the Rule for assessing taxes in Massachusetts ? 45. What is the rule in the State where you reside ? 46. Are there other taxes besides town or city, county and state taxes ? 47. What is an Excise Tax ? 48. What is a Stamp Tax? . 49. What is the United S"tates Internal Revenue ? 50. What **iabor-saving" way is sometimes employed by asses- sors in computing taxes ? 51. In calculating a tax-list, what . description of table is found very useful ? 52. What kind of taxes are called Customs or Duties ? 53. For what are these taxes laid ? 54. What are the only ports in the United States where goods brought from foreign countries can lawfully be landed, called ? 55. What is established at each port of entry? 56. What is smuggling? What are persons engaged in it sub- ject to? 57. What is Tonnage ? 58. How many kindi of Duties are there, and what are they called ? 59. What is an ad valorem duty ? 60. What is a specific duty ? 61. What is an invoice ?- ^2. How are ac? ??aZorew duties computed ? Give aft example. 63. On what only are specific duties computed ? 64. What is Leakage ? Breakage ? Draft or Tret ? 65. What is Tare ? Gross weight ? Ket weight ? 6^. How are specific duties computed ? Give an example. ARITHMETIC. ' 35 SECTION XYII. EXCHANGE ANI> EQUATION OF i»AYMENTS. 1. What is Exohange, in "commerce? 2. What is a Draft or Bill of Exchange ? 3. Who is the Makier or Drawer of a Bill ? 4. Who is the Drawee of a Bill ? The Payee ? 5. Explain the operations of Exchange. 6. How are bills made payable .P 7. When is a bill payable. aj5 siglii^ When for any specified' time ? 8. Define the word negotiable. 9. What is a negotiable bill or note ?. 10. When a person sells a bill of exchange, what is the person who" buys it called ? 11. What is the person possessing a bill at any time called ? . 12. Who is the indorser of a bill ? 13. In what manner is the indorser responsible for a hWi? 14. What is it to accept a bill ? 15. What is it to. protest a Bill ? 16. What officer is employed to protest bills ? 17. When should a bill regularly be presented for payment ? 18. If the bill is not paid when presented, what is neeessary to hold indorsers ? 19. What are Exports? 20. What are Imports ? 21. When is the balance of trade in our favor? When against us ? 22. By what must the deficiency or debt, on either side, be made up ? 23. How does the balance of trade affect Exchange ? '24. If the United States buy more of England than they sell to England, will exchange on England be at a premium or discount ? How, if they sell more than they buy ? 25. Will the variation of Exchange, that is, the premium or dis- count, ever be very great ? Why ? 26. Are time bills of exchange subject to discount? 27. What is the exchange value, in the United States, of a pound sterling ? The intrinsic or commercial value of a pound sterling ? 28. Write a draft or bill of exchange in proper form. 36 ARITHMETIC. 29. An order or bill of excliange payable In a country where It is drawn, is called what ? What if payable in a foreign coimtiy ? 30. How is the cost of a bill of exchange found ? Give an exam- ple. 31. How is the face of a bill found which a given sjim In United States Money will buy ? Give an example. How, when United States Currency is at a discount ? 32. Define the word equation, 33. What is Equation of Payments ? 34." What is meant by the equated time ? 35. What is meant by the term of credit ? 36. What Is meant by the average term of credit ? 37. How is the average term of credit for several bills found ? Give an example. 38. How, from the average term of credit, is the equated time found ? 39^ How Is the time most conveniently expressed in solving such examples ? 40'. Explain a method of equating payments by means of reckoning the interest. The reason for it. 41. What Is a short way of finding the interest on any sum for two months ? How then for one month ? , 42. What is done with fractions of a day In equation of payments ? 43. Which of the above methods of equating payments is prefer- able, and why ? 44. In finding the average date of debts of equal terms of credit, from what date may the interest be reckoned ? From what date is it most conveniently reckoned ? Why ? 45. What is the date from which the interest on several bills is reckoned called ? 