University of California • Berkeley The Theodore P. Hill Collection of Early American Mathematics Books t THE MISSIONARY ARITHMETIC : OH ARITHMETIC MADE EASY^ m A NEW METHOD : DESIGISTED TO DLMINISH THE LABOR OF THE TEACHER, INCREASE THE IMPROVEMENT OF THE LEARNER. ACCOMMODATED TO ** THE PRESENT ERA OF BENEVOLENT ENTERPRIZE, AND ADAPTED TO THE USE OF LANCASTERIAN AND OTHER SCHOOLS. BY WILLIAM R. WEEKS. UTICA:* PUBLISHED BY MERRELL & HASTINGS, No, 40 Genesce-Sti'eet. William VViUlams, Pmiier^ JK^ortheni IHstnct of J^e^v-York. ss* BE IT REMEMBERED, That on the eighth day of February, in the forty sixth year of the Independence of the United States of America, A. D. 182?, Williara R. Weeks, of the said District, has deposited in this office the title of a Book, the right whereof he claims t s author, in the words following, to wit : " The Missionary Arithmetic; or Arithmetic made easy, in a new method. Designed to diminish the labor of the teacher, and increase the improvement of the learner ; accommodated to the present era of benevolent enterprize, and adapted to the use of Lancasterian £nd other schools. By William R. Weeks." In conformity to the act of the Congress of the United States, entitled " An act for the encouragement of learning, by securing ihe copies of maps, charts, and books, to the authors, and proprietors of suth copies, during the time thereiQ mentioned ;" and also to an act entitled *' An act supplementary to an act entitled an act for the encouragement of learning by securing the copies of maps, charts, and books, to the authors ahd proprietors ot" such copies, during the time therein mentioned ; and extending the benefits thereof to the arts of designing, engraving and etching historical and other prints." RICHARD R. LANSING. Clerk of the JVorthern District ofJS^exV'York, PREFACE, IN" presenting to the public a new Arithmetic, the Author ^v ill floubtless be expected to oifer soiae reasons for such a publication. He has been several years employed in the instruction of schools, and has found much inconvenience in teaching: arithmetic from the books in common use, and much interruption to the oiher business of the school, from the study of it in the common way. The obscure Kianner in which tlie rules are usually expressed, and the want of sufficient illustration, render them extremely difficult to be comprehended by beginners. This discourages the learner from committing them to memory, and makes him impatient to go forward and attempt to perform the various questions which are inserted tnr practice in the rules. In doing this, he meets with many difficulties, owing to his ignorance of the rules, and is perpetually running to the instructor for assistance. The instructor must take time to work out his questions for him, or spend still more in giving explanations which are seldom reriiembered. And if he does not stop all the other business of the school, to attend thus to every learner, he is complained of for his neglect, and the learner is discouraged from attempting to comprehend what appears so dark and perplexing. None but those who have had experience in teaching schools, can appreciate the trouble, aad vexation, and interruption, which con- tinually arise from this source. To furnish an Arithmetic, and to point out a mode of instruction, which shall remedy tiiese evils, is the design of thi^ work. The improvements attempted in it, are the following : 1. The rules are expressed in terms, and accompanied by expla- nations, more easy to be understood. 2. Under every rule, one question or more is performed at full length, and every step of the operation explained at large, that the meaning and application of the rule may be clearly seen. 3. Under the first rules, a number of examples, prepared in the Lancasterian manner, are given, for classes to be exercised in by the help of a monitor. 4. A large number of exercises are inserted, which consist of short and easy questions, w ith answers annexed, to be performed mentally, and answered extemporaneously, to a monitor j designed to quicken the attention of learners, and render all the usual operations ii^ arithmetic perfectly familiar. 5. A vset of questions are inserted, on the nature of each rule, without answers, in the manner of the modern improvements in teaching geography, that the scholar may examine the rule itself^ and find out the answers ; intended alsQ for the use of classes. iv Frejace. 6. The ^eat mass of questions for the practice of learners, are expressed more in the form in which they will naturally arise in the transaction of business, anr^ are set down by themselves in the second part of the work, without answers ; and after a few olthe hrst, ihey are not arranged in the order of the rules ; so that the learner, in order to perform them, must understand his rules, and pay attention to the nature of the questions themselves. The numb- r of them is also greater than usual, that the learner may have, in going through the book, abundant ex<^ret-e in all the rules, and in all -iorts of questions that will be likely to arise in the transaction^! of active life. 7. In forming the questions for practice, a large number of useful and interesting facts are embraced, which will not only serve the purpose ofextrcising the learner in the rules, but willConve} to him much important information respecting the great enterprizes of Christian benevolence which distinguish the present age, and render this work a useful auxiliary, in training the rising generation to es- teem the privilege, and practise the duty of doing good. 8. The rules for extracting the roots, are expressed in a new f©rm, more easy to be understood and remembered. 9. Under the head of Mensuration, easy rules are given for finding the content of the various solids, the capacity of difiierent vessels, the measurement of heigiits and distances, and the surveying of land, as far as is necessary for the common purposes of the tarmer, without the aid of mathematical instruments. 1 0. The whoile is adapted to the use of schools, in such a maimer that all classes of learners may receive the requisite attention and instruction, with very little trouble to the instructor, and very little interruption to the other business of the school. These iuiprovemeuts have been the result of several years' atten- tion to the subject ; and most of them have had the test of experi- ence, in schools under the direction of the Author, long enough t© demonstrate their utility. For the valuable hints with which the Author hai been obligingly furnished by various literary ^tntlemen, he woiild begtfeem to accept his thanks. He is particularly indebted to the Rev. Joseph Emerson, of the Byfield Seminary ; Mr. Professor S rong, of Hamilton College ; Mr. John Randel, Jun of Albany, Surveyor; Mr. Luther Jackson, of New -York, Teacher; and the publiiatiotts of Mr. Joseph Lan- caster. As some errors are almost inseparable from a first impression of a work of this nature, those who may discover any, will confer a favor on the Aiithor, by transmitting to him, or to the publisher-, a state- ment of them, that they may be corr^icted in a subsequent editioa. Paris, February, 1822, ARITHMETIC MADE EASYo PART T. Arithmetic is tiie science of Numbers, and the art of using them. JVotation teaches how to express any number by the fol- lowing characters, called figures; 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. JWimeration teaches how to read, in the proper words, any number expressed by these figures. When the figures stand one by one, their value is as fol- lows : 1, one, 2, two, 3, three, 4, four, 5, five, 6, six, 7, seven, 8, eight, 9, nine, 0, nought ; which is called tiieir simple val- ue. Besides their simple value, they have another, when two or more of them are joined together, which depends on the place in which they stand, and which may be called their lo- cal value. The places are counted from right to left, as follows : first, units; second, tens; third, hundreds; fourth, thousands; fifth, tens of thousands ; sixth, hundreds of thousands ; se- venth, millions; eighth, tens of millions ; ninth, hundreds of millions : tenth, thousands of millions ; and so on. Thus, 3, standing alone, is in the first place, and denotes three units, or three. With a cypher at the right hand of it, thus, 30, the 3, standing in the second place,' denotes three tens, or thirty. With two cyyjhers, thus, 300, it stands in tlie third place, and denotes three hundred. With three cyphers, thus, 5000, it stands in the fourth place, and denotes" three thou- sand. With four cyphers, thus, 3 000, it stands in the fifth place, and denotes three tens of thousands, or thirty thou- sand. AVith five cyphers, thus, 300000, it stands in the sixth place, and denotes three hundreds of thousands, or three hundred thousand. With six cyphers, thus, 3000000, it stands in the seventh place, and denotes three millions. With seven cyphers, thus, 30000000, it stands in the eighth place, and denotes diree tens of millions, or thirty millions. As in the following table : AS JS^umeration, r^ ^ a H H ffi 5 3 3 3 3 3 3 Three. Thirty. Three hundred. Three thousand. Thirty thousand. Three hundred thousand. Three millions. Thirty millions. From which it is plain, that every remove a figure makefe from the right hand towards the left, increases its value ten- fold ; its value in the column of tens, being ten times as much as in the column of units ; and in the column of hundreds, ten times as much as in the column of tens, and so on. The cypher has no value of its own, but only serves to show^ the local value of other figures to which it is annexed. 8o also, when other figures are put in the places of these cyphers, each figure has a value according to the place in v.hich it stands. Thus, in 21, the 1, standing in the first place, denotes 1 unit, or one, and the 2, standing in the se- cond place, denotes 2 tens, or twenty ; and taken together, they are to be read, twenty-one. In 32 1, the 1, standing in the first place, is one unit, or one ; the 2, standing in the se- cond place, is two tens, or twenty ; and the 3, standing in t]w. third place, is 3 hundreds, or three hundred; and taken to- gether, they are to be read, three hundred and twenty one. See the folio wino; table : o k^= § U tr^ 1 One. 2 I Twenty-one. 3 2 1 Three hundred and twenty -one. 4 3 2 1 Four thousand, three hundred & twenty -ene. 5 4 3 2 1 Fifty-four thousand, 3 hundred and 21. 6 5 4 3 2 1 Six hundred & 54 thousand, 3 bund. & 21, 7 6 5 4 3 2 1 Seven million, 654 thousand, 3 hund. & 21, 8 7 6 5 4 3-21 Eighty-seven million, 654 thous. 321. 987654321 Nine hund. ^ 87 million, 6 Imnd. & 54 thoTis. S2I, JSTumeration^ QUESTIONS ON THE FOREGOING. How much does a ftgui'e increase its value, by every remove from right to left f How do you read the figures 1, iJ, 3, 4, when placed so that 1 shall stand in the first place, 2 in tlie second, 3 in the third, and 4 i * the fourtli ? How do you read the figures, !♦ 2, 3, 4, 5, 6, standing 1 in the first place, 2 in the second, and so on ? How, tlie figures, 1, 2, 3, 4 5, 6, 7, after the same an-angement ? How do you read the same figures, when 7 stands in the first place, 6 in the second, 5 in the third, and so on ? What is Aritlfmetic ?. What does Notation teach ? What does Numeration teach ? 'What is meant by the simple value of any figure ? What by its local value ? Where do you begin to count the places of figures, in N^umeration? What is the name of tlie first place P the second f third ? fourth ? fifth ? sixth ? seventh ? eighth ? ninth f tenth ? What is the value of tlie figure 3, standing in the third place, with cyphers at tlie right hand of it ? What, in the second place P the fifth ? fourth ? eighth ? seventh ? sixth ? J\''ote. To use the following exercise, let the class be seated with their slates, but without any books. Let the Monitor take a book, and read the Avords, twenty-one ; and let every boy write down, at the top of his slate, the iigures which express that number. Then let the Monitor examine all the slates, and if any one has not written it right, show him how to do it. i hta let him read the words thirty-two^ and let them write it as before ; and soon, placing the numbers under each other, as they stand in the table, units under units, tens under tensj &c. When the slates are filled, let them read their iigures in words, the first boy reading the first number, and the second the itext, and so on ; the Monitor looking over the words ik the table, to see if ihey read them right. Exercise 1. 21 Twenty- one. 32 Thirty-two. 524 Five hundred and twenty- four. 78 Seventy- eight. 169 Oae hundred and sixty -nine. 436 Four hundred and thirty- six. 1234 One thousand, two hundred and thirty-four. 3451 Three thousand, four hundred and fifty- one« 89 Eighty-nine. 643 Six hundred and forty -three. 4326 Four thousand, three hundred and twenty six. 235 Two hundred and thirty-five. 6478 Six thousand, four hundred and seventy- eight. £1564 Twenty-one thousand, five hundred &, sixty -four. 987 Nine hundred aYid eighty seven, 99 Ninety-nine. 34567 Thirty four thousand, five hundred and 67. 2348673 Two million, 348 thousand, 6 hundred and 73. 6542 Six thousand, five hundred and forty two. 129834 One hundred and 29 thousand, 8 huadred and 34. • J^meration* J^'oU, When ♦he tlass are sufficiently practised in this exereisei let then* take exei-cis'S t and 3 After they have had a little practice in them, let questions be given ftw.n all promiscuoasly. And lot no boy proceed, to the nrxt rule, till he is tboroughlv acquainted with ail that goew before, and can write (lowji correctly, from the mouth of the Instructor, any number he shall ilictatc, and read correctly any number he shall write. Exercise 2. 12 Twelve. 201 Two hundred and one. 2020 Two thousand and twenty. 2200 Two thousand, two hundred. 2002 Two thousand and two. S0303 Thirty thousand, three hundred and three. 303030 Three hundred and three thousand and thirty. 330303 Three hundred & 30 thousand, 3 hundred & 3. 3033003 Three million, thirty -three thousand and three. 31091 Thirty one thousand and ninety one. 4040 vO Four hundred and four thousand and forty. 4014 Four thousand and fourteen. 4004041 Four million, four thousand and forty-one. 15115 Fifteen thousand, one hundred and fifteen. 505^-5 Fifty thousand, five hundred and five. 6606060 Six million, six hundred and 6 thousandand 60. 60006 Sixty thousand and six. 9901019 Nine million, nine hundred and 1 thousand and 19. 109090 One hundred and nine thousand and ninety, rroori Seven hundred and seventy thousand and 71. Exercise 3. 101 One hundred and one. 110 One hundred and ten. 1001 One thousand and one. 1010 One thousand and ten. Ill One thousand, one hundred. 1011 One thousand and eleven. 11001 Fiieven thousand and one. 10100 Ten thousand, one hundred. 10010 Ten thousand and ten.- 100001 One hundred thousand and one. 101100 One hundred and one thousand, one hundred. 11010 Eleven thousand and ien. 10011 Ten thousand and eleven. 100101 One hundred thousand, one hundred and one; 1101 1 Eleven thousand and eleven. 10111 Ten thousand, one hundred and eleven. I Bxplanation of Characters. 101011 One hundred and one thousand and eleven. 1 1101 Eleven thousand, one hundred and one. 1100001 One million, one hundred thousand and one. 1101101 One million, one hundred and one thousand, 101. EXPLANATION OF CHAltACTERS. Two parallel horizontal lines signify equality, as 100 centt = 1 dol. that is, 100 cents equal 1 dollar. Ir}- A cross, made by a horizontal line and another perpen- dicular to it, is the sign of addition, as 2+4 =s6, that is, the sum of 2 and 4, is equal to 6. A horizontal line, is the sign of subtraction, and shovi^s that the number which stands after it is to be taken from the number which stands before it, as 6 — 2=4, that is, 6 diminished by £, is equal to 4. A cross, like the Roman letter X, is the sign of multipli- cation, as 3x6z= 1 8, that is 3 times 6 is equal to 18. t-^ A horizontal line, with a point above and below it, is the sign of division, and shows that the number wh'ch stands before it is to be divided by that which stands after it, as 24 —6 =4, that is, 24 divided'by 6, is equal to 4. : : : Points standing one above another like colons, arc used to signify proportion. That is, when four num- bers are placed in succession, with one colon between the first and second, two colons between the second and third, and one colon between the third and fourth, they signify that the first number has the same propor- tion to the second, that the third has to the fourth ; thus, 2 : 4 : : 8 : 16, that y$, as 2 is to 4, so is 8 to 16. •• Two points, standing beside each other, are used in this work to separate different denominations ; as, ^ 2 •• 6 •• 8. that is, 2 pounds, 6 shilling:^, and 8 pence. • A single point is used in decimal fractions, to separate the whole numbers from the decimal parts, as 8*5, that is 8 and 5 tenths. It is also used to separate dollars from cents and mills, because cents and mills are de- cimal pans of a dollar; as, SS'65r, that is, 3 dollars, 65 cents and 7 mills. J One number written over another, with a line between, is called a vulgar fraction ; as J one half, i one third, J three fourthi. 10 Addition. JVote 7velL The learner should be careful not to make any of these mai^ts «pon his slate or paper, for any other purpose, or with any other meaning, flian is here directed; and to make no unnecessary marks whatever. Questions on the foregoing. What is tlie sign of equality ? of ad- For what is a single point used ? dition ? subtraction ? multipli- Why is it used for tlie laf ter ? cation ? division ? proportion ? flow are vulgar fractious written ? "What marks are used to separate What cant on should the leai'uer oV different denominations ? serve about mai-ks ? FUNDAMENTAL RULES. There are four rules which are called the Fundamental Rules, because all operations in arithmetic are perforjued bj the use of them. Thej are Addition, Subtraction, Multipli- eation, and Division. ADDITION, Is putting together two or more numba's, so as to find their total amount, which is called their sum. It is calleti Simple. Addition, when the numbers to be put together are all of the same denomination. Rule. 1. Write down the several numbers under each other, s« that units shall stand under units, tens under tens, &c. 2. Draw a line under the lowest number, to separate the ^iven numbers from their sum, when it shall be found. 3. Take the right hand column, or row of units, begin at the bottom, and add up. If the sum of that column is les» than ten, that is, if it is but one figure, it is units, and you must set it down under the column of units, and proceed to the next column. If it is ten or more, that is, if it is more than one figure, set down the right hand figure, which is a unit, under the column "of units, and carry the rest, which will be tens, and add them to the column of tens. 4. Add up the column of tens, and when you have found the sum, if it is but one figure, it is tens, and you must set it down under the column of tens ; but if it is more than one figure, the right hand one is tens, and must be set down under the column of tens, and the rest will be hundreds, and must be carried and added to the column of hundreds. 5. Proceed in like manner through all the columns to the last, where you must set down the whole amount of that •olumu. I Mdition* li Proof. Begin at the top, and add downwards, and if the total is the same as the first total, the work is probably right. Example. Find the sum of S21, 43b, 372, and 647. First, I write down the numbers under each other, so that units stand under units, tens under tens, &c. ; then I draw a line under, and add as follows : 321 I begin at the right hand column, at the bottom, and 436 say, 7 and 2 is 9, and § is 15, and 1 is 16. llie right 372 hand figure 6, being units, I set down under the co- 647 lumn of units ; and the other 1 , being a ten, I carry to the column of tens, and say, 1 to 4 is 5, and 7 is 12, 1776 and 3 is 15, and 2 is 17. This being the sum of the column of tens, is 17 tens, or one hundred and seventy. The right hand figure 7, being tens, and denoting 7 tens or 70, I set down under the column of tens, and cairy the other 1, being a hundred, to the column of hundreds, and say, 1 to 6 is 7, and 3 is 10, and 4 is 14, and 3 is 17. This being the sum of the column of hundreds, is 17 hundreds, or one thou- sand seven hundred. The 7, denoting seven hundreds, I set down under the column of hundreds ; and the i, denoting one thousand, I set down in the place of thousands, there being no column of thousands to which to carry it. And the answer is, one thousand, seven hundred and seventy-six. Proof. 321 I begin at the right hand column at the top, and say, 436 1 and 6 is 7, and 2 is 9, and 7 is 16. Set down 6 un- 372 der the T, and carry 1 to the next column. 1 and 2 is 647 S, and 3 is 6, and 7 is 13, and 4 is 17. Set down 7 under the 4, and carry 1 to the next. 1 and 3 is 4, 1776 and 4 is 8, and 3 is 11, and 6 is 17. Set downaz. And the total is 1776, the same as before ; so 1 con- clude the work is right. JYote 1 o use the following example^, let a c^ass be seated with their slates, and let thfc Monitor take the book ar»d read the first number, and let it be taken down and examined. Then let him read the second number, and see that that is taken down correcth, and placed under the first, so that units shui- stand under units, tens under tens, &c W hen aU the numbers are correctly tak< n down, and a line drawn under, let him read the work as it is set down under the question, repealing it slowly and distinctly. Whilu he readfej, let each boy fijlow him up tlie column, pointing to each figiu'e as the monitor names it, and taking notice of th':; amount which it makes ; and when the Monitor tells what figure to set down at the bottom of the column, \*ti each boy set it dowxi ; and so on, till the -vrhole is finished. And when 12 Jtdditisn. #ie Monitor reads the apiount, let each boj read it after him, from his slate. When this is done, let thai work bo rubbed out, and another example per- formed in the same inrinner When all the examples havejaeen several times repeated in ihis Avay, let th^ Monitor vary the process, in this manner; let Mm nnme the fiist and second fij^^m es, and instead of reading from his hook iB^hat th'7 amount to, iet the firat bo} tell , tlien let the Monitor repeat the amount, and n nie the next fiij;uie, and the second b(,y tell the'amouut, and BO Oi), till the whole is finished. If one boy tells wrong, let it be put to the mext, and if no oiie cau teU right, let the Monitor ttll. JSTo, 1. 27935 Work, Take the right hand column, and begin at 3963 the bottom. 7 and 9 is 16, and 3 is 19, and 5 is 24 ; 8679 set down 4 under the 7, and carry 9. to the next. 143. 7 Second column. 2 and 2 is 4, and 7 is 11, and 6 is — — — 17, and 3 is 20 ; set down under the 2, and carrj 54904 2 to the next. Tiiivd cUumn, 2 and S is 5, and 6 is 11, and 9 i» SO, and 9 is 29 ; set down 9 under the 3, and carry 2 to the next Fourth column. 2 and 4 is 6, and 8 is «4, and 3 is 17, and 7 is 24 ; set down 4 under the 4, and carry 2 to the next. Fifth column. 2 and is 3, and 2 is 5 ; set down 5. Total, in figures, 4904; in words, fifty-four thousand, nine hundred and four. JVo. 2. 12345 Work, first column. 5 and 1 is 6, and 6 is 12, and 678 > 6 is 15, and i^ is 27, and 5 is 32; set down v, and S2356 cany 3. 7890 Second column. 3 and 4 is 7, and 9 is 16, and 5 is 13456 21, and 9 U 30, and 5 is 35, and 8 is 43, and 4 is 47 ; 7891 set down 7, and carry i. 2845 Third column. 4 and 3 is 7, and 8 is 15, and 4 is 1 9, arid 8 is 27, and 3 is 30, and 7 is 37, and 3 is 40 ; 83072 set down 0, and carry 4. Fourth column. 4 and 2 is G, and 7 is 13, and 3 is 16, and 7 is 23, and 2 is 25, and 6 is 31, and 2 is 33 ; set down 3, and carry 3. Fifth column. 3 and 1 is 4, and 3 is 7, and 1 is 8 ; set down 8. Total, in figures, 83072; in words, eighty -three thousand, ftftd seventy -tw». "i, Addition. 15 b. 5, 56784 90235 45676 81237 45988 76549 32131 45802 72343 56784 92345 Work ; first column. 5 and 4 is 9, and S is 12, and 2 is 14, and 1 is 15, and 9 is 24, and 8 is 32, and 7 is 39, and 6 is 45, and 5 is 50, and 4 is 54 ; set down 4, and carry 5. SecoKd column, 5 and 4 is 9, and 8 is 17, and 4 is 21, and 3 is 24, and 4 is 28, and 8 is 36, and 3 is 39, and 7 is 46, and 3 is 49, and 8 is 57 ; set down 7, and carry 5* Third a ! imn. 5 and 3 is 8, and 7 is 15, and 3 is 18, and 8 is 26, and 1 is 27, and 5 is 32, and 9 is 41, and 2 is 43, and b is 49, and 2 is 51, and 7 is 58 ; set down 8, and carry 5. 695874 Fourth column, 5 and 2 is 7, and 6 is 13, and 2 is 15, and 5 is 20, and 2 is 22, and 6 is 28, and 5 is 53, and 1 is 34, and 5 is 39, and 6 is 45 ; set down 5, and carry 4. Fifth column. 4 and 9 is 13, and 5 is 18, and 7 is 25, and 4 is 29, and 3 is 32, and 7 is 39, and 4 is 43, and 8 is 51, and 4 is 55, and 9 is 64, and 5 is 69 ; set down 69. Toia^, in figures, 695874; in words, six hundred and nine- ty-five thousand, eight hundred and seventy-four. JVo. 4. 5678 Work ; first column. 4 and 3 is 7, and 5 is 12, 9123 and 6 is 18, and 5 is 23, and 7 is SO, and 6 is 36, and 4567 8 is 44, and 7 is 51, and 3 is 54, and 8 is 62; set 21 98 down 2, and carry 6. 3456 Second column. 6 and 4 is 10, and 6 is 16, and 7 1987 is 23, and 4 is 27, and 8 is 35, and 5 is 40, and 9 is 2345 49, and 6 is 55, and 2 is 57, and 7 is 64; set down 9876 4, and carry 6. 8765 Third column. 6 and 2 is 8, and 7 is 15, and 8 is 1203 23, and 3 is 26, and 9 is 35, and 4 is 39, and 1 is 40, 2044 and 5 is 45, and 1 is 46, and 6 is 52 ; set down 2, ■ ' - ~ and carry 5. 5 1 242 Fourth column, 5 and 2 is 7, and 1 is 8, and 8 is 16, and 9 is 25, and 2 is 27, and 1 is 28, and 3 is 31, and 2 is 33, and 4 is 37, and 9 is 46, and 5 is 51 ; set down 51. Total, in figures, 51242 ; in words, fifty-one thousand, two hundred and forty -two. B 1^ Mdiiion. JSTo. 5. 3456 Work ; first column. 9 and 7 is 16, and 6 is tft, 7890 and 8 is SO, and 5 is 35, and S is 58, and 4 is 4ii, and 128 6 is 48, and 9 is 57, and 6 is 63, and 8 is 71, and 9 907 is 80, and 7 is 87, and 7 is 94, and 8 is 102, and 6 is 4017 108 ; ^et down 8, and carry 10. 8969 Second column. 10 and 8 is 18, and 7 is 25, and 9 798 is 34, and 2 is S6, and 5 is 41, and 8 is 49, and 7 is 1476 56, and 7 is 65, and 9 is 72, and 6 is 78, and 1 is 79, 5079 and 2 is 81, and 9 is 90, and 5 is 95 ; set down 5, 8986 and carry 9. 754 Third column* 9 and 7 is 16, and 6 is 22, and 9 9023 is 3 1, and 8 is 39, and 7 is 46, and 9 is 55, and 4 i» 805 59, and 7 is 66, and 9 is 75, and 9 is 84, and 1 is 85, 998 and 8 is 93, and 4 is 97 ; set down 7, and carry 9, 1676 Fourth column. 9 and 6 is 15, and 2 is 17, and 1 2007 is 18, and 9 is 27, and 8 is 35, and 5 is 40, and 1 is 6789 41, and 8 is 49, and 4 is 53, and 7 is 60, and 3 is 63 | set down 63,- 63758 Total, in figures, 63758; in words, sixtj-tlirec thousand, seven hundred and fifty -eight. Questions. 1. The number of ordained missionaries among the hea- then in the year 1821, was as follows : From England, 255 ; Scotland, 7; United States, 39; Denmark, -Z; Moravians, (diff'erent countries,) 68 : how many in all ? Ans, 351. 2. In the year 1820, the American Board of Commissitmers for Foreign Missions, had the following missionaries and as- sistants among the heathen, to wit : In Eastern Asia, 25 ; Western Asia, 2 ; Sandwich Iblands, 17 ; American Indians, 44 : how many in all ? ^ns. 88. 3. The disbursements from the treasury, for expenses, du- ring the same year, were as follows : For the Bombay mission, 7221 dollars ; Ceylon, 7135 ; Cherokee, 9967 ; Choctaw, 10414; Arkansaw, 1150; Indian missions generally, 252; Palestine, 2348 ; Foreign niission school, 3350 ; Sandwich island, 10330 ; travelling expenses of members of the Board, tac. 457 ; salary of Secretary, 500 ; salary of Treasurer, 600 ; clerk hire, postage and stationary, 1143; printing, 1558; agents to collect funds, 261; expenses of meetings, 84; transportation of articles, 107; bad bills, 184; other con- tingencies, 84 : how jnuch in all? ^ns. 57144 dollars. Mdition. U J/ote. For further questions io exercise the learner* see Part III ; and he thould proceed to perform some of them immediately* Questions on the foregoing. What are the fundamental rules of \rithmetic ? Whv are they so called ? What is a Irlition ? Wli'3«i is it culled simple addition ? Wha! is to he observed in writing down the nnnbers to be added ? Which ■.•olamn do vou add first i Where lo you bej^in to adl ? When y .'U have adiled up the co- lumn of units, what do you do with th.; a^nor.it, if it is one fissure ? What, if it is more than one ? When you have added up the column of tens, and the amount oi it is two fissures, what is the value of the rij^ht hand Ha^ure ? What of the other ? VVhat do you do with them ? When tbe amount of the column of hundreds is two figures, what is the Value of each ? VVhat do you do with them ? When you have added up the last column, what do you do with th« amount ^ How do you prove addition ? ^ote. To use the following exercise* let a class be seated without slates or books. an I must add as much to the lower, and so I carry 1 to the columia Subtraction. If ©f tens, and say, 1 to 3 is 4, and 4 from 5 lea.ves 1 ; set down 1 , and proceed to the next column. 2 from 4 leaves 2 ; set down 2, and proceed to the next. 1 from^ ajeaves 2 ; set down 2. And the remainder, thus found, is 2218. Add the remainder to the less number, and if^ the work i* right, their sum will equal the greater number. J\ote, The following examples are to be perfoi^med by a class with a mo- nitor, as those in addition. JSTo. 1. From 2345 Work, Begin at the right hand at the bot- Take 1452 torn, and say, 2 from 5 leaves 3 ; set down 3. Second column. 5 from 4 I cannot ; borrow 1 0, Rem. 893 which added to the 4 is 14 ; 5 from 14 leaved 9 ; set down 9. Third column. For the 10 that I borrowed in the last column, carry I to this ; I and 4 is 5 ; 5 from 3 I cannot ; borrow 10, and say, 5 from 13 leaves 8 ; set down 8. Fourth column. Carry 1 to 1 is 2 ; 2 from 2 leaves 0. Remainder, in figures, 893 ; in words, eight hundred and ninety-three. JS^o. 2. From 98764 Work. 5 from 4 I cannot; borrow 10, and Take 34985 5 from 14 leaves 9^ set down 9. 1 carried to 8 is 9 ; 9 from 6 I cannot ; borrow 10, and 9 Rem. 63779 from 16 leaves 7; set down 7. 1 carried to 9 is 10 ; 10 from 7 I cannot ; borrow 10, and 10 from 17 leaves 7 ; set down 7. 1 carried to 4 is 5 ; 5 from 8 leaves 3 ; set down 3. 3 from 9 leaves 6 ; set down 6. Remainder, in figures, 63779 ; in words, sixty- three thou- sand, seven hundred and seventy- nine. ^0. 3. From 91234 Work. 1 from 4 leaves 3 ; set down 3. Take 51301 from 3 leaves 3 ; set down 3. 3 from 2 I can- not ; borrow 10, and 3 from 12 leaves 9 ; set Rem, 39933 down 9. 1 carried to lis 2, 2 from 1 I can- not; borrow 10, and 2 from 11 leaves 9 ; set down 9. 1 carried to 5 is 6 ; 6 from 9 leaves 3 ; set down 3, Remainder, in figures, 39933 ; in words, thirty -nine thou- BStnd, nine hundred and tiiirty-three^ 2» Subtraction^ JVo. 4. From 98r6545 Work, 4 from 3 I cannot, but 4 from IS Take 987654 leaves 9. 1 to 5 is 6; 6 from 4 I cannot, . but 6 from 14 leaves 8. 1 to 6 is 7 ; 7 from Rem. 8888889 5 I cannot, bat 7 from 15 leaves 8. 1 to 7 is 8 ; 8 from 6 I cannot, but 8 from l6 leaves 8. 1 to 8 is 9 ; 9 from 7 I cannot, but 9 from 17 leaves 8. 1 to 9 is lU ; 10 from 8 I cannot, but 10 from 18 leaves 8. I to is 1 ; I from 9 leaves 8. Remainder, 8888889. Questions. 1. If the population of the world is 820000000, and the number of nominal christians is 214000000, how manj are destitute of the gospel ? Jlns. 606000000. 2. Our Lord and Saviour, previous to his ascension, in the year 33, commanded his disciples to preach the gospel among all nations, and the London Missionary Society was formed in the year 1795 ; how long between ? Ans. 1762 years. 3. The canon of scripture was completed in the year 97, and the British and Foreign Bible Society was formed in the year 1804 ; how long between ? Ans, 1707 years. Questions on the foregoing. What is Subtraction ? When is it called SimpleSubtraction? How many numbers are employed ? How must they be written down ? Where do you begfin ? What is to be done if the upper figure is less tlian the lower ? When you have borrowed 10. how many must you carry, and where ? Why is 1 carried, while 10 was bcw- rowed ? How 18 subtraction proved ? Exercise 7. From Take Ans. Prom Take Ans. From Take Ans. 12, 3+ 4, 5 n, 4+3+2, 2 20, 6+5 + 3, 6 15, 4 7, 4 ir. 5 3 4, 5 18, 9 3 4, 2 18, 9 6, S 19, 7 2 3, 7 19, 4 5 7, S 20, 7 12, 1 21, 9 6 S, 2 ir. 5 1 4, r 25, 9 12, 4 16, 4 3 7, £ 16, 4 3 6, 3 U, 3 5, 6 23, 5 3 4, 11 15, 1 2 3, 9 12, 1+2+6, 3^ 24, 6 9 3, 6 19, 2 4 7, 6 MULTIPLICATION, Is a short way of performing addition, and teaches how to find the amount of a number when added to itself a certain number of times. It is called Simple Multiplication, when the number to be iaaultiplied is of one denomination, Multiplication. 21 m The number to be multiplied is called the multiplicand s the number to multiply by, is called the multiplier ; and the number found, or total amount, is called the product. The multiplicand and multiplier are both called factors. J>tote. Before pre)ceeding to any operatious in multiplication, it is necessary to learn perfectly the following table. Multiplication Table. % times 1 4 times 6 times 2 is 4 2 is 8 2 is 12 Sis 6 3 is 12 Sis 18 4 is 8 4 is 16 4 is 24 .5 is 10 5 is 20 5 is 30 6 is \2 6 is 24 6 is 36 ris 14 7 is 28 7 is 42 Sis 16 8 is 32 8 is 48 9 is 18 9 is 36 9 is 54 10 is 20 10 is 40 10 is 60 1 1 is 22 i 1 is 44 11 is 66 12 is 24 12 is 48 12 is 72 3 times 5 times 7 times Sis 6 2 is 1 2 is 14 3 is 9 3 is 15 3 is 21 4 is 1£ 4 is^O 4 is 28 5 is 15 5 is 25 5 is 35 6 is 18 6 is 30 6 is 42 7 is 21 7 is 35 7 is 49 8 is £4 Sis 40 8 is 56 9 is 27 9 is 45 9 is 63 10 is 30 10 is 50 10 is 70 1 1 is 33 1 1 is 55 11 is 77 12 is 36 12 is 60 12 is 84 8 times 2 is 16 3 IS 4 is 5 is 6 is 7 is 8 is 9 is 10 is U is 12 is 24 32 40 48 56 64 72 80 88 96 9 times 2 is 18 3 is 4 is 5 is 6 is 7 is 8 is 9 is 10 is 11 is 27 36 45 54 63 72 81 90 99 12 is 108 10 times 2 is 20 3 is 4 is 5 is 6 is 7 is 8 is 9 is 10 is 100 11 is 110 12 is 120 1 1 times 2 is 22 3 is 4 is 5 is 6 is 7 is 8 ih 9 is 10 is no 11 is 121 12 is 132 33 44 SB 66 77 88 99 12 times 2 is 24 3 is 36 4 IS 48 5 is 60 6 is 72 7 is 84 8 is 96 9 is 108 10 is 120 11 is 132 12 ia 144 Case I. To multiply any given number by a single figure, or by any number not over 12. Rule. 1. Set down the multiplier under the units figure, or right hand place of the multiplicand, and draw a line underneath. 2. Begin at the right hand figure, and Hiultiply. If the product of the units figm^e of the multiplicand is but one figure, set it down in the place of units. If it is more fig- ures than one, set down the right hand figure, or units, and «arry the rest to be added to the product of the ten». 2t JIultiplication* S. Multiplj the tens, and to the product add what wai earned from the product of the units. Set down the right Sland figure, and carry the rest to be added to the product of the hundreds. And so on, to the end, setting down the whole m the last place. JVbie. To prevent mistakes, it will be well to make t miaute of what is to be earned each time. Example \, Multiplj 5678 TTork. 4 times 8 is 52 ; set down 2, and Bj 4 carry 3, 4 times 7 is 28, and 3 I carried is Prod. 22712 31 ; set down I, and carry 3. 4 times 6 is 24, and 5 I carried is 27; set down 7 and carry 2. 4 times 5 is 20, and 2 I carried is 22 ; set down 22. Product, in figures, 22712; in words, twenty-two thousand seven hundred and twelve. JVbtf*. Observe, th;jt I do not carry the tens of the product of the units t« the tens of the multiplicand, before I multiply them, but to their product j Ihe reason of which will appear, by varying the process, as follows : 5678 4 ' Here, 4 times 8 is 32, 4 times 70 is 280, 4 times 32 600 is 2400, and 4 times 5000 is 20000, which added 280 together, is 2271 2, as before. From which itappears, 2400 that the carrying is only done in the addition of the 20000 ser^ral products together. 22712 Example 2. Multiply 34567 Work. 7 times 7 is 49: set down 9, By 7 and carry 4. 7 times 6 is 42, and 4 I car- ried, is 46 ; set down 6, and carry 4. 7 Prod. 241969 times 5 is 35, and 4 I carried, is 39 ; set down 9, and carry 3. 7 times 4 is 28, and 3 I carried, is 31 ; set down 1, and carry 3. 7 times 3 ia 21, and 3 I carried, is 24 ; set down 24. Prod. 241969. Example S. Multiply 9876 Work. 12 times 6 is 72 ; set down 2, By 12 and carry 7. 12 times 7 is 84, and 7 I car- ried is 91 ; set down 1, and carry 9. Prod. 118512 12 times 8 is 96, and 9 I carried, is 105 ; set down 5, and carry 10. 12 times 9 is 108, and 10 I carried is 118 ; set down 118. Prod. 1185 U* Multiplication. tt Case II* To multiply by a number consisting of eevcral figures. Rule* 1. Set down the multiplier under the multiplicand, so that units shall stand under units, tens under tens, &c. and draw a line underneath. 2. Multiply the whole of the multiplicand bf the first or units figure oi the multiplier, and set down the product, a» in case first. S. Multiply the whole of the multiplicand by the second figure of the multiplier, and set down the product in the game manner, only placing each figure of the product one remove to the left. 4. Multiply in the same manner by the third figure of the multiplier, and place the figures of the product two removes to the left. And so on, to the end, placing the figuies of each product so that the first shall always stand under the figure by which you are multiplying. 5. W hen you have nmltipliett by all the figures of the multiplier, add the several products together, and their sum will be the answer, or whole product required. Examyle 1. Multiply 3456 Work. First fgure, 5 times 6 is 20 5 By 3-i5 set down 0, and cany S. 5 times 5 is 2 ', and 3 1 canitd, is 28 ; set down 8, nnd 17280 carry 52. 5 times 4 is 20, and 2 1 carried 6912 is 22 ; set down 2, and carry 2. 5 times 10368 Sis 15, and i2 J carried, is 1 . , srt down 17* Second fgure, 2 times 6 is 1^; set Prod. 112S20O down 9 under the 2, and carry i. 2 times 5 is 10, and 1 1 carried is 1 I ; ^et down 1, ami carry 1. 2 times 4 is 8, and 1 1 earned is 9; set down 9. 2 times 3 is 6 ; set down 6. Third figure. 3 times 6 is 18; set down 8 under the 5, and carry 1 . 3 times 5 is 15, and 1 I carried is 1 6 ; set down 6, and carry 1. 3 times 4 is 1£, and 1 1 carried, is 13 ; get down 3, and carry 1. 3 times 3 is 9, and I 1 carried is 10 ; set down 10. Having obtained the product of the multiplicand by each figure of the multiplier, those products are now to be iidded. Draw a line under and add. 24 Multiplication^ First eolufnn, is ; set down 0. Second column, 2 and 8 is 10: set down 0, and carry 1. Third column. 1 carried to 8 is 9, and I is 10, and 2 is 12 ; set down 2, and carry 1. Fourth column. 1 carried to 6 is 7, and 9 is 16, and 7 is 23 ; set down S, and carry 2. Fijih column. 2 carried to 3 is 5, and 6 is 11, and 1 is 12 ; set down 2, and carry 1. Sljjth column. 1 carried to is 1 ; set down 1. Seventh column. 1 is 1 ; set down I. Total Prod. 1 12S200. J\ oife. 1 he reaon for placing the fiict lUrure of ihc second product one re- move to the left, or in the place of tens, is, that it is ilie product of the units of the multiplicand by the tens of the muitiplien and is the efore tens, and should hf^ put in the place of tern. Fo the same reason, the first figure of* cf the tliird product is hundreds, and of the fourth 4 housa ds» and soon. Accordingly* in the abov example, 17280 is the product of 3456 by 5. But 6912» with the 2 standir *< iu \hf place of tens, is th^- same as 691 £0 and is the product of 3456 by 20. ' s uith th;- K stand in ibe place of hundreds, is the same as 1036801 produci. of 3456 by 300. Example ^. ' Multiply 98765 Work. First figure. 8 times 5 is 40 } By 678 set down under the 8, and carry 4. 8 times 6 is 48, and 4 I carried, is 52 ; 790120 set down 2 and carry 5. 8 times 7 is 691355 56, and 5 I carried, is 61 ; set down 1, 592590 and carry 6. 8 times 8 is 64, and 6 I carried, is 70 ; set down and carry 7. Prod. 66962670 8 times 9 is 7£, and 7 I carried, is 79; set down 79. Second figure. 7 times 5 is 35 ; set down 5, under the 7, and carry 3. 7 times 6 is 42, and 3 1 carried, is 45 ; set down 5, and carry 4. 7 times 7 is 49, and 4 1 carried is 53 ; «et down 3, and carry 5. 7 times 8 is 56, and 5 1 carried, is 61 ; set down 1, and cany 6. 7 times 9 is 63, and 6 1 car- ried, is 69 ; set down 69. Third figure. 6 times 5 is 30 ; set down 0, under the 6, and carry 3. 6 times 6 is 36, and 3 1 carried, is S9 ; set down 9, and carry 3. f times 7 is 42, and 3 1 carried, is 45 ; set down 5, and carr/ 4. 6 times 8 is 48, and 4 I carried, is 52 ; set down 2, and carry 5. 6 times 9 is 54, and 5 1 car- ried, is 59 ; set down 59. Next, draw a line under and add. is ; set down 0. 5 and 2 is 7 ; set down 7. 5 and 1 is 6 ; set down 6. 9 and 5 is 12; set down 'z, and carry i. 1 to 5 is 6, and I is 7, and 9 is 16 ; set down 6, and carry 1. 1 to 2 is 3, and 9 is 12, and 7 is 19 ; set down 9, and carry 1. 1 to 9 is iO, and 6 is 16 ; set down 6, and carry 1. 1 to 5 is 6 ; set down 6. Total product, 66962670^ Multiplicati9n, 25 Case. III. When ther« arc cyphers at the right hand of one or botk the factors, or between the other figures of the mtiltiplier. Rule. 1 . If the cyphers are at the right hand of the numbers, multiply the other figures as in case I or 2, and annex to the product as many cyphers as were neglected. 2. If cyphers are between other figures in tlie multiplier, neglect them also, only take care to place the first figure of every product exactly under its multiplier. Example. Multiply 304050 Work, Neglect the three cyphers By 20 SCO at the right hand of the factors, and begin with 3 and 5. S times 5 is 91215000 IS ; set down 5, under the 3, an- 60810 ' nexing three cyphers to it, and car- — — . ry 1. 3 times is(), and I I carri- Prod. 6172^^15000 ed, is 1 ; set down i. 3 times 4 is \2; set down 2, and carry 1. 3 times is 0, and I I carried, is I ; set down i. 3 times 3 is 9. The next figure of the multiplier being a cypher, neglect that, and take 2. 2 times 5 is 10; set down 0, under 2, and carry 1. 2 times is 0, and 1 I carred is i ; set dtwn I. 2 times 4 is 8 ; set down 8. 2 times is ; set down 0. 2 times 3 is 6 ; set down 6. Next, draw a line under, and add. is ; set down 0. is ; set down 0. is ; set down 0. 5 is 5 ; set down 5. 1 is 1 ; set down 1. and 2 is 2 ; set down 2. 1 and 1 is S ; set down 2. 8 and 9 is 17 ; set dov/n 7, and carry 1. 1 to is 1 ; set down 1. 6 is 6 ; set down 6. Total ^ro^we^, 6172215000. J\ote. The reas^. for annexing to the product Dieycphfrs which had been neglected at the rii^ht hand of the factors, is, tliat in the above exaruplf , the- product of the first sigrilficant figures is not reaif.y the ])roduct of 5 Wy 3, bus, of 50 bv 300. wljicli is not 15, I.mi 1. HK). A;u] the rerso :- r i-c'-i^ihe first figure of the xjroduct of 5 by il it; Hic place of hundreos c- tho sands, is, that' it is not really the product of 5 by ±, hvX of 50 by 20000, w hich is not 10, but? 1000000. Proof. First meikocL Multiply the multiplier by the multiplicand, and the product will be the same as before, if the work is right. C 2^ Multiplicatiou. Exnmple, Take the first example under Case I. and invert the factors. Multiply 4 Work. 8 times 4 is 32; set down 31. 7 By 5678 times 4 is .28 ; set down 28, so that the 8 shall stand under the 7. 6 times 4 is 24 ; set down 24, so that the 4 shall stand under the 6. 5 times 4 is 20 ; set down 20, so that the shall stand under the 5. Next, add, 2 is 2 ; set down 2. 8 and 3 is 1 1 ; set down 1, and carry 1 . 1 to 4 is 5, and 2 is 7 ; set down Prod. 22712 7. and 2 is 2 ; set down 2. 2 is 2 ; set down 2. l^otal product, 22712, which is the same as before ; so I conclude the work is right. Second method. Divide the product by the multiplier, and the quotient will be equal to the multiplicand, if the work is right. This is the method usually practised by experienced arithmeticians, as the safest; but as the learner is not sup- posed to be as yet acquainted with division, let him prove his work by the ftrst method, or let him make use of the Third method, 1. Cast the 9's out of both the factors, and place the remainders at the opposite ends of a dotted line. 2. Multiply those remainders together, cast the 9's out of their product, and set down the remainder above the dotted line. 3. Cast the 9's out of the product you wish to prove, and set down the remainder under the dotted line. If the work is right, the figures above and below the dotted line will be alike. Example,. Proof, Multiply 34562 1 By 5 Md, 2 5 Mr. 1 Product, 172810 Fred. Explanatiov., I first take the multiplicand, and say, 3 and 4 is 7, and 5 is 12; cast away 9, and 3 remains. 3 and 6 is 9 ; cast it away. 2 is left, as the remainder of the multipli- cand, which I set down at the left hand of the dotted line. I then take the multiplier, which being but 5, there is no 9 to cast away : so 1 set down 5 at the right hand of the dotted line, as the remainder of the multiplier. 2. I multiply these twn remainders together, saying 2 times 5 is 10 ; cast away 9, and 1 remains, which I set down above the dotted line. Multiplioatiou, m % I take the product, and say, 1 and 7 is 8, and 2 is 10; «ast away 9, and 1 remains. 1 and 8 is 9 ; cast it away. 1 and is I , whicli being less than 9, I set it down under the dotted line, as the remainder of the product. And the figures above and below the dotted liae being alike, I conclude the work is right. jstote. This method of proof is not infallible, because the right figures may stand in the product, and not stand in the right order ; or two wrong figiu-et may amount to the same, when added together, as the two right ones would. But as this method will usually detect a mistake, and is shorter than the other methods, it is thought useful to he retained. Questions. 1. If a child eats 6 cents worth of fruit, nuts, &c. every week, how many cents could he save in a year, being 52 weeks, and give for the education of heathen children, by de- nying himself those indulgencies ? •^HS. 312. 2. If a child should employ his play hours in working^for the education of heathen children, and should earn S cents a day, by so doing, how many cents would he earn in a year for that purpose, there being S13 working days ? Ans, 939. 3. If the population of the United States is 9630000, and every person should earn or save 3 cents a day for doing good, liow many cents would that be in a year ? Jin^, 9042570000* Questions on the foregoing. rather than under the figure muU tiplied ? When you have multiplied by all th« figures of the multiplier, what do you do next ? What is the third case ? What do you do with cyphers at th« right hand of the factors ? Why do you annex th^m to the right hand of your product ? What do you do with c\ phers be- tween other figures ©f your multi- plier ? What is the first method of proof I second ? Which is the safest ? What is the? third method of proof ? In the third method, what is the first thing: to be done ? the second } the third ? When do you conclude the work '\% right ? Is this pi^thod of proof certain } Why so \ Wh^ then is it retaiaed ? What is multiplication } When is it called simple multiplica- tion ? How many numbers are employed ? What is each one called ? What are they called together ? What is the answer called ? What is to be done, before you be- gin to perform questions in multi- plication ? What is the first case } How must the numbers be written down } Where do you begin, & how proceed? When you carry, tp what do you add the figure carried ? Why so ? What is the second case ? How many figures of the multiplier do you use at a time ? When you have multiplied the first figure of the multiplicand by the second figure of the multiplier, where do you begin to set it down P Why do you get it under that figure, 2$ Multiplication, Exercise 8. Tell the amount of 5x2x2, Ms, 123X3X5, Ans. 45 5x3x2, Ms . SO 3 3 2, 18 3 4 5, 60 5 2 4, 40 2 2 4, 16 3 2 6, 36 5 5 4, 100 2 3 3, 18 3 2 4, 24 6 2 2, 24 2 3 4, 24 3 3 3, 27 6 3 2, 36 2 2 5, 20 4 3 3, 36 6 4 2, 48 2 3 5, 30 5 2 2, 20 6 4 3, 72 Exercise 9, Tell the amount of 2X0X10, Ms. 10 4x2x4, Ms. 32 8x 5x1 Ms . 40 2 3 8, 48 5 2 5, 50 9 4 2, 72 3 4 8, 96 5 2 2, 20 9 10 2, 180 5-6 2, 6Q 6 5 2, 60 2 1 9, 18 5 5 2, 50 7 2 1, 14 3 3 9, 81 S 5 5, 75 8 3 2, 48 2 5 9, 90 3 4 4, 48|8 4 3, 96 4 10 3, 120 Exerc JISE 10. From take ./Itw. From take ^ns* From take ^nsi 24, 4x 5, 4 36, 9X 3, 9 37, 7X 4, 9 28, 3 5, 13 45, 3 12, 9 41, 6 5, 11 20, 2 7, 6 24, 4 3, 12 29, 7 3, 8 19, 3 4, 7 28, 4 4, 12 36, 8 4, 4 14, 5 g, 4 32, 5 3, 17 19, 3 4, 7 >1, 7 4, 3 18, 3 5, 3 25, 8 2, 9 ..r, 2 8, 11 22, 5 3, 7 38, 2 11, 16 Exercise 11. From take »^7is. From ta ke ^7is. From take .fns. iOO, 8x 8, 36 7^, CX12, 24 83, 6x12, 13 96, 4 12, 48 67, 8 b, 19 94, 8 9, 22 72, 7 9, 9 98, 6 11, Sz 105, 6 11, 39 66, 9 5, 21 102, 9 8, SO 86, 9 5, 41 48, 6 5, 18 55f 6 r, 13 74, 7 6, 32 84, 5 12, 24 48, 4 7, 20 ' 65, 5 7, 30 #r. 7 7, 18 79, 7 8, 23 47, 7 5, 12 Division. 3§ DIVISION, Is a short method of performing subtraction, and teaches to find how often one number is contained in another. It is called Simple Division, when the number to be di- vided is of one denomination. Tlie number to be divided, is called the dividend ; the number to divide by, is called the divisor ; and the result, or answer, is called the quotient, j\ote Before proceeding to perform operations in division, let the student learn the multiplication tab e in an inverted order, as t'ollows : 2 in 4 i« twice, 2 in 6 is 3 times, 2 in 8 is 4 times, kc. Rule. 1. Set down the dividend first, and the divisor at the left hand of it, separated by a curve line ; and leave a place at the right to set the quotient, separated also from the dividend by a curve line. 2. Take the fewest figures of the dividend, beginning at the left hand, that will contain the divisor, and see how many times they will contain it, and place the figure denoting ihat Dumber of times, for the first figure of the quotient. 3. Multiply the divisor by that quotient figure, and place ' the product under those figures of the dividend before men- tioned. 4. Subtract this product from that part of the dividend under which it stands, and set down the remainder. 5. Bring down another figure of the dividend, and place it at the right hand of the remainder, and then divide as be- fore this number so increased ; but if, when one figure is brought down, the divisor is not contained in it, set down a cypher in the quotient, and bring down another figure of the dividend ; and so on, till all the tigures of the dividend are brought down and divided. Eocample, Divide 561720 by 24. Mivis. Divid. Quot. 24)561720(23405 Work. First, write down the divi- 48 dend, and at the left hand of it, the — divisor, separated from it by a curve 81 line, and put another curve line at the 72 right, to separate the quotient. — Next, consider how few figures of 97 the dividend will contain the divisor ; 96 the first one, 5, will not, but the two — - first, 56, will ; and 24 in 56 is 2 times; 120 set down 2 in the quotient, and multi- 120 "dIv the divisor bv it- 2 times 4 ii; 8 ^ so Division. set down 8 under the 6. 