^wi THE SLATED ARITHMETIO. Entered accorjio? to Act of Confess in the year Is6«, by A. S. BARNES & CO., in the Clerk'* OfBco of the [iistrict Court of the United St:ite« for the Southern District of New York. SILICATE BOOK SLATE SURFACE, P«t«nt«,i February 24, 1857; January 15, issr, and AnguU 3ith, 1868. UNIVERSITY ARITHMETIC,. EMBIIACING THE SCIENCE OF NUMBERS, GENERAL RULES FOR THEIR APPLICATION. By CHARLES DAVIES, LL.D., AUTIIOB OF PRIMARY, INTELLKCTPAL AND SCHOOL ARITnMKTICS; ELEMENTARY ALOEBEA ; KLKMENTAKY GEOMKTP.V; PItAOTIOAL MATHEMATICS; ELEMENTS OF 8UKVEVING ; ELEMENTS OF ANALYTICAL GEOMETRY; nESCRIPTlVK GEOMETRY; SHADES, SHADOWS AND PEP^PECTIVE; DIFFERENTIAL AND INTEGRAL CAL- CULUS, AND LOGIC AND UTILITY OF MATHEMATICS. NEW YORK : A. S. BARNES & Co., Ill & 113 WILLIAM STREET. BOSTON: WOOLWOIITH, AINSWORTH & Co. 1868. ADVERTISEMENT Th2 attention of ■ T^acliers U respectfully invitefl to the Revised EWTIONS of MiMn' griilmeHtal §tm% FOR SCnOOLS AND ACADEMIES. 1. DAYIES' PRIMARY ARITHMETIC. 2. DAVIES' INTELLECTUAL ARITHMETIC. 3. DAVIES' PRACTICAL ARITHMETIC. 4. DAVIES' UNIVERSITY ARITHMETIC. 5. DAVIES' PRACTICAL MATHEMATICS. The above Works, by Charles Davies, LL.D., Author of a Complete Course of Mathematics, are designed as a full Course of Arithmetical Instruction necessary for the practical duties of busi- ness life; and also to prepare the Student for the more advanced Series of Matliematics by the same Author. The following New Editions of Algebra^ by Professor Davies, are commended to the attention of Teachers: 1. DAVIES' NEW ELEMENTARY ALGEBRA AND KEY. 2. DAVIES' UNIVERSITY ALGEBRA AND KEY. 3. DAVIES' BOURDON'S ALGEBRA AND KEY. Eatorcd according to Act of Congress, in the year ono thousand eight hundred and sixty-four, BY CHARLES DAVIES, In Uio Clerk's Office of the District Court of the United States for 'the Southern District of New York. TO THE TEACHERS OF THE UNITED STATES, TREATISE ON ARITHMETIC, TIIB LAST OF A SERIES OF "WORKS DESIGNED TO LESSEN THE LABOB AND IMPROVE THE SYSTEMS OF TEACHING, RESPECTFULLY DEDICATED, BY THE AUTHOR. IT IS OFFERED AS A TOKEN OF HIS GRATEFUL APPRECIATION OF THE INDULOENOB •WITH wniCn IlIB OTTlIEn WOBKS HAVE BEEIT BlOBIVTa), ASOt AS A TESTIMONY OF HIS REGARD FOR THOSE WITII WHOM HE HAS LONG BEEN A 00- UUBOUJER IN THE WORK OF PUBLIC INSTBUOTIOK PREFACE Science, in its popular signification, means knowledge re- duced to order ; that is, knowledge so classified and arranged as to be easily remembered, readily referred to, and advan- tageously applied. More strictly, it is a knowledge of laws, relations, and principles. Arithmetic is the science of numbers, and the art of applying numbers to all practical purposes. It is the foundation of the exact and mixed sciences, and an accurate knowledge of it is an important element either of a liberal or practical education. It is the first subject, in a well-arranged course of instruo- tion, to which the reasoning faculties of the mind are applied, and is the guide-book of the mechanic and man of business. It is the first fountain at which the young votary of knowledge drinks the pure waters of intellectual truth. It has seemed, to the author, of the first importance that this subject should be carefully treated in our Elementary Text- books. In the hope of contributing something to so desirable an end, he has prepared a series of arithmetical works, em- bracing four books, entitled. Primary Arithmetic ; Intellectua Arithmetic ; Practical Arithmetic ; and University Arithmetic— the latter of which is the present volume. Primary Arithmetic. This first-book is adapted to the capacities and wants of young children. Sensible objects are employed to illustrate and make familiar the simple combina- tions and relations of numbers. Each" lesson embraces ono combination of numbers, or one set of combinations. 6 PRE}"ACE. Intellectual Arithmetic. This work is designed to present a thorough analysis of the science of numbers, and to form a complete course of mental arithmetic. I have aimed to make it accessible to young pupils by the simplicity and gradation of its methods, and to adapt it to the wants of advanced students by a scientific arrangement and logical connection, in 11 the higher processes of arithmetical analysis. Practical Arithmetic. Great pains have been taken, in the preparation of this book, to combine theory and practice; to explain and illustrate j^rinciiDles, and to apply them to the com- mon business transactions of life — to make it emphatically a practical work. The student is required to demonstrate every principle laid down, by a course of mental reasoning, before deducing a proposition or making a practical application of a rule to examples. He is required to fix and apprehend the unit or base of all numbers, whether integral or fractional — to reason with constant reference to this base, and thus make it the key to the solution of all arithmetical questions. It is hoped, that the language used in the statement of principles, in the definition of terms, and in the explanation of methods, will be found to be clear, exact, brief, and comprehensive. University Arithmetic. This work is designed to answer another object. Here, the entire subject is treated as a science. The pupil is supposed to be fAmillar with the simple operations in the four ground rules, and with the first principles of frac- tions, these being now taught to small children, either orally or from elementary treatises. This being premised, the language of figures, which are the representatives of numbers, is care- fully taught, and the different significations of which the figures themselves are susceptible, depending on the manner in which they are written, are fully explained. It is shown, for example, that the simple numbers in which the value of the unit increases from right to left according to the scale of tens, and the De- nominate or Compound numbers in which it increases according to a varying scale, belong to the same class of numbers, and that both may be treated under tke .same rules, Heace, the rules for Notation, Addition, Subtraction, Multiplication, and Division, have been so constructed as to apply equally to all numbers. This arrangement, which the author has not seen elsewhere, is deemed an essential improvement in the science of ^I'ithmctic. ;^ .. In developing the properties of mmaberSj from their elemcn xnry to tlicir highest combinations, great labor has been be- stowed on classification and arrangement. It has been a lead- ing object to present the entire subject of arithmetic as form- ing a series of dependent and coimected projyosUions : so that the i)upil, while acquiring useful and practical knowledge, may at the same time be introduced to those beautiful methods of reasoning which science alone teaches. Groat care has been taken to demonstrate every proposition — to give a complete analysis of all tlie .methods employed, from the simplest to the most difficult, and to explain fully tlie reason of every rule. A full analysis of the science of Numbers has developed but one law; viz., the law which con- vects all the numbers of arithmetic with the unit one, and which points out the relations of these numbers to each other. In the Appendix, which treats of Units, Weights, and Meas- ures, &c., the methods of determining the Arbitrary Unit, as well as the general law which prevails in the formation of numbers, are fully explained. I cannot too earnestly recom- mend this part of the work to the special attention of Teach- ers and pupils. In fine, the attention of Teachers is especially invited to this work, because general methods and general rules are employed to abridge the common arithmetical processes, and to give to them a more scientific and practical character. In the present edition, the matter is presented in a new form ; the arrange- ment of the subjects is more natural and scientific ; the metliod;^ have been carefully considered; the illustrations abridged and simplified ; the definitions and rules thoroughly revised and cor- 8 rPwEFACE. rected ; and a very large number and variety of practical ex- amples have been added. The subjects of Fractions, Propor- tion, Interest, Percentage, Alligation, Analysis, and Weights and Measures, present many new and valuable features, which are not found in other works. A Key to the present work has also been published for the use of such Teachers as may desire it, — prepared with great care, containing not only the answers and solutions of all the examples, but a full and comprehensive analysis of the more difficult ones. The author has great pleasure in acknowledging the interest which Teachers have manifested in the success of his labors : they have suggested many improvements, both in rules and methods, not only in his elementary, but also in his advanced works. The recitation-room is the final tribunal, and the intel- ligent teacher the fiiml judge, before which all text-books must stand or fall. CoLUJNrBiA College,) May, 1864. J CONTENTS-. FIRST FIVE RULES. PAQX Definitions 13 Ex-pressing Numbers 14 Notation and Numeration 14-24 Formation and Nature of Numbers 24 Scales 25-28 Integral Units 28 Properties of the 9's 29 Eduction 80-34 Addition 34-44 Subtraction 44-54 Multiplication 54-C8 Division 68-87 Practice 87-90 Longitude and Time 00-94 Applications in tlie Four Rules 9 1-103 Properties of Numbers 103 Divisibility of Numbers 103-105 Cancellation lOG-109 Least Common Multiple 109-111 Greatest Common Divisor 111-115 COMMON FRACTIONS. Definition of, and First Principles 115-119 The six Kinds of Fractions 119-120 Six Propositions 120-124 llwduction of Common Fractions 124-133 Reduction of Denominate Fractions 132-135 Addition of Common Fractions 135-140 Subtraction n\' CnirTSKm Frp/^tions 140-145 10 CONTENTS. PACK Multiplication of Common Fractions 145-148 Division of Common Fractions 148-152 Complex Fractions 153 Applications in Fractions 153-155 DUODECIMALS. Principles and Rules 155-1G4 DECIMAL FRACTIONS. Definition of Decimals, &c.. 1G4 Decimal Numeration Table — First Principles, &c 1G5-170 Addition of Decimals.. 170-173 Subtraction of Decimals , . 173-175 Multiplication of Decimals 175-177 Contractions in Multiplication 177-179 Division of Decimals 179-184 Contractions in Division 184-186 Reduction of Common Fractions to Decimals 180-190 Reduction of Denominate Decimals 190-191 Repeating Decimals — DejSnition of, &c 191-195 Reduction of Repeating Decimals 195-201 Addition of Repeating Decimals 201 Subtraction of Repeating Decimals 201 Multiplication of Repeating Decimals 202 Division of Repeating Decimals 203 CONTINUED FEACTIONS. Definitions and Principles ' 203-206 RATIO AND PROPORTION. Ratio Defined 206 Compound Ratio 207 Simx)le Proportion 209 Direct and Inverse Proportion , 211 Single Rule of Three 214-220 Double Rule of Three 231-235 Partnership 225 -2n0 CONTENTS. l^t PEBCENTAGE. • PAQK Percentage Defined and Ill-ustrated ...iitj*tii¥fitfti:,..i... 331-230 Profit and Lobs ..^.v.^^^^....... ... 237-241 Commission 242 244 Interest ^^» ^. , ^ 245-253 Partial Payments ,.., . ,^. .^. . . . . .,. . . .^^,^. . . 253-25G Problems in Interest.. :...;..::;:?. .■Tr'^'l'^.^^T:"'^'..';:. ..^ 250-257 Compound Interest 258-259 Discount \. . , . i .'i . i; H; 200-201 Banking ...,.^»^.,.,.. ..."....... ... 202-204 Bank Discount « tu.. 204-206 Stocks ....^«...... 207-272 Insurance ..««.»».,•- 273-275 Liie Insurance It^v: V.'.^^ 275-277 Endowments 278 Annuities 279-281 AI'FLICATIOKS. Assessing Taxes .-...".. 281-284 Equation of Payments .'.v.v.'.v.-. 284-294 Alligation ..'............ 295 Alligation Medial. 295 Alligation Alternate. .'. 290-301 Custom-house Business 202-200 Tonnage of Vessels .' ". 300-307 General Average '."..'. 308-310 Coins and Currencies. .. .'.'.V .".'/. v. 311 Exchange ............... ;<>.¥. Im .*Vw^ 312-319 Arbitration of Exchange. ...... ...;;::.-;;.-t;;8i;^. 820-332 POWERS AND ROOTS. Involution 322 Evolution 323 Extraction of Square Root 324-333 Cube Root 333-339 ARirnMETICAL PROGRESSION. Definition of, &c 330 Different Cases a40-3^4 12 CONTENTS. GEOMETRICAL PROGRESSION. Different Cases 345-348 PAGK Definition of, &c... : 344 ANALYSIS. Analysis and Promiscuous Examples 348-869 MENSURATION. Mensuration of Surfaces. 370-375 Mensuration of Volumes 375-380 Gauging 381-383 Mechanical Powers 384-393 Questions in Natural Pliilosophy 392-398 APPENDIX. Different Kinds of Units 399-403 Abstract Units 403 Units of Currency 404-408 Linear Units 408-409 Units of Surface 410 Units of Volume 411-413 Units of Weight 414-41G Units of Time 416-418 Units of Circular Measure 419 Miscellaneous Table 419 Books and Paper , 420 Metric System of Weights and Measures 421-432 Answers 433-466 UNIVERSITY ARITHMETIC. Definitions. 1. A Unit is a single tiling, or one. 2. A Number is a unit, or a collection of units. 3. Science treats of the properties and relations of things : Art is the practical application of the principles of Science. 4. Arithmetic is the Science of Numbers, and also the Art of applying numbers to practical purposes. 5. A Proposition is something to be done, or demonstrated. 6. An Analysis is an examination of the separate parts of a proposition. 7. An Operation is the doing of something with numbers. 8. A Rule is the direction for performing an operation. 9. An Answer is the result of a correct operation. Operations of Arithmetic. 10. There are, in Arithmetic, five fundamental operations : Notation and Numeration, Addition, Substraction, Multiplica- tion, and Division. 1. What is a Unit? — 3. What is a Number? — 3. Of what does Scionco treat? What is Art? — 4. What is Arithmetic?—^. What is a Pn^posi- tion?— G. What is an Analysis? — 7. What is an Operation ?— 8. AVlmt is a Rule? — 9. What is an Answer? — 10. IIow many fundamental oper- ations are there in Arithmetic? What are thoy? 14 NOTATION AND NUMERATION. Expressing Numbers. 11. There are three methods of expressing numbers: 1. By words, or common language ; ; ,2. By letters, called the Roman method ; '3. By figures, called the Arabic method. Expressing Numbers by "Words. 12. A single thing is called One and one more Two and one more Three and one more Four and one more Five and one more Six and one more Seven and one more Eight and one more Nine and one more Each of the words, one, two, thr number, and denotes how many units are taken are generally called numbers ; though, in fac the names of numbers. One, Two. TJiree. Four. Five. Six. Seven. Eight. Nine. Ten. ee, four, &c., expresses a These words they are but NOTATION AND NUMERATION. 13. Notation is the method of expressing numbers, either by letters or figures. Numeration is the art of reading, correctly, any number expressed by letters or figures. There are two methods of Notation : the one by letters, the other by figures. The method by letters is called the Roman Notation; the method by figures is called the Arabic Notation, NOTATION AND NUMEEATION. 15 Roman Notation. 14. In the Roman Notation, seven capital letters are used. They express the following values : I Y five, X ter L fifty, c one hundred, D five hundred. M All other numbers arc expressed by combining these letters, according to the following principles : 1. Every time a letter is repeated, the number which it de- notes is repeated. 2. If a letter denoting a less number be written on the right of one denoting a greater^ the number expressed will be the stem of the numbers. 3. If a letter denoting a less number be written on the left of one denotmg a greater, the number expressed will be the difference of the numbers. 4. A dash ( — ), placed over a letter, increases the number for which it stands, a thousand times. I II III IV V VI VII VIII IX X XX XXX XL L LX LXX Roman Table . One. LXXX . Eiglity. Two. xc . Ninety. Three. c . One hundred. Four. cc . Two hundred. Five. ccc . Three hundred. Six. cccc . Four hundred. Seven. D . Five liundred. Eight. DC . Six hundred. Nine. DCC . Seven hundred. Ten. DCCC . Eight hundred. Twenty. DCCCC J . Nine hundred. Thirty. M . One thousand. Forty. MD . Fifteen hundred. Fifty. MM . Two thousand. Sixty. V Five thousand. Seventy. X . Ten thousand. ion was used by the Ro mans hence its name. numbering chnpteip, pages, &c. It Id NOTAI.ON AND NUMEEATION. Ezamples Express tlie following numbers in Roman Notation : 1. Eleven. 23. Eighty-one. 2. Fourteen. 24. Eighty-seven. 3. Sixteen. 25. Eighty-nine. 4. Seventeen. 26. Ninety-four. 5. Nineteen. 2t. Ninety-five. 6. Twenty-two. 28. Ninety-seven. t. Twcnty-eiglit 29. Ninety-nine. 8. Twenty-nine. 30. One hundred and fifteen. 9. Thirty-three. 31. Seven hundred and fifty. 10. Thirty-seven. 32. One thousand and sixty. 11. Thirty-eight. 33. Two thousand and forty. 12. Forty-three. 34. Five hundred and sixty. 13. Forty-seven. 35. Nine hundred and sixty, 14. Forty-nine. 36. Six hundred and ninety. 15. Fifty-six. 3t. One thousand and fifty. 16. Fifty-eight. 38. Four thousand and four. n. Fifty-nine. 39. Six thousand and nine. 18. Sixty-five. 40. Nine thousand and nine. 19. Sixty-nine. 41. Eight hundred and six. 20. Sixty-seven. 42. Six hundred and eight. 2L Seventy-five. 43. Eight thousand and six. 22. Seventy-six. 44. Two thousand and one. 11. How many methods are there of expressing numbers? Wliat aro they ? 12. What does each of the words, one, two, three, &c., denote? What are these words generally called? What are they, in fact? 13. What is Notation? What is Numeration? How many methods f Notation are there ? What are they ? 14. How many letters does the Roman notation employ ? "SVhich are they? What value does each represent? What is the efiect of repeat- ing a letter ? What is the number, when a letter denoting a less number is placed on the right of one denoting a greater? AYhat is the number, when a letter denoting a less number is placed on the left of one de- noting a greater? What is the efi(3Ct. of placing a dash over a letter? NOTATION AND NUMERATION. IT Arabic Notation. 15. Arabic Notation is the method of expressing numbers by figures. Ten figures are used, and they form the Alphabet of the Arabic Notation. They are, 01234 56^89 naught, one, two, three, four, five, six, seven, eight, nine. The naught, 0, is also called cipher. It denotes no number )ut the absence of a thing. Thus, if there are no apples in a basket, we write, the number of apples in the basket is The other nine figures are called Significant Figures, or Digits Orders of Units. 16. We have no single figure for the number ten. We therefore combine the figures already known. This we do by writing on the right hand of 1: Thus, 10 which is read, ten. This 10 is equal to ten of the units expressed by 1. It is, however, but a single ten, and may be regarded as a unit, ten times as great as the unit 1. It is called, a unit of the second order. 17. When two figures are written by the side of each other, the one on the right is in the place of units, and the other in the place of tens, or of units of the second orde". Each unit of the second order is equal to ten units of the first order. When units simply are named, units of the first order are always meant. Units of the second order are written thus : One ten, or ... . . 10 Six tens, or sixty, . 60 Two tens, or twenty, . 20 Seven tens, or seventy, . 70 Threti tons, or thirty, . 30 Eight tens, or eighty. . 80 Four tens, or forty, . . 40 Nine tens, or ninety. . 90 Rve tens, or fifty, . . 50 One hundred, . . . . 100 1-8 KOTATIOl^ AND >TtrMERATION. 18. To express ten imits of the second order, or one hundred^ we form a new combination : Thus, . 100 by writing two ciphers on the right of 1. This number is read, one hundred, and is a unit of the third order. We can now express any number less than one thousand. In the number two hundred and fifty-five, there are 5 units, 5 tens, and 3 hundreds. Write, there- g «; .^ fore, 5 units of the first order, 5 units of the second ^ ^ | order, and 2 of the third ; and read from the right, 2 5 5 units, tens, hundreds; and from the left, two hundred and fft7j-five. In the number five hundred and ninety-five, there g » ^ are 5 xmits of the first order, 9 of the second, and ,§-2 3 five of the third ; and it is read from the right, units, 5 9 5 tens, hundreds. In the number six hundred and four, there are § g .^ 4 miits of the first order, of the second, and 6 of ^ ^ 3 the third. 6 4 The right-hand figure always exjjresses units of the first order; the second, units of the second order ; and the third, units of the third order, 19. To express ten units of the third order, or one thousand^ we form a new combination : Thus, 1000 by writing three ciphers on the right of 1. This num])er is read, one thousand, and is a unit of Wiq fourth order. We may. now form as many orders of units as we please : A single unit of the first order is expressed by ... . 1 A unit of the second order by 1 and 0; thus, . . . . 10 A unit of the third order by 1 and two O's ; . . . . 100 A unit of the fourth order by 1 and three O's ; . , . . 1000 A imit of the fifth order by 1 and four O's ; . . . . 