46. How is the equated time found when all the terms of credit are equal, but begin at different times ? Give an example, and analyze it. 47. What is the maturity of a bill or note ? 48. When the terms of credit are unequal, and begin at different times, bow is the equated time found ? Give an example, and analyze it. 49. How does the last case differ from the one which precedes it? • 60. How is the equated time found for paying the balance of an ARITHMETIC. 37 account which has both debit and credit entries ?' Give an example, and analyze it. 51. Where the larger interest arises on the smaller side of the ac- count, what is the result ? What must then be done ? 52. What determines the choice of focal date ? 53. What is the principle upon which Equation of Payments is based? SECTION" XVIII. PROFIT AND LOSS, AND PARTNERSniP. 1. Explain the term ** Profit and Loss." 2. What is absolute gain or loss ? 3. What is percentage of gain or loss, and on what is it comput- ed? 4. How is absolute gain or loss found ? Give an example, and analyze it. 5. How is the per cent, of gain or loss found when the cost and selling price are given ?^ Give an example, and analyze it. 6. How can the selling price be found when the cost and gain or loss are known ? Give an example, and analyze it. 7. How can the cost be found when the selling price and per cent, of gain or loss are given ? Give an example, and analyze it. 8. How is the per cent, of gain or loss found, when it is proposed to sell goods at any given price ? How when only the per cent, gained or lost, if sold at a given price, are known? Give an example, and analyze it. 9. How may the marking price of goods be found, so that the merchant may fall a certain per cent., and yet, sell the goods at cost, or at a certain per cent, gain or loss on the cost price ? Give an ex- ample, and analyze it. 10. What is Partnership ? 1 1. AVhat is a Partnership Company often called ? 12. What is meant by an active and what by a silent partner? 13. What is the Capital or Stock ? 14. How are the profits and losses divided? 15. What is the difference between a Partnership Company and an Insurance, Manufacturing, or Railroad Company ? 4 38 ARITHMETIC. 16. By what authority is this difference established, and what au- thority defines the privileges and responsibilities of each ? «. 17. What is a c/iar^er ? 18. Can you tell the object of compelling certain companies to ob- tain charters of incorporation ? 19. How is each partner's share of the gain or loss found? What other method ? Give an example. 20. What is the proof of the above process ? 21. How is each partner's share of gain or loss ascertained, when , their capital is employed unequal times? Give an example. 22. Upon what principle is the above computation based ? 23. Can all problems which come under the head of Percentage be performed by analysis ? What is the advantage of special rules in the various cases ? SECTION XIX. RATIO AND PROPORTION. / 1. AVhat is Ratio ? 2. How is the ratio between two numbers usually expressed? 3. What are the quantities compared called ? 4. What is the first term called? The second? The two to- gether ? 5. Which term is considered the divisor ? 6. Is the other term ever considered the divisor? 7. Which is sometimes called the English and which the French method ? 8. Which is called a direct and which a reciprocal ratio ? 9. Considering one term of the ratio as the divisor and the other as the dividend, what general principles apply to them as affecting the quotient? 10. How, then, is the ratio affected by multiplying the antecedent [dividend]? 11. How is the ratio affected by dividing the antecedent [dividend] ? 12. How is the ratio affected by multiplying the consequent [divisor] ? 13. How is the ratio affected by dividing the consequent [divisor] ? ARITHMETIC. 99 14. ITow is tlie ratio affected by multiplying or dividing botli antecedent [dividend] and consequent [divisor] by the same nmnber ? 15. How are the antecedent, consequent, and ratio related to each other .^ 16. What is a simple ratio ? 171 What is a compound ratio ? ISy^Vhat is the value of a compound ratio ? ~^ ly. What is Proportion ? 20. IIow is a proportion indicated ? How read ? 21. When are four numbers in proportion ? 22. What are the first and last terms called? 23. What are the two middle terms called? 24. A¥hat are the first and third terms called ? 