2 times ^ is 4 ; set down 4. Tlien subtract. 8 from 6 I cannot, but 8 from 16 leaves 8 ; set down 8. 1 carried to 4 is 5 ; 5 from 5 leaves nothing. The remainder is 8. To this bringdown the next figure of the dividend, which is I, and it makes 81, Then divide again. 24 in 8 1 h 3rtimes ; set down 3 in the quotient, and multiplj. S times 4 is 12 ; set down 2 under the 1, and carry 1. 3 times 4 is 6, and 1 I carried is 7 ; set down 7. Then subtract. 2 from 1 1 cannot, but 2 from 1 1 leaves 9 ; set down 9. 1 carried to 7 is 8 ; 8 from 8 leaves 0. The remainder is 9, to which bring down 7, and it makes 97. Then divide again. 24 in 97 is 4 times ; set down 4 in the quotient, and multi- ply. 4 times 4 is 16 ; set down 6 under the 7, and carry 1. 4 times 2 is «, and 1 I carried is 9 ; set down 9. Then sub- tract. 6 from 7 leaves 1 ; set down 1. 9 from 9 leaves 0. The remainder is I, to which I bring down 2, and it makes 12. Then divide again. 24 in 12 is timeB ; set down in the quotient, and bring down the next figure of the dividend, which is 0, and annex it to the 12, and it makes 120. Then divide again. 24 in ISO is 5 times ; set down 5 in the quo- tient, and multiply. 5 times 4 is 20 ; set down under the 0, and carry 2. 5 times 2 is 10, and 2 1 carried is 12 ; set down 12. This product being equal to the number from which it is to be subtracted, there is no remainder ; and there being no more figures of the dividend to bring down, the work is finished, and the quotient is 23405. Proof. Multiply the quotient by the divisor, and to the product add the remainder, if any ; and the amount will be equal to the dividend, if the work is right. Contractions in Division. \ 1. Division by a number not exceeding 12, may be expe- ditionsly performed, by multiplying and subtracting in the mind, omitting to set dow i the work,^ excepting only the quotient, which may be set down immediately below the di- vidend. 2. When the right hand figure or figures of the divisor are cyphers, cut them off, and also cut off as many figures from the light hand of the dividend; then divide the remaining figures of the dividend by the remaining Hgures of the divisor. If any thing remains after this division, annex to the right of it the figures eut off from the dividend, and the whole will be JDivmon. 51 i the true remainder ; if nothing: rc^rnains from the division, the iigiuef^ cut oft' will be the remainder. J\'ote 1. This method is us(m1 fluly u) aYoiU the needless repetition of cy- phers which \v< aid haj)pt:n in tin- conmion 'way. vVr^i'tf 2. The ijuojei way of selling do"n a remainder after division, is t® place it a: tht i i^lit hand of the quotien , with iJie divisor under it, and a line between, m the form of a vulgar fraction. Questions. 1. If there arc 1189 chapters in the bible, and a child should read 2 chapters a day, how many days would it take him CO read it ihrough ? Jns, 5 94 J. 2. If a Cliris'ian school for 50 heathen children, can be* kept ill Ceylon for 2 00 cents a yeai% according to the state- ment of the missionaries there, how many cents is that for each child ? dns, 48. S. If the population of the United States is 9650400, and one minister of the gospel is necessary for every 800 souls, how many ministers are necessary in the United States ? Alls. 12038. Questions on the foregoing. What is division ? When is it called simple division f What is the dividend ? divisor P quo- tient ? [down ? How ar*^ the numbers to 1)6 written W iicav do ou begin to divide ? How many figures of the dividend do you take P V/here do }Gu set your quotient figure ? Wliat do }ou do next ? From whnt do you subtract ? What do you do after subtracting ? How many figures do you bring down at a time i* What do you do, when the divisor is not contained in the remainder so increased ^ How do you prove division ? What, is the first method of contract- ing division p the secoiid ^ In the second method, what is the true remainder P How should the remainder be set down P What is tlie result of an operation in addition called ? in subtraction ? in multiplication ? in division ? Tell how many times Ajis, 5 in 2^, 4 in 20, 6 in 3 6, Tin 42, 8 in 56, 8 in 24, 5 m S7, Exercise 12 4 in 32, dns. 8 9 in 108, 3 in 33, 1 i 6 in 30, 5 in 100, 20 5 in 45, 9 in 54, 6 9 in 72, 7 in m, 9112 in 96, 2 in 120, 10 9 in 63, 5 in 20, 4 il in 99, Ms, 12 5 9 8 3J| Exercius* Tell how many times 2 in 4x 4, Ms. 8 S in :^ 4 in 2 5 in 10 4 in 8 4 in 6 6 in 3 6, 10, 4, 2, 2, 8, Exi RCISE 13» 2 in 5x 6, Jlns. V 3 in 9 ; in 8 o in 4 12 in 6 3 in 2 5 in 3 6, 4, 6, 15, 10, 6 1 2 3 10 6 3 in 6x 4, 4 in 5 8, 8 in 4 12, 6 in b 4, 8 in 4 6, 3 in 2 9, i; m 3 12, 6 6 3 6 4 Tell how many times 7 in 3x21, Jins, 5 in 2 15, 8 in 3 16, 6 in 4 12, 8 in 6 12, 5 in 4 15, 6 in 9 10, Exercise 14. !. 9 7 in 3x14, Ayis, e 6 in 2x18, Atis, 6 ( 9 in 4 18, 8 4 in 2 14, 7 6 3 in 4 12, 16 8 in 2 20, 5 8[9 in 2 2r, 6 7 in 3 2 i , 9 9'5 in 3 ^0, 12 8 in 4 16, 8 12 9 in 3 15, 5 9 in 2 18, 4 15 8 in 2 16, 4 6 in 5 12, 10 Tell how many times Jtiis, 6in3x2xS, 3 8 in 2 2 4, S 10 in 2 2 5, 2 8 in 2 3 4, 8 6 in ^^ 3 5, 5 Exercise l5. 6 in 3x2x4, 4 in 3 2 6, 4 in 2 2 5, 12 in 6 4 3, 8 in 5 2 4, Ans. 4 9 5 6 5 4 in 5x2x2 6 in 5 7 2, 8 in 5 4 4, 6 in 4 5 3, 6 in 4 3 3, An9. 5 7 10 10 6 12 in 3 4 5, 5|]0in5 5 4, 10112 in 8 2 6, 8 9 in 3 2 6, 4 12 in 3 2 4, 2110 in 5 6 3, Exercise 16, Tell the sum of 7+2 + 4+5. 9 12 3, 8 1, 2, 1, 4, 6, 18 15 17 16 15 20 17 9+2+1+4, 7 8 3 1, 2 8 2 7 3 5 9 6 4, 2, 3, 4, Am. 16 19 17 16 21 19l4 5+4+9+2, 8 3, 3 2, S 5 3, 1419 6 1, 2, 3, 3, Ans, 20 19 16 20 If 18 25 Ejcereises, 3$ Exercise IT. Tell the sum of Am, Jim* Am* Ans. 24+33, 5T 35 + 51, 96 66+24, 901 44+52, 96 75 26, 101 45 24, 69 51 32, 83 33 53, 86 vj 18, 45 34 41, 75 36 48, 84 47 $6, 73 >2 13, 105 27 49, 76 24 47,' 71 22 34, 56 ^'9 S4, 103 39 28, 67 55 35, 90 54 21, 75 \: Exercise 18. Tell the sum of Ans. Alls. Ans, Am. S7+28, 115 65+41, l(/6 54+55, 19 72+4,5, iir 65 4S, 108 96 70, 166 71 36, 107 84 ^ 47, 131 91 42, 133 67 36, 103 83 42, 125 69 56, 125 82 27, 109 78 87, 105 64 59, 123 8.5 b4. 149 73 %7, no 62 76» 138 78 58, 136 74 66, 140 Exercise 19. Tell how many times fin 13+ 15, Ans. 4 8 in 19+29, Ms. 6 9 in 57+35, Jm. 8 6 iu 21 15, 6 9 in i5 23, 4 7 in 1 1 \7, 4 5 in 11 16, 9 8 in 45 27, 9 6 in 33 15, 8 4 in 13 19, 8 7 in 55 29, 12 5 in 41 19, 12 8 in 27 £9, 7 6 in 26 40, 1 1 7 in 21 f8. r 5 in 21 14, 7 5 in 13 32, 9 8 in 17 23, 5 7 in 25 31, 8 8 in 27 13, 5 Exercise 20. 4 in 20 32, IS From take, aid tell the Ans. From take , and tell the An9» 27, 3, half, 12 66, 3 ninth, f 26, 5, third, 7 73, 7, eleventh. 6 21, 6, fifth, 3 5^9, 5, sixth. 4 21, 5, fourth, 4 35. 7, fourth, 7 36, 6, fifth, 6 43, 8, seventh. 5 -1, 5, third, 12 52, 4. srah, 3 51, 9, sixth, 7 57, 15 six^h. 7 62, 14, eighth, 6 63, \5 eighth. 6 79, 7, sixth, 12 54, 9 third. 15 i^4, 7, sey€ nth, 1 1 64, 8 , seventh, 8 34 ReduGiion. REDUCTION, Is the changing of numbers from one name or denomina- tion to another, without altering their value. Before proceeding to operations in reduction, the following tables should be committed to memory, as far as Table 15. REDUCTION TABLES. 1. Federal Money. 10 mills, (m,) make 1 cent, ct. 10 cents, 1 dime, d, 10 dimes, or 100 cents, 1 dollar, S, or D. 10 dollars, 1 Eagle, E. JStote, In stating any sum in Federal Money, eagles and dimes are usually aeglected, and dollars and cents only arc mentioned, 100 cents making a dollar. I'he dollar is considered the money unit, and the lower denomina- ^ons as decimal parts of a dollar » 2. Sterling Money, and old Currencies of the several States, 4 farthings, (q.) make 1 penny, d, 12 pence, ^ 1 shilling, s. 20 shillings, 1 pound, £, or L JS^ote. Farthings are usually written as fractional parts of a penny, «• Xd* iS 1 farthing ; jcf. i» a halfpenny, or 2 farthings ; %d. is f farthings. 3. Troy Weight. 24 grains, (gv,) make 1 pennyweight, dwt, 20 pennyweights, 1 ounce, ox, 12 ounces, 1 pound, Ih, J^ote, A grain is equal to *7 ten thousandths of a solid inch of pure water. This weight is used for gold, silver. jeAvels, electuaries, and liquors. The Ineness of gold is tried'hj fire, and is reckoned in carats. Jf it loses nothing in the trial, it is said to be Zi carats fine. If it loses 2 twenty -fourths, it is said to be 22 carats fine* which is the stfmdard for gold. Silver, which lose* nothing in the fir«, is said to be 12 ounces fine. The standard for silver coin, 88 II 02. tidivt, of pure silver, and ISd-wt. of copper, melted together. 4. Avoirdupois Weight. 16 drams, (dr.) make 1 ounce, o^, 16 ounces, 1 pound, lb. 28 pounds, 1 quarter, qr. 4 quarters, 1 hundred, Cwt, or C, 20 hundreds, 1 Ton, T. 1 quintal of fish, is equal to 1 Cwt Reduction. Jsiote. This weight is used for provisions, groceries, hay, iron, lead, and, in general, for alT coarse and bulky articles. 192 ounces Avoii'dupois are equal to 175 ounces Troy ; and l44/ds. Avoirdupois, to 175lbs. Troy; and lib. Aveirdupois, to 7000 grains Troy, or to 27-664 solid inches of pure water, 5. Apothecaries' Weight. 20 grains, (gr,) make 1 scruple, sc, or ^ 3 scruples, 1 dram, dr, or 31 8 drams, 1 ounce, oz. or 3^ 12 ounces, 1 pound, lb. JVote. This weight is used for compounding medicines. A pound of this weight is the same as a pound Troy. 6. Cloth Measure. 2:J inches, (in») make 1 nail, na, 4 nails, 1 quarter, qr. 4 quarters, 1 yard, yd. 5 quarters, 1 ell Flemish, E. FL 5 quarters, 1 ell English, E. E. 6 quarters, 1 ell Frencb, E, Fr, 7. Long Measure. 6 points, (pf.j make . 1 line, /. 4 lines, 5 barley corns, 12 inches, 3 feet, 16§ feet, or 5 j yds. 4 rods, or 66 feet, 10 chains, or 40 rods, 8 furlongs, or 52 80 ft. 5 miles, 69§ statute miles,(nearly) 1 degree, deg, 060 degrees, I great circle of the earth, cir. J^fote. The chain is divided into 100 links. In measuring the height of liorses, 4 inches make I hand In measuririj^ depths, 6 feet make 1 failion. Long measure is used to measure distances, or any other thing in which length is considered without regard to hreadth. 8. Square Measure. 144 square inches, make 1 square foot, /f. 9 feet, 1 yard, yd, 30^ yards, or %7^\ feet, ' '" 40 rods. barley corn, &. c. 1 inch, in. 1 foot,/^ 1 yard, yd. \ rod,pole,orperch, r. or|?. 1 chain, c/i. 1 furlong, /wr. I mile,'m. 1 league, L. 4 roods, 16 rods, or iOCOO links, to chains, 640 acres, I rod, pole, or perch, n or p. 1 rood, R. 1 acre, J., 1 chain, eh. 1 acre. A- 1 mile, m 36 Reduction, JVo^e. This measure is used to ascertain the quantity of aay thing whick has length and breadth, without rega.id to tliickness, as the floor of a room, the content of a piece of land, &c. 'I'he length and breadth being multiplied together, to make the area, oi superficial content. Hence, 144 inches make 1 foot; because \2 inches in length, and 12 inches in breadUi, being multi- plied together, make l44 square inches in one square foot. 9. Solid or Cubic Measure. 728 solid inches, make 1 solid foot, ft 27 feet. 1 yard, yd. 40 feet of round timber, or 50 feet of hewn timber. \ ton, or load, T. 128 feet of wood, 1 cord, c. J^ote. 1 his measure is used to ascertain th^ quantity of any thing whiek has length, breadth and thickness, and to regulate all measures of capacity, (^ Tiv^hatever forni. A solid foot contains 1728 solid inches ; because it is 12 inches loi 1 SlabbatK day's journey, Reduction. ^reek, 4 fingers, 1 doron ; 4 dora, 1 foot, equal to 120875 English inches ; 18 fingers, 1 pugme. or smaller cubit ; 24 fingers, 1 pechus, or larger cubit ; 4 larger cubits, 1 orguia, or pace ; 100 paces, 1 stadium ; 8 stadia, 1 mile. Roman, 4 fingers, I palmus minor; 4 pa'mi, 1 foot, equal to 11-604 En- glish inches ; 24 fingers, 1 cubit ; 40 fingers, 1 gradus ; 5 feet, 1 pass\is j 625 feet, 1 stadium; lOOO passus, 1 mile. Square Measure. Greek or Egyptian aroura* 10000 square cubits ; Greek plethron, 2 arour« ; Roman juger'am, 2 English roods, 10 poles, and 25005 ieet. Cubic Measure. Hebrew, 10 avoirdupois ounces of rain water, 1 cotjift; 10 cotylse. 1 gomer, or omer ; 10 omers, 1 bath, epbst, or metretes, equal to 6® English ^vine pints, and 15 solid inches ; 3 baths, 1 rebel ; 5 baths* 1 le- theck; 10 baths, 1 cor, or homer. 6 eggs^ or betzahs, 1 log, or rebah ; 4 logs. 1 cab ; 3 cabs. 1 hin ; 2 bins, I seah ; 3 seahs, 1 bath. Greeks for liquids. 5 cochlearia, 1 concha; 2 conchse, 1 cyathns; 6 cyathi. 1 cotyle ; 2 cotylse, 1 xestes ; 6 xest^, 1 chous ; 12 choi, 1 metretes, equal to 82 English wine pints, and 19-626 Solid inches. For things dry» 3 cotylae, 1 choenix ; 48 chcenices, 1 medimnus, equal to 4 English pecks, S[nd 205-101 solid inches. Romany for liquids. 41igulse. 1 cyathus; 12 cyathi, 1 sextarius; i sex- tarii, 1 congius ; 4 congii, 1 urna ; 2 urnce, 1 amphora, equal to 57 English wine pints, and 10* 66 solid mcheis; 20 amphorse, 1 culeus. For things dry, 16 sextarii, 1 modius, equal to 1 English peck, and 7-68 Solid inches. 18. Specific Gratities, The following table, (taken chiefly from Enfield's Philosophy,) shows the weight in avoirdupois ounces of a solid Foot of each substance. Platina, (pure,) Fine s:old. Standard gold, Mercur}^, Lead, Fine silver. Standard silver. Copper, Gun metal? !Fine brass. Steel, Ii'on, Pewter, Cast iron, Load stot^e. Diamond, 23000 19640 18888 14019 11325 11091 10535 9000 8784 8350 7850 7645 7471 7425 4930 3517 White lead, Marble, Rock chrystal. Greet! glass, Flint stone. Brick, Ivory, Sulphur, Chalk, Alum, Lignum vitfe, Coal, Ebony, Mahogany, Cows' milk. Boxwood, ■oz.. 3160 2705 2658 2600 2570 2000 1825 1810 1793 1714 1327 1250 1117 1063 1034 1030 GZ, Sea water, 103d Rain water, 1000 Red wine, 993 Proof spirits, 925 Dry oak. 925 Olive oil. 913 Ice, 908 Living men, 891 Spts. ofturpentine,864 Alcohol, 850 Elm and ash, 800 Ether, 732 White pine. 569 Cork, 240 Common air. 114 Inflammable air, •12 19. Miles of different Countries* 11 Irish equal 14 English ; I Scotch equals I and a half English • I Indian equals 3 Fit^glish ; I Dutch, Spanish and Polish, equals 3 and a half English ; 1 German equals 4 English ; I Swedish, Danish, and Hungarian, equals 5 and a lialfEnglish; I Russian verst equals 3 quarters of a mile English, (j^ Sound moves 1142 feet in one second of time. Light flies from the sun to the earth, which is nearly 94 millions of miles, in about 8 minutes of time, which is nearly 195834 miles per second ; so that for any short distance it may be considered instantaneous. 40 Reduction, REDUCTION, Is performed by multiplication and division. Rule. 1. When the given number is to be reduced from a higher denomination to a lower, multiply the given number of the highest denomination by as many units as it takes of the next lower to make one of that higher ; to this product add the number, if any, which was in this lower denomination before, and set down the amount. And so on, through all the denominations, till you have brought the number into the denomination required. 2. When the given number is to be reduced from a lower denomination to a kigher, divide the given number by as many units as it takes of that denomination to make one of the next higher, and set down what remains, as well as the quotient. And so on, through all the denominations, till you have brought the given number into the denomination re= quired. Example 1. Reduce ^1234 .. 15 .. 7, to farthings. Operation. Eocplanation. Here, because 20 shil- L s, d. lings make 1 pound, I multiply the 1 234 1234 .. 15 .. 7 pounds by 20, to bring them into shillings, 20 and have S4680 shilling;*, to which 1 add the 15 shillings of the given number, and have 24695 shillings. Then, because 12 pence make 1 shilling, I multiply the 24695 shillings by 12, to bring them into 24695 s. pence, and have 296540 pence, to which 12 I add the 7 pence of the given number, and have 296347 pence. Then, because 4 farthings make 1 penny, 1 multiply the 296347 pence by 4, to bring them into farthings, and have 1185388 farthings ; 296347 d, and as there were no farthings in the 4 given number, 1 have nothing to add, and the answer is 1 185388 farthings. 1185388 q* Answer. Reduction, 41 Example 2. Reduce 1185388 farthings to pounds. Operation. — - 4)1185388 q. 12)296347 d. dei2S4.. 15, dnswer. Explanation. Here, because 4 farthings make 1 penny, I divide the given farthings by 4, to bring them into pence, and have 296347 pence. Then, because ISpence make 1 shilling, I divide the pence by 12, 20)24695 s. 7d. to bring them into shillings, and have 24695 shillings, and 7 pence remains, 7, which I set down. Then, because 20 shillings make 1 pound, I divide the shillings by 20, to bring them into pounds, and have 1234 pounds, and 15 shillings remains. So the answer is 1234Z. 15s. 7rf. Questions. 1. In the year 1819, the receipts of the British and Fo- reign Bible Society were 9S0SSL 6s. Td., how many farthings was that ? Ms. 893 1 1996. 2. The time from the creation to the flood was 14 cen- turies and 56 years ; how many calendar months was it ? Ms. 17472. S. The highest mountain in the known world is Himalaya, in India, which is estimated to be 952632 barley corns above the level of the sea ; liow many miles is that ? Ms. 5m. 62/f. 4. The next is Chimborazo, in South America, which is estimated to be 4m. 330/f. ^ how many inches is that ? / ^«s. 257400. Exercise 2^1. the pence in 7s. Sd. Ans. 87 Ss. Sd. Ms. 101 14.. 2, 170 9.. 9, 117 15.. 6, 186 11.. 11, 143 13.. 4, 160 13.. 6, 162 5.. 11, 71 10.. 5, 125 10.. 10, 130 16.. 0, 192 9.. 8, 116 12.. 6, 150 r.. 6, 90 21.. 0, 252 ^•. 10, 82 H.. 8. 17^ D2 42 Eeduction. Exercise 23. Tell the farthingj 3 in 2s. Sd. ^n5. 176 SS. 6cZ. ^ns. 168 2.. 5, 116 2.. 3, 108 4.. 4, 208 4.. 3, 204 r.. 6, 360 2.. 1, 100 6.. 2, S96 3.. 4, 160 5.. 6. 264 4.. li. 198 3.. 6, 164 1.. u. 53 2.. 6, 120 3.. 3^, 158 4.. 5, 212 2.. 4i. 114 COMPOUND ADDITION, Is the addition of several numbers of difterent denomina- tions, but of the same general nature. Rule. 1. Write down the numbers in such a manner that those of the same denomination may stand directly under each other, and the lowest denomination at the right hand, the next lowest next, and so on ; and draw a line underneath. 2. Add up the numbers of the right hand column, as in simple addition ; and find, by reduction, how many units of the next higher denomination are contained in their sum. Set down the remainder under that column, andxarry those units to the next column. 3. Proceed in the same manner through the several deno- mination-, to the highest, the sum of which, together with the several remainders, at the foot of the other columns, will be the answer sought. " Work, Begin with the right hand co- lumn, at the bottom, and gay, 15 and 11 is 26, and 10 is 36, and 9 is 45, and 8 is 53. This is 53 ounces; and as 16 ounces make 1 pound, divide 53 by 16. 1 6 in 53 is 3 times, and 6 remains. Set down 5 ounces, and carry 3 to the columrs of pounds. Second column. 3 carried to 1 3 IS 1 6, and If is 33, and 27 is 60, and 21 is 81, and 15 is 96. This is 96 pounds ; and as 2Slb. make 1 quarter, diyide 96 by 28. 28 in 96 is 3 times, and ; 2 remains. Set down \2lb. and carry 3 to the next. Third column, 3 carried to 1 is 4, Example. Cwt qr. lb. oz. S..S..15. . 8 4..2..21. . 9 5.. 1..27. .10 4..3..17. . 11 5.. 1.. 13. . 15 21.. 1.. 12. . 5 Answer. Compound Addition* 43 and 3 is 7, and l is 8, and 2 is 10, and 3 is 15. This is 13 quarters; and as 4 quarters make i Cwt. divide 13 by 4. 4 in 13 is 3 times, and 1 remains. Set down \qr. and carry 3 to the next. Fourth column, 3 carried to 5 is 8, and 4 is 12, and 3 is 15, and 4 is i9, and 3 is 2^Z. This is 22 Cwt, ; and as this is the last column, set down 22 Cwt. And the answer is 22 Cwt. \qr. I'zlh, Soz, Questions. 1. The government expenses of the United States for the year 1819, were estimated ab follows: Civil, diplomatic, and miscellaneous, D.l6l9836vn ; military and Indian, D. 8666- 252'8.5 ; naval, D.3802485*60 ; public buildings, and roads, . D.326644 ; public debt, D.lOOOOO^iO ; erecting custom hous- es, &c. D. 100000 : what is the amount ? ^ws. 24515219-76. 2. The receipts of the B. and F. Bible Society for the first eleven years, were as follows : First, 5592L J Os. 5d, ; se- cond, 8827L iOs. Sid, ; third, 6998L 19s. Td.; fourth, 10039/. 12s. O^d.; fifth, 11289Z. I5s. 3i.; sixth, 23337^ Os. ^id ; seventh, 25998Z. 35. Id.; eighth. 4353^Z. 12s. 5^d^, ; ninth, 76455/. Is.; tenths 87216/. 6s. 9d.; eleventh, 99894/. 15s. 6d. : how much in all ? Am. 399182/. 6s. 7d. 3. Tell the whole weight of the following parcels of me- dicine, to wit : First, 3/6. 5o%. 7dr. 2sc. ; second, 6oz. 5dr. isc. l6gr. ; third, 5/6. loz. 6dr. 2sc. lOgr. ; and the fourth, 9oz. 3dr. 19gr. Ana. 9/6. l\oz. 7 dr. Isc. 5gr. 4. Add the following distances, and tell the amount : first, I27yd. I ft. Sin. 26. c. ; second, \2yd. \Oin. 16. c. ; third, 2ft. II in. ; fourth, 9yd. Tin. 26. c. ; and fifrh, I2yd. 9.ft. 16. c. Ans. ]62yd. iff. Win. 5. Five vessels of wine contain as follows : first, 6 1 gal. 2qt. ; second, 30 gal. Iqt. Ipt. Sg. ; third, 48 gal. Sqt. Sg. ; fourth, 57 gal. \pt.; fifth, i hlid, 60 gal. Ipt.Sg.: what is the whole quantity ? Ans. 5 hhd. 6 gal. Ipt. Ig. COMPOUND SUBTRACTION, Is the subtraction of one number from another, when those numbers are made up of different denominations, but of the same general nature. Rule. 1. Set down the less number under the greater, in suoh a laam^r that those parts which are of the same denominati©n 44 Compound Subtraction. may stand under each other, and the lower denomination at the right hand of the higher. 2. Begin at the right hand, and subtract each part in the lower line from that above, and set down the difference. 3. But if any part in the lower line is gi-eater than that above it, borrow as many of that denomination as makes one of the next higher, and add to the upper part, and then sub- tract the lower part from the upper one thus increased, and set down the remainder. 4. Carry 1 to the next higher denomination in the lower / line, as an equivalent to what you borrowed in the upper, I and proceed as before ; and so on, till the whole is finished, f Then the sereral remainders, taken together, will be the whole difference sought. Example. Work* Begin with the farthings at L s. d, the right hand, and say, a halfpenny, or From 345.. 19.. 6| 2 far Slings from 3 tarthings, leaves Take 97.. i0..8j one farthing, or i of a penny. Set n down ^flf. Pence. 8 from 6 I cannot, (j £48 .. 8 .. 10 J As 12 pence make 1 shilling, I bofrow •Answer. 12 and add to the 6, and it makes 18; 8 from 18 leaves 10. Set down 10, and carry 1. Shillings, Having added 12 pence, or 1 shil- ling, to the upper line in the column of pence, I must now carry or add 1 shilling to the low^er line in the columa of shillings, as an equivalent. 1 to 10 is 11 ; 11 from 19 leaves 8. Set down 8. Pounds. 7 from 5 1 cannot. This being the last and highest denomination, I cannot borrow as many as makes one higlier, but must proceed as in simple subtraction. Borrow 10, and add to the 5, and it makes 15 ; 7 from 15 leaves 8. Set down 8, and carry 1. 1 to 9 i* 10 ; 10 from 4 I cannot. Borrow 10, and loTrom 14 leaves 4. Set down 4, and carry 1, 1 t6 is 1 ; I from 3 leaves 2. Set down 2. And the answer is 248^ 8s. lOid. Questions. 1. The total expenditure of the B. and F. Bible Society for the first seventeen years, was 908248L 10s. 6^^., of which 828687/. 17s. were expended in the first sixteen years ; what was the expenditure of the seventeenth ? dns. 79560Z. 13s. 6d. 2. The Baptist missionaries at §erampore had expended in translating and printing the scriptures, from 1799 to 1809, S36445-7>^, and had received for that purpose, S39574-17: what was then unexpended ? dns. S3128'45o Compound Multiplication. 45 3. A jeweller bought 7lb. Soz. 14dwt» ll^r. of silver, and made up into spoons Sib. 7 ox. ISdwt. 19gr. ; how much was left? Jtns. 3lb. Toz. IBdwt. Ugr. 4. Bought 5 Cwt, \7lb. 5oz. Gc^*". of sugar, and sold 3 Cwt. 2qr. 2llb. 3oz, IQdr, ; how much was left ? Ms. 1 Cwt. Iqr, 24lb. loz. I2dr. COMPOUND MULTIPLICATION, Is when the multiplicand consists of different denomina- tions. Rule 1. 1. Set the multiplier under the lowest denomination of the multiplicand. 2. Multiply the lowest denomination of the multiplicand by the multiplier ; find how many units of the next higher denomination are contained in that product, set down the remainder, and carry those units to the product of the next denomination. 3. Multiply the next denomination of the multiplicand by the multiplier, add to that product what was carried from the product of the denomination below, find how many units of the next higher denomination are contained in this product so increased, set down the remainder, and carry those units to the product of the next denomination. 4. Proceed in this manner to the highest denomination, and set down the whole of that product, which, together with the several remainders, will be the answer. Work. Begin v/ith the farthings, at the right hand column, and say, 7 times 1 farthing is 7 farfliings, which is 1 penny and 3 farthings. Set down 3, and carry 1 to the pence. Prod. 250 .. 10 .. 4| Pence. 7 times 9 is 63y and 1 I car- ried is 64. 64 pence is 5 shillings and 4 pence. Set down 4, and cany 5. Shillings. 7 times 15 is 105, and 5 1 carried is IK*. 110 shillings is 5 po* nds and 10 shillings. Set down 10, and carry 5. Pounds. 7 times 5 is 35, and 'J I carried is 40. Set down 0, and carry 4. 7 times 3 is 21, and 4 I carried is 25. Set down 23 1 and the answer is 250^. los. 4%d. Example 1. /. s. d. Mul tiply 35.. .15. ..QA By 7 46 Compound Divismi. Example 2. Ih. oz. dwt gr. Mult. 21.. 1.. 7.. 13 By 4 Prod.84.. 5.. 10 ..4 Example 3. M.fiir. p. yd. ft 24 .. 5 .. 20 .. 4 .. 2 5 1£2.. 1..24.. 1.. 1 Example 4. el. hr. min, see. .6.. 20.. 35.. 51 6 23.. 6 ..3.. 35, Rule 2. Reduce the multiplicand to the lowest denomination of wliich it consists, and then multiply as in simple multiplica- tion ; reducing the product, when found, to a higher deno- mination, if required. Example. Multiply 35L 15s. 9id. by 7. Work, I first reduce the 35^ 15s. 9 Id, to farthings, and find it to be 34357 farthings. I then multiply those 34357 farthings by 7, and the product is 940499 farthings. I then reduce these 240499 farthings of the product to pounds, and have 250Z. 10s. 4Jci. for the answer, as in the first example. COMPOUND DIVISION, Is when the dividend consists of different denominations. Rule 1. 1. Write down the dividend and divisor as in simple di- vision. 2. Find how many times the divisor is contained in the highest denomination of the dividend, and put that amount in the quotient, as a part of the answer of the same denomi- nation. 3. If there is any remainder after the division of the high- est denomination, reduce that remainder to the next lower denomination, and add to it the number (if any) which is al- ready in that denomination. 4. Divide again, as before, and so on, till the last denomi- nation has been divided ; and the several numbers of the quotient, taken together, will be the answer. Compound Division. 47 Example 1. Sd. by 16. Explanation. I begin with the pounds, and say, 16 in SO is once. Set down 1 in the quotient, in the place of pounds. After multiplying and subtracting, as in. simple division, 1 find 14 remains. This being 14 pounds, is to be reduced to shillings; and there being 20 shillings in a pound, I multiply 14 by 20, and the product i3 280, which is 280 shillings. I then add the 18 shillings of the dividend, and it makes 298 shillings. I then di- vide again. l6 in 29 is once, and 13 re- mains. Set down 1 in the quotient, ia the place of shillings, and bring down the 8. 16 in 138 is 8 times, and 10 re- mains. Set down 8 in the quotient, also in the place of shillings, which makes 18 shillings. The remainder, being 10 shil- lings, is to be reduced to pence; and there being 12 pence in a shilling, I multiply 10 by 12, and the product is 120 pence. I then add the 8 pence of the dividend, and it makes 1 28 pence. I then divide again. 16 in 128 is 8 times. Set down 8 in the quotient, in the place of pence. After multiplying and subtracting, I find there is no remainder, and I have divided the last denomination. So the answer is l^. I8s. Bd. Divide 30L 18s. Operation. 16)30.. 18.. 8(U. 16 14 20 280 18 298(1 8s. 16 138 128 10 12 120 8 128(8d. 128 Example 2. lb. oz. dwt.gr. J>rJ)Z3"7'.G"12,Bd. Q$, 3 .. 4 .. 9 .. 12 Example 3. lb. ox. dr. scgr. 12)13" 1- 2 "O-O 1" 1..0-2-10 Rule 2. Example 4. yd. qr. na^ 47)571 .. 2 .. 1 12.. 0..2-(- Reduce the dividend to the lowest denomination of which it consists, and then divide as in simple division ; reducing the answer, w^hen found, to a higher denomination, if re^ quired. 48 Compound Division, Example. Divide 30?. 18s. 8(Z. hy 16. Work. I first reduce the 30^. 1 8s. Sd, to pence, and find it to be 7424 pence. I then divide 7424 by 1 6, and find the quotient to be 464, which is 464 pence. I then reduce 464 pence to pounds, and have 1^. IBs. Sd. for the answer. Questions on the fouegoing. "What is Reduction ? By what other rules is it performed ? \Vhen the reduction is from a high- er denomination to a lower, how is it performed ? "When from a lower to a higher, how is it pei foraied ? What is Compound Addition ? How does it differ from Simple Ad- dition ? How are the numbers to be written down ? Which column is to be added first ? What is to be done with its sum r For how miany of one denomination must you carrj'^ 1 to the next high- er ? What is Compound Subtraction ? How does it differ from Simple Sub- traction ? How are the numbers to be written down ? What is to be done when the lower number is greater than that above it? What is Compound Multiplication ? How does it differ from Simple Mul- tiplication ? How must the numbers be placed ? W^here do you begin to multiply f What must be done with tliat pro- duct ? For how many in the product of a lower denomination do you carry 1 to the product of a higher .' What is the second rule »' What is Compound Division ? How does it differ from Simple Di- vision ? How are the dividend and divisor to bt placed f Where do you be^in to divide ? When you have divided « higher de- nomination, what do you do v/ith the remainder ? W^hat do you do next ? What is the second rule ? Exercise 23. From take. and tell the ^ns. From take» and tell the Jlns. 97, 25, twelfth. 6 87, 15, eighth. 9 101, 26, fifteenth. 5 112, 4o, third, 22 89, 26, ninth, 7 28, 7, seventh. 3 67, 12, fifth. 11 89, 25, eighth. 8 53, 13, fourth. 10 27, 3, eighth. 3 65, 11, sixth. • 9 38, 11, third. 9 74, 11, ninth. 7 44, 8, sixth. 6 29, 2, third. 9 57, 12, third. 15 36, 8, fourth. 7 48, 13, fifth. 7 103, 49, sixth. 9 59, 11, sixth. 8 96, 12, twelfth. 7 63, 9, ninth. 6 ' Ea^ercises. 49 EXERC] [SE 24. Tell what is the Jtns. Tell what is the Am* Sd of a half of 18, 3 3d of a 5th of 45, S 4th of a 3d of 24, 2 5th of a half of 50, 5 3d of a half of 36, 6 5th ofa half of 20, 2 half of a 3d of 24, 4 3dof a6thof 36, 2 3d of a 4th of 48, 4 4th of a 5th of 40, 2 4th of a half of 40, 5 4th ofa 3d of 48, 4 Sd of a 3d of 27, S Sd ofa 1 2th of 72, 2 4th of a 3d of 36, 3 half of a 12th of 96, 4 3d ofa 4th of 60, 5 5th of an 8th of 80, 2 5th of a half of 30, 3 4th ofa 4th of 48, 3 6th of a 3d of 72, 4 3d of a 3d of 36, 4 Exercise 25. Tell what is the ^ns. Tell what is the AlU, 4th of a 5 th of 40, 2 Sd of an 8th of 96, 4 3d of a 6th of 72, 4 8th ofa 3d of 72, ^ 3 oth of a half of 60, 6 4th of a 6th of 120, 5 3d of a 6th of 54, 3 5th of a 7th of 105, S 4th of a 5th of 100, 5 8th of a 9th of 360, 5 5th of a 6th of 120, 4 3d ofa 7th of 63, 3 3d of an 8th of 72, 3 4th of an 8th of 96, 3 halfofa 9th of 36, 2 8th ofa 3d of 120, 5 Sd ofa 7th of 42, 2 4th ofa 4th of 1 60, 10 5th ofa 4th of 60, 3 5 th of a 3d of 90, 6 Oth of a 7th of 84, 2 4th of a 7th of 140, 5 ExERC isE 26. Tell what is *^ns. Tell what is Ans. 2 fifths of 25, 10 9 tenths of 90, 81 3 fifths of 35, 21 7 eighths of 56, 49 4 fifths of 75, 60 3 sevenths of 42, 18 3 sevenths of 21, 9 4 fifths of 55, 44 2 ninths of 45, 10 3 eighths of 40, 15 6 sevenths of 42, 36 3 fifths of 60, 36 5 eighths of 64, 40 4 sevenths of 5^, 32 7 eighths of 96, 84 6 sevenths of 49, 42 6. sevenths of 35, 30 5 ninths of 72, 40 6 sevenths of 49, 35 4 fifths of 60, 48 5 ninths of 45, 30 7 eighths of 64, 5^ E 50 Exen lists. Exercise 27. Tell what is ^ns. Tell what is v^/W. 3 times 5 times 11, 165 6 times 3 times 2, 36 4 6 12. 288 8 3 7, 168 5 6 10, soo 9 4 5, 180 5 6 9, 270 7 5 3. 1Q5 6 7 8, 336 6 6 3, 108 6 7 4, 168 2 5 11, 110 7 5 2. 70 3 6 12. 216 6 6 2. 72 4 6 10, 240 4 9 8. 288 3 4 9, 108 5 9 7. 189 5 4 8, 160 S 6 4, 48 3 5 12, 180 Exercise £8. Tell what is ^?is. Tell wha tis Ans* 7 times 3 times s. 63 6 times 8 tim es7. 336 8 4 64 3 6 7. 126 6 S 4, 72 4 7 3, 84 7 5 2. 70 5 9 4, 180 4 5 6, 120 4 4 4. 64 5 6 7, 210 5 5 5, 125 6 7 8, 336 6 6 6. 216 7 8 9, 504 3 4 11. 132 S 8 9, 216 4 5 12, 240 4 7 8, 224 6 7 4, 168 5 8 9. 360 5 8 3, 1^0 Exercise 29. Tell what is Ans, Tell what is Ans, 5 times 25, 125 3 times 39, 117 4 26, 104 4 49, 196 £ 84, 168 5 59, 295 3 17, 51 6 61. 366 4 19, 76 7 14, 98 5 16, 80 S 16. 48 6 14, « 84 4 18. 72 7 IS, 91 5 17. 85 S IS. 120 6 13, 78 9 21, 189 7 17, 119 S 29, 58 8 18, 144 Eivercises» 51 Exercise 30. From take. ^ns. From take, Jns 20, 2 thirds of 18, 8 18. 6 sevenths of 14, 6 18, 3 fourths of 12, 9 29. 8 ninths of 27, 5 16, 2 fifths of 25, 6 17, 5 sixths of 12, 7 17, 3 sevenths of 2 8, 5 31, 6 sevenths of 21, 15 21, 4 fifths of 15, 9 24, 9 tenths of 20, 6 30, 5 sixths of 18, 15 27. 3 fifths of 25, 12 52, 2 thirds of 21, 18 18, 3 sevenths of 14, 12 28, 6 sevenths of 14, 16 23, 3 eighths of 24, 14 12, 3 fourths of 8, 6 26, 4 ninths of 27, 14 14, 5 eighths of 16, 4 EXERC 19, ISE 3 5 sevenths of 21, 4 From I take, ^is. From take. ^115. 40, 6 sevenths of 42, 4 32, 3 fifths of 40, ■' 8 50, 5 eighths of 64, 10 80, 6 sevenths of 70, 20 90, 7 eighths of 96, 6 75, 5 eighths of 96, 15 36, 6 sevenths of 35, 6 39, 8 ninths of 36, 7 40, 6 ninths of 45, 10 47, 5 sixths of 42, 12 50, 4 fifths of 60, 2 26. 3 fifths of 40, 2 36, 3 sevenths of 42, 18 31, 4 fifths of 3 5, 3 26, 9 tenths of 20, 8 42, 4 ninths of 36, '^^ 45, 3 elevenths of 55, 30 57. 5 twelfths of 96, 17 35, 6 sevenths of 28, 11 39. 3 eighths of 48, 21 J^^'iite, The author of this work is of the opinion, that the most scientific aiTanj^emcnt of the several rules, requires Fractions to precede the Kule of I'aree, and the other rules which usually follow that. But if any instructor slionld think it most for the advantag;e of particular pupilc to attend to those rules before Fractions, he can direct them to pass on accordingly, and omit those parts of them in which fractions are used. VULGAR FRACTIONS. A fraction is an expression for the part or parts of a quan- tity, which quantity is denoted by unity. A vulgar fraction is denoted by two numbers, one placed above another, with a line between them. The number placed below the line is called the denomina- tor, and that above it the numerator, and both are called terms. The denominator show^s how many equal parts the unit or quantity is divided into; and the numerator shows how iftiany of those parts are taken. 2 Vulgar Fractions, Thus, if a pound sterling is divided into £0 equal parts or shillings, and 5 of those parts, or 5 shillings, are given to A, 7 to B, and 8 to C, the share of each, expressed in the manner of a vulgar fraction, is, A's,2^oof a pound ; B'8,2^oof a pound ; and C's, 2^0 ^^ ^ pound.* The denominator represents the divisor, in division, and the numerator the remainder. A proper fraction, is one of which the numerator is less than the denominator, as J, or J. An improper fraction, is one of which the numerator is equal to, or greater than the denominator, as f , or 5 . A single fraction, is a simple expression for any number of the parts of a unit, as 5. A compound fraction, is the fraction of a fraction, as J of ^"* A mixed number, is composed of a whole number and a fraction together, as 3^. A whole number may be expressed like a fraction, by writing 1 under it for a denominator, as^, which is the same as S. A common measure of two or more numbers, is that num- ber which will divide each of them without a remainder. A common multiple of two or more numbers, is that num- ber wliich can be divided by each of them without a remain- der. Problem I. To find the greatest common measure of two or more numbers. Rule. 1. If there are two numbers only, divide the greater by the less, then divide the divisor by the remainder ; and so on, dividing always the last divisor by the last remainder, till nothing remains ; and the last divisor will be the greatest common measure sought. 2. If there are more than two numbers, find the greatest common measure of two of them, as before ; then do the same for that common measure and another of the numbers ; and so on, through all the numbers ; and the greatest common measure last found, will be the answer. * The denominator, instead of being put under the numerator, is some- times written after it, separated by a hyphen, tJius, 1^2 Q7ie half, i3-4 t/o'ce- foitrthSf 10-20 ten t-weritieths. I Vulgar Fractions, 5S "j^ote. If it happens that the common measure thus feund is 1, the numbers are said to be incommensurable, or not having any common measure. Example. What is the greatest common measure of 336, 720, and 1736? Operation. First, I take 336 and 720, and find the great- est common measure of these, as follows : ^36)720(2 Having divided 720 by 336, I find 48 re- 67^ mainder ; and having divided the last divisor 336, by this remainder 48, 1 find no remain- 48)336(7 der; consequeutly 48 is the greatest com- 336 mon measure of 336 and 720. Next, I am to find the greatest common measure of 48 and the third given number, 1736 ; and I pro- ceed in the same way. 48)1736(36 144 Here, having divided 1736 by 48, I find 8 remainder ; and having divided 48 by this, I find no remainder ; consequently 8 is the greatest common measure of 336, 720, and 1736. 8)48(6 48 Problem 2. To find the least common multiple of two or more numbers* Rule. 1. Divide by any number that will divide two or more of the given numbers ^vithout a remainder, and set down the quotients and the numbers not divided in a line below. 2. Divide the second line in the same* manner ; and so on, till there are no two numbers that can be divided without a remainder. 3. Multiply the numbers in the lower line and the several divisors continually together, and the product will be the least common multiple required. E^^VMPLE* What is the least common multiple of 3, 4, 8, and 12 ? Operation. Here, first, I perceive that 4 will divide 4)3, 4, 8, 12 three of the numbers, to wit, 4, 8, and 12, without a remainder. I thereiore divide 3)3, 1, 2, 3 them by 4, and set down their quotients, ' * 1, S, avul ?. u^ii]i:r ihp'j} rrKiipcfivolv. and 1 1 o 1 - •• 54 Seduction of Vulgar Fractions. I perceive that 3 will divide two of the numbers, to wit, S and 3, without a remainder. Accordingly, 1 divide them, and set down their quotients, and the numbers not divided, as before ; and the third line is 1, 1, 2, 1, which being mul- tiplied together, and by the divisors, 4 and 3, gives 24, as the answer. REDUCTION OF VULGAR FRACTIONS, Case 1. To reduce a fraction to its lowest terms. Rule. Divide both terms of the fraction by any number which will divide both without a remainder, and those quotients again in like manner ; and so on, till you can proceed no further, and the last quotients will be the fraction in its low- est terms. Or : Find the greatest common measure of the two terms, and divide them both by it, and the quotients will be the frac- tion in its lowest terms. Example. Reduce |~|-|- to its lowest terms. Operation, Here, I first divide both terms by 7) 2, and it gives | § , and these I divide 2)|-||= 9^^==T4 ^^n5. again by 7, and it gives f-|-, which I cannot divide again ; and consequent- ly |-|- is the answer. Case 2. To reduce fractions of d'tferent denominators to other fractions of the same value, having a common denominator. Rule. ^lultiyly rich numerator into all the denominators except s own, lor the new numerators ; and all tiie denominators gelher for a common denominator. Example. Reduce J, 3, and |, to a comn\on denominator. Operat^'on. ' Here, I take 1, the nu- 1x3x4 = 12^ the new merator of the first fraction, ? X 2 X 4 = 1 r^ I nu mera- and multiply it by 3 , and 4, o X 2 X 3 =in J tors. the denominators of the se- cond and third fractions. 2x3x4=^4 new denomiBator. and it gives 19., for the new numerator of the first frac- Tlon. Reductioji of Vulgar Fractions. 55 Then, I take 2, the numerator of .the second fraction, and multiply it by 2 and 4, the denominators of the first and third fractions, and it gives 16 for the new numerator of the second fraction. Then I take 3, the numerator of the third fraction, and multiply it by 2 and 3, the denominators of the first and se- cond fractions, and it gives 18 for the new numerator of the third fraction. Lastly, I take 2, 3, and 4, the denominators, and multiply them together, and it gives 24 for the new denominator ; and the new fractions are -^f, ^, and |^. A''otc. Wh«n the denomii^ator of one fraction is a multiple cf the denomi- nator of j\nolher, they may be reduced to the sarae denominf.tor, by multi- plying boih the terms of that fraction whose denominator is the smaller, by such number as will make its denominator equal to that of the other. Example. Reduce § and-j\ to a common denominator. Operation. Here, 12, the denominator of one fraction, is S times 4, the denominator of the other. Therefore, f may have both its terms multiplied by 3. Now, S times 3 is 9 ; and 3 times 4 is 12. Therefore -j^ is equal to 3, and its de- nominator is the same as that of j^> ^^'^ other fraction. So that-j^2 and -j^ are the fractions, having the same denomina- tor. This method will sometimes very much shorten operations in addition and subtraction of vulgar fractions. Case 5. To reduce a mixed number to its equivalent improper fraction. Rule. Multiply the whole number by the denominator of the fraction, and add the numerator to the product, and this will be the numerator, under which write the denominator, and it will be the improper fraction required. ^ . Example. Reduce 23 1 ^o ^n improper fraction. Here, I first multiply 23, the whole number, by 3, the de- nominator of the fraction, and it gives 69, to which I add 2^ the numerator of the fraction, and it makes 71, for the new numerator, under which 1 write 3, the denominator ', and -^ is the improper fraction required. 56 EeducHon of Vulgar Fr actons. . Case 4. To reduce an improper fraction to its equivalent whole or mixed number. Rule. Divide the numerator bj the denominator, and the quotient will be the whole or mixed number sought. Examples. Reduce ^3*- to its equivalent mixed number. 5)7 1 Reduce ~^^ to its equivalent whole number* — 7)56 8 Ans. Case 5. To reduce a compound fraction to an equivalent single one. Rule. Multiply all the numerators together for a numerator, and all the denominators together for a denominator ; and they will form the single fraction required, which reduce to its lowest terms. JVote. This operation may frequently be contracted. When the same number is both among the numerators and the denominators, it may be struck out of both. When a number in one set of terras will dividet without a re- mainder, any number in the other set of terms, the quotient may be substi- tuted for the dividend, and the divisor be struck out. Or. when one in each set of terras can be divided by the same number without a remainder, the quotients may be substituted in their stead. EXAMPE. Reduce 3- of |^ of ^ of-| of y, to a single fraction* Operation. I first multiply all the numerators, 2, 3, 4, 5, 6, together, for a new numerator, and it is found to be 720; 8nd then I multiply all the denominators, 3, 4, 5, 6, 7, toge- ther, for a new denominator, and it is found to be 2520. And the single fraction required is "25^%> which reduced to its lowest terms, is y, which is the answer. Contraction. This operation may be contracted, by striking out those figures which are the same in both sets of terms, and which, in the following example, are marked with an asterisk : •|of|of|of|off * * * ■* "Where all the i^i^rnf^ra^-^rs c^re stri^rk fi\:t except the ?, anJ alithe denc-'^*^ ' -''.k y ;^..\.\ :h- Reduction of Vulgar Fractions. 57 J\'otc. The reasons for this contraction are, that when any number is mul- tiplied by another, and afterwards divided by tlie same, it is brought back ?o what it was : and in a ccwnpouad tVaction, the upper set of terms are multi- pliers, and the lower aet of terms are divisors. So that, in the above ex^ ample, if you take 2. the first of the upper terms, and multiply it by 3, the second, it makes 6 ; but you afterwards have to divide it again by 3, whicii brings it back to 2, as it was. Both tlie multiplication by 3. and the divisioa by 3, may tlierefox*e be omitted, and the two 3's struck out. And so of the rest. Case 6. To reduce a fraction from one denomination to another. Rule. Consider how many of the less denomination make one of the greater : then, if the reduction is to a lower denomination^ multiply the numerator by that number ; if to a higher, mul- tiply the denominator ; and then reduce the fraction so form- ed to its lowest terms. Example !• Reduce -f of a pound to the fraction of a penny. Operntion. Here, I consider that 240 pence make a pound; and as the reduction is to a lower denomination, I multiply the numerator by it, and it is, 2 times 240, that is 480 which, is the numerator of the fraction, and 9 is the denominator : and -* 9 ^, being reduced to its lowest terms, is -^-|^, which is the answer. Example 2. » Reduce g of a penny to the fraction of a pound. P ^ Operation. Here, I consider that 840 pence make a pound j and as the reduction is to a higher denomination, 1 multiply the denominator by it, and it is 6 times 240, that is 1440, i\ which is the denominator of the fraction, and 5 is the nume- l rator -, and-^ ^^ ^, being reduced, is2"^ g, which is the answer. tl , Case 7. % To find the value of a fraction of a higher denomination, I in whole numbers of a lower. f- Rule. Consider how many of the lower denomination make one ^^ of the higher 5 multiply the numerator by these, and divide [ji by the denominator. Example. What is the value of -| of a pound ? . Operation. Here, 1 consider that 20 shillings make a I pound ; so 1 multiply 2, the numerator, by 20, and it makes 40, v/hich I divide by 3, tiie denominator, and it gives 13 -hillings and ^. Again, I consider that 12 pence make 1 53 Reduction of Vuls:ar Fractions shilling ; so I multiply 1, the numerator of this fraction, t 12, and it makes 12, which 1 divide by 3, the denominate and it gives 4 pence. And the answer is 13s. 4d, Gase 8. To reduce any given value or quantity in a lower denom nation, to the fraction of a higher. Rlle. Reduce the given quantity to the lowest name in it, for numerator, and one of the higher denomination to the san nam.e, for a denominator, which reduce to its lowest terms. Example. Reduce 2 feet, 8 inches, 11 barley corns, to the fraction a yard. Operation. ?Iere, I am first to reduce this quantity to t lowest name in it, which is the fifth of a barley corn, for numerator ; and then to reduce a yard to iiflhs of a barl corn, for a denominator ; which is done as follows : ft, in. h. c, 2..8.. i^ 1 yard, 12 3 32 inches, 3 feet, 3 12 97 bavlev corns, 36 inches, 5 " 3 486 fifths of a b. e, 108 barley corns, numerator. 5 540 fifths of h. c, denominate And the fraction is y-f^, which being reduced to its lowe terms, is j^-, which is the answer. ADDITION OF VULGAR FRACTIONS. Rule. Reduce compound fractions to single ones, mixed nut bers to improp^.T fractions, these of different denominatio to the same denomination, and all to a common denomin tor; and the sum of the nume^-ators, being written over t common denomirator, and that fraction reduced to its low^ terms, will give the answer. Addition and bubtractton, oj yutgar rractiom, 59 Example. Add -f , f of J, and 9^, together. Operation. Here, 1 first reduce the compound fraction, f- wf J, to a single one, and it makes X5"* I then reduce the 'mixed number, 9^, to an improper fraction, and it makes ^^ And the question stands, ^ + - *5 +-yV • Next, I reduce these S fractions to a common denominator, and they are iVoV+ AVo + ^"W^- '^^^J ^^^ "«^ P^^" pared for addition, and the snm of the numerators is found to be 15025, which being written over the common denomi- nator, is ^ 1 5 Q§ , and this reduced to its lowest terms, is 10 6^0' ^vhich is the answer. SUBTRACTION OF VULGAR FRACTIONS. Rule. Prepare the fractions as in addition, and subtract the nu- ^ merator of the less from that of the greater ; and the diffe- I rence placed over the common denominator, and that frac- i tipn reduced to its lowest terms, will be the answer. Example. From 14i, take |of 19. Operation. Here, the greater number, 14^, is a mixed number, and is to be reduced to an improper fraction, which being done, it is^^. The less number, §^ of 19, is a compound fraction, and is to be reduced in a single fraction, which being done, it is ^. The question then becomes this, from ^^ take ^. The next thing to be done, is to reduce these two frac- tions to a common denominator, which being done, they are ^i^^nd ^^y ; and the question becomes this, from ^i^tak^ ^1^; and »52, the numerator of the less, being taken from 171 , the numerator of the greater, leaves 19, which being pla- . ijed over the common denominator, is |-| , and this reduced to its lowest terms, is I j^, which is the ansv/en MUL'tlPLICATION OF VULGAR FRACTIONS. Rule. Reduce compound fractions to single ones, mixed numbers to improper fractions, and those of different denominations to the same denomination ; then multiply the numerators to- gether for a new numerator, and the denominators together ^^>r a new deaomiaator, and redi^ce it to its lowest terms. 60 Multiplicatmi and Division of Vulgar FradlonB. Example. Multiply 12^ by § of 7. Operation, Here, the first is a mixed number, and is to be reduced to an improper fraction, which being done, it is ^5*. The second number is a compound fraction, and is to be re- duced to a single one, which being done, it is 7. The nu- merators t 3 and 7, being multiplied together, give 44 ^ for a new numerator ; and the denominators 5 and 3, being multi- plied together, give 15 for a new denominator ; and the pro- duct is *ix"> which being reduced, is ^-f^, or 29 f, which is the answer. DIVISION OF VULGAR FRACTIONS. Rule. Prepare the fraction as in multiplication ; invert the terms «ftlie divisor, and proceed as in multiplication. Example. Divide 4 q by ^ of 4. Operation, Here, the dividend is a mixed number, and is to be reduced to an improper fraction, which being done, it is 9*g^ . The divisor is a compound fraction, and is to be re- duced to a single one, which being done, it is -^9^. Now, 9- divided by -^9^, is the same as ^9* multiplied by y^, the terms of the divisor being inverted. And -9^ multiplied by ^, is it 0» which, being reduced, is 2 2^0 , which is the answer. Questions on the foregoing. How do you find the greatest com- mon measure of two numbers f How of more than two ? How do you find the least common multiple ? How do you reduce a fraction to its lowest terms ? How do you reduce fractions to a common denominator ? How do you reduce a mix^d number to an improper fraction ? How do you reduce an improper fi'action to a whole or mixed num- ber ? How do you jreduce a compound fractitm to a single one f How may this operatiou be contract- ed? What is a fraction ? What is a vulgar fraction ? What is the denominator ? What is the numerator ? What are they both called ? W^hat does the denominator show ? What does the numerator aliow ? Wh^t does each represent ' What is a proper fraction ? What is an improper fraction ? What is a single fractian ? What is a compound fraction ? What is a mixed number ? How may a whole number be ex- pressed like a fraction ? What is meant by a common mea- sure ? Wliat by a comraon multiple ? Decimal Fractions. 