24. In a square room which is calculated to accommodate 100 boys, how many must sit on a single bench? 25. Suppose 400 scholars should wish to form themselves into a solid phalanx,1 or square body, how many must stand in each rank' and file ? A. 20. 26. What is the cubic or solid content of a regular cube 10 inches long, 10 inches wide, and 10 inches thick? 27. What is the 10 and 1000 each called? A. The 10 is called the cube root of the 1000, and the 1000 the cube of 10. 28. What then is the cube of 2, and why? A. 8, because 2 times 2 are 4, and 2 times 4 are 8. 29. What is the cube of 3? What is the cube of 4? 30. What is the cube root of 8, and why? 2 are 4, and 2 times 4 are 8. A. 2, because 2 times 31. What is the cube root of 27 of 125 of 1000? 32. What is the length of each side of a cubical block which contains 1000 solid or cubic inches? What is the cube of 10? of what number? of what number? of which is 9? of which is 8? of 12?of which is 11? 33. 6 is of what number? 10 is 34. 12 is of what number? 11 is 35. What is that number which is 10? of which is 36. What number is that 37. What number is that 38. What number is that of which is 63 of which is 10? must be 5. A. 15. of which is 24? Find first. A. 32. of which is 16?-3 of which is 30? of which is 60? of which is 72? 39. If of a barrel of flour cost 4 dollars; what will of a barrel cost? What will the whole barrel cost []? 40. A man bought of a load of hay for 6 dollars, what was the whole load worth at that rate? Find the value of first. of his brother's age is first. of all the money he had 41. Henry's age is 14 years, which is How old is his brother? Find how much 42. A man lost 15 dollars, which was how much had he? How much had he left? 43. A man, who owed a certain sum of money paid 12 dollars, which was of the debt; how much remained unpaid? 44. A man, who lent a certain sum of money, could collect only 8 dollars, which was of it; how much did he lose? 45. If a man, having a quantity of flour on hand, sells 20 barrels, which is of it; how much will he have left? 46. Suppose a man sells of a barrel of flour, for 14 dollars, what will the remainder of the barrel bring at that rate? 1 PHALANX. A square battalion or body of soldiers, formed in ranks and files close and deep; any body of troops or men formed in close array, or any combination of people distinguished for their intrepidity and union. 2 RANK. A row or line; men standing side by side in a line; a line of things; degree class; order; degree of dignity. 3 FILE. A thread, string or line; a bundle of papers tied together with the title t each indorsed; a roll, list, or catalogue; a row of soldiers ranged one behind another from front to rear 47. A boy having a stick, broke it into two parts, one of which was 2 feet long, or of the length of both parts; what was the length of the stick before it was broken? 48. If and of a number are 16; what is tha ́ number? NOTE-Say and are which is 16; then is of 16, which is 4, and is 5 times 4, which is 20. A. 20. what is that number? 49. If and 3 of a number are 10; what is that number? 50 If and of a number are 70; 51 There is a pole erected so that stands in the mud, in the water, and the rest which is 10 feet above water; what is the entire length of the pole ? 52. There is a pole, above water, in the water; and 8 feet in the mud; what is the entire length of the pole? 53. Four men A, B, C, and D purchased a sloop together. A took of it, B., C. and D, the rest, which cost him 100 dollars. What was D's part? What did the other parts severally cost? What did the whole sloop cost? 54. The fifth part of an army was killed; of it taken prisoners, and 1000 fled; how many were there in the army? How many were killed? How many were taken prisoners? 55. In an orchard of fruit trees, of them bear apples, bear plums: 8 bear peaches, and 2 bear cherries: how many trees of each sort are there in the orchard? How many trees does the orchard contain? 8 trees and 2 trees are 10 trees which are }. 56. In a certain school of the pupils study Arithmetic, study Grammar and 10 only read and spell: what is the number of scholars in the school? What is the number in Arithmetic? What is the number in Grammar? 57. A man having 16 oranges would divide them so that his own son Samuel may have 4 more than his neighbor's son George; How many must he give to each? NOTE―Give 4 to Samuel first then divide the rest equally between the two? A. George 6, and Samuel 10. 58. A gentleman bought a horse and carriage for 240 dollars, paying 40 dollars more for the horse than for the carriage; what did each cost? 59. A man and a boy were both hired for 20 dollars a month, the man receiving 4 dollars a month more than the boy; what would the wages of each amount to in a year? 60. A man, woman, and boy were hired a week for 21 dollars; the woman to receive 5 dollars more than the boy, and the man 5 dollars more than the woman; at that rate what would the wages of each amount to in one month. 3 ARITHMETIC. PART SECOND. AS CONSISTING BOTH IN THEORY2 AND PRACTICE3 QUANTITY AND NUMBER. IX. 1. QUANTITY is any thing that may be increased or diminished; as, a sum of money, a line, weight. 2. A QUANTITY is ascertained to be great or small, much or little only in comparison with a known quantity of the same kind, which is either greater or smaller. 3. For example, ten thousand hogsheads of water is a great quantity, compared with one gill of water, but quite a small quantity, compared with the water in the ocean. 5 4. A UNIT, which represents a single thing; as, 1 hat, 1 ounce, &c. is fixed upon as the criterion' or known quantity by which to measure all other quantities of that kind. 5. Thus 2 would express a quantity 2 times as great as 1, that is 2 units; 3, 3 times as great, or 3 units, and so on. 6. QUANTITIES then of every kind are properly expressed by NuмBERS; as 5 bushels of rye, 5 oranges, &c. 7. A CONCRETE NUMBER has reference to some particular object or objects; as, 1 man, 2 dollars, 3 benches. 8. AN ABSTRACT NUMBER has no reference to any object whatever; as 1, 2, 3. IX. Q. What is Quantity? 1. How is a quantity ascertained to be great or small? 2. Give an example? 3. What is the criterion for estimating different quantities? 4. Illustrate it? 5. How are quantities expressed? 6. What is a Concrete Number? 7. An Abstract Number? 8. 1 ARITHMETIC, [G. Arithmetike.] Reckoning by numbers; calculating. 2 THEORY, [F. theorie. L. theoria.] Speculation; a system, plan, scheme; opposed practice. 3 PRACTICE, [F. pratique.] Habit, use, dexterity, method. 4 QUANTITY, [Quantitas.] Any thing that may be increased or diminished; bigness; bulk, weight; measure. 5 UNIT, [L unus.] One; a word denoting a single thing. 6 REPRESENT. To show; to exhibit; to describe. 7 CRITERION, [G. kriterion.] A standard of judging; a distinguishing mark. 8 CONCRETE, [L. concretus.] United in one mass; a compound; a term involving both the thing and its quality; as, a white fence; 2 mellons; 1 cent. 9 ABSTRACT, [I.. abstractus.] Separate distinct; expressing only quality or num ber; as, whiteness; 1, 2, 3, &c 9. DENOMINATION' is a name given to units or things of the same sort or class; as, 4 dollars, 5 oranges, 10 pigeons. 10. A SIMPLE NUMBER is composed of units of the same value or denomination. 3 11. A COMPOUND NUMBER is composed of two or more simple numbers of different denominations, but of the same genus, kind, or general class. 12. Thus 2 pounds is a simple number, so is 5 shillings; but 2 pounds 5 shillings taken together, forms a compound number, for it has one denomination of pounds, another of shillings; but both are of the same kind, or general class, viz. money. 13. ARITHMETIC treats of numbers: as a SCIENCE,' it explains their properties; and as an ART,' it teaches the method of computing by them. 14. ARITHMETIC has five principal rules for its operation, viz. Numeration, Addition, Substraction, Multiplication, and Division, which are often called the fundamental' or ground rules of Arithmetic, because they are the foundation of all the other rules.10 Q. What is meant by denomination? 9. What is a Simple Number? 10. Compound Number? 11. Give an example of a compound number. See 12. What is Arithmetic? 13. When is it regarded as a Science? 13. When as an Art? 13. How many rules has it for its operations? 14. What general name have these rules, and why? 14. 1 DENOMINATION, [L. denomino.] The act of naming, a name; a vocal sound, a class, sort, or name of a species. 2 COMPOUND. Composed of two or more ingredients; united in one. 3 GENUS, [L. genus.] A general name for several species; a class of greater extent than species; thus animal is a genus; embracing a great variety of species; as man, horse, beast, bird, reptiles, &c. 4 SPECIES. A kind, sort, class; a subdivision of a general sum called a genus, thus, things that resemble each other in several particulars form a species; when several species are compared together and we observe several particulars common to the whole, they form a genus; a species then is one class of a genus. 5 TREAT, [F. traiter.] To handle; to use; to discourse on; to entertain without expense; to negociate; to manage in the application of remedies. 6 SCIENCE, [L. Scientia.] Knowledge, a system comprising the theory and reasons without any practical application; and therefore stands opposed to Art. 7 ART, [L. art.] Human skill; a system of rules; skill; dexterity. 8 COMPUTING. Counting; numbering; reckoning, estimating. 9. FUNDAMENTAL, [L. fundament, seat.] Relating to the foundation or basis. 10 Addition alone may not inappropriately be styled the sole or fundamental rule o Arithmetic, for all the other rules are easily resolvable (11) into this single one. 10 Thus 4 in 20, 5 times, because 5 times 4 are 20. Division then involves the principle of Multiplication. 10 Again, 4 times 5 are 20, because 5 and 5 and 5 and 5, that is 5 added 4 times, makes 20. Hence Multiplication may be performed by Addition. 10 Subtraction too is virtually performed by Addition, for 5 from 20 leaves 15 only because 15 and 5 are 20. 11 RESOLVABLE, [L. resolvo.] That may be reduced to first principles. NUMERATION.' X. 1. There are three methods, as we have seen, by which num bers are represented; viz. by words, by single letters, and by charac ters usually termed figures1 and sometimes digits." 2. In the method by letters; which is called the Roman method. ecause the Romans invented it; are employed seven letters only. iz., I, V, X, L, C, D, M. 3. This method possesses some advantages over that by figures; but it has become nearly obsolete," being confined principally to the numbering of chapters, hymns, &c. 4. The method by figures; which is called the Arabic method, because the Arabs invented it; is, taken as a whole, by far the shortest and best method ever devised." 5. In this method are employed ten figures only; viz. 2, 3, 4, 5, 6, 7, 8 two three four five 9 0 ; six seven eight nine cipher 6. The first nine figures, which have each an absolute value, are called significant figures, to distinguish them from the cipher; which has no value in itself, being used merely to fill a vacanto place. The cipher is also called naught or zero.' 7. By variously combining11 these ten characters, no number can be conceived of too great to be represented by them, as will appear by the sequel.12 X. Q. How are numbers represented? 1. Describe the Roman method? 2. Is it still used and to what extent? 3. Which is the best method? 4. Describe it? 4. What characters does it employ? 5. Of what use is the cipher? 6. What other names has it? 6. What names have the other characters and why? 6. Can a large number be represented by so few characters? 7. 1 Figures were introduced into Spain, by the Arabs, in the 8th century (13) and from Spain into England about the middle of the 11th century; most eight hundred years ago. On the continent their use had become quite extensive: they are now so common, that if you were to visit China, for instance, you would recognize (14) at once their numerals, (15) without understanding a word of their language. 2 DIGIT, [L. digitus, a finger.] The measure of a finger's breadth, or the fourth of an inch. Figures were so called trom counting the fingers in reckoning. 3 The character 0 is called a cipher, from the Arabic word tsphara, which signifies A blank or void. The uses of this character in numeration are so important, that its nam cipher, has been extended to the whole art of Arithmetic, which has been called to cipher eaning to work with figures. 4 NUMERATION, [L. numeratio.] Numbering; the act of numbering, 5 INVENTED, [L. inventus.] Found out; devised; contrived; forged. 6 OBSOLETE, [L. obsoletus. Gone into disuse; disused, neglected. 7 DEVISED, [F. deviser.] Given by will; bequeathed; contrived; invented. ABSOLUTE, [L. absolutus.] Complete; positive; unconditional; independent 9 SIGNIFICANT, [L. significans.] Having meaning; expressive; important. 10 VACANT, [F. from L. vacans.] Empty; not filled; exhausted of air; unoccupied 11 COMBINING, [F. combiner.] Uniting closely; joining; confederating in purpose. 12 SEQUEL, [F. sequelle.] That which follows; consequence; event. 13 CENTURY, [L. centuria.] A period of one hundred years. 14 RECOGNIZE, [L. recognitio.] To know again; to revise. 15 Numerals, [L. numeratio.] Characters used for representing numbers |