25. What are the second and fourth terms called? 26. To what is the product of the extremes equal? 27. What is meant by the " Rule of Three" ? 28. How many terms of a proportion must be given ? 29. From the given terms, how can the remainder of the proportion be found ? 80. The product of the extremes divided by either mean, will give what ? 31. The product of the means, divided by either extreme, will give what ? 32. Give ai| example, and illustrate the above. 33. Which ^air5 of terms of a proportion may be multiplied or divided by the same number without destroying the proportion ? 34. What is the effect of multiplying or dividing the four terms of a proportion by the same number ? 85. In how many orders may the four terms of* a proportion be written, and the numbers still be in proportion ? Give |n example il- lustrating this. 36. What is meant by a mean proportional ? 37. How is a mean proportional foun^l? Give an example. 38. What is a third proportional ? How found ? 30. Of what kind must two of the three numbers given in a simple proportion be ? Of what kind the other ? 40. How is an example in simple proportion performed ? Explain by giving an example. 41. By what other method can every example in proportion be solved ? Solve an example by proportion and then by analysis. 40 ARiTrorETic. 42. AVhat IS Compound Proportion ? 43. How is an example in compound proportion p'^fformcd? Explain, by giving an example. 44. Can every example in compound proportion be solved by simple proportion ? Give an example. 45. By what other method than those mentioned can all examples in compound proportion be solved ? Give an example, and solve it by compound proportion, by simple proportion, and then by analysis. 46. Why are the methods given in proportion used in preference to analysis .»* SECTION XX. ALLIGATION. 1. What is Alligation? Of what two kinds? 2. What is Alligation Medial ? Give an example. 3. How is the price of a mixture found, when the quantities and prices of the articles are given ? Give an example. 4. What is Alligation Alternate ? 5. When the prices of several kinds are given, how is it ascer- tained how much of each kind may be taken to form a compound of a proposed medium price ? Give an example, and explain the process by analysis. 6. May different answers be obtained and yet all be correct? 7. Explain a method where the quantities are at first assumed, 8. When the price of each of the simples, the price of the coin- poimd, and the quantity of one kind are given, how is the quantity of the other simples, which may be taken, found? Give an example. 9. When 4he prices of the several simples, the price of the com- pound, and the entire qua,ntity in the compound are given, how is it ascertained how much of each simple may be taken? Give an example. SECTio:Nr XXL INVOLUTION AND EVOLUTION. 1. What' is a Power of a Number? 2. What is Involution ? Give the derivation of the word. ARITHMETIC. 41 3. The number to be involved is what power of itself? What is it in respect to other powers of itself? 4. What is the index or exponent of a power ? 5. How is a number involved to any required power ? Give an example. 6. Plow is the involution expressed? Give an example. 7. How is a common fraction involved? A mixed number? 8. How is a decimal fraction involved? How many deeimal places in any required power of a given decimal ? 9. What is the sign of inequality, and its use ? 10. What are the powers of 1 ? In respect to size, what relation do powers of numbers greater than unity bear to the numbers ? The power of numbers less than unity ? 11. In involving a number what determines how many multiplica- tions are required ? 12. To what power is the product of two or more given powers equal ? 13. How is a quantity involved that is already a power? Give an example. 14. How is the power of any number divided by any other power of the same number ? 15. The product of two given numbers consists of how many figures ? How is this shown ? 16. The square of a number consists of how many figureiS ? The cube or third power, of how many ? The fourth power ? ' 17. The square of units may consist of how high an order of figures ? The square of tens, of how high an order? How low? 18. What is Evolution? Define the word, and give its derivation. 19. What is given and what required in Involution ? 