61 How do you reduce a fraction from a higher denomination to a lower ? How from a lower to a higher ? How do you fiiid the value of a frac- tion ot a higher denomiuatioa in whole numbei-a in a lower ? riow do you reduce a giren value or quantity to tlie fraction of a higher denomination ? What is the rule for the addition of vulgar fractions ? For subtractioa ? multiplication ? division ? DECIMAL FRACTIONS. When the denominator of a vulgar fraction is 1, with any number of cyphers annexed, as 10, 100, 1000, &c. it is called a decimal fraction ; and instead of writing the denominator under the numerator, the numerator only is set down, with a point at the left hanti of it ; thus, ^^, is written '5, f-£-^ is writ- ten '25, and yW^ is written 'S75> But if the numerator has not as many places as there are cyphers in the denominator, cyphers must be prefixed to make up that number ; as, xVVo> must be written '075, and joWo o ^^^t be written •001£4. Cyphers at the right hand of decimals make no alteratioa in their value ; for, '5, and '50, and ^oOO, are decimals of the same value, and signify 5 tenths, or 50 hundredths, or 500 thousandths. But if cyphers are placed on the left hand of the significant figures, and after the decimal point, they de- crease the value of those figures in a tenfold proportion ; thus, -5 is ^0, but -05 is only y|o, and -005 is joVo* In numerating decimals, begin at the decimal point, and proceed towards the right hand. The first place is tenths, the second hundredths, the third thousandths, the fourth ten thousandths, the fifth hundred thousandths, the sixth mil- lionths, and so on. The numeration of decimals and of whole numbers is similar, only that whole numbers proceed from right to left, and decimals proceed from left to right. Read, in words, the following decimals : .6 •05 •1 26 •005 •a •845 •065 •01 •42 •00078 •001 257' •0708 •0101 •3.567 •0^567 •OllOl •467 •00001 •lOlOl •3506 •00011 •lion •9 •00026 •011011 •98 •00678 •lOlOlOl •9876 •002789 •oioroi •6r •0102034 F •011001 ^^ JDecimal Fractions. Write down in figures the following decimals ; Thirty-four hundredths. Four hundred and sixty-one thousandths. Five thousand and eleven ten thousandths. Seven tenths. Eight thousandths, Nine hundredths, l^wenij ei^ht ten thousandths. Sixty eight millionths. One thousandth. One hundred and one millionths. One thousand one hundred and one ten thousandths. Two hundred and thirty -four thousand three hundred and five millionths. One hundred and one thousand and one ten thousandths. One thousand one hundred and one millionths. One ten millionth. One hundred and one ten thousandths. Twenty-one ten millionths. One hundred thousandth. One millionth. A mixed number is made up of a whole number and some decimal fraction, the one being separated from the other by the decimal point, hence called the separatria;, Ihus, 123-4, is one hundred and twenty- three, and four tenths ; 12-34, is twelve, and thirty- four hundredths ; 1-234, is one, and two hundred and thirty-four thousandths. J^'oiC' In stating results, after operations liave been performedj it has been found more perspicuous to write the denominator under, in the manner of a Tad§ar fraction. ADDITION OF DECIMALS. Rule. Place the numbers so that the decimal points shall stand exactly under each other, and then proceed as in addition of whole numbers, putting the decimal point in the sum exactly under those in the numbers added. Example. Whatisthesumof4jO-f3l-47+376-004+P08+456+'7'€ +•05 ? k Subtraction and Multiplication of Decimals. 6S Operation. Here, I first set down the 450 ; and as this is a 450* whole number, I put the decimal point after it. 31-47 Next, I set down the 31*47, so that the decimal 376*004 point in it stands under the other decimal point, 1*08 and I, the unit of the whole number, stands un- 456* der 0, the unit of the first whole number ; and 4^ *76 the first decimal, occupies the first place of deci- •05 miils. In like manner, all the rest are set down, — and added according to the rule* SUBTRACTION OF DECIMALS. Rule. Place the less numlter under the greater, so tiiat the deci- mal points shall be one Under the oth^r, and proceed as in whole numbers, only putting the decimal point in the re- mainder under those of the other numbers. Example. "What is the difference between 100-17 and 84-476 ? Operation* From 1 00- 1 7 JSTote, As there is no figure above the 6, Take 84-476 from which to subtract, you must suppose a cypher. Rem. 15*694 dns. MULTIPLICATION OF DECIMALS. Rule. Proceed as in whole numbers, only poi?.t off, in the pro- duct, as many dt^cimal places as there are in both multiple cand and multiplier together. Example. Multiply -00345 by -25. Operation, Here, I multiply the significant figures 345 by •00345 25, and get 8625 for the figures of the product ; but •25 as there were five decimal places in the multipli- cand, and two in the multiplier, there must be 1725 seven in the product; so I prefix three cyphers 690 to make up the number. ^0008625 Ans. €4 Bivision and Reduction of Decimals DIVISION OF DECIMALS. KULE. Divide as in whole numbers, and point ofi as niany place* for decimals as the decimal places in the dividend exceed those in the divisor. JSf^ote. When there is a remainder after division» or when the decimal places of the dividend are not so many as those of the divihor, then annex cyphers to the dividend, and carry on the operation as far of shall be thought requisite. Example. Divide -0008625 by -00345. Operation. Here, I divide 8625, the significant '=OO345)-00O8625(-25 figures of the dividend, by 545, the 690 ^ns, significant figures of the divisor, — — and get 25 for the figures of the 1725 quotient. Tcr know whether these 1725 are decimals or not, I count the de- cimal places of the divisor, and find them five ; and then count the decimal places of the divi- dend, and find them seven, that is, two more than those of the divisor. There must, therefore, be two decimal places iu the quotient, aad I place the point before 25 accordingly. REDUCTION OF DECIMALS. Case 1. To reduce a vulgar fraction to its equivalent decimal. Rule. Divide the numerator by the denominator, annexing as Hiany cyphers as may be necessary, and the quotient will h^ the decimal required. Example. Reduce | to its equivalent decimal. Operation, Here, as I cannot divide 3 by 4, I annex a 4)3-0(-75 ^ns. cypher to the 3, and it makes 30, and say, 4 ' 28 in 30, 7 times, and 2 remains. Again, 1 an- — nex a cypher to the 2, and it makes 20, and 20 say, 4 in 20, 5 times, and nothing remains. 20 The quotient, then, is 75. And as I annexed two cyphers, or decimal places to the 3, and there were no decimals in the divisor, I point off two decimals in the quotient, and the answer is -75. Med action of Decimah. 65 Case 2. ^ To reduce a quantity consisting of different denominations, to its equivalent decimal value. Rule 1. Write the given numbers in order, from the lowest deno- mination to the highest, in a perpendicular column, and di- vide each of them by as many units as it takes of that deno- mination to make one of the next higher. Set down the quo- tient of each division, as decimal parts, on the right hand of the dividend next below it, and the last quotient will be the decimal required. Example 1. Reduce l5s. 9^^^. to the decimal of a pound. Operation. Here, I set down, in a perpendicu- 4)3' q* lar column, 3 farthings, 9 pence, and 12)9*75 d* 15 shillings. Next, as 4 farthings £0) 10-8125 s. make 1 penny, I divide the 3 by 4, •790625 £, Jins. and it makes -ISd,, which I set down at the right hand of 90^. the next di- vidend. Next, as 12 pence make 1 shilling, I divide 9*7 bd. by 12, and it makes •SlSos., which I set down at the right hand of the 15s. Then, as 20 shillings make 1 pound, I di- vide 15-8 125s. by 20, and it makes •79G625iS., which i& the answer. Example 2. Reduce Zqvs, 12lb, 6oz» 14-592 drams, to the decimal of a Civt Opernt^o7i, Here; I set down, as before, 14-592in • fi 5 '^) 14*592 dr. 6oz, 1 2lb. and 3 qrs. in a column, with ^ ^ 4)[3-6 i8] a little space between them. Next, be- C 4) 6-912 oz, cause 16 drams make an ounce, I am ^4)[l-7^^ffl to divide \4'599.dr. by 16 ; but since 5 4)12-432 lb. 4x4 is 1 6, if I divide by 4, and that I 7)r3-lo8] quotient again by 4, it will be the same 4) 3*444 qrs. as dividing by 16, and will be more •861 Civt. convenient. So, 1 divide 14*592 by 4, %Bns, and the quotient is 3-648, which I set ilown under the drams, enclosing it in brackets, for the sake of distinction ; and tiien divide that quotient, 3-048, again by 4, and it makes •9'2o,^., which I «et dcwn at the right hand of the %oz. In like manlier, I proceed throughout } and the answer is 'BGl €wt* 66 Meduct'ion of Dechnals, Rule 2. Reduce the whole quantity to the lowest denomination of which it consists, for a numerator ; and reduce one of that denomination in which you wish your answer to be, to the same, for a denominator ; and then reduce this fraction to a decimal, according to Case 1. Example. Reduce 1 5s, 6cL to the decimal of a £. Operation. 1 5s. 6d. l£. 12 20 186 d 20 s. numerator. 12 240 cL denominator. And the fraction is -Jff of a £,, which is reduced to a deci- mal as follows : 240) 18 6-0 (-775 Mswer. 1680 1800 1680 1200 1200 Case S. To reduce a decimal fraction to its value, in terms of the lower denominations. Rule. Multiply the decimals given, by as many units as it takes of the next lower denomination to make one of the denomination given, and point off as many places for decimals as there are in the multiplicand ; the rest will be whole numbers of that lower denomination. Examplf. Reduce '775 of a £. to its value. Gjjerat'on. Here, I first multiply *775 by 20, because there '775 are 20s. in a £,, and the product is 15*500, the 20 three last figures being decimals, because there were three decimals in '775, the multiplicand ; and the other two figures, to wit, i5, are so many shillings. Next, I multiply the decimals of this pro<^luct, to wit, *500, by 12, because there are rf. 6-000 i2 pence in a shilling ; and the product is 6-000, Reduction of Decimals. 67 of which the 6 is 6 pence ; and there being no more decimals to reduce further, the work is done, and the answer is 15s. 6c?. i Questions on the foregoing. Hiat is a decima' fraction i* lu what manner are decimil fractions written ? What effect have cyphers placed at the right or left hand of the signifi- cant figures of a decimal ? In numerating decimals, where do yo\i hegin ? Which way do you proceed ? AVhat is the value of the first place ? 1 he second ? third ? fourth ? fifth? sixth ? "yi/'hat difference is there in the nu- meration of whole numbers and of decimals .•' What is a mixed number, in decimal fractions ? What is the separatrix ? In what manner are results usually stated ? Why are they so written ? What is the rule for the addition of decimals p For subtraction ? mul- tiplication ? division ? How do you reduce a vulgar fraction to a decimal f How do you reduce a quantity of different denominations to its de» cimal value, Ly the first rule ? How by the second ? How do you reduce a deciraal frae=« tion to its value ? Tell what is the half of 2 thirds of 45, Sd of 3 fourths of 60, 4th of 4 fifths of 60, 5th of 5 sixths of 72, 6th of 6 sevenths of 49, 7th ofr eighths of 56, 8th of 8 ninths of 99, 9th of 9 tenths of 100, 8th of 8 ninths of 45, half of 2 thirds of S6, T*l'1 what Is 6 fifths of 55, 7 thirds of 24, 8 thirds of 12, 9 thirds of 27, 1 1 fourths of 24, 10 sevenths of 77, 9 fifths of So, 8 sevenths of 49, 9 fifths of 60, 9 sevenths of 2 8, Exercise 32. .Hns. Tell what is the ^ns. 15 7th of 7 eighths of 48, 6 15 3d of 3 fourths of 48, 12 12 6th of 6 sevenths of 35, 5 12 5th of 5 sixths of 48, 8 7 4th of 4 fifths of 55, U 7 6th of 6 sevenths of 56, 8 11 , 4th of 4 fifths of 45, 9 10 Sd of 3 fourths of 44, 11 5 5th of 5 sixths of4g. 7 12 8th of8 ninths of 54, 6 EXERC isE 33. .^ns. Tell what Is An^^ 66 10 thirds of 60, 200 56 1 1 fourths of 44, 121 32 12 thirds of 36, 144 81 1 1 fourths of 1 6, 44 66 15 thirds of 9, Ao 110 9 fifths of 55, 99 45 8 thirds of 18, 48 56 10 fourths of 48, 120 lOS 12 fifths of 45, 108 3^ 9 sixths of 42, 63 6S Exercises. Exercise 34. From lake Ans. From take A71S, 35 and 46, ££, 59 22 and 17, 33, 6 £4 and 13, 19, 18 21 and 33, 29, 25 33 and 12, £8, 17 37 and l6, 41, 12 ' 37 and 19, 41, 15 24 and 19, SO, IS 26 and 18, S3, 11 19 and 14, 23, 10 19 and 33, 45, 7 87 and 18, 56, 49 £1 and i2, 29, 4 73 and 17, 63, 27 j 69 and 14, 47, 36 61 and \l, 46, 26 i 53 and H, 39, 26 54 and 13, S7, 40 i 44 and 14, 49, 9 69 and 15, 56, 28 f! 1 Exercise 35. From take Ans. From take A71S* I £9, 1 5 and 6 and 7, 1 63, 25 and 13 and 4, 21 J SO, 12 and 11 and 3, 4 77, 1 1 and 5 and 6, 55/ 32, 5 and 6 and 13, 8 S7, 5 and 1 1 and 12, 29 li 45, 7 and 8 and 19, 11 46, 6 and 11 and 13, 161 43, 17 and 4 and 11, 11 48, 5 and 7 and 9, &7\ 19, 7 and 3 and 4, 5 73, 11 and 12 and 6, 44 ; SO, 8 and 5 and 7, SO 64, 9 and 12 and 21, -i 49, 8 and 5 and 13, 23 25, 7 and 6 and 5, 70, 10 and 11 and 12, 37 39, 11 and 13 and 5, 10 ^ £5, 3 and 7 and 1 1 , 4 £8, 7 and 4 and 6, 11 EXERC isE 36. From take Ans. From take Ans, 45 and 63, H and 12, 85 27 and£l, 1 1 and 7, 30 , 17 f 3S ' 22 and 37, 13 and 15, 31 33 and 28, 19 and £5, 45 and 11, 2£ and 13, 21 67 and 31, 47 and t3. 77 and 8, S3 antl 1 6, 36 £2 and 13, 19 and 12, 4 64 and 13, 11 and 17, 49 74 and 12, £1 and 32, 33 21 and 33, 18 and 9, 27 85 and 11, S7 and £1, 58 46 and 15, 6 and 21, 34 96 and 15, 74 and 25, 1£ S7 and 17, 19 and 6, £9 87 and 17, 26 and 31, 47 gl and 49, 14 and 27, £9 45 and 52, IS and 49, 35 / 43 and 14, 16 and 27, 14 57 and 51, 24 and 19, 45 ExERC ISE 37. Tell what is the Am. Tell what is the Anc. 4th of 5 and 8 and 3, 4 4th of 2 and 9 and 1, 3 5tb of 7 and 4 and 9, 4 7th of4and8and 2, £ 7th of 8 and 9 and 4> S 4th of 6 and 7 and 3, 4 Exercises. 69 5 1 2 o 3 5 £ 5 S 4 10 Tell what is the Sd of 5 and 3 and 7, 4th of 6 and 8 and 2, 5th of 3 and 10 and 12, Sd of 4 and 1 1 and 6, 6th of 9 and 2 and 7, 7th of 5 and 9 and 7, 3d of 1 and '8 and 9, Tell what is the 8th of a 12th of 96, 10th of a 12th of 240, 8th of an 8th of 128, 6th of a 6th of 108, 5th of a 5th of 125, 12th of a 12th of 288, 4th of a 20th of 400, 6thofan8thof 144, 5th of a l6thof 320, 3d of a I5thof450, Tell what is the half of 2 thirds of SO, 4thof 4fifthsof 100, 3d of 2 thirds of 72, half of 2 fifths of 40, 3dof3 fifths of 45, half of 3 fourths of 40, 4th of 3 fourths of 16, 4th of 3 fourths of 48, Gthof2 thirds of 36, 4th of 3 fifths of 80, Tell what is 2 thirds of 3 fourths of 48, 24 12 Tell what is the 9th of 3 and 1 1 and 4, 8th of 5 and 7 and 12, 6th of 8 and 11 and 5, 4th of 6 and 7 and 1 1, 4th of 7 and 5 and 8, 6th of 7 and 3 and 2, 6th of 3 and 1 1 and 4, Exercise 38. .dtis I Tell what is the 4th of a 12th of 240, 5th of a 10th of 500, 5th of an 18th of 360, 4th of a l6thof 320, 3d of a 10th of 330, 5th of a 15th of 300, 4th of a 20th of 800, 8th of a 10th of 400, 3d of a 9th of 270, 3d of an 8th of 240, Exercise 39. Tell what is th€ 3d of 9. thirds of 45, 4th of3 fourths of 96, 5th ofS fifths of 100, 7th of 5 sevenths of 49, 12thof 7eighthsof 96, 4th of 5 eighths of 64, 3d of 4 fifths of 75, 7thof 3fifdisof35, 4th of 3 fifths of 60, 8th of 4 sevenths of 56, Exercise 40. ^ns. Tell what is 3 fifths of 5 sixths of 54, Ans, 10 20 16 8 9 15 S 9 4 12 Ans, 2 3 4 6 5 2 Ans. 5 10 4 5 11 4 10 5 10 10 Ans. 10 18 12 5 7 10 20 s 9 4 AH9. 27 3 fourths of 4 fifths of 20, S fifths of 5 sixths of 30, 15 2 thirds of 3 fifths of 45, 18 3 fourths of 4 fifths of 45, 27 2 thirds of 3 fourths of 24, 12 2 thirds of 3 fourths of 36, 18 4 fifths of 5 sixths of 48, 32 5 sixths of 6 sevenths of 28, 20 3 fourths of 4 fifths of 35, 21 2 fifths of 5 sevenths of 42, 12 2 thirds of 3 severxths of 49, 14 3 fourths of 4 fifths of 60, 36 3 fifths of 5 sevenths of 35, 15 2 sevenths of7 ninths of 72, 16 3 sevenths of 7 eighths of 56,21 2 ninths of 9 elevenths of 99,18 5 ninths of 9 twelfths of CO, 25 5 sevenths of 7 eighths of 32^24 Exercises. Tell what is the 6th of an 8th of 12 tunes 12, 4th of a 20th of 5 times 80, 6th of a 6th of 9 times 12, 10th of a 12th of 6 times 40, 5th of a 16th of 8 times 40, 3d of a 15th of 9 times 50, 5th of an 18th of 8 times 45, 4th of a 16th of 8 times 40,. 5th of a 15th of 5 times 60, 4th of a 10th of 6 times 40, Exercise 41. AnsATfAl what is the 3 3ii of an 8th of 4 times 60, 5 3d of a 7th of 6 times 70, 3 8tfi of a 3(i of 6 tunes 12, 2 5th ora4th of 5 times 12, 4 5th of a 6tli of 10 times 12, 10 3d of a 6th of 12 tiy:ies 9, 4 51h of a 7th of 5 times 21, 5 8th of a 9th of 9 times 40, 4 1 5th of a 6th of V2 times 20, 5l3dofau8th of 6 times 20, Tell what is the 4th ofa 5th of 27 and 13, 3dof a6'hof 15aftd57, 5th ofahalfof39 and 21, 3d ofa6thof 19aMd35, 4th ofa 5th of 87 and 13, 5th of a 6th of 65 and 55, 3d of an 8th of 39 and 33, half ofa 9th of 25 and M, 3d ofa 7th of 19 and 23, 5th ofa 4th of 46 and 14, Tell what is the 3d of 13 and a 3d of 24, 3d of27 and a 4th of ^8, 5th of 35 and a 6th of 48, 4th of 24 and a 3d of 33, 8th of 40 and a 5th of 45, 3d of 36 and a 7th of 35, half of 1 8 and an 8th of 32, 5th of 35 and a 9th of 45, 7th of 49 and a 6th of 48, 8th of 56 and a 3d of 45, Exercise 42. 14 16 15 17 14 17 13 From the half of 26, 3ddf27, 3d of 33, 4th of 48, 5th of 45, 6th of 42, 7th of 28, half of 50, 3d of 66, 4th of 44, take the 8th of 3^, 9th of 27, 4th of 32, eth of 66, 9th of 36, 8th of 24, 8th of J 6, 3d of 66, 6th of 48, 5th of 40, Ans. Tel! what is the 2 6th of a 7th of 38 and 46, 3d ofan8thofl9and 77, 8th ofa 3d of 27 and 45, 4th ofa 6th of 87 and 33, 5th of a 7th of 63 and 42, 8th of a 9th of 195 and 165, 6thofan8thof97 and 47, 5thofa5thof 83and42, 7th of an 8ih of 84 and 28, 4ihofa5lhof53 and 27, Exercise 4:^. Tell what is the lIthof44 and a half of 56, 9th of 63 and a 3<1 of 96, halfof64anda3dof27, 3d of21 and a 3d of 99, 5th of 60 and an 8th of 96, 9th of 72 and a 2d of 72, „ 8th of 160 and a 3d of 120, 12 9th of 270 and a 5th of 250, 15J8th of 320 and a 7th of 140, 22 half of 150 and a 3d of 45, Exercise 44. Ans. 9 6 3 1 5 4 2 3 14 3 From 3d of 63, 4th of 32, 3d of 27, half of 48, 3u of 69, 4th of 88, 5th of 60, 6th of 54, 7th of 84, 8th of 96, take 6lh of 54, 7 th of 35, 8th of 61, 5th of 100, 4th of 72, 5th of 45, 7th of 35, 8th of 48, 9th of 81, 12th of 96. i^- Exercises, 7i EXER ciSE 45. Tell wliat multiplied Tell what multiplied is the bj the Jns is the by the Jim. Balfofie 3d of 21, 53 8th of 48, 7th of 56, 48 4th of 20, half of 12, 30 4lhof36, 6th of 42, 63 6th of 42, 3dof», 21 3d of 33, 9th of 54, 66 nu of 28, 5th of 30, 24 half of 18, 8th of 48, 54 8th of 40, 7th of 35, 25 5th of J5, half of 24, 84 9th of 45, 3d of 33, 55 8th of 32, 3d of 33, 44 10th of 80 9th of 27, 24 7th of 21, 4th of 20, 15 12th of 36 4th of S6, 27 12th of 108, 7tiJof21, 27 6th of 54, 8th of 56, 63 3d of 36, 5th of 45, 108 7th of 42, 5th of 35, 42 9t|iof63, ISE /l6. 6th of 42, 49 PjXERc Tell what divided TcU w!>.at divided is the by the Ans. is the by the J71S, half of 24, 8th of 16, 6 8th of 160, 9th of 45, 4 3d of 36, 4th of 24, 2 5tbof200, 6tii of 30, 8 5th of 60, 9th of 27, 4 - 4th of 120, 7th of 35, 6 4th of 72, 8th of 24, 6 9th of 108, 7th ol 28, 3 6th of 48, 9th of 18, 4 3d of 240, 12th of 96, 10 7th of 63, 7th of 21, 3 6th of 96, 11th of 22, 8 7th of 56, 12th of 24, 4 7th of 84, 7th of 21, 4 8th of 96, 9th of 36, 3 9th of 81, 6th of 18, 3 9th of 81, 3d of 9, 6th of 72, 9th of 27, 4 6th of 90, 8th of 24, 5 8th ©f 112, llth of 22, 7 EXERC ISE 47. Tell the least coramon multip eof Tell the least common mulupl eof 2, 3, and 4, Ans . 12 3, 6, and 8, Ans 24 8, 3, and 2, 24 2, 5, and 6, 30 2, 4, and 6, 12 4, 6, and 9, 36 6, 4, and 10, 60 8, 10, and 4, 40 12, 9, and 6, 36 S, 6, and 9, 18 2, 3, and 9, 18 4, 8, and 6, 24 6, 10, and 20, 60 2, 3, and 6, 12 3, 9, and 15, 45 3, 4, and 8, 24 2, 5, and 10, 20 4, 6, and 3, 12 3, 5, and 6, SO 6. 7. and 3, 42 PROPORTION, Is the relation which one qiiantitj has to another. Arithmetical Proportion will be treated of hereafter. Geometrical Proportion is that relation of two quantities •of the same kind, which arises from considering what part the one is of the other, or liow often one is contained in the #ther. 72 Froportion. Four quantities are said to be proportional, when the first is the same part or multiple of the second, that the third is of the fourth. Thus, 2, 4, 3, 6, are proportional, because the first is one half the second, and the third is one half the fourth. To denote this proportion, they are written thus : — 2 : 4 : : 3 : 6 ; that is, as 2 is to 4, so is 3 to 6. Direct proportion is when one quantity increases in the same proportion as another increases. Inverse proportion is when one quantity increases in the same proportion as another diminishes. If four quantities be in geometrical proportion, the product of the two means will be equal to the product of the two ex- tiem.es. Hence, If the product of the two means be divided by either ex- treme, the quotient will be the other extreme ; which is the foundation of the following rule. SIMPLE PROPORTION, OR THE RULE OF THREE, Is a composition of multiplication and division, which teaches how to find the fourth term of a proportion from three that are given. 1 . Observe that two of the given numbers are of the same name, but one greater than the other. 2. Observe that tlie other given number is of the same name with the number sought. Rule. 1. Set down for your second or middle term, that number which is of the same name with the answer, or number sought. 2. To know how to place the other two numbers, consider whether the answer should be greater or Less than the middle term. If greater, put the greater number at the right hand, , for the third term ; if less, the less ; and put the other num- ber on the left, for the first term. 3. Multiply the second and third terms together, and di- vide their product by the first ; the quotient will be the an- swer. JVote. If the first and third terms consist of different denominations, re- duce them both to the same ; and if the middle term is a compound number, reduce it to the lowest denomination mentioned ; then the answer will be of the same denomination. If, after division, Uiere is any remainder, and it can be reduced to a lower denomination, reduce it, and divide again, and so on. And if the answer, when found, can be reduced to a higher denomina- tion, reduce it accordingly. Example 1. If 7 Cwt 1 qvn of sugar cost £'ZQ •. 10 .. 4, what will 43 Cwt* 2qr, cost? Froportion. 73 Here, I first consider of what name the answer will be, and conclude it will be pounds, shillings and pence. So I set down £^6 .. 10 .. 4, for the second or middle term. Then, to determine where to place the other two terms, I consider whether the answer will be more or less than the middle term. I conclude it must be more, because 43 Cwt. will cost more than 7 Cwt So I put 43 CwU ^2qr, on the right hand for my third term, and 7 CwU \qr. on the left, for my first, thus : First. Second. Third. Cwt, qr. £ s. d, Cwt qr. 7.. 1 : 26.. 10.. 4 : : 43 .. 2 The next thing is to multiply the second and third terms. But, as they consist of different denominations, they must first be reduced to the lowest mentioned, and the first term must also be reduced to the same denomination with the third, This is done as follows : CwU qr. =g s. d. Cwt qr, 7.. I g6., 10.. 4 43.. 2 4 20 4 S9 qrs. 530 s. 174 qrs, 12 6364 d. And the question stands, qrs. d. qrs^ 29 : 6364 : : 174 : I have now to multiply the second and third terms toge- ther, and divide by the first, and the quotient will be the an- swer in pence. 29 : 6364 :: 174 174 25456 44548 6364 The quotient is 38184 pence, which being re- duced to pounds, gives ^159 ..2, for the an- swer. 1107336(38184] 87 237 24$ 232 232 5^ 116 29 116 24 G 74 Froportion* Example 2. If 48 men can perform a piece of work in 24 days, how many men can do it in 192 days ? days. men. days. 192 : 48 : : 24 : .24 192 96 192); 152(6 mc», Ajis. 1152 5. If ^100 in 12 months gain a certain interest, what sum will gain the same in 8 months ? ^ns. ^150. 4. If S ounces of silver cost IT'S., what cost 48 oz. ? Ans.£\^..12. Questions on the foregoing. iWbat is Proportion ? What is Geometrical Proportion ? When are four quantities propor- tional ? How is that proportion denoted ? What is direct proportion ? What is inverse proportion ? What is the foundation of the Rule of Three? What is th« Rule of Three ? How many numbers are given ? What is the first observation respect- ing the given numbers ? What is the second ? Which number do you set down fir? ^ and for what ? How do you find out how to set down ihe others I When do you put the greater at the right handr and when the les« ? When you have your question stated, which numbers do you multiply together ? By which do you divide ; When your first and third terms consist of different denominations, what is to be done ? What, when your middle term con- sists of different denominations ? Of what denomination will the quo- tient be ? If there is any remainder after divi- sion, what is to be done ? If the quotient is in a lower deno- mination, what is to be done ? COMPOUND PROPORTION, OR THE DOUBLE RULE OF THREE, Is that which embraces such questions as require two or more statings in the Single Rule of Three. The number of terms given will always be odd, that is, there will be five, or seven, or nine, &c. If more than five terms be given, work by several statings in the Single Rule of Three. If five numbers be given to find a sixth, three of them will be a supposition, and tw© a demand. Thea work by the fal- lowing Froportioru T5 Rule. !. Write down the three terms of supposition, so that the principal cause of gain or loss shall possess the first place ; the meansy that is, the distance, time, &c. the second ; and the effect produced, that is, the gain, loss, &c. the third ; then place the other two numbers under those of the same name. 2. If the blank place falls under the third term, multiply the two first together for a divisor, and the rest for a dividend. S. But if the blank falls under any other term, multiply the third and fourth terms together for a divisor, and the rest for a dividend. And the quotient will be the answer, or term sought. JYote. If any of the terms consist of different denominations,^ they must lie reduced, as in the single rule of three. Example 1. If 3 men, in 4 days, eat 6lb» of bread, how much will suf- fice 6 men for 12 days ? Here, to know how to state the question, I consider what are the terms of supposition, and find them to be, that 3 men, in 4 days, eat 6lb. of bread. These, then, are to be the three first terms. Next, to kaow in what order to place them, I consider which is the operating cause, and find it to be 3 men. This, then, is the first term. I then consider which is the means, or time, and find it to be 4 days ; therefore this is the second term. The 6lb* of bread is the effect, raid is therefore the third term. And they stand thus : men. days, lb. 3 4 6 I then wish to know where to place the other two ; and the rule is, under those of the same name. And the whole stands thus : men. days. lb. 3 4 6 6 12 Now, to know which terms to multiply for a divisor, and which for a dividend, I consider where the blank falls ; and it being under the third term, I multiply the first and second for a divisor, and the rest for a dividend. T€ Compound Proportion, I)ivisor, 3 4 Dividend* 6 6 12 36 12 12)432(36 Quotient 36 72 And the answer is 56lh 72 Example 2. If 7 men ean reap 84 acres of wheat in 12 clays, how manj men can reap 100 acres in 5 days B Stated thus: men, days, acres* 7 12 84 5 100 Here, because the blank falls not under the third term, I multiply the third and fourth terms together for a divisor,, and the rest for a dividend. Divisor, Dividend. 84 7 5 12 420 84 100 4£0)8400(20 Quotient. 840 And the answer is 20 men* Contractions. Questions in this rule may be contracted several ways. 1. When the same number is found^in both divisor and divi- dend, strike it out of both. 2. If a number in one will divide a number in the other without a remainder, strike out both, and put the quotient of the one divided in its place. 3. If both divisor and dividend can be divided by the same number without* a remainder, their quotients may be substituted.— The reason for which contractions is, that when a nuriaber is multiplied by any figure, and the product divided by the same figure, the number remains just as it wa3 at first. Compound Proportion* 77 Take the preceding examples. men, days, lb. *3 *4 6 6 6 m 6 36 Ans, Here, 3x4 = 12. This being the divisor, and fliere being \1 also in the dividend, the 3, 4, and 12, marked with an as- terisk, may be struck out; and there remains only 6 and 6, parts of the dividend, to be multiplied together, which makes 36 ; and there being none of the divisor left, 36 is the answer. In the second example, men, days, acres, '^r n2 ^84 5)100 5 100 ^^ ' 20 Ans. 84 and 5 form the divisor, and 7, 12 and 1 00 the dividend , rxl2 is 84, which being struck out of the dividend, and 84 being struck out of the divisor, there remains 100 to be di- vided by 5, which gives 20 for the answer. This method of contraction may be used also in the single rule of three, and in division, and in any other operations of a similar nature. Questions on the foregoing, What is the Double Rule of Three ? If more than five nunibers Skve gives, how must you proceed ? If five arfe given, what will they be ? How ilo you slate questions in this rule ? How do yoa proceed after your ques- tion is stated } What if any of the numbers consi t of difTcrcnt denomiriutions ? Ho ;* may questionsr in this rule be f onlracted ? On what reason are such contractioii. founded I Can such contractions be made in any other rule ? THE SINGLE RULE OF THREE; IIS VULGAR FRACTIONS. Rule. 1. State the question as in the Rule of Three in vvhok. numbers. 2. Prepare the fractions as in multiplication of vulgar fractions. 3. Invert the terms of the divisor, and proceed as in mul- tiplication. A'^ote, Th^ operation ^lay frequently be contracted, as in Case 5 of Rc' * i^ction of vSear Fractions. 78 Single Rule 'of Tliree in Vulgar Fractions. Example. If f of a yard cost /^ of a £., what cost y\ ol a yard ? Operation. Here, in order to state the question, I consider whether the answer needs to Ije more or less ; and finding it must be less, I say, yd, £. yd, f * T T • • t\ • i^^^ Answer. Next, I consider whether the fractions need any reduction, to prepare them for the further operations, and find that they do not. Then, I invert the terms of that which will be the divisor, |, and it makes f ; and multiply them all together, as follows: which being reduced^ is |^ of a £. or 3s. Ad., which is the an- swer. Contraction. This operation may be contracted as follows : * * * 3 '^ 1 5 ^ 1 4 b • * # *r 3 2 Here, there being a 3 in the line of numerators, and ano- ther in the line of denominators, they are both struck out. Next, the denominator 15 is divided by the numerator 5, and the quotient 3 substituted for the 15, and both 1 5 and 5 are struck out^ Then, the denominator 14 is divided by the nu- merator 7, and the quotient 2 is substituted for the 14, and the 14 and 7 are struck out. Now, all the numerators being struck out, I put down 1 for a numerator ; and there being nothing left of the denominators but 3 and 2, I multiply them together, and it makes 6, and the ansv/er is }, as before. THE DOUBLE RULE OF THREE IN VULGAR FRACTIONS. Rule. 1. State the question as in the double rule of three ia whole numbers. 9. Prepare the fractions as in multiplication of vulgar fractions. 3. Invert the terms of those w^hich will form the divisor, and proceed as in multiplication. Double Ruh of Three in Vulgar Fractions. 75 Example. When 12 persons use llib. of tea per naontli, how much , should a famii J of 8 persons providefor ^ a year ? Operation, Here, I first state the question according to the rule in whole numbers : 2:iersons» yr. lb. 12 tV H 8 I Next, T prepare the fractions as in multiplication, and it becomes : persons.^ yr. lb. 1 12 8 . AX The blank falling und^r the third term, I conclude the first and second will form the divisor ; and these being inverted, all are multiplied together, as follo^vs : 1 2 '^ 1 '^ 8 '^ 1 '^ 2 1 9 2'* The product, being reduced, is f, or 4^, which is the answer. This may be contracted as follows : •THE SINGLE RULE OF THREE, AND THE DOUBLE RULE OF THREE, IN DECIMALS. Questions in these rules are stated and performed as m whole numbers, regard being had to the placing of the deci- mal point, according to the rules for the multiplication and division of decimals* Example 1. If 3*5?&, of tea cost U^. what cost 5-25i^.? Operation, lb. £. lb. 3-5 : 1-4 : : 5-25 : 1-4 5'5)7-35G(2-10^, Ms. 70 35 S9 The Rule of Three in Decimals. Example 2. If 4*3 bushels of wheat serve 1-6 men 2-5 months. how inj bushels will serve 5 men 3-2 months ? Operation. men. months. 5ms/?. dividend. 1-6 2-5 4-2 3-2 5 S'2 4-2 divisor. — > 1-6 128 150 13-44 m 5 4-00)67-20^^l6-8 400 *^ns. 27 m- 2400 Mow 3200 3200 Questions on the forxgoing. How do you perform questions in the Double Rule of Three in Vulgar" do 3'OU state questions in the Single Rule of Three in V^ulgar Fractions ? How do you prepare the fractions ? How do } ou proceed then ? How can the operation be contracted? Exercise 48, Flections f How do you perform questions in tlie Single and Double Rule of Three Decimals ? [f ^Ih. cost 5s. what cost 9/5. ? Ms. 15s. 5 . . 4 . . . 15 . . . . 12 7 . . 9 . . . 35 . . . . 45 6 . . 8 . . . 24 . . . . 32 7 . . 8 . . , 21 . . . . 24 5 . . 9 . . . 20 . . . . 36 9 . . 6 . , .- 45 . . . . 30 5 . . 8 , , . \5 . . . , 2i 6 . . 4 . . . 24 . . . , 16 7 . . 6 . . . 21 . . . . 18 8 . . 10 . , . 40 . . . . 50 9 . . 4 . . . 27 . . . . 12 11 . . 12 . . . 55 . . . . 60 7 . . 3 . , • 42 ... . 18 4 . . 9 . . . 16 . . . . 56 K (\ 6d.isi 85 .. IS amount at 4s. ic/, is j\ 10 .. 14 amount at 6^. 17.. 10 amount at ^d. ^1381.. 3.. 10 am't at ^3.. 4.. 6§,Jns. * Here, 1 first multiply 428, the quantity, by £3, the highest ^denomination of the price, and it makes ^i^84, which is the amount of the whole quantity at £3. Then, because 4s. is I of a pound, I take J of ^6428, which is 1,85 .. 12, for the amount at 4s. Then, because 66^. is i of 4s. 1 take | of jL85 .. 12, which is LiO.. 14, for the amount at ed. Lastly, because id. is j\ oi6d. I take Jj of LIO ..^14, which is 1 7s. lOd. for the amount at id. All which, added together, gives X1381 .. 3 .. 10, for the answer. Case 3. When both the price of one, and the quantity, are of diffe- rent denominations. Rule. Multiply the price by the highest denomination of the quantity, and take proportional parts for the rest. Or, work by the Rule of Three, which will usually be the better way. u Fractice^ Example. iVhat C04 17 Cwt. Sqr. mb. at L2 ,. 2.. 6 per Cwtt Operatiorio £. s. d. 2gr. i« \ 2.. 2., 6x5 12+5=17 12 25 .. 10 .. amount of 12 Cwt. 10 .. 12 .. 6 amount of 5 Cwt. l^r. isf 1 .. 1 .. 3 amount of 2 qrs. 16(6. is 1 2(6. is \ 1(6. is X 10 .. 7^ amount of 1 qr. 6 .• 0§ amount of ] 6 Ih, 9 amount of 2 lb. 4J amount of 1 lb. jess .. 1 .. 6| amount of 17 Cii;f. S^rs. 19(6. The reasoA why I multiply by 1^^ and by 5, is, that 13 and 5 is 17, and multiplying ^82 .. 2 .. 6 by 12 and bj 5, and add- ing their products, is the same as multiplying it by 1 7. In taking the parts, when I say 1 6(6. is 4, it is not 4 of Iqr. but ^ of 1 Cwt, and the upper line is to be divided by 7, and not the line opposite to which it stands, as in the rest. TARE ANDTRETT, Are allowances made to the buyer on some particular commodities. Tare is the weight of the barrel, box, bag, &c. Trett is an allowance of 4(6. per 104(6. for waste and dust. Gross is the weight of the goods, together with that in which they are contained. JS*eai is the weight of the goods, after all allowances are deducted. Case 1. When the tare is so much in the whole gross weigW. 'Rule. Subtract the tare from the gross, and the remainder is the neat. Example. What is the neat weight of 56 Cwt. Iqr. 19(6, of tobacco^^ the tare being 15 Cwt. ^r. IS (6. ? Cwt. qr. lb. 56 .. 1 .. 1 9 gross. 15.. 2.. 13 tare. 40..3r» 6 neati Ans* Tare and Trett ts Case 2. When the tare is so much per barrel, box, &c. Rule. Multiply the number of barrels, boxes, &c. by the tare, and the product will be the whole tare^ which subtract from the gross, the remainder is the neat. Example. What is the neat weight of i 6 hhds. of tobacco, the gross being 86 Cwt. 2gr. 14^6. and the tare being lOOlb, per hhd. ? Tare. Cwt qr, IL 16 86 .. ^.. 14 gross. 100 14.. I .. 4 tare. 4) ^ ^ 2S) \e0()lbs.{57qrs. 4lb. 72 .. 1 .. 10 neat, Arts. 140 200 196 14.. 1.. 4 Cwt* qr. lb. Case S. When the tare is so much per Cwt. Rule. Deduct from the 2;ross such proportional part of it, as the tare is of a ( wt. and the remainder will be the rseat Or, multiply the pounds gr^JbS by the tare per Cwt. and divide the product by 112, the quotient will be the tare, which deduct as before. Example. In 12 butts of currants, each 7 Cwt. Iqr. lOlb., tare per Cwt. iQlb., how much ;ieat ? Cwt. qr. lb. 7.. I.. 10 12 I6lb. is 88 .. .. 8 gross* \ 2 .. 2 .. 9 tare. 75.. I ..^7 neat, Ans. Or, according to the second method, as follows : H S6 Tare and Trett. Cwt qr. lb. 7.. 1.. IG 12 88.. 0.. 8 §864 J&. gross. 4 1409/6. tare. 352 S8)8455 lb. neat. 28 84 (301 qrs. 2824 55 4)301 704 28 — — 75 .. 1 9864 i&. gross. 27 76. 1 6 lb. tare per Ct«^. . JItns. 75 Cwt. Iqr.'Sink 59184 9864 112)157824(1409 i5. tareo 112 458 448 1024 1008 16 Case 4o When trett is allowed with tare. Rule. Deduct the tare as before, and the remainder is called suttle ; which, divided by ^^^6, (which is i of 1 04,) will give ^he trett, and that being subtracted from the sutUe, the re- ijaainder will be the neat. Example. In BCwt. Sqr. QOlb. gross, tare 38i6. trett 4 li. per 104 Ib^ kow man J lbs. neat ? Tare and Treit* $T €wtqr. lb. 8 .. 3 .. 20 4 55 28 ^2 lb. suttle, 37 lb. trett. 300 70 925 lb. »eat, w ICGO IL gross. 38 IL tare. 6)962 lb. suttle. 78 (37 lb. 182 182 trett fc^5. Another allowance, called CloJ] is sometimes made, afQlb. for t'Xery 3 C-wt., which may be found by Case 3. Questions on the foregoing. What is practice ? What is the first case ? What is the rule ? The second case ? the rule ? The third case ? the rule ? What i;i tare ? What is tiett ? What is gross weight i What is neat weight ? What is suttle ? What is the first case ? the rule ? The second ? the rule ? The third? the rule ? The fourth ? the rule ? Why do you divide by 26 ? INTEREST, Ts an allowance made fer the use of money. The Frincipal is the sum at mterest, or the sum for the use of which the allowance is made. The Rate per cent is the interest of j8iOO, or glOO, or the allowance made for the use of it, for one year. The Amount is the sum of the piincipal and interest. SIMPLE INTEREST, Is that v/hlch arises from the principal only. Case 1. To find the interest, when the principal, time, and rate per cent, are given. Rule 1. Say : As 100 is to the rate per cent, So is the principal to the interest for one year. Tlien, S8 Simple Interest. Multiply the principal bj the rate per cent, and divide the product bj 100. The quotient will be the interest for one year. JK'otel. To divide by 100, point off the two right hand figures, and the other figfures will be the quotient. JVote 2 To multiple by a mixed number, as 6 J, or 5§, multiply by the Tfhole number, and take proportional parts for the fraction. Example 1. What is the interest of £S7 .. 14 .. 5 for one year, at 6 per eent ? Here, I multiply the principal, LS7 .. 14 .. 5, by the rate per cent, 6, and the product is JL526 .. 6 .. 6. The pounds being divided by icO, give L5 interest, and L-6 remainder. I set down L . in the answer, and multiply the 26 by 9.0, to bring it into shillings, adding in the 6 shillings, and it makes 526 shillings. This, again^ being divided by ? 00, gives 5s. interest, and 26s. remainder. I set down 5s, in the answer, and multiply the 26 by 12, to bring it into pence, adding in the 6 pence, and it makes 318 pence. This again being divi- ded by 100, gives 3 J. interest, and \Sd, re- mainder. 1 set down Sd, in the answer, and 5 multiply the 18 by 4, to bring it ml o far- things, and it' makes 72 farthings, which being less then 100, cannot be divided by it, and makes only y\2_ of a farthing, and is of no importance in this place. So that the answer is L5 .^5 .. 3. Becond method. Retluce the principal to its lowest deno- mination ; then multiply by the rate per cent, and divide by 100 ; and then reduce the answer to a higher denomination. Take the same example : L. s. d. ST.. 14.. 5 20 1754, 12 21053 principal in pence 6 rate per cent* 12 63- 18. 87.. 14 d. .. 5 6 5'£6.. 6 20 ..6 5-26 12 3*18 4 •72 Jins. L5 .. 5.. Simple Inierest S^ The interest, therefore, is 12636^., which being reduced to pounds, is L5 .. 5 .. 3, as before. Example 2, What is the interest of S37-45 for one year, at 5| per cent ? 837*45 This product being divided by 5 J 100, or the decimal point being re- moved two figures to the left, is £•- 187-25 product by 5. 597^,. or 2 dollars, 5 cents, 9 mills, 18«72§ product by J. and 7§ tenths of a mill, which is the answer, 205-97^ product by 5^. EXAMPLE 3. What is the interest ot X£0 .. 16 .. 6 for one year, at 6| per cent ? L. s. d.' 20.. 16..6 6| 124.. 19.. product by 6. 10.. 8.. 3 product by §. 5 .. 4 .. 1 J product by i* 1-40 .. 11 .. 4j product by 6^. 20 — Ms. L\ .. 8.. li. 8-11 12 Having found the interest for one — year, find it for any other time, bymul- 1*36 tiplication, the rule ofthree, orpractice, 4 as the case may require. 1-46 Rule 2. i" To find the interest for any time at 6 per cent, multiply the principal by half the time in months, and divide by 1 00, Example 1. What is the interest of §9 1 6'72 for one year and 4 months^ at 6 per cent ? Operation, g9 16*72 prmcipaL 8 half the number of months. So the answer is 73 dt)llars, SS 7Z'^376 cents, 7 mills, and j\ of a inilL H2 90 Simple Interest. Example 2. ^ What is the interest of 1,342 .. 1 6 .• 6 for one year and months, at 6 per cent ? £. s. d. S42.. 16.. 6 7| half the number of months. 2399 .. 15 .. 6 product bj 7. 171 .. 8.. 3 product by j. 25-7 1 .. 3 .. 9 product by 74. 20 14-23 12 — Ans. ^25.. 14.. 2 J. 2-85 4 S-4(> Rule 3. Multiply the principal by the number of days, and that product by the rate per cent, and divide by 36500 y the quo- tient will be the interest. Example. What is the interest of 752 dolls, for 101 days, at 7 per cent ? Ojjeration. 752x101=75952, and 75952x7=531664 j and 53 .664-r-S6500=S 14-366+, which is the interest. JSTote 1. The reason for dividing by 36500, is, that there are 365 days in a year, and dividing by 36500 is the same as dividing by 365 and by 100. J\ote 2. Instead of dividing by 36500, you may multiply by the decimal '0000274, T/hieh is f nearly J the quotient of 1 divided by 36500 ; and the result will be the same. Or, in Fe^leral moneys multiply by 274, and point off the seven right hand figures for decimal parts of a dollar. Take the same question : 752x101=75952, and 75952x7=531664, and 53 1664 x •0000274=1 i-5675936, or S 14-5675936, which is the interest- Case 2. The method pursued by the banks. As the interest of 1 00 dollars at 6 per cent is just 1 dol- lar, or 100 cents, for two months, the interest of any number of dollars for that time, is the same number of cents. Hence the following Simple Interest 9 1 Rule, t down the principal in dollars and decimal parts of a dollar, and remove the decimal point two figures to the left, and you have the interest of that sum for two months at 6 per cent. For any greater or less time, work by multiplicatioa^ or practice. IR JSTote 1 If you wish to find the interest at 7 per cent, add to the interest at 5 per cent, I sixth of itself JVote 2 The banks reckon 30 days a month, and 360 days a year ; so that by this method tliey gain tlie ii^terest of 5 days in a year. Example i.. What is the interest of Dr856'43 for 3 months, at 7 per eent? Operation. 6)78'5643 interest for 2 months at 6 per cent* 13-09405 i to be added. 2)9 1*65835 interest for 2 months at 7 per cent. 45*82917 interest for l month at 7 per cent. 8137*48752 interest for 3 months at 7 per cent, t^ns. Example 2. What does a bank gain in a year, by pursuing the above method, on 500000 dollars, at 7 per centB Operation. 6)5000-00 int. for 60 days at 6 p. c> 833-33J I to be added. 5 days is J3 of 60 days...t2)5833-33j int. for 60 days at 7 p. c» •11-f- Ans. Case 3. To find the sum due on an obligation, when there are se- veral payments* Rule 1. 1. Find the amount of the principal for the whole time.^ 2. Find the amount of each payment, computing the inte- rest on it from the time it was made to the time of settle- ment. 3. Subtract the amount of all the payments from the amount of the whole sum, and the remainder will be the sum due. 9x! Simple Interest*' Example. A gave a note to B, dated Jan. 1, 1780, for 8^000, payable on demand, with interest at 6 per cent, on which are endorsed the following payments : i. April 1, 1780, S2^ 2. August 1, 1780, 4, What is due on this note, June 1, 1784? 3. December 1, 1780, 6. 4. February 1, 1731, 60. 5. July 1, 1781, 40. Operation. 1. The whole time from Jan. 1, 1780, to June 1, 1784, is 4 years and 5 month-. The interest of S 000 for that time, is g26o ; consequently the amount is %\2C5, 2. The first payment is 8:^4. It was made April 1, ITSO, From which to the time of settlement, is 4 years and 2 months. The interest of 824 for that time, is g6, which added to 24, the amount of the first payment is gSj. 3. The second payment is 84, made Aug, 1, 1780. Its time is 3 years and lo months 5 and its interest 92 cents, which makes its amount 84-92. 4. The third payment is 86* made Dec. 1, 1780. Its time is 3 years aiad 6 months, and its interest Sl*26, which makes its amount 87-26. 6. The fourth payment is 860, made Feb. 1, 1781. Its time is 3 year^j and 4 months, and its interest 81^/ which makes its amount 872. 6. The-fifth payment is 840, made July 1, 1781. Its time is Z years and II months, and its interest 87, which makes its amount 8 '7. 7. The whole amount of the payments, then, is 8 161-18; which being subtracted from the amount of the whole sum, gl265, gives the sum due Si 103-82, which is the answer. Rule 2. The method established by the courts of law in Massachu- Beit:-, is the following : Cast the interest on the whole sum up to the time of the first payment; and if the payment exceeds the interest then due, deduct that excess from (he principal, and con-ider the remainder as thfe new principal, and cast the interest on that up to the time of anotlier payment, and so on. But if the first payment does not exceed the interest due when it was made, cast the interest on the whole sum from the date of the W)ligatiQn up to the second payment, and see if the first and I Simple InterrsU 9S second payments, taken together, exceed the interest due at the time of the second payment. If they do, deduct their excess from the principal, as before ; but if they do not, cast the interest again upon the whole sum up to the time of the third payment, or the fourth, or till such time as the payments taken together do exceed the interest then due, and then de- ^ duct as before. I Take the preceding example. f Operation. 1. From Jan. 1, 1780, the date of the note, to April 1, 1 780, the time of the first payment, is 3 months. The interest of glOOO for ^^ months at 6 per cent, is Dl5. The payment made at that time, 1)24, exceeds the interest then due by D9, J which is therefore to be subtracted from the principal, which being done, leaves D991, to form the new principal. 2. From April 1, 1780, the time of the first payment, to Aug. 1, 1780, the time of the second, is 4 months. The in- terest of D991 for 4 months is D19-82. The payment then made is D4, which is not so much as the interest then due. So there is nothing to be deducted from the principal at that time. 3. The third payment is made Dec. 1, 1780, But as the second payment did not exceed the interest due at the time it was made, I go back and compute from April 1, (780, the . time of the first payment, to Dec. I, 1780, which is 8 months. The interest of D99] for 8 months is D39-64. The third payment is D6 ; and this added io Dl, the second payment, \ is DlO, which is less than the interest due at the time of the ; third payment. So, there is no deduction to be made at that time. 4. The fourth payment is made Feb. 1, 1781. But as the second and third payments did not exceed the interest due at the time when they were made, 1 must still go back, and compute from the time of the first payment, April 1, 1780. ; From that time to Feb. 1, 178i, is 10 months. The interest |ofD991 for 10 months is D49*55. The fourth payment is V D60, which, with the two preceding, D4 and D6, is D70. ' This exceeds the interest now due, by D20-45 ; which being deducted from D991, the principal, leaves D970'55, for the w principal. 5. The fifth payment is made July 1, 1781, which is 5 months from the time of the fourth. The interest of D970'55 for 5 months, is D24*£6 ; and the payment is D40, which ex- 94 Simple Liter est eeeds the interest then due by Di5-r4, and that being de- diu ted from 1)970^55, the principal, leaves Do .'Si, i&r the new principal. 6. Froir the time of the fifdi payment to June 1, 1784, the tiir.8 yif settlemeat, is ': ye rs and 1. '- months. The interest of \) ■ -b- i for 2 year:3 .md 1 months, is D; 67*09 ; which being added to the priacipid, makes the sum due D)i2l*90, Vv h 1 c h is; t lip a n ^ w e r. JYote. The medu/^l " ~ ' rul ■ is ntore fir orableto the (L bfor, as ap- peft s b} t e foreji-oi(, . he .'; t^ enct' in ihe Hip.oiint being §18 08. But 'he mrthdd or r: i ;^ eq-iitnble. B( cause it u&y so ha'M^en, h) xlnt iii':"* h^ , ..liKt i'l a course of yetirs, ihe obli'- gat'on niMY be tancc.. hj'idcr (A' .h ^ note b..-';ught in debt ta tii< ;>;iver, by tiie pa} :r^' t , st oni) , '.vr)i -ut any part of the priii- ti:-:.! being paid ; :-3 apiior. 