10000 And so on, for units of higher orders. NOTATION AND NUMERATION. lU Hence, tlie following principles : 1st. The same figure expresses different units according to the place which it occupies: 2d. Units of the first order occiqyy the place at the right; units of the second order, the second place ; units of the third order, the third place; and the unit of any figure is deter mined hy the number of its place: 3d. Ten units of the first order make one of the second ^ ten of the second, one of the third; ten of thd third, one oj the fourth ; and so on for the higher orders: 4th. When figures are written hy the side of each other, ten units in any one place maJce one unit of the place next at the left. Examples in Writing the Orders of Units. 1. Write 7 units of the 1st order. 2. Write 8 units of the 2d order. 3. Write 9 units of the 4 th order. 4. Write 3 units of the 1st order, with 9 of the 2d. 15. What is the Arabic Notation? How many figures are used? What do tliey form? Name the figures. What does express? WTiat are the other figures called? IG. Have we a separate character for ten? How do we express ten? To how many units 1 is 1 ten equal ? May ten bo regarded as a single unit ? Of what order ? 17. When two figures are written by the side of each other, wliat place docs the right-band figure occupy? The figure on the left? When units simply arc named, wliat units are meant? 18. How do you write one hundred? To how many units of the second order is it equal ? To how many of the first order ? How may it be regarded ? Of what order ? How many units of the third order in 200? In GOO? In 900? 19. To what arc ten units of the third order equal? How do you write it ? How do you write a single unit of the first order ? How do you write a unit of the second order? Of the third? Of the fourth? Ten units of the first order, make what ? Ten of. any order, make what ? When figures are written by the side of each other, how many unita of any place make one unit of the place next to the left ? 20 NOTATION AND NUMEEATION. 6. Write 9 units of the 3d order, with 6 of the 2d, and 1 of the 1st. 6. Write units of the 2d order, 8 of the 1st, with 4 of the 3d, and 1 of the 4th. I. Write 8 units of the 6th order, t of the 4th, 9 of the 5th, of the 3d, 2 of the 2d, and 1 of the 1st. 8. Write 8 units of the 8th order, 6 of the Tth, of the 1st, 3 of the 2d, 4 of the 3d, 9 of the 4th, of the 6th, and 2 of the 5th. 9. Write 4 units of the 10th order, 8 of the 1th, 3 of the 9th, 2 of the 8th, of the 6th, 3 of the 1st, 6 of the 2d, of the 3d, 1 of the 4th, and 2 of the 5th. 10. Write 3 units of the 2d order, 2 of the 1st, 9 of the 3d, of the 4th, 9 of the 9th, 6 of the 8th, t of the tth, of the 6th, and 4 of the 5th. II. Write 3 units of the 11th order, of the 10 th, 8 of the 4th, of the 5th, 2 of the 6th, of the Ith, 3 of the •8th, 4 of the 9th, 1 of the 3d, 2 of the 2d, and 3 of the 1st. 12. Write 3 units of the 12th order, 6 of the 11th, 3 of the 8th, 7 of the 6th, 2 of the 4th, and 1 of the 2d. 13. Write 5 units of the 13th order, 8 of the 12th, of the 9th, 6 of the Tth, 8 of the 3d, and 12 of the 1st. 14. Write T units of the 14th order, 5 of the 13th, 6 of the 12th, 5 of the 10th, 1 of the 8th, 9 of the 6th, 5 of the 4 th, and 8 of the 1st. 15. Write 9 units of the 15th order, 4 of the 13th, 8 of the 9th, 2 of the 6th, 1 of the 3d, and 2 if the 2d. 16. Write 6 units of the 16th order, 9 of the 12th, 1 of the 9th, 4 of the 1th, of the 6th, 8 of the 4th, 9 of the 5th, and 2 of the 2d. 11. Write 8 units of the 20th order, the 13th, 4 of the 11th, 9 of the 9th, the 5th, and 9 of the 3d. 18. Write 6 units of the 10 th order, the 6th, of the 4th, and 1 of the 1st 5 of the 18th, 6 of 1 of the mh, 4 of 5 of the 8th, 9 of NOTATION AND NUMEEAITON. 21 19. Write 9 units of the ISth order, and then dhnuiish the figure of each order by 1 till you come to and mclude ; then the figure of each order by 1, till you reach the first order. increase order ; and then read each Numeration Table 7th Period. Quintillions. 6th Period. Quadrilliona. 6th Period. Trillions. 4th Period. Billions. Sd Period. Millions. 2d Period. Thousands o j Ct-I O 00 (=1 w i-l EH H W W 'S « 2 a -a H 70,804,21G,G3G,806,304,625 Notes. — 1. Numbers expressed by more than three figures are writ- ten and read by periods, as shown in the above table. 2. Each period always contains three figures, except the left-hand period, which may contain one, two, or three figures. 3. Thfe unit of the first, or right-hand period, is 1; of the second period, 1 thousand; of the third, 1 million; of the fourth, 1 billion; and so on, for periods, stiU to the left. 4. To QuintiUions succeed Sextillions, SeptiUions, Octillions, Nonil- lions, Decillions, Undecillions, Duodecillions, &c. 5. The pupils should be required to commit, thoroughly, the names of the periods, so as to repeat them in their regular order from left to right, as well as from right to left. 6. Formerly, in the English Notation, six places were given to Millions. They were read. Millions, Tens of Millions, Hundreds o Millions, Thousands of Millions, Tens of Thousands of MlUions, Hundreds of Thousands of Millions. This method produced great irregularity in the Notation, as it gave three places to the units of the first two periods (viz.; units and thousands), and six places to the next denomination. The French method, which gives three places to the unit of each period, is fully adopted in this country, and must soon become universal. ^2 NOTATION AND NUMERATION. Notation and Numeration. Rule for Notation. I. Begin at the left hand and write each period in order, as if it were a period of units: TI When the number, in any period except the left-hand period, can he expressed by less than three figures, prefix one or two ciphers; and ivhen a vacant period occurs, fill it with ciphefs. Rule for Numeration. I. Separate the number into periods of three figures each^ beginning at the right hand: II. Name the unit of each figure, beginning at the right: III. Then, beginning at the left hand, read each period as if it stood alone, naming its unit. Examples for Practice. Express the following numbers in figures. 1. Six hundred and twenty-one. 2. Five thousand seven hundred and two. 3. Eight thousand and one. 4. Ten thousand four hundred and six. 5. Sixty-five thousand and twenty-nine. 6. Forty millions two hundred and forty-one. 1. Fifty-nine millions three hundred and ten. 8. Eleven thousand eleven hundred and eleven. 9. Three hundred millions one thousand and six. 10. Sixty-nine billions three millions and two hundred. Let the pupil point off and read the following numbers ; then wi'ite them in words : 11. 91 16. 3204560*1 21. 184236104 12. 326 It. 90464213 22. T403026054 13. 3302 18. 47364291 23. 21104080495 14. 65042 19. 4031902169 24. 21896120421 15. 142604 20. 91046302 25. 8140290308091 NO-mTION AND NUMEBATION. 23 26. 85046804GY023 21. 90403040t20l5G 28. 172304136893210 29. 30461214302704 30. 167320410341204 31. 2164032189765421 Let each of the above examples, after being written on the blackboard, be analyzed as a class exercise ; thus — 1. In how many ways may the number 97 be read? 1st. The common way, ninety-seven. 2d. We' may read, 9 tens, arid *l units. 2. Im how many ways may 326 be read? 1st. By the common way, three hundred and twenty-six. 2d. Three hundred, 2 tens, and 6 units. 3d. Thirty-two tens, and six units. 3. In how many ways may the number 5302 be read? 1st. Five thousand three hundred and two. 2d. Five thousand, three hundred, tens, and 2 units. 3d. Fifty-three hundi'ed, tens, and 2 units. 4th. Five hundred and thirty tens, and 2 units. 4. In 65042, how many ten thousands? How many thou- sands ? How many hundreds ? How many tens ? How many units ? 6. In 742604, how many hundred thousands ? How many ten thousands ? How many thousands ? How many hundreds ? How many tens? How many units? Let the pupil express the following in figures : 32. Forty-seven quadrillions, sixty-nine billions, four hundred and sixty-five thousand, two hundred and seven. 33. Eight hundred quintillions, four hundred and twenty-nine millions, six thousand and nine. 34. Ninety-five sextillions, eighty-nine millions, eighty-m'n housand, three hundred and six. 35. Six quintillions, four hundred and fifty-one bilhons, sixty five millions, forty-seven thousand, one hundred and four. 36. Nme hundred and ninety-nine billions, sixty-five millions, eight Imndred and forty-one thousand, four hundred and eleven. 24 FORMATION OF NUMBWiS. Formation of Numbers. 20. One refers to any single tJdng, and has no reference to kind or quality. It is called an Abstract Unit. One foot refers to a single foot, and is called a Denominate or Concrete Unit. 21. An Abstract Number is one whose unit is abstract (hus, three, four, six, &c., are abstract numbers. 22. A Denominate or Concrete Number is one whose unit is denominate or concrete ; thus, three feet, four dollars, five pounds, &c., are denominate numbers. 23. A Simple Number is a single unit, or a single collection of units, either abstract or denominate. Two numbers are of the same denomination when they have the same unit ; and of different denominations when they have different units. 24. A Compound Denominate Number is one expressed by two or more different units ; as, 1 yard 2 feet 6 inches. Laws of the Units and Scales. 25. We have seen that when figures are written by the side of each other, thus, 6 18 9 4, the language implies that ten units, of any place, make one unit of the place next to the left. When figures are written to express English Currency, thus, £ s. d. far. 4 17 10 3, the language implies, that four uniti of the lowest denomination 20. To what does one refer? What is it called? To wliat does one foot refer? 'What is it called?— 21. What is an Abstract Number?— 22. What is a Denominate Number ?— 23. What is a Simple Number ? When are two numbers of the same denomination ? When of different denominations ?— 24. Wliat is a Compound Denominate Number? F^kMATION OK NUMKKliS. 25 make one unit of vlie next liig-lier; twelve of the second, oue of the tliu'd ; and twenty of tlie third, i ■ of the fourtli. When figures arc written to expres.; Avoirdupois weight, thus, T. cwt qr. lb. oz. dr. 27 11 2 24 11 10 be language implies, that IC units of the lowest denomiuatio make one unit of the next higher; 16 of the second, one o the third ; 25 of the third, one of the fourth ; 4 of the fourth, oue of the fifth; and 20 of the fifth, oue of the sixth. All the other compound denominate numbers are formed on the same principle : hence, We pass from a lower to the next higher denomination by considering how many units of the lower make one unit of the next higher, 26. A Scale is a series of numbers expressing the law of relation between the different units of any number. There are two kinds of scales — Uniform and Varying. A Uniform Scale is one in which the law of relation between the units, at any step of the scale, is the same. A Varying Scale is one in which the law of relation between tlie units is different, at different steps of the scale. The Units of a Scale, at any step, are denoted by the num- ber of units of the louver denomination which make one unit of the next higher. 25. When several figures are written by the side of each other, what docs the language imply ? In the English Currency, how many units of the lowest denomination make one t»f the next higher ? IIow many of tlie second make one oi th(5 thliii? IIow many of the third, one of the fourth? In Avoirdupois weight, how many units of tlie lowest denomination make one of the next higher? How many of the second, one of the tliird? 20. What is a Scale ? IIow many kinds of scales are there ? Name them. What is a Uniform Scale? What is a Varying Scale? 26 FORMATION OF NUMBERS. Uniform Scale of Tens. 27. If wc write a row of I's thus : 2 S . I'S -^a. -ga. -g^g -g f-itSfl JnSS *-if-iS fH r^Rg '^'^.S nnt-i^ 'C!.^ §gg §§^ §g2 gg-a W^M W^S WHH WHt^ 111, 111, 111, 111, the language of figures expresses that the unit of each place increases from right to left, according to the scale of tens. This is called the decimal system of numbers, and the scale is uniform. United States Currency. 28. United States Currency affords an example of a system of denominate units, increasing according to the scale of tens : thus, ^ ^ S - • a & 1 1 in which ten units of any denomination make one unit of the next higher. The dollars are denoted by $, and separated from the dimes, cents, and mills by a period (.), called the decimal point. Varying Scales. 29. If we write the well-known signs of the English Cur- -ency, and place 1 under each denomination, we shall have £ s. d. far. 1111 27. If several figures are written by the side of each other, what does the language express? Wliat name is given to this system of numbers? What is the scale ? — 28. How do the different units compare with each other in United States Currency ? 1 1 1 1 1 FOliMATION OF NUMBERS. 27 The signs, £ s. d. and far., denote the value of the unit 1 in each denomhuitiou ; and they also determine the relations be- tween the different units. For example, this simple language expresses the following ideas : 1st. That the unit of the right-hand place is 1 farthing ; of the place next at the left, 1 penny ; of the next place, 1 shilling ; of the next place, 1 pound : and 2d. That 4 units of the lowest denomination make one unit of the next higher ; 12 of the second, one of the third ; and 20 of the third, one of the fourth. Hence, 4, 12, and 20 are the numbers which make up the scale. 30. If we take the denominate numbers of Avou-dupois weight, we have T. cwt. qr. lb. oz. dr. 111111 in which the units increase in the following manner : viz., count- ing from the right, 10 units of the lowest denomination make 1 unit of the next higher ; 16 of the second, 1 of the third ; 25 of the third, 1 of the fourth ; 4 of the fourth, 1 of the fifth ; 20 of the fifth, 1 of the sixth. The scale, therefore, for this class of denominate numbers, varies according to the above law If we take any other class of denominate numbers, as the Troy weight, we shall have a different scale, and the scale will continue to vary as we pass from one class of numbers to another. But in all the formations, we shall recognize the application of the same general principles. 31. There are, therefore, two general methods of forming the different systems of integral numbers, from the unit one. The first consists in preserving a uniform law of relation between the different units. If that law of relation is expressed by 1 0, we have the system of decimal or common numbers. 29. Is tho scale uniform or varying in the English Currency ? Namo the units of the scale at each change of denomination. — 30. Name tlio units of the scale, at each step, in the Avoirdupois weight. Name then also in the Apothecaries weight? 28 INTEGRAL UNITS. The second method consists in the application of known, though varying laws of change in the units. These changes in the units, produce different systems of denominate numbers, each of which has its appropriate scale, Integral Units of Arithmetic. 32. The Integral Units of Arithmetic are divided into eight classes : 1. Units of Abstract Numbers ; 2. Units of Currency ; 3. Units of Length, or Linear Units ; 4. Units of Surface ; 5. Units of Yolume, or Cubic Units ; 6. Units of Weight ; 7. Units of Time ; 8. Units of Angular Measure. First among the units of arithmetic is the abstract unit 1. This is the primary base of all abstract numbers, and becomes the base, also, of any denominate number, by merely naming the particular thing to which it is applied. Of the Signs. 33. The sign =, is called the sign of equality. When placed between two numbers, it denotes that they are equal ; that is, that each contains the same number of units. The sign +, is called plus, which signifies more. When placed between two numbers, it denotes that they are to be added together. Thus, 3 + 2 = 5. The sign — , is called minus, a term signifying less. When placed between two numbers, it denotes that the one on tlie right is to be taken from the one on the left. Thus, 6 — 2 = 4. 31. How many general methods are there of forming numbers from the unit one? What is the first? What is the second? — 32. Into hov many classes arc the Units of Arithmetic divided ? Name them. PROPERTIES OF THE 9's. 29 The sign X, is called the sign of multiplication. When placed between two numbers, it denotes that they are to be mul- tiplied together. Thus, 12 x 3, denotes that 12 is to be multi- I>Iied by 3. The parenthesis is used to indicate that the sum or difference )f two or more numbers is to be regarded as a single number. Thus, (2 + 3 + 5) X G, sliows, that the sum of 2, 3, and 5, is to be multiplied by G. And (5 — 3) X 6, denotes that the difference between 5 and 3, is to be multiplied by 6. The sign -^, is called the sign of division. When placed between two numbers, it denotes that the one on the left is to be divided by the one on the right. Thus, 4 -f- 5, denotes that 4 is to be divided by 5. Properties of the 9's. 34. In any number, written with a single significant figure, as, 4, 40, 400, 4000, &c., the excess over exact D's is equal to the number of units in the significant figure. For, any such number may be written thus, 4 = 4. Also, 40 = (9 + 1) X 4, 400 = (99 + 1) X 4, 4000 = (999 + 1) X 4, &c., &c., &c. Each of the numbers 9, 99, 999, &c., contains an exact num- ber of 9's ; hence, when multiplied by 4, the several products w^ill contain an exact number of 9's ; therefore, 03. What is the sign of Equality? What is the sign of Addition? Wliat of Subtraction? AVhat of Multiplication? For what is the pa- renthesis used ? Wliat is the sign of Division ? ol. Wliat will be the excess over exact 9's in any number expressed by a single significant figure? How may the excess over exact 9's be found in any number whate.vcr ? 30 REDUCTION OF The excess over exact 9's, in each number, is 4 ; and the same may he shown for each of the other significant figures. If we write any other number, as 6253, we may read it, 6 thousands, 2 hundreds, 5 tens, and 3. Now, the excess of 9's in the 6 thousands, is 6 ; in 2 hundreds, it i? 2 ; in 5 tens, it is 5 ; and in 3, it is 3 : hence, in them all, ii is 16, which is one 9, and 1 over : therefore, t is the excess over exact 9's in the number 6253. In Hke manner, The excess over exact 9's, in any number whatever, is found by adding together the significant figures, and rejecting the exact 9' s from the sum. Note. — It is best to reject or drop the 9, as soon as it occurs : thus, we say, 3 and 5 are 8 and 2 are 10; then, dropping the 9, we say, 1 to 6 is 7, which is the excess ; and the same for all similar operations. 1. What is the excess of 9's in 48t01 ? In 6t498 ? 2. What is the excess of 9's in 9412021 ? In 2704962 ? 3. What is the excess of 9's in 87049612? In 4987051? REDUCTION. 35. Reduction is the operation of changing a number from one unit to another, without altering its value. 36. Reduction Descending is the operation of changing a Qumber from a greater unit to a less. 37. Reduction Ascending is the operation of changing a number from a less unit to a greater. 38. If we have 4 yards, in which the unit is 1 yard, and wish to change to feet, the units of the scale will be 3, since 3 feet make 1 yard ; therefore, the number of feet will be 4 X 3 = 12 feet. 35. What is Reduction? — 30. What is Reduction Descending? — o7 What is Reduction Ascending? DENOMINATE NUMBERS. 31 If it were required to reduce 12 feet to inches, the units of the scale would be 12, since 12 inches make 1 foot : hence, 4 yards = 4 x 3 = 12 feet = 12 x 12 = 144 inches. If, on the contrary, we wish to change 144 inches to feet, and then to yards, we would first divide by 12, the units of the scale in passing from inches to feet ; and then by 3, the unit of the scale in passing from feet to yards. Ilence, 1st. To reduce a number from a higher unit to a lower Multiiiily the units of the highest denomination by the number of units in the scale, and then add to the product the units of the next lower denomination. Proceed in the same manner through all the denominations till the number is brought to the required denomination. 2d. To reduce a number from a lower unit to a higher: Divide the given number by the number of units in the scalCf and set down the remainder, if there be one. Divide the quo- tient thics obtained, and each succeeding quotient in the same manner, till the number is reduced to the required denomina- tion: the last quotient, with the several remainders annexed, will be the answer. Examples. 1. Kednce £3 14s. 4d. to pence. We first multiply the £Z by 20, which gives 60 shillings. We then add 14, makmg 14 shillings : we next multiply by 12, and the product is 888 pence : to this we add 4d. and we have 892 pence, which are of the same value as £>Z 14s. 4d. If, on the contrary, we wish to change 892 pence to pounds, shiUings, and pence, we should first divide by 12 : the quotient is 14 shillings, and 4d. over. We next divide by 20, and the quotient is i£3, and 14s. over: hence, the result is £Z 14s. 4d., which is equal to 892 pence. The reductions, in all the denominate numbers, arc made in the «amc manner. 