20. What is given and what required in Evolution ? 2 1 . What is the Root of a number ? 22. What is the relation of powers and roots to each other? 23. How many methods are there of indicating a root, and what are they ? 24. In what consists the process of Evolution or extracting roots ? 25. How is a number involved, or raised to a certain power ? How evolved, or the root extracted ? 26. How are fractional indices found ? 27. How is the power and root of a number indicated at the same time ? Any other way ? Give examples. 4* 43 AEITHMETIC. 28. Can all num'bers be involved to any required power? Can all numbers be evolved'? * 29. What is a perfect power ? An imperfect power ? 30. What is a rational number ? An irrational, radical, and surd number ? 31 . What is every root of 1 ? In respect to size, what relation do the roots^ of numbers greater than unity bear to their powers ? The roots of numbers less than unity ? SECTioN^ xxn. SQUARE AKD CUBE ROOT. 1. What is it to extract the Square Root of a number? 2. Of how many figures does the square or second power of a number consist? Of how many the square root of a number? Explain. 3. Of what order of figures must the square of units be? Of tens ? Give an example. 4. In a number of three figures, of how many figures will the root consist ? 5. How is the number of figures of which the root will consist ascertained and indicated ? 6. In what order of figures must the square of tens be found ? 7. What, then, is the first step in extracting the root of a number of three figures ? 8. Having found the tens figure of the root, how is the number, from which the unit figure is obtained, found ? How is the unit fiw? 73. The same reasoning applying to all cases of extracting the cube root, give the full directions for extracting the cube root of any number. 74. What IS the proof of this process ? 75. Can the cube root of decimals, mixed decimal numbers, com- mon fractions, and mixed numbers, be extracted? How? 76 . When are rectangular bodies similar ? What is the ratio between similar bodies ? 77. Give the full directions for extracting a root of any degree. SECTION xxm. PROGRESSIONS, ANNUITIES, AND PERMUTATIONS. 1. When is a series of numbers said to be in Arithmetical Pro- gression ? Give an example. 2. How many kinds of series are there in Arithmetical Progres- sion, and what are they ? 3. What are the terms of a series ? 4. What are the first and last terms of a series called? What the other terms ? 5. AVhat is meant by the common difference ? 6. How many particulars are considered in examples in Arith- metical Progression ? What are they ? 46 ARITHMETIC. 7. How many of tlie terms must be given in order to find tlie others ? ' 8. How is an ascending series formed ? A descending series ? 9. How is the last term found, when the first term, common differ- ence, and number of terms are given? Give an example. 10. How is the common difference found, when the extremes and number of terms are given ? Give an example. 11. How is the number of terms found, when the extremes and common difference are given? Give an example. 12. How is the sum of the series found, when the extremes and number of terms are given ? Give an example. 13. What constitutes a series in Geometrical Progression ? 14. How many kinds of series in a Geometrical Progression, and what are they ? 15. How many particulars are considered in problems in Geomet- rical Progression ? What are they ? How many of them must be given to find the others ? 16. How is an ascending series formed ? A descending series ? 17. How is the last term found when the first term, ratio, and number of terms are given? Give an example. 18. How is the ratio found when the extremes and number of terms are given? Give an example. 19. How is the sum of the series found when the extremes and ratio are given ? Give an example. ^^ 20. What is an Annuity ? A certain Annuity ? A perpetual Annuity ? 21. What is an annuity in arrears ? ' ij 22. What is meant by the amount of an annuity ? '^ 23. How is the amount of an annuity in arrears, at simple interest, found ? Give an example. 24. How is the amount of an annuity in arrears, at compound in- terest, found ? Give an example. 25. How is the present worth of an annuity certain^ at compound interest, found? Give an example. 26. How is the present worth of an annuity perpetual found? 27. What is Permutation? How is the number of permutations of a certain number of things found ? Give an example. 28. How is the number of arrangements that can be made of a certain number of things taken in sets, as 2 and 2, 3 and 3, &c., found? 29. How is the number of comhiyiations of any number of things, in sets of 2 and 2, 3 and 3, &c., found? AKITHMETIC. 4:7 SECTIOI^J" XXIV. MENSUIIATION. 1 . What is Mensuration ? 2. What are parallel lines ? 3. What is an oblique angle ? 4. AVhat is a triangle ? Its ha^e ? Its altitude ? 5. How is the area of a triangle found? 6. What is a quadrilateral or quadrangle? Of how many kinds? 7. What is a trapezium ? Trapezoid? Parallelogram? 8. What is the diagonal of a figure ? 9. What the altitude of a trapezoid? Of a parallelogram? 10. How is the area of a trapezium found ? 1 1 . How is the area of a trapezoid found ? 12. How is the area of a parallelogram found ? 13. AVhat is a polygon ? The perimeter of a polygon ? 14. What are the names of some of the different polygons? 15. How is the area of a circle found? ^ 16. What is a prism ? 17. What is a cylinder ? 18. How is the surface of a prism or cylinder found ? 19. How are the solid contents of a prism or cylinder found ? 20. AVhat is a pyramid ? Its vertex ? Slant height ? 21. What is a cone ? The altitude of a pyramid or cone ? 22. How are the solid contents of a pyramid or cone found ? 23. What is the frustum of a pyramid or cone? How are its solid contents found ? 24. What is a sphere or globe ? Its diameter? 25. How is the area of the surface of a sphere found when the cir- cumference and diameter are known ? 26. How is the volume or solid contents of a sphere or globe found when the surface and diameter are known ? THE END. EATON'S SERIES "of ARITHMETICS. "The High-School Arithmetic Is all that could reasonably be de- sired." — J. D. PiiiLBRiCK, Superintendent of Public Schools, Boston, **The Common-School Arithmetic seems to combine all the essen- tial requisites of a model text-book for teaching both the science of numbers and its practical application." — Ibid, **The Intellectual Arithmetic is unquestionably a work of rare excellence." . . . ** I am fuUy convinced that it has no superior and no equal." — Ibid. **The Primary Arithmetic is happily adapted to lead the young pupil to a knowledge of the rudiments of numbers. — Ibid, *' Your Committee think Eaton's the best series of Arithmetics to be had at the present time." — Boston Text-Book Committee, June, 1864. ** Eaton's.Aritlimetics are found to meet all the wants of the schools, and are working well, — Ibid, June, 1865. ** I can truly say that it is the best treatise on the subject I have e^'.er used." — S. AV. Mason, Master of Eliot School, Boston, ** The books deserve the wide-spread popularity they have attained." — H. II. Lincoln, Master of Lyman School, Boston, **It fully meets the expectations which I entertained when it, was introduced." — W.T. Adams, Master of Boioditch School, Boston, **I consider them superior to any series of the kind with which I am acquainted." — E. T. Quimby, Professor of Mathematics, Dartmouth College, " I believe Eaton's Treatise (High-School) far surpasses any other work of the kind." . , . ** Its merits cannot be easily overrated." — Albert C. Perkins, Principal of High School, Lawrence, Mass, ** Eaton's works give entire satisfaction in our schojols." — L. E. NoYES, Superintending School Committee, Abington, Mass, "Eaton's series was introduced last October (1-864) into all the schools." . . . ** The satisfaction so far is complete." — J. Wiii> Belcher, Superintending Scfiool Committee, Randolph, Mass, "The State Board substituted Eaton's Common School for Rob';i- son's Practical, because, in thGir opinion, it was better adapted to%he wants of our schools."— ^ Hon. John Swett, State Superintender^of Schools, California. 4 **We think Eaton surpasses any o-iithor we have ever used, and believe his mathematical series is better adapted than any other to the wants of our schools." — John F. Colby, Late Principal of Stetson High School, Randolph, Mass, ** I have never used a work so complete in all its parts as the Com- mon School." . . . *' This bookstands high with California teachers." — E,. P. Foss, Principal of City Grammar School, Sonoi^a, Col, *' Your Eaton's series of Arithmetics, so far as examined, receive the miquahfied approbation of all teachers." — S. M. Shearer, Prin- cipal of San Juan Public Schools. "Eaton's Arithmetics are everywhere received with great favor." — J. C. Pelton, Superintendent of Schools, San Francisco. * * After a careful examination of all the recent works on this sub- ject, they unanimously recommended . . . that Eaton's Common- School Arithmetic be substituted for GreenleaPs. The schools have steadily improved in Arithmetic ever since the change." — Extract from Report of Worcester School Committee, * * It surpasses all others with which I am acquainted in the following particulars." — Ephraim Knight, Professor of Mathematics, New London Literary and Scientific Institution. * * Altogether the most satisfactory that has fallen under our notice for practical use." — Rev. Lyman Coleman, D,D., Philadelphia, ** The Board of Education recommended, at their late meeting, the introduction of Eaton's Arithmetics into the schools of New Hamp- shire." — J. W. Patterson, Secretary of the Board of Education for New Hampshire. **The Common-School Arithmetic is just the book for teaching written Arithmetic in all District and Grammar Schools. — J. D. Phil- brick, Superintendent of Boston Public Schools. ** The Intellectual Arithmetic is giving complete satisfaction.'^ — C. G. Clarke, Master of Bigelow School, Boston. ** I consider them the best works on this subject now extant." — F. F. Preble, Snh-Master Adams School, Boston. A valuable treatise, well adapted to the wants of public schools." — J. N. Camp, Superintendent of Schools for the State of Connecti- cut and Principal of State Normal School, Connecticut. *' The substitution of Eaton for Greenleaf was voted without a sin^'- dissenting voice." — Rev. J. D. E. Jones, Superintendent of Schools, Worcester, Mass. E A.. T o isr ' s STANDARD SERIES OF ARITHMETICS IS RECOMMENDED IN HIGH TERMS BY HUNDREDS OF THE BEST EDUCATORS AND TEACHERS OF THE COUNTRY, AMONG WHOM ARE THE FOLLOWING :.- Hon. John D. Philbrick, Superintendent of Public Schools, Boston. Prof. J. P. Fisk, Beloit College, Beloit, Wisconsin. Hon. John Swett, State Superintendent of Schools of California. Hon. James W. Patterson, Secretary of N. H. State Board of Education. William H. Wells, Esq., Superintendent of Schools, C|iicago, Illinois. Prof. A. Jackman, Norwich University, Norwich, Vt. A. P. Stone, late President of American Institute of Instruction, Portland, Me. M. T. Brown, Superintendent of Schools. Toledo, Ohio. Hon. David N. Camp, late State Superintendent of Schools of Connecticut. FREDERICK A. Sawyer, Principal of High School, Charleston, South Carolina. Prof. E. T. Quimby, Professor of Mathematics, Dartmouth College, N. H. Emory Lyon, University Grammar School, Providence, R. I. Rev. Charles Anthon, D.D., Columbia College, New York City. Rev. Lyman Coleman, D.D., Seminary, Philadelphia, Pa. Thomas Sherwin, Principal of Boston English High School, and author of several works on Mathematics. Rev. J. D. E. Jones, Superintendent of Schools, Worcester, Mass. J. C. Pelton, Superintendent of Schools, San Francisco, California. J. W. EwiNG, Superintendent of Schools, Perrysburg, Ohio, Prof. J. V. N. Standish, Lombard University, Galesburg, 111. Prof. S. F. Newman, Principal of Normal School, Milan, Ohio. J. E. Dow, Superintendent of Schools, Burlington, Iowa. Geo. W. Perry, Superintendent of Schools, Tiffin, Ohio. J. B. Roberts, Superintendent of Schools, Galesburg, Illinois. Rev. J. F. Dudley, St. Paul, Minnesota. Albert C. Perkins, Principal of High School, Lawrence, Mass. Isaac F. Cady, Principal of High School, Warren, R. I. Augustus Morse, Principal of Grammar School, Hartford, Connecticut. C. F. Emery, Principal of High School, Troy, N. Y. Levi Cass, Principal of High School, Janesville, Wisconsin. Rev. Arthur Little, late of North Haven Academy, Connecticut. Horace Day, Esq., late Superintendent of Schools, New Haven, Connecticut. Rev. Walter S. Alexander, Superintendent of Schools, Pomfret, Connecticut John G. W. Martin, Principal of High School, Elizabethtown, Pa. C. P. Barrows, Teacher, Caroline County, Va. B. P. Chenoworth, Teacher, Moore's Hill, Indiana. S. W. Mason, Master of Eliot School, Boston, Mass. Henry Freeman, Commissioner of Schools, Illinois. 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