1 ov\ing qu«^stioii : Suppose a liote was give.!! Jan ■ 5 800, fm^ ':ii'. "t at 6 per ce. t, and thil f^5 a year is pr. ! on .he Rrot of J year, till Jan. 1, 1850; how does ths ace ii I o. n jitand a st ruJe ' ih<^ pri;;cipa' is ^3^**^' ' '' '•-''■'' -f^nst on it N-,6 a year, for 50 years, is D30v> ; J.,^ G4. i h.^ Juterest of the teco' d for"48 \ears is D17-28, and its auioiint Is I)'23 t28 and so on F. ora compitUni^ which, it appe.irs, that tlie ;»nciOu'.t rif di several pa tnents. viJh the inte' est on them, is 't>7.'35. So that at the time of s:tt!einent, 35. \Vh rea , in equit;' , n< thing has been paid but the interesCas it pr pe' i- bo. a ,,:- ^^ :•■, 9i;o )io;>e of th- principal l;as ever been paid, 'i ';f . CO;.'! r 'le ;s |!erh;r s, as equitable as Chn cor.veiieiitly be made, but is not e:\'-tC(ly so Because the interest '>iight to b<:^ paid at the nd of every year, i^ it is paid sooner, there is an advantage to the creditor ; if later, to the debtor. • Case 4. To find the rate per cent, when the amount, time, and principal, are given. Rule. As ihe principal is to the interest for the whole time, so h LlOO to its interest for the same time. Having found the interest of jL 00 for any given time, the interest ef it for ons year may be toun I by division, multiplication, or practice, as circumstances recpire. ' Example. At what rate per cent will X500 amount to L725 in 9 years ? Here, I wish first to find the interest; and knowing the principal and the amount, I subtract the principal, 1*500, from the amount, X725, and the remainder, L225, is the in- terest. 1 then say : As L500 is to L2'^i5, so is L.OO to its interest for 9 years. This proportion being worked out, gives X45 as the interest Simj)le Interest 95 «f jLIOO for 9 years ; and this being divided by 9, gives L5 as the interest of L 100 for one year, that is, 5 per cent, which is tiie answer. Casf. 5. To find the time, when the principal, amount, and rate per cent, are given. Rule. Divide the whole interest by that of the principal for one year, and the quotient will be the time. Example. In what time will 7>500 amount to L79.5^ at 5 per cent ? Here, I first find the interest of jL500 for one year at 5 per cent, which is L25 ; and then I find, by subtracting, as before, what is the whole iHt^rest, which is L225 ; and then I divide L225 by 25, and it gives 9 years as the answer. Case 6 — Discount. To find the principal, when the amount, time, and rate per «ent, are given. This case is the same as the rule of Discount. Discount is an abatement made for the payment of money before it becomes due, by accepting as much as, being put ©ut at interest, would amount to tlie whole debt at the time it becomes due. The present worth is the amount so accepted ; and is the Same as the principal found by this case. Rule. As the amount of LlOO, at the rate and time given, is to XlOO, so is the amount or whole debt to the principal or |)resent worth. , To find the discount, subtract the present worth from the whole debt, and the remainder will be the discount. Proof. Find the amount of the present worth at interest for tlie same rate and time, according to Case 1, and it will equal the whole debt, if the work is \:ight. Example. ; What is the discount of L400 for 6 months, at 6 per cent ? (hp^eration. Here, I first find what LlOo will amount to,^t interest for 6 months at 6 per cent, which is L 03. 1 then say, as LIOS is to LlOO, so is L^OO to the present worth ; which propor- tion being worked out, gives jLS88 .. 6 .. I IJ, for the present worth ; which being subtracted from X40<% the whole debt, gives 1/ 1 1 .. 13 .. Oi, as the discount, which is the answer. \ S6 Compound Interest. Insurance, Commission, and Brokage, Are allowances made to insurers and factors, or other agentSj at a stipulated rate per cent ; and the amount of such allowance is the same as the simple interest for one jear at the same rate per cent, and is found in the same manner. COMPOUND INTEREST, Is that which arises from a principal increased from time to time by the addition of the interest to the principal, as often as the interest becomes due. Rule. Find the first year's amount by Simple Interest, which will be the principal for the second year ; and the amount for the second year will be the principal for the third year, and so on. From the last amount, subtract the first principal, and the remainder will be the compound interest. Example. What is the compound interest of jL450. for 3 years, at 5 per cent ? Here, I first find the interest of La^50 for one year at 5 per cent, which is i22 .. i 0. I then add this to the principal, and the amount for one year is L-^7'' .. 10, which is the principal for the second year. 1 then find the interest of L47'^-> 10 for one year, which is I>23 .. 12 .. 6 ; and this being added to , its principal, the amount for the second year, is 496 .. 2 .. 6, 5 which is the principal for the third year. I then find the in- ' terest of M96 .. 2 .. 6 for one year, v/hich is X2 i .. 16.. IJ ; and this being added to its principal, the amount for the third year is jL:'20 .. 18 .. 7|. Then subtracting the or'ginal principal, M50, from this last amount, 1 have LjQ„ 18 .. 7^, ^ as the compound interest, and answer. JVote. In this example, the^ interest is supposed to be pa\ able annually. If it is pa^able more or less frequently, the iiteiesi must be calculated up to : the time when it is due, and then added to its principal, to foi^ffi a new prin« \ cipal. Questions on the foregoing. What is the second method of find* ing thfc interest r Having fend the interest for one yeai , how do you find it for more than a j ear ? What is interest ? What is the principal ? "What is the rate per cent ? What is the amount ? Wliat is simple interest ? What is th^ fifst case ? What is the first rule ? How do you divide by 100 > How do > ou mvdtiply by ft mixed How for less than a year ? What is the second rule f What is the third rule ? f Why »io you divide by 36500 ? Wiuit ia ^« f¥«oAa ci^e f the iruW t Exercises, ^7 What is the thu*d case? the first rule ? the second ? Which is the more favorable to the debtor ? Which is the more equi- table f What is the fourth case ? the rule ? The fifth case ? the rule ? The sixth case ? W hat is discount ? What is the present worth ? How do you find the present worth ? How the discount ? How do you prove the operation ? What are insurance, commission, and brokage ? How are they found ? What is compound interest ? What is the inile ? When you have found the amount for any given time, how do you find the Qompound interest ? What if the interest is not payable annually ? Exercise .^0. Tell the interest, at 6 per cent, jins. Of 81*^0 for 12 months, 8900 8-00 21 00 IfrOO 3-00 22-50 7-50 13-50 6-00 10-00 Tell the interest, at 6 per cent. Of 8250 for 6 months, 200 8 700 e 800 4 600 1 750 6 100 15 300 9 400 3 600 4 125 225 275 300^ 350 400 475 425 650 12 6 6 4 6 8 6 12 4 7-50 7-50 6-75 8-25 600 10-50 16-00 14-25 25-50 1100 Exercise 51. Tell the interest, at 5 per cent, Ans, Of gl20 for 12 months, D600 2-50 6-25 4-25 11-50 7-25 20-00 6-25 16-50 7-75 Tell the interest, a( 5 per cent. Am. Oft)7C0 for 6 months, Dl7-oO 2(i0 3 250 6 340 3 4€0 6 580 3 600 8 500 3 440 9 620 3 140 260 380 900 860 180 240 560 640 3-50 9-75 4-75 1500 21-50 4-50 7-00 7-00 24-00 EQUATION, Is the reduction of several stated times at which money is payable, to one time, which shall be equal in value. Rule. Multiply each payment by its time, and divide the sum of all the products by the whole debt, the quotient will be the mean or equated time. Proof. The interest of the sum payable at the equated time, will be equal to the intexest of the several payments at their re^ spective times. I ^8 Barter.-^Loss and Gain, Example. A owes B LI 00, of which i^50 is payable in two months, and L50 at four months ; what is the equated time ? O^eratwn. 50x2=100 Here,I multiply each payment 50x4=200 by the number of months be- fore which it becomes due, 100)300(3 months, Jlvs. and add their products, which makes 300. T then divide this sum by the whole debt, and the quotient is 3 months, for the equated time. JVote This is the common method, hut it is not exactly equitahle, hecause the interest is allowed instead of the discount on the payment which is made before it would faU due. BARTER, Is exchanging one commodity for another, by duly pro- portioning their quantities and values. Rule. Work by multiplication, the rule of three, or practice, as occasion requires. Example. How much sugar at ^d, per Ih, must be bartered for 6| Cwt of tobacco at lAd, per Ih. ? Operation. Here, 1 first find the amount of the tobacco at l Ad., which is I0l92ei. ; and then find how much sugar at 9d, that sum w^ill buy, and it is 10 CwL 12.1b, and |^ of a ^6. which is the answ^er. LOSS AND GAIN, Is a method of computing the profit or loss on the purchase and sale of goods. Rule. "Work by the rule of three, or practice, as occasion requires. Example. Bought 9 Cwt of cheese at i^2 .. 16 per Cwt., and retailed it at 7d, per lb, ; what is gained or lost in the whole ? Operation. I first find how much was paid for the cheese, which was Z^5 .. 4 ; and then how much was received, which was jL29 .. 8 ; and the gain is i>4 .. 4. J^'ote. If the gain or loss per cent is required, it is found by the rule of nree, as follows : Make the sum of money employed, tlie first term; the gain or loss, the second ; aad 100, the third. Thus, in the preceding ex- ample, the sum employed was 251. is , and the gain was Al. 4^., which K r;#UT)d to be W. ids.. Ad.^^T lOOl. or IQ and 2 thards per eeaC* Fellowshijj. 99 FELLOWSHIP, Is the rule for adjusting the several sliares of gain or loss in any joint business. Case 1. When the stocks of the several partners continue for the gam« time. Rule. As the whole sum or stock is to the whole gain or loss, so is each partner's share of stock to his share of the gain or loss. Proof. The sum of the several shares must be equal to the whole gain or loss. Example. A and B bought a parcel of goods, for which A paid LS, and B LT, The goods being sold, there was again of 25s. What is the share of each ? Operaiion. I first add the stocks, and find the sum to be LlO; and tlien say : As LlO is to 25s., so is Ls to A's share, and L7 to B's share ; which proportions being worked out, give A 7s> M,, and B 17s. 6d. Case 2. When the stocks continue unequal terms of time. Rule. Multiply each man's stock by its time ; then, as the sum of the products is to the whole gain or loss, so is each particular product to its share of the gain or loss. Example. A and B traded as follows : A put in L50 for 6 months^ and B L75 for 3 months, and they gained. 1,25.. What is the hare of each ? Operation,. Here, I first multiply A^s stock, L50, by its time, Gmonths, and A's product is 300. Then I multiply B's stock, L75, by its time, 3 months, and B's product is 225. The sum of these products is 525. I then say, as 525 is to L'io, so is 300 to A's share, and 225 to B's share ; which proportions being worked out, give A L14 .. 5 .. 8^, and about half a fiir- thiug ; and B, LiO .. 14.. 3^", and about half a farthing ; which is the answer. EXCHANGE, Is the rule by which the money of one state or country ii> brought into that of ano58-33SJ, which is the an- swcr. Or, 96 and 56 may first be divided by the common divisor 8, and th ir quotients, 12 and 7, used in their room ; lOOxT =700, and 700-T-l2=i.5S-333^, as before. Rule 2. 1. Consider whether you are to add to, or subtract from the given currency, in order to find t]\(i amount in the cur- rency required When the dollar is fewer siiillings in the given currency than in the required, you are to add ; when it is more, you are to subtract. 2. To find how much you are to add or subtract, fin(\ the diff'erence between the value of a dollar in the two currencie^o and see what part that is of a dollar in the given currency. 3. Take such part of the given sum, and add it to, cr subtract it from itself, as occasion requires. J^X AMPLE 1. In Z>100 N. England currency, how much Pennsylvania ? 1. The value of a dollar in New-England currency i fewer shillings than in Pennsylvania ; I am therefore to add to the given sum* 12^ 102 Exchange. Q. The value of a dollar in New-England currency is Gs., and in Pennsylvania 7s, 6d. The difference is Is. 6c?., which is I of 6.-^. I am therefore to add'^ of the given sum to itself. 3. I find what is ^ of 1.100, the given sum, and it is L2.j ; and this added to the given sum, makes 1.125, which is the answer. Example 2. In JLIOO New-York currency, how much New -England ? Operation. 1. The value of a dollar in New- York currency is more shillings than in New-England ; I am therefore to subtract. 2. The value of a dollar in New-York currency is 8s., and in New-England 6s. The difference is 2:s., which is -^ of 8s» I am therefore to subtract i of the given sum from itself. 3. I find what is i of LlOO, the given sum, and it is I/'ES, and this subtracted from the given sum, gives L75, for the answer. By the tw^o preceding rules, the follow^ing table is con- structed. The parts to be added or subtracted, are found by the second rule, and the multipliers and divi&ors by the first. It is best to add or subtract, when the numerator of the fraction to be added or subtracted is 1. When that is Bot the case, it is best to make use of the multipliers and divisors. Tabular Rules for reducing vayious Curreneies to others. Look for the given currency in the left hand column, and then look along the top for the currency required, under which, and opposite to the given currency, is the part of the o-iven sum which is to be added or subtracted. Where it is more convenient to multiply and divi being 1 seah for a shekel of silver ? Exchange. 105 Operation. By the tables, I find the seah to be § of the epha, and the epha 60 pints, wine measure, and l5 solid inches. The seah, therefore, is 20 wine pints and 5 solid inches, that is, 582 J solid inches. A shekel of silver is 4 drachmae, of \2-^£jCts, each, that is 50|| cents; and a bushel is 2150| solid inches^ Therefore, 5S2^in. : 50||cfs. :: 2l50|in. : the answer ; which proportion being worked out, gives Dl^Sr^^f , for the answer, J\*ote 1. Sometimes the rate of exchange is stated at a certain sum per cenL That is, ZlOOin one country are worth' so much more than //lOO in the other. Wiien this is the case, consider in which currency lAOO is worth the most. If in the currency required, add the given rate per cent to ilOO, and make it the first term of a proportion ; make XlOO the second term; and the sum in the given currency the third ; and proceed as in the rule of three. JVote 2. When the exchange is in favor ot the given currency, make ZlOO the first term ; 7^100 added to the rate per cent, the second ; and the given currency the third. Example. Philadelphia is indebteii to London, |g 1400, Pennsylvania currency ; how much is that in sterling money, when the exchange is at 64 per cent in favor of London P Operation. Here, I consider, that since the exchange is 64 per cent in favor of London, £ 1 00 sterling is equal to ^164 Pennsylva- nia ; and so I say, as =gl64 is to ^10;/, so is i61400 to the answer; which proportion being worked out, gives ^853 .. 13 .. 2, and a little more, for the answer^ Questions on the foregoing. What is equation ? How do you find the equated time ? How do you prove equation ? Is this method of equation exactly equitable ? Why so? "What is barter ? By \vliat rnle do you woik questions in b.'irter ? \\ hut is loss ar.d »fiin ? How do you work questions \% loss lAii] gain i' Wliut IS fellowship? "Wliat IS the first case, and rule ? The second, and rule ? What is ihe method of proof ? What is exchange I What is j_)af ? How do you reduce pounds to dol- lars ? Mow (loyou reduce dolliars to pounds? What is the shorter method of redu- I V'"r? p'J'ii'ds sterling to »lollars ? What is ihc rrasoii of this rule ? How do you rcduec the currency of one stule to that of another, by the first method ? How, 1)' the "second method How do you reihice ih«^ weights, mea- sures, and coins of one country, to tliose of another ? When the rate of exchange is a cer- tain sum percent aiul in favor of the required currercy^how do you proceed ? How, wh^ n it is in favor of the jjivea fiurj'Ciicy ? 106 JBuodecimals, DUODECIMALS, Are fractions so called because thej decrease by twelves^, inches being twelfths of a foot, which is the whole number ; and seconds being twelfths of an inch, tiiirds twelfths of a second, and so on. They are chiefly useful to ascertain the superficial or solid content of such things as are measured by feet, inches, &c. Addition and Subtraction of Duodecimals are performed as in Compound Addition and Subtraction. MULTIPLICATION OF DUODECIMALS. Rule. 1. Place the multiplier under the multiplicand, in such manner that the feet of the multiplier stand under the lowest denomination of the multiplicand. 2. Begin with the lowest term of the multiplier, and pro- ceed to the left, placing the product of the lowest term of the multiplicand under its multiplier, and so on through all the terms, carryirg 1 for every 1^2, 3. Take the second term of the multiplier, and proceed in the same manner ; and so on through all the terms ; and the sum of the products will be the answer* Example 1. Multiply Sft. 6m. 9 sec. by 7fU 3in. 8 sec. Operation. Ft. in. sec. th. /. Here, I first place the multiplie]? 8 .. 6 .. 9 so that 7, the feet, may stand under 9, the seconds of the multiplicand, and the other denominations of the multiplier in order towards the right. I then begin witli 8 and 9, the lowest denominations of the multipliei: and multiplicand ; and 62 .. 6 .. 7 .. 9 .. Jns. say, 8x9 is 72, which being 6 times 12, I set down under the multi- plier &, and carry 6 to the next. Then, I say, 8x6 is 48, and 6 1 carried is 54, which being 4 times 12 and 6 over, I set down 6 and carry 4. Then, 8x8 is 64, and 4 I carried is 68, which is 5 times 12, and 8 over. Set down 8, and carry 5. But as there is no other term to multiply, I set down the 5 in the next place. I then take 3, the next term of the multiplier, and say^ 3x9 i:: 27^ which is 2 time^ 12, and 3 over. Set down 3 \ic 7. .3. .8 5 . .8. .6. .0 o ^ .. I . .8. .3 59, .. 11 . .3 Duodecimals, 107 der the multiplier 3, and carry 2 to the next. In like man- ner, I proceed till 1 have multiplied all the terms of the mul- tiplicand by 3. I then take 7, the last term of the multiplier, and say 7x9 is 63, which is 5 times 12, and 3 over. Set down 3 under the multiplier 7, and carry 5 to the next. And so on, till I have multiplied all the terms of the multiplicand by 7. I then add up these several products, and the answer is 6^ft 6in, 7 sec. 9 thirds. Example 2. How many feet of wood are there in a load that is 8/^ 6in* long, 2/f. Sin, wide, and 2ft, din, high ? Operation, ft. in. sec. th, 8 .. 6 length. 2 .. 3 w idth. 2.. 1 ..6 17..0 19. 1 .. 6 3 .. 9 height. 14..4.. 1 ..6 57.. 4.. 6 71 .. 8 .. 7 .. 6 Ms, Questions on the foregoij^g. What are Duodecimals ? For what are they useful ? How are addition and subtraction of duodecimals performed ? Ih multiplication of duodecimals, how do you set down the multiplier ? Exercise 52, Where do you begin to multiply ? For how many do you carry 1 ? Where do you set down the product of each figure of the multiplicand ? What will be the answer ? Tell what is the least common muiti- pie of Am. pie of Ans, 5, 4, and 5, 60 3, 7, and 6, 42 6, 7, and 8, 168 7, 8, and S, lr,8 7, 8, and 9, 504 5, 7, and 8, 280 S, 6, and 7, 210 4, 5, and 7, 140 S, 5, 8. and 10, 120 5, 3, and 9, 4S 2, 7, and 14, 28 4, 6, and 7, 84 6, 10, and 8, 120 6, 7, and 12, 84 §, 3, and lU 66 3, 5, and 8, 120 Tell what is the least ©ommon multi- loa Exercises. Exercise 53. Eeduce to a common denominator, | Reduce to a common denominator, I and I. ^Ins. 1 and | j and |, ^ns. 9 and II ^ and 1 '2> f and f fand 1. S.4 3 'and If i and i, fandf 1 and h l\ and y\ 1 and h A and T^-j *and i. 3 6 and j\ fahd h ^V and If fand I. If and^V Exercise 54. Tell the sum of .4ns. 1 ell the sum of Ans* ^ and i. s. 4 f and !. ■* 1 .5 ^ and §. 6 fand 1. Hf i and f. H f and h 1* i and i A f and i. If fand f. h\ 1 and Exercise 55. h U From take v^i*. From take Aii^. 2. JL i 3 Z .1-3 4> ^f 2 4» T> 28 f> i. i 1. # h JL I 5 4> i. * f. h 3V h h A 1. h • tV 1 1 7 1 3 1 2» 7» T8 ' 2» Exercise 56. 7> T? Multiply by JltlS. Multiply by v5;?*. I 2"» ^. i h |. f !' 1 i i> "t* t'o t' I. t\ h 1.- 1 1 2 2 1 5 5 t> 3> 15 -2> -6» IT 4 3 ? X 2 a i» ?» 5 3J Exercise 57. ^' 2T MuIiWy by ^?w. Multiply by Atis, 1, 1. f H. -J. i 2, 1. J| 2§. *, i 3. 1. 2f Si. il. if 4, f. 2| 1 4|. h 2* 5, 1 "2> 2| 1 5|. -h 4A Exercise 58. Sivide by .^ns. Diride fey JtM* 1. ^. 2 4, 4. H i. h 1: i. i. i 4. h f i. h f }. h i i. h 1 i. h H i. h If ^o'erciseso \ Exercise 59. Tell the sum of Ans, s. s. d!. £. s. d. *. s. d. 2, .. 3 .. 6 and 1 .• 4 .. 3, 3. . 7.. 9 3. .. 15 .. 8 and 2 .. 3 .. 5, 5. .19.. 1 2. .. 10.. 1 and 3.. 15.. 11, 6. . 6.. 2, ..13.. 6 and 1.. 16.. 8, 4. .10.. 2 10, ..10.. 10 and 2.. 2.. 2, 12. .13.. 5. . 5 .. 5 and 6 .. 6 .. 6, 11 . .11.. 11 13, .. 13.. Band 14.. 14.. 4, 28. . 8.. 5. .. 6.. 7 and 8.. 9.. 10, 13, . 16 .. 5 4, ,. 5 .. 6 and 5 .. 6 .. 7, 9. .12.. I 3. .. 5.. 4 and 4.. 3.. 5, 7. . 8.. 9 6. .. 4 .* 3 and 3 .. 4 .. 6, 9. .. 8.. 9 3. ,. 7 .. 6 and 4 .. 3 .. 6, Exercise 60, Tell the sum of 7. ..11.. s. d. s. d, s. d. .€. s. d. 3, ,. 6 and 4 ..,6 and 5 .. 6, 13.. 6 2. .. 4 and 3 .. 4 and 4 .. 4, 10.. 3. . 5 and 4 .. 6 and 5 .. 7, 13.. 6 7. ,. 6 and 8 .. 8 and 9 .. 9, 1 . . 5.. 11 3. .4 and 5 .. 6 and 7.. 8, 16.. 6 8. .. 7 and 6.. 5 and 4.. 3, 19.. 5 6. . 5 and 6 .. 6 and 6 .. 7, 19.. 6 4. ,. S and 5 .. 6 and 6 .. 4, 16.. 6 3. ..5 and 3 .. 9 and 3.. 11, 11.. 1 il . .. 3 and 9 .. 3 and 5 .. 3, 1. . 5.. 9 6. .. 8 and 7.-6 and 8 .. 4, 1. . 2.. 6 2, .. 7 and 6 .. 3 and 4 .. 5, Exercise 61. 13.. 3 From take Ans. €• s. d. £. s. d. £' S, d. 1.. 1.. 0, 13.. 6, 7.. 6 2.. 3.. 6, 19.. 0, 1 . . 4.. 6 18.. 5, 13.. 6, 4.. 11 2.. 2 .. 2, 1 .. 3 .. 5, 18.. 9 1.. 17.. 8, 18.. 9, 18.. 11 3.. 18.. 0, 1.. 19.. 6, 1. .18.. 6 1.. 12.. 10, 18.. 6, 14.. 4 1.. 1.. 1, 19.. 9, 1.. 4 2.. 2.. 2, 1.. 11.. 11, 10.. 3 3.. 3.. 3, 2.. 19.. 0, 4.. S 4.. 5.. 6, 3.. 4.. 5, 1. . 1.. 1 5,. 4 «• 3j 3 •• 4 •• 5f K 1. . 19 .. 10 109 110 Exercises^- ExERC isE 62. Tell what is ^?w. Tell what is An3. S. d. £.' s. d. S, d. £' s, d. 3 times 6 ..^8, 1 . 0..0 3 times 9 .. 9, 1 . 9.. 3 4 times 2 .. 3, 9..0 5 times 8 .. 8, 2, . 3..4 5 times 10.. 6, 2 . 12.. 6 6 times 2 .. 3, 13 ..6 € times 5 .. 6, 1 .. 13.. 5 times 4 .. 8, 1 . . 3.. 4 7 times 4 .. 6, 1, . 11 .. 6 9 times 5 .. 5, 2 . 8..9 8 times 8 .. 8, 3 . 9..4 3 times 10.. 10, 1 ..12..6 7 times 7 .. 7, 2 . 13.. 1 8 times 6 .. 7, 2 . 12.. 8 4 times 6 .. 6, 1 . 6..0 ExERC 7 times 9.. 10, ISE 63. 3 .•8.. 10 Tell what is the ^71S. Tell what is the Ans. £. s. d. 5. d. £. s. d. S. d. 4th of 1 .. 8.. 0, 7..0 10th of 9.. 2, 0.. 11 Sd of 1 .. 4 .. 6, 8.. 2 6th of 1 .. 5 .. 6, 4.. 3 5tli of 3 .. .. 0, 12. .0 4thof 1.. 12 ..8, 8 .. 2 6tk of 1 .. 4 .. 6, 4.. 1 5th of I .. 15 .. 1(J , 7.. 2 7th of 2 .. 2 .. 0, 6..0 7th of 2 .. 2 .. 7, 6.. 1 3d of I .. 1 .. 3, 7.. 1 8th of 1 .. 14 .. 8, 4.. 4 8th of 2 .. 8 .. 0, 6..0 3d of 13 .. 9, 4.. 7 9th of 1 .. 7 .. 9, 3.. 1 ExERC 4th of 17.. 8, ISE 64. 4.. 5 Tell how many times Ans. Tell how many times Am. s. d. s. d. s. d, s. d. 1 .. 2 in 4 .. 8, 4 3 .. 5 in 10 .. 3, 3 2 .. 6 in 12 .. 6, 5 1.. 3in 6.. 3, 5 1.1 in 6 .. 6, 6/ 2 .. 5 in 9 .. 8, 4 2 .. i in 6 .. 3, 3 1 .. 5 in 8 .. 6, 6 1 .. 5 in 5 .. 8, 4 1 .. 7in 7 ..11, 5 1 .. 2 in 7 .. 0, 6 2 .. 9 in 8 .. 3, 3 2 .. 2 in 6 .. 6, 3 1 .. 1 1 in 7 .. 8, 4 3 .. 1 in 12.. 4, 4 2.. Sin 11 ..3, 5 INVOLUTION, Is the raisitig of numbers to powers. A power ^ is a number produced by multiplying a smaller number by itself a certain number of times. The smaller number so multiplied to produce a power, is called the root of that power. Thus, 3x3=9. Here, 9 is the power, and 3 is its root. 4X4X4=64. Here, 64 is the power, and 4 is its root. 5x5x5x5=625. Here, 625 is the power, and 5 is its root. Involution. Ill When a number is multiplied by itself once, the product s called the second power, or square, of that number ; and the number multiplied is called the square root of that pro- duct. As, 3x3=9 5 here, 9 is the second power or square of 3 ; and 3 is the square root of 9. Again, 5x5=25 ; here> 25 is the square of 5, and 5 is the square root of 25. When a number is multiplied by itt^elf, and the product so produced is multiplied again by the same number, the second product is called the third power, or cube of that number ; and the number is called the cube root of that second pro- duct. As, 4X4x4=64 ; here, 64 is the third power of 4, and 4 is the cube root of 64. Again, 3x3x3=^27; here, 27 is the cube of 3, and 3 is t|ie cube root of £7. W' hen a number is multiplied by itself, and that product again by the same number^, and the second product again by the same number, the third product is called the fourth pow- er, or biquadrate of that number ; and the number is called the fourth root, or biquadrate root of that third product. As, 2x2x2x2 = 16; here, 16 is the fourth power of 2, and 2 is the fourth root of 16. Again, 3x3x3x3=81; here, 81 is the fourth power of 3, and 3 is the fourth root of 81. Rule. To raise a given number to any given power, multiply that number by itself, and that product by the given number, and so on, till the number of multiplications shall be one less than the number of the given power. Example. Find the sixth power of 2. Operation, 2 2 4 square. 2 8 cube. 2 16 biquadrate. 2 33 5th power. ^ 2 64 6th power* Ans. 112 Evolution* A vulgar fraction is involved to any power required, by involving the numerator to that power,*^ and then involving the denomioator to the same power. Thus the square of § is f, and the cube off is -/. EVOLUTION, Is extracting or finding the roots of given powers. Numbers, with respect to their roots, are rational, or irra- tional. A rational number is one of which the exact root can be found. An irrational number, or surd, is a number of which the exact root cannot be found. The same number may be rational with respect to one of its roots, and irrational with respect to another. Thus, 4, with respect to its square root, is rational ; but with respect to its cube root, it is irrational ; for the square root of 4 can be found, but the cube root of 4 cannot be found exactly. Table of Powers and Roots. Roots, 2 3 4 5 6 7 8 9 Squares, I 4 I 8 9 27 16 25 36 49 64 81 Cubes, 1 64 125 216 343 512 7&9 4th pow. 1 16 81 256 1024 625 3125 1296 2401 4096 6561 5th pow. ] 32243 7776 16807 32768 59049 6th pow. ] I 64 739 4096 15625 46656 117649 262144 531441 To extract the Square Root. Rule. 1. Distinguish the given number into periods of two figures each, beginning at the place of units, and marking every second figure from the place of units to the left, in whole numbers, and from the decimal point to the right in decimals, if any, annexing a cypher to the decimals, if necessary to make an even number of places. 2. Begin with the left hand period, and find by trial the greatest square it contains, and set its root on the right hand of the given number, separated from it by a curve line, for the first figure of the required root. Evolution, 1 1 3 * 5. Subtract the square thus found from the said period, and to the remainder bring down and annex the next period, for a dividend. 4. Double the root already found, for a defective divisor. 5. Find how often the defective divisor is contained in the dividend, exclusive of its right hand figure; and the figure denoting that number of times, will be the next figure of the root, probably, 6. Complete the divisor, by annexing at the right hand of it the last figure of the root. 7. Multiply the divisor so completed by the last figure of the root, and subtract the product from the dividend. 8. Bring down another period, and find another figure of the root in the same manner ; and so on, through all the pe- riods. JS'ote 1. The reason for dividing the given number into periods of two figures each, is, that the square of any one figure is never more than two figures.* Heiice^ there will be as many places of whole numbers in the root, as there were periods of whole numbers in the square. JVote 2m It will sometimes happen, that on multiplying the complete divi- so^ by tliejast figure of the root, the product will be greater ithan the divi- dend. In that case, you mwst try the next lower figure, and if that prove too great, the next, ami so on. J^foteS, When you have proceeded through all the periods of the given number, and there is a remainder, the operation may b^* continued further, if required, and another figure of the root found, by annexing two cyphers to that remainder, and proceeding as before. If thai remainder consisted of whole numbers, the first figure found by annexing a period of cyphers, will be the first decimal, and so on. JYote^. The square root of a vulgar fraction may be found by reducing it to its lowest terms, and extracting the root of each term. Or, the numerator and denominator of the given fraction may be multiplied together, and the square root of that product, being extracted, may be made the numerator to the denominator of the given fraction, or tlie denominator to the numerator of it, for the answer But if the exact root of it cannot be found by either of these methods, an approximation to it may be found, by reducing the vulgar fraction to a deci- mal and extracting its root to as many places as shall be thought necessary, JVote 5 A mixed number in vulgar fractions must be reduced to a mixed liumber in decimals, and the root extracted as before. Proof. Squire the root, when found, and add in the remainder, if any ; and the sum will be equal to the given number, ii the work is right. Example. What is the square root of 552*25 f K2 U4 SiSoluiioni « >■ Operation. Explanation, ... 1. I distinguish the given numbei' 552'25(2S'5 Ans. into periods of two figures each, be- 4 ginning at the place of units, and — marking ever j second figure each way 5 43)152 and have three periods, the first con- 129 sisting of 5, the second of 52, and the third of -25. 465)2325 2. I take the left hand period, which 2325 is 5 J and try how great a square num- ber I can find in it, which is 4, the root of which is 2. So I set down 2 far the first figure of the re- quired root. 3. I set down 4, the square thus found, under 5, the first period, and subtract; and the remainder is !. To this re- mainder, I bring down and annex the next period, 52 ; and I have 1 52 for a dividend. 4. T double 2, the root already found, and it makes 4, Which I set down at the left of the dividend, for a defective divisor. 5. I seek how often 4, the defective divisor, is contained in 15, the dividend with the exception of the right hand figure ; 4 in 15 is 3 times. So I set down 3 for the second figure of the root. 6. I now complete the divisor, by annexing at the right hand of 4, the defective divisor, 5, the second figure of the root ; and it makes 43. 7. 1 multiply 43, the divisor so completed, by 3, the secotid figure of the root; and it makes 129, which 1 subtract from the dividend, 152 ; and the remainder is 23. 8. To this remainder, ^3, 1 bring down and aiinex the next period, 25 ; and it makes 2325, for a dividend. 9. I double the root already found, 23, for a defective di- visor ; and it makes 46. 10. I seek how ofteii 46 is contained in 232, which is the dividend excepting its right hand figure ; and find it 5 times. So I set down 5 for the third figure of the root. 11. I now complete the divisor, by annexing at the right hand of 46, the defective divisor, 5, the third figure of the root ; and it makes 465. 12. I multiply 465, the divisor go completed, by 5, the l-hird figure of the root ; and it makes 2325, which being sub- iFacted from the dividend, there is no remainder. Evolution* 115 The figures of the root, therefore, are 235. But as I had in the given number only two periods of whole numbers, I must kave only two whole numbers in the root. So I insert a decimal point between the second and third figures, and the answer is 23*5. To prove the operation, I square the root so found ; that is, I multiply 23*5 by 23-5, and it makes 552'QS ; which be- ing equal to the given number, I conclude the work is right. JSfote 6. The areas of similar figures ere to each other as the squares of their similar dimensions. ^^K Example. '^^ If a certain field, one side of which measures 50 rods, con- tains 6 acres, how much does another field contain, of the same shape, the similar side of which measures 50 rods ? Operation, 30 50 30 50 900 : 2500 : 6 : 6 : 900)15000(16- 900 66+ Ans 6000 5400 6000 5400 6000 5400 600 J\'ote 7. The square of the longest side of a right angled triangle is equal ta the sum of the squares of the other two sides. Example. The height of a certain wall is 17 feet, and there is a ditch at the foot of it 20 feet wide ; how long must a ladder be, to Teach from the outside of the ditch to the top of the wall ? 116 Evolution. Operation. height of wall. 20 width of ditcfi* 17 20 1 1 9 400 square of width. 17 289 square of height. 289 square of height. 689 sum, and square of the length of the ladder ; the square root of which being extract- ed, is 26-24-1- feet, which is the answer. Problem. The sum and product of two numbers being given, to find the numbers. Rule. From the square of the sum, subtract four times the pro- duct, and the square root of the remainder will be the diffe- rence of the numbers. Example. A and B make up 1000 dollars, and trade till they have gained two hundred per cent, which gain is to be divided in proportion to the snare of each in the capital stock. A's gain is the most, and is such a sum, that if multiplied by B's, the product would be 960000 dollars. What was each man's share of the capital stock ? Operation. Here, the gain, being 200 per cent on the capital stock, is 2000 dollars ; and the product of the two parts is 960000 dol- lars. Therefore, 2000 960000 2000 4 4000000 is the square of the sum. 3840000 3840000 is 4 times the product. •>.«.> -. -. - 160000(400 is the difference, 16 0000 Having found the difference, I add half the difference t© half the sum for the greater, and subtract for the less ; and each person's share of the gain is, A's, 12()0 dollars, and B's, 800 dollars ; consequently their shares of Stock were^ A'e^ 600 dollars, and B'g, 400 dollars. Evolution. IIT Questions on the foregoing. What is involution f What is a power ? What is a root ? What is a square ? The square root ? What is a cube ? The cube root ? What is a biquadrate ? The biquad- ratic root ? What is the rule for finding any given power ? How is a \-u!gar fraction involved ? What is evolution ? What is a ratid' al number ? What is a surd ? In extracting the square root, what is the first thing to be done ? Where do you begin to point off? How many figures do you put in a period, and why ? How many whole numbers will there be in the root ? How do you find the first figure of the 'root ? Of what does the first dividend con- ^ sist ? Of what does the defective divisor consist ? When you have found tlfc defective divisor, how do you find another figure of the root ? How do you complete the divisor ? By what do you multiply ? What do you do next ? What if the product to be subtract- ed is larger tiian the dividend, from which it is to be subtracted ? How do you form another dividend ? When you have extracted the root of the given number, and there is a remaiT.dfc:r, how can you extend the operation fuither ? What isthi^ first method of extracting the square root of a vulgar fractiouj The second ? The tliird ? How do you extract tlie square root <)f a m'xed number in vulgar frac- tions f How do you prove the operation ? How do you know the work is right ? What proportion have the areas of similar figuies to each other ? What proportion have the sides of a right angled triangle to each other? When the sum and product of two numbers are given, how do you find '. he diflTerence ? When you have found the difference^ how do you find the numbers ? To extract the Cube Moot. Rule. 1. Distinguish the given number into periods of three figures each, beginning at the place of units, and marking every third figure to the left in whole numbers, and to the right in decimals, if any, adding cyphers to the decimals, if necessary, to make out the last period. 2. Begin with the left hand period, and find by trial the greatest cube it contains, and set its root on the right hand of the given number, for the first figure of the required root. 3. Subtract the cube thus found from the said period, and to the remainder bring down and annex the next period, for a dividend. 4. Take three times the sti[uare of the root already found, for a divisor. 5. Divide the dividend by the divisor, so far as to find one quotient figure, which will be the second figure of the root^ pvobablif. 118 Evolution. 6. Subtract the cube of th^se two figures of the root from the two left hand periods of the given number, and to the remainder bring down and annex the third period, for a se» cond dividend. 7. Find another divisor, and another figure of the root, in the same manner, and so on, always subtracting the cube of the root found from as many of the left hand periods of the given number, as you have found figures of the root. J\''ote h The reason for dividing the given liumber into periods of tlA-ee figures each, is', that the cabe ^f any one figur*' is never more than three figures. Hence, there will he as many pbces of whole numbers in the root, as there were y)eriods of whole numbers in the given number. JSi^oie 2, It Aviil sometimes happen, that tin- cube of the figures placed in the root will be found greater than the pej iods from whicli it h to be sub- tracted. In that case, the last number placed in the root is too large, and must be made smaller. JVote 3. When you have prrceeded through all the periods of the given number, and there is a remainder, the operntion may be continued further, if required by annexing three cyphers to that remainder, and proceeding as before. JVote i The cube root of a Tulgar faction may be found by reducing it to its lowest terms, and extracting the root of each term. But if the exact root of each term cannot be found, an approximation may be made towards it, by reducing the fraction to a decimal, and extrf^cting its root, to as many places as shall be requisite. A^ste 5 The cube root of a mixed number in ^ulgar fractions, may be found, by reducing it to a mixed number in decimals, and then extracting the root. Proof. Cube the root, when found, and add in the re- mainder, if any ; and the sum will be equal to the given number, if the work is right. Example. What is the cube root of 34328-12') ? Operation, Eocplanation* 1. I di>tinguish the given S4328'125(32'5 Ans, number into periods of three 27 figures each, beginning at the — place of units, and marking 27) 7 "28 first dividend, every third figure each way ; and have three periods, the first S2768 cube of 32. consisting of 34, the second of 328, and rhe third of • i 25. 3072)1560125 second dividend. 2. I take the left hand period, 34328 125 cube of 525. which is 3^, and try h(*w great a cube I can find in it, which IS 27, the root of which is 3 ; so 1 set down 3 for the first figure of the required root Evolution. 1 i 9 0. I set down S7, the cube thus found, under 34, the first period, and subtract, and the remainder is 7. To this re- mainder, 1 bring down and annex the next period, 328, and I have 7328 for a dividend. 4. 1 square 3, the root already found, and it makes 9 ; and multiply that by 3, and it makes 27, which I set down at the left hand of the dividend, for a divisor. 5. I seek how often 27, the divisor, is contained in 7328, the dividend, and find the first quotient figure to be 2. So I set down 2 for the next figure of the root. 6. [ cube 32, the root already found, and it makes 32768, which I subtract from 34328, the two left hand periods of the given number, and the remainder is 1560, to which I bring down and annex 125, the third period, and it makes 1560125 for the second dividend. 7. I square 32, the root already found, and it makes 1024 ; and multiply that by 3, and it makes 3072, for the second divisor. 8. 1 seek how often 3072, the second divisor, is contained in 1560125, the second dividend, and find the first quotient figure to be 5. So I set down 5 for the third figure of the root. 9. I cube 325, the root found, and it makes 34328125, which I subtract from 34328125, the given number, and no- thing remains. So that 325 are the figures of the root. But because there were two periods of whole numbers, and one of decimals, in the given number, there must be two whole numbers and one decimal in the answer ; and I place the decimal point accordingly, and the answer is 32-5. To extract any root. The rule for extracting the cube root will serve for ex- tracting an^ root, with a little variation, as follows : 1. Distinguish the given number into periods of as many figures each, as is the root to be extracted ; that is, for the fourth root, into periods of four figures each ; for the fifth, five, &c. 2. Begin with the left hand period, and find by trial the greatest power of the same name with the required root, that is, the fourth power for the fourth root, the fifth power for the fifth root, &c. and set the root of that po%ver on the right hand of the given number, for the first figure of the root. 3. Subtract the power thus found from the said period, and to the remainder bring down and annex the next period for a dividend. 120 Evolution. 4. If the required root is the fourth, take four times the cube of the.root already found, for a divisor ; if it is the fifth, take five times the fourth power ; if the sixth, take six times the fifth power, &c. 5. Divide the dividend by the divisor, so far as to find one quotient figure, which will be the next figure of the root, probably. 6. Raise these two figures of the root to the power which is of the same name with the required root, and subtract the said power from the two left hand periods of the given num- ber, and to the remainder bring down and annex the third period, for a second dividend. 7. Find another divisor, and another figure of the root, in the^ same manner ; and so on, always subtracting the power found from as many of the left hand periods of the given number, as you have found figures of the root. Example 1. What is the 5th root of 51 53632 ? Operatioti. 5153632(22 Ms. 32 80)1953632 first dividend. 5153632 5th power of 22. Example 2. What is the fourth root of 2215534565= Operation. • • • 221533456(122 ^m. 1 4)12153 first dividend. 20736 4th power of 12. 6912)14173456 second dividend. 221533456 4th power of 122* Mte, The olMervationB in the notes under the rule for extracting the cube root, will apply to this rule for extracting any root, with vEriatioB» similar to those in the rulet. (luestions. Questions on the foregoinj&. l^H ■ lift extracting the cube root, what is the first thine to be done ? Where do yon'oegin to point off? How many figures do you put in a period, and why ? How many whole numbers will there be in the root ? How do you find the first figure of the root ? Of what does the first dividend con- sist ? Of what does the divisor consist ? When you have found the divisor, how do you find another figure of the root ? How do you form a second dividend ? How do you find a third figure of the root ? From how many periods of the ,e;i- ven number, .must you always ExurisE 65 subtract ? What if the cube of the figures pi a*, ced in the root is .greater than the periods from which it is to be sub- tracted ? When you have extracted the cube root of the given number, and there is a remainder, how can you extend the operation further ? What is the first method of extract- ing the cube root of a vulgar frac • tion ? What is the second method ? How do you extract the cube root of a mixed number in vulgar frac- tions ? How do you prove the operation ? How do you know the work is right' What is the rule for extracting any root ? Tell what is the ^iis. Square of 3, 9 Cube of 2, 8 Square root of 16, 4 Cube root of 27, 3 Square of 9, 81 Cube of 4, 64 Cube root of 8, 2 Sq uare root of 64, 8 Cube of 5, 125 Square of 11, 121 Cube root of 64, 4 Square root of 25, 5 Square root of 144, 12 Square of the suna of 2 and 3, 25 Square of the square of 3, 8 1 Tell what is the ^%is. Sura of the squares of 5 and 6, 61 Cube of the square of 2, Sum of the cubes of 2 and 3, Difierence of the square and cube of 2, Difierence of the square and cube of 3, Sum of the square &■ cube of 2, Sum of 3 and the square of 3, Square root of the cube of the square of 2, Pro(hict of the square and cube of 2, Quotient of the cube of 2 by the square of 2, 64 35 18 12 12 8 32 2 Exercise 66. Tell what is the JlnJi. Square of the sum of 4 and 5, 81 Sum of the squares of 4 and 5, 41 Difference of the cubes of 2 and 3, 19 Sum of the cubes of 2 and 4, 72 Sum of the square roots of 64 and 100, 18 Square root of the sum of the square roots of 36 & ? 00, 4 Square of the difierence of tHe square roots of 64 & 25, 9 Cube of the difference of the square roots of 49 and l6, 2T Add 1 to 5x7, and tell the square root, 6 Square root of the sum of the squares of 3 and 4, 5 Sum of the square of 8 and cube of 2, 72 L 12!^ £jcercise3. Exercise 67. Add 4 to 5x9, and tell the square root, Jlns. 7. Add 1 to 7x9, and tell the square root, 8 Subtract 20 from 7x8, and tell the square root, 6 Add 2 to the square of 5, and tell the cube root, 3 Subtract 8 from the cube of 4, and tell the seventh^ 8 Subtract 5 from 6x9, and tell the square root, 7 Add 5 to the square of 5, and tell the tenth, 3 Subtract 4 from the square of 5, and tell the cube of 1 seventh, 27 Add 2 to the cube root of 8, and tell the difference between its square and cube, 48 Subtract 1 from the square root of 4, and tell the sura of its square and cube, 2 Tell 5 times the sum of the squares of 3 and 4, 125 One third of the sum of the cubes of 2 and 4, 24 Half the difference of the cubes of 3 and 2, 9| Subtract the square of 5 from the cube of 5, and tell the square root, 10 Subtract the square of 4 from the cube of 4, add 1, and tell the square root, 7 Add the square of 3 to the square of 4, subtract 5, £c tell the 4tb, 5 Tell one half of one tenth of the square of 10, ^^ Onehalf oftwo thir^Ts ofthe square of 6, 12 Two thirds of one half of the square of 12, 48 One half of two thirds of three fourths of the square of 10, 25 Add the odd numbers below 10, and tell the cube of the square root, 125 Exercise 68. Add 3 and 4 and 5 to 5 times 5, Jns. 37 Take 9 and 8 and 7 from 6 times 6, 12 Multiply 2 and 3 and 4 by 3 times 3, 81 Divide 9 and 5 and 10 and 20 by 11, 4 Add the sum of 6 and 11 to their difference, 22 From the sum of 8 and 13, take twice their difference, 11 Multiply the sum of 8 and 14 by half their difference, 66 Divide the sum of 22 and 14 by half their difference, 9 Add the sum of the squares of 2 and 3 to twice their difference, 23 From the sum of the squares of 3 &• 4, take twice their difference, 11 Multiply the sum of the squares of 4 and f>, by one third of their difference, 123 Divide the sura of the square and cube of 3 by 4, and tell the square root, 3 Subtract 4 from the product of the square roots of 49 and 16, and tell the square of one half of it, 144 Subtract the square of 4 irom the square of 5, and tell the square of it, 81 Multiply 1 and 2 and 3 and 4 and 5, by one fifth of it, 45 ^ Add the square of 1 to the cube of 1, and tell the square of one oneth, 4 Exercises, ISS Exercise 69. I Tell the ../f/zs. TeU tiie difference of the squares of square off, 4 9 i and J, Ans. 3^ cube of J, 1 8 i and :^, 1 6 square of i, t\ } and J, tIt cube of i, oV Product of tlie squares of square of |, if i and 1, J and i. jh cube of -|, H Too 3um of the squares of i and ^-, i and i, 5 T6 i and -], 4^0 i and -J, A3 iandi, T-L i and |, o 9 4 1 and 1, 2 a;-, Tell ^vhat is the sum of 2 thirds af 18 '3c 3 ib^.irths of 16, 24 2 fifths of 35 ik. 5 sixths of 48, 54 3 fourths of 40 6c2thh'dsof 24, 46 4 fil'ciJS of 35 Sc 3 fit*ihs of ,40, 52 2 fifths of 15 &• 3 fiitbs of 30, 24 3 sevenths of 21 & 2 thirds of 27, 27 Exercise From take 2 thirds of 24, 3 fourths of 32, 2 fifths of 55, 3 fifths of 55, 3 thirds of 33, 2 fifths of 60, 2 ninths of 36, 3 tenths of 80, 2 sevenths of 28, .3 fifths of 100, 2 sevenths of 140, Exercise 70. Jiii.'i. Tell what is the sum of Ans. 2 iff hs of 25 & 1 third of 33, 21 3 iburthb of 28 &c 2 thirds of 18, 33 3 eighths of 32 6c 2 fifths of 15, 18 3 sevenths of35^ 2 fifths of 40, 31 2 fifths of 25 6c 2 eighth, of 96, 34 3 fifths of 30 6c 2 sevenths of 77, 40 n. ISfuItipiy 2 thirds of 9, 3 fifths of 15, 4 fifths of 25, 2 sevenths of 21, 3 fifths of 25, 2 thirds of 1 2, 3 fourths of 8, 2 fifths of 20, 5 sixths of 30, 3 sevenths of 28, 3 fourths of 12, 2 fifths of 15, 2 thirds of 12^ 2 sevenths of 28, 2 ninths of 27, 3 sevenths of 21, 2 fifths of 15, 2 ninths of 45, 2 ninths oi2T, 4 fifths of 60, S eighths of 96, Exercise 72 by 3 fourths of 12, 2 thirds of 18, 1 tenth of 30, 1 twelfth of 21, 2 thirds of 9, 3 fifths of 15, 2 thirds of 1 2, 2fifthsof 15, 2 sevenths of 21, 2 ^nthscfsr; 7 18 6 25 16 15 2 14 2 12 4 54, 108 60 J2 90 48 4S 150 Exercises. Exercise 73. Divide by ,lfi^. 2 thirds of 36, 2 sevenths of 21 , 4 3 fourths of 24. 2 ninths of 27, 3 4 fifths of SO, 3 fourths of 16, 2 5 sixths of 18, 1 ninth of 45, 3 2 fifths of 60, 3 eighths of l6. 4 3 fourths of 48, 3 sixths of 1 8, 4 4 fifths of 60, 2 ninths of 27, 8 3 sevenths of 42. , 3 eighths of 1 6, 3 2 thirds of 72, 2 sevenths of 42. t 4, 3 fourths of 48, 3 eighths of 24, Exercise 74. 4 Divide the sum of by J?i.?, 2 thirds of 24 and 1 half of 18, 5, 5 2 thirds of 50 and 3 fourths of 32, 4. 11 1 fourth of 48 and 2 thirds of 24, 7, 4 3 fourths of 16 and 2 fifths of 60, 6. 6 4 fifths of 20 and 2 fifths of 30, 7. 4 2 sevenths of 28 & 3 sevenths of 1 4, 2, 7 3 eighths of 40 and 3 sevenths of 35, 6, 5 2 thirds of 36 and 3 fourths of 24, 7. 6 Exercise 75^ From take and divide by Alts. 2 thirds of 36, 1 fifth of 40, 4. 4 2 thirds of 24, 1 fourth of 24, 5. 2 2 fifths of 45, 2 ninths of 27, 4, 3 3 fourths of 44, 3 sevenths of 21, s. 8 3 eighths of 64, 3 eighths of 24, 5/ 3 2 ninths of 108, 1 ninth of 36, 4, 5 3 sevenths of 77, I seventh of 35, 4, 7 4 fifths of 95, 4 ninths of 45, Exercise 76. 7. 8 From take and mu iltiply by Ans. 2 thirds of 45, 3 fifths of 20, 3, 54 3 fourths of 16, 5 sixths of 12, 9, 18 6 sevenths of 42, 7 eighths of 32, 5, 40 5 sixths of 36, 6 sevenths of 28, 9, 54 3 fifths of 55, 4 filths of 35, 5, 25 8 tenths of 50, 3 tenths of 60, 3, 66 6 sevenths of 28, 5 ninths of 36, 7, ■ 28 7 eighths of 48, 6 sevenths of 42, 9, 54 Sfifilisof 35, 3 eighths of 32, 12, 24 3 fourths of 44, 8 ninthspof 27, 6. 54 4 fifths of 30, 6 sevenths of 21, 8, 48 Jrithmetica I Froportioyu 125 ARITHMETICAL PROPORTION, Is the relation between two numbers with respect to their difference. Four quantities are in arithmetical proportion, when the difference between the^firstand second is equal to the diff'e- rence between the third and fourth. Thus, 4, 6, T, 9, are in arithmetical proportion, because the difference between 4 and 6, the first and second, is 2 ; and the difference between 7 and 9, the third and fourth, is also 2. ARITHMETICAL PROGRESSION, Is a continued arithmetical proportion, or it is a series oi numbers which increase or decrease by a common difference, as, 2, 4, 6, 8, 10, • 2, &c. ; or, 20, l6, 12, 8, 4, &g. The first and last terms are called extremes. In any series of numbers in arithmetical progression, the sum of the two extremes is equal to the sum of any two terms equally distant from them, or to twice the middle term, if the number of terms is unequal. Case 1. To find the sum of all the terms, whjsn the first term, the last term, and the number of terms, are given. liULE. Multiply the sum of the two extremes by half the number of terms, and the product is the sum of all the terms. Example. How many strokes does a clock strike in 1^ ;iiours ? Operation, 1 first term. 12 last term. 13 sum of the two ext^'^mes. 6 half the number of terms* 78 Ms. Case 2. To find the number of terms, when the first and last terms, dnd common difference, are given. Divide the difference of the extremes by the common dif- ference, add I to the quotient, ana it wiii be the number of terms. L2 12S Arithmetical Prognssion. Example. If a man gave his 3roungest child 20 dollars*, the next 40, and so on, increasing to the eldest^ who had 1 00, how many children had he ? Operation. 100 last term. 20 first term. com. diff. 20)bO difference of the extremes* 4+1=5, Ms. Casp', 3. To find the common difference, when the first and last terms, and number of terms, are given. Rule. Divide the difference of the extremes by one less than the number of terms, and the quotient will be the common diffe- rence. Example. A man had 10 sons, whose ages differed alike ; the youngest was 3 years old, and the eldest 48. What was the common difference ? Operation. number of terms, 10 48 greater extreme. 1 3 less extreme. divisor 9 )45 difference of the extremes. 5 Ms. Case 4. To find the last term, when the first term, the common difference, and number of terms, are given. Rule. Multiply the common difference by that number which is one lens than the number of terms ; then, if the series is in- creahing, add the first term to that product, and the sum will be the last term ; but if the series is decreasing, subtract that prv)duct from the first term, and the remainder will be the last term. Example 1. If a man travels 4 miles the first day, and 7 ihe second, and so on, increasing 3 miles each clav. ho^v far will he travel the 20th dav ? Geometrical Frogresnon, 1 S7 3 oommon difference. 19 one less than the number of terms. ' 57 product. 4 first term. 61 Ans* Example 2. If a man travels 6 1 miles the first day, and 58 the second, and so on, decreasing 3 miles each day ; how far will he travel the 20th day ? 3 common difference. €1 first term. 19 one less than the number of terms«> 57 product* 57 product. 4 Ans. GEOMETRICAL PROGRESSION, Is a series of numbers which increase or decrease by a ommon multiplier or divisor, called the ratio, as 2, 4, 8, 16, 32, 64, &c. which increase by the common multiplier 2 ; or, 486, i62, 54, 18, 6, 2, which decrease by the common divisor 3. In any series of numbers in geometrical progression, the product of the two extremes is equal to the product of any two terms equally distant fiW them, or to the square of the middle number, if the number of terms is unequal. Case J. To find the last, or aay other remote term ; the first term, he number of terms, and the ratio, being given. Rule. Involve the ratio to that power which is one less tbanthe lumber of terms, and multiply the power so found by the iirst term, and the product will be the term required. Example. A man hired a laborer for one year, promising to give him 2 dollars for the first month, 4 for the second, 8 for the third, and so on ; what was his wages for the last month ? Operation, The ratio is 2, which is to be involved to its 11th I'ower, because 12 is the number of the term sought. The 11th power ot*2 is 2048, which being multiplied by 2, the first term, gives 1096 for the 12th term, or tUe wages of the last month. Case 2. To find the sum of the series ; the first term, the last term, and the ratio, being given. 128 Mlgalion. Rule. Multiply the last term by the ratio, from the product sub- tract the first term, and divide the remainder by that number which is one less than the ratio, and the quotient will be the sum of all the terms. Example. According to the terms of the preceding ques- tion, what is the wages of the laborer for the whole year ? Operatioji. The first term is 2, the ratio 2, the number of terms 12, and the last terra 4096, as found by Case 1. Now, I multiply 4096, the last term, by 2, the ratio, and the product is 8192 ; tVom this I subtract 2, the first term, and the remainder is 8190. This is to be divided by that number which is 1 less than the ratio ; but as the ratio is 2, the number which is 1 less than it, is 1 ; and 8190 divided by 1, gtves 8190 oollars for the answer. Questions on the foregoing. What is arithmetical proportion ? What 'S arlthmetiQal progression ? What are the extremes ? i'o what is the sum of the two ex- tremes equal '' How do you find, the sum of all the terras ? JIow do you find the number of terms ? How do you find the common diffe- rence ? How do you find the last term ? What is geometrical progression ? To what- is the product of the ex- tremes equal ? What IS the rule for finding the last term ? How do you find the sum of the se- ries ? ALLIGATION, Is a rule which teaches how to mix together several in- gredients of differeat values or qualities, so that the mixture maj be of some intermediate value or quality. Case 1. To fmd the value or quality of the mixture, when the quantities, and values, or qualities, of the several ingredients ef which it is composed, are given. Rule. Multiply the quantity of each ingredient by its value or quality ; then say: As the sum of the quantities of the seve- ral ingredients, is to the sum of the several products ; so is any given quantity of the mixture, to its value. Example. A mixture being made of 5 lbs. of tea, worth f.v a ih, ; 9lbs, worth bs, 6d, a lb,; and I4^l&s. worth 5s. lOt^. a lb, : what is a lb. of it worth ? Operalion, lbs. (L 5 X 84=== 420 9 Xl02= 918 14-5X ro=ioi5 t?um of the -™« quantities, 28-5 )2353 sum of the product.«i> Alligation. IS^ Ih. d. lb. ! Therefore, 28-5 : 2353 :: \ i 82J(£.=65. lOJ+rf. dns. Explanation. I set doivn the quantity or number o^Bs. 5, 9, and 14§, ui a convenient coll mn for addition, expressing the halt* lb, by a decimal, for greater convenience. Opposite to each quantity, I set its value, reduced to pence, because the value of one sort was partly pence; 84, 102, and 70. I then set, in a third column, the several products ibrmed by multiplying each quantity by its value, 420, 918, and 1015. 1 then find the sum of the quantities, which is 2S'5lbs.f and the sum of the products, which is 2353 ; and say, a^ 28*5, the sum of the quantities, is to*?353, the sum of the products ; so is 1, the given quantity, to its value. Which proportion being worked out, gives 82rf. 2q., or 65. 10(/. 2q, for the value of a lb. ; which is the answer. v Case 2. When the values of several ingredients are given, to find how much of each will make a mixture of a given value. Rule. 1. Set the values of the ingredients in a column under each other, and the value of the mixture at the left, 2. Consider which of the ^;alues of the ingredients are greater than that of the mixture, and which less ; and connect each greater with one less, and each less with one greater. 3. See what the difference is between the value of eack ingredient and that of the mixture, and set down that difte- rence opposite to the value with which such ingredient is connected. 4. Then, if only one*difference stands against any value, that will be the quantity belonging to that value ; if more than one, their sum will be the quantity. JVote I. If all the given values of the ingredients are greater or less than that of the mixture, they must be linked with a cypher, JVo^e 2 Questions of this kind will admit of as many answers as there can be different modes of connecting the values, or of dividing them by a common divisor, or multiplying them by a common multiplier; for which reason they *re called indeterminate ^ or unlimited problems. Example. How much oats, a't 2s. 6^^., barley at 3s. %d** corn at 4s., and rye at 4s. %d. per bushel, must be mixed to- gether, that the compound may be worth 3s. lOrf. per bushel ? Operation, hush. 10 oats, ^ 2 barley, f ^^^^ 2 corn, f ^ \ ^e,^ 16. rye, J 130 %Blligation, Explanation, 1. I set dowo 46r?. the value of the mixture, and at the right of it, in a column, 3i)i. the vahie of the oats, 44d. that oi* the barley, 48(i. -|hat of the corn, aivJ 56d. that of the rye. 2. I con5>ider v'.^hich of them are greater, and which less than 46, the value of the inixt\ire, and connect them accordingly, each great- er with one less, and each less with one greater ; that is, 48, a greater, with 44, a less, and 56, a greater, with 30, a less. 3. I see what the difference is betvveen the value of each ingre- dient, and that of the mixture ; and I find that 16 is the difference between 30 and 46. So I >et down 16 opposite to 56, with which 30 is connected. 2 is tht difference between 44 and 4^ ; and I set down opposite to 48, with which 44 is connected. 2 is the diffe- rence between 48 and 46 ; and I set down 2 opposite to 44, witli which 48 is connected. 10 is the differenee between 55 and 46 ; and I set down 10 opposite to 30, with which 56 is connected. 4. Now, as I have only one difference opposite to each value, that is the quantity belonging to that value ; that is, there must be IG bushels of oats, 2 of barley, 2 of corn, and 16 of rye. Again: The operation ijaay be varied, and a different answei produced, by connecting the values in a different manner, as follows: d. d. hush. rso-^ 2 oats, "^ 44 ^ 10 barley, 16 corn. > Ans, J 6^ 2 rye, ^ And again, as follows : d. d. bush. f30-s:>v 10+S=12 oats, "^ [^56-^ 16 =16 rye, J And so on indefinitely. Proof. Case 1 and 2 prove each other. Case 3. When the whole mixture is to consist of a certain quantity Rule. Find the quantity of each ingredient, by Case 2 ; and then 3ay, as the sum of the quantities thus found, is to the giver quantity; so is the quantity of each ingredient thus found, U the quantity required of eacli. Example. How much oats, at 2s. 6d,, barley at 3s. Sd. corn at 4s., and rye at 4s. Sd. per bushel, must be mixed to- gether, to form a mixture of 90 bushels, worth Ss. 10c?. pei bushel ? Mligation. • r oucstions of this kind call* ^'d, rji^ why '. Of 'ow r.-'any arswei? do tl ey ad- mi> ? What is the ijiethod of proof? What is the third case ? The rule ? The fourth case ? i he rule i POSITION, Is a tnethod of perfornfiing certain questions, by the sup- position ot false numbers, by working with ^'i^iich/^the true numbers are found. SINGLE POSITION, Is that by which a question is performed by means of one supposition only. J\i"ote. Questions which have their results proportional to their suppositions, belong to this mle. P.ULE. Take any number, and perform the same operation with it, as is described in the question ; and then say, as the result of said operation is to the number taken, so is the result in the question, to the number sought. Example. A person, after spending | and i of his money, has ^6o left ; what had he at first ? - Operation. Suppose he had £ 1 20. Then, £. £. J of 120 is 40 and i of 120 is SO their sum is ^TO, which beingtaken from dei£0, leaves £'50. Then, 50 : 120 :: 60 : 14^£. dns. Proof, J of 144 is 48 i of 144 is 36 their sum is ^684, which being snbtract- ed from ^144, leaves ^60, as by the question. Position. 133 DOUBLE POSITION, Is that by which a question is performed by means of two suppositions. JYote Questions whicli have their results not propprtional to their suppo- sitions, belong to this rule. Rule. 1. Take any two numbers, and proceed with each of them separately according to the conditions of the question, as in ■ single position ; and find how much each result is different from the result mentioned in the question, calling these dif- ferences the errors, noticing also whether the results are too great or too little. 2. Multiply tlie first supposition by the last error, and the last supposition by the first error. 3. If the errors are like, divide the difference of the pro- ducts by the difference of the errors ; but if unlike, divide the sum of the products by the ^rn of the errors ; and the quotient will be the answer, or true number sought. A'^ote. The errors are said to be like, Avhen they are both too ,J^r(?at, or IboUi too little ; but ujilike^ when one is too great, and the other too little. Example 1. What nuir^ber is that, which being multiplied by 6, the product increased by 18, and the sum divided by 9, the quotient will be iiO .^ Operation. 1. Suppose it to be 18. Then, 18x6 is 108, and 18 added to 108 is 126, and 126 divided by 9 is 14. But instead of 14, it ought to be 20, according to the terms of the question ; therefore the error is 6 too little. Again : Suppose the number to be 30. Then, 30x6 is 180, and 18 added to 180 is 198, and 198 divided by 9 is 22. But it ought to be 20 ; therefore the error is 2 too great. 2. Next, r multiply 13, the first supposition, by 2, the last error, and the product is S6 ; and I multiply 50, the last sup- position, by 6, the first error, and tlie product is \ 80. 3. To know whether to take the sum or dlilerence of these products and errors, for division, I consider whether the er- rors are like or unlike. As one was too great and the other too little, they are unlike ; and 1 take the sums. The sum of 36 and 180, the products, is 216, which is the dividend ; and the sum of 6 and 2, the errors, is 8, which is the divisor. And 21 6 divided by 8, gives 27 for the true number sought. Trnof, 2*^x6 is l62, and 18 added to 162 is 180, and 180 divided by 9 is 20, according to the terms of the question. M 134 Fermutation. Example 2. A man left 10000 dollars to his two sons, one aged 11, and the other l6, to be divided in such a manner that their respective shares being put out at simple interest at 4 per cent per annum, should amount to equal sums when they come of age. " What are the shares ? Operation. 1. Suppose the youngest to have 4000 dollars ; then the eldest will have 6000. The interest of 4000 dollars, at 4 per cent, for 10 years, is 1600 dollars ; which makes the sum of the youngest 5600 dollars. The interest of 6000 dollars for 5 years, is 1200 dollars ; which makes the sum of the eldest 7%00 dollars. The sum of the youngest, therefore, is 1600 dollars too little ; which is the first error. Again : Suppose the youngest to have 4500 dollars ; then the eldest will have 5500; and the amount of their shares will be 6300 and 6600, which makes the sum of the youngest still too little by 300 dollars. 2. Next, the suppositions multiplied by the errors, are,. 40(0x 00=1200000, and 4500x 1600=7200 00 ; and the difference of the products is 6000000, which being divided by 1300, the difference of the errors, gives 8461 5-3846, for the share of the youngest ; and this subtracted from Si 0000, gives So384'6l54, for the share of the eldest. . Froof, To prove the operation, the interest of S4615-3846 for 10 years, at 4 per cent, is gl846*15o8, which added ta the principal, makes the amount S946 1-5384 ; and the in- terest of ]S5384'6 1 54 for 5 years at 4 per cent, is S 1076^9^30, which added to its principal, makes the amount S6461'5384 ; so that the sums are equal, according to the terms of the question. Questions on the foregoing. What is position ? What is single position ? W hat questions belong to single po- sition ;' \\ ]\Rt is the rule ? W'hat is d-mble pcsitton ? What questions l)(,4ong to this rule ? What is the first thing to be done ? When are the errors said to be like or unlike ? When you have found the errors, wliRt is to be done next ? If the errors are like, what is your divisor ? Your dividend ? It they are unlike, \Aiht ^ PERMUTATION, Is the changing of the position or order of things, or the showing of how many diSerent ways they may be placed. Combination, 135 Rule. .Multiply all the terms of the natural series together, from 1 up to the givea number, and the last product will be the answer. Example. How many days can 7 persons be placed in a different position at dinner ? Operation, 1x2 is 2, and 2x3 is 6, and 6x4 is 24, and 24x5 is 120» and 120x6 is 720, and 720x7 is 5040 ; which is the answer. m^ COMBINATIONT, IP Is the showing of how many diiFerent ways a less number of things may be combined out of a greater. Rule. 1. Multiply all the terms of the natural series together, from 1 up to the number to be combined, and make this pro- duct the divisor. 2. Take another series of numbers, of as many places, be- ginning with the number out of which the combination is to !)e made, and decreasing continually by 1 ; and multiply them t(»gether for a dividend. 3. Divide the dividend by the divisor, and the quotient will be the answer. Example. How many combinations can be made of 6 let- ters out of 10? Operation. 1x2x3x4x5x6=720, divisor; 10x9x8x7x6x5==15l£00, dividend. Then, 151200 divided by 720, gives 2 10, for the answer. Questions on the foregoing. What is permutation ? 1 1 How do you form your dmsor ? What is the rule j" |j How, your dividend \ What is combination ? What is the answer ? Exercise 77. Divide 2 iuto 3 such parts, that the sum of their squares shall be 1^, Ans. 1, §, and §. Divide 1 into 3 such unequal parts, that the sum of their squares shall lack | of being i. Jlns, |, J, and |. Divide 2 into 3 such parts, that they shall have the ratio of 5, 2, and 1. Jins, \\, |, and ^. Divide 2 into 3 such parts, that they shall have the ratio of 6, 3, and 1. Ms, U, }, and j. Divide 2 into 3 such parts, that the sum of their squares «hall lack ^ of If. Ms, 1, f , and §. V * 136 Simple Interest by JDeciniah, Divide 2 into 3 such unequal parts, that the sum of their squares shall be | more than H. Jlns, 1, f, and ^. Divide 1 into 3 such unequal parts, that the sum of their squares shall lack J^ of i, Jlns, f , a, and i. Divide 1 into 2 such part«, that the difference of their squares shall be §. Ans, |, and §. Divide I into 2 such parts, that the difference of their squares shall be \, Jlns. |, and ^. Divide 5 into 2 parts, in the ratio of 1, and 2. Jlns, If, and 3 J. There are 2 numbers, and the difference between their sum, and the sum of their squares, lacks y^ of being i ; what are the numbers ? Jins, i, and %, Divide a shilling into 2 parts, so that one part shall be one farthing more than the other. Jlns, ^\d, and 5|6?. Divide 3s. Ad, into 2 parts, so that one part shall be 2\d, more than the other. Ans. ]s, 9^d and Is. 6fc?. Divide a dollar into three parts, so that the largest shall be 8 cents more, and the smallest 7 cents less, than the mid- dle part. Jins. 41 cents, 33 cents, and 26 cents. SIMPLE INTEREST, BY DECIMALS. Mdtio is the simple interest of ^1. or i dollar for 1 year, at any given rate, expressed as the decimal of a ^. or a dol- lar. Thus, 5 per cent, is -05 ; six per cent, '06 ; six and a half per cent, '065, &c. Case 1. The principal, time, and ratio given, to find the interest. Rule. Multiply the principal, time, and ratio, continually toge- ther, and the product will be the interest. Example. What is the interest of 365^. 5s. for 10 years and 6 months, at 6 per cent ? Operation, The principal is £365*:25, the time is 10'5z/r. and the ratio is '06, which being multiplied continually to- gether, tiie answer is ^230-1075, or ^230 .. 2 .. 1|. J\ ate. To find the amount, add the interest to tlie principal. Case 2. The amount, time, and ratio given, to find the principal. Rule. Multiply the time by the ratio, and add 1 to the product for a divisor, by which divide the amount, and the quotient will be the principal. Simple Interest by Decimals. 137 Example. What principal will amount to SSIOL, in 6 years, at 4^ per cent ? Operation* The time is 6, and the ratio is '045 ; and their product is -27 ; to which I being added, the divisor is 1'27. And the amount, ^38 1(, being divided by 1-27, the quotient is ^3000, which is the answer. JYote. This case is the same as discount. The principal found, being the same as the present worth. Case 3. The amount, principal, and time given, to^ find the ratio. Rule. Subtract the principal from the amount, the remainder will be the interests Divide the interest by the product of the time and principal, and the quotient will be the ratio. Example. At what rate per cent will 543 J. amount to 705^. 18s. in 5 years ? Operation, Principal, 543 amount, 705*9 time, 5 principal, 543* 2715) Int. l62-90(-06 ratio of 6perceiit 162-90 Ms. Case 4. The amount, principal, and rate per cent given, to find the time. Rule. Find the interest, and divide it by the product of the prin- cipal and ratio ; the quotient will be the time. Example. In what time will 543d£. amount to 705^. 18^, at () per cent ? Operation. Principal, 543 amount^ 705*9 ratio, '06 principal, 543* 32-58) Int. 162-90(5 years. Ms. 162-90 COMPOUND INTEREST BY DECIMALS. Ratio is the amount of ^l, or SI, for 1 year, expressed m a decimal form. Case 1. The principal, rate, and time given, to find the amount. Rule. 1. Involve the ratio to such a power as is the same witk flie number of years. M2 138 Vomponml Interest hj IJccimah. 2. Multiply the power so found, by the principal, ar^l the product will be the amount. J\oie. Having t'oui^d the amount, subtract the pi mcipul from it, v. ^ Iht remainder will be the compound inteiesi. Example. What Is the amount of ^3^-1 for 4 yeai - it 5 per cent, compound interest ? Operation. The ratio is 1*05 ; of whitL the -th power (be- cause 4 is the nuniber of years) is l'2l550^> 5 ; and this being multiplied by 300, the principal, gives ^^36 4* 65 1875, for the amount, which is the answer. ^Case2. The aiiiount, rate, and tiiue given, to find the principal. EULE. Divide the aBiouat given by the ratio, involved to such power as is the same as the given number of years, and the quotient will be the principal. Example. What principal, at 5 per cent, compound inte* rest, for 4 years, will amount to^364*65 1875 ? Operation. The ratio is L\'05, and its 4th power Ll- •21550625, and :^64-651875 divided by i-£l 550625, gives SOoA. for the answer. r' J^te Tliis case is the same as discount at compeund iriterest, the principal found, being the same as the present wortli* Cas:3. The princ pal, rate, and aitiount given, to find the time. Divide the amount by the principal, then involve the ratio till it equals the quotient, and the number (if involutions will be the same as the number of years. Example. In what time will jL4oO amount to X520 93125^, at 5 per cent, compound interest ? Operation. The amount, L520«93125, being divided by X450, the principal, gives M 67625, for the quotient. The ratio, 1*05 involved to the 3d power, is 1-157625, which, equals the quotient* So the arlsw^er is 3 years. Case 4. The principal, amount, and time given, to find the rate per cent. Rule. '' Divide the amount by the principal, and extract such root of the quotient as is denoted by the number of years; which "oot will be the ratio. Example. At what rate percent will 45(DX. amount to /.o'20'93i25j in three years ? Annuities at Simple Interest 1 39 Operation. 5':>0-93I26 divided by 4.'.0, gives M57625; vA the cube root ot M576i5, is l'05, the ratio of.') percent, QlESTIONS ON THE FOKEGOING. In couipOMud interest by decimals, what is meai.i by the 7'atio ? How d»}ou find the amount? The compouiid interest ? i he principal ? the time ? the rate per cis it ? Which case is the same as discount, and why ? In sim])le interest by decimals, what is meant by ratio '^ How do \ou find the i.iteiest ? How, the pricipal \ flov, the ratio ? The time ? Which case is the same as discount, and why ? ANNUITIES. An annuity is a sum of nionej payable every year, for a number of years, or forever* When the annuity is not paid as it becomes due, it is said to be in arrears. AVhen the annuity is not to begin till after a certain time has elap ed, it is said to be in reversion. The sum of all the annuitiers in arrears, together with the interest due upon each, is called the amount. If an annuity is to be bought otf, or paic^ all at once, the price which ought to be paid for it, is called the present worth. ANNUITIES AT SIMPLE INTEREST. Case 1. To find the amount of an annuity at simple interest. Rule. 1. Make the first term, and 1 the common difference, of a series of numbers in arithmetical progression, and make the number of terms one less than the number of years ; and find the sum of the series. 3. Multiply that sum by one year's interest of the annuity, and the product will be the whole interest* 3. Multiply the annuity by the number of years, and add the whole interest so found, and the sum will be the amount sought. J\%te. The reason for making the number of terms in the arithmetical ae- rie s o),e less ihai^ the number ot years, is, that there is no interest due upon the last ) ear's an; uity Example. What is the amount ^of an annuity of 700 dol- lars for 6 years, allowing simple interest at 7 per cent ? Op, ration. 1+24-3+4+5=15 sum of the series. 49 i te. e^t of 7(''0 dollars for I year. 7^->5 whole interest. 6x7QO= »=42eo six annuities. S4935 amount. 148 Annuities. J\''ote. The reason of this operation will appear, if we eonsider that at the end of G years, there is due tlie first year's annuity, 700 dollars, and its in- terest for 5 years, that is 5 times 49 doilars, wh)ch is '245 dollars ; the second year's annuity, 700 dollars, and its interest for 4 years, 196 dollars j ttie third year's annuity, 700 dollars, and its interest for 3 years, 147 dollars; the fourth year's annuity, 700 dollars, and its interest for 2}earf, 98 dollars; the fifth year's annuitj , 700 dollai-s, and its interest for 1 year, 49 dollars; and the sixth year's annuity, 700 dollars : all which added together, makes 4935 dollars, as before. Case 2. To find the pres«it worth of an annuity at simple interest. Rule. Find, as in discount, the present worth of ea«h payment by itself, allowing discount to the time it becomes due, and the sum of all these will be the present worth sought. Example. What is the present worth of an annuity of 100 dollars, to continue 5 years, at 6 per cent per annum, simple interest r Operation. [annuity. 94'SS96, present worth of the 1st year's 89' 857, do. 2d do. 84-74 5 r, do. 3d do. 8-6451, do. 4th do. 76-9230, do. 5th do. 106 : 100 : : 100 ll^Z : 100 : : 100 118 : 100 : : 100 124 : 100 : : 100 ISO : 100 : : 100 Ans, S425-9391, present worth of the whole. ANNUITIES AT COMPOUND INTEREST. Case 1. To find the amount of an annuity at compound interest. Rule. 1. Make 1 the first term of a series of numbers in geome- trical progression, and the amount of L\ or gl for 1 jear, at the given rate per cent, the ratio, expressing it in decimals. 2. Extend the series to as many terms as the number of years, and find its sum. 3. Muldply the sum thus found by the given annuity, and the product will be the amount required. Example. What is the amount of an annuity of 200 dol- lars, for 5 years, allowing compound interest, at per cent ? Operation, The first term is I; the ratio is 1-05. The first term being multiplied by the ratio, gives f05forthe second term; and that being multiplied by the ratio, gi^es 1-025 for the third term ; and that being multiplied by the ratio, gives )-15762;) for the fourth term ; and that being multiplied by the ratio, gives 1-21550625 for the fifth term. •Annuities. 141 The sum of these five terms is 5-52563125, which being mul- v tiplied by 200, the annuity, gives 81105' 12625 for the answer. J\''ote. '1 o find the amount for additional parts of a year. Having found j the amount for the whole years, find the interest of that amount for the givea parts of a year, and add it. Case 2. To find the present worth of an annuity at compound in- 1 terest. I Rule. 1. Take the amount of Xl or gl for 1 year, at the given rate per cent, and involve it to that power which is the same as the number of years, for a divisor. 2. Divide the annuity by this divisor, subtract the quo- I tient so found from the annuity, and set down the remainder X for a second dividend. » S. From the amount of L\ or Si for 1 year, subtrac t 1 and take the remainder for a second divisor. 4. Divide the second dividend by the second divisor, and the quotient will be the present worth required. Example. What is the present worth of an annuity of 40 dollars, to continue 5 years, discount at 5 per cent per an- num, compound interest ? Operation, 1^ The amount of 1 dollar for i year at 5 per cent, is 1*0 5. This being involved to the 5th power, because 5 is the number of years, is 1-3762815625, which is the first divisor. 2. The annuity, 40 dollars, being divided by 1-2762815625, gives 3 1-S4 104, for the first quotient ; which being subtracted from 40, the annuity, leaves 8-65896, f&^r the second dividend. 3. From 1-05, the amount of 1 dollar for 1 year, I subtract 1, and the remainder is -05, which is the second divisor. 4. The second dividend, 8-65896, being divided by -05, the second divisor, gives S 173* 175 2 for the present worth, which is the answer. JSfote. To find ihe present worth for additional par>0341+. Consequently, A's is the most valuable by 75 dollars, 93 cents, 1 mill, and 7 tenths of a mill. Questions on What is an annuity ? When is it said to be in arrears f When, in reversion ? What is the amount ? What is the present worth ? What is the first case of annuities at simple interest ? In working that case, what is the first thing to be done ? The second ? The third ? *i Of how many terms do you make 1 your arithmetical series ? I Why so? What is the second casr: ? the rule ? What is the first case of annuities at compound interest ? In working that case, what is the first THE FOREGOING. thing to he done ? The second? The third ? How do you find the amount for ad- ditional parts of a year ? What is the second case ? In working the second case, what is the first thing to be done ? The second ? the third ? the fourth ? How do you find the present worth of an annuity in rcA ersion ^ How, for additional parts of a year ? What are perpetuities ? ^^ hat is the first case ? The rule ? What is the second case ? In working the second case, what is the first thing to be done ? The second ? the thiid ? MENSURATION. A superficies, or surface, is that which has length and breadth, but not thickness. It is called a, plane supnficies, when the surface is even, without any curvature ; that is, when it is such, that if you take any two points in the surface, and draw a straiglit line from one point to the other, the whole of that straight line will be in the said surface. The area is the whole surface enclosed. 144 Mensuration. A'^ote. In measuring the length of a road, the chain or measuring rod must fee kept parallel with ihe surface of the ground, however irregular; because the traveller cannot move horizontally, but must go up and down all the hills. But in measuring land, the measuring rod must be kept parallel to the hori- zon, or upon an exact level, the reason for which is. that all the calcula- tions of the quantity of land, are calculations of the areas of plane figures. And they ought in justice to be so ; because, although a piece of ground which has a hill in it, has, in reality, more surface than if the hill was re- moved, and it was reduced to an exact level , yet nothing more can grow upon it; for the stalks of grain always shoot up in a direction perpendicular to the horizon, and not perpendicular to the surface of the soil. A plane triangle is a figure bounded by three straight lines. Every piece of ground, bounded by right lines, may be di- vided into triangles. B Example. Let ABCDEF, be an irregular field of six sides. Draw a line from A to C, from C 1o E, and from E to A, and it will be uivided into four triangles. Problem 1. To find the area of a triangle* Rule 1. Measure one side of the triangle, and also mea- sure the perpendicular distance from that side to the opposite angle. £. Multiply these together, and half the product will be the same. JVote I he most convenient instrument for measuring land, is the chain. As 160 ^uaie poles make an acre, the chain is made 4 poles, or 66 feet in' length ; and so, 10 square chains make a>i acre i he chain again is divided into 100 links And the account being kept in chains and decimals of a chain, is reduced to acres and decimals of an acre, by removing the decimal point one figure to the left. Example. Let ABC be a triangular field. Let the side AB be ^5 chains and 75 links, and the perpendicular CD 1 6 chains and 25 links. How many acres does the field contain ? Operation. 25-75x 16-25=418,4575, and half of that, or 209-21875 is the area in chains ; which gives 20-921875 for the area in acres, M^iich is the answer. Where a chain cannot be had, a pole may be used, 16j feet long, which is the length of a pole, rod, or perch. The Mensuration, 145 fbove field being measured by such a p©le, the side AB Would be 103 poles, and he perpendicular CD 6s poles, i These multiplied together, make 6695 ; and the half of that i is 3347 1, which is the area in poles: which, divided by 160, / the number of square poles in an acre, gives, as before, ) 20*921775 acres, or 20 acres, 3 roods, and 27^ poles. .' J\/ote. I o finO the point i 1 the side \B, wheiv the perpendicular from I the a'tgle C will taU : Maki- a cross of two pu;c s f wood, which shall cross \ cacli otlierat eight angles, (thus f ;) whicli may eSsily be done by the help of H coinmoii carpv-nter's square Lay the cross ujion the line AB. so that ot«eof the piece> -hall coincide with ihat line; and then move it along that line, tiil the oihe- piece shall point to 'he angle C ; and the point where the cross stands, will be the point where the |te!j)C' dicnlar falls. 1( is not sup- posed iliat lh;s method wid give the point exaciiy, hiu near enough for coni- iiion ])in'poses. Where a perpendicular cannot be conveniently measured, work by Rule 2. 1. Measure all the sides of the triangle, add them together, and take half the sum. !2. Subtract the sides, one by one, from the half sum S9 found, and note the three remainders. 3. Multipl;y the half sum and the three remainders all to- - 'gather, and the square root of the last product will be die area. Example. C A Let ABC be a triangle, of whicli the side AB B is 103 pole^, BC 77 poles, and CA 9. poles ; what is the area ? Operation. 1034-77+90 is 270, half of which is 135% Next, 103 from . 35 leaves 32 ; 90 from 15 leaves 45, and 77 from h 35 leaves 58. I hen, 135x32x45x58 is 11^75^200; the square root of which is 3357'8, for the area in poles, or 20 acres, 3 roods, and ST*8 poles. Problem 2. To find the area of a field which has four sides parallel to each other. Rule. Multiply the length by the breadth, and the product will be the area. Example. ABE F Let ABCD, or EFGH, be a field of four parallel sides, to find the area. 13 I H If the angles are right ^ngle^, as in the figure ABCD, AB or CD will be the length, N s/ru 1 46 Mensuration* and AC or BD the breadth. But if the angles are not rio;ht angles, as in the figure EFGH, measure a perpendicular from one side to the other, as EI, and this will be the breadth. Let AB be S.. chains, and AC 14 chains ; what is the area? Operation- 32x14=4^8 chains, or 44* acres. Jins, Again, i et EF be 32 chains, and El 14 chains; the an- swer is the same. A^'ote If the sides are ?i#t parallel, divide the field into triangles, as be- fore; for if J oil multiply the length by the breadth, you will not have the true area. Problem 3. To find the art' a of a circle, i-ULE. Square the diameter, and multiply that square by the de- cimal '7854, and the product will be the area. Example. If a rope, 3 rods long, be tied one end to a horse 'ji head, and the other end lo a stake, how great an area of grass can he eat? Operation, As 3 rods is ihe radius, or distance from the centre to the circumft'ren<:e, 6 rods is tht dian»eter of the circle. This squared s 36 ; and that n ulHplied by v854, is 2b'Jsi744 rods or polt s, the area of the Ciicle. Problem 4. To faid the diameter, or the circumference, one from the (Aher, RuLp. As lis is to 355, so is the diameter to the circum- ference. Example. What is the circumference of a circle, whose diameter is 315 rods r Operation. 113 : 355 : : 315 : 98;>6-f Jns. pROBi.EM 5. lb fmd the relative proportion of similar figures. Rule. The areas of similar figures are to each other a& the ; quares of their similar dimensions. Example. If a rope 3 rods long allow a horse to graze 28* 744 rods of ground, how long must a rope be to allow him to graze an acre r Operation, 28-t744 : 9 :: 160 : 50-9 94. That is, as the first area is to the square of the length of the rope, or ra- dius of the circle, so is the second area, to the square of its radius; which being found to be 50*9 9 , its square root is 7* ! 3-f rods, which is the length of the rope required. Problem 6. To measure the ; ei^ht oj a tree, or other object. Rule. Set up a pole perpendicularly, the lengTh of which above the ground is krtown. Go to the foot of the tree, and aaake amark in it at the height of your eye from the ground. Mensuration. 147 and make a mark in ole and the top of the tree, and also in a range with the marks in the pole and in the tr^ee. Measure the d stance from that station to the toot of the pole, and also to the foot of the tree. And then say, as the distance from your station to the foot of the pole, is to the he ght of the pole above the mark ; so is the distance from your station to the foot of the tree, to the height of tie tree above the mark. Then, add to the height so found, the distance from the mark to the ground, and the sum will be the true height of the tree. Example. ^Kk B Let AB be a tree, the height of which ^R^^^ yj is to be measured. Let CD be an up- ^^^^K> y/^' right pole, 15 feet above ground. Let ^HHK-rv y/^ E be the mark in the tree for the height ^^^ ^y^ of your eye, ^\hich suppose to be 5 feet; ■K' j/\ and F the maik in tlie pole, of the saute height. Thtn, FD, the part of the pole above the mark, will be 1 feet. Let G be the place of your eye, which is in a range with D and B, the top of tlie pole and the top of the tree ; and let II be the s^t^tion, or place where you >tand. Let tlie distance from H to C mea- sure 5 feet, and from H to A 45 tV et. Then, I ,. HC : F!) : : HA I 15 : 10 : : 45 : \ /the tree above the mark; and ! height of the tree. J\^ote. ff the tree is not perpendicular, but leans, let ilie pole be placed paral'el to it. and tiie same process will ^ive its 1 ngtb, ^H^ Problem 7. To i/ieasitre the breadth of a river, ^^E' FULE. ^H B Take any station, A, on one side; and se- ^H; A lect any object, B, on the other side^ oppo- ^^' ^ site to A. Measure back any distance,- to C, in a range with BA, and note the dis- tant e. At any distance from A, take a E/ _ j point D, and also such a point, E, as shdll be in a range with BI), and so that EC shall be parallel te DA. From D, make DF parallel to AC, and it will be of the same length ; and measure EF and DA. Then say, EF : FD : : DA : AB, which is the breadth of the river required. EB, that is, > , which is the height of 30-f-5 is 35 feet, the true Da 148 Mensuration, Example. Let AC be 20 rods, and of course FD is 20 rods. Let EF be 12 rods, and DA 18 rods. Then, EF : Fi) : : DA : AB ; thiit is, 12 : 20 : : 18 : SO rods; which is the breadth of the river. Definitions. A solid is that which has length, breadth, and thickness. A euhe is a solid bounded by six equal squares. Problem 8. Tujtnd the solid content of a laad of wood, HuLE. Multiply the length by the breadth, and that pro- duct by the height, and the last product will be the content. J^Tote The solid content of a stick of squ;ired timber, the height and breadtk of which do not vaiy froni'oue end to the other, is found in he satno manner. But if it tapers regularly throug)i»«ut, it is the frustum of a p)ramiii ; and its solid content may be found by the following Rule. Add into one sum the areas of the two ends, and the square root of th ir product ; and take one third of that sum for the mean area, which being multiplied by the length of the frustum, will give the solidity. Example. A stick of squared timber measures as follows: At the butt end, 14 inches by U' ; at the small end, 10 inches by 8 ; and the length is 20 feet. What is the solid content ? Operation, 14 x 12=168 inches is the area of the butt end, and 8x10=80 inches is the area of the small end. I'38x80 58=1.3440, and the square root of that is 1 1 5*93. And t^8-f- 80-1- 1 1 5*9 ) is 363-93, of which one third is 2 '31, which is the mean area in inches. This multiplied by the length, 240 inches, gives 29 1 14-4 inches, or 16 feet, 1466*4 inches, for the solid content. Pkoblem 9. To find the superficial content of a right cone. Rule. Multiply the circumference of the base by the slant height, and to half the product add the area of the base, and it will give the superficial content. JVote The superficial onttnt of a ri}2:ht pyramid, is foiind in the same jnanner, the slant height h«*ing measured oo a line let down from the vertex })erpttndicularly upon the base of the triangle which forms one side of the pyVamid. Pro B lem 1 0. To jind the solid content of a right pyram id, or cone. Rule, Find the area of the base, and multiply that area by the perpendicular height, and one third of the product will be the solid content. Problem li. To find the relaiive proporti^xn of similar $$lid^ II Mensuration, 149 Rule. Similar solids are to each other as the cubes of their similar dimensions. Example. If a cone, the diameter of whose base is 3 feet, contains 100 solid fe^t, how many solid feet Mill a similar cone contain, the diameter of whose base is 6 feet ? Operation, The cube of 3 is 27, and the cube of 6 is £16 ; therefore, 27 : 216 :: 100 : 800 feet, Jns, ProblExM 12. Tojind the solid content of a cylinder. Rule. Multiply the area of the base by the height, and the product will be the solidity. J^''ote. A stick of round timber, ot'tlie same diameter throughowt, is a cy- linder. If the stick tapers then it is ihe frustum of a cone ; and its soliri •ontent may be found in the same mantier as the solid content of the frus- tum f)f a pyramid See note to prohkni 8. Problem 13. Tojind the surf ace af a globe or sphere. Rule. Multiply the circunjference by the diameter, and .the product will be the superficial content. ^ ** Problem 14. To find the solidity of a globe or sphere. Rule. Multiply the surfaoe by the diameter, and one sixth of the product will be the solid content. Problem 15. To find the capacity of a cask of the usual form. Rule. Add into one sum 39 times the square of the bung diameter in inches, 24 times the square of the head diameter, and 26 times the product of those diameters ; multiply that sum by the length of the cask, and that product by '00034 5 and the last product divided by '?, will give the content ia wine gallons, and by 1 1, in ale gallons. Example. What is the capacity of a cask, of which the head diameter is 27 inches, the bung diameter 33 inches, and the length 36 inches ? Operation. 33x33 = 1089, and 1089x39=42471 27x27= 729, and 729x24= 7496 33x^7= 891, and 89lx2*=23166 83133 And 83133x36=i?992788, and ^9927fc8x-0C0S4=l0I7 •54792, and 1017-5479 '^--9 = 1 13'060S8. So, the answer is 1 13'0"088 gallons, wine measure. Problem 16. To find the tonnage of a ship. Rule. Multiply the length of the keel in feet, by the breadth of the beam, and that product by half the breadth of the beam ; and divide the last product by 95 ; the quotient will be the number of tons. N2 lot Mensiiratioiu Problem 17. To find the solid content of an irregular body. Rule. Put it into any cylindrical or cubical vessel, and fill the vessel with water, sand, or any other convenient sub- stance. Then take out the body, and measure the space left empty in the vessel by its removal, according to the preceding rules. J\^ote. It was by the help of this rwle that. Archimedes discovered the cheat that was practised upon Hiero, km^ ofSyiacuse, respecting- his crown. He hud dii-ected a crown of pure gold to oe tr^aoe ; but suspected the woik- man Ijad mixed alloy with it. He therefore requested Archimedes to ascer- taii) the fact, without injuring the c.own. Archimi-des took a muss of pure goUl, and another of alley, eacli equal i - weight to the crown ^ and puiting e^ch separately ii.t<: a vessel fil'ed with waer, observed the quantity of water cxpi-lled by each ; from which he ascei'taii-ed their respective bulks, and the qua;. lit) of gold and alioy which w^^re mix< d in the crown. Supp<'^e the weight of the crown and of each nr.-ss to be 10/^,9. ; and \].itt, on being put into watei-, the allo} expelled -O'i/i?) , the tjoltv, '5'Zlb ^ arWJ the compound, -Gi/o. Then, by case 3d of ailigation, the proportion of gold and liltoy may be found, as foiiows . •92. -12 of alloy, } =i*40 ; but there ought robe •52^ -28 of gold, > !0^6.; therefore, :: 12 : SlL of alloy, and 28 : 7^&. ofgold, S ^^'^' Problem 18. To find what weight may be raised by any poiver with a lever. Rule. As the distance between the weight and the prop, is to the distance between the prop and the power, so is the power to the weight it will balance. Example, if a man, weighing ioOlhs. rest on the end of a lever lO feet from the prop, what weight will he balance at the other end of it, 1 foot from the prop ? 1 : 10 :: l/>0 : l5 '0 Lbs. Ans. sJVote No allowance is here madf; for the weight of the lever, whicli ought to be done in order t© obtain the exact answer Problem \ 9- To find what weight may be raised by any power, with the wheel and axle. Rule. As the diameter of the axle, is to the diameter of tlie wheel, so is the power to the weight it will balance. Problem ^.0. To find what weight may be raised by any poiver, with a screw. Rule. As the distance between the threads of the screwy is to the circumference described by the end of the lever, so is the power to the weight. JS'^Qte. One third of the eifect ©i'this macliine skould be abated for frictiw. Mensuration. 151 Problem 21. To find what weight may he raised by any jpowevt with a pulley. Rule. If the pulley is fixed, the power and the weight arc equal ; but if the pulley is movi?ble, as l to the number of ropes, so is the power to the weight. Problem 2:2. To measure any height, by the time a heavy body would fall from it to the gTGiind. Rule. In the tirst sec jnd, it would fall l6 feet, and in the next 48, and so on, with a velocity uniforn.ly increasing. Therefore, as I is to i 6, so is the scjuure of the number of seconds, to the number of feet through which the borly fulls. Example, if a bullet falls from the t)j» ot a steeple in S seconds of time, what is the height of the sleepier 1 : 16 : : 9 : J44 feet, Ans. Question, How deep is a chasm, into which, if you drop a stone, it will be 10 seconds before you hear it strike the bot- tom ? Operation, Part of the time is occupied by the f^illitig of the stone, and part by the return of tlie sound after the stune strikes the bottom. To ascertain which, I work by double position, as follows : First, I suppose the depth to be 1250 feet. Then, by the above rule, l6, : 1 :: 12i0 : the square of the time occu* pied by the stone's falling ; which proportion being worked out, gives r?>*l25 for the square of the time ; and the square root being extracted, is 8*838-f- seconds, for the time of fall- ing. This being tak^m from 10, the whole time, leaves 1* i ( 2-f for the sound to return. And as sound flies 1-12 feet per second, I multiply 1*1 62 — by 1'a% and it gives 1327*004 — feet, tbr the distance it returns. But it ought to be *»nly 1250, by the supposition ; consequently the first error is 77-004— too great. Secondly, 1 suppose the depth to be 1260 feet, and proceed in the same manner, and find 7S'75t for the square of the time of the fall ; and 8-874+, its square root, is the time; which taken from 10, leaves 1- 25-f, for the time of the return of the sound; and this multiplied by 1142, gives 1284-75+, for the distance it leturns, which ought, by the supposition, to be only 1260 ; so, the second error is 24-75+ too great. Next, 1250x24-76 is 30957-5, and 1260x77-004 is 97025-04, and their difterence is 66087-54, which being divided by 52-254, the difference of the errors, gives 1264*73+ fett, for the answer or depth cf the chasm. 152 jfppendix to JSTumeration. "^l?? I J ■§ S f 5 •sp^apimii t^ I To 1 1 « cr '2 ^-^ COO •SpUUSUOIl^ ^ 3 53 ^ 2 .gcc-«QiC:ra3^ .spTivjsnoii^ |() sua} ^ "^ * ? 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"^ i 2 «• nl S -inoojo -nori} |0 su,a -? ^ i 1 1 -t^ 'i . -^ . - c« - w 'jpoojo -nom j«> punq ^ '*'~~i a^ 5S 'SiiotutuQ^i Q^ ^?2^«.2i5i ''•^'^•*'' '"""? •^" P"""^ - ^ ;:?^ ^ -^ ! I PART IT. J\1ote. The questions which follow, are intended for the practice of thofie who are put suing the studv of Arithmetic. As soon as the leafier has been sufficiently extrcised in the questions in Pari I. under simple additio. , in ihe mannei* there directedi he should be put upon performing these; cure beij>^ taken however, that he should proceed in learning ihe rules in Part I. as fast as he proceeds in performing the quesiious in this Ihe first 600 ques- tions go through all the rules in the book, being qutstio^s ot ihe most simple form. 'l"he Instructor of the school should fur- ish himself with a Key, con- taining the answer to each qu' stion annexed to its number, so that wb n a question is performed by any scholar, he can see at f»uce whether the a; sh r is right If not right, the scholar should be s t to wotk it out 'gain, and not be told how, till he hss mad^^^ sufficient trial of his own skill If the school is large, and the txamination of questions should be tr«u^ les- me to the Instruc* tor, Monitors may be appointed to do it ; ai^d the dfff. rent parts of the Key may be put into their hands for that pui-pose I'o prevent scholars front copying their answers from ^^ach other they should be pt evented riom kecp- ingthetii, and be directed to show the work of each question to the Instructor or Monitork ^should th« Instructor think aecessar>, he can diiect a scholar to go over the first 600 q«estions a second time, before !ie iroceed furtiier. The questions from No. 601, to the end, are on all the rules promiscuously, s©me of the ino»t difficult being placed towards the last. QUESTIONS. No. 1. The Old Testament is divided in the following manner : The Pentatf^uch, containing five books ; other his- torical books, 12 ; the Hagiographa, 5 ; and the Prophets, i7. How many books in all ? 2. The New Testament contains, the Gospels, 4 books ; Acts of the Apostks, 1 ; Epistles of Paul, 14 ; of thi- other Apostles, 7 ; and the Revelation, 1. How many in ail ? 3. How many books in the whole Bible ? 4. In the Pentateuch, there are i 87 chapters; other histo- rical books, ^^49; Hagiographa, 243 ; Prophets, 250. How many chapters in the Old Testament ? 5. The Gospels contain ^9 chapters ; Acts, ?8 ; Epistles, 121 ; Revelation, 22. How many chapters in the New Tes- tament ? 6. How many chapters in the whole Bible ? 7. In the year 1821, the number of onlained missionaries employed among the heathen, was as follows : By the Society in England for propagating the Gospel, 1 ; Soc. for promoting Christian knowledge, 3 ; Danish Mission College, 2 ; Mo- ravians, 68 ; English Methodist Missionary boc. 74 ; English 154 ^vesfions. Baptist Miss. Soc. 28 ; London Miss. Soc. 85 ; Scotch Missi, Soc. 7 ; Eng;lish Church Miss. Soc. 32 ; London Jews' 8oc. 6 ; En^ish Soc. for conversion of Negro Sldves, 6 ; Amerscaa Board for Foreign Missions, 24 ; American Baptist Do. t ; United For. Miss. Soc. 7 ; American Methodist Miss. Soc. 1. How many in all ? 8. The number of graduates at the Colleges in New-Eng- land, in the year U>}0, was as follows: Harvard, 6i ; Yale, 54; Dartmouth, 26 ; Williams, 28; Brown, ; Burlington, 17 ; Middlehury, 9 ; Bowdoin, ir.. How many in all? 9. From the creation to the flood, was 165 years; from tliat to the call of Abraham, 427 ; from that to the departure of the Israeiitesout of Egypt, 480; from that to the building of the temple, 479 ; from that to the founding of Rome, 266 ; from that to the birth of Christ, 74-8 ; and from that to tl\e eonnnencement of the Christian era, 4 ; and from that to the pre?ent year, ' h22. How long since the creation ? 10. The follow ng sums were subscribed to the American Bible Society, in a few days after its formation, to wit : By Elias Boudinot, ten thousand dollars ; John Langdon, 400 ; Robert Oliver, 300-; Matthew Clarkson, .0; Ann ISancker, 10 . ; and John Jay, 1 50. How much was subscribed by these six persons ? 1 1. The founders of the Antlover Theological Semii^ary had, in the year i 820, given to it as follows : Samuel Abbot, on^ hundred thousand dollars; William Bartlet, 90000; Mrs Norri*^, 30( 0«. ; Moses Brown, .S:- 00 ;. William Phdips & Son, 15000 ; and J^.hn Norris, lOOOO. How ^^ach in all ? J 2. The army* of Bonaparte, when he invudc Ku?.sia, con- sisted of 25000(* French, ^0 00 Pole.s, ^0 00 Haxof^s, M^ 00 Austriaus. 3^ u- Bavarians, 2'20 Prussians, .;.:•: «^'^ W^st- phalians, 8 '00 Wirtembergers, 19 0^* subjects of ihe smaller German Princes, 20000 Neapolitans an ; South America, 1 5 millions. How many in all ? -« 1 i lie number of nomiiaal Chriltians is estimated a-, fol- lows: [n Europe, 175f>65 00 ; America, ?^8 .>000 ; Asia, 300000 ; Africa, 40! 0000. How many inall ? 1 5. The number of Mahometans is estlm'd< copies ot the New 'le^tament, of which 10o434 were during the last year ; how many before that r 24. The Connecticut Bible Society distributed, in their first 4 yt*ars, 7644 Bibles, of which 2341 were in the fourth year ; how many in the other three r 2^. In eight years, they had distributed 18056. Bibles; kow many m the second four years ? 156 ^uesthns, 26. The first Sabbath school was established at Gloucc^terj England, by Robert Raikcs, in the year 7S'^. ; and Robert M<»rrjsoa, who receved his first reliuious impressions at a Sabbath school, and afierwirds became a tnissiouary, finish- ed kis translation of the Bible into Chinese in the year I8i9. How long between ? *^7. The Russian Bible Society was formed in 1813, and that was 36 year-, after the Emperor Alexander was born, and that was 5t years after Russia bscame an empire, and that was 740 years after Christianity was introduced into Russia ; in what years did each of these take place ? 28. In one month of the year 181 P, the number of negro slares brought into Cuba, was 1 728 ; how many would thai be in a year, at the same rate ^ 29. In consequence of the numbers that were brought in, the price had fallen to 450 dollai's each; what would the number for a year amount to ? 3;-. The number of slaves transported from Western Africa for 25 years, ending in 18 i 9, was stnted to be such as w^ould average sixty thousand a year ; what was the whole number in that time? 31. What would be the whole value, at the above price in Cuba.?^ 3^. In 1815, the sum received by the two London theatres, was *?tated nt L>00 sterling a night ; suppose this to be con- tinued 5 months, 25 ni^ihts in a month, what would be the auKunt r S3. In December, 1814, the nine Paris theatres were stated to have received L\ Hi^oo sterling ; if this continued 5 months, "what was the amount ? 34. In 18 1 8, the London Hibernian Society supported 480 charity schools in Ireland, which averaged about 98 scholars each ; how many poor children were receiving an education from this charity r 3.5. The same year^ the T^ondon Sabbath school society for Ireland, assisted 634 schools, which averaged about 1 2 scho- lars each ; how many poor children were receiving aid from this charity } r6 A ujan who depended on his dady labor to. support himself and family, appropriated the earnings of one half day each month tocharifable purposes, and in one year the amount wah ten dollar-. The militia rolls of the United States con- tain the n.imes of -7o828 men. If each of these should "go a»d do likew ifce," how aiuch would be thus raised annually ? Questions, 157 37. The Royal Mission Chapel, built bj King Pomarre, in Otaheite, and dedicated in May, 18 1 9, is stated to be 712 feet long, and 54 wide ; how many square feet does it contain? 38. If 4 square feet be allowed to each person, how many persons would it accommodate ? 39. The whole number of schools in Scotland in 1820, was 3556, in which were taught 1 76303 children ; what is the average number for each school ? 40. In the year 18)^;, the number of schools in Ireland, under the patronage of the London Hibernian Society, was 534-, and the number of scholars in them 54520 ; how many to each school ? 4t.' In the year 1665, Connecticut contained abount 9000 inhabitants, and had 21 ministers of the gospel ; how many souls to each minister ? 42. In 1713, the inhabitants w^ere 17000, and the ministers and licensed preachers 45 ; how many souls to each ? 43. The annual expense to the inhabitants of Boston for the support of their theatre, was estimated, in 1820, at 75000 dollars; how many missionaries would that sum support among the heathen, at 5oo dollars each ? 44. How many would the receipts of the London theatres support, at ^125 sterling each ? (See No. 32.) 45. How many would the receipts of the Paris theatres support, at the same rate ? (See No. 33.) 46. The New England Tract Society' was formed in the year 1814, and in seven years had published 2708000 tracts ; what is the average number per year ? 47. In the western part of Virginia, there was stated to be, in 1821, a district containing l7o(X)0 souls, and only 8 edu- cated ministers of the gospel ; how many souls to each mi- nister ? 48. The population of London, in l81 1, was 1039000, and it is estimated that there are 212000 strangers constantly there ; how many in all ? 49^. If half these attend public worship, how many churches are necessary, each accommodating 80 persons ? 50. In I8i6, there were the following places of religious worship in London: EpivScopal, 166; Dissenters, 136; Dutch and German, 19; Catholic, 13; Jews, 6; Quakers, 6. How many in all ? 51. How many more wore needed? O 158 Questions. 52. By official returns, it appeared, in 1820, that there were, in England and Wales, 37382 schools, and 1571372 children taught in them ; what is the average number to a school ? 53. At the same time, in France, 1075500 children were learning to read and write, under the care of 28000 masters; how many scholars is that for each master ? 54. The following were the receipts of the principal reli- gious charitable societies in England, in the year 1 820, to wit : 13. & F. Bible Society, ^89154 sterling; Christian Know^- ledge Soc. ^53100; Church Miss. Soc. ^31200; London Miss. Soc. ^56174 ; Methodist Miss. Soc. =g225@0 ; Baptist Miss. Soc. jei3200; Soc. for propagating the Gospel, ^13000; Soc. for conversion of Jews, £ I tr780 ; National Soc. for Education, L8./00; Religious Tract ^oc. jLr56l ; Hibernian Soc. L7049; Moravian Missions, LsOOO; Navul and Mili- tary Bible Soc. £2348 ; Br. & For, School Soc. ^£034 ; Prayer Book and Homily Soc. il993. How much in all ? 55. The first missionaries landed at Otaheite in the year 1797, and idolatry was abolished in I8l5 ; how long between? 56. The missionaries sailed for the Sandwich Islands in the year 1819, which was 2 years after the establishment of the Cherokee Mission, and that was 5 years after the first missionaries sailed from America to India, and that was 2 years after the American Board for Foreign Missions was formed, and that was 2 years after the Theological Seminary was established at Andover ; in what year did each of these take place ? 57. It is computed, that in the year 1813, at least 800000 men died in war, and 200(00 more were maimed for life, and rendered useless ; and it is reckoned that the pecuniary loss to the public, from the death of an able bodied man, is j^rO dollars ; if so, what is the whole loss, in this way, o£ that one year ? 58. it is computed that the United States lost 17000 men in the late war with Great Britain ; what is the amount of that loss, on the same principle ? 59. The English Sabbath School Union, in the year 1820, had, in the schools connected with it, 237584 scholars ; that of Scotland, S4000 ; and that of Ireland, 84174. How many in all ? 60. The Inquis'tion was established in Spain in the year 148', and abolished in 1808 ; how many years did it exist there?* Questions. 159 61. During that time, 32382 persons were burnt alive by its order ; what is the average per year? 6:2. During the same time, 291450 persons were imprison- ed, and their goods confiscated, by its order ; what is the average per year ? 63. It is stated that the following sums were paid by the inhabitants of Charleston, in the year 1820, for the support of the poor, to wit: Orphan Asylum, §22000; Poor House, D24000 ; Marine Hospital, D6000 ; Ladies Benevolent So° ciety, D2000. How much in all ? 64. Of the above expense, the following is stated to be rendered necessary in ccmsequence of the intemperate use of ardent spirits, to wit : Orphan Asylum, D \ 4000 ; Poor House, DliJOOO; Marine Hospital, D4000 ; Ladies Benevolent So- ciety, DlOOO. How much are the inhabitants of Charleston annually taxed, to support drunkards and their families? 65. The first American missionaries to Jerusalem, sailed in the year 1819, which was 602 years after that city was taken by the Turks, and that was SO years after it was re- taken from the Crusaders by Saladirr, and that was 88 years after it was taken by the Crusaders, and that was 463 years after it was taken by the Saracens, and that was 22 years after it was taken by the Pei*sians, & that was 484 years after it was rebuilt by Adrian, and that was 60 years .after it was destroyed by Titus ; in what year did each of these take place ? 66. The number of Lancasterian schools in France in the year 1820, was 1340, containing^ 154000 children; how many is that for each school ? 67. The number of students, professors of religion, and charity scholars, at 12 of the Colleges, in the year l821, was stated as follows : Stud. prof. ch. sch. Stud. prof. ch.sch. Yale, 316 97 46 Princeton, 116 25 11 H;i«-vard, 291 17 15 Bowdoin, 101 23 7 Union, 255 66 32 Middlebury, 100 48 22 Blown, 151 59 18 Hamilton, 92 48 34 Dartmouth, 146 65 43 Williams, 83 42 24 N. Carolina, 135 10 Burlington, 35 9 1 How many students, how many professors of religion, and how many charity scholars, are in these 1 2 colleges ? 68. In the year 1 8 19, the London Religious Tract Society issued 5626674 tracts, which was 1583353 more than they had issued the preceding year ; how many were issued in 18li>? l60 Questions, 69. In 1820, the number of graduates at several of the col- leges, was as follows : Union 65, Harvard 56, Yale 54, Brown 29, Dartmouth 24, Middlebury 22, Pennsylvania 17, Hamilton 14, Bowdoin 1 1, Burlington 9 ; how many in all ? 70. in 1821, as follows : Union 67. Harvard 59, Yale 67, Brown ^0, Dartmouth 17, Middlebury 23, Pennsylvania 35, Hamilton 18, Bowdoin 21, Burlington 5, Columbia 30, Princeton 40, Georgia 3 ; how many in all tf 71. At the close of 1816, the Connecticut Miss. Soc. had sent S579d books to the new settlements, of which 5589 had been sent in that year ; how many before ? 72. In 1821, the English Methodist Miss. Soc. had among the heathen 150 missionaries and assistants, with 2700 con- verts under their care ; how many is that for each ? 72. In 1819, the Baptist missionaries in India had under their care, 92 schools for heathen children near Serampore, 1 1 at Cutwa, 3 at Moorshedabad, and 5 at Dacca ; and in these, about 10000 native children: what is the average number for each school ? 74. The General Assembly of the Presbyterian Church in the United States, was formed in the year 1787, which was 144 years after the Assembly of Divines met at Westmin- ster, and that was 83 years after Presbyterianism w^as esta- blished in Scotland by John Knox, and that was 26 years after the Reformation commenced in England, and that was 1 7 years after the Reformation was begun in Germany by Luther, and that was 157 years after the opposition to Popery was made in England by Wicklitfe, & that was 754 years after the Pope v^as acknowledged Universal Bishop by the Emperor Phocas, and that was 10 years after Christianity was intro- duced into England by Augustin, and that was 164 years after St. Patrick began to preach in Ireland, and that was 119 years after Christianity was established in the Roman Empire by Constantine ; in what year did each of these take place ? 75. The property belonging to the Choctaw mission at Elliot, in Dec. 1S20, was valued as follows : Sixty acres of improvements, D900 ; a horse mill, D200 ; shops, tools, and stock, D600 ; twenty-two other buildings, D3000 ; farming utensils, D400 ; seven horses, D420 ; two yoke of oxen, Dl60 ; two hundred and twenty neat cattle, D1760 ; sixty swine, Dl50; provisions, D 1758 ; groceries, LS60 ; house- hold furniture, D500 ; cloth, D250 ; library, D.)20 ; boat, B400 ; fifty thousand brick, D300. What is the whole value ? ^uestions> l6i 76: The Danish mission to Tranquebar, fit ate, in the year 1736, that in 29 years, they had received into their churches 3^39 converts from heathenism ; what was the average per year ? 7/. In 24 years afterwards, the number of converts added was 8267 ; what was the average per year ? 78. In 1813, the Moravians had 31 missionary stations, as follows : South Africa, 2 ; S. America, 4 ; N. Aroer^a, 7 ; Greenland, 3 ; and the rest in the West Indies ; how many were the last ? 79. At the same time, they had 157 missionaries and as- sistants at their stations ; what is the average to each station? 80. In 1817, the number of converted negroes under the care of the Methodist missionaries in the West Indies, was stated as follows : In Antigua, 3552 ; St. Christophers, 2552; St Eustatius, 513 ; vSt. Vincents, 2760 ; Bahamas, 584 ; St. Barfs, 447 ; Bermuda, 62 ; Dominica, 638 ; Grenada, 171 ; Nevis, 1183 ; Trinidad, 267 ; Tortola and Virgin Islands, 1664; Jamaica, 4126; Barbadoes, 44: Tobago, 140, How many in all ? 81. These were under the care of 38 missionaries and as- sistants ; what is the average to each ? 82. The number baptized by the Baptist missionaries in India in the year 1814, was 129; and the whole numbei* from the commencement of their mission, 765 ; how many before that year ? 83. The number of ordained missionaries among the hea- then in the year 1821, was 351, in the following countries, to wit : iVfrica, 45 ; Isle of France, 2 ; Malta, 3 ; Ionian Islands, 1 ; Polish Jews, 5 ; Turkey in Europe, 1 ; Turkey in Asia, 5 ; Russia in Asia, 17 ; China, 1 ; India beyond the Ganges, 10 ; India within the Ganges, 78 ; Ceylon, 26 ; Indian Archipelago, 7: Australasia, 2; Polynesia, 17 ; Spa- nish and Portuguese America, 14 ; Blacks of the W^est In- dies, 66; Indians in the United States, 23; Labrador, 19; and tlie rest in Greenland ; how many were the last ? 84. If 50000 missionaries should be sent to the heathen^ ?ind 1 jiOOO heathen should be allotted to each missionary, how many of them would be supplied at that rate ? 85. It is estimated that the annual income of tlie people of the United States, is three hundred millions of dollar^ ; ii one tenth of this should be devoted to the support of mis- sipnaries, how many would it furnish, at 500 dollars eack ? 02 i 62 (Itustions, 86. In 1821, Dr. Carey and his associates had translated and printed 'the whole Bible in five of ihe languages of the East, the New Testament in ten more, and parts of the latter in sixteen more ; this was £6 years after they began the work, and that was 84 years after the Tamul Testament was published by Ziegenbalg, and that was £6 years after Elliot's Indian Bible was |rriiited in America, and that was 7^ years after the present English version was published byKingJames, and that was 74 years after the first English editioit of the Bible was authorised by Henry the eighth, and that was 13 years after the brst English Testament was published by Tyndai, and that was 4 years after Luther published his Testament in German, and that was 78 years after the first books were printed From metallic types by Faui^t and others, and that was 14 years after prmting with wooden types was invented by Lauren tius in Holland, and that was 70 years after Wicklitte translated the Bible into English ; in what year did each of these take jilace ? 87. In 1820, the number ot Christian pilgrims to Jerusalem was as follows: Greeks, i600; Armenians, i SOO ; Copts, 150; Roman Catholics, 50; Abyssinians, 1; Syrians, 30. How many in all ? 88. In 1810, the English Soc. for promoting Christian Knowledge, distributed 10-224 bibles, 16^^42 testaments and psalters, 20555 prayer books, 20908 other bound books, and 145123 tracts ; how many in all ? 89. In 1814, asfollovv*s: 2676^ bibles, 48018 testaments and psalters, 65492 prayer books, 5 1525 other bound books, and 6 3501 tracts ; how many in all ? 9c. How m^ ny more in the last year, than in the otiier ? 91. The American Bible Soc. in 18^0, (their 5th year,) issued 29{ 00 bibles, and 30000 testaments ; and the total of copies of the whole bible, or parts of it, issued by them, was S3 552 ; how m;iny in the first four years ? 9S. The Hegira, or flight of Mahomet from Mecca to Medina, was in the year 622 ; and Constantinople was taken by the Turks in 1453 ; in what year of the Hegira was ic ? 93. The New-England Tract Society was establi^hed in 181.S, and in 1821 had published 2708000 tracts ; how many is that for each year ? 94. In i8iO, the Maine Miss. Soc. received 1081 dollars and 38 cents ; how many mills is that? 95. In 18 8, the receipts of the English Christian Know- fedge Soc. were £\S9%3 .. 9 .. 5 ; bow many farthings is that? Qiiestions, 1 6S In 1815, the amonnt was £50226 .. lo .. 1 ; how manj pence is that ? 9r. In 1817, the amount was 56885012 farthings ; how many pounds is that ? 98. In 1819, the receipts of the London Jews Soc. were 107 i:. 961.' farthings ; how many pounds is that? 99. How many grains in 367 lbs. Troy ? 100. How many drams in 3 tons ? 101. How many grains in 45 lbs. Apothecaries' wt. ? 102. hi 50 miles, how many barley corns ? 103. How many lbs. Apothecaries' wt. in 13337791 grains? 104. In 356 yds. how many nails ? 105. In 18 lbs. how many scruples? lOt^. In 64960 lbs. how many tons? 107. in 4 1 60 poles, how many acres ? 108. How many pints in 21 hhds. wine measure ? 109. In 4976 pints, how many bushels ? 110. In 2279772 barley corns, how many miles? 111. How many yds. in 105ii nails ? 112. How many tons in 20563/12 drams? 113. In 2 401600 seconds, how many weeks ? 114. In 5 hhds. wine measure, how many gills? 115. How many poles in 456 acres ? 116. In ^13096 grains, how many lbs. Troy? 1 i7. How many pints in 786 bushels ? 118. In 363 days, how many seconds? 119. In 14 tons, how many lbs. ? 120. How many miles in 34665840 inches? 121. In 319 nails, how many yds, ? 122. How many ounces in 5 tons ? 123. How many lbs. Troy in 245678 grains ? 124. The expenditures of the Charitable Soe. of Hillsbo- rough County, N. H. for the year 1818, were as as follows : For bibles, 0^96-77; domestic missions, D34*20 ; foreign missions, 1)126*86 ; education of pious youth, Dft09'34 : how much in al* ? }S5. The General Committee of the Moravians, received for their several missions, in 1818, as follows : Collections from congregations and friend?, ^1545.. 2.. 10 steiling; benefactions, chiefly in England & Scotland, ^4035 .. 10 .. 8; legacie?, Lf.SS .. 13 .. 2 ; balance from West Indies, ^240 .• .. 5 ; gained by exchange, £b,» 17 .. 6 : how much in all ? 1 64 (luestions. 126. Their |Miyments for missions were as follows : — Greenland, L7i2.. 10.. 7; Labrador, (besides what was supplied from other sources,) ^105 .. 5 .. 11 ; N. American Indians, ^218.. 4.. 4; W. Indies, jL288l ,. 9 .. 2 ; South America, ^190.. 10.. 11 ; S. Africa, X1124.. 12.. 2 : how much in all ? 127. Their other expenses were as follows : Pensions to superannuated missionaries, ^748 .. 1 1 .. 2 ; widows of mis- sionaries, ^317.. 10.. S; education of sixty-three children of missionaries, L853.. 15.. 7; sundries, X.787.. 14: how much in all ? 128. What was the whole amount of expenditure for that year ? 129. In the year 1820, there was raised in the county of Otsego, N. Y. i 25 bushels, 4 quarts of corn, on one acre ; 1^20 bush. 2 pecks, on another ; 1 18 bush. 4 qts. on another ; 117 bush, on another; 11 1 bush, on another; 95 bush. 4 qts. on another ; and 90 bush. 2 pecks, 6 qts. on another : how much on seven acres ? 130. One piece of cloth contains 37 yd. 3 qr. 3 na. ; ano- ther, 28 yd. 2 na. ; another, 39 yd. 2 qr. ; another, 9 yd. 3 na.: how much in all ? 131. Boughtof A, 76 acres, 3 roods, 27 poles; ofB, 26 acres, 57 poles; of C, 19 acres, 3 roods; of D, 11 acres, 2 roods, 17 poles : how much in all ? 132. Journeyed on different days as follows : 36 miles, 3 furlongs, 21 poles; 21 miles, 37 poles; 34 miles, 7 furlongs, 28 poles ; 56 miles, 6 furlongs ; 47 miles, 27 poles : how far in all? 133. Sold A, 3 Cwt. 2 qr. 57 lb. of flour ; B, 4 Cwt. 3 qr. 19 lb. ; C, 5 Cwt. 2 qr. 19 lb. ; D, 4 Cwt. 21 lb. ; E, 9 Cwt. 3 qr. 18 lb.: how much in all ? i34. In 1820, the American Bible Soc. received D49578 •34, and expended D47759-60; what is the difterence ? 135. The American Board tor Foreign Missions received D39334-51, and expended D57420-93 ; how great was the deficiency? 136. The United Foreign Mission Soc.receivedD15263-3jf, and expended D 14010; what sum remained unexpended ? 137. The receipts of the Ame ican Education Soc. for 1819, wereDl9330; for 1820, Dl5l4ri'80; how great was the falling off? 138. In 1821, its receipts were D13108*97 and its expen- ditures D 10018-72; what is the difference ? ^uestions^ 1 64 139. In 1819, the British and Foreign Bible Soc. received /X9S053.. 6.. 7 sterling, and expended jLi23.47'.. 12.. 3 ; what was the excess of expenditure ? J 40* The Church Missionary received ZSOOOO sterlings and the London Miss. Soc. 2-25406 .. 16.. 4; what is the difference ? 141. In 1820, the London Missionary Society received Z26174.. 4.. 3 sterling, and expended JL27790 .. 1 7 .. I ; what was the excess of expenditure ? 142. The London Jews Soc. received L10789 .. 18.. 2 sterling, and expended X13137.. 16.. 1 ; what was the ex- cess of expenditure ? 143. Bought 2 tuns of wine, and sold 3 hhds. 25 gals. 1 qt. ; how much is left ? 144. Bought 642 lb. 9 oz. 8 gr. of silver, and sold 537 lb. 6 oz. 10 dwt. ; how much is left ? 145. Borrowed 46 Cwt. 3 qr. 16 lb. of hay, and returned 10 Cwt. 1 qr. 26 lb. ; how much remains to be returned ? 146. From 6 lb. 9 oz. 1 sc. 19 gr. of medicine, take 5 lb. 1 1 oz. 7 dr. 10 gr. ; how much is left ? 147. If a man earns 1 doll. 12^ cents a day, and should devote to the Lord the earnings of one day every month, what would be the amount in a year ? 148. If the fees of a physician average D3*25 every week day, and he should be under the necessity of attending pa- tients on the Sabbath to half that amount, and should devote the proceeds^ of all his Sabbaths to Him who is Lord of the Sabbath ; what would be the yearly amount ? 149. If a journeyman mechanic can earn 9 cents an hour, and perform his day's work in 9 hours, how much can he earn in a year for doing good, by working one hour extra each day, there being 313 working days in a year ? 150. If an apprentice can earn 6 cents an hour, how much can he earn in a year for doing good^by the same method ? 15 1. If a young woman can earn with her needle, 4 cents an hour, how much can she earn in a year for doing good, by the same method ? 152. If a little girl can earn by knitting, 5 mills an hour, how much can she earn in a year for doing good, by the same method? is 3. If a little boy should raise 12 chickens in a year, which, when full grown, should weigh i? lb. 8 oz. each, and should sell them for 5 cents a lb. and devote the avails to the 166 Questions, 1 education of heathen children, what would be the annual amount ? 154. If a man drinks half a gill of ardent spirits every day, how much is that in a year ? " 1^5. If be makes use of half a pint every day for himself and friends, how much is that in a year ? 156. How much is it in 20 years, allowing 5 leap years ? 157. What cost 1 ; 9 cords of wood, at D^'67 a cord ? 158. What cost 1 2 lb. of tea, at Ts. 6d. a lb. ? 159. What cost 96 bushels of rye, at 6s. 9d. a bushel ? 160. What cost 11 Cwt. of flour, at Ll .. 4 .. 6 per Cwt. ? 161. Sold to 19 persons, each, 17 Cwt. 3 qr. 2l lb. 14 oz. 15 dr. ; how much in all ? 162. Bought of 25 persons, each, 9 lb. 10 oz. 17 dwt. 21 gr. of silver; how much in all? 163. Mixed 16 sorts of medicine, of each 2 lb. 3 oz. 5 dr. 1 sc. 18 gr. ; what is the weight of the whole mass ? 164. Sold to 24 persons, each, 8 quarters, 7 bush. 3 pks. 1 gal. 3 qt. 1 pt. of wheat ; how much in all ? 165. Bought 13 parcels of wood, each 13 cords, 127 ft. 1727 inches ; how much in all ? 166. Bought 44 pieces of land, each 16 A. 3R. 36p. ; how much in all ? 167. Sold 35 pieces of cloth, each 25 yd. 3 qr. 3 na. ; how much in all ? 168. If 1 68 yds. cost X40 .. 12, whatis that per yd. ? 169. If 55 proprietors bought a tract of 40000 acres, what is the share of each ? 170. If 13 persons joined in purchasing 3 hhds. of wine, what is the share of each ? 171. If 2S6lb. 10 oz. 6dr. 2 sc. 18 gr. of medicine be made up into 16 equal parcels, what weight will be in each ? 172. If 25 persons were joint purchasers of 35 Cwt. 3 qr. 21 lb. of sugar, what is the share of each ? 173. In the year i819, the receipts of the American Edu- cation vSoc. were D{93^'0, and the number of young men as- sisted was 161 ; how much would that average to each, if the whole had been distributed ? 174. In 1820, the receipts of the Western Education Soc. were 1)1755-61, and the expenditures Dl601'62 ; whatis the difference ? 175. They had 56 young men under their care ; what would the amount expended be for each ? 176. 'the receipts of the Baptist Missionary Soc. of Ma ft- Questions. 167 sachusetts, were D2.575-68 ; how many weeks missionary labor would it pay for, at D8 a w *ek ? 177 In the year ;80^ there were 11 societies in the U. States for the support of missions in our own country, and their receipts were DlOl ; what was the average for each? 178. In 18 0, the year the American Board for Foreign Missions was form<^d, the receipts of the same societies were D 10721 ; what was the average for each ? 179. In 1818 the receipts of these same societies were D23675 ; what was the average for eaeh ? ISO. What was tht^ average yearly increase of each so- ciety for the ^ years before the Board was formed ? IP . What, for the 8 years after the Board was formed ? 182. In 1P21, the American Education Soc. had under its care '25f beneficiajies, and distributed among them D9093 ; what is the average for each ? 183. The donations to the Massachusetts Miss. Soc. for 18 8, were L356.. 0.. 11, N. Eng. currency ; if that sum paid for 196 weeks missionary labor, how much would it be per week ? 184. In 1820, there was received for the aid of charity students at the Theoloicical Seujinary at Princeton, D2855 •40 J, and the number of students was 7 i. If one half of these received aid, how much would it be for each? 185. In i8l5, the number of Hottentots belonging to the settlement at Bethelsdorp, was about 1200. The same year they paid in taxes to government, D3500 ; contributed for miijsions, D 32 80 ; collected for their own poor, D i 77 ; and were building a school room, and printing office, 70 feet by 80, estimated to cost at least D6 14-20 ; what does the whole amount average for each individual ? 186. The number of missionaries and assistants employed by the American Board in 1820, was 88, and they had 3000 heathen children under instruction ; what is the average for each ? 187. The disbursements for the several missionary stations were 0*8565 ; how much is that for each missionary and his scholars ? 188. The Rev. Joseph Emerson received for his astrono- mical lectures in Boston, in 1819, D643, and his expenses were Dl i 7. If the remainder was divided among 14 young ladies, to assist in qualifying them for instructing schools, how much would it be for each ? 168 ^utstions, 189. In 1820, it was estimated that there were in New- England, 250000 young men, between 15 and 35 years of age. If 70000 of these give 2") cents each per annum ; 100000, 75 cents each ; 50000, D2 each ; 20000, D5 each ; and lOOOO, DtO each, to the American Education Soc, what will be the annual amount ? 190. How many young men would that assist in preparing for the ministry, at Dl2:> each per annum ? 191. What is the greatest common measure of 82 & 124 ? 192. What is the greatest common measure of ! 64 & 248 ? 19 j. What is the least common multiple of 3, 4, and 5 ? 194. ( f ', 5, 6, and 7? 19 . Of 2, 3, 5, and 12? 196. Reduce ifl to its lowest terms. 197. Reduce i, |, and |, to a common denominator. 19^. Reduce 12J to an improper fraction. 199. Reduce y to a mixed number. 200. Reduce f of | of f to a single fraction. 201. Reduce | of a penny to the fraction of a pound. 202. Reduce f of a Cwt. to the fraction of a lb. 203. Reduce | of a L. to its value. 204. Reduce "?Jd. to the fraction of a shilling. 205. Add f and f . 206. Add I and |. 207. Add I of a shilling, and j^j of a penny. 208. From |, take 4. 209. Take^o from'}. 210. Tell the product off by f. 211. Off by f. 212. Divide V by f . 213. Divide f by/_. 214. Tell the sum of 29-0146+qi46-5+2109+-624l7-+ 14- 6. 21 >. Tell the difference between 91*73, and 2-138. 2 1 6. Tell the product of 79-347 by 23-15. 217. Of5i-3 by 1 000. 218. Tell the quotient of 27 by -2685. 219. Of 2 7-3 by 100, 220. Reduce ^V ^^ a decimal. 221. Reduce 9d. to the decimal of a Z. 22 \ Reduce I dwt to the decimal of a L. 223 Reduce 15s. 9f d. to the decimal of a L, 224. Tell the value of -625 of a shilling. Questions, 1 1)9 225. Of -8635 of a L. 226. Of 62j of a Cwt. 227. If 4 yds cost 1 2s., what cost 8 yds. 9 228. if 7 yds. cost I 5s , what cost 9 yds. ? 22 ^ if :;6 men cm build a wall in 2i days, how many men can do it in '^ i days ? 230. if 6 Cwt. i qr. of sugar cost 1^1 8 .. 1 6 .. 4, what cost 3 Cwt. I qr. 27lb. ? QM. If J 8 yds. costnOs,, what cost 31 yds. ? 232. If 50 men can perform a piece of work in 12 days, how many can do it in 4 days ? ^33. What will 12 yds. of lace cost, at the rate of Xr56 for 96 yds.? 231. If, when the price of wheat is ^s. 6d. a bushel, the penny loaf weighs 8 oz., wiiat must it weigh when the price of wheat is 6s. a bushel ? 235. How many men must be employed 12 days, toper- form the work which 4 men can do in 48 days ? 236. If l^ yds. cost 9s., what cost 2 I yds. ? 237. A lent B T^O dollars for 8 months; how long must B lend A 500 dollars to be equivalent ^ 238. How many yds. can be bought for Ll4.. 8, when l6 yds., cost 1: s. ? .<39. A giddsmith bought 14 lb. 3 oz. 8 dwt. of gold, for 2035 dollars, what is that per ounce ? 240 If a staiT feet high, casts a shade, on level ground, 5 f -et long, how high is that steeple, the shade of which, at the same ti-me, measures -41 5 feet? k 2^1. In how mny days can 12 men perform apiece of 1 work, wliich 1 8^ men can do in '»0 days ? I 242. In i8i 8, the number in the Moravian societies was f stated to be ; 6000, and that they then had 170 missionaries & f a!?sistants among the heathen. If the rest of the nominally Christian world did as well as the Moravians, how many ' missionaries would now be in the tield ? (See No. 14.) r 24 >. How nia;»y of those destitute of the gospel, would fall to tu' lot of e.u'h missionary? (See Nos. 13 and 14.) :^44 If the whole number of Protestants is 60 millions, and they had all done as well as the Moravians, how many missionaries^ would they have in the field? 24 •. How many would fall to the lot of each missionary, in that case ? 246. The Moravians reckoned their converts from hea- i thenisni to be 60200 ; if the rest of the Protestaat world had P \ 170 ({uestions. done as well, what would have been the whole number oi Gonverts ? 247. The Baptist mission at Serampore, was begun in 1794; and in 1818, they had 60 native preachers, and 20 churches of converted natives ; of which, that at Chittagong was estimated at 150 members ; that at Jessore, 95 ; that at Dinagepore, 105 ; that at Serampore and Calcutta, 190 : how many in these four ? 248. If the other 16 contained half as many in proportion, what would be the whole number ? 249. The number of slaves imported into Havana from Africa, from Dec. 1, 1816, to July Si, 1817, was i 1161 ; how many would that be in a year, at the same rate ? 250. If 4 men, in 12 days, can reap c 6 acres of wheat, how many acres can 6 men reap in 18 days ? 2:> . If 6 men can reap 72 acres in 12 days, how many men can reap 96 acres m h days ? 252. If the wages of 6 men for 42 weeks be 60 dollars, what will be the wages of 14 men 52 weeks ? 255, If 40 shillings be the hire of 8 men for 4 days, how many days must '60 men work for ^20 ? 254. If 1 2 oxen eat 24 acres of grass in 1 5 days, how many acres will serve 24 oxen 45 days ? 255. If the interest of 350 dollars for 8 months is 18 dol- lars, what sum in a year will gain 6 dollars ? 2.6. If 100 dollars in a year gain 6 dollars, what will /i25 dollars gain in 254 days ? 257. If LlOO gain Zr6 in 12 months, what sum will gain is 6 in 6 months ? 258. An auxiliary missionary society, composed of 100 slaves, at Berbice, S. America, raised, in 8 months, LS5 ster- ling ; what would that be for each member per annum ? 259. Iff of a yd. cost | of a £., what cost y^o of a yd. ? 260. If j\ of a ship is worth ^273 .. 2 .. 6, what is -^^ of it worth ? 261. If the penny loaf weighs 6y\oz. when wheat is 5s. a bushel, what ought it to weigh when wheat is 8s. 6d. a bush.? 262. If 3 men; in 6 days, spend ZilOf, what will 20 men spend in 30 days ? 263. if 1-5 oz. of silver costs 7*8s., what cost 29'1 lb.? 264. What cost S-4 lb. at ^£4-5 for 1-47 Cv/t. ? 265. If the wages of i men for y.4»6 days, be ^18-9, what will be the wages of 8 mea tor 16*4 days ? Questions. 171 £66. 267. 268. 269. 270. 271. 272. 273. 274. 2^5. 276. 289, 290. 29 i. 292. 293, Find the cost of ' 7612 lb. at id. 6812 4712 15344 7672 9424 8652 1218 8612 7813 1218 6002 Find the cost of 121 lb. at ^s, . 8 per lb. per lb* 278. id. 279. 471 5s. |d. 280. 191 6s. Id. 281. 242 8s. id. 282. 345 6s. 8d. Id. 283. 678 13s. 4d. lid. 284. 567 13s. 2d. 2id. 285. 825 ^3.. 6 .. 8 IJd. 286. 676 14.. 17.. 9i 3id. 287. 346 12.. l9.. Hi 8d. 288. 488 14.. 8.. 6 llid. Find the cost of ^^7lb. lOoz. Troj, at ^1 .. 6 Of 13 lb. 10 oz. 12 dwt. 8 gr.at £2 .. 3 .. 4 Of 476 A. 3 R. 28 p. at ^4 .. 12 .. 8 per acre. Of 957 A. 3 R. 16 p. at ^3 .. 7 .. 1 1 per acre. F.nd the neat weight of 856 Cwt. 1 qr. 19 lb. of to- bacco, tare in the whole, 17 Cwt. 2 qr. 13 lb. 294. Find the neat weight of 1 3 hhds. of tobacco, each weighing 6 Cwt. 2 qr. 27 lb. gross ; tare in the whole, 9 Cwt. S qr. 17. lb. 295. Find the neat weight of 6 casks of sugar, weighing, gross, 4 C\vt. 3 qr. 12 Ib/cach ; tare \B lb. each. 296. What do they come to, at £9, .,4. .7 per Cwt. ? 297. Find the neat weight of 12 casks of raisins, each weighiRg S Cwt. 3 qr. 26 lb- gross ; tare> 20 lb. per cask. 298 What is the value, at^:2 .. 14 .. 6 per Cv/t. ? 299. Find the neat weiglit of 50 kegs of rigs, gross 86 Cwt. k 6 qr. 14 lb., tare 14 ib. per Cwt- \ 300, What do- they ccuie to, at I9s. 8d. per Cwt. ? 301. In 28 bags of coifee, each - Cwt. 3 or. 12 lb. gross, e 14 lb. per Cwt-, trett 4 lb. per 104 lb-, how much aeat ? 302. Find the value, at dg% .. 18 .. 9 per Cv^t. 303 In 9 Cwt. 3 qr. 27 U). gross, tare 38 ib, trett 4 lb. \ per 104 lb., how many lb. neat .' 1 304-. Find the value at 8id [)er lb. 30 "5. Tell the interest of Li 57 for 1 year, at 6 per cent 30*-:. (if 7^25 i, at 4 percent. 3^7- Of 'L5678, at 7 per cent ? 308. Of ^234, at 7-J per cent. 309. Of S 158, at S per cent. 3 1 J. Of S789, at 5}y per cent V 31 1. Of Z/2340, for~i4Tionths, at 6i per cent 72 Questions- S12. Of L600 for 8 months, at l^f per cent. 313. Of Z6740 for 7 montlis, at ej per cent. n4. Of jL56 .. 12 .. 8 for I J years, at 5 per cent il5. Of L65 .. 19 .. 6 for 5 years, at 6 per cent. SI 6. Of L66 .. 10 .. 6 for 6 years, at 6^ per cent. 317. Tell the amount of 8624-25, for 130 days, at 6 per ct. SIS. iff 8786-30, for 235 days, at 6i per cent. 319. Of g687-34, for 320 days, at? per cent S20. Of 2.62S .. 13 .. S3^ for 5 y. IH nio- at 6 per cent 32 1. What is the insurance of an East India ship and <^rgo, valued at L8406 .. 1 8 .. 6, at 5^ per cent. 322. What principal, at interest for 8 years at 5 per cent, will amount to i>720 ? 323. At what rate per cent will Z600 amount to jL9ii4,in 9 years ? 324. In what time will X700 amount to L940, at 5 per ct. ? 325* A merchant has sold goods on commission, to the amount of S3 45 600 ; what is his commission, at 2| per ct. ? 326. What sum at interest for 9 years and 6 months, at 7 percent, wdl amount to Sl456'87i v 327. If 2i per cent is allowed for commission, what must be paid on 1/1234.. 17.. 8? 328. In 1817, t^ngland agreed to pay Spain X400000 ster- ling, for which Spain agreed to abolish the slave trade, after May, i 820 ; what is the annual interest of that sum, at 6 per cent ? 329. In 1818, the Connecticut school fund was §1608673 •39 ; what would it yield annually at 6 per cent? 330- The amount distributed from the common school fund of New-York, was §1^0000 ; what must the fund have been, to yield that amount at 7 per cent? 331. The expenditures on the United States armory at Springfield, from 179.0 to I8;i0, had been 82072676; if this had been put into a fund for religious purposes, what would it annually yield at 6 per cent ! 532. How many bibles would it annually furnish for cha- ritable distribution, at 60 cesits each ? 333. How many young men would it assist in obtaining an education, at gl25 each ? 334. How many missionaries would it support, at S500 each per annum ? 33^. In 18:^0, the permanent fund of the Connecticut Missionary Soc- was D33405-55J- ; what sum will it annually yield, at 6 per cent ? ^iiesiionB' 173 m » 3S^. How many weeks missionary labor will it pay for ' ever/ year, at 8 dollars a week '' S r . In 1 820, the amount ol interest received by the Ame- rican Board for Foreign Missions, was D2 154-60 ; what must have been the principal of their permaneat fund, to yield that amount, at 6 per cent ? 338. What is the amount of L720, for 4 years, at 5 per cent, compound interest? 339. What is the amount of L50, in .'i years, at 5 per cent, compound interest? 340. What is the compound interest of 1/370, for 6 years, at 4 per cent ? 341. What is the compound interest of i/450, for 7 years, at 5 per cent ? 342. Tell the present worth of L80.. 15, for 19 months, discount at 5 per cent. 343. Tell the discount of jL159I .. 2 .. 4, for 1 1 months, at 6 per cent. o 14. Sold goods for LS97 .. 15 .. 7, to be paid in 4 months ; what is the present worth, at 3| per cent ? 345. A owes B -L2408, of which L1234 is payable in 6 months, and the rest in 1 months ; but they agree to reduce them to one payment ; when must that be ? 346. A merchant has owing to him LIOOO, of v/hich Irl50 is due now, LI 50 in 2 months, L200 in 4 months, and the rest in 6 months ; what is the equated time ? 347. C owes D 7.480, payable 5 months hence ; but is willing to pay L80 now, if D will wait longer for the rest ; to what time must he vvait ? S'iS. What quantity of tea, at 20s. per lb. must be gives, in barter for 1 Cwt- of chocolate, at 4s. per lb ? 3 v9. How much flour, at 5t)s. per Cwt- must be given for 7 Cwt- of raisins, at 5d. per lb- ? 350. A has 24 sheep, at 1 6s. 8d. each, for v»'hich B is to pay L12 in cash, and the rest in potatoes, at 2s. a bushel ; how many bushels of potatoes must A receive ? 351. B delivered 6 hhds. of molasses, at 6s. 8d. a gallon, to C, for 2.S2 yds. of cloth ; what was the cioth per yard ? 352. How much coflee, at 25 cents per lb. can I have for 56 lb. of tea, at 80 cents per lb. ? ^^5S. A delivered to B 9Sii bushels of corn, at 50 cents a bushel, and received 55 Cwt. 2 qr. of cheese, at 4 dollars per Cwt. ; how much money must A receive in addition, to pa for his corn ? P2 IT4 Questions' 364. How much wine, at Dl'^28 per gal. must I liave for 13 Cwt. 1 qr. 7 lb. of raisins, at D9'444 per Cwt. ? 355. If I buy candies at '9 cents a lb., and sell them at 23 cents a lb., what shall 1 gain per cent ? 356. Bought indigo at DM0 a lb., and sold it at 90 cents a lb. ; what was lost per cent ? 557. Bought 74 gals, of wine, at DM0 a gal., and sold it for DcSO; what was gained or lost per cent? 558. Bought hats at 8s. each, and sold them at 9s. 6d. each ; what was gained per cent ? 559. If 1 buy wheat at Dl-25 per bushel, how must I sell it, to gain 18 per cent ? 560. If a hhd. of rum cost 50 dolls, for how much must it be sold, to lose iO per cent ? 361. If 60 lb. of steel cost^^JLS .. 10, how must I sell it per lb. to gain 15 J per cent ? 862. A and B join stock, and make up D600. A yiuts in B225, and B the rest. They gain D150 ; what is the share of each ? ^ 563. A man dying, left 5 sons, as follows : A, 184 dolls., B 155, and C 96 ; but when his debts were paid, there were but 368 dolls, left; what is the share of each r 304. A and B companied ; A put in irl35, and took | of the gain ; what did B put in ? 365. A, B, and C, entered into partnership. A put in D170 for 8 months, B Dl20 for 10 months, and C 1)2 lO for 5 months; and they lostD82; what is each man's share of the loss ? 366. A, B, and C, hold a pasture in common, for which they pay L40 a year. In this pasture, A had 80 oxen 76 days, B 7S oxen 50 days, and C 100 oxen 90 days ; what must each man pay ? 567. in I8l7, the London Hibernian Society had in its schools in Ireland, under gratuitous instruction, 52000 poor children, at an average expense to the Society of 5s. sterling each ; what is the amount in Federal money ? 368. The return oi Bonaparte to France from Elba, and his subsequent measures to the battle of Waterloo, are stated to have cost the French nation i02l millions of francs ; how much is that in Federal money ? 569. In 18 i 8, the London African Institution for promo- ting the abolition of tl^e slave trade, expended LS05 .. 15 .. 9 vsterling ; how much is that in New- York currency ? Questions. 175 H^fpO. The bank of England is said to have a capital of ^^T!m60O500 sterling ; how much is that in ^iew- England currency ? 371. In 18)9, the London Prayer Book and Homily Soc. expended i.!iC06 .. 11 .. 4 sterling ; how much is that in Penn- sylvania currency ? oTz, The London Hibernian Soc. expended 7 8387 .. 16 .. 8 sterling; how much is tiiat in fcouth-Carolina cunency ? 373. The receipts of the Biitish Naval and Military Bible Soc. were L'^'i69. sterling ; how much is that in Canada cur.? 374. in I8l9, the exports of Russia were to the amount of 43559343 rubles more than their import:?, which were 167 millions of rubles; what is the amount, in Federal money, of their exports? 375. The expenditures of the Scottish Miss. Soc. for 1819, were L4599 .. 11 .. 1 1 sterling ; what is that in >Jew-Jersey currency ? 37^^. The receipts of the. New Hampshire Miss. Soc. for 1820, were D^537-21 ; what is that in sterling/ 377. The receipts of the Boston Jews Soc. were D 1 195*67 ; liQ^w much is that in Virginia currency ? 378. The expenditures of the British and Foreign School Soc. were jL2432 .. 3 .. 3 sterling; hov/ much is that in North Carolina currency ? 379. The Berbice Auxiliary Miss. Soc. composed of slaves, contributed, in 1820, 420 guilders; how much is that in Federal money ? 380. The total expenditure of the British and Foreign Bible Society in 17 years, was jL908248 .. 10.. 6 sterling ; how many livres is that ? 381. How much in Federal money ? 382. The receipts of the Maine Miss Soc. in 1820, were D20.58-47 ; how much is that in Georgia currency ? 383. The receipts of the Hampshire Missionary Soc were D I 590-59 ; how much is that in Canada currency? 384. The receipts of the New- York Miss. Soc in 1807, were D 1.360*47 ; how much is that in Irish currency ? 335. The receipts of the Vermont Missionary Soc in 1812, were D652-67 ; how many livres is that ? 386. The receipts of the Connecticut Miss. Soc- in 1816, w^ere D6019-32; how many rubles is that ? 387. The receipts of the Connecticut Bible Soe. iu 1816^ w^ere D2 1 77*20 ; how many rials of plate is that? 176 ^estions. 3S8. The receipts of the Massachusetts Miss. Socin 181S, were D3120'04 ; how much is that in sterling '' o89. The receipts of the Massachusetts Society for pro- pagating the Gospel, in 1810, were D2477'8G; how much is that in New-England currency? 390. The expenditures of the Massachusetts Society for promoting Christian Knowledge, in 1813, were Di 407*62; how much is that in Delaware currency ? 391. The receipts of the Berkshire and Columbia Miss. Soc. in 1 813, were Dsi 1*44 ; how much is that in New -York currency ? 392. The receipts of the American Board for Foreign Mis- sions the first year, were Dl3»9*52 ; how much is that in South-Carolina currency ? 393. The second year, D139j304; how much is that in Irish currency ? S94. The third year, Dl 1436* 18; how much is that in Canada currency ? 395. How much, in Federal money, did David give Arau- nah for his threshing floor and oxen, (2 Sam. 24- 24,) it being 50 shekels of silver ? 396. How much, for the whole place, (1 Chron. 21. 25,) it feeing 60(J shekels of gold ? 397. What was the avoirdupois weight of Absalom's hair, (2 Sam 14. 26,) it being 200 shekels ? 398. How many bushels of flour, and how many of meal, were required daily for Solomon's table, (l Kings 4. 2S,) it being 30 cors of flour, and ^^O of meal ? 399. What were the dimensioni<, in feet, of Solomon's temple, it being 60 cubits long, 20 wide, and 30 high ? 400 How many wine gallons did the brazen sea contain, being 3000 baths ? 4 1. How many miles was Bethany from Jerusalem, being 15 Hebrew furlongs ? 402. How many acres of land were assigned to the Le- ^ites as glebes, (Lev. 35. 3 — 5,) being lOOo cubits square on each of the 4 sides of each of the 48 Levitical cities ? 403. George W^ashington was born Feb- 22, 1732, and died Dec. 14, 1799 ; how old was he ? 40',. The p(»pulation of the New -England states, at each census, was as iEbllows : (luesiions. irr Vermont, N. Hampshire, Miiiue, Massachusetts, Rhode Island, Connecticut, New York, New-Jersey, Peunsjlvariia, Delaware, Maryland, Dist. Columbia, In 1810. In t8'20. 2 '7895 23; 7 04 2 ^460 21-4 -61 228705 29^-^335 47 040 52^-287 76931 83059 261942 275248 In 1790. In 1800. 85589 154449 141885 183858 9r>540 151719 3787 7 422530 68825 69122 237946 251002 What was the number of inhabitants in New England at each census ? 405. The population of the Middle States at each census, was as ft)llows : 340 1 20 586050 184139 2 H49 434 373 602545 69094 64273 319728 3i96&2 8124 AVhat was the number of inhabitants in the Middle States, at each census r 406. The population of the Southern States at each census, was as follows : 886149 478(03 S45591 162686 959049 1372812 245562 277575 810091 1049P^8 72674 72749 380546 407350 2402 5 33039 747610 393751 239073 82548 974622 555500 415115 252433 lO'^S.^e Virginia, K. Carolina, S. Carolina, Georgia, Alabama, What was the number of inhabitants in the States, at each census ? 407. The population of the Western States at each census, was as follows : ^38829 502741 340989 127901 Southern Kentucky, 73677 22095 4065 119 664317 Tennessee, 35691 14260 261727 4:22813 Ohio, 42179 230760 581434 Mississippi, 40352 75448 Louir^iana, 76556 153407 Indiana, 4875 24520 147178 Illinois, 12282 55211 Missouri, 20845 66586 Michigan Ter. 4762 8896 Arkansaw Ter 14273 What was the number of inhabitants in the Western States, at each census ? 408. What was the whole number of inhabitants in the Wnited States, at each census ^ 178 (luestiotis, 409. The contributions to the American Board for Foreign Missions, for the year ending Aug. 31, 1820, were, from the several states, as follows : Massachusetts, $1466l'7l ; Con- necticut, $6036-68 ; New York, $379i-41 ; Vermont, $1843 •69; New-Hampshire, S1464.-82; Maine, | 451-83; New- Jersey, $1443-51 ; Georgia, 81280*52 ; S. Carolina, $764-30 ; Pennsylvania, $702-30 ; Maryland, $68- -50 ; N. Carolina, $r}64-3£; Ohio, $392 91; Tennessee, $251-17; Virginia, $209 ; Louisiana, $200 ; Rhode-Island, $.11-56; Delaware, Si 05-44; Mississippi, $£0 ; Dist. of Columbia, $10; Choc- taw Nation, Sl69; Cherokee Nation, $8 ; places unknown, $4 1-97. How much in all ? 4 lO. How much less than half the whole, was contributed by Massachusetts ? 4H. How much more than half the whole, by Massachu- setts and Connecticut ? 4 1 *. If the amount contributed by Massachusetts, were divided equally among the inhabitants of that state, accor- ding to the census of » 8 20, (See No. 404,) how much would it be for each ? 413. If the amount contributed by Connecticut, were so divided among the inhabitants of that state, what would it be for each ? 41 4. If New- York had contributed in the same proportion as Massachusetts, how much would have becH raised in that state ' 415. If all the states had contributed in the same propor- tion as Massachusetts, what suui would have been raised iu the whole ? 416. If the sum which was contributed, enabled the Board to support 88 missionaries and assistants among the heathen, what is that for each ? 417. If all the states had contributed in the same propor- tion as Massachusetts, how many would it have enabled the Board to support, at that rate ? 418. In 1810, the number of blacks in the United States, was as follows: Northern states, 3 1 687 slaves 91317 free; Southern spates, 1159677 slaves, 95129 free: how many more slaves than free ? 419. In 181'^, there were living 1336 ministers of the gos- pel, graduated at the following colleges, to wit: Harvard, Yale, C9lum-)ia, Brown» Dartmou^^h, Carlisle, Williams, UBion, Bowdein, Middlebury, South-Carolina, Transylvania, Questions. 179 William & Mary ; allow 130 more for Princeton, and 84 for other colleges in America and abroad ; and how many mi- nisters are there in the United States, who have been educa- ted at college ? 420. If the number of ministers who have obtained a suffi- cient education without going to college, is half as many ; ,how many educated ministers are there in the U. Stales ? 421. If 900 of these are in New- England, how many more are wanted there, to make one for every 800 souls ? (See No. 4« 4. ^22. How many are wanted in the other states, at the same rate ? (See No. -t08.) 4:1^. How many souls are there in New-England, to one educated minister ? 424. How many in the other slates ? 425. The population of the United States is said to have doubled once in 'iS years ; if it should continue to do so, what wdl it be in the year 958 ? 4-6. The number of ministers educated at college in the United States, has doubled once in TO years ; suppose t to continue to double once in 69 years, and the numbei of edu- cated ministers in 1820 to be 25l'0 ; what will it be in the year 1958? 427. If one minister is necessary for every 800 souls, how many will then be destitute ? 4. 8. How many will be destitute, for one that is supplied? 429. Mrs. Norris left §30000 to the American Board for Foreign Missions ; what is the annual interest, at 6 per cent? 430. How many missionaries will that support continually, at $500 each per annum ? 431. If each of these missionaries should be instrumental in the conversion of 20 heathen souls every year, what will be the whole number in a century ' 4S2, If that interest should be applied to the board and education of heathen children in the mission families at Cey- lon, at 12 dolls, each, how many would it constantly support? 433. How many heathen children would thus receive a Christian education, in a century, allowing each to be 4 years at school ? 434. If it should be applied to the school expenses of such children as are fed and clothed by their parents, which at Ceylon amount to 48 cents each per annum, how many such children would it keep at school continually ? 180 Questions. 435. How many heathen children would be educated bv it, in this way^ in a century, allowing each to be 4 years at school r - ,S5. The exhibition of West's picture of Christ healing the sic^, produced the sum of '|4l33'4o to the Pennsylvania hospital, in ;8i8, froui 165 5 visitors; how much is that from each, in Pennsylvania currency ? 437. In the year i274, the price of a small bible, neatly written, was ^35 sterling; and the wages of a laboring man were i Jd. per day : huw many years, of 313 working days Cach, must h:*, have labored to pa^ for a bible ? 4 >8. If a laboring man now earns 50 cents a day, how Kiany bibles woul:; t le same labor pay for. at bO cents each ? 4.^ . If oiie person in >() in the United States, wears watch tiTiikets that cost 5 doll* s, how much might be saved for doing good by that class (f ^eutlv-men, by doing without these useless articles / 44.0. If one in 200 should save 10 dolls, in this way, what would be the anKmnt . '4 . If ')ne in 1 should save ^^0 dollars in this way, "what woii'l be tne arie.)iiii ? 44 . If one in (:• should save 50 d.rllars in this way, whai wo*. Id 'Ml the am'»unt r 4 3. If one in ^3!j'^0ii should save 100 duliars in tliih way, what would be the amount ? 4 44. What is the whole amount ^hat might be saved bj these fiV5 classes of gentlem vn ' 4.5 If clas-es of L dies, ecjiudly rjumerous, v/ear orna- ments to half that a*n :«;.ni, \vh r :\n^' ih'.'V ^^ave ? 446 A!g an educa^tum for the gospel oiini^iry, at A ^25 each per an ? 4 8. How many heathen children niight be ed'icated by It in a ceiury, at the rate mentioned in N s. i.s l. and 4 j 4 19. How many billies would it annually aiford for cha- ritable distrib I tion, at 60 cents each ? 4 50. The first society in P — , N. Y. the. population of whi^ h does not exceed 450 souls, besides supporung the g^js- pel a- home, contributed for the spread of the j^o-pel ahiui^d, in the year 1821, the fdlowing sums, to wit: For ioni-stic Bftissious, DiS-i^o; cash to the American Board, Dibo-dT; Questions, ISi clothing, 5cc. for Indian missions, D341'35 ; Auburn Semi- ndry, iJ39 ; EJu':ation a.id Bible societies, estiuiated at D25 ; board and tuit.oii ot a charity student, Do8 ; clotUmg for do. estimated at 030 : how jiiuch in all ? 451. it' all the people m the United States should contri- bute in tha same proportion every year, what would be the amount, and how many missionaries would it support, at 500 dollars each ? 452. In Oct. 1816, the money divided to school societies in Connecticut, from the school fund, was D^OoOS'TS, and in March, IBIT, the sa^ne sum ; and in trie same time, there was paid to citizens, as colli^tors^ fees, Dlo395-82; in t^e same time, the amount of the state tax payable into the trea- sury, was 048645*81 ; how much did the people of Con- necticut receive from the treasury that year, more than they paid in ? 453. A druggist mixes togetlier several simples, as follows: First, 2 oz. 3 dr. 1 sc; second, 3 oz. ^ dr. 1 S gr. ; third, 4 oz. 7 dr. 2 sc. ; fourth, 1 1 oz. 6 dr. 19 gr. : what is the weight of the whole coiiiposition ? 454. A, B, and C, each owe me 230 dolls. ; D, E, and F, each twice that sum ; what is the amount ? 455. Borrowed of A £'25 .. I6 .. 6^, of B £27 .. 16 .. 8, of 0^54.. 6.. ri; and paid A sgl4.. 19., 10, B =glO.. 19.. 9, C ^48 .. 1 .. 1 1 ; how much do 1 still owe ? 456. What cost 1945 bbls. flour, at D6'25 per bbl.? 457. What cost^l3 lbs. at 4s. 6d. per lb. ? 458. if a man who fails in trade, owes 3765 dollars, and is able to pay 45 cents on the dollar, what sum will his creditors lose ? 459. In 36 weeks and 4 days, how many seconds ^ 460. Bought 3 horses for £l6 .. 17 .. 7 each,, and 2 cows for £5 .. 14 .. 7 each, and 3 bushels of wheat for IBs. lO^d. ; what is the whole amount ? 461. Divide jLll.. 11.. 3 by 3. 462. If a debtor pays 7s. 6d. on the pound on L5S78, how much will his creditor receive ? 463. If 6 men have LS .. 10 for 4 days work, how much inu^t 36 men receive for 18 days work ? 464. Tell the amount ot D507'25 in 3 mo. at 7^ per cent? 465. In 1764 nails, how many ells Flemish ? 466. Divide L7 .. 1 .. 9 by :^7* 467. What cost 7121 lbs. at 15d. per lb. ? 182 Questions. 468. What cost 2345 acres, at L2 .. 3 *. 6 per acre r 469. Bought 2 i bales of cloth, in each bale l3 pieces, and in each piece 25 ells English, 4 qrs. 3 na. ; how much in all ? 470. Find the greatest common measure of 246 and 372. 471 . Reduce m to its lowest terms. 472. Reduce |, |, and J, to a common denominator. 473. Reduce 14/^;^ to aa improper fraction. 474. Reduce *||2 to its equivalent number. 475. Reduce | of f of |f , to a single fraction, 476. Reduce f of a farthing to the fraction of a £* 477. Reduce f of an acre to its value. 478. Add 1,71, and i off. 479. From y\,"take ^\. 480. Multiply i, |, and 3, continually together. 481. Divide ^f byf. 48^. If i of a ship is worth L7S .. 1 .. 3, what part of it worth XS.iO^ ? 483. If a man performs a certain journey in 35| days, tra- velling 13f hours a day ; how many days would he be, tra- velling 1 1 j% hours a day ? 484. What quantity of cloth, f yd. wide, will line 9^ yds. that is 2i yd. wide ? 485. If L2 ..5..} .. 2| be the interest of L9.5 for | of a year, in what time will lAS^ gain -LItV ^ 486. Tell the difference between 1-9 r85 and 2*73. 487. Tell the productof 3 by •3. 488. Tell the quotient of -48624097 by 179. 489. Reduce | to a decimal. 490.* Reduce 5 oz. 12 dwt. I6gr. to the decimal of a lb. 491. Tell the value of •u0994o of a mile. 492. What cost 6»25 hhds. of wine, at l-2s. a pint ? 493. If 2 persons receive 4-625s. for one day's labor, how much should 4 persons receive for 5*25 davs' labor ? 49 i. Tell the product of 4 ft. 7 in. by 9 ft. 6 in. 495. Of 12 ft. 5 in. by 6 ft. 8 in. 496. Of 35 ft. 4|r in. by 1 2 ft. 3 in. 497. Find the content of a load of wood 6 ft. 4 in. wide, 4 ft. high, and 7 ft. 8 in- long. 498. How many solid feet in a bale of cotton 7 ft. 6 in. loiflig, 3 ft. B in. wide, and 3 ft. 3 in. thick ? 499. Find the solid content of a stick of squared timber, P,0 ft. 3 m. long, 1 ft. 2 in. broad, and 1 U in. thick. 500. Find the cube of 29. (Questions. 183 )1. The square of 624. 502. The cube of 101. 503. The square of 4* 16. 504. The cube of 3-5. 505. The square of |. 506. The cube of |. 507. The 4th power off. , 508. The square root of 29506624. 509. The cube root of 48228-044. 510. The 4th root of 194481. 511. The square root of IJ. 512. The cube root of 39304. 513. The cube root of 4^|. 514. The square root of f-^\. 515. Bought 13 yds, of cloth, at 2(1. for the first ydi, 4d. for the second, 6d. for the third, and so on ; what was the amount? 5 i6. Bought 100 acres of land, at 1 doll, for the first acre^ 2 dolls, for the second, 3 dolls, for the third, and so on ; what ^vvas the amount? 5 i 7. Fourteen persons bestowed charity upon a beggar, the first giving 3d., the second 6d., the third 9d., and so on ; what did thelast person give, and what did the beggar receiver 518. A debt is to be discharged at sixteen several pay- ments, the first to be t/6, and the last to be L200 ; what is the common diiferencc of the payments, and what is the whole debt ? 519. Sold 10 yds. of clolh, the first for 3s., and the last for L2 .. 8 ; what was the common difierence ? 520. A certain person married off his daughter on New- Year's day, and gave her 5 dollars towards her portion, pro- mising to double it on the first day of every month through the year; what is the amount ? 521. Bought a horse, with 4 shoes on, and S nails in each f^hoe, at 1 cent for the first nail, 2 for the second, 4 for the third, and so on ; v,^hat was the price of the horse ? 522. Sold 30 acres of land; at 2 hob-nails for the first acre, G for tiie second, 18 for the third, and so on ; and sold the nails for a farthing per 100 ; how sraicii did I gain, if I gave L50 per acre for tlie land ? 52.^. Mixed 20 bushels of wheat at 5s., S6 of rye at 3s., und 40 of barley at 2s. ; how much i:^ a busliel of the mix^ tiire worth ? 184 (luestions, 624. A merchant would mix wines, at 17s., 18s., and 22s. per gallon, so that the mixture may be worth 20s. a gallon; what quantity of each must be taken ? 525. It is required to mix brandy at 8s., wine at 7s., cider at Is., and water at per gallon, so that the mixture may be worth 5s. per gallon ; what quantity of each must be taken ? 526. What number is that, which being increased by i, i, and I of itself, the sum will be 125.^ 527. What number is that, which beingmultiplied by 7, and the product divided by 6, the quotient will be 14 ? 528. One being asked his age, said, if | of the years I have lived be multiplied by 7, and | of them be added to the pro- duct, the sum will be 292 ; how old was he r 529. A son aking his father how old he was, received this answer : Your age is now one fourth of mine ; but 5 years ago, it was only one fifth of mine. What was the son's age? 530. How many days can 9 persons be placed in a diffe- rent position at dinner ? 531. How many different ways can the 6 vowels be put together ? 532. How many combinations can be made of 5 letters out of 20 ? 533. Tell the interest of ^365-123, for 7'S years, at 5 per cent. 534. Of ^325-5, for S'5 years, at 6 per cent. 535. Tell the amount of jL35-7, in 6*2^ years, at 7'5 per cent. 536. Of ^123-65, in 10 years, at 6} percent. 537. W^hat principal will amount to £565, in 4 years, at 5 per cent ? 538. What is the piesent worth of ^^390, payable in 5 years, at 6 per cent discount ? 539. At what rate per cent will ^300 amount to L428*25, in 9'5 years ? 540. In what time will jL4C0 amount to i491, at 6*5 per cent? 541. In what time will ^525 amount to X603-75, at 5 per cent ? 542. What is the amount of L2S0, lor 3 years, at 6 per cent, compound interest ? 543. W hat is the compound interest of ^350, for 5 years, at 4 per cent ? 544. What principal, at 5 per cent, compoand interest, f©r 4 jears, will amount to D972'4©5 ? ^nest. 185 545. WKat principal, at 4 per cent, compound interest for 5 years, will amount to L6691-5909632 '? 546. In what time will d^ 17500 amount toL22334-927343r5 at 5 per cent, compound interest? 547. At what rate per cent will ^225 amount to X263 •218176, in 4 years? 548. At what rate per cent will L1234 amount to L1469 •713744, in 3 years? 549. What is the amount of an annuity of D600, for 5 vears, allowing simple interest at b per cent ? 550. What is the amount of an annuity of D 5 00, for 4^ years, allowing simple interest at 7 per cent ? 551. Find the present worth of an annuity of D200, to continue 4 years, at 5 per cent, simple interest. 552. Find the amount of an annuity of D 100, for 4 years, allowing compound interest at 6 per cent. < 553. What is the amount of an annuity of L200, for 3 years, allowing compound interest at 4 per cent? 554. What is the present worth of an annuity of Z. 100, to continue 4 years, discount at 4 per cent, compound interest ? 555. Find the present worth of an annuity of D300, to continue 3 years, discount at 6 per cent, compound interest^ 556. Find the present worth of a perpetuity of DlOO, dis- counting at 4 per cent, compound interest. 557. What must i give for an annuity of L50, to continue forever, discounting at 6 per cent, compound interest ? 558. What perpetuity can be purchased for D 1 200, al- lov/ing discount at 5 per cent, compound interest? 559. What must I give for a perpetuity of D80, to com- mence 4 years hence, discounting at 4 per cent, com. int. ? 560. Which is the most valuable, and how much so, au annuity of D200 a year for 10 years, or a perpetuity of D400, to commence lO years hence, discounting at 4 percent, compound interest ? 561. Find the area of a triangular field, one side of which measures 186 poles, and the perpendicular upon it, from the opposite angle, 24 poles. 562. Find the area of a triangular field, the three sides of which measure 126 poles, 100 poles, and 86 poles. 563. Find the area of a field of six sides, which being di- \Tided into 4 triangles, the bases and perpendiculars measure as follows : triangles, bases. perpend, I triangles, bases, perpend^ No. 1 48 poles, 14 poles, No. 3 47 poles, 18 poles, 2 35 13 4 80 10 186 (:luestiGns, 564. What is the area of a circle, whose dianaeter is 10 poles ? 665. What is the circumference of a circle, whose diame- ter is 25 rods ? 566. Find the diameter of a circle, whose circumference is £78 rods. 567. If a piece of ground of 7 sides, one of which is 78 rods, contains 12 acies, what will be the content of another piece of the same shape, the similar side ut which is 25 rods ? 668. if the diameter of one circle is 20 rods, what must be the diameter of another, to contain S times as m.uch ground ? 569. What is the height of a tree, when, it jou set up a perpendicular pole 'zX) feet above ground, and take such a suUoii that }our eje is? iii a ran^e with the top of the tree and the top ot tl»e |julf, a our eye is 6 feet from the ground, and your station 10 tVet hum the pole, and 6' from the tree ? 67t. What \h the breathh of a river, accoroing to Problem 7s of Mensuration, when Eb is 15 rods, FD 35 rods, and DA 135 rods? 571. Find the number of solid feet is a stick of squared timber, which measure;- at Oie butt end, l6 inches by 14, and at the small end, 14 by 10, and is 45 feet long? .^7'2. Find the superficial content of a pyramid, the base of v/hich is 1^ feet scpiare, and the slant height 25 feet. 57 S. Find the solid content of a cone, the diameter of whose base is 10 teet, and the height 30 feet. 574. if a cone, the diameter of whose base is 6 feet, con- tains 150 solid feet, what must be the diameter of the base of a similar one, that shall contain 300 feet ? 575. What is the solid content of a stick of round timber, ■whose diameter is 15 inches, and the length 50 teet, the dia- meter being the same throughout ? 576. What is the superficial content of a globe, whose diameter is 25 inches ? 577. In 600/. Canada, how much New-York ? 578. Reduce -^}^ to its equivalent nun»ber. 579. V^ hat cost 7 100, at ef d. each ? 5 SO. Tell the least common multiple of 7, 8, 9, and 10. 581. Reduce |f| to its lowest terms. 582. Reduce | of an inch to the fraction of a yard. 583. Reduce | ot an English guine i to sterling. 554. In 40/. sterling, how much Kentucky ? 5^§. T^U ^le value ©ft ©f a £. ^msttons^ i97 586. In 400 f, sterling, how many livresr 587. Multiply 2-714 by 100, ^88. Multiply -3 by -3. 589. Divide -6 by 6, 590. Add f , f, and 4. 591. Tell the superficial content of a globe, whose circum- ference is 6 feet, 592. Tell the number of solid inches in a globe, whose diameter is 3 feet. 593. What is the number of wine gallons which a cask will contain, of which the head diameter is 36 inches, the bung diameter 40 inches, and the length 06 inches ? 594. Find the number of tons a ship will carry, whf n the length of the keel is 150 feet, and the breadth of the beam 80 feet. 595. How many solid inches in an irregular stone, which being put into a cubical vessel, and the vessel filled with water, and the stone being taken out, the space left empty measures 15 inches long, l5 wide, and 4 deep? 596. If a man weighing 100 lbs. rest on the end of a lever, 12 feet from the prop, what weight wdl he balance at the other end of it, 8 inches from the prop, no 'allowance being made for the weight of the lever ? 597. If a power weighing 50 lb. be applied to the end of a rope, round a wheel whose diameter is 4 feet, what weight will it balance suspended to a rope which goes round an axle 6 inches in diameter, no allowance being made for friction ? 598. What is the height of a steeple, from the top of which a bullet being let fall, strikes the ground in 4 seconds ? 599. Salisbury steeple, in England, is supposed to be 400 feet high ; how long w^ould it take a bullet to fall from its top to the ground ? 600. How deep is a chasm, into which, if you drop a stone, it is 8 seconds before you hear it strike the bottom ? 601. The population of North America is estimated as follows : United States 9640000, British possessions 1420000, Indians in both 510000, Floridas 25000, Mexico 75410000, Guatimala 1200000, British West Indies SiOOOO, Spanish do. 900000, French 150000, Dutch 80000, Danish 40000, Hayti 600000 ; how many in all ? 602. Of South America as follows: New- Grenada 1 600000, Caraccas 900000, Peru 1500000, Chili 900900, Buenos Ayres 1100000, Portuguese Brazil 2000000, GuiaEa 50000©, In- 188 ^iiessions, dians in Brazil 2000000, in Amazonia 3000000, in Fatogo- nia, ^c. i,iG 000 ; how many in all ? (303. Of Europe asfoaows: Great Britain and Ireland 1 6816^00, Sweden i 77000, Norwuj i^i^Oou, Denmark 164o000, Russia 3;>829000, Prussia 9737000, Holland 20n2:.00, Is ethedands 4140000, France 267750'. 0, Austria £4646j() ', German s^tates 150000 0, Switzerland 1768000, Spain 10396000, Portugal 3559000, Italy l6n7Oo0, Turkej 988200©, various islands 275ooo ; how many in all ? 604. Of Asia as follows: Russia 14 millions, Turkey If, Persia ^2, Arabia 11, Hindoostan loo, Burmah :2o, Siam, Malacca and Lacs lo, Japan 3o, Chinese empire !si-o, Indian Archipelago icio, Independent Tartary 2, Australasia 5, Po- lynesia ?, Ceylon and other islands 2 ; how many in all ? 605. The population of Africa is estimated at 98945ooo, of which the empire of Morocco is said to contain '4886ooo, Algiers 15ooooo, Tunis 3 mdlions, Tripoli 1 million, Egypt 3500000, Nubia 25ooooo, Abyssinia 2 m llions, Madagascar 4 millions, British colonies ISo-ooo, British, Spanish and Por- tuguese islands oooooo ; and the rest are savages ; how many ^re th last? 606. The seven most populous cities of Asia, are Pekin, containing 3 millions ; Nankin, 2 millions ; Canton, 2 mil- lions ; Fo-han, Hang-tchau, King-te chmg, and Jeddo, each 1 million : how many people in these seven cities? t)07. The next ten cities in Asia, are, Calcutta, containing 650000 inhabitants, Surat 6ooo©o, Miaco 5^9726, Benares 500000, Patna 5ooooo, Susa 5ooooo, Ispahan 4ooooo, Madras Sooooo, Erzerum 27oooo, and Aleppo 25oooo ; how many less in these ten, than in the foregoing seven ? 608. The seven most populous cities of Europe, are, Lon- don, containing 1 £00000 inhabitants, Paris 715595, Constan- tinople 500000, Naples 412439, Lisbon S5oooo, Moscow 300000, and Fetersburgh 271137 ; how many less in these, than in the seven first cities of Asia ? 609. The next ten, are, Vienna, containing 252o49, Am- sterdam 217o24, Dublin l9oooo, Berlin 169ooo^ Madrid 156672, Palermo l5oooo, Barcelona 147ooo, Ediiiburgh and its port 138235, Venice 13724o, and Rome I29ooo; how many less in these ten, than in the preceding seven ? 610. The seven most populous cities of Africa, are, Fez, containing 38oooo inhabitants, Cairo 3ooooo, Morocco v7oooo, Tanis 150000, Mequinez 11 0000, Sennaar looooo, and AI- Questions. 1 &9 pers 80000 ; how many less in these, tlian in the seven first cities of Europe ? 61 1. The seven most populous cities in America, are, Rio Janeiro, containing 15oooo inhabitants, Mexico ISrooo, New^ York 143706, Philadelphia 118630, St. Salvador ilOooo, Potosi 1 00000, and Buenos Ayres 7oooo ; how many less in these, than in the seven first cities of Africa? 612. The Netherlands contain i75oo square miles, and 41 4o255 inhabitants ; how many is that to a square niile ? 613. The Italian possessions of Austria contain 17453 square miles, and S82ol28 inhabitants; how many is that to a square mile ? 6i4. The earth contains 199 millions of square miles, of which I60 millions are sea and parts unknown; how many square miles compose the habitable world ? 615. Of the habitable world, America contains m, Asia ^f f , Africa 3^^, Europe 3Y0, and New-Holland the rest ; how many square miles in the last ? 616. According to Melish's map of the United States, the number of square miles within their limits, from the Atlantic to the Pacific, is 2256955. If the population should increase to the amount mentioned in No. 425, how many will it be to a square mile ? 617. According to the map of the land of Canaan, as di- vided by Joshua, it contained 8362 square miles ; from which, deduct the territory of the Sidonians, 5o miSes long and 8 broad, and that of the Philistines, 4o long & 15 broad ; and how many square miles were left for the Israelites ? 618. The Israelites, in the time of SoloBion, are supposed to have been seven millions ; if so, how many is that to a square mile ? 619. In l§ol, the population of Great Britain was stated as follows : England 8331434, Wales 541546, Scotland 1599o68, Army and Navy 47o598 ; how many in all ? 620. In 181 l,as follows: England 94994oo, WaJes 6o738o, Scotland 18o4864,Army & Navy 64o5oo; how many in all? 621. What was the increase per cent in those Jo ytars? 622. Allowinj*: the same increase per cent for the next ten years, what would be the population in 1821 ? 623. If England and W ales had increased at that rate per cent, what would their proportion be in 182 1 ? ' 1 9o (^iiestions. 624. It is 'estimated that there are, in England and Wales, I0434 Episcopal clergymen, and as many dissenters. If the whole population were equally divided among them all, how many souls would constitute the charge of each minister ? Gis. The number of Synods, Presbyteries, and Ministers in the established church in Scotland, in 18o3, were as fol- lows : SvnGcls.' Presb. Min. Lotiiian and Tweeckle, 7 116 Merse and Teviotdaie, 6 66 Dumfries, 5 54 Galloway, 3 m Glasgow and Ayr, 7 130 Perth and SterliDg, 5 SO Fife, 4 71 Angus and Mcarns, G 81 How many Presbyteries, and how many ministers in all ? 626. Suppose there were loo Burgher ministers, So Anti- burghers, and £5o of other denominations; and add the mi- nisters of the established church, and divide among them the whole population of Scotland, as it was in I80I, (See No. 619 ;) and how many souls will be to each ? 62r. The number of Synods, Classes, and Ministers, in the established cliurch in Holland, in i8o3, was as follows S 1/710 ds. Aberdeen, MorAy, Ross, Sutherland & Caithness, Ar^ie, Glenelp:, Orkney, Presb, Min. 8 101 ■ 7 54. 3 23 ;s3, 3 23 5 41 5 • 29 4 SO Synods. Classesi- JlLi. Synods. Classes. JS'lin. Gueideriund, 9 245 Friesiand, 6 207 South Holla.id, 11 331 Overyssel, 4 8* North Holland, G 220 Drente, 3 40 Zealand, 4 163 Groningen, 7 161 Utrecht, 3 79 On the i;land of Ameland, How many Classes, and how^ many Mini-lers in all? 628. There were also, of oti er denominations, as follows : Walloon Calvhiists, 5o ministers ; Kn*.';ii-;h Presbyterians, 7; Scotch do. 1 ; P^piscopalians, 2 ; Catholics, 4oo { Lutherans 7o; Remonstrants, ^5 ; Baptists, 25 i; Khinsburghers, So ; Armenians, I : how many in all ? 6!i9. Deduct 50000 Jews from the population of Holland, as stated in No. 6o3, and divide the reominder among all the ministers ; and how many souls, will there be to each ? 6. o. Of the population of Russia in 819, it was stated that -.1262000 were ot the established church ; and the num- ber of clergymen in that church in lHo5, was stated as fol- lows : Protoires, Prieijvts, and Deacons, 441^7; Readers and Sacristans, 64'?39. If this population w^ere divided equally among all the clergy, how many souls would there be to each ? ({iiestions. 191 GS 1. The number of Methodists in 18o5, was as follows : in Great Britain lol9l5, Ireland 23321, Gibraltar 4o, West Indies 2265o, America o2328 ; how many in all ?' 632. In 1819, the number in the United States was as follows : Ohio Conference 29 i 34, Missouri 4764, Tennessee 2o6r6, Mississippi ^'>71, S. Carolina 32646, Yirginia 22585, Baltimore 3^7:6, Philadelphia 32796, New-York 22638, New-Encrland 15312^ Genesee 23913: how many in all? 633. Tiie number of travelling preachers was 812; how many were under tlie care of each, on an average ? 634. The number of local preachers was staged to be more than looo ; suppose them to be 1 188 ; and how many mem- bers would there be to a preacher ? 635. The same Tear, the number under the care of the British and Irish Conferences, was 242459 ; how many were in all the world ? 656. The following statement of the number of Baptists in the United States, is compilfed from the returns made to their Gen. Convention in 1 82 1. Wliere there were blanks in those returns, they are filled by estimates, and marked with an asterisk. The number in New-England is stated as fol- lows : min. C/.A-9. inem. min. chhs. mem.. Maine, 129 171 9740 Massachusetts, 105 109 10078 N. Hampshire, 34 44 2765 Rh. Island, 41 54 r.052 Vermont, 97 121 9978 Connecticut, 59 60 6946 How many ministers, how many churches, and how many i.x3mbers? 637. In the Middle states, as follows : f New-Y(irk, 304 426 35200 .Delaware, 7 7 55S j N.Jersey, 22 23 £225 :vL,rylar.d, 19 36 $06 j *Pemuylvan!a, 76 81 597r)IColumbla Dist. 11 16 1511 ]. How many ministers, how many churches, and how many 'members'? 638. In the Southern states, as follows : »\irginisi, 157 300 21000 Geortria, 101 181 1457^ *N. Carolina, 198 233 11924!"A'abama, 42 61 252? *:5. Carohua, 95 190 14401 ^Mississippi, 36 68 2006 How many ministers, how many churches, and how many members ? 639 In the Western states, as follows : *rennessee. 123 170 9707 *ln(liana. 61 102 3886 ^Kentucky, 240 395 27299 tllinois, 19 15 332 *Ohio, 87 150 5557 * Missouri, 27 37 726 \ How many ministers, how many churches, and how many members ? 1§2 Questions, 64.0. Take the total of the ministers, churches, and mem- bers, in the preceding four que-^tioos, and add the seventh day Baptists, esti'-iateil at 15 in'misters, 2o churches, and 2ooo members ; and what is the amount ? 64}. tlow many more churches than ministers? 64^^'. In l^-it J the General Assembly of the Presbyterian Church in the United' States, had under iis care ^2 i^.v.>by- ieries, of which 5o reported l3oo co igregations, 7 ''4 or iain- ed, and lo3 licensed preachers; if those wliich .iui not re- port, contained each two thirds as many in proportion as those which did, how many congregations, and how many preachers in kll ? 643. If every preacher supplied a congregation, how many congregations v/ere 2o, the several Congregational associations of Kew- Hampshire, reported as follows : min. c'^hs. mm. chh^. Deeifild, 8 9 Orange, 8 12 Haveihill, 6 8 Pisc^Uqua, IG 15 Holies, 4 5 Plvtnouth, 3 6 Hopkiuton, 11 12 Union, 7 Mofiadiock, 14 17 Coos. 3 10 If there are lo ministers and lo churches not connected v/ith these associations, and 8 licensed preachers ; what is the whole number of preachers and churches of this order ia New Hampshire / 647. In 81 of these churches, there were 8843 members; if the rest. averaged the same number each, what was the whole number ot members ? 648. In i8lo, the Congregational ministers and licentiates in the several associations in Vermont, were as follows : Windham, 13 | Royaltoa, 11 Rutland, 19 7 Orange, 13 5 Adlison, 13 2 j N. Wester*, 8 How many in all ? 6;9. In 18-4, the number was 79 settled ministers, 1© unsettled, & 8 licentiates; what was the increase in 4years 65o. At the same rate of increase, what would be the BLUmber ia l82o ? ((uestwns. 19S 651. If there were 140 churches of 109 members in each;, how many members were there in all ? ^^5^. la 18:0, the ministers and churches connected with the Congregational Associations of Connecticut, were as follows : min. cJihs. min. chhs.. Ilartfoid North, 21 20 Fairfield East, 8 VZ Haitford South, 16 16 V^indham, 20 2$ New-Haven West, 16 19 Litchfitld North, 18 18 New Haven East, 11 1.'3 Litchfield South, 15 18 New-London, 15 21 Middlesex, 15 15 Fairfield Wt^st, 10 i; lolland. 15 15 Besides which, there were 2 1 unsettled ministers, and 9 licensed preachers ; how many ministers, how manj church- es, and how many members, if the churches average lOy each? 653. The number of settled Congregational ministers in Massachusetts, was stated, in 816, to be 3'23. Suppose thefn, in :820, to be 330, and the number of vacant churches to be 5o ; suppose the unsettled ministers and licentiates to be ^0, and the churches to average 1* 9 members ; and what is the number of Congregational ministers, of churches, and of members, in Massachusetts ? 654. The number of settled Congregational ministers in lihode -Island, was stated, in 1816, to be b. Suppose them, in 1820, to be 9, and the number of vacant churches to be 4 ; supj>ose the unsettled ministers and licentiates to be 3, and the churches to average I'^t' members ; and what is the num« ber of ministers, of churches, and of members, in Rhode- Island ? 65 -. Suppose t^ ere are, in the states out of New-England, 110 Congregational ministers, and 150 churches of i09 mem- bers each, and in Maine the same as in Vermont ; and what is the whole number of ministers^ churches, and members, m the United States r 656. The number of ministers and churches of the Asso- ciate Reformed Presbyterians, in 18 16, was as follows : Synod of New-York, 2 Min. 32 Chhs.; Pennsylvania, 9 IMin. 23 Chl»s. ; Sciota, 23 Min. 53 Chhs. : how many minis- ters, how many churches, and how many members, if thej average at 109 members each ? 6:*. Suppose the Associate Presbyterians to be 20 minis- ters and 3o churches, and the Cumberland Presbyterians to be 25 ministers and 40 churches, and that these churches av^erage 109 members each ; and how many ministers., churches, and members, in these two bodies ? H 194 %iestions, 658. The Dutch Tleformed Church, in 1 820, contained tht following classes, ministers, and churches : Classes of min. chhs. Classes of min. chhs. Nev.-Yoi'k, 16 15 Albtniy, 10 16 New-Brunswiek, 9 10 Washington, 4 9 Bergtn, 9 13 Poughkeepsie, 8 11 *Paramus, 8 11 Ulster, 8 20 *Long-lsland, 7 9 * Montgomery, 11 14 Pliiladelphia, 7 6 *Rensselaer, 5 6 «^^7^. emig. In Pennsylvania, 50 246 Ohio, 5 52 >taiylancl, 7 34 Virginia, 4 33 Besides which, there were 4 licentiates reported, and pro- bably as many more not reported ; if so, how many ministers, and churches, in that body ? 659. Of these churches, 67 reported 902S communicants ; if the rest averaged 1(9 each, what is the whole number ? 66i;. The German Reformed Church, in 1821, reported as follows : min. cong. N. Carolina, . 1 28 S. Carolina, 8 Tennessee, Kentncky, & elsewhere, (estimated,) ' ^13 99 Besides which, there were 10 licentiates; how many mi- nisters, how many congregations, and how many communi- cants, if the congregations average 30 each ? 66 1. Add the ministers, churches, and members, as above stated, of the General Assembly, the Congregationalists, the Associate Reformed, the Associate and Cumberland Presby- terians, the Dutch anu German Reformed Churches; and find how many ministers, how many churches, and how many members in communion. 662, The number of Episcopal clergymen in the United States in 'v8l7, was stated as Ibllows : New-'flampshirp, 4 y Massachusetts, 13; Vermont, 4; Rhode-Island, 4; Con- necticut, 35; NeW'York, 67; New-Jersey, 11; Pennsyl- vania, 25 ; Delaware, 3; Maryland, 36; Virginia, 33 ; N. Carolina, 3 ; S. Carolina, 17 ; how many in all r (563. In 1821, there were 9 bishops, 200 presbyters, and 124 deacons; what was the increase in 4 years ? 664. Of these, there were, in the diocese of New-York, 77 ; Maryland, 5 '■ ; Virginia, 33 ; Ohio, 5 : how many in the ether live dioceses ? 665. If there are 500 churches, with 80 communicants in each, what is the whole number ? 666. If the number of Lutheraii ministers is 100, and their churches 150, with 80 members in each; what is the number of their communicants ? * EstbiiRtes, Questions. 195 667. What is ihc whole number of preachers of all the jove denonnnations, according to Nos. 633, 634, 640, 661, 663, and 666 ? 668. If 500 of these ministers are employed as officers of colleges, instructors of academies, or are, from various causes, not employed as ministers ; and the rest, except the Metho- dists, are all engaged in supplying each one church ; how many churches are destitute? 669. How many more preachers are wanted, that every 800 souls in the United States may have one to supply them? 670. What is the whole number of communicants of all ihii above denominations ? 671. How many are not communicants of any of these denominations, for one that is ? 67-2. The expenditures of the Worcester county charitable society in 1817, were, for education, S703*48 ; foreign mis- sions, $2i)4-9^ ; feeble churches, $280 ; bibles, &c. $57*45 ; ontingent expenses, §5*05 : how many guilders in the whole? 673. The expenses of the Deaf and Dumb Asylum at Hartford, in the year )8l8, were, for buildings and lands, .^'S860-85 ; for tuition, S3283-87 ; boai\l of pupils, $8398-80: how many rubles in the whole ? 674. The receipts were, donations, g7528'48 ; paid by pupils, . $5843*20 ; contributioQS from churches in Connecti- cut, |2546-J 2 ; interest of fund, $1018-42 : how many mill- eas in the wliole ? 675. The payments of the Female Society in Boston foi he conversion of the Jews, in 1818, were, for board, &c. of N. Myers, D40; sent the Society at London, 1)444*44; for the education of Jewish children at Bombay, 1)100 ; printing, D23-50; freight, D2-35 ; exchange, D6.'91 ; deposited in bank, D129 : how many marcs banco in the whole ? 676. In Liberty county, Georgia, there was said to have been contributed, in i8l8, for charitable and religious pur- poses, by 75 persons, asioliows: For free schools, D 1600 ; bible society, D20v}0 ; clergymen, D3000 ; female asylum, D650 ; missionary, tract, and education societies, DlOOO. if 37 persons paid Do each, what w^as the average to the rest?. 677. A laboring man in V(r.aont, saved the following amount in one year, for charitable purposes : By working on the 4th of July, 7 5 cents ; by not wearing a cravat, one dol- lar ; by doing without ardent spirits, one dollar ; by having his cloth only colored, but not dressed, l)i**;5; by wearing, 196 Questions. himself and family, thick shoes, 4 dollars : what was the amount ? 678. If every tenth person in the United States would " go and do likewise," how great a fund would it annually raise ? 679. In 1831, Rev. Mr. Ward collected for the Baptist Mission College at Serampore, in New-York, D2467*19 ; Boston, D]8b0-62; Philadelphia, D]202-6!2; Baltimore, D420 ; Washington, D2r 1 ; Princeton, DM2 ; New- Haven, D406-50; Hartford, D28( -06; Providence, D3 12-68 ; Alex- andria, D40 ; Newark, D9S-1 9; Pawtucket, D59 ; Middle- town, D103; Schenectady, Dl90; Worcester, Dlb:0-37 ; Cambridge, Dl81 ; Salem, 0200-72; Portland, D24I-06; North- Yarmouth, D85-73 ; Portsmouth, D84-42 ; eleven other towns in Massachusetts, D701'04 : how much in all ? 680. Kean, the playactor, visited several of our principal cities near the same time, and received for himself about D50000, for his winter's wages; how many missionaries would that support for six months, at DsOO a year each ? 681. The expenses of the Richmond theatre for the two last seasons, were stated, in 1821', to have exceeded the re- ceipts by D 19884 ; how many bibles would that furnish for distribution, at 60 cents each ? 682. The receipts of the Domestic Missionary Society of Massachusetts, for 1820, were D6 19-63; the balance on hand from the former year, was D380-15 ; and the expendi- tures of that year were D644-48 : what was left ? 683. The receipts of the Connecticut Missionary Society, for 1818, were, donations and interest, D5052-2I§ ; contri- butions, D32l3'24J ; and the expenses wxre D7244-57 : what was the excess of expenditure ? 684. The productive property of the Massachusetts Mis- sionary Society, in 1821, was D5 327-28 ; what is the annual interest, at 6 per cent .^ 685.. If the expense of the American Board is D57144, and should be equally divided among the inhabitants of viassachusetts, how much would it be for each ? (See No. 104.) 686. In 1803, the Society in Scotland for propagating the gospel at home, had sent out in all, 100 missionaries, and had received in all, X2683 .. 5 .. 1 1, and paid for tracts, 1/356 ..0 .. 9 ; what was there left for each missionary ? 687. In 1793, there were 4 missionaries in India; in 1809, Questions, 197 there had been added 5 Episcopalians, 14 Baptists from Eu- rope, 3 Hindoo Baptists, 1 Presbyterian, 6 Independents, 2 Lutherans ; besides which, there were 3 missionaries in Ceylon, and 1 in China ; how many in all ? 688. In a certain town in the interior of New-England, containing 3000 inhabitants, there was raised, m 1803, for schools, D800; the poor, DlOOO; taxes, D900 ; support of ministers, D670 ; highw^ays, D3000 ; incidental expenses, D 1 000 : how much is that for each inliabitant ? 689. In the same year, there were retailed in the town, 10230 gals, of N. E. rum, at 6i cents a gaL ; 5900 gals. W. I. rum, at Dl a gal. ; 1500 gals, brandy, at Dl'50 a gal. ; and rso gals, gin, at 1) 1*50 a gal. : what is the expense for each iihabitanf ? 690. What quantity of ardent spirits for each ? 691. In a certain district of Bengal, in two months of 1812, 70 widows were burnt alive on the funeral piles of their deceased husbands ; how many would that be in a year? 692. These 70 left 1 84 orphan children ; how many were left by the whole, at the same rate ? 693. Within 30 miles of Calcutta, there were 275 widows burnt alive in 1803; if that extent contains 785000 inha- jitants, how many would be burnt in the whole of Hindoos ill, at the same rate ? (See No. 604.) 6y4. If there are 35000 widows annually burnt alive in Ji'^ whole of Hindoostan, as is supposed to be the fact, how many orphans are thus annually made, at the above rate? 695. There are 12 pilgrimages annually made to the single ernple of Juggernaut in Orissa, at each of which from 100000 cO 600000 persons attend, of which a vast proportion (some think a large majority) never return home, but die, from want, fatigue, fevers, &,c. Suppose the average attendance to be 300000, and that of these, only one in five die ; what is ihe number of lives annually -sacrificed to this one Idol ? 69^>. In the wars kindled by the ambition of Bonaparte, from 1800 to 1815, it is estimated that the following lives were lost: In Hayti, 160000; in the war with England, du- ring 12 years, 200000; in the invasion of Egypt, 60000; in the winter campaign of 1805 and 160old cojitribute one cent a week, how ma >y tru'ts ol ; 'i pages eacii, would it pay for annually, at 1 nri* a |»a«,t T 78 ^. if every person in the Un;red Siates should contri- bute one cent a week, for doing gtiod, how much would be thus raised annually ? 7H4. How many missionaries would the half of it support, at 500 dollars a year each ? 78 . How many young men would the quarter of it assist in preparing tor the ministry, at *25 dollars a year each ? 786. How many bibles would'the eighth of it pay for, at 60 cents each ? 787. How many tracts, of 12 pages each, would the rest pay for, at I mill a page ^ 788. According li, 5 S West Tennessee, 6 16 Missouri, 4 *Shiloh, 7 10 I How many ministers, and how many congregation'^^ •S • Estimates, 206 Questions, 798. In the Synod of North-Carolina, as follows : Presbytery of JMiiu Co7ijf, Concordi 16 68 Prestytery of JWn» Cong, Orange, 10 22 Fayetteville, il 32 How many ministers, and how many congregations ? 799. In the Synod of S. Carolina & Georgia, as follows : Harmony, 19 «8 I Hopewell, 6 15 S. Carolina, 15 30 j How many ministers, and how n^any congregations ? 800. Take the whole number of ministers, and the whole number of congregations in the preceding eleven questions, and add 104 licensed preachers, not ordained ; and tell the whole number of authorised preachers, and of congregations, in connection with the General Assembly, in the year 1819. 8' a. If the whole number of communicants in all the churches in the United States, should be equally divided into 8 classes, how many would be in each class ? 802. If the first class, comprising the most wealthy in our large towns, should contribute each 50 dolls, a year for cha- ritable purposes, what would be the annual amount? 803. If the second class, comprising the most wealthy in the country villages, give each SO dollars, what would be the amount ? 804. If the third class, comprising those less wealthy, give each lO dollars, what would be the amount? 805. If the fourth class, comprising farmers, mechanics, &c. give each 5 dollars, what would be the amount ? 806. If the fifth class, comprising young men, give each 3 dollars, what would be the amount ? 807. If the sixth class, comprising young w^omen, give each 2 dollars, what would be the amount? 808. If the seventh class, comprising the poorer sort, who enjoy health, and can labor, give each one dollar, what would be the amount? 809. If the eighth class, comprising the ag;ed, the infirm the feeble, &c. give each 25 cents, what would be the amountr 810. What would be the whole amount annually contri- buted by these eight classes ? 811. if they should all continue their contributions for 10 years, W'hat would be the amount ? 812. If this was made a permanent fund, what annual in- terest would it yield, at 6 per cent ? 813. If half that interest should be applied to the educa- tion of young men for the gospel ministi y, how many v/ould it assist, at g 1 25 each ? Questions, 207 "^4. If the other half should be applied to the support of issioiiaries, how many would it maintain, at 8500 each ? 815. In 1817, the receipts of the British National School Society for educating the poor, on the Lancasterian system, had been, for 6 years, SS25QI. sterling ; how much is that in federal money ? 816. The number of children in their schools in that year, was 155000; if one fourth of the above sum was expended in tbeir education in that year, how much is that for each child? 817. It is estimated, that there are 64 millions of children in the world at a time, who ought to be at school ; and if each child can be schooled, on the British system, at 30 cents a year, what would it cost for the whole ? 818. What, for 22 years ? 819. If 5 persons are allowed to a family, how many fa- milies are there in the whole world ? (See No. i3.) 820. If 5 millions of families are already supplied with bibles, and a bible costs 60 cents, how much would it cost to supply the rest of the world with one to a family ? 821. If a preparation of 8 years, at an expense of 203 dollars a year, is necessary' for a missionary ; what is the whote amount ? 822. If one missionary is v/anted for every 3000 of those who are not nominal Christians, what would it cost to edu- cate the requisite number? (See Nos. 13 and 14.) 823. If it should cost as much to convey each one to his station, as the passage from i^merica to India, which is 250 dollars ; v/liat would it cost to convey the whole number to their stations r 824. What would it cost to maintain the wli(4^r"22 years at .500 dollars a year each ? ' 825. If 40000 additional ministers are wanted to supply the destitute in Christian countries, what would it cost to educate them, and support them 2'^ years, at the above rates? 826. Wliat would it cost to scliool all tlie cluklren in the world, for 22 years ; to supply the whole world with bibles ; and to educate and support for 22 years, a supply of ministers iid missionaries for all the destitute in the world ? 827. During the war consequent upon the French revoiu tion, from 1793 to 1815, a period of 22 years, the war expense of Great Britain is calculated to have been 3200 millions of dolls.; France, 3130 millions ; Austria, 1000 millions; U. £08 %iestwns. States, (3 years,) 120 millions ; other powers of Europe, es- timated at 4550 millions : what was the whole expense of that war to the nominally Christian vvorld i* 828. If they had been willing to expend one third as much to inform, moralize, and chri^^tianize the world, how much more than sufficient would that sum have been, to accomplish all these objects, during the same period, according to the preceding statements ? 829. If, instead of an appeal to arms, the nations had esta- blished a general congress of all the Christian powers, for the settlement of all difficulties between nations ; and the above surplus had defrayed its expenses for the same period, how much would it have been for each year ? 830. In the preceding estimate of the expense of a single war, it is probable that nothing is included but the sums ac- tually paid out by the respective governments ; and that the loss of productive labor, the loss of lives, and the destruction of private property, are omitted. The military w^r establish- ment of Europe is stated at 3908000 men. Suppose only 2 millions were actually under arms during those 22 years, and 9. millions more were employed m preparing and conveying arms and stores ; what is the loss of productive labor, reck- oning these men to have been able to earn only 30 cents a day, at some useful employment, excluxling the Sabbaths, ana including the additional days for leap years ? 831. If the number of lives lost in the first seven years of that war, was in the same proportion as in the last 15, (see No. 696,) how many lives were lost in all ? 832. If the pecuniary loss to the public, from the death of m able bodied man, is S1500 ; what is the amount of this item of loss by that war > 833. What is the total amount of the loss of labor and the loss of lives? 834. Among the numerous idolatrous festivals of the Hin- doos, it is computed, that the annual expense of one to the inhabitants of Calcutta, is 500000/. steiling ; what would it be to the whole of Hindoostan, at the same rate ■? (See No* 60;, mr.) 8 5. How n^any missionaries would it support, at II SL each ? 836. If the whole population of Hindoostan were divided Hmong them, how many souls would it be to each ? # ^uestianh 2bd 837. What was the solid content, in feet, ot Noah's ark, ^being 300 cubits long, 50 wide, and 30 high ? ' ' 838. It is stated by the learned, that there are about 150 kinds of quadrupeds, 200 kinds of birds, and 40 kinds of reptiles, that must have been preserved in the ark. Suppose there were £00 kinds of quadrupeds, of which 20 were clean; 300 kinds of birds, of which 20 were clean ; and 50 kinds of reptiles ; and that 7 of every kind of clean animals, and 2 of every kind of unclean ones, and 8 persons, were preserved ; how many living creatures would there be ? 839. Suppose the quadrupeds to average the size of a two year old steer, and to require stalls that should give each 8 feet in length, 5 in width, and 6 in height ; the birds to ave- rage the size of a hen, and to require lofts that should give each 2 cubic feet of room ; and the reptiles to have i cubic foot each : how many cubic feet would all these animals occupy ? 840. Allow Noah and his family, 1 kitchen, 20 ft. long, 20 wide, and 10 high ; i parlor, and 1 store- room for provisions* of the same dimensions ; and 4 lodging rooms, each 10 ft. long, 10 wide,& 10 high ; how many cubic feet would they occupy ? 841. If one third of the unclean beasts and one fourth of the unclean birds were carnivorous, and 2 sheep a day were allowed for 6 beasts and 7 birds, and each sheep required 62 'cubic feet of room ; how many cubic feet would be occupied by the sheep necessary in a year ? 842. If each of these sheep was allowed a quart of grain a day, and an equal quantity of water, till it was killed, how many cubic feet would be necessary to store that grain and water ? 843. If each of the rest of the beasts was allowed a peck of grain, and an equal quantity of water per day ; and each of the rest of the birds, one eighth of a quart of grain, and an equal quantity of water per day ; how many cubic feet would be necessary to store that grain and water ? 844. If each of the sheep kept for food was allowed :J of a cubic foot of pressed hay per day, till it was killed ; and each of the other beasts not carnivorous, was allowed 2 cubic feet per day, through the year ; how many solid feet would be necessary to store that hay ? ^4i. Allow Noah's family to use 50 gas ale measuje, water per day j how many cubip feet would be necessary to store it ? 1S2 210 i^uestions. ^^ 846. Allow Noah 3 rooms, each 20 ft. long, 20 wide, and 10 high, to store farming utensils, and other necessary arti cies ; how many cubic feet would they occupy ? 84/. Allow JO ft. wide, at one end of the ark, the whole breadth and height, for stair cases; and 10 feet wide diroisgl the whole remaining length, and the whole height, for pas- sages ; and how many cubic feet would be occupied by these 848. Add the preceding nine items together, and how many cubic feet wouJd be left for the thickness of the walls, floors, partitions, and other purposes ? 849. The contributions to the funds of the Americar Board for Foreign Missions, for the year ending Aug. 31, 1821, were as follows : From Massachusetts, 19820-66 dolls.; Connecticut, 7874-08 ; New York, 6424-96 ; Vermont, 1912 •96: N. Hampshire, 1699*40; Maine, 1429-76; N.Jersey, 1384-74; Pennsylvania, 1312-99 ; Georgia, 1052-36; Ohio, 506-10; S. Carolina, 495-06 ; Virginia, 400-62; Maryland, 393-50; Kentucky, 369-31; N. Carolina, 251-30; Rhodes Island, 65-56 ; Tennessee, 63-00 ; Delaware, 36^00 ; Mi- chigan, 26-75 ; Columbia Dist. 15-00; Indiana, 8-43 ; Choc- taw Nation, 74-25; Cherokee Nation, 31-00; Switzerland, 212 00; England, 40-00; S. America, 3 C^O ; West Indies, 3-00 ; places unknown, 491 '89 : how much in all ? 850. How much less than half the whole, was contributed by Massachusetts ? 85 . . How much more than half the whole, by Massachu- setts and Connecticut ? 852. How much is the amount contributed by Massachu- setts, for each person in the state ? (See No. 404.) 853. How much is the amount contributed by Connecticut, for each person in the state I 85 1. If New York had contributed in the same proportion as Massachusetts, what sum would have been raised in that s^ate? 8.^5- If all the states had contributed in the same propor* tion as Massachusetts, what sum would have been raised in the whole r 856. If the extra expense of the thanksgiving dinner, to the inhabitants of New-England, reckoning time and money, is 30 cents each person; andif they should satisfy themselves witb. a common dinner on that day, and testify their gratitude to God by devoting that amount to his service ; how much •would be thus raised annuallv? m Questions. 211 857. If all the people of the United States should content themselves with a dinner, once a week, that should cost one cent less than ordinary, for each person, and should pay that amount into the treasury of the Lord ; how niuch would be tlius raised annually ? 85b. Take the receipts of the principal charitable societies in En<:;land, as stated in No, 54, and suppose all the other charitable societies in Europe to do as much as the British and Foreign Bible Society, and what is the annual amount, in federal money, contributed in Europe for spreading the gospel ? 859. During the year ending in 1821, the American Bible Society received 49578 dolls. ; the American Board for Fo- reign Missions, 463^8 dolls. ; the Baptist Board for Fore:gn Missions, 18000 dolls. ; the United Foreign Mission Society, 15263 dolls. ; the American Education Society, 13109 dolls.; and 30 smaller societies, about 60388 dolU. Add 60000 dolls, more, for all other societies ; and what is the annual amount contributed in the United States for spreading the gospel ? 860. How long would the people of the United States need to abstain from ardent spiiits, to save that amount, supposing the same quantity to be consumed now, that was in 1810? (See No. 734.) 861. How long, to save as much as is contributed in both Europe and America ? ' 862. Nearly all the societies in the world for the spread of the gospel, have been formed within 30 years, and most of them within half that period. It is calculated, partly from documents, and partly from estimates, that the following sums have been raised for that purpose, by the principal so- cieties in England, during that period, to wit : By the British and Foreign Bible Society, 403^658 dolls. ; by 13 others, 7704222. Suppose all the other societies in Europe to have done as much as the B. & F. Bible Society ; and what is the whole amount v\ lich has been raised in Europe for the spread of tlie gospel, within Hie last 30 years ? 863. What is the average amount per year ? 864. it is thought, that if we allow to the American So- cieties an average income for 20 years, which shall bear the same proportion to their present income, that the above stated average income of the European Societies bears to their present income, it will equal th« whole amount of what 21^ Questions. has been done in the United States for the spread of the gospel in 30 years. If so, what is the amount? 86^. What is the whole amount of what has been done for that object, in Europe and America, during 30 years ? 866. How long would the people of the United States need to do without ardent spirits, to save that amount ? (See So. 734.) 867. It is computed, that 200 bushels of potatoes, or some- thing equivalent, can be raised in a missionary field, by labor equal to 36 days' work. If this is so, and one person in 4 of the whole population of the United States should labor S days every season for that jiurpose, and the potatoes should sell for b sixteenths of a dollar per bushel ; what amount would be thus raised annually ? 868. If one person in ten of the whole population of the United States, sleeps an hour later every morning than is necessary, and his labor is worth 8 cents an hour ; what an- nual amount is thus lost, which might be saved for doing good, reckoning 313 working days '? 869. If a youth can read an octavo page in 2 minutes, and is in the habit of spending 3 hours each evening in idleness, from the first of November to the first of March in each year ; how many volumes, of 300 pages each, could he read, in his winter evenings for 10 years ? 870. In the year 1821, it was supposed there were .2500 dram shops kept in the city of New York. If the rent of ea* h of these is 70 doli^., and the labor of one man to attend each, is worth 200 dolls ; what is the amount of these two items of the expense of these public nuisances ? 871 If each of these shops sells liquor to the amount of only 2 dolis. per day, what is the annual amount ? 872. In 1820, there was stated to be SvOO paupers in the city of New- York. If the expense of supporting them ave- rages 1 doll, per week, what is the annual amount ? 873. It is believed, that three fourths of the expense of supporting paupers is occasioned, directly or indirectly, by intemperance. Take | of the last amount, and add to the two preceding, and tell the total- 874. During the same year, the amount expended for pub- lic and private schools, was $14759*41 ; and there were 63 ministers of the gospel employed in the city. If these had a salary of $1500 each, what amount was expended for reli- gious and literary instruction ? ^ueshon^, 213 By a report made to the legislature of Massachusetts In 182 I, It appeared, that the number of paupers in that state was about one sixty-feixth of the whole population ; and that the average expense of supporting them was one doHar per week. If it is the same in all the other states, and three fourths of the expense i» occasioned by intemperance, what amount would be annually saved in this way, by the disuse of ardent spirits ? 71 876. In 1812, Mr. H. Campbell estimated the poor rates in England and Wales, at 1 64 j26o 6/. sterling, and the number of paupers at 20794S2 ; what is that, in federal money, for each pauper ? 877. In i8£0, it was stated that there were 14000 paupers in Liverpool, supported by parish rates paid by 20000 indivi- duals ; if each pauper cost as stated in the last question, what had these individuals to pay on an average ? 878. The Connecticut Missionary Society paid for mis- sionary services, in 1816, 85466*38, of which $45.::'6l were for services rendered in Kentucky, jg 1328*38 in New- York, $6lo-85 in Missouri, $o-ll in the sod thern part of Ohio, ^394-93 in Pennsylvania, $367 in Tennessee, S294-28 in Vermont, $150 in Indiana, and the rest in New Connecti- cut ; hbw much was the last ? 879. In 1730, the number of graduates at Yale College for 29 years, had been 235, of which 118 became ministers of the gospel ; how many ministers does that average yearly ? 880. How many did not become ministers, to one that did? 881. In the next 20 years, to 1750, there were 385 gradu- ates, of which 162 became ministers; how many ministers yearly? I 882. How many did not become ministers to one that did ? v^ 883. In the next 20 years, to 1770, there were 648 gradu- ates, of which 201 became ministers ; how many ministers yearly ? 884. How many did not become ministers, to one that did? 885. In the next 20 years, to 1790, there were 723 gradu- ates, of which 177 becatne ministers; how many ministers yearly ? 886. How many did not become ministers, to one that did? 887. In the next 20 years, to 1810, there were 790 gradu- ates, of which 160 became ministers ; how many minister^ yearly ? 888. How many did not become ministers, to one that did? 889. In 1767, the graduates at Princeton College for %0 214 Questions, years, had been S§1, of which 130 became ministers ; how many ministers yearly ? 890. How many did not become ministers, to one that did ^ 891. In 1807, the graduates for 20 years' had been 525, oi which 50 became ministers ; how many ministers yearly ? 892. How many did not become ministers, to one that did ? 893. Harvard University, from 1719 to 1741, furnished an average of 13 ministers annually ; how many in the whole in that time ? 894. From 1800 to 1810, only 6 annually; how many in that time ? 895. Dartmouth College, from 1780 to 1800, furnished an average of 8 ministers annually ; how many in that time ? 896. From 18G0 to 1810, only 5 annually; how many in that time ? 897. The proportion of graduates at the principal colleges, who entered the ministry, from 1800 to iSi), was one sixth ; and Harvard, Yale, Union, and Princeton, together, sent out about 200 graduates a year ; how many ministers did they furnish in that time? 898. U all the other colleges in the United States fur- nished as many more ; what is the whole in those 1 years ? 899. If the number of educated ministers in the United States, in 1820, was 2390 : and there should be 500 gradu- ates annually at the colleges, of which one sixth should be- come ministers, and half as many more should obtain a sufR- cient education without going to college, and none should die ; how many would there be in the ^ear 1 843 ? 900. But if the life of a minister averages 25 years after entering the ministry, only 2 twenty-fifths of the old ones, and about 1 6| twenty -fiftlis of the new ones, will then be alive ; how many will that be ? 901. If the population of the United States should double in *t3 years, and one minister is necessary for every bOO souls, how many would then be destitute of an educated mi- nistry ? 902. In the next 23 years, let 801 be graduated antmally, and one fifth of them enter the ministry, and half as man3r more without going to college ; how many ministers would there be then, after deducting the deaths as above ? 903. If t'iie population doubles again, how many will be destitute in the year 1 866 ? 904. In the next 23 years, let lOOO be graduated annually, and one fifth of them enter the ministry, and half as many Questions, 215 wore without going to college ; how many ministers would /there be then, after deducting the deaths as above ? 905. If the population doubles a^ain, how many will be destitute in the year 1889 ? 906. If the whole number of Protestants is 60 millions, and only 50000 missionaries are wanted to send to the hea- then, and if the number of Protestants in the United States is 93750^0, how many ought they to furnish ? 907. The donations to the Massachusetts Missionary So- ciety, for 18S0, were $2371v^T, of which g 1341-33 was for the permanent turd ; how much, in N. England currency, was for current expenses ? 908. The expenditures of the Massachusetts Christian Knowledge Society, for 1820, were $1456*3:>; in 18 l6 they were $2622-33 ; how many guilders was the decrease? 909. The receipts of the Lf>ndon Jew^' Society, in 1817, were 1-10091 sterling ; how many rials of plate is that ? 910. The VernunJ Bible Soc. received, in i8l4,$l462'13; how many English guineas is that ? 911. The Connecticut Education Society, in 1817, receiv- ed ^'1 370*43 ; how many millrees is that ? 912. The Female Education Society of New-Haven, re- ceived $351*18 ; how many rupees is that ? 913. The Connecticut Domestic Missionary Society re- ceived 81263-63; how many rubles is that? 914.. The permanent fund of the Connecticut Missionary Society, was S3 1583 65 ; how many Hebrew shekels of silver is the annual interest at 6 per cent ? 915. I'he Philadelphia Education Society, in 1890, re- ceived $2039-8^ ; how many Greek oboli is that ? 91G. In 1819, the Female Missionary Society of the Wes- tern District of N- York, received $1352-38: how many Roman sestertii is that ? 917. In 1820, the General Assembly of the Presbyterian Church appropriated for their theological Seminary, $47i2 •055, of which were expended as follows : Salaries and house rent oi 2 professors, $4000 ; half year's salary of as- sistant teacher of languages, $200 ; printing, stationary, &c. $40-07i ; travelling expenses of one director, $21 ; treasu- rer's commissions on the above, $42-61 : how many drachmae of silver in what was unexpended ? 918. In 1818, the Western Education Society received g^;028'67, and expended $1531-53 ; how many Hebrew she- kels of gold in the difiereuce ? ^16 Questions. #'9. In 1819, the N. York Religiaus Tract Society re- ceived B849*95 ; how many Greek staters of silver is that ? ^ 920. In 1 8 19, the collections for the Theological Seminary of the Dutch Reformed Church, were $3730'04 ; how many Roman denarii is that ? 921. The funds of their General Synod were as follows : Van Bunschooten fund, 1 14730 ; professoral fund, $120.52 '57; permanent fund, S9133-05: how many Hebrew mi nae of sdver is that ? 9 2. How many 'drachmae of gold is the annual interest at 7 per cent ? 923. One of their theological professors has $1750 a year, and the other ^ . 5 sO ; how many lioinin asses in the whole > 924. In 1821, the Board of Missions of the General As- sembly appointed missionaries to labor m the destit iie set- tlements, 168 weeks ; how many Attic minse of silver would th;it ain unt to at %) a week ? 9 3. The collections reported for missionary purposes, amounted to S^'l^'i 51 ; how many weeks labor would that pay for, at %9 a week ? 9 26. The collections reported for the Commissioners' fund, ^iMC 3i4t6*24 ; how many miles travel would that pay for, at o J cents per mile ? 9^7. In 1821, the permanent fund of the American Educa- ti.'ni 8.>ciety was S t)87o*50 : how many rials of velion in tueaiii'Ml interest at 6 per cent •' 9:is In i'3 K the receipts of the Young Men's Missionary Society of N. Vork, were S237j'77; how many pagodas is that ? ' , 9-29. In 1813, the receipts of the Hampshire Miss. SoCc were %1527''25 ; how many sequins is that ? 9 >0. The Leeds and Liverpool canal is 139 ndles long, and cost 800 .'wO/. sterling ; what iS that per mile ? 931. Tb^ canal of Languedoc is 180 miles long, and cost Xd 40000 ; what is that per mile ? 932. The Middlesex canal, in Massachusetts, is 28 miles long, and cost 4780J0 dolls. ; what is that per mile r 933. rhe Erie canal, in New York, is expected to be 36S mUes long, and to cost 457 1814 dolls. ; what will that be per mile ? 934. Tf one million of tons should pass through it annually, and the tt)ll should be a cent and a half per ton per nule;, what will be the amount ? Qiiestions. 2\T 5. If only half that amount be produced, and S500000 be annually required for repairs, officers, &c. 5 what will be the annual net proceeds to the s'ate - 936. If the whole expense should be at simple interest at 7 per cent, three years before the canal is opened, and one year more before a year's toll can be applied to pay it, and. tho net proceeds be applied, at the end of every year, to ex- tinguish the debt ; how much will re n:iin to the state at the end of the third year afier the canal is opened ? 937. If 500000 tons pass through the canal annually, and the transportation by land was 90 dolls, per ton, and on the canal is S^'^J^, and the toll is as stated a'jqve ; what is the annual saving" to the public? 938. If the toll and transportation together are 2 and a .half cents per mile, and a million of tons sli )uld be annuiUy transported on the canal ; what would be the annual saving ^ to the public, on eveyj mile the canal shouhl be shortened ? 939. It is estimated, that if tlie canal cv)uld be made di- rectly from Schenectady to x\lbany, instead of going round , by the Cohoes, the distance would be shortened I > mdes, and I ^ other ijdvanta.^es gained e'[ual to a saving of 5 miles more ; Z if so, what vvould be the annual saving to the public, on toll ' and transportation alone; and wiiat capital would this be equivalent to, on the principle of perpetuities, discounting at 5 per cent compound interest ? 940. If the width of the canal is 28 feet at the bottom, and 40 feet at the top of the water, and the water is 4 feet deep; how many cubic feet of water are in every foot in length ? • 941. In 1316, the permanent annual expenses of the go- vernment of the United States, were estiuiated, by the Se- cretary of the Treasury, to be i^3o00u00 doils. ; of which, : 1765313 dolls, were civil, diplomatic, and miscellaneous; 64596^6 dolls, were military; S9866o9 dolls, were naval ; 28^3:20 i dolls, were incidental ; and the rest to pay the inte- rest, and reduce the prnicipal of the public debt : how much „ 'is the last ? 94^2. It is estimated, tliat the annual expense of militia trainings in the United States, i, 5 millions of dollars. Add that to t!\e uulitary and naval expenses of the government, a» above stated, and tell the asnount. 943. If that amount shouhl be appropriated to spread th« gospel of pea e, how many bibles would one eighth of it aii- iiualiy furnish, at 60 cents each ? T 218 Questions. 944. How many missionaries would one fourth of it sup- port, at 500 dolls, each ? 945. How many young men would one eighth of it assist in their education for the ministry, at 1 25 dolls, each ? 946. How many tracts, of 12 pages each, would one six- teenth of it annually furnish, at one mill per page ? 947. How many children would the rest of it keep at school, at the rate mentioned in No. b i 7 ? 948. Congress reserved of the public lands in Alabama, 460u0 acres for a university, which is estimated at li dolls, per acre ; if it should be sold for that, and the money placed in a fund which yields 6 per cent, what will be the yearly amount ? 94i^. The representation in Congress, according to the sensus of i8£0, is to be as follows : New- York, 34 represen- tatives; Pennsylvania, 26; Ohio, 14; Massachusetts, i3; Maine, 7 ; Connecticut, 6 ; NewJersey, 6 ; New-H impshire, 6 ; Vermont, 5 ; Indiana, 3 ; Rhode-Island, 2 ; Delaware, 1 ; Illinois, 1 : how many in the Northern states ' 950. In the Southern states, Virginia, .2 ; North- Carolina, 13 ; Kentucky, 12; Tennessee, 9 ; feouth-Carolina, 9 *, Mary- land, 9 ; Georgia, 7 ; Louisiana, 3 ; Alabama, 3 ; Missouri, 1 ; Mississippi, 1 : how manj in these ? 961. How many in all ? 952. 'Ihe value of domestic articles exported from the United States in the ;^ear 1819, was as follows : Produce of the sea, 82024000 ; of the forest, ^4927000 ; of agriculture, g4 140 2000 ; manufactures, S-*;dV4lOu ; uncertain, gbSOOOO ; how much in all / 953. The value of foreign articles exported, was 18OC8029 dollars; what is the total value of exports in that year? 964. If one per cent of this was devoted to charitable pur- poses, how many livres would it be ? 955. The mean temperature of Boston in 1819, according to observations made three times a da}?, was as follows : Ja- nuary, 30 degrees of Fahrenheit's theruiometer ; February, 31 ; March, 29; April, 41 ; May, 52 ; June, 67 ; July, 71 ; August, 69; September, 64; October, 53 ; November, 40'; December, 31 : what was the mean temperature of the whole year ? 956. The number of inches of rain which fell, was as fol- lows : January, 1'05 ; February, 2-27 ; March, 6*51 ; April, 3-74; May, 3-06; June, 3-o6; July, 2-02; August, 4*38 ; Questions. ^ 219 September, 5'27 ; October, 1-40; November, 1'22; Decem- ber, 1*29 : how many inches in the year ? 957. From July 18:6 to June 1817, the mean degrees of heat at Bombay, were as follows : July, 80 ; August, 78i ; September, 79i ; October, S:.^ ; November, 82 J ; December, 79J ; Januarv, 78^ ; February, 76i ; March, 79 ; April, h3| ; May, 86^ ; /une, 82^ : what was the mean heat of the whole year ? 958. The population of Massachusetts in 1820, was stated as follows ? males. females. Whites, under ten years old, 70993 69265 from ten to sixteen, 38573 38303 from 16 to twenty -six, 49506 52805 from 26 to forty five, 544U 57721 from 45 upwards, ^^'6^^ 46171 Blacks, 3308 3560 What is the whole number, and how many more females than males ? 959. In ^819, the number of revolutionary pensioners was stated as follows: In N. Hampshire, 1142; Maine, 1824; Massachusetts, 25l4 ; R, Island, 249; Connecticut, 1373 j Vermont, 1296: how many in New England? 960. In N. York, 3196 ; N. Jersey, 467 ; Pennsylvania, 1090; Delaware, 41; Maryland, 575; Dist. of Columbia, 51 : how many in the Middle states ? 961. In Virginia, 69i; N. Carolina, 212; S.Carolina, 180; Georgia, 46 ; Alabama, 5 ; Mississippi, 6; Louisiana, 1 : how many in the Southern states ? 962. In Kentucky, 474 ; Tennessee, 114 ; Ohio, 647 ; In- diana, 96 ; Illinois, 4 ; Missouri, 6 ; Michigan, 3 : how many in the Wes ern states ? 963. How many in all the states ? 964. What would be the amount of pensions, at 8 dollars a month each ? * 9:>5. The amount of the funded debt of Great Britain and Ireland in 1.^12, was £906939589 .. 16 .. 8, and unfunded, =g5639r848 .. 16.. 10; what is the whole in federal money ? 966. What was the value of the golden candlestick and its appendages, being a talent of gold ? 9S7. What was the whole value of the gold and silver of the tabernacle, being 29 talents, 730 shekcb of^old,ancl 100 talents, 1775 shekels of silver ? 220 Quest iQUs. 968. What was the value of a chariot from Egypt, in Sor- lomon's time, being 6(0 shekels of silver ? 969. What, of a horse, being 150 shekels of silver ? 970. If the price of redemption of a field sown with a homer of barley, was 50 shekelj* of silver, (Lev. 27. l6,) what was that, in federal money, for the ground sow n w ith a bushel? 97 1. How many quarts, dry measure, w^as the allowance ©f manna for each person, being an omer ? 972. How many galkms of wine did our Lord make by his miracle at Cana, if each of the 6 water pots contained 2 J baths ? 973. What was the value, in federal money, of the oint- ment which Mary poured on the feet of Jesus, being SOO Roman pence, or denarii i* 974. How long is the side of a cube of fine gold, which weighs a ton ? 975. What is the solid content of 2 Cwt. of steel ? 976. if the diameter of the Earth is 7928 miles, and that of the Sun, 885248 ; how many times larger than the Earth is the Sun? 977. How many wine gallons in 10 C«vt. of proof spirits ? 978. How many solid inches in a living man who weighs 170 lb. avoirdupois ? 979. A and B travel the same way, as follows : A, 55 miles, 6 furlongs, 27 poles ; S6 m, ; 27 m. 28 p. ; 29 m. 3 fur. : B, 36 m. 3 fur. 29 p ; i^7 m. ; 28 m. 27 p. ; 34 m. 7 fur. : and then A travels back 24 m. 3 fur- i7 p. ; 24 m. 6 fur. : liow far asunder are they ? 98u. There are two numbers, the less is 8967, and their difference three times as many ; what is the greater number? 981. There are two numbers, the greater of which is 79 times 209, and their difference 2t) tmies ^9 ; what is their sum ? 982. An apothecary mixed 5 sorts of medicines, each 3 lb. 11 oz. 7 dr. 2 sc. 13 gr. ; and 7 sorts, each 11 oz. 5 dr. i sc. 12 gr. ; how^ much in all ? 983. What is the amount of twice twenty five added to tv/ice five and twenty ? ^ 984. Bought of A, S?' gol. 3 qt. of wine ; of B, three times as much, and .^ g^irj^r^. more ; of C, as much as of A and B both, and 7 gal. 3 qt, 1 pt. more : sold T>, 7 gal- 1 pt. ; E, 5 gal. ; F, as much as the difftrence between D and E, and 2 gal. 1 c^t, more ; how much is unsold ? 985. Add 333 eagles, 333 dolls. 333 dimes, 333 cents, and 333 mills, together. ■ 986. How many lb. Troy, in f of | of i of 4 of ^^ off oi 800 lb. avoirdupois? 987. What day of the year was the last day of June 1816'^ 988. By what must Jl30.. 8.. 6^ be divided, to give .- quotient of ^6.. 17.. 31? 989. If a debtor who owes L2000, pays but ^625, how much is that on the pound ? 990. Tell the product of i off off off of 1500, by -| of ,^j of 90. 991. From 9 lb. 7 oz. 10 dwt. of silver, how many spoons can be made, each weighing 2 oz. 15 dwt. ? 992. How many solid feet in a ton of cork ? 993. Bought two parcels of flour, which together weighed 19 Cwt. 3 qr. 4 lb., for ^97.. 17.. 6; their difference in weight was 3 Cwt. 1 qr. 4 lb., and in price, =g8 .. 13 .. 6 ; what was the weight and value of the greater parcel ? 994. Received ^62000.', and paid 16 persons ^54 .- 10.. "6 feach, and 3 persons ^ 1 93 .. 6 .. 4 each ; what is } of the rest? 995. A was born August 6, 17?^3, and B, Oct. 21, 1815; how old will A be when B comes of age ? 996. There are two numbers, of which the greater is 76 cimes 1 1 1, and their difference is 1 8 times 27 ; what is th^ir product ? 997. From 19 chests of tea, each 3 Cwt. 2 qr. 7 lb., how many canisters of 7 lb. each can be filled ? 998. Sent to the bank Si eagles, 9| dolls. 9^ dimes, and V}, cents ; and drew clieoks for g.36-25, g27-27^and g 19-34 1 how much is left ? 999. One planet has moved through 9 si2:ns, 29 degrees, £9 minutes, 25 seconds of its orbit ; and another has moved 5 signs, »5 7. in 40i)l. sterling, how much Maryland? 1058. Reduce 44V25 to its lowest terms. 1059. Tell the t^quare root of jVt- 1060. Reduce | of a penny ster. to the fraction of a dollar. 1061. Find the difterence between ^ of 13^2_^ ^n^j j of ^ of of 67^5_, 1062. Reduce |d. N. York, to the fraction of an English guinea. 1063. Tell the difterence between f of a lb. avoirdupois, and If of an ounce. 1064. Tell the square root of jV 1065. What is the 4th root of 4 96. 1066. Tell the difterence between H^ aind y^^s, 1067. Tell the difference in sterling between f/. sterling, and f of a dollar. 1068. What is the value of r^^ of a year ? 1069. When 6 persons use 1| lb. of tea in 2 months, how much will suftice 8 persons | a year ? 107 . Reduce y3j of a day to its value. 1(^71. Find the cube root of the square root of 262144, 1072, Tell the difference between 2714 and -916, ^26 ^uestions^ 1073. In 100/. Irish, how much Connecticut? 1074. Tell the sum of i^T6+54-321 + li2+'65+l2-5 + •0463. 1075. Reduce f of a crown to the fraction of an English guinea. 1076. Divide 234-70525 by 64-25. 1077. In lOOZ. Iri^h, how much Georgia? 1078. Reduce f of i crown to the fraction of an English shilling. 1079. Tell the difference between | of 2s. 6d. & f of 5s. 8d. 1080. Divide 14 by -7854, 1081. In IGOL Irish, how much sterling? 1082. How many feet is the side of a square containing ^11 of a square mile ? 1083. Reduce 3 ft. 8 in. to the fraction of a mile. 1084. Reduce J of ^ of |, & 3^, to a common denominator. 108.5. Divide 5-1 6 by 1000. 1086. Reduce 3 qt. li pt. to the fraction of a hhd. 1087. Tell the difference, in federal money, between f of a guilder, and ^ of 3^ livres. 1088. Tell the difference, in federal money, between 2/. lis. N. England, and | off of lof dolls. IoSj. If 2 ships sail from the same port, one north 76 leagues, the other east 58 leagues ; how far are they asunder? 1090. Reduce 4 lb. 3 oz. 6 dr. to the fraction of a Cwt. 1091. Divide 6 by -6. 1092. Tell the value of -3375 of an acre. 1093. Reduce 3s. 5Jd. ster. to the fraction of a guinea. 1094. In 4oo livres, how mucn Nova-Scotia? 1095. Add ^L fs. and yVl. 1096. Add J, f , 1, and | of f of 5. 1097. Reduce ^//^ to a decimal. 1098. In 4oo livres, how much Irish ? lo9i:'. Tell the difference between 5^, and f of 4^. 11 00. Tell the deference between | off of 7 Cwt. 3 qr. 12 lb., and ^ ot J of j% of 5 Cwt. 3 qr. 27 lb. 1 lol. Tell the difference between ^l. and | of fs. 1 lo2. Reduce -6875 yd. to its value. 1103. it a ship of -^oo tons has 8o feet keel, what must be the keel of another of the same shape, to carry 5o tons ? 1 lo4. Add 7| off, and f of 4 of 7. and 5f, and j\. 1 loi. FinrI the side of a cubical bi»x containing 12 bushels. II06. Add ^s. and 1^. Questions. 227 m HI ilo7. Add f of a farthing, and j\ of a shilling. lloS. i'cll the difference between f of o^L, and |s. I lo9. In 600 guilders, how much Canada ? 11,0. Add f dwt- and j\ lb. Troy. 111'. What cost 564, at 6/. 13s. 4d. each ? 1112. Add ^L ^s. and fq. 1113. Tell the difference, in sterling:, between j% of |f of 5 Eng. guineas and | of 4 ot } off of 2o crowns. li . k If a gh)be of sUver, 5 inches in diameter, is worth 465 iolls., what is the value of one 2 feet in diameter ? I I lo. Reduce '056 of a pole to the decimal of an acre. 1116. In 600 guilders, how much Irish ? 1 1 i7. Multiply |, 3^, 5, and f off, continually together. 1 . 8. Ketiuce -21 pt. to the decimal of a peck, n i9. Add I of a year, f of a day, and | of an hour. 1 1^0. Reduce 1 1 minuter to the decimal of a day. 1 1- I Multiply J, f , and 4y\, continually together. 1 ££. Add I of a crown to | of a dollar. 1 2.>. Add f hlid. and f gal. 1124. Multiply |- by | off. 1 2o. Divide | of ^ by 4 of 7|. 11.6. Add ^L ster. to f of an Eng. guinea. 112,. Add |/. sterling, | of an Eng. guinea, f of a crowB^ and I of a dollar. 1 128. What cost 12Cwt. ?qr. 14lb. at 7L lo s. 9d. per Cwt.? 1129. Reduce I of} of i of lo, to a single fraction. I >3o. In 5|s N. York, 5|s. Vermont, 5§^. M.Jersey, 5|s. eorgia, 5|s. sterling, and 6| dolls., how many dollars ? 1131. Tell the difference between f of /y of lo inches, and I of I of ,^0 ot feet. il;i2. What cost 9 Cwt. 2 qr. 26 lb. at 71. los. 9d. per Cwt. ? 1 1:j3. Reduce J of f of f of | of 7, to a single fraction. 1134. In |4. ster., :a*31s. Vermont, and g5i, how many dollars ? 1 135. Reduce ^ of | of i of 7h to a single fraction. 1 136. Add 1-2347. and |s. and f of a farthing. 1137. If a board is '7.3 of a foot wide, what length of it will make 24 square feet ? 1138. Tell the difference between f off of 21 hhds., and S-789 gallons. 1139. A cubical stone contains 42873 solid inches ; what is the superficial content of one of its sides ? ^28 Questio}i9' 11 40, At 41. \Ts. per Cwt. what cost 17 lb«? 1141. Nedure | of a Cwt. to the fraction of an ounce* 11-2. Tell the ditfeience between^ of 4'689 yds., and y\ of f of iir^*67 nails. 114:3. Multiply 5, |., # of |, and 4J, continually together. 1 i 44. Multiply I of I of 1 1 j\, by ? of | of f of 2o. 1145. Multiply 23-678 by f off of ^ of l5f 1 Kb. Multiply J of •/. las. 6d. by f of J of 19|. 11 4 i'. What cost o hhd. tobacco, each 4 Cwt. ^ qr. 7*4 lb., at 8s. ^•3d.f.r4'£lb. ? 1148. A person, after spending '» and i^, and ^\ of his money, has l6o dolls, left ; how nvuch had he at Hist ? 1 149. Multiply I of f off ot ; 5 dolls, by .-7689. I l5o. Suppose 3oo stones were laid 3 v(l^. from each other in a line, and a basket was placed yds. from \he first ; how fiir nuist a person travel, togath^-r them one by one into trie baket? llol. Multiply I of 5 crowns by 3-6789, and tell the amount in sterling. 1132. A person, after spending i of | off of his monej, and \ of J off of the remainder, has Sdo dolls, left; what hud he at first r I I y3, Mul ipJy I of 1 of -?- of s Cwt- 3 qr. £6 lb. by 10-6o 1 13 4. Sold, it* j(U. if clo.h, Reduce l5s. 6d. to the decimal of a pound. i204. Tell the difference between -J of ^ of J ot l6 yds. and J of -f f of 4i rods. 1205. If a ladder, 80 feet long, be so placed as to reach a window 40 feet from the ground, on one side of the street, and without being moved at the foot, will reach a window 22 feet high on the other side, how wide is the street? 1206. If 00 apples were placed in a straight line, 6 feet apart, and a basket placed 6 feet from the first, how far must a person travel, to gather them one by one into the basket ? 120T. A merchant sold ?50 yds. of cloth, at Is. for the first yd. ^s. for the second, 5s. for the third, and so on ; wha^t did he gain or lose, if he gave 3/. per yd, 1208. If 9-5 yds. cost 8^5-75, what cost 435-5 yds ? 1209. Bought >4 yds, the first at 2 dollars, and the last at 16 1 dollars, in arithmetical progression ; what was the com- mon difference -^ 12 0. Tell the present worth of an annuity of S.50 dolls, to continue 6 years, at 7 per cent, simple interest i 2 i 1. Find the area of a triangular field, of vvhich one side measures 43 poles, and the perpendicular upon it 15 poles 1 2 1 2- What is the content of a triangle, whose sides are all equal, and measure 34 rods each ? 1213. If a stick of round timber, whose diameter at the butt is 2 s inches, contains 100 solid feet, what must be the butt diameter of a simil r stick, to contain 68 feet ? 1214 How many days can 6 persons seat themselves dif- ferently at dinner ? 12 i 5- If a triangle whose base is 27 chains, contains 35 acres, what will another of the same shape contain, whose base is 50 chains i* 1216. Mixed 10 bush, wheat, at gl'25 a bushel, 12 bush. rye at 70 cents, 13 bush corn, at 60 cents, 20 bush, barley, at 40 cents, and 30 bush, oats, at 30 cents : what is a bushel of the mixture worth ? 1 17. Find the content of a four sided field, which being divided into two triangles by a diagonal line, that line meas- ures 41 chains, and the perpendiculars upon it from the op- posite angles 25 chains and 19 chains. 12i 8. Find the content of a field, which being divided in- ^32 Questions. to 6 triangles, the sides and perpendiculars upon them mea« ure as follows : Triangles. Bases. Perp. Triangles. Bases. Perp. No. 1, 34 rods, 19 rods. No. 4, 4.3 rods, 25 rods. 2, 25 15 5, 49 S3 S, 35 22 6, 50 34 1219. Mixed the following quantities of sugar, worth the following prices per cwt. to wit: 2 cwt. at 9 dolls. 4 cwt. at iO dolls. 5 cwt at 12 dolls, and 6 cwt at 14 dolls, what is S cwt. of the mixture worth? 1220. Find the length of a slanting tree, when if you set up a pole parallel to the tree, 18 feet long from the ground, and take such a station that your eye is in range with the top of the pole and the top of the tree, and also in a range with a mark on the pole and another on the ix^ct, each 5 feet Irom the ground, your station is i2 feet from the pole, and 55 from the tree. 12:;- 1. If £0 oz. of gold, at 5 2. per oz. 12 lb. of silver, at 5L 10s. per lb. and 50 lb. of copper, at 5s per lb. be mixed, together ; what is 20 lb. of the mixture worth ? 12^2. Find the present worth of a perpetuity of 800 dolls, per annum, discounting at 4 per cent, compound interest. i 223. How long must be the side of a cubical box, to con- tain 20 bushels ? 1^24. Tell the compound interest of 347 dolls for 4 years, at 5 per cent. 1225. Tell the number of solid feet in a stick of squired timber which is l6 by 18 inches in diameter throughout, and 65 feet long. 1226. What principal, at 5 percent compound interest, will amount to 643t. os- !• i778i. in 6 years ? 1227. Find the area of a circle whose circumference is 340 rods. 1228. What must be given for a perpetuity of 250 dolls. to commence in 6 years, discount at 5^ per cent, compound interest ? 1^29. In what time will 600 dolls, amount to g71 4-6096, at 6 per cent, compound interest ? 1230. With 1 5 gals brandy, at 14s. a gal. I mixed 14 gals, of whiskey at 5s. and 10 gals of water at 0; at what rate per gal- must I sell it to gain 12 per cent ? 1231. A man has a square garden, one side of which mea- sures 23 rods, and a circular fish ]>ond in the center, the di- ameter of which ife 9 rods ; how much ground has he ? 1232. How much gold, at 4/. per oz. and silver, at 12s. questions. ^33 per oz. must be mixed with 20 lb. of copper at 6s. per lb* that 60 lb of the mixture may be worth ^240 ? 1233. Find the solid content of a pyramid whose base is an equilateral triangle, the circumference of which is 33 feet, and the perpendicular height 40 feet. 12 >4. Which is the most valuable, and how rpuch so, an annuity of ^300 for 6 years, or a perpetuity of gSOO to commence after 6 years, discount at 5 per cent compound in- terest ? I 23d. At what rate per cent, compound interest, will 23 Id^ amount to -49^ l6s. i 904d. in 2 years ? 1236. If a pyramid, whose perpendicular height is 15 feet, contains 192 solid feet, what must be the height of another of the same shape, to contain half as much ? 1237. If the diameter of one circle is :7 rods, what must be the diameter of another to contain ^ as much ground? 1238. Twenty -two persons bestowed charity on a beggar; tlie first gave id. the second >d. the third, 7d, and so on ; what did the last one give ^ 1239. Two faniil es set out, at 9 o'clock in the forenoon, each in their own carriage, to go to a place 20 miles distant The first travels at the rate of 6 miles an hour, and the other at the rate of 5 miles an hour. At the half way house, the first stops 15 minutes, and then goes on to the end. The second stops 1 > minutes at the end of 8 miles, and then goes on. When the first gets through, it stops 20 minutes, and the other not coming, the empty carriage goes back, at the rate of 8 miles an hour. The second carriage breaks down at the end of 12 miles, and after a delay of 15 minutes, the party walk forward at the rate of 4 miles an hour. The first carriage meets them, and takes them up, an;l proceeds with them at the rate of fi miles an hour, to the end. At what O'clock does the second family reach the place of des- tination • 1240 If a cone 15 feet high, contains 100 solid feet, how far from the base must it be cut, to divide it into two equal parts ? 1^41. Tell the difference between f of f of 9f acres, and 4 of f of 1 ^S^^^ poles. 1242. Multiply -f of i of | of ''^f of 25£. \6s. by 3-98. 1243. Divide f of f of 21, by f of j\ of 3|. 124f.. If I of /. of S '^ P«iy'fo«'i off of,j9^of iOO acres, . iiOW much can be bnu«?hi lor '?- of X of 123*4.^/. N. 8rofia ? ^54: Questions, 1^45. If y\ ot 25 Eng. guineas pay for a. of | of f of 556-34 yds., what will f of f of | of 325^ 13s. 6(1/N. York, pay for ? 1246. If f of 1 of I of 235 guilders pay for | of | of J of 39fi|i yds., how many dolls, will pav for f of -/^ of | of of 23-678 yds. ? 1247. If I of 1 of 3 lb. of tea, serve 9 persons f of | of 91 months, how many persons will J of i of ^ of 10^ lb. serve ^ of J of 15 months ? 1248. If 34'56L ster. be the interest of 356 Eng. guineas, for 1 of I of 345-67 days, how much sterling will be the in* terest of 3867f dolls, for i of f of 4 of 234^4 days ? 1249. Divide 3^ by ^ of |-"of | of J of>f, and tell the difference between the square and cube of the quotient. 1 250. If the walls of the temple of Solomon, had been 5 feet thick when finished, and one inch of the outside and one inch of the inside had been fine gold, how many talents would it have required, allowing one foot high on the inside to have been occupied by the floor ? (See No. 399.) 1251. Ascending bodies ar-e retarded in Ihe same ratio hat descending bodies are accelerated ; therefore, if a ball discharged from a gun perpendicularly into the air, returned lo the earth in 10 seconds, how high did it ascend ? l'^52. Find the number of solid feet in a load of wood, which is 4 ft. 6 in. high, 3 ft 10 in. wide, and 9 ft. 4» in. long. 1253. A laborer was hired for 50 days, upon condition that lor every day he labored he should receive 8s., and for every day he was idle he should forfeit 3s. At the end of the time, lie received 9/. lis. ; how' many days did he labor ? 1254. Tell the area of a circle whose radius is 14 chains. 1255. Find the difference in th6 depth of two wells, into which a bullet let fall, reaches the bottom in 4 seconds and 6 seconds respectively. 1256. If I of I of I of 10^ lb. of tea serve 5 persons 3| months, how much will serve 7 persons -| off of | of ^ mo. ? 1257. Tell the area of a circle whose circumference is 25 chains. 1258. What mu>^t be paid for a perpetuity of 100 dollars, to commence in 4 years, discount at 4^ per cent compound interest ? 12 .9. If a man travels 34-56 miles in 4| days, when the days are 13f hours long ; how far can he travel in 10-45 days^, when they are 15'37 hours long ? QuesVoiTS. 235 1260. Tell the difference between SO square rods, and 30 :ods square. 126i. In what time will 90Ci. amount to 23U. 10s. 6d. at 5 per cent compound interest ? 1£62. Tell the superficial content of a pyramid, the base of which is an equilateral triangle, each side measunng 13 feet, and the slant height 35 feet. 1263. Find the solid content of a cone, the diameter of vhose base is 17 feet, and the perpendicular height 35 fpet. 1264. What is the circumference of a circle, whose dia* meter is 36 rods ? 126,5. Multiply i of | of the square of 45, by f of |^ of the square root of 144. 1266. What number is that, which being multiplied by 18, and the product divided by 12, the quotient is 30o ? 1267. Add f of f of the square of ||4, to ^ of | of | of the sq u a r e roo t of ^ ^V* 1268. What number is that, wliich being increased by J, ■ , \, and ^ of J of itself, and the sum divided by 20, the quo- ient will be 77? 1269. Add the square of || to the cube of f|-, and tell what is i of I of f of the sum. 1270. Tell the solid content of a cone, the circumference of whose base is 45 feet, and its perpendicular height 34 feet. 1271. Find the difference between the simple interest of 375 dolls, for 6 year*, at 7 per cent, and the discount of the same sum, at the same rate and time. 1272. Add the cube of || to the square of |f, and tell what is ^ of ^ of its square root. 1573. AVhich is the most valuable, and how much so. an annuity of 500 dolls, for 8 years, or a perpetuity of 500 dolls, to commence after 8 years, discounting at 6 per centcom-r pound interest ? 1274. From ^t ^ake 4 of ^ of itself, and tell what is the Bum of the square and cube of the remainder. 1275. At what rate per cent, compound interest, will 200 dolls, amount to S262-4*;5392, in 4 years ? 1 276. A man sold | of his sheep, and 12 more, at one time ; And at another, J of the remainder, and i5 wiore, and had 137 left ; how many had he at first ? 1277. Find the content of a circular ring 2 rods broad, round a circulai* fish pond of 3 acres ? 236 Questions. 1978. A person who owned | of a ship, sold f of his share for 500 do is. ; what was the ship worth ? 1 2. 9. In an orchard, |- are apple trees, ^ pear trees, ^ cherry trees, i peach trees, and lO plum trees; how many trees in all ? 12H). What is the content of a circular ring 3 rods broad, the inner circun^ference of which measures 346 rods ? 1281. If § of i of I of a ship is worth -} off of | of J of her cargo, that part of the cargo being worth 1 200/. ; what is the value of the ship and cargo together •' \^S^. Tell the amount of an annuity of 650 dollars, for 7 years, at 6 per cent, simple interest ? 12s3. A, B. and C, purchased a vessel in company; A paid |, B |, and C loo dolls ; what was the whole ? 1284. tell the cube root of i of f of | of 8. 1285. Which is the larger, and how much so, a square of 450 rods circumference, or a circle of the same circumfe- rence ? 1285. What sum of money will produce as much interest in 4^ years, as 345 dolls, would in 7§ years ? li;87. Tell the cube root of ^of i of | of 21. 1 2^-;8. What part of 2s. 6d. is |- of f of | of Is. 6d. ? 1 2 89. if a cone whose slant height is 20 feet, contains 2 1 6 solid feet, how far from the base on the slant height, must it be cut, by a section parallel to the base, to be divided into two equal parts ? 1290. Find the cube root of the sum of the square roots of 529 and 1936. 129 . What number is that, from which, iff of | is taken, the remainder is ^ of|? 1 ::92. Tell the superficial content of a globe which is 3 feet in diameter. 1293. Teil the cube root of the diiference of the square roots of 3 1 66 and 5776. 1294. What number is that, to which, iff of|^of|ff be added, the sum will be 1 ? I v95. What number is that, which being multiplied by | off of 2, the product will be 1 ? r29fv. A, B,.and C, bought a ship, which cost 2500 dolls. B paid lOt; dolls, more than A ; and C, iOO dolls, more than A and B both. They furnished a cargo which cost 250 dolls. less than twice the vaiue of the ship ; and the expense of fitting out the vessel was | of f of | of j\ of the value of both %iestions. 237 ship and cargo- The profits of the voyage were 25 per cent on the whole, which were to be shared according to the in- terest ot* each in the ship : what was A's share of the gain ? 5 297. A man received for his wages, one graki of wheat for the fir t day, 4 for the second, 6 for the third, and so on; what is the amount of 30 days' labor, if the wheat is worth 1 doll, a bushel, and 7680 grains of wheat make a pint ? 129^, What number is that, from one half of the square- of which, if I, |, |, f, and ^ of the number, and 123 more, be subtracted, the square root of the remainder is 294 ? 1299. Two merchants, A and b, began trade with the same capital. A was successful, and gained f off of |^ as much as his stock ; but B lost | of f of ^ of ^^AO dolls, and 20 dolls, more ; and then their capitals were in the propor- tion of 5 to 3 : what capital did each begin with ? 1300. A, B, and C, joined stock n trade, and made up a capital often thousand dollars. B furnished ^25 dolls, more than A ; and C, 600 dolls, less than A and B together. At the end of 6 months, A took out his stock; and at the end of 10 months, B took out his. At the year's end, the gain was a sum equal to | of y^^ of |^ of 3 times the stock of C : what was each man's share of the gain ? 1301. A and B bought 300 acres of land for 600 dollars, of wliich they paid equal sums. One part of the lot prov^ig better than the other, A says to B, if you will let me have my choice, your land shall cost you 75 cents an acre less than mine. If B agrees to this proposal, how much land will he have ? 1302. 1 wish to fence a circular piece of ground, with rails which shall be so long as to make ' feet to each length, and the fence to be 5 rails high, and to have as many acres of ground as 1 have rails ; how many acres will there be ? 1303. Suppose the frustum of a right pyramid to be 4 feet square at the base, and one foot square at the top, and the slant height .0 feet ; and a rope -2 inches thick, to be wound round it, so as to cover its sides from the bottom to the top ; how long is the rope ? 1304. A, B, C, and D, purchased a grindstone in compa- ny ; and the sums they paid respectively, were such that A, B, and C together, paid 3 dolls. 9 cents ; A, B, and D to- gether, 1/. '3s. 3-36d. N. York cur; B, C, and D together, ll, .s. li-52d. N. England cur.; and A, C, and D together, M. i s. 3d. sterling. The grindstone was 3 feet in diameter, and 5 inches thick ; and there was a hole in the middle, 4 £38 ^uestians, inches in diameter. A is to have it first, and grind off in proportion to the sum he paid ; then B, C, and D respectively^ I demand the breadth of the circular ring which each is to grind off. 1305. A party of 17 persons wish to go to a place 25 miles distant. They have but one carriage, which will carry 5 persons. At 8 o'clock in the forenoon, the carriage sets out with 5 of the party, and goes on at the rate of 6 miles an hour, till half past 9, when it stops 1 5 minutes. Finding the roads worse, it then goes at the rate of 5i miles an hour, till a quarter past 10, and then 5 miles an hour to the end, having made one more stop of 20 minutes. At the end, it stops 18 minutes, and then goes back, with one person to drive, at the rate of 7 miles an hour. The rest of the party set out on foot, 10 minutes after the carriage, and w^alk at the rate of 4 m:les an hour, till a quarter before 10, when they stop 20 minutes. After this, they walk on at the rate of Sf miles an hour, till half past 10, when they stop 15 minutes. They then walk on at the rate of 3 miles an hour, stopping once more 15 minutes, till they meet the carriage. The carriage takes up 4. of them, and goes on to the end, at the rate of 5 miles an hour. After resting 15 minutes, it goes bnck again, at the rate of 6 miles an hour. The remainirig 8, after resting 20 minutes, walk on at the rate of 2^ miles an hour, till they meet the carriage again. Then the carriage takes up 4 more, and carries them to the end, at the rate of 5 miles an hour, and returns without stopping, at the rate of r> miles an hour. The remaining 4 walk on at the rate of 2^ miles an hour, 1 11 they meet the carriage, and then ride to the end, at the rate of 5 miles an hour. How far does the carnage travel in all, and at what o'clock do the last of the party reach the place of destination ? 306. Suppose A, B, and C, can do apiece of work in 165 days ; B, C, and D, in 220 days ; A, B, and D, in 18 days: A, C, and D, in 198 days : how long will it take each one to do it separately ; and how long, if they all work together ? 1307. Divide 15 into 2 such parts, that their product shall Ha 7 4 8 4 130^. If 6?i bushels of oats aref sufficient for 12 horses for 4 weeks, and 236^ bushels are sufficient for 21 horses for 9 weeks ; how many horses will 627^ bushels suffice for I S weeks, proceeding in the same ratio i* Questions* ^39 1309. A and "B join in trade, and make up a capital of such a number of dollars, that if it were diminished b^ ^ of J of f of itself, and then by | of | of f of the remainder, there would be 648 dolls, left. Their gain was 50 per cent on their capital, and is to be divided in proportion to their shares of the capital. A's share is to be the most, and to be such a sum, that if multiplied by B's, and that product mul- tiplied by the whole capital, and that product divided by ^ of I off off of the capital, the quotient would be 324000 dols. What is each man's share of the gain ? 1310. Six persons, to amuse themselves, threw upon the ground a sum of money, to see how much each could pick up ; and the first time, they gathered as follows : A, | off of f of a pound sterling ; B, f of | as much, and | of a dollar be- sides ; C, I of § of I as much as B, and ^^ of a penny, and f of 7 livres besides ; D, f of |- of f as much as C, and 8d. and ^ of a farthing, and J of | of 4 rubles besides ; E, | of f of i as much as D, and iff of a penny, and f of f of 5i crowns besides ; and F, the rest, which was | of | of | of | as much as E, and J of J of | of | of 2| guilders besides. Then the money was thrown down again, and the second time they gathered as follows : A, | of 4 of f of | of what he had the first time ; B, f of f of -^ of ^ of J as much as A ; C, I of I of f of 45 times as much as B ; D, f of f of ^^^ of | of J of 1 92 times as much as C ; E, | of -/^ of | of | of f of 25 times as much as D ; and F, the rest. Now, as F had so much the most, he threw down his again ; and A got | of f of ^ of | of it, and \ shilling, and g^r^ of a shilling besides ; B, | of |^ of f of J of 12 times as much as A ; C, 1^ of ^ of ^ of J of f of 35 times as much as B ; D, |^ of ^ of I of j\ of i 60 times as much as C ; F, | of | of ^ of -J of 24 times as much as D ; and E, the rest. Then E gave to B | of a marc banco, to C 3 livres, and to ¥ half a crown ; and A gave to B § of a rupee, and to F one rial of plate. How much, in sterling money, was the sum thrown down ; and how much had each person at last ? MJSrn OF FART JL Page 26, line 3j for 31 read 32. 32, ,, 27 right hand c.lumn, insert 9 under 8. ;, 43, ,, 15 after i?is. insert D. „ 48, „ 55, *or 112, 40. read 112, 46 ;, 49, „ 38, for 6 s» venths, rea'l 5 sevenths, 39, for 5 nitiths, r- ad 6 ninths ., 60, ,, 14, for fraction, read fractions, 20, ff.r 94^1 read 4_i J, G8, „ 31, read 64 and 13, 11 at.d 17, 49. ,, 70, „ 34 right band colun.n, reau 80 „ 79, „ 28, f .r 14 A. read 1-4/.. „ 95, ,, 40, for Z03 r ad Xl03 „ 115, „ 18 undei 30, read 900 „ 134, „ 26, for ^9461- &c , read ^6461- &c. „ 144, „ 25, lor bame, < tad area. ., 145, „ 6, rt.ad. 20-92! 875 acres. „ 149, „ 34, for 7496, read 17496. „ 150. , 23ik24, •bi-12i«28, tead-12&'28, ,, 151, „ '17, for 1 1624- read 1 162— 5, 154, „ 9, r ad 5 own, 20 2il, read, Clarkson, 150, 40, r^ad Ann ricn. 29815000. V, 155, „ 13, read rear 1799 14, read i^s';e(i 2967000. -> 156, „ 2, r ad v. ar 1782. 29, read. jClSOOO. 43, read 8758 8. ,, 160, „ 13, for 2700 read, 27000. „ 168, „ 38, read, of 2t7 3. 41, for A. ead ih ,5 177, „ 31, for 22095 r( ad, 220959. and fcr 4065119, read 40651L j5 188, „ 4, for 1377000 read 1377000. 36, for ^52049, read 232049. „ 190, „ 6, for were, r^ ad was. „ 17. for 30, read 80. „ 25, for 245, read 285 „ 42, for 44187, read 44487. 5, 205, „ 25, for 29, read 19. ., 211, „ 33, read 4036658 „ 2>3, „ 7, f r 335 '-4, read^65 1-4 5, 224, „ 31, veaa 2000. *) f\