32 REDUCTION OF 2. Id £5 5s., how many shil- lings, pence, and farlliings ? £5 5s. 20 105 5 shilh'ngs added. 12 1260 4 5040 Here the reduction is from a greater to a less unit. 4. In 34 T. 16 cwt. 3 qr. 19 lb., how many pounds? 3. In 5040 farthings, how many pence, shillings, and pounds ? 4) 5040 farthings. 12 ) 1260 ponce. 2|0 ) 10|5 shillings. i25 5s. In this example, the reduc- tiop is from a less to a greater unit. 5. In 69694 lb., how many tons, cwt., qr., and lb. ? 25 )69694 4)2m qr. . 19 lb. 2|0 )69|6 cwt. . 3 qr. 34 T. . . 16 cwt. Ans. 34 T. 16 cwt. 3 qr. 191b. 69694 lb. I 6. In $426, ho^7 many cents ? How many mills ? 7. In 36 eagles 8 dollars and 6 dimes, how many cents ? 8. In 8150 mills, how many dollars and cents? 9. In 43 eagles 3 dollars and 5 mills, how many mills ? 10. In £31 9s. 8d., how many pence ? 11. In 1569 farthings, how many pounds, shillings, pence and f ir things ? 12. In IT. 14 cv\t. 1 qr. 20 lb. Avoirdupois, how man} pounds? 13. In 15445 lb. Avoirdupois, how many tons, cwts., qrs., and lbs.? 34 20 696 4 16 cv,-t. added. 2187 25 3 qr. added. 13954 55U 19 lb. added. DENOMINATE NUMBEliS. 33 14. IIow many grains of silver in 4 lb. G oz. 12 dwt. and T-r.? 15. TIow many pounds, ounces, pennyweights, and grains of old in T04121 grains? IG. In 5lb 1 3 13 13 2 gr. Apothecaries' weight, how n I an j^ grains ? JT. In 114947 grains, how many pounds, ounces, drams, .-^(.ruples, and grains? 18. In G yards 2 feet 9 inches, how many inches? 19. In 5 miles, how many rods, yards, feet, and inches? 20. In 2730 inches, how many yards, feet, and inches? 21. In 56 square feet, how many square yards? 22. In 355 perches, or square rods, how many acres, roods, and perches ? 23. In 45G square chains, how many acres ? 24. In 3 A. 2 R. 8 P., how many perches ? 25. In 14 tons of round timber, how many cubic inches? 2G. In 31 cords of wood, how many cubic feet? 27. In 5G320 cubic feet, how many cords? 28. In 157 yards of cloth, how many nails? 29. In 192 ells Flemish, how many yards? 30. In 97 yd. 3 qr., how many ells English ? 31. In 4 hlid. wine measure, how many quarts ? 32. In 75G0 pints, wine measure, how many hogsheads? 33. In 7 hogsheads of ale, how many pints ? 34. In 74304 half-pints of ale, how many barrels? 35. In 31 bushels, dry measure, how many pints? 36. In 2110 pints, dry measure, how many bushels? ' 37. In 2 solar years of 365 d. 5 h. 48 ra. 48 sec, each, how many seconds ? 38. How many months, weeks, and days in 254 days, reckon- ing the month at 30 days? 3* 3i ADDITION. ADDITION. 39. Addition is the operation of finding tlie sum of two or more numbers. The Sum of two or more numbers, is a number containing as many units as all the numbers taken together. Operations of Addition. The operations of Addition depend on four principles, viz.: 1. A single number expresses a collection of like units. 2. Like units alone can be added together; that is, units must Ix added to units, tens to tens, dollars to dollars, &c. 3. Every number expressed by two or more figures, is the sum ot its various units. 4. The smn of sev^cral numbers is equal to the sum of all their parts. 1. What is the sum of 769 and 487 ? operation. Analysis. — Write the numbers, so that the like units 7 6 9 may fall in the same column, thus: 4 8 7 Sum of the units 16 Sum of the tens 14 Sum of the hundreds .... 1 1 Entire sum 12 5 6 The example may be done in another way, thus : Set down the numbers as before: then say, 7 and 9 are IG : set down 6 in the units' place, and the 1 ten operation. under the 8 in the column of tens. Then say, 1 to 8 7 6 9 are 9, and 6 are 15. Set down the 5 in the column of f i tens, and the 1 hundred in the column of hundreds. ■ We then add the hundreds, and find their sum to be t -^ o 12 : hence, the entire sum of 1256. Note. — 1. Observe, that units of the same value are always written In the same column. 2. When the siun in any column equals or exceeds the units of th Bcale 10, it produces one or more units of a higher order, wliich belong to the next column at the left. In that case, write down the excess, and add the higher units to the next column. This is called carrying to the next column. The number to be carried, should not, in practice be written under the column at the left, but added mentally. Al>L»iTiuN. (2; (3) (4) 85468 672143 4783614 9104 79161 504126 379 8721 872804 94951 760025 6160544 oo 5. What is the sum of 35 dollars 4 dimes 6 cents 5 mWh i dollars 7 mills, and 97 cents 3 mills? Analysis. — Write the figures expressing units of the same value in the same column, separating the dollars from the cents and mills by a period: then add the columns as in simple numbers. OPERATION. $35,465 4.007 .973 $40,445 OPBRATION. £ B. d. far. 14 7 8 3 6 18 9 2 21 6 6 1 6. Let it bo required to find the sum of i£14 7s. 8d. 3far.; and £(j 18s. 9d. 2far. Analysis. — Write the numbers, as before, so that units of the same order sball fall in tlio same column. Beginning with tho lowest de- nomination, we find the smn to l)e 5 farthings. But since 4 farthings make a penny, we set down the excess, 1 farthing, and carry one penny to tho column of pence. Tho sum of the pence then becomes 18, which is 1 shilling, and G pence over. Set down the pence, and carry the 1 shilling to the column of ehillings, the sum of which becomes 20 ; that is, 1 pound and G shillings. Setting down the G shillings, and carrying 1 to the column of pounds, we find the entire sum to bo £21 Gs. Gd. Ifar. Rule. I. Write the numbers so that units of the same value shall fall in the same column: II. Add the units of the lowest denomination, and divide their sum hy so many as make one unit of the denomination next higher: set down the remainder, and carry the quotient to the next higher denomination. Proceed in the same man- ner through all the denominations, and set down the entire turn of the last column BQ ADDITION. Proof. 40. The proof of an operation, in Addition, consists ia show- ing that the answer ontains as many units as there arc in all the numbers added. There are three methods of proof. I. Begin with the units' column and add, in succession, all the columns in an opposite direction. If the work is right, the residts will agn : II. Divide the giccn numbers into parts, and add the p)0.rls separately : then add together the partial sums : if the work is right, the residts will agree: III. Find the excess of 9's in each number, and place it at the right (Art. 34). Add these numbers, and note the excess of 9's in their sum. This excess should be equal to the excess of 9's in the sum of the numbers. Note. — The third method of proof applies only to simple numbers.^ 1. What is the sum of 182t96, 143274, 32160, and 4t04t? and what the proof? 1st Method. 182196 143214 32160 41041 405211 2d Method. 182196] 143214 ) 32160] 41041) 326010 19201 405211 89. "What is Addition ? What is tlie sum of two or more numbers ? On how many principles do the operations of Addition depend ? What is the first principle ? What the second ? What the third ? What the fourth ? What is the Paile for Addition ? 40. How many methods of Proof are there for Addition ? AYhat is the process in the first method? What ifi the second? What in the third ? 41. What is the process of reading? IIow does it differ from epelliug? ADDl'l 'ION. 37 3d Method of Proof. 182706 • . . 6 excess of D's, 143274 . 3 " <( 32160 . 3 " << 47047 ..7 4 " (( 405277. 16 . 7 excess of 9's. Sum Reading. 41. The pupil should be early taught to omit the iniermediale words in tlie addition of columns of figures. Thus, in the above example, instead of saying, 7 and are 7 ; 7 and 4 are eleven ; 11 and 6 are seventeen ; he should simply say, seven, eleven, seventeen. Then, in the column of tens, he should say, five, eleven, eighteen, twenty-seven ; and similarly, for the other columns at the left. This is called reading the columns. Let the pupils be often practised in the readings, both separately and in concert in the class. Examples. (1) (2) (3) (4) 94201 - 80032 98800 10304 46390 • 4291 10926 67491 37467 2376 321 1324 4572, 840 479 46 0. VvHiat is the sum of 1376, 38940, 8471, 23G07, 891? 6. What is the sum of 3480902, 3271, 507321, 91243, 6001, 169? 7. Wliat i.s tlie sum of 42300, 6000, 347001, 525, 47? (S) (9) (10) (11) (12) dayi bushels. rods. minutes. gftllons. 1276 47917 9003 67321 760324 3718 12031 1881 4702 18720 9024 5672 6035 1067 5762 1028 728 3176 377 1082 9131 47 2004 99 47209 J8 ADDITION. (13) (14) ' (15) (16) (17) miles. furlongs. pouuds. dollars. casks. 1600 47468 76389 1602 40506 2588 69012 1036 9614 37219 9101 23419 2671 4732 50170 6793 15760 5132 5675 32614 8267 27900 6784 8211 73462 4572 12317 1672 (20) 4455 (21) 10001 (18) (19) (22) a75.365 $30,365 $180,000 $300.40 $4802.279 278.056 28.779 489.007 167.275 1642.107 420.96 10.101 76.119 18.197 3026.267 76.125 9.08 16.423 29.94 125.093 41.04 ) 7.14 9.011 10.08 42.75 (23 (24) (25) (26) £. 8. d. far. lb. oz. dwt fij I 3 t). oz. dr. 14 11 3 1 174 11 19 17 11 7 17 15 12 17 18 ] LO 2 75 10 13 94 10 6 29 32 10 29 7 6 642 3 10 60 9 2 84 10 9 42 14 ] LI 3 125 7 5 42 3 9 14 3 7 17 10 1 62 16 12 6 40 9 9 84 1 39 1 4 98 7 5 76 4 7 16 19 8 2 176 10 15 127 1 18 11 15 (27) ( 28) (29) (30) cwt. qr. lb. yd. qr. na. E. E. qr. na. L. rr.I. fur. 174 2 20 74 3 3 14 4 3 17 2 7 820 1 14 60 1 2 75 1 2 10 1 4 136 3 23 14 1 84 3 1 7 6 47 12 45 2 3 17 2 5 2 3 84 1 24 69 1 10 2 25 1 90 2 9 11 19 1 1 36 2 2 7 3 5 30 3 1 29 3 2 40 1 ADDITION. 39 ( 31) ( ;32 ) (33) (34 ) fd. ft in. A. E. P. Tun. hhd. gal gal. qt. pt. 174 1 11 77 3 39 714 3 56 14 3 1 2G0 2 64 2 37 626 1 48 74 2 1 150 2 10 16 1 29 320 29 96 1 126 1 9 72 18 156 2 31 47 2 I 90 7 36 2 20 225 1 42 22 1 72 1 4 42 2 14 84 17 65 1 8 2 6 11 3 7 96 1 34 19 ( 35 ) {' 36) (37) (38) chal. bu. qt yr. T\k. da. da. hr. min. qr, lb. oz. 14 31 6 127 9 2 140 12 27 44 21 14 25 14 2 320 10 3 340 16 40 14 16 12 36 29 .7 146 8 1 227 20 56 22 10 11 42 24 3 75 6 102 13 25 36 19 7 39 32 1 70 11 2 67 21 37 51 13 9 56 19 5 54 7 1 14 9 10 30 22 11 14 20 4 27 4 3 10 19 46 16 15 15 39. The population of the United States and Territories, in 1850, was as follows: White population, 19553068; Free Col- ored population, 434495 ; Slave population, 3204313 ; Indians, 400674 : what was the entire population? 40. In the j^ear 1850, the expenditures of the United States amounted to 43002168 doHars; in 1851, "to 48905879 dollars; in 1852, to 46007893 dollars : what were the expenditures of the United States for these three years? 41. A man of fortune bequeathed to each of his three sons, 10492 dollars; to each of his two daughters, 5976 dollars; to bis wife, the remainder of his property, which exceeded tlie amount bequeathed to his children by twelve hundred dolhirs : find the amount of his property. 42. A stage goes in one day 27 miles 3 furlongs 36 rods; 40 ADDITION. the ucxt, 32 miles 10 rods; the next, 3G miles 2 furlongs; tlie next, 25 miles G furlongs 38 feet : how far did it go in 4 days ? 43. Bought a barrel of flour for eight dollars and seventy- five cents ; a ton of plaster for five dollars sixty-two and a half cents ; a hat for three dollars twelve cents and five mills ; fifty pounds of sugar for four dollars fifty cents and nine mills : what was the amount of my bill ? 44. A lady bought a bonnet for $5,375 ; some silk for $12.03 ; some ribbon for 80.8t5 ; a shawl for $9.46 : what did the whole amount to ? 45. A wine-merchant taking an invoice of his liquors, finds that he has 5 hhd. 36 gaL 2 qt. of wine ; 3 hhd. 15 gal. 1 qt. 1 pt. of rum ; 1 hhd. 2 qt. of gin ; 40 gal. 1 pt. of whiskey : how much liquor in all? 46. Tea was imported into the United States, in the year 1851, to the value of $4798005; in 1852, $7285817; in 1853, $8224853 : what was the value of the tea imported during these three years ? 47. The United States exported tobacco, in the year 1851, to the amount of $9219251; in 1852, $10031283; in 1853, $11319319: what was the entire value of tobacco exported in these three years ? 48. A man sold his house and lot for $25840, which was $3186 less than he gave for them; how much did they cost him? 49. A speculator bought three city lots : for the first he paid $2870.43 ; for the second, $2346.75 ; for the third, $1563.82. He sold the same at an average profit upon each of $476.25: what amount did he receive for the lots? 50. "What is the fortune of a merchant who has $79650 in real estate, $25640 in merchandise, $9654 in furniture and librarj', $16835 in stocks, $12642 in debts due him, and $5685 in cash? 51. The churches of the United States and Territories, in l850, were : Baptists, 9375 ; Congregationahsts, 1706 ; Presby- ADDITION. 41 tcrians, 4824 ; Methodists, 13280 ; Universalists, 529: what was the whole number of churches belonging to these five denom- inations ? 52. In the same year, the value of the church property owned by the Baptists in the United States and Territories was $11020855 ; by the Congregationalists, $7970195 ; by the Presbyterians, $14543789 ; by the Methodists, $14822870 ; ])y the Universalists, 11752316 : what was the entire amount? 53. During the year 1853, there was coined in the United States, $51888882 of gold; $7852571 of silver; and 867059 of copper ; what was the amount of money coined in the United States in 1853? 54. A farmer sends to market the following quantities of butter: 18cwt. 2 qr. 161b.; 1 ton 5 cwt. 211b.; 2 qr. 141b.: how much did he send in all? 55. A man having 84 acres 3 roods 26 perches of land, buys 120 acres 14 perches more: how much did he then have? 56. Suppose a father divides his estate equally among his three sons, giving each twenty-five thousand dollars seven dimes six cents and five mills : what was the value of the estate ? 57. A farmer has three fields of grain : The first yields 1375 bushels ; the second, 1810 bushels ; the third, 1265 bushels ; he values his entire farm at $2975 more than the number of bushels of grain raised from these three fields : what was tlie value of his farm ? 58. Bought a silver teapot weigiiing lib. 6 oz. 12dwt.; a cream-cup, weighing 10 oz. 18 dwt. 20 gr.; a poninger, weighing lloz. 16gr.; a dozen large spoons, weighing lib. 14 dwt. 1 2 gr. : what was the weight of the whole ? 59. The whole number of adults in the United States and Territories, over twenty years of age, who could not read and write, in 1850, was as follows : Of whites, males,. 389G64 ; females, 573234 ; free colored, males, 407^2 ; females, 49S00 : wh:it was tlie whole number ? 42 ADDITION. 60. Caesar was murdered b. c. 43, and Washington died a.d. 1799. How many years elapsed between tlie death of these great men? 61. A forwarding merchant had in his store-room, at one time, Y500 bushels of corn; 12865 bushels of wheat; 4680 bushels of oats ; 3296 bushels of barley ; and had room enough left to store 4000 bushels of oats : how many bushels of grain frould the storehouse hold ? 62. A man engaging in trade, had 85164.50 in cash; 111810.25 in goods ; $3004 in notes. His net profits aver- iged 12384.16 a year, for 3 years : what was the total value jf the property at the end of the three years? 63. A person paid two eagles for a coat ; four dollars and ?ix dimes for a hat ; two dollars and sixty-three cents for a rest ; eight dimes seven cents and five mills for a knife : what was the amount of his bill? 64. From a piece of cloth, 12 yd. 2 qr. were cut at one lime; 16 yd. 1 qr. 3 na. at another, when there were 10 yd. 1 qr. 1 na. remaining : how much was there in the whole piece ? 65. A farmer purchased a plough for $91 ; a wagon, for $451 ; a horse, for $110J ; a load of hay, for $12^ ; a harrow, for $31 : what was the cost of the whole ? 66. If a certain warehouse be worth $12540.31^, and one- fourth the contents is valued at $5632.108 : what is the value of the warehouse and the whole of its contents? 67. An English gentleman wishing to possess a certain horse, offers in exchange another horse, valued at £25 13s. 6d., a carriage valued at £15 8s. 9d. 2far., and £IS in cash. The offer was accepted : what did he pay for the horse ? 68. In 1850, the State of New York produced 13121498 bushels of wheat ; Pennsylvania, 15367691 bushels ; Yirginia, 11212616-busheIs; Ohio, 14487351 bushels ; Missouri, 2981652 bushels ; Illinois, 9414575 bushels : what was the whole num- ber of bushels produced by those States in that year? ADDITION. 43 69. A farmer sold his wheat for $825.8t J ; his barley for $67.12J ; his pork for 880.10 ; his apples for $46 : how much did he receive for the whole ? 70. Three persons enter into copartnership : The first put iu 7825 dollars capital ; the second put in 1250 dollars more than the first ; and the third put in as much as the other two : wliti"- arts, we find that each part is 24 dollars. If Z_l_ each of these parts he now divided into 3 equal parts, 8 tliere will then be 30 parts in all, each equal to 8 dollars : here, tlie unit of the result is the same as thxit of tJic dividend. Ilcnco, we may regard division under two ix)ints of view : 1. As a process of reduction, in ivhich the unit of each suc- ceeding dividend is increased as many times as there are units in the divisor: 2. As a process of separating a number into equal parts ; in ichich case the unit of a part will be the same as that (f the dividend. Hence, when the divisor is a composite number: Divide the dividend by one of the factors of the divisor ; then divide the quotient, thus arising, by a second factor, and so on, till every factor has been used as a divisor: the last quo- tient will be the answer. 9'1. Hdt^ do yoii divicte when the divisor is a composite number? M CONTRACTIONS IN Examples. Divide the following numbers by the factors of the divisors 1. 2322 by 6 = 2x3. 2. 31152 by 24 = 4 X 6. 8. 19152 by 36 = 6 X 6. 4. 38592 by 48 = 4 X 12. 5. 1145592 by 12= 8x9. 6. 185160 by 96 = 8 X 12 1. 115716 by 64 = 8x8. 8. 463104 by 144 = 12 x 12. Note. — ^When there are remainders, after division, the operation is to be treated as one of reduction. 92. How to find the true remainder. 1. Divide the number 3611 by 30 = 2 x 3 x 5. 2 )3611 3 ) 1835 . . 1 = 1st rem. ... 1 6 )611 . . 2 = 2d rem. 2x2 = 4 12 2 . . 1 = 3d rem. 1x3x2 =_6 Ans. 122 J J. For remainder, 11 i^TALYSis.— Dividing 3071 by 2, we have a quotient 1835, and a remainder, 1. After the third division, the quotient is 122, and the re- mainder, 1. Now, it is plain, from the first analysis, that, 1. The Tmit of the first quotient is as many times greater than the tmit of the dividend, as the divisor is times greater than 1 ; and similarly for all the following quotients. 2. The unit of the first remainder is the same as the unit of the dividend ; and the unit of any remainder is the same as that of the corresponding dividend. 3. The unit of any dividend is reduced to that of the preceding dividend, by multiplying it by the preceding divisor. Hence, to find the remainder in units of the given dividend is simply a case of reduction in which the divisors denote the units of the scale : therefore. To the first remainder, add the products lohich arise hy mxdtiplying each of the following remainders hy all the pre- ceding divisors^ except its own : the snm will he the true remainder: DIVISION. 85 Examples. Divide the following numbers, and find the remainders : 315 = 7x9x5. 462 = 3 X 2 X 7 X 11. 4 X 8 X 9 X 12. 3x5x7. 416705 by 804106 by 756807 by 3456 = 8741659 by 105 = 947043 by 385 = 5 X 7 X 11. 4704967 by 1155 = 11 x 7 X 5 x 3. 71874607 by 7560 = 8x7x9x5x3. 93. When the divisor is 10, 100, 1000, Ac. 1. Divide 3278 by 1000 = 10 x 10 x 10 Analysis.— We divide 3278 by 10, by simply cutting off 8, giving 327 tens, and 8 units remainder. We again divide by 10, by cutting off the 7, giving 32 hun- dreds, and 7 tens remainder. We again divide by 10, by cutting off the 2, giving a quotient of 3 thousands, and 2 hundreds remainder. The quotient then is 3, and a remainder of 2 hundreds, 7 tens, and 8 units, or 278: hence. Cut off from the right of the dividend as many fir/ures as there are cijohers in the divisor^ considering the figures at the left^ the quotient^ and those at the rights the remainder OPERATION. 10 )3 2 7|8 10 )3 2|7 . 10) 3|2 . . 3 . . qj2 7_8_ 8 rem. 7 rem. 2 rem. Ans, 94. When any divisor contains significant figures, with one or more ciphers at the right hand. 1. Divide 875896 by 32000. An Aiorsis.— The divisor 32000 = 32 X 1000. Dividing by 1000, gives a quotient 875, and 896 remainder. Then dividing by 32, gives a quotient 27, and 11 remain- der, which gives the hence, OPERATION. 32|000)875|896(27 64 235 224 Ans 11896 rem. 27MS8S- 86 CONTRACTIONS. Cut off, hij a line, the ciphers from the right of the divisor, and an equal number of figures from the right of the cUvi- dend: divide the remaining figures of the dividend by the remaining figures of the divisor, and to the remainder, if any, annex the figures cut off from the dividend, and the result will form the true remainder. Examples. j Divide tiic following numbers : 1. 1972654 by 420000. 2. 1752000 by 12000. 3. 73199006 by 801400. 4. 11428729800 by 72000. 5. 36981400 by 146000. 6. 141614398 by 63000. 95. When the divisor contains a fraction. 1. Divide 856 by 4J. AiTALTSis. — There are 5 fifths in 1 ; orERAxioN. hence, in 4 there are 20 fifths ; therefore, 3 ) 4 2 8 4i = 21 fifths. In the dividend 856, there w \ -, . n r> « r *• txea •* i *v * 7)1426.2 rem. are 5 times as many fifths as units 1 ; that is, 4280 fifths ; therefore, the quotient is 2 3.5 rem. 4280 divided by 21, equal 203lf. Hence, ^^^^^ 203?-^. wlien the divisor contains a fraction, Ilcduce the divisor and dividend to the fractional unit of the divisor^ and then divide as in integral numbers. Examples. Find the quotients in the following examples : 1. 3245 ^ 161. 5. 87317 -f- 9f. 2. 47804 -T- 151 6. 87906 ~ 12f 3. 870631 ~ 141 7. 95675 -^ 15|-. 4. 37214 -^ 511 8. 71096 ~ 17f. 92. How do you find the true remainder? 93. How do you divide when the divisor is 10, 100, &c, 94. How do you divide Avlien the divisor contains significant figures, «\'ith ciphers at the right? 95. How do you divide when tli6 divLsof Contains a fraction? PKACTICE. 87 Applications. 96. Division has three applications. 1. Given the number of things and their cost, to find the price of one thing. 2. Given the cost of a number of things, and the price of one thing, to find the number of things, 3. To divide any number of things, into a given number of equal parts. Rules. I. Divide the number dcnotiiKj the cost hy the number of things: the quotient icill be the price of one: II. Divide the number denoting the cost by the price of one : the quotieiit icill be the number of things : III. Divide the number denoting the things by the number of parts into which they are to be divided: the quotient icill be the number i?i each 2)(trt. PRACTICE. 97. Pkactice is an easy and short method of ajjplying the rules of arithmetic to questions which occur in trade and business. 98. An Aliquot Part of a number is any exact divisor of it, whether integral or fractional. Thus, 3 months is an aliquot part of a year, being one-fourth of it, and 12^ cents is an aliquot part of 1 dollar, being one-eighth of it. Aliquot Farts of a Dollar. $1 =100 cents, i of a dollar = 50 cents. A of a dollar = 33J- cents. J of a dollar = 25 cents. ^ of a dollar = 20 cents. 96. How many applications has division? Wliat are they? Give tlic rules. 97. What is Practice ?— 98. What is an aliquot part of a number? i of a dollar = 121 cents. j\ of a dollar z=z 10 cents. I'iT of a dollar — 6} cents. A of a dollar z= 5 ceiils. -h of a cent = 5 mill>:. 88 PllACTICE. Aliquot Parts of a Pound. £1 = 20 shillings. J of a pound = 10 shillings, i of a pound = 6s. 8d. J of a pound = 5 shillings. 4 shillings. i of a pound •i- of a pound = 3s. 4d. ■J of a pound = 2s. 6d. ^■2 of a pound = Is. 8d. 1 year ^ of a year J of a year Aliquot Parts of a Year. 12 months. 6 months. 4 months, 1 of a month = 15 days, i- of a month =10 days, days. J of a month ♦2 i rV of a year = 3 months of a year = 2 months of a year = 1 month. Fa Month. i i rV of a month = 6 days, of a month = 5 days, of a month = 3 days. 1. What is the cost of 3*16 yards of doth, at $1 75 a yard ? Analysis. — At $1 per yard, $376 = cost at $1 per yard. 376 yards cost $376. Separat- ing 75 cents into 50 cents, an < aliquot part of one dollar, and < 25 cents, an aliquot part of 50 cents, we take one-half of $376, and obtain $188, the cost of 376 yards at 50 cents per yard. Since 25 cents is one-half of 50 cents, we take ^ of $188, and thus obtain $94, the cost at 25 cents per yard. The sum $658 gives the cost at $1.75 per yard. 188 = cost at 50 cts. 94 = cost at 25 " '$658 = cost at $1.15 2. What is the cost of 196 yards of cotton, at 9d. per yard? A'ALYSis.— 9d. = 6d. + 3d. The. cost of 196 yards at ls.= 196s. Since 6d. = is., the cost at 6(1. = J of 196s. = 98s. The cost at 3d. = ^- as much as at 6d. ; h of 98s. = 49s. Tlie cost at 9d. = the sum = 147s. = £7 7g. 196s. = cost at Is. per yard. 1 1 6 i 3 i 20 ) 147s. = cost at 9d. £1 7s., entire cost. 98s. = cost at 6d. 49s. = cost at 3d. PRACTICE. 89 Examples. 1. What is the cost of 425 yards of calico, at Is. Gd. per yard ? 2. What is the cost of 415 yd. of tape, at Id. Ifar. per yard ? 3. What is the cost of 354 yards of cord, at IJd. per yard ? 4. At 12J cents = $| a yard, what will be the cost of 4756 ards of bleached shirting ? 5. At 2s. 6d. — £i per pair, what will be the cost of 3154 pairs of gloves ? 6. If wheat is 3s. 6d. a bushel, what will be the cost of 5320 bushels ? t. If broadcloth costs £1 Is. a yard, what will be the cost of 435 yards ? 8. If linen is 2s. 6d. = 2Js. a yai:d, what will be the cost of 660 yards? 9. What will be the cost of 40 lb. of soap, if 1 pound costs 6 J cents? 10. What will be the cost of 148 yards of cloth, at $3.15 a yard? 11. If one bushel of apples cost 62 J cents, what will be the cost of 816 bushels? 12. What will be the cost of 1000 quills, if every 5 quills cost 1^ cents ? 13. If 1 yard of extra-superfine cloth costs $9.50, what will be the cost of 85 yd. 2qr.? 14. What will 6J yards of cloth cost, at $3.15 a yard? 15. What will 8J boxes of lemons cost, at $1.25 a box ? 16. What will 151 pieces of calico cost, at $20.15 a piece 11. If one ton of iron costs $124, what will be tlic cost of 3T. 15cwt. 2qr. 151b.? 18. What will be the cost of 350 bushels of potatoes, at 3s. 6d. a bushel, English Currency? 90 LONGlTUDa AND TIMS. LONGITUDE AND TIME. 99. The equator of the earth is divided into 360 equal parts^ ?v^hich are called degrees of longitude. 100. The sun apparently goes round the earth once in 24 iours. This time is called a day. Hence, in 24 hours, the sun apparently passes over 360° oJ longitude ; and in 1 hour, over 360° -^ 24 = 15°. Since the sun, in passing over 15° of longitude, requires 1 hour, or 60 minutes of time, in 1 minute he will pass over 15 -^ 60 = 15' of longitude; and in 1 second of time, he will pass over 15' -f- 60 = 15" of longitude : Therefore, 15° of longitude require 1 hour of time. 15' " " 1 minute of time. 15" " " 1 second of time. Hence, I. If the loJigitude, expressed in degrees^ minutes, and seconds, be divided by 15 = 3x5, the quotient wiU be hoursy minutes, and seconds of time. II. If time, exj^ressed in hours, minutes, and seconds, be multiiMed ^y 15 = 3 x 5, the product will be decrees, minutes^ and seconds of longitude. Examples. 1. Reduce 45° 31' 45" of longitude to time. 5 )45° 31' 45 " Analysis. — We divide by 15, as in com- . - pound numbera ; giving us 3 hr. 2 m. 7 sec 3 hr. 2 m. T sec. 90. How is the equator of the earth divided? too. How long is the sun in apparently going round the earth? What is this time called? How many degrees of longitude does the sun pass over in a day ? How much in 1 hour ? How much in 1 minute? How much in 1 second? How is longitude reduced to time? How is time reduced to longitude? LONGITUDE AND TIME. 91 2. Reduce 8 hr. 16 m. 40 sec. of time, to longitude. Analysis. — We multiply the seconds, operation. minutes, and hours, each by 15, carry- 8 hr. 16 m. 40 sec. ing from one to the other as in the l£ multiplication of compound numbers. 124° 10' 00" 3. If the difference of time between two places be 42 m 16 sec, what is the difference of longitude? 4. What is the difference of longitude between two places, if the difference of time is 2 hr. 20 m. 44 sec. ? 5. When it is 12 m. at New York, it is llhr. 6 m. 28 sec. at Cincinnati : what is th«ir difference of longitude ? 101. Which place has the earlier time. When the sun is on the meridian of any place, it is 12 o'clock, or noon, at that place. And since the sun apparently goes from east to west, it will be past noon for all places at the east, and before noon for all places at the west. If, then, we find the difference of time between two places, and know the exact time at one of them, the corresponding time at the other will be found by adding the difference, if that other be east, or by subtracting it, if west. 102. Knowing the longitude of two places, and the time at one place, to find the corresponding time of the other. 1. The longitude of Albany is t3° 42' west, and that of Buf- falo '18° 55' west : what is the time at Buffalo when it is 10 o'clock A. M. at Albany ? , Analysis.— The difference of lonffi- operation. tude is found by subtraction, and is lo urj» 5^ 13'. This difference is changed into *^ ^^ the time 20 m. 53 sec, by dividing by 15) 5° 13' diff. lonjf. 15. Since Buffalo is west of Albany, Diff. time, 20 m. 52 sec. this difference must be subtracted from 10 hr.. the time at Albany, and the ^^^^^' ^ ^' ^ ^'^^' iL-maiudcr shows the time at Bufiklo i >l ^. to he 9 hr. 39 m. 8 sec. 9 hr 39 ra. 8 sec. 92 LONGITUDE AND TIME. Hence the following Rule. — I. Reduce the difference of longitude to time: II. Add the result to the given time, when the jylace at which the time is required lies east, and subtract it, ivhen west. Note. — If the longitudes are both east or botli west, the difference of longitude is found by subtraction. If one is east and the other west, the difference of longitude is expressed by the sum. Examples. 1. The longitude of New York is 74° 1' west, and that of Springfield, Illinois, 89° 33' west : what would be the time at New York when it is 12 m. at Springfield ? 2. The longitude of Philadelphia is 75° 10' west, and that of New York 74° 1' west ; what is the time at Philadelphia, when it is 3 o'clock p. m. at New York ? 3. Washington is in longitude 77° 2' west ; New Orleans in 89° 2' west. When it is 9 o'clock a. m. at Washington, what is the time at New Orleans ? 4. The difference of longitude between St. Louis and New York is 15° 35'. In travelling from New York to St. Louis, will a watch, keeping accurate time, be fast or slow at St. Louis, and how much ? 103. The time at each of two places, and the longitude of one, being known, to determine the longitude of the other. 1. New York is in longitude 74° 1' west. In what longitude is that place whose tkne is 10 o'clock a. m., when it is 2 o'clock p. M. at New York ? , 101. What is the time, at any place, when the sun is on the me- ridian? How will the time then be for any place at the east? IIow "vfill it be for any place at the west? If you have the difference oi time of two places, and know the time at one of them, how do you find the time at the other when it is east? When it is west? 102. Knowing the longitude of two places, and the time at one, how do you find the corresponding time at the other? 103. If the time at each of tlie places be known, and the longitude of one, how do you find the longitude of the other? 14 hr, 10 OPEtti . Om. ^TIOX. time diff. diff. > at N. Y. at place. 4 = 15 of timp. 60° U ~0' = 1 of long. LONGITUDE AND TIME. 93 AjfALYSiS. — The difference of time is 4 hr., equal to G0°, T/llich is the dif- ference of longitude. Since the time at New York is later in the day than that of the required place, New York must be east of that place, and the longitude is found by adding G0° to IP 1 which gives 134° 1' west, — the required longitude. Hence we have 134° 1' the following Rule. — I. Reduce the difference of time to difference of longitude : II. Add the result to the given longitude, when the place at which the longitude is required has the earlier time, and sub- tract when it has the later time. Examples. 1. Philadelphia is in longitude 75° 10' west. In what longi- tude is a vessel, whose chronometer indicates 11 hr. 30 m., a. m., Philadelphia time, when it is 2 hr. 15 min., p. m., on board the vessel ? 2. The longitude of St. Louis is 90° 15' west. A person at that place observed an eclipse of the moon at 10 hr. 40 m., p. ir. ; another person, in a neighboring State, observed the same eclipse 22 m. 12 sec. earlier; what was the longitude of the latter place, and the time of observation ? 3. Oxford, in England, is in longitude 1° 15' 22" west. What is the longitude of that place, whose local tune is 9 o'clock p. m., when the time at Oxford, as shown by an accurate chronome- ter, is 101 o'clock, p. M. ? 4. In going from London, whose longitude is 0, to Orego City, an accurate timekeeper was found to have gamed 8 hours In what longitude is Oregon City? - '6. A captahi observed an eclipse of the moon at 11 hr. 18 m. 15 sec, p. M., which was seen at Greenwich, according to the Kautical Almanac, at 12 hr. 50 m. 19 sec, p. m. In what longi- tude was the vessel? 94 APPLICATIONS. Applications in the Fundpcmental Rules. 1. What will it cost to build a wall 96 rods long, at $1.33 J a rod? 2. A farmer wishes to put 1066 bush. 2 pk. of potatoes into 4-74 barrels : what quantity must he put into each barrel ? 3. How many barrels of apples, each containing 2J bushels, can I buy for $36, at 45 cents a bushel? 4. The quotient arising from a certain division is 1236 ; the divisor is 375, and the remainder 184 : what is the dividend ? 5. The Croton Water Works of New York are capable of discharging 60000000 gallons of water every 24 hours : what is the average amount per minute? 6. The population of the United States in 1850 was 23191876. It is estimated that one person in every 400, dies annually from intemperance : how many deaths may be attributed annually to this cause in the United States? 7. If a quantity of provisions lasted 25 men 2 mo. 3 wk. 6 da., how long would it have lasted 10 men? 8. If a man's salary is $1200 a year, and his expenses are $640, how many years will be required to save $6720? 9. How long will it take to count 20 millions, at the rate of 80 per minute? 10. If 3160 barrels of pork cost $47400, how many barrels can be bought for $11475 ? 11. What will be the cost of 6 firkins of butter, each con- taining 96 pounds, at 12 J cents a pound? 12. What will 1000 quills cost, at i cent apiece? 13. What will be the cost of 85i yards of cloth, at $9^ a yard? 14. What will be the cost of 1 hhd. 2 gal. 3 qt. of brandy, at 56-J- cents a quart? APIMJCATIOXS. 05 15. What will b«j the cost of 196 yards of cotton goods, at Is. 6d. per yard ? 16. At 23. 8d. per bushel, what will 1246 bushels of oats cost? 17. If 112 lb. of cheese cost £2 16s., what is that per pound ? 18. What will be the cost of 1426 pounds of hay, at $9.15 per ton? 19. How much must I pay for the transportation of 3840 pounds of iron, from Albany to Buffalo, at $4.50 per ton? 20. Bought 124 bbl. of potatoes, each containing 2} bush., at 33 J cents a bushel : what was the cost ? 21. There are three numbers, whose continued product is 16200 ; one of the numbers is 25 ; another, 18 : what is the third number? 22. If 1 pwt. of gold is worth 92 cents, what would be the weight of $10059.28 in gold? 23. A man sold his house and lot for $4200, and took his pay in railroad stock, at 84 dollars a share : how many shares did he receive ? 24. A person bought 640 acres of land, at 15 dollars an acre. He afterwards sold 160 acres, at 20 dollars an acre ; 240 acres, at 18 dollars an acre ; and for the remainder he received $4560. What was his entire gain, and what did he receive per acre on the last sale? 25. A piece of ground, 60 feet long and 48 feet wide, is in- closed by a wall 12 feet high, and 2^ feet thick : how many cubic feet in the wall ? 26. What will be the cost of transportation from Montreal vo Boston of 325640 feet of lumber, at $2.37J per thousand? 2T. Bought 684 pounds of hay, at $12.40 a ton : what did it cost me? 28. At $2.12-J a hundred, what will 186 feet of lumber cost ? 96 APPLICATIONS. 29. How many shingles will it require to cover the roof of a building 40 feet long and 26 feet wide, with rafters IG feet long, allowing one shingle to cover 24 square inches ? 30. If 14 lb. 8 oz. 12 pwt. 3 gr. of silver be made into 9 tea- pots of equal weight, what will be the weight of each ? 31. A man bought 320 barrels of flour for $2688 : at wha rate must he sell it to gain $1.60 on each barrel? 32. A farmer has a granary containing 449 bush. 1 pk. 2 qt. of wheat ; he wishes to put it into 182 bags : how much must he put into each bag ? 33. A trader bought ^150 barrels of flour, for which he paid $48*15 ; he sold the same for $7.25 a barrel ; what was his profit on each barrel? 34. How many sheep, at $1.62J a head, can be bought for ^169? 35. How many canisters, each holding 31b. 10 oz., can be filled from a chest of tea containing 58 lb. ? 36. In 26 hogsheads the leakage has reduced the whole amount to 1358 gal. 2qt.; if an equal quantity has leaked out of each hogshead, how much still remains in each ? 3t. A man bought a piece of land for $3415.25, and sold it for $3801.65, by which transaction he made $3.40 an acre : how many acres were there? 38. The whole amount of gold produced in California in the year 1855, was as follows : $43313281, sent to the Atlantic States ; $6500000, sent directly to England ; and $8500000 retained in the country. In 1854, the total product of gold iu California was $5tU5000 ; how much more was produced iu 1855 than in 1854? 39. If the forward wheels of a carriage, are 12 feet in circum- ference, and the hind wheels, 16 feet 6 inches, how many more times will the forward wheels turn round than the hind wheels, iu running a distance of 264 miles ? Ai'i'LlCATIONS. 07 40. If a certain township is 9 miles long, and 4 J miles wide, how many farms of 192 acres each does it contain? 41. The total number of land warrants issued durmg the year ending September 30, 1855, was 34337, embracing 4093850 acres of land : what was the average number of acres to each warrant ? 42. The longitude of Philadelphia is 75° 10', and that o Kew Orleans 89° 2', both west : when it is 12 m. at Philadel- phia, what is the time at New Orleans? 43. The sun passes the meridian at 12 m., the moon at 8 hr. 30 m. p. M. : what is the difference in longitude between the sun and moon ? 44. Two persons, A and B, observed an eclipse of the moon ; A observed its commencement at 9 hr. 42m. p.m.; B was in longitude 13° 20', and observed its commencement 23 minutes earlier than A : what was A^s longitude, and B's time of ob- servation ? 45. If in 11 piles of wood there are 120 cords t cord feet 5 cubic feet, how much is there in each pile? 46. If 16cwt. 2 qr. 111b. 10 oz. of flour, be put into nine barrels, how much will each barrel contain? 47. A miller bought a quantity of wheat for $625.40, which he floured and put into barrels at an expense of $110.12^ : what profit did he make by selling it for $900? 48. America was discovered October 11, 1492 : how long to he commencement of the Revolution, April 19, 1775 ? 49. From a hogshead of wine, a merchant draws 18 bottles each contaming 1 pt. 3 gi. ; he then fills three 6-gallon demijohns, and 4 dozen bottles, each containing 2 qt. 1 pt. 3 gi. : how much Ibmafaied in the cask? 50. In 753689 yards, how many degrees and statute miles? 51. In 189 m. 3 fur. 6 rd. 1ft., how many feet? 98 APF LIGATIONS. 62. If 24 meu can build 168 rods of wall in 1 day, how many rods can 48 men build in 9 days ? 53. A certain number increased by 1164, and the sum multi- plied by 209, gives the j^roduct of 1913516 : what is the number? 54. If a man travels 146 mi. 1 fur. 14 rd. 14 ft. in 5 days, how much is that for each one half-day? 55. If 325 acres of land costs $11112.50, how many acres can be bought for 1545? 56. A merchant having $324, wishes to purchase an equal number of yards of two kinds of cloth ; one kmd was worth 4 dollars a yard, the other was worth 5 dollars a yard : how many yards of each can he buy? 51. From one-fourth of a piece of cloth, containing 68 yd. 3 qr., a tailor cut 5 suits of clothes : how much did each suit contain ? 58. A manufacturer having £6 10s., distributed it among his laborers, giving every man 18d., every woman 12d., and every boy lOd. ; the number of men, women, and boys was equal : what was the number of each ? 59. It is estimated that 1 out of every 1585 persons in Great Britain is deaf and dumb. The population, according to the census of 1851, was 20936468 : how many deaf and dumb per- sons were there in the entire population? 60. A grocer, in packing 6 dozen dozen eggs, broke half a dozen dozen, and sold the remainder for 1|- cents a piece : how much did he receive for the eggs? 61. How much time will a man save in 50 years, beginning with a leap year, by rising 45 minutes earlier each day? 62. During the year 1855, there were shipped to Great Britam from the XJnited States, 408434 barrels of flour ; 2550092 bushels of wheat ; 1048540 bushels of corn. Supposing the flour to have sold for $10.25 a barrel, the wheat for $2.1 2 J a bushel, and the corn for $0.94 a bushel, what was the value of the whole 7 APPIJCATIONS. 99 63. Richard Roe was born at 6 o^clock, a. m., June 24th, 1832: what was his age at 3 o'clock, r. m., on the 10th day of January, 1858 ? 64. A man dying without making a will, left a widow and 4 children. The law provides, in such cases, that the widow shall receive one-third of the personal property, and that the remaind(5r shall be equally divided among the children. The estate was Falued as- follows : Stocks worth $5000 ; 5 horses, at $85 each ; a yoke of oxen, at $110 ; 25 cows, at $22 each; 150 sheep, at $2 each ; some lumber, at $45 ; farming utensils, at $174 ; house- hold furniture, at $450 ; grain and hay, at $380 : what was the share of the widow and each child ? C5. How many shingles will it take to cover the two sides of the roof of a building, 55 feet long, with rafters 16| feet in length, allowing each shingle to be 15 inches long and 4 inches wide, and to lay one-third to the weather? 6G. The longitude of St. Petersburgh is 30° 45' east, and that of Washington t1° 2' west: what is the diflference of longitude between the two places ; and what is the time at St. Petersburgh when it is 6 o'clock a. m, at Washington ? 67. A vessel sails from New York to Liverpool. After a number of days, the captain, by taking an observation of the sun, finds that his chronometer, which gives New York time, differs 1 hr. 44 m. from the time, at the place of observation. If his chronometer shows the time to be 3 hr. 12 m., p. m., what is the time at the place of observation, and how far is the vessel east of New York ? 68. A cistern containing 960 gallons, has two pipes ; 45 gal- ons run in every hour by one pipe, and 25 gallons run out by he other : how long a time will be required to fill the cistern ? 69. The whole number of gallons of rum manufactured in the United States in 1850, was 6500500 : if valued at 50 cents a gallon, how many school-houses could be built, worth $750 each, with the proceeds ? • JOO APPLICATIONS. 10. A speculator sold 840 bushels of wheat for $2180, which was 1^500 more than he gave for it : what did it cost him a bushel ? 11. A farmer sold a grocer 30 bushels of potatoes, at 31 J cents a bushel, for v/hich he received 6 gallons of molasses, at 45 cents a gallon ; 60 pounds of mackerel, at 6 J cents a pound ; and the remainder in sugar, at 10 cents a pound: how many pounds of sugar did he receive ? 12. If a man travels 12 mi. 3 fur. 20 rd. in one day, how long will it take him to travel 114 mi. 1 fur. at the same rate? 13. If a man sells 2 bar. 12 gal. 2 qt. of beer in one week, how much will he sell in 12 weeks? 14. A hquor merchant had 550 pint bottles, 400 quart bottles, 350 two-quart bottles, 315 three-quart bottles, and 150 jugs hold- ing a gallon each : how many barrels of wine will fill them ? 15. How many yards of carpeting, one yard wide, will it take to cover the floors of two parlors, each 18 feet long and 16 feet wide ; and what will it cost, at $1.33J a yard ? 16. How many rolls of wall-paper, each 10 yards long and 2 feet wide, will it take to cover the sides of a room 22 feet long, 16 feet wide, and 9 feet high ? *l*l. Two persons are 1 mi. 4 fur. 20 rd. apart, and are travel- ling the same way. The hindmost gains upon the foremost 5 rods in travelling 25 rods : how far must he travel to overtake him ? 18. A man sold 500 bushels of wheat at 11.15 a bushel, and took his pay in sugar at 5 cents a pound. He afterwards sold one-half of the sugar : what quantity had he left ? 19. A man bought 1 barrels of sugar, at $12.81^ a barrel; he kept two barrels for his own use, and sold the remainder for what the whole cost him : what did he receive per barrel ? 80. A flour merchant bought a quantity of flour for $18150, and sold the same for $26250, by which he gained $3 a barrel : how many barrels were there ? APPLICATIONS. 101 81. Three men rented a farm, and raised 964 bush. 2pk. 4qt. of grain, which was to be divided in proportion to the rent paid by each. The first was to have one-half the whole ; the second, one-third the remainder ; and the third what was left : how much did each have? 82. A vessel, in longitude tO° 25' east, sails 105° 30' 56" west hen 46° 50' east, then 10° 5' 40" west, then 39° 11' 36" east: *n what longitude is she then, and how many days will it take her to sail to longitude 17° west, if she sails 3* 20' each day? 83. A privateer took a prize worth $25000, which was divided into 125 shares, of which the captain took 12 shares; 2 lieuten- ants, each 5 shares ; 6 midshipmen, each 3 shares ; and the re- mainder was divided equally among 85 seamen : how much did each receive? 84. If the longitude of Boston is 71° 4', and a gentleman, in travelling from Boston to Chicago, finds that his watch is 1 hr. 5 m. 44 sec. too fast by the time of the latter place : what is the longitude of Chicago, provided his watch has kept accurate time ? 85. What time would it be in Boston if it was 8hr. 2T m. 30 sec, A. jr., in Chicago ? 86. What time would it be at Chicago if it was 12 m. at Boston ? 87. Two places lie exactly east and west of each other, and by observation it is found that the sun comes to the meridian of the latter place 1 hour and 16 minutes after the former : how far apart arc they in minutes and degrees of longitude? 88. In 12 bales of cloth, each bale containing 16 pieces, and each piece containing 20 ells English, how many yards ? 89. A speculator gave $8968 for a certain number of barrels of flour, and sold a part of it for $2618, at $7 a baiTcl, and by so doing lost $2^ on each barrel : for how much must he sell the remainder, to gain $1060 on the whole? 102 APPLICATIONS. 90. How many eagles can be made from 24 lb. 4 oz. 6 pwt. 18 gr. of gold, making no allowance for waste, if each eagle weighs 11 pwt. 9gr. ? 91. A man paid $3284.82 for some wheat. He sold T40 bushels at 2 dollars a bushel ; the remainder stood him in $1.42 a bushel : how many bushels did he purchase ? 92. A man sold 105 A. 2 R. 20 P. of land for as many dollars as there were perches of land, payable in instalments, at the rate of 1 dollar an hour. If the contract was closed at 12 o'clock, m., April 1st, 1856, what length of time will be allowed the pur- chaser to pay the debt, reckoning 365 days 6 hours to the year ? 98. The sum of 2 numbers is 98, and their difference is 46 : what are the numbers? 94. A farmer paid $76 dollars more for a horse than for a cow J he paid $190 for both: what was the value of each? 95. How many days intervene between March 5th and August 21st, both days inclusive? 96. A merchant buys 810 barrels of flour, at $9.50 a barrel. He finds one-half of it injured, and is willing to lose one- quarter on the value of that part : how much loss was that on each half barrel? 9t. Three merchants. A, B, and C, are engaged together in business, and gain in one year $24612. This amount is to be equally divided among them, after paying A $6t5, and B $812, for extra services. How much did each receive ? 98. Four merchants are in partnership. Their apparent profits during the year amount to $56895 ; but they have expended for clerk hire, $6750 ; for rent, $3500 ; for insurance, $156 ; and for incidental expenses, $364. The first is to have $250 for extra services ; the second, $175 for travelling expenses ; and the third, $95 for various articles furnished by him to the concern. What was th^ share of profit of each, after paying these expenses ? PROPERTIES OF NUMBERS. 103 PROPERTIES OF NUMBERS. £jzact Divisors — Prime Numbers. 104. An Exact Divisor of a number, is any number, except 1 ind the number itself, that will divide it without a remainder. 105. One Number is divisible by another, when the remainder is 0. 106. An Odd Number is one not divisible by 2. 107. An Even Number is one divisible by 2. 108. A Prime Number is one which has no exa ¥> To> 9> t* 137. Give an example in writing, reading, and analyzing fractions 138. How many kinds of fractions are there? Name and describe each. 120 COMMON FRACTIONS. 2. An Improper Fraction is one* whose numerator is equal to, or exceeds the denominator/ The following are improper fractions : 3. A 6 8L 9. J_2 IJL 1_0 2' 3> 5> 7» 8» 6» 7» 7* Note. — Such a fraction is called improper, because its value equals or exceeds 1. 3. A Simple Fraction is one whose numerator and denomina tor are both whole numbers. The following are simple fractions : i35.8.98.6X ^ 4> 2> 6» 7' 2> 3> 3> 6* Note. — A simple fraction may be either proper or improper. 4. A Compound Fraction is a fraction of a fraction, or several fractions connected by the word of. The followmg are compound fractions : 1 of i, 4 of 1 of 4, 1 of 3, ^ of i of 4. 5. A Mixed Number is made up of a whole number and a fraction. The following arc mixed numbers: 3|, 41, Ql 6f, ^, 31 6. A Complex Fraction is one which has a fraction in one or both of its terms. The following are complex fractions; (i) A ii) IM FUNDAMENTAL PEINCIPLES. 139. Let it be required to multiply f by 4. Analysis.— In f there are 3 fractional operation. units, each of which is ^, and these are to |. x 4 = -'^- = ^. be taken 4 times. But three things taken 4 times, give 12 things of tbe same kind; that is, 12 eiglitlis ; hence, the product is 4 times as great as the multiplicand; tlierefore. Proposition I. — If the numerator of a fraction he multiplied by any number, the value of the fraction will be multiplied as many times as there are units in the multiplier. COMMON FRACTIONS. 121 Examples. 1. Multiply I by G, by 1. 2. Multiply I by 4, by 9. 3. Multiply /y by 11, by 12. 6. Multiply ^ by 3, by 4. 6. Multiply lA by 7, by 9. t. Multiply 41 by 5, by 10. 4. Multiply ^j by 12, by 14. | 8. Multiply §J by 3, by 11. 140. Let it be required to multiply yj by 4. Analysis. — In y'^ there are 5 fractional operation. 5 V 1 — 5 units, each of which is j\. If we divide s v> >i — s — __ the denominator by 4, the quotient is 3, and the fractional unit becomes ^, which is 4 times as great as ^^; because, if ^ be divided into 4 equal parts, each part will be ^j. If we take this fractional unit 5 times, the result, |, will be 4 times as great as j\; therefore, Proposition II. — If the denominator of a fraction he divided by any number^ the value of the fraction will be midliplied as many times as there are unUs in the divisor. Hence, to multiply a fraction by any number, divide its de- nominator. Examples. 1. Multiply U by 8, by 4, by 2. 2. Multiply ^ by 2, 3, 4, 6, 8. 3. Multiply 3^ by 6, 5, 10, 15. 4. Multiply i| by 2, 3, 4, 6, 8. 5. Multiply A by 4, 5, 10, 20. 6. Multiply ^V by 1, by 5. Y. Multiply f^ by 21, 6, 1, 3, 2. 8. Multiply if by 3, 4, 6, 9, 12. 141. Let it be required to divide ^j by 3. Analysis. — In y\ there are 9 fractional operation. units, each of which is yV» ^^^ thesQ are _9 -i. 3 — 9±1 — s to be divided by 3. But 9 tilings, divided 139. What is proved in Proposition I.? — 140. What is proved in PropoBition II.? 122 COinfON FRACTIONS. by 3, gives 3 things of the same Jchid for a quotient ; hence, the quo- tient is 3 elevenths, a number which is one-third of yy; hence, Proposition III. — If the numerator of a fraction he divided hy any number, the value of the fraction will he divided into as many equal parts as there are units in the divisor. Examples. 1. Divide i| by 2, 4, 8, 16. 2. Divide -fA by 2, 1, and 14. 3. Divide f § by 2, 5, 4, and 10. 4. Divide f f by 5, 6, 10, 15, 20. 5. Divide if by 2, 3, 6, and 9. 6. Divide f ^ by 3, 6, 8, 12. 7. Divide f| by 3, 9, and 21. 8. Divide f| by 6, 9, 21, 54. 142. Let it be required to divide jj by 3. 9 TT 3 = Analysis. — In y\ there are 9 fractional orEUATioN. units, each of which is y^-. Now, if we multiply the denominator by 8, it becomes 83, and the fractional unit becomes ^'j, which is one-third part of y\. If, then, we take this fractional unit 9 times, the result, ^^3, is just one-third part of ^j; hence, we have divided the fraction y^ by 3 : therefore, we have Proposition TV. — If the denominator of a fraction he multi- plied hy any numher, the value of the fraction will he divided into as many equal parts as there are units in the multiplier. Hence, to divide a fraction, multiply the denominator. 1. Divide f by 6, T, and 8. 2. Divide f by 5, 4, and 9. 3. Divide if by 3, 4, and 12. 4. Divide ff by 6, 8, and 11. Examples. 5. Divide |f by 1, 5, and 3. 6. Divide if by 1, 8, and 6. 7. Divide f | by 3, 1, and 11. 8. Divide fj by 8, 4, and 10 141. What is proved in Proposition III.? — 143. What is proved iu Proposition IV. ? COMMON FRACTIONS. 123 143. Multiply both terms of the fraction f by 4. Analysis. — In f, the fractional unit is |, and it operation. is taken 3 times. Bj multiplying the denominator 3x4 12 by 4, the fractional unit becomes ^V* *h® value of 5 X 4 "^ 20 which is one-fourth of |. By multiplying the D jmerator by 4, we increase the number of fractional units taken, 4 times ; that is, we increase the number of parts taJcen just as mamj times as wo diminish the value of each part; hence the value of the fraction is not changed: therefore, Proposition Y. — If both terms of a fraction be multiplied by the same number, the value of the fraction will not be changed. Examples. 1. Multiply both terms of the fraction J by 4, by 6, and by 5 2. Multiply both terms of ^\ by 5, by 8, by 9, and 11. 3. Multiply both terms of jf by 1, by 8, and 9. 4. Multiply both terms of ^ by 5, 8, 6, and 12. 6. Multiply both terms of f | by 2, 3, 4, and 5. 144. Divide both terms of the fraction j^ by 3. Analysis. — In yj, the fractional unit is ^j, and operation. it is taken times. By dividing the denominator 6-^3 2 by 3, the fractional unit becomes |, the value of 15-7-3 5* which is 3 times as great a^ yj. By dividing the numerator by 3, we diminish the number of fractional units taken 3 times ; that is, we diminish the number of parts taken just as many times as we increase the value of the fractional unit: hence, the value of the fraction is not changed ; therefore. Proposition YI. — If both terms of a fraction be divided by the he same number^ the value of the fraction will not be changed. 143. What is proved in Proposition V.?— 144. What is proved la Proposition VI.? 124 REDUCTION OP Examples. 1 . Divide both terms of f by 2 and by 4. 2. Divide both terms of f by 3. 3. Divide both terms of || by 2, 3, 4, 6, and 12. 4. Divide both terms of ff by 2, 4, 8, and 16. 5. Divide both terms of ^ by 2, 3, 4, 6, and 12. 6. Divide both terms of ^ by 2, 3, 4, 6, and 36. REDUCTION OF FRACTIONS. 145. Reduction of Fractions is the operation of chaLging a fractional number from one unit to another without altering its value. 146. The lowest terms of a fraction are when the numerator and denominator are prime to each other. CASE I. 147. To reduce a whole number to a fraction having a given denominator. I. Reduce 17 to a fraction whose denominator shall be 5, Analysis. — To reduce 17 to such a fraction is operation. the same as to reduce 17 to fifths. In 17 there 17 X 5 =85 are 17 times as many fifths as there are in 1. In 17 — _8_5 1 there are 5 fifths; therefore, in 17 there are 17 times 5 fifths, or, 85 fifths; hence, Rnle. — Multiply the whole number by the denominator, and ivrite the product over the required denominator, 145. What is reduction of fractions ? 146. Wliat are the lowest terms of a fraction ? 147. How do you reduce a whole number to a fraction having a given denominator? COMMON FRACTIONS. 125 Examples. 1. Change 18 to a fraction whose denominator shall be *l. 2. Change 25 to a fraction whose denominator shall be 12. 3. Change 19 to a fraction whose denominator shall be 8. 4. Change 29 to a fraction whose denominator shall be 14. 5. Change 65 to a fraction whose denominator shall be 37. 6. Reduce 145 to a fraction having 9 for its denominator. 7. Reduce 450 to twelfths. 8. Reduce 327 to a fraction having 36 for its denominator. 9. Reduce 97 to a fraction having 128 for its denominator. 10. Reduce 167 to eighty-ninths. 11. Reduce 325 to a fraction whose denominator shall be 75. CASE II. 148. To reduce a mixed number to an equivalent improper fraction. 1. Reduce 12^ to its equivalent improper fraction. Analysis. — Since in any num- operation. ber there are 7 times as many 7ths 12x7 = 84 sevenths, as units 1, there will be 84 sev- add 5 sevenths, enths in 12 : To these add 5 sev- gives 12f- = 89 sevenths, enths, and the equivalent fraction ^^^^ _ sj^ becomes 89 sevenths. Hence, Rule. — Multiply the whole number by the denominator: to the iwoduct add the numerator, and place the sum over the given denominator. Examples. 1. Reduce 39|- to its equivalent improper fraction. 2. Reduce 112j®o to its equivalent improper fraction. 148. How do you reduce a mixed number to an equivalent improper fraction ? • 120 REDUCTION OF 3. Reduce 42TJi to its equivalent improper fraction. 4. Reduce 67 Off to an improper fraction. 5. Reduce 367 jf 4- to an improper fraction. 6. Reduce Silj-fj to an improper fraction. 7. Reduce 67426fff to an improper fraction. 8. How many 200ths in 6751|J? 9. How many 151ths in 187 ^Vt ? 10. Reduce 149f to an improper fraction. 11. Reduce 375ff to an improper fraction. 12. Reduce 1*74949 1|-|9^ to an improper fraction. 13. Reduce 4834|-|- to an improper fraction. 14. Reduce 1789f to an improper fraction. 15. In 125|- yards, how many sevenths of a yard ? 16. In 375J feet, how many fourths of a foot? 17. In 464if hogsheads, how many sixty-thirds. 18. In 96JJq acres, how many 640ths of an acre? 19. In 984jY2 pounds, how many 112ths of a pound? 20. In 353^0- years, how many 366ths of a year ? 21. How many one hundred and thirty-fifths are there in the mixed number 87y3j? 22. Place 4 sevens in such a manner that they shall express the number 78. 23. By means of 5 threes write a number that is equal to 334. CASE III. 149. To reduce an improper fraction to an equivalent whole or mixed number. 1. In ^-p- how many entire units ? 149. How do you reduce an improper fraction to an equivalent mixed number? COMMON FliACTIONS. 127 Analysis. — Since there are 5 fifths in 1 unit, operation. there will be, in 278 fiftlis, as many units 1 as 5 is 5 )2t8 contained times in 278, viz., 55 and ^ times. Hence, 55f • the following Rule. — Divide the numerator by the denominator^ and the (Quotient will he the equivalent ivhole or mixed number. Examples. Reduce the following fractions to whole, or mixed nnmbers. 1. Reduce J^. 2. Reduce ^^^-. 3. Reduce VaV- 4. Reduce i|?g^. 6. Reduce Jy'/- pounds. 6. Reduce — Jf- days, t. Reduce -^^ yards. 8. Reduce 4f^. 9. Reduce ^Wtr^ acres. 10. Reduce Vir- 11. Reduce ^^^^^. 12. Reduce ^M|^. 13. Reduce ^^^. 14. Reduce VVr- 15. Reduce ^\%%^. 10. Reduce ^J^. CASE IV. 150. To reduce a fraction to its lowest terms. 1. Reduce -^^^ to its lowest terms. Analysis. — By inspection, it is seen that 1st operation. 5 is a common factor of the numerator and ^)tW ^^ sT* denominator. Dividing by it we have ^j. We then sec that 7 is a common factor of */35 — s- 14 and 35 : dividing by it, we have f. Now, 2 and 5 are prime to each other; therefore, the fraction f is in its lowest terim. 2d. The greatest common divisor of 70 and 175, is 35 (Art. 119); if we divide both terras of tfie fraction by it, we obtain |. The value of the fraction 2d operation. is not changed in either operation, since the 35) 7 o _. 2^ nnraGrator and denominator are both di- vided by tlic same number (Art. 144) : hence, 128 REDUCTION OF Rule. I. Divide the numerator and denominator ^ successively, by all their common factors: Or, II. Divide the numerator and denominator by their greatest common divisor. Examples. Keduce the following fractions to their lowest terms : 12. Keduce ^%*q. U. Reduce m. 1. Reduce -^q. 2. Reduce ^^y. 3. Reduce JJf i 5 Reduce iff|. 14. Reduce t'AV 15. Reduce 8343 9T4y 16. Reduce T^- n. Reduce an- 18. Reduce T%V 19. Reduce mil 20. Reduce 5%V5- 21. Reduce sffjo- 22. Reduce .^"o- Reduce f^. 6. Reduce f|J. 7. Reduce t^tV.. 8. Reduce f f by 2d method. 9. Reduce fjf *' 10. Reduce ^2^ " 11. Reduce yVA " CASEY. 151. To reduce a compound fraction to a simple fraction. 1. What is the equivalent fraction of f of f ? Analysis. — Three-fifths of 4 is three times operation. I of ^ : 1 fifth of I is 3^ (Art. 142) ; and 3 3x4 12 times a^j is ^f (Art. 139) ; hence, § of | = if ; hence. 6 X T - 35 Rnle. I. If there are mixed numbers, reduce them to improper fractions : II. When there are common factors in the numerators and de7iominators, cancel them: 150. How do you reduce a fraction to its lowest terms? 151. How do you reduce a compound fraction to a simple one? COMMON FRACTIONS. 129 III. Multiply the numerators together for a new mtmer- ator, and the denominators together for a new denominator. Examples. 1. Reduce f of J of f to a simple fraction. 2. Reduce f of J of J to a simple fraction. 3 Roduce J of | of 2J to a simple fraction. 4. Change | of | of f of 3 J to a simple fraction. 5. Change ^q of f of J of j\ to a simple fraction. G. What is the value of | of J of J of 12| ? 7. What is the value of f of | of 4J ? 8. What is the value of j% of ^ of 5y\ ? 9. Reduce |- of 9J of 6y of 2 J to a whole or mixed number. 10. Reduce ^j of -Jj of 21| to a whole or mixed number. 11. Reduce J of f of | of j^j^ of /j to a simple fraction. 12. Reduce ^^ of j'g of ^Yt of y to a simple fraction. 13. Reduce 3|- of ^ of ^^j of 49 to a simple fraction. CASE VI. 152. To reduce fractions of different denominators to eqaiv alent fractions that shall have a common denominator. 1. Reduce f, |, and f to a common denominator. Analysis.— Multiplying both terms operation. of the first fraction by 20, the pro- 2x5x4 = 40 1st num. duct of the other denominators, 4 X 3 X 4 = 48 2d num. gives f °. Multiplying both terms 3 X 5 X 3 = 45 3d num. of tlie second fraction by 12, the 3 X 5 X 4 = 60 deuom. product of the other denominators, gives jf. Multiplying both terms of the third by 15, the product of the other denominators, gives |^. In each case, both terms of the fraction have been multiplied by the same number; therefore, the value is not changed (Art. 143) : hence, Rule. — I. Reduce mixed niimhers to improper fractions, and compound to simple fractions, when necessary 130 KEDUCTION OF TT. Multiply the 7nimerator of each fraction by all the de- nominators except its own, for the new numerators, and all the denominators together for a common denominator. Note — Wlien tlie numbers are small, the work may be performed mentally ; thus, 112 Vromp 2 L5 24 . n,.pi 2 1 2 become i^ 2.0 so Examples. Reduce the followhipr fractions to common denominators : 1. Reduce f, 5|, and f. 2. Reduce f , f, \, and \ of 5. 3. Reduce ^, 4i, 2f, and |. 4. Reduce f, |-, |, \, and 2 J. 5. Reduce ^ of 3, f , f , and f . 6. Reduce 2i of 3i and 4|. 7. Reduce f of J, and 6f. 8. Reduce 4f, 2J, 5^, and 6. 9. Reduce 5}, f , 3^, and 3|. 10. Reduce J of 5|, and 4f 11. Reduce 4i of 31 and ^. 12. Reduce 61 of 2, f , 5f , and J. Note. — We may often shorten the work of multiplying the nimier- ator and denominator of each fraction by such a number as will make the denominators the same in all. Reduce the following fractions to common denominators. 1. Reduce f, /j; h ^'^^ f to a common denominator. 2. Reduce f, ^j, and -J to a common denominator. 3. Reduce 4 J, ^q, and 7 J to a common denominator. 4. Reduce lOf, |, and 7| to a common denominator. 5. Reduce 6 J, J, and 7| to a common denominator. 6. Reduce f, |-, 141, and 3 J to a common denominator. 7. Reduce j'^, f, 2f, and If to a common denominator. 8. Reduce f, i if, and J to a common denominator. ■9. Reduce -fj, f, ^, and ^ to a common denominator. 10. Reduce 21, 51, j%, and 4y^2 to a common denominator. 152. How do you reduce fractions to a common denominator? COMMON FKAGTIONS. 131 CASE VII. 153. To reduce fractions to their least common denominator. The least common denominator is the least common multiple of the denominators. I. Reduce f, f, and J, to their least common denominator. Analysis. — If there are mixed numbers or compound fractions, thej must be reduced. We then find the least common multiple of the denominators 4, 6, and 9, which is 36. This number is divided by each denominator, to ascertain by what the terms of the fraction must be multiplied to reduce it to 86ths. OPERATION. 2)4 3 )2 . . 3 . . 9 2 . . 1 . . 3 LELiST COMMON DENOMINATOR. 2x3x2x3 = 36. (36 (36 (36 4) X 3 = 2t 1st numerator. 6) X 5 = 30 2d numerator. 9) X 4 = 16 3d numerator. Therefore, the fractions, reduced to their least common denom- inator, arc U, ^l a^^ if. Rule. I. Find the least common multixile of the denominators: this will he the least common denominator of the fractions : II. Divide the least common denominator by the denomina- tor of each fraction^ separately; multiply the quotient by the numerator and place the product over the least common de- nominator ; the results icill be the neio and equivalent frac- tions. 153. How do you reduce fractions to their least common dcnoniina' tor? 132 REDUCTION OF Examples. 1. Keduce |, j, and y\ to their least common denominator. 2. Reduce y^, f, and if to their least common denominator. 3. Reduce 2f , -f^, and ^ to their least common denominator. 4. Reduce 5|, 43-^2* ^^^ 24 ^^ *^^^^ ^^^^^ common denominator. 6. Reduce 8j^3-, -|, and g'^g- to their least common denominator 6. Reduce 9j 5-, 2^, and /g- to their least common denominator T. Reduce 2J, S^\, and -^-^ to their least common denominator 8. Reduce 3^^, |-, ^, and y% to their least common denominator. 9. Reduce f , ^y, and 5^ to their least common denominator. 10. Reduce 4y^, 12^, and -^^ to their least common denominator. 11. Reduce 6f, Bj^q, and 2^^^ to their least common denominator. 12. Reduce ^^y, 2^, and 1^ to their least common denominator. 13. Reduce 6 J, 6^, g^g, and -^ to their least denominator. DENOMINATE FRACTIONS. 154. A Denominate Fraction is one whose unit is denominate. Thus, f of a yard is a denominate fraction. CASE VIII. 155. To change a denominate fraction from a greater unit to a less. 1. In J of a yard, how many inches ? operation. Analysis. — Since 3 feet make a yard, 1r I yd. = I of ^ feet; and since 12 inches _>/_><_— — make one foot, I yard == | of f of V | ^ ^ ^ inches = ^ = Zll inches. ~ .SU Rule. Multiply the fraction hy the units of the scale, in succession, till you reach the required unit. 154. Wliat is a denominate fraction? — 155. How do you change a denounnate fraction from a greater to a less unit ? COMMON FRACTIONS. 133 CASE IX. 156. To change a denominate fraction from a less unit to a greater. 1. Reduce I of a pound to a fraction of a ton. Analysis. — Since one pound is ^\ of a quarter, | lb. = ^ of o'- qr.; ?.iul since one quarter is | of a cwt., f lb. =f of ^s of I cwt; and Bince one cwt. is -^^ of a ton, 9 ^^- = 9 ^^ 25 ^^ 4 ^^ ^ = 4500 ^'^ 5 Rule. — Divide the fraction, that is, multiply the denomina- tor by the units of the scale, in succession, till the required unit is reached. CASE X. 157. To find the value of a denominate fraction in integers ol lower denominations. 1. What is the value of I lb. opekation. Troy ? 1 12 Analysis. — I lb. = f of V" = V = 9)84 91 oz. : i oz. = f of V = V = n oz 9 3 pwt. : f pwt. = I of 2-4 = 144 ^ 16 2^ gr. : hence, 9JgO Rule. — 3fuUiply the numerator W^- ^ • -^ of the fraction by the units of the scale, and divide the product by the denominator ; if there is a remain- der, treat it in the same way, till the Inquired denomination is reached. Hie quotients of the several operations will form the answer. 24 9 )144 grTlG Ans. 9 oz., 6 pwt., 16 gr 15G How do you cliango a denominate fraction from a less te a greater imit? — 157. How do you find tlie value of a denominate frac- tion in integers of lower denominations. 134 REDUCTION OF Examples. 1. Reduce £^ to the fraction of a farthing, 2. Reduce f ton to the fraction of a pound. 3. Reduce ^ week to the fraction of a minute 4. Reduce /g lb. Troy to the fraction of a grain. 5. Reduce | inch to the fraction of a rod. 6. Reduce f inch to the fraction of a yard. 7. Reduce ij of a second to the fraction of a degree. 8. Reduce ij of a cubic foot to the fraction of a cord. 9. What is the value of £^^ ? of £-^^ ? 10. Find the value of J mile : the value of f mile. 11. What is the value of |- furlong? 12. Reduce f penny to the fraction of a guinea. 13. Reduce ^ farthing to the fraction of 6 guineas. 14. Reduce j^j hour to the fraction of 5 seconds. CASE XI. 158. To reduce a compound denominate number to a fraction of a given denomination. 1. Reduce 3oz. 14pwt. 15 gr. to the fraction of a pound. Analysis. — 3oz. 14pwt. 15gr. = 1791gr. In lib. there are iJTCOgr. ; therefore 3oz. 14pwt. 15 gr. is i^lJlb. Rule. — Reduce the compound number and the unit of the given denomination to the lowed unit named in either, and then divide the first result by the second. Examples. 1. Change 7 fur. 28 rd. 2 yd. to the fraction of a mile. 2. Reduce 17s. 6d. 2 far. to the fraction of a £. 3. Reduce lOcwt. oqr. 161b. to the fraction of a ton. 158. How do you reduce a compound number to a fraction of a given donomiualiou? COMMON FRACTIONS. 135 4. Reduce 9oz. 5|pwt. to the fraction of 1 lb. Troy. 6. Reduce 5 da. 161ir. 40 m. to the fraction of a week. 6. Change 3pk. Tqt. Ipt. to the fraction of a bushel. T. Change 3qr. 3na. linch to the fraction of 1 yard. 8. Change 18s. 8d. 3 far. to the fraction of £1 9s. 6d. 9. Change Js. to the fraction of £|. 10. Change ijd. to the fraction of £^. ADDITION. 159. Addition of Fractions is the operation of finding the sum of two or more fractions. 160. The sum of two or more fractions is a number which contains tlie same fractional unit as many times as it is con- tained in all the fractions taken together. CASE I. 161. When the fractions have the same unit. 1. What is the sum of J, f, |, and § ? Analysis. — In this example the unit of the fraction is 1, and the fractional unit ^. There is 1 half in the first, 3 halves in the second, 6 in the third, and 3 in the fourth ; hence, there are 13 halves in all, equal to 6|. 2. What is the sum of £^ and je§ ? Analysis. — The unit of loth frac- operation. tions is £1. In tlie first, the fractional £\ = £\ unit is £^, and in the second, £.}. j£| = £^ These fractional units, being different, jgs ^ jgj. _ . ^i_ _ £\\ cannot be expressed in one collection. But £i = £,\ and £\ = £|, in each of which expressions tlie fractional unit is £\ : hence, their sum is £J = £1|. 150. What is Addition of Fractions?— 1 GO. What is the sum of two or more fractions? — 161. How do you add fractions which have the Btuuo unit? opekation. 1+3+6+3 = 13 hence, J^ = 6^ sum. 136 ADDITION OP Rule. 1. W/ie?i the fractions have the same denominator^ add their numerators^ and place the sum over the common de- nominator : II. When they have not the same denominator^ reduce compound fractions to simple ones, and then reduce cdl to a common denominator, and add as before. Note. — 1. Reduce each fraction to its lowest terms before adding. 2. After the addition is performed, reduce every result to its simplest form ; that is, improper fractions to mixed numbers, and the fractional parts to their lowest terms. 162. When each of two fractions has 1 for a numerator. 1. What is the sum of \ and -J-? Analysis. — Reducing to a common operation. denominator, we find the fractions to 11 ^ \^ ■'^ bo -^^ nnd ^^, and their sum to be ^f. 5 7 35 35 35' That is, tlie sum of two fractions whose numerators are each 1, is equal to the sum of their denominators divided ly their product. 2. What is the sum of i and J? of ^ and J ? of | and J? of J and^V? 3. What is the sum of ^ and -^q ? of Jj and j^g ? of J and J ? of i and \ ? 163. When there are mixed numbers. 1. What is the sum of 12f, llf, and 15f-? OPERATION. WhoU IN'um'bera. Fractions. 12 + 11 + 15=38 f +f +f-TV5+A'5 +T¥5=fSf =li8l : then, 38 + lfgf=39i§f. Ans. 162. What is the sum of two fractions when each has a numerator 1 ?— lO-J. How do you add mixed numbers ? COMMON FKACTIONS. 137 When there are mixed numbers, add the ivhole numbers and the fractions separately, and then add their sums. Examples. 1. Add J, j\, /«, and ff 2. Add 1, /^, if, tt, and M- 3. Add f, f, j% ^S, and if. 4. Add J^, f, i and |. 5. Add 1 41 and f. 6. Add t\, a, ¥, and §. 7 Add -9- -*- ^ and ^ 8 Add If, 3i and i of 7. 9. AddSff, 7f, Hand 2] J. 10. Add 2|, 4J, and J of 5jV 11. Add 12J, 9|, f of 61. 12. Add /_ of 6|and| of IJ. 13. Add \ of 9} and f of 4f . 14. Addf,i%of ^\of8,and2i. 15. Add 4f, Vt of i of l^i- 16. Add 3f, 4f, and J of 16. 17. Bought a cord of wood for 2|- dollars ; a barrel of flour for $9| ; and some pork for 15 J : what was the entire cost ? 18. A person travelled in one day 35| miles ; the next, 28^ miles ; and the next 25^y miles : how many miles did he travel in the three days ? 19. A grocer bought 4 firkins of butter, weighing respective- ly 64f, 65f, 51/g, and 50|^ pounds : what was their entu*e weight ? 20. I paid for groceries at one time ^-^ of a dollar ; at an- other, 3 J dollars ; at another, 7f dollars ; and at another, 5^ dollars : what was the whole amount paid ? 21. A merchant had three pieces of Irish linen ; the first piece contained 22| yards ; the second 20J yards ; and the third 21 J yards : how many yards in the three pieces ? 22. A man sold 5 loads of hay ; the first weighed 18j^^ cwt. ; the second lOJi cwt. ; the third 19§- cwt. ; the fourth 21-J J cwt. ; and the fifth 20 if cwt. ; what was the weight of the whole ? 23. A farmer has three fields ; the first contains 17f acres ; 138 AUDITION OF the second 25-| acres ; and the third 463^5 ^cres : how many in the three fields ? 24. A man sold 112f bushels of wheat for 250 J dollars ; 9j^2 bushels of corn for 62|- dollars ; 225-/j bushels of oats for 104 J dollars : how many bushels of grain did he sell, and how much did he receive for the whole? CASE II. 164. When the fractions have different units. 1. What is the sum of f lb. and f oz. ? Analysis. — In operations. fib. there are ^oz- f 1^- = t X 16 oz. = ^/ oz. (Art. 155.) Then, 6_4 ^z. + f oz. = 2^V^ oz. + if oz. the units of the = 2_7_i ^z. z= ISfJ oz. fractions being the same, viz., 1 oz., we reduce to a common denominator and add, and obtain 18^| oz. Second Method. —Three-fourths of f oz. = f X tV 1^. = g^lb. an ounce is equal | lb. + /^-Ib. ^ffflb. + 3V0 lb. = JJJlb. to ^\ lb. (Art. 15G.) Then, by adding, we 2 ti i^ = 13-U oz. =: 13 oz. 84 dr. find the sum to be 320 ''^ — -^"20 Third Method. | lb. = f X 16 oz. = ^3* oz. = 12 oz. 12f dr. —Find the value 3 q2. = s ^ 16 dr. = \« dr.=r 12 of each fractional Sum . . . . 13 8*^ part in terms of integers of the lower denominations (Art. 157), and then add. Rule. I. Reduce the given fractions to the same unit, and then add as in Case I. Or, II. Reduce the fractions separately to integers of lower de- nomijiations, and then add the denominate numbers. COMMON FRACTIONS. 139 Examples. 1. Add I of yard to § of an inch. 2. Add together J of a week, J of a day, and \ of an hour. 3. Add fewt., V ^^-j 1^ oz., fcwt., and lib. together. 4. Add J of a pound troy to J of an ounce. 6. Add ^ of a ton to j^ of a hundredweight. 6. Add f of a chaldron to f of a bushel. I. What is the sum of f of a tun, and f of a hogshead of wine? 8. Add I of J of a common year, f of J of a day, and J of § of f of 19^ hours, together. 9. Add I" of an acre, f of 19 square feet, and f of a square inch, together. 10. What is the sum of -f of a yard, | of a foot, aid ^ of an inch? II. What is the sum of |- of a £, and f of a shilling? 12. Add together J of a mile, f of a yard, and f of a foot. 13. What is the sum of | of a leap year, ^ of a week, and J of a day ? 14. Add I lb. troy, A oz. and | pwt. 15. Add together y'^ of a circle, 3|- signs, J of a degree, and f of 5-}- minutes. 16. What is the sum of |-yd., f of f qr. and 3Jna. ? 17. Add y\ of a cord, f cubic feet, and | of i of 24f cubic eet. 18. What is the sum of f of i of 4 cords, | of ^ of 15 cord feet, and | of 31 J cubic feet? 19. Add I of 3 ells English to y\ of a yard. 164. How do you add fractions wheii they have difTerent units? 140 SUBTRACTION OF 20. Add together f of 3A. IR. 20 P., f of an acre, and f of 3 K 15 P. 21. What is the sum of ^^ of a ton, -j^^ of a cwt., and ^- of an ounce? 22. What is the sum of ^ of f of a mile, f of a furlong, ^j of a rod, and J of a foot ? SUBTRACTION. 165. Subtraction of Fractions is the operation of finding the difference between two fractional numbers. 166. The difference between two fractions is such a num- ber as added to the less will give the greater. CASE I. 167. When the unit of the fractions is the same. 1. What is the difference between J and ^ ? Analysis. — The unit of both fractions is the operation. same, being the abstract unit 1. The fractional 3 12 unit is also the same, being ^ in each; hence, 4 4 4 the difference of the fractions is equal to the difference of the fractional units, which is f. 2. What is the difference between |- lb. and f of a pound ? Anai^ysis.— The unit in both orERATiON. fractions is lib. The fractional 4 2 12 10 2 unit of the first is ]lb., and of the 5 ~ 3 ~ 15 ~ 15 ~ 15 second ^Ib. Reducing to the same fractional unit, we have }f lb. and xI^^m *^® difference of which is 7^5 lb. ; hence, 165. What is subtraction of fractions ?— 166. What is the difference between two fractions ?— 167. How do you subtract when the unit of the fractions is the same? COMMON FRACTIONS. 141 Rule. I. If the fractional unit is the same in both, subtract the less numerator from the greater, and i^lace the difference over the common denominator, H. When the fractional units are different, reduce to a common denominator ; then subtract the less numerator from the greater, and place the difference over the common denom- inator. Examples. 1. From f take \, 2. From \^ take -jj. 3. From Jf take ^|. 4. From joj take ij|. 5. From f take f. 6. From \^ take |f. 7. From j| take |J. 8. From 37 }i take J of 5J-. 9. From f take |. 10. From J take jV 11. From 25 take jj. 12. From /^ of 3 take J of |. 13. From i of f of 7 take f. 14. From 3f take f of f 15. From f of 15 take | of 3. 16. From 7 J of 2 take J off. 17. To what fraction must I add | that the sum may be f ? 18. What number added to 1|^, will make 5 ? 19. What number is that to which if 7 J be added the sum will be 17f ? 20. From the sum of 3|- and 10|- take the difference of 25} and 17iJ. 21. What number is that from which if you subtract J of I of a unit, and to the remainder add f of J of a unit, the sum will be 9 ? 22. If I buy f of I of a vessel, and sell J of .| of my share, how much of the whole vessel will I have left? 23. A man bought a horse for ^ of | of ^^ of $500, and sold him again for f of J of f of $1680 : what did he gain by the bargain ? 143 SUBTRACTION OF 24. Bought wheat at 1|- dollars a bushel, and sold it for 21 dollars a bushel : what did I gain on a bushel ? 25. From a barrel of cider containing 31^ gallons, 12f gal- Ions were drawn : how much was there left ? 26. Bought lOJ cords of wood at one time, and 24| cords t another; after using 16 J cords, how much remained? 2Y. A merchant bought two firkins of butter, one contain ing 54j% pounds, and the other 56}J pounds ; he sold 43}! pounds at one time, and 34 J pounds at another : how much had he left? 28. A man having $50 J, expended $15y^^ for dry-goods, and 12|- for groceries : how much had he left? 29. A boy having f of a dollar, gave J of a dollar for aa inkstand, and J of it for a slate : how much had he left ? 30. Bought two pieces of cloth, one containing 27f yards, the other 32 1- yards, from which I sold 40^ yards : how much had I left? 168. When each fraction has the numerator 1. 1. What is the difference between J and l? Analysis. — Reducing both frac- operation. tions to a common denominator ^ — l- = -A. — -^ = -^. and subtracting, we find the difier- ence to be -^^ ; that is, The difference between two fractions, each of whose numer- ators is 1, is equal to the difference of the denominators divided by their product. 2. From J take yV- 3. From Jj take jL. 4. From jL take ^, 5. From Jy take ^. 168. What is the difference when the numerator of each fraction is 1? COMMON FRACTIONS. 143 169. WTien there are mixed numbers. 1. What is the difference between 16J and 3J ? Analysis. — Since we cannot take y% from operation. •^\, we borrow 1 = y| from the whole num- 16J = IGjV ber of the minuend, which, added to fV, gives 3J = 3j\. |-f : then j^j from -}-| leaves j§. We must now T24^. carry 1 to the next figure of the subtrahend, and say 4 from 16 leaves 12. Hence, to subtract one mixed num- ber from another, Subtract the fractional part from the fractional part, and the integral part from the integral part. 2. What is the difference between 144- and 12^%? 3. What is the difference between 115f and 39 J? 4. What is the difference between ^^y& ^^^ ^'h'^- 5. What is the difference between 48 jV and 41J-|? 6. What is the difference between 287^ and 104/^^^ ? CASE II. 170. When fractions have different imits. 1. What is the difference between J of £ and J of a shil- Hng ? ,. Analysis. — Reducing to the common unit Is., we find the difference to be V^- = ^s- V^- — ¥- = %°s. — §s. = ^S- 8d. Second Method. — Reducing to the common unit £1, we find the difference to be £|^ = _ ^2_9 _ gg^ gj^ 9s. 8d. TniRD Method. — Reduce the £1 = lOs. fractions to integral units, and Js. — 4d. then subtract as in denominate "^sTScL numbers. OPEEiHON. ^ = i X 20s. : Vs- - Js. = t»s. . -|s.= = 9|s. = 9s. 8d. 4s.= Jx£,V = £A- £h- JE^ = £f ? -^- 144 SUBTRACTION OF Rule. I. Reduce the fractions to the same unit, and then subtract as in Case I,: Or, II. Flyid the value of each fraction in units of lower de- nominations, and then subtract as in denominate numbers. Examples. 1. From f of a pound troj, take f of an ounce. 2. From f of a ton, take f of J of a pound. 3. From -| of f of a hogshead of wine, take f of J of a quart. 4. From f of a league, take f of a mile. 5. What is the difference between Ifs. and f of tjd? 6. What is the difference between §i of a degree and f of j- of a degree. 7. From i| of a square mile, take 36J acres. 8. From f of a ton, take f of 12 cwt. 9. From If lb. troy, take \ of an ounce. 10. From 2f cords, take f of a cord foot. 11. From 1 of a yard, take f of an inch. 12. From i of J of a pound, take f of J of a dram, apoth- ecaries' weight. 13. From a piece of ground containing 2^^% acres take lA. IP. and 9 square yards. 14. A pound avou'dupois is equal to 14 oz. llpwt. IGgr. troy : what is the difference, in troy weight, between the ouncG avoirdupois and the ounce troy? 1G9. How do you subtract when there are mixed numbers? 170. What is the rule wlien tlie fractions have different units. COMMON FK ACTIONS. 145 MULTIPLICATION. 171. Multiplication of Fractions is the operation of taking one number as many times as there are units in another, when one or both are fractional. I. If 1 pound of tea cost f of a dollar, what will ^*of a pound cost. Analysis. — The cost will he equal operation. to the price of 1 lb. taken as many $|. X y = g ^ | = $|-|* times as there are units in the multi- plier (Art 84). One-seventh of a pound of tea will cost one-seventh as much aa lib. Since lib. cost $f, | of lib. will cost } of $f = $5^ (^rt. 142). But 3 sevenths of lib. will cost three times as much as I ; that is, $/g^ x 3 = ^| (Art. 139). Hence, to multiply one frac- tion by another: Rule. Cancel all factors common to the numerator and denom- inator ; then multiply the numerators together for a new numerator, and the denominators together for a new denom- inator, 172. Principles of the operation. 1. When the multiplier is less than 1, we do not take the wkole of the multiplicand, but only such a part of it as the multiplier is of 1. 2. When tbe multiplier is a proper fraction, multiplication does not increase tbe multiplicand, as in the multiplication of whole numbers. The product is the same part of the multiplicand as the multiplier is of 1. 8. ^Vhen either of the factors is a whole number, write 1 under it for a denominator. 4. \Vhen either of the factors is a mixed number, reduce it to an improper fraction. 171. Wliat is multiplication of fractions? What is the rule?— 173 What is the first principle of the operation? What is the second? 7 146 MULTIPLICATION OF 1. Multiply f by 8. 2. Multiply j\ by 12. 3. Multiply Jf by 9. 4. Multiply If by 15. 9. Multiply 6T by 9^2- 10. Multiply 842 by TJ. 11. Multiply 360 by 12f. 15. Multiply f by 8. 16. Multiply 15 by f. 11. Multiply 1J by 8. 18. Multiply 91 by 18|. 19. Multiply 3|- by 4if. Examples. 5. Multiply I of A by 35. 6. Multiply If of 2 i by 16. 1. Multiply 21 of f by TO. 8. Multiply 4| of 8 hj 36. 12. Multiply 4G0 by llf. 13. Multiply 620 by lOf. 14. Multiply 1340 by 8|. 24. Multiply ^ of f by f of j%. 25. Multiply f by 16. 26. Multiply 28 by y\. 21. Multiply 8/o by 15. 28. Multiply ^t of f by if. V¥ by 9. 20. Multiply 21. Multiply I by |. 22. Multiply J of f by |. 23. Multiply ^^ by 2^^ of 33. Multiply j\, If, and -,-, 2 7* 46 29. Multiply 5J by | of 3. 30. Multiply 8421 by IJ. 31. Multiply f by f. 32. Multiply j% by 1j\. together. 34. Multiply ^f ^^, ^, aud ff together. 35. What is the product of }f by f of 17. 36. What is the product of 6 by | of 5. 3t. What is the product of J of J of 3 by 15| ? 38. What is the product of f of f by f of 3f ? 39. What is the product of 5, f , | of f , and 41 ? 40. What will 1 yards of cloth cost, at $f a yard ? 41. What will 12| bushels of apples cost, at If- a bushel ? 42. If one bushel of wheat costs |1|, what will | of a bushel cost ? COMMON FRACTIONS. 14? 43. If one horse cats J- of a ton of hay in one month, how much will 18 horses eat iu the same time? 44. If a man earns ^f| in one day, how much can he earn in 24 days ? 45. What will 3J yards of cloth cost, at J of a dollar a yard ? 46. At $16 a ton, what will \^ of a ton of hay cost? 47. If one pound of tea costs $1J, what will 6J pounds cost? 48. What will 3f boxes of raisins cost, at $2J a box ? 49. At 75 cents a bushel, what will fi of a bushel of corn cost ? 50. If a lot of land is worth $75j®j, what will j\ of it be worth ? 51. What will 17 J yards of cambric cost, at 2 J shillings a yard? 52. Bought 15f barrels of sugar, at $20 J a barrel : what did the whole cost ? 63. If one bushel of corn is worth f of a dollar, what is | of a bushel worth ? 54. If I own ^ of a farm, and sell ^ of my share, what part of the whole farm do I sell ? 55. I bought a book for ^q of a dollar, and a knife for ,*^ the cost of the book : how much did I pay for the knife ? 56. At } of fj of a dollar a pound, what will J^ of |f of a pound of tea cost? 57. If hay is worth $9|- a ton, what is J of 3 J tons worth? 58. If a man can dig a cellar in 22 J days, how many day ould it take hun to dig f of it ? 59. If a railroad train runs 1 mile in ^ of an hour, how tuiig will it be in running 106 J miles ? 148 DIVISION OF 60. A owned I of a farm and sold f of his share to B, who sold f of what he bought to C, who sold f of what he bought to D : what part of the whole did D have ? 61. A owned f of 200 acres of land, and sold f of his share to B, who sold J of what he bought to C : how many acres had each ? DIVISION. 173. Division of Fractions is the operation of finding how many times one number is contained in another, when one or both are fractional. 1. What is the quotient of J divided by y ? Analysis. — How many times is ^/ operation. contained in | ? If I be divided by 1 ^ y = J ^ ^-, 14, the quotient will be .g^ = J_. = _5_ j^^g^ Since the true divisor is but } of 14, the divisor used is 5 times too large; hence, the partial quotient y'g, is 5 times too small. Multiplying this by 5, we have the true quotient, = ■^\. This result is produced by inverting the terms of the divisor and multiplying. Rule. — Invert the terms of the divisor, cancel, and proceed as in multiplication. 174. Directions for the operation. 1. If either the dividend or divisor is a whole number, make it fractional, by writing 1 under it for a denominator. 2. Cancel all common factors. 3. If the dividend and divisor have a common denominator, they will cancel, and the quotient of the numerators will be the answer. 173. What is division of fractions? What is the rule?— 174. What is the first direction for performing the operation? What the second? What the third? What the fourth? W^hat the fifth? COMMON FRACTIONS. U9 4. When either term of the fraction is a mixed number, reduce to the form of a simple fraction. 5. If the numerator of the dividend is divisible by the numerator of the divisor, and the denominator by the denominator, divide with- out inverting. 1. Exai] Divide f} by 1. aple 26. 3. Divide jf by 4. 2. Divide ^ by 6. 27. Divide Jf by 5. 3. Divide if by 9. 28. Divide fa by 8. 4. Divide Jfg by 40. 29. Divide i^} by 48 5. Divide ff by 13. 30. Divide j\^^ by 21 6. Divide 5 by ^^. 31. Divide 36 by j%. 1. Divide 27 by -J. 32. Divide 420 by |. 8. Divide i by i. 33. Divide /^ by f. 9. Divide j% by f . 34. Divide iJ by j\. 10. Divide U ^y tV 35. Divide f of f J by |f. 11. Divide f of f by f of f 36. Divide I by if. 12. Divide l of | by | of f . 37. Divide f of f by f of f . 13. Divide 1 of f by f of |. 38. Divide i of! off by! of |. 14. Divide 56 by ji 39. Divide 650 by i^f 15. Divide 1000 by -^j. 40. Divide 1273 by i|. 16. Divide 725 by Jf 41. Divide 4324 by Jff. n. Divide 4|- by 5. 42. Divide 6f by 8. 18. Divide 9xV by 12. 43. Divide 12f by 42. 19. Divide J of 16J by 41. 44. Divide 3i by 9|. 20. Divide 9J by J of 7. 45. Divide 100 by 4f. 21. Divide f of 50 by 4^. 46. Divide 443V by |i§. 22. Divide 300/^ by 6i 47. Divide 111 J by 33^. 23. Divide 4 of 3f by ij of 7^. 48. Divide 191} by 159}. 24. Divide 9| by SJ. 49. Divide 5f by f of 1^. 25. Divide 1 of -j^ by 6^. 50 Divide 5205} by f of 90. 160 DIVISION OF 51. At i of a dollar a pound, how much butter can be bought for I J of a dollar ? 52. At f of a dollar a yard, how much cloth can be bought for I" of a dollar ? 53. If a bushel of potatoes cost | of a dollar, how many bishels can be bought for ^^ of a dollar? 64. If I of a ton of hay will feed 1 horse one week, how many horses will -j^ of a ton feed, the same time ? 55. If f of a bushel of apples cost f of a dollar, what will 1 bushel cost ? 56. What will a barrel of flour cost, if ^g- of a barrel cost I of a dollar ? 57. If f of a bushel of apples cost f of a dollar, what will 1 bushel cost ? 58. How much molasses at f of a dollar a gallon, can be bought for 1^ dollars ? 59. A man sold fj of a mill, which was J of his share : what part of the mill did he own ? 60. What number multiplied by f, will give a product of 15f? 61. What number multiplied by 5 J, will give a product of 146? 62. The dividend is 620 J, and the quotient 36/o : what is the divisor ? 63. What number is that, which if multiplied by f of f of I5|-, will produce f ? 64. If 71b. of sugar cost |-f of a dollar, what will 1 pound cost? 65. If 10 J lb. of nails cost f of a dollar, what is the price per pound ? COMMON FllAGTIONS. 151 C6. If f of a yard of cloth cost $3, what will 1 yard cost ? 67. A family consumes 165f pounds of butter in 8J weeks : how much do they consume in 1 week ? 68. At $9| a barrel, how much flour can be bought for nssf ? 69. If a man divides $3|- equally among 8 beggars, how much docs he give them apiece ? 70. If 8 pounds of tea cost $7f, what is the price per pound ? 71. If J of a ton of hay sells for $10 J, what is the price of 1 ton ? 72. If J of an acre of ground produces 84/^ bushels of potatoes, how many bushels will 1 acre produce? 73. What quantity of cloth may be purchased for $5j'g, at the rate of $6f a yard ? 74. How long would a person be m traveling 125|- miles, if he traveled 31y^ miles per day? 75. How many bottles, each holding If gallons, can be filled from a barrel of wine, containing 31 J gallons? 76. How long will it take 11 men to do a piece of work, that 1 man can do in 15| days? 77. If f of a barrel of flour costs 6 dollars, what is the price per barrel ? 78. Eighty-one is f of how many times 8 ? 79. Five-eighths of 48 is f of how many times 9 ? 80. How many times can a vessel, containing | of a gallon, be filled from J of a barrel of 31^ gallons? 81. If 5^ lb. of tea cost $4f, what is the price of 1 pound? 82. If f of f of a ship is worth $2540, what is the whole vessel worth? 83. If f of f of a barrel of flour willlast a family 1 week, how long will 9j\ barrels last them? 152 COMPLEX FRACTIONS. COMPLEX FRACTIONS. 175. A Complex Fraction is only another form of expression 7. for the division of fractions : thus, 5., is the same as ^ divided ^y f ; and may be written, 5. — 4 2 6 — 45' 176. To reduce a complex fraction to a simple fraction. 6i 1. Reduce _3 to a simple fraction. Analysis. — Eeducing th e divisor and dividend each to OPERATION. 6} = 2_o^ and 11 = f. a simple fraction, we have ^ 2_o _u 8 _ 2_o v 1 — 3_5 — and f . Then %' divided byf ^ • i — 3 ^ s — e — is equal to %° x ^ = V = H- Rule. — Beduce both terms of the fraction to simple frajo- lions: then divide as in division of fractions. Examples. Reduce the following to simple fractions : t. Reduce HA. 87 1. Reduce «. i 2. Reduce X. if 3. Reduce It. 4. Reduce ^^^. 6. Reduce X. 6. Reduce ?i. 12 8. Reduce 20 4 ' T 9. Reduce f of 1^\ xVofnf 0. 1. Reduce Reduce 26A 1 of 17' 55A 12. Reduce f of /^ o^^- 8 10 jg 175. Wliat is a complex fraction? — 176. How do you reduce a complex to a yhiiple fraction? APPLICATIONS. 153 Applications in Fractions. 1. What will 5^- cords of wood cost, at i of f of | of 150 a cord? 2. A farmer sold | of a ton of hay for $Gf : what would be the price of a ton at the same rate ? 3. A person walks 7tf miles in 10 J hours : at what rate is that per hour? 4. From the product of f and 11 J, take ^, and multiply the remainder by 20 J. 5. How much greater is f of the sum of J, J, 1, and J, than the sum of J, J, and J ? 6. If I of a ton of hay is worth $Ti what is 2| tona worth ? T. If f of a dollar will pay for | of a yard of cloth, how many yards can be bought for $llf ? 8. What is the value of 3 J cords of wood, at $4§ a cord ? 9. At J of a dollar a peck, how many bushels of apples can be bought for $6}? 10. AVhat is the difference between J of a league and ^^^ of a mile ? 11. What is the sum of 4^^ miles, |- of a furlong, and | of 1 J yards ? 12. At $1J per day, how many days^ labor can be obtained for $3Gf ? 13. Bought 5 J yards of cloth at $4 J a yard, and paid for it in wheat at ^l-f a bushel : how many bushels were required ? 14. What number must be taken from 2*r|, and the re- mainder multiplied by 14f , that the product shall be 100 ? 15. Three persons. A, B, and C, purchase a piece of prop- erty for $6300 ; A pays f of it, B J, and C the remainder : what is the value of each one's share ? 154 COMPLEX P^KACTIONS. 16. What number diminished by the difference between f and f of itself, leaves a remainder equal to 34 ? IT. What is the sum of f of ^£15, £^, J of f of f of £1, and f of f of a shilling? 18. If ^ of John's marbles is equal to J of James\ and together they have 66, how many has each? 19. A person owning f of 2000 acres of land, sold | o. his share : how many acres did he retain ? 20. A boy having 240 marbles, divided them in the follow- ing manner : he gave to A, J, to B, j^, to C, J, and to D, J, keeping the remainder himself : what number of marbles had each? 21. A man having engaged in trade with |3t40, found, at the end of 3 years, that he had gained $156 J more than J of his capital : what was his average annual gain ? 22. Two boys having bought a sled, one paying | of a dollar, and the other J of a dollar, sold it for ^^ of a dollar more th'in they gave for it : what did they sell it for, and what was each one's share of the gain ? 23. A farmer having 126f bushels of wheat, sold f of it at |2i a bushel, and the remainder at Ufa bushel : how much did he receive for his wheat ? 24. A man having $19^, expended it for wheat and corn, of Q-dch nn equal quantity ; for the wlieat he paid 81| a l)u.^]iel, nn ! foi" tlic corn || a bushel : how mucli of each did he buy? 25. Two jiersoiis engage in trade: A furnislied /.-j -'' '■ .'.'ipltal, and 13, -j\ : if B had furnished *492| ui • shiires would have been equal : how much did each furn; -^i . 26. A man being asked how many sheep he had, said, he had them in three fields : m the first he had 63, which was |- of what he had in the second ; and | of what he had in the -ecoiid was 4 times what he had in the third : how many had he in all ? DUODKCIMALS. 155 DUODECIMALS. 177. DuoDEcmALS are a system of numbers, which arise from dividing a unit according to the scale of 12. The units divided arc, the foot in length, the square foot, and the cubic foot. If the unit 1 foot be divided into 12 equal parts, each part is called an inch or prime, and marked '. If a prime be divided into 12 equal parts, each part is called a second, and marked". If a second be divided, in like manner, into 12 equal parts, each part is called a third, and marked '" ; and so on for divisions still smaller : hence, y^^ of a foot —- 1 inch, or prime, 1'. tV 0^ T2 ^^ ^ ^^^^ = T4T ^^ ^ ^o^*> ^^ 1 second, . 1''. tV ^^ tV ^^ tV ^^ ^ ^0^^ = TT2T 0^ ^ ^oo*» ^^ ^ third, . 1'" If the square foot, and the cubic foot, be divided according to the same scale, the primes, seconds, thirds, &c., will have the same relation to the unit and to each other, as in the foot of length. Table. 12'" make 1" second. 12" " 1' inch or prime. 12' " 1 foot. Hence : Duodecimals are denominate fractions, in which the primary unit is 1 foot, and the scale uniform, the units of the scale, at every point, being 12. Notes. — 1. The marks', ", '", &c., which denote the fractional units, are called indices. 2. Duodecimals are chiolly used in measuring LengtJis, Surfaces, Volumes, or Solids. 177. Wliat are duodecimals? What are tlie units divided? If the unit 1 foot be divided into 12 oi[ual partes what is each part called? 156 DUODECIMALS. ADDITION AND SUBTRACTION. 178. The operations of Keduction, Addition, Subtraction, Multiplication, and Division of Duodecimals, correspond so nearly with those of denominate numbers, that additional rules are deemed unnecessary. Examples. 1. In 86' how many feet? 2. In t50" how many ft.? 3. In 3t000"'howmanyft.? 4. In 6V how many feet? 5. In 4Y0'" how many ft.? 6. In 375" how many ft.? 1. What is the sum of 8 ft. 9' V and 6 ft. Y 3" 4'"? 8. Find the difference between 32 ft. 6' 6" and 29 ft. 1'" 9. Add together 9 ft. 6' 4" 3'", 12 ft. 2' 9" 10'", 26 ft. 0' 5", and 40 ft. 1' 0" 3"\ 10. What is the sum of 125 ft. 0' 6", 45 ft. 11' 0" 2"', and 12 ft. 6'? 11. What is the sum of 84 ft. r, 96 ft. 0' 11", 42 ft. 6' 9" 10''', and 5' 1" 11'"? 12. From 12Ut. 3' 6" 4'" 11"", take 40 ft. 0' 10" V" 5"". 13. What is the difference between 425 ft. 9' 10" and 107 ft. 10' 9" 8'" ? 14. What is the sum and difference of 325 ft. 7' 6" 2'" and 217 ft. 10' 9"? MULTIPLICATION. 179. Multiplication of Duodecimals is the operation of find- ing the superficial contents and the contents of volume, when the linear dimensions are known. If 1 inch be divided into 12 equal parts, what is each part called If the seccaid be divided in like manner, what is eacli part called What are indices? For what are duodecimals used? — 178. How are the fundamental operations performed? — 179. Wliat is multiplication of duodecimals ? How are the areas of figures found? How are the •contents of volume found ? MULTIPLICATION. 157 The superficial contents, or area of figures, are found by multiplying the length and breadth together. The contents of volume or cubical contents, arc found by mul- tiplying together the length, breadth, and height. 180. Principles of the Multiplication. 1. Feet multiplied by feet, give square feet. 2. Feet x Primes = 1 ft. x j\ ft. = j\ sq. ft., or primes. 3. Primes x Primes = yV ft. x i ^t* = tIi sq. ft., or seconds. 4. Primes x Seconds = yV ft. x yii ^** = tt*2^ s^* ^*-> ^^ thirds. 5. Seconds x Seconds = y|j ft. x j\^ ft. = g^TSff ^q. ft., or fourths. From the foregomg, we have the following principles : 17ie index of any product is equal to the sum of the in- dices of the factors. Note. — The denominator of primes is 13, of seconds, 144, of thirds 1728, of fourths, 2073G, &c. 181. To find the square measure, or area of a surface. 1. Find the square measure of a floor that is 9 feet long and G feet wide. Note. — A Square is a figure bounded by four equal sides at right angles to each other. Analysis. — Draw an horizontal line and lay off 9 equal parts, each de- noting a foot. Then draw a second horizontal line perpendicular to it, and lay off 6 equal parts, each denot- ing a foot. Through the points of division of the first lino draw paral- 9 els to tlio second, and through the points of division of the IbO. \Vhi\t are the five principles of multiplication? What is the rulo for the indices? What is the rule for the multiplication of duo- decimals?— 181. What is the rule for finding the square measure of a surface? 158 DUODECIMALS. second line draw parallels to the first: there will thus be formed number of small squares. The number of squares in the first row will be equal to 9, the number of linear units in the first line; and the number of rows v>i]l be equal to six, the number of units in the second line : there- fore, the whole number of squares will be equal to 9 x 6 = 54. Hence, to find the area, or measure, IfliiUiply the length by the breadth, and the product will 6e« the number of square's. Note. — The square which is the unit of surface, is the square de- scribed on the unit of length. If the unit of length is a foot, the unit of surface is 1 square foot — if 1 yard, the unit of surface is 1 square yard, &c. 182. To find the Cubic Measure of a Volume or Solid. 1. What is the cubic measure of a block of marble that is 9 ft. long, 6 ft. wide, and 4 ft. thick ? Note. — A Cube is a figure bounded by six equal squares at right angles to each other, called faces; and the sides of the squares are the edr/cs of the cube. Analysis. — The face on which the block stands, is called its dase, tlie area of which is equal to 9 x 6 = 54 sq. ft. If now you take 54 equal cubes, of 1 foot each, they can be placed side by side on the base, and will form a block of marble 9 ft. long, 6 ft. wide, and 1 foot thick. If you place a second tier, the block will be 2 feet thick ; a third tier will make it 8 feet thick, and so on, for any number of tiers: hence, the contents of the block, that is four feet thick, are 9 X G X 4: — 21Q cu. ft. 18!^. How do you find the cubic measure of a volume or solid? 1st OPEIlxVTION. 8 ft. 9' 5" 3 6' 4 4' 8" 6'" 26 4 3 MULTIPLICATION. 159 Bnle. — Multiply the length, breadth, and thickness together. 183. When the Dimensions are in feet and 12ths of a foot. Multiply 8 ft. 9' 5" by 3 ft. 6', and theu the product by 2 ft. 6'. Analysis.— First multiply 8 ft. 9' 5" by 6'. Since 5" =y^j ft., and 6' = ft, or 30 thirds. Since 12"' = 1", 30'" ^12 = 2" and G'" over, which write down. Then 9' x G' = y"^ x ii = ui sq. ft., or 54", to which add the 2" found ill the last product, making 56". Then, 30 sq.ft. 8' 11" 6'" since 12" = 1', 56" -M2 = 4' and 8" over, which write down. Then 8 fectxG' = 8 ft. x -^.j ft. = *§ sq. ft, to wliidi add !ie 4' from the last product, makuig 52'. Then, since 12' = 1 ijuare foot, 52' H- 12 = 4 sq. ft* and 4', both of which set down. We next multiply, in the same manner, by 3 feet, giving a product of 2G sq. ft 4' 3". The sum of the partial products, 30 sq. ft. 8' 11" 6'", is the first required product Now, nmltiply by 2 ft. 6'. First, G'" X C' = y^\e sq. ft x j% ft := :.%, cu. ft = 36"" cu. ft =3"'. [.^u 11" X 6' = ,VV sq. ft X {^ ft f'U. ft. and three added from o liwt product gives 60'" = 5" and " over, which write down. riirli S' X G' = ^% sq. ft. X -i'V ft -= ,V,- cu. ft = 48'', to which add 76cu. ft. 10' 4" 9'' 5" from the last product, gives 53" = 4' and 5" over, whicli v.'rite down. Then, 30 sq. ft x -/'^ ft. = W - 180', to whicli add 4 from the last product, making 184' :=r 15 cu. ft and 4'. Next, multiply by 2d OrERATION. 30 sq. ft. 8' 11" 6' rf 2 6' 15 4' 5" 9' ff 61 5 11 160 DUODECIMALS. 2 feet, giving the partial product 61 cu. ft. 5' 11"; and the sum 76 cu. ft. 10' 4" 9"' is the entire product, in cubic feet and 12ths of cubic feet. Rule. — I. Place the multiplier under the multiplicand, so that units of the same order shall fall in the same column : II. Multiply the mxdtiplicand by each term of the multiplier in succession, beginning with the lowest unit of each, and make the index of each product equal to the sum of the in- dices of the factors : III. Reduce each product^ as it arises, to the next higher unit; write down the remainder, and carry the quotient to the next product : ly. Find the sum of the several products. Examples. 1. How many cubic feet in a stick of timber 12 feet 6 inches long, 1 foot 5 inches broad, and 2 feet 4 inches thick ? 2. Multiply 9 ft. 6' by 4 ft. V. 3. Multiply 12 ft. 5' by 6 ft. 8'. 4. Multiply 35 ft. 4' 6" by 9 ft. 10'. 5. What is the product of 45 ft. 4' 3" by 12 ft. 2' 9" ? 6. What is the product of 140 ft. 0' 2" 4'" by 20 ft. 10'? t. What is the product of 219 ft. 10' 6" by 8' 4" ? 8. What are the contents of a board 14 ft. 6' 3" long, and 2 ft. 9' wide ? 9. How many square feet in a floor 18 ft. 9' long, and 15 ft. 10' wide? 10. How many square yards in a ceiling tO ft. 9' long, and 12 ft. 3' wide ? 11. How many square feet are there in a ceiling whose ength is 15 feet, and width 42 feet ? 12. How many square yards are there in a lot of ground wliose length is 118 feet, and width 25 feet? 13. How many square feet are there in a board whose length is 18 feet, and breadth 14 inches? MULTIPLICATION. 161 14. What is the cost of painting the side of a house that is 2T feet high and 22 feet wide, at 40 cents per square yard? 15. How many acres are there in a field whose length is 45 rods, and width 3Y rods? 16. What is the area of a piece of ground that is 112 ft. 5 in. long, and 27 ft. 9 in. wide ? 17. How many flagstones, that are 4ft. Gin. by 4ft., will be required to cover a walk which is 6 ft. 9 in. wide and 2G4ft. long? 18. What will be the cost of paving a yard 64 ft. 6' square, at 5 cents a square foot ? 19. What are the cubic contents of a block of marble 6 ft. 9' long, 4ft. 8' wide, and 2ft. 10' thick? 20. There is a room 97 feet 4' around it ; it is 9 feet 6' high: what will it cost to paint the walls, at 18 cents a square yard ? 21. What* is the cubic measure of a pile of wood that is 18 ft. long, 7 ft. high, and 4 ft. wide ? 22. How many cords are there in a pile of wood that is 48 ft. long, 9 ft. high, and 3 ft. 6 in. wide ? 23. A gallon contains 231 cubic inches : how many gallons of air are contained in a room, which is 21 ft. 6 in. long, 15 ft. wide, and 10 ft. high? 24. A common brick is 8 in. long, 4 in. wide, and 2 in. thick : how many bricks are there in a pile, whose height is 12 ft. 4 in., width 8ft., and length 15 ft. 9 in., supposing no waste space ? 25. A ditch surrounds a plot of ground which is 240 ft. long, and 164 ft. wide. The ditch is 3 ft. 6 in. wide, and 6 ft. 9 in. deep. What is the cubic measure of the ditch ? 26. How many cubic feet of wood in a pile 36 ft. 5' long, 6 ft. 8' high, and 3 ft. 6' wide? 27. What will a pile of wood 26ft. 8' long, 6ft. Gin. high, and 3ft. 3' wide cost at $3.50 a cord? 28 How many cu])ic yards of earth were dug from a cellar 162 DUODECIMALS. which measured 38 ft. 10' long, 20 ft. 6' wide, and 9 ft. 4' deep ? 29. x\t 16 cents a yard, what will it cost to plaster a room 22 ft. 8' long, 18 ft. 9' wide, and 11 ft. 6' high? There are to be deducted 8 windows, each 6 ft. 4' high, and 2 ft. 9' wide ; 2 doors, each 7 ft. 6' high, and 3 ft. 2' widej and the base moulding, which is 1 foot wide? DIVISION. 184. Division of Duodecimals is the operation of finding from two duodecimal numbers a third, which multiplied by the first, will give the second? I. The floor of a hall contains 103 sq. ft. 4' 5" 8'" 4'', and is 6ft. 11' 8" wide : what is its length? Analysis. — The operation. anits of the dividend ft. sq. ft. ft. are square feet and 6 11' 8") 103 4' 5" 8'" 4*^(14 9' 11" fractions of a square 9t *7' 4" foot. The units of c q/ i /r Q^fti the divisor are linear ^, „ ,„ feet and fractions of - a linear foot 6' 4" 8'" 4" First, consider how ' 6' 4" 8'" 4'^ often the first two parts of the divisor are contained in the first part of the dividend. The first two parts of the divisor are nearly equal to 7 feet, and this is contained in 103 sq. ft. 14 times and something over. Multiplying the divisor by this term of the quotient and subtract- ing, we find the remainder 5 ft. 9' 1", to which bring down 8'". Next, consider how many times the first two parts of the divisor, ecjuai to 7 feet nearly) are contained in the first two parts of the remainder, reduced to the next lower unit; that is 5 ft. 9' = CO'. Multiplying the divisor by the quotient figure 9', and making the -ubtraction, we have 6' 4" 8", to whicli bring down 4". 184. Wliat is the division of duodecimals? How is it performed? I DUODECIMALS. 163 Consider, again, how often, nearly^ 7 feet is ^contained in 6' 4" :z^ 70". Multiplying the divisor bj the quotient 11", we find a [)roduct equal to tlie last remainder. Hence, The process of division is the same as that in other de- nominate numbers, except in the manner of selecting the quotient figure. 185. Principles of the operation. Notes. — 1 If the integral unit of the dividend and divisor is ilu- same, the unit of tJie quotient will le abstract. 2. If the unit of the dividend is a superficial unit, and the unit of the divisor a linear unit, the unit of the quotient will be liintu S. If the unit of dividend is a unit of volume, and the unit ol ' divisor linear, the unit of the quotient mil be superficial. 4. If the unit of the dividend is a unit of volume, and tlu- i. the divisor superficial, the ludt of the quotient wUl be linear. Examples. * 1 D.vido 29 sq. ft. 0' 4" by 6 ft. 4'. _. Divide 50 sq. ft. 0' 10" 6'" by 9 ft. 6'. :]. Wiittt is the length of a floor whose area is iUoxp ti 1 (>", and breadth 24 ft. 3'? 4. A load of wood, contaiuiug 119cu.ft. 2' G" 8'", is 3 ft. 4' high, and 4 ft. 2' wide: what is its length? 5. In a granite pillar there are 105 cu. ft. 5' V (j'" ; it is 3 ft. 9' wide, and 2 ft. 3' thick : what is its length ? 6. There are 394 sq. ft. 2' 9'' in the floor of a hall that is 10 ft. 7' wide: what is its length? 7. A board 17 ft. 6' long, contains 27 sq. ft. 8' 6'': what is its width? 8. From a cellar 42 ft. 10' long, 12 ft. 6' wide, were thrown 158 cu. yd. 17 cu. ft. 4' of earth: how deep was it? 9. A block of marble contains 86 cu. ft. 2' 7" 9'" 6". It is 4ft. 8' wide and 2ft. 10' thick: what is its length? 185. What are the principles of the operation ? 164 DECIMAL FKACTIONS, DECIMAL FRACTIONS. 186. There are two kiuds of Fractions in general use : Com- mon Fractions and Decimal Fractions. A Common Fraction is one whose unit is divided into any number of equal parts. A Decimal Fraction is one whose unit is divided according to the scale of tens. 187. If the unit 1 be divided into 10 equal parts, each part Is called one-tenth. If the unit 1 be divided into one hundred equal parts, or each tenth into ten equal parts, each part is called one-hun- dredth. If the unit 1 be divided into one thousand equal parts, or each hundredth into ten equal parts, the parts are called thou- sandths, and we have like expressions for the parts, when the unit is further divided accordinir to the scale of tens. 'O These fractions may be wri Three-tenths, Seventh-tenths, - Sixty-five hundredths, 215 thousandths, 1275 ten-thousandths, ten thus t'o- tV- 100' 1000- 1275 loooo- From which we see, that the fractional unit of a decimal is one of the equal parts arising from dividing the unit 1 accord- ing to the scale of tens : hence, it is one-tenth, one-hundredth, "ne-thousandth, &c. 188. A Decimal Number, or decimal, is one which contains a decimal unit. 189. A Mixed Decimal, is one composed of a whole num- ber and a decimal. NOTATION AND NUMERATION. 165 Notation and Numeration. 190. Tlae denominators of decimal fractions are seldom written. The fractions are expressed by means of a period, placed at the left of the numerator, called the decimal point (.). Thus, y^o .is written .3 Too -"^ 2 15 II OIK TodS -^^^ ^i-2JL5_ ... " 1275 Toooo .A^»u The denominator, however, of every decimal, is always un- derstood : It is the unit 1, with as many ciphers annexed as there are places of figures in the decimal. The place next to the decimal point is called the place of tenths, and its unit is 1 tenth ; the next place, at the right, is the place of hundredths, and its unit is 1 hundredth ; the next is the place of thousandths, and its unit is 1 thousandth ; and similarly for places still to the right. Decimal Numeration Table. (A ;5 =0 ' - S r^ ^ i- fl •5 'cJ a -V "^ 2 1 1 g 2 S gS « a illions. undreds o jns of tho lousands. undreds. ii 'S 42031451.2043018 191. Principles. 1. That the denominator belonging to any decimal fraction is 1, with as many ciphers annexed as there are places of figures in the decimal. 2. That the unit of any place is ten times as great as the unit of the next place to the right — the same as in whole numbers: hence, whole numbers and decimals may be wri^jtcn together, by placing the decimal point between them. 186. How many kinds of fractions are there? What are they? What is a common fraction ? What is a decimal fraction ? — 187. Wlien the unit 1 is divided into 10 equal parts, what is each part called? What is each part called when it is divided into 100 equal pans? When into 1000? Into 10,000? &c. How are decimal fractions formed? — 188. What is a decimal number? — 189. What is a mixed decimal? — 190. Are the denominators of decimal fractions generally set down ? How are the fractions expressed ? Is the denominator understood ? What is it ? What is the place next the decimal point called? What is its unit? What is the next place called? What is its unit ? What is the third place called ? What is its unit ? Which way are decimals numerated and read? — 191. What are the two prin ciples of decimals? NOTATION AND NUMERATION. 167 192. Rule for Writing Decimals. Write the decimal as if it icere a ichole number, prefixing as many ciphers as are necessary to make its right-hand Jig- lire of the required name. 193. Rule for Reading Decimals. Bead the decimal as though it were a whole number^ adding the denomination indicated by the lowest decimal unit. Examples. Write the following common fractions decimally : (1.) (2.) (3.) (4.) (5.) 6 17 5 2 7 47 ToT Tft TBoT Toll TinFS- (6.) (t.) (8.) (9.) (10.) 6AV lioVs 9rfi» lOm 12-W. Write the following numbers in figures, and numerate them : 1. Twenty-seven, and four-tenths. 2. Thirty-six, and fifteen-thousandths. 3. Ninety-nine, and twenty-seven ten-thousandths. 4. Three hundred and twenty thousandths. 5. Two hundred, and three hundred and twenty millionths. 6. Three thousand six hundred ten-thousandths. T. Five, and three-millionths. 8. Forty, and nine ten-millionths. 9. Forty-nine hundred ten-thousandths. 10. Fifty-nine, and sixty-seven ten-thousandths. 11. Four hundred and sixty-nine ten-thousandths. 12. Seventy-nine, and four hundred and fifteen millionths. 13. Sixty-seven, and two hundred and 2t ten-lOOOths. 14. One hundred and five, and ninety-five ten-millionths. 15. Forty, and 204 thousand millionths. 192. What is the rule for writing decimals ?— 193. What is the rule for reading decimals? 168 DECIMALS. UNITED STATES MONEY. 194. The denominations of United States Money correspond to the decimal division, if we regard one dollar as tJie unit : For, the dimes are tenths of the dollar, the cents are him dredths of the dollar, and the mills, being tenths of a cent, are thousandths of the dollar. Examples. 1. Express $31 and 26 cents and 5 mills, decimally. 2. Express $lt and 5 mills, decimally. 3. Express $215 and 8 cents, decimally. 4. Express $2t5 5 mills, decimally. 5. Express $9 8 mills, decimally. 6. Express $15 6 cents 9 mills, decimally, t. Express $21 18 cents 2 mills, decimally. 8. Express $3 5 cents 9 mills, decimally. ANNEXING AND PREFIXING CIPHERS. 195. Annexing a cipher is placing it on the right of a number. If a cipher is annexed to a decimal it makes one more de- cimal place, and therefore, a cipher must also be added to the denominator (Art. 190). The numerator and denominator will therefore have been 194. If tlie denominations of Federal Money be expressed decimally, what is tlie miit? What part of a dollar is 1 dime? What part Oi a dime is 1 cent? What part of a cent is a mill? What part of a dollar is 1 cent? 1 mill? — 195. Wlien is a cipher annexed to a num- ber ? Does the annexing of ciphers to a decimal alter its value ? Why not? What does five-tenths become by annexing a cipher? What by annexing two ciphers? T'iree ciphers? DKCIMAL FUACTIONS. 1G9 multiplied by the same number, and consequently the value of the fraction will not be changed (Art. 143) : hence, Annexing ciphers to a decimal does not alter its value. Take as an example, .5 = ^o. If we annex a cipher to the decimal, wo at the same time aimex one to the denominator ; thus, .5 becomes .50 = ^^q by annexing one cipher. .5 becomes .500 = -f^Q by annexing two ciphers. .5 becomes .5000 = to°oVo ^J annexing three ciphers. 196. Prefixing a cipher is placing it on the left of a number. If ciphers are prefixed to a decimal, the same number of ciphers must be annexed to the denominator ; for, the de- nominator must always contain as many ciphers as there are decimal places in the numerator. Now, the numerator will re- main unchanged while the denominator will be increased ten times for every cipher annexed ; and hence the value of the fraction will be diminished ten times for every cipher pre- fixed to the decimal (Art. 142) : hence. Prefixing ciphers to a decimal diminishes its value ten times for every cipher prefixed. Take, for example, the decimal .3 = ^. .3 becomes .03 = y^^ by prefixing one cipher ; .3 becomes .003 = y^o'Q ^7 prefixing two ciphers ; .3 becomes .0003 = yo^irff ^7 prefixing three ciphers : in which the fraction is diminished ten times for every cipher prefixed. 196. When is a cipher prefixed to a number ? When prefixed to a decimal, does it increase the numerator ? Does it increase the denom- inator? What effect, then, has it on the value of the decimal? 8 170 ADDITION OF 197. Analysis of decimals. Analyze 62.25. It is composed of 6 tens, 2 units, 2 tenths, and 6 hundredths ; or it is composed of 62 units and 25 hun- dredths ; or of 622 tenths and 5 hundredths ; or 6225 hun- dredths. Note.— Let it be remembered that a fractional unit of any one place is y\j of the unit of the place next on the left, or yl^ of the miit wliich is 2 places to the left, or y^Vff ^^ ^^^ fractional unit which is three places to the left. ADDITION OF DECIMALS. 198. Addition of Decimals is the operation of finding the sum of two or more decunal numbers. It must be remembered, that only units of the same value can be added together. Therefore, in setting down decimal numbers for addition, figures having the same unit value must be placed in the same column. The addition of decimals is then made in the same manner as that of whole numbers. 1. Find the sum of 87.06, 327.3, and .0567. OPKRATION. Analysis. — Place the decimal points in the same 87 06 column: this brings units of the same value in the qo>7 q same column: then add as in whole numbers: hence, .0567 Rule. 414.4167 I. Set down the numbers to he added so that figures oj the same unit value shall stand in the same column : 198. What is addition ? What parts of a unit may be added to- gether? How do you set down the numbers for addition? How will the decimal points fall ? How do you then add ? How many decimal places do you point off in the sum ? DKCIMALS. 171 II. Add as in simple numbers, and point off in the sum, from the right hand, a number of 2)laces for decimals equal to the greatest number of places in any of the numbers added. Proof. — The same as in simple numbers. Examples. 1. Add G.035, 763.196, 445.3741, and 91.5754 together 2. Add 465.103113, .78012, 1.34976, .3549, and 61.11. 3. Add 57.406 + 97.004 + 4 + .6 + .06 + .3. 4. Add .0009 + 1.0436 + .4 + .05 + .047. 5. Add .0049 + 49.0426 + 37.0410 + 360.0039. 6. Add 5.714, 3.456, .543, 17.4957 together. 7. Add 3.754, 47.5, .00857, 37.5 together. 8. Add 54.34, .375, 14.795, 1.5 together. 9. Add 71.25, 1.749, 1759.5, 3.1 together. 10. Add 375.94, 5.732, 14.375, 1.5 together. 11. Add .005, .0057, 31.008, .00594 together. 12. Required the sum of 9 tens, 19 hundredths, 18 thou- sandths, 211 hundred-thousandths, and 19 millionths. 13. Find the sum of two, and twenty-five thousandths, five and twenty-seven ten-thousandths, forty-seven, and one hun- dred twenty-six millionths, one hundred fifty, and seventeen ten- millionths. 14. Find the sum of three hundred twenty-seven thousandths, fifty-six ten-thousandths, four hundred, eighty-four millionths, and one thousand five hundred sixty hundred-millionths. If). What is the sum of .5 hundredths, 27 thousandths, 476 ladred-thousandths, 190 ten-thousandths, and 1*279 ten-mil- lionths ? 16. What is the sum of 25 dollars 12 cents 6 mills, 9 dol- 172 DECIMALS. lars 8 cents, 12 dollars t dimes 4 cents, 18 dollars 5 dimes 8 mills, and 20 dollars 9 mills ? It. What is the sum of 126 dollars 9 dimes, 420 dollars 15 cents 6 mills, 317 dollars 6 cents 1 mill, and 200 dollars 4 dimes 1 cents 3 mills? 18. A man bought 4 loads of hay, the first contained 1 ton 25 thousandths ; the second, 99 1 thousandths of a ton ; the thu'd, 88 hundredths of a ton ; and the fourth, 9876 ten- thousandths of a ton : what was the entire weight of the four loads ? 19. Paid for a span of horses, $225.50 ; for a carriage, $127,055 ; and for harness and robes, $75.28 : what was the entire cost ? 20. Bought a barrel of flour, for $9,375 ; a cord of wood, for $2.12J ; a barrelof apples, for $1.62J ; and a quarter of beef, for $6.09 : what was the amount of my bill ? 21. A farmer sold grain as follows : wheat, for $296.75 ; corn, for $126.12^; oats, for $97.37J ; rye, for $100.10; and barley, for $50.62^ : what was the amount of his sale ? 22. A person made the following bill at a store : 5 yards of cloth, for $16,408; 2 hats, for $4.87}; 4 pairs of shoes, for $6 ; 20 yards of calico, for $2,378 ; and 12 skeins of silk, for $0.62 J : what was the amount of his bill ? 23. What is the sum of. $99 87 cents 5 mills; $87 6 cents 18 mills ; $59 42 cents 20 mills ; $60 49 cents 16 mills ; and $21 29 cents 13 mills? 24. What is the sum of $97 4 mills ; $25 19 mills ; $65 95 cents 6 mills ; $4 87J cents 3 mills ; and $55 14} cents 9 mills ? 25. Mr. James bought of Mr. Squires, the grocer, the fol lowing articles : a bag of coffee, for $37,874 ; a chest of tea, for $50,009 ; a barrel of sugar, for $19 4 cents and 6 mills ; and 9 gallons of wine, for $27 69 cents and 15 mills : what was the amount of his 1)111 ? DECIMALS. 178 SUBTRACTION 199. Subtraction of Decimals is the operation of finding tlic difference between two decimal numbers. 1. From 6.304 take .0563. Analysis. — In this example a cipher is annexed orEiiATioN. o the minuend to make the number of decimal n ^040 places equal to the number in the subtrahend. 0563 This does not alter the value of the minuend (Art. 195): hence, 6.2477 Rule. I. Write the less number under the greater, so that figures of the same mat value shall fall in the same column: II. Subtract as in simjjle numbers, and point off the deci- mal jilaces in the remainder, as in addition. PnooF. — Same as in simple numbers. Examples. 1. From 875.05 take .0467. 2. From 410.0591 take 41.496. 3. From 7141.604 take .09046. 4. Required the difference between 57.49 and 5.768. 6. What is the difference between .3054 and 3.075 ? 6. Required the difference between 1745.3 and 173.45. 7. What is the difference between seven-tenths and 54 ten- thousandths ? 8. What is the difference between .105 and 1.00075 ? 9. What is the difference between 150.43 and 754.365 ? 10. From 1754.754 take 375.49478. L 199. What is subtraction of decimal fractions? How do you set down tlie numbers for subtraction ? How do you then subtract 1 How many decimal places do you point off in the remainder ? What js the proof? Vti DECIMALS. 11. Take ^5.304 from 175.01. 12. Required the difference between 17.541 and 35.49. 13. Required the difference between t tenths and 7 mil- lionths. 14. From 396 take 67 and 8 ten-thousandths. 15. From 1 take one-thousandth. 16. From 6374 take fiftj-nine and one-tenth. 17. From 365.0075 take 5 millionths. 18. From 21.004 take 98 ten-thousandths. 19. From 260.3609 take 47 ten-millionths. 20. From 10.0302 take 19 millionths. 21. From 2.03 take 6 ten-thousandths. 22. From one thousand, take one-thousandth. 23. From twenty-five hundred, take twenty-five hundredths. 24. From two hundred, and twenty-seven thousandths, take ninety-seven, and one hundred twenty ten-thousandths. 25. A man owning a vessel, sold five thousand seven hun- dred sixty-eight ten-thousandths of her : how much had he left? 26. A farmer bought at one time 127.25 acres of land ; at another, 84.125 acres ; at another, 116.7 acres. He wishes to make his farm amount to 500 acres : how much more must he purchase ? 27. Bought a quantity of lumber for $617. 37J, and sold it for $700 : how much did I gain by the sale ? 28. Having bought some cattle for $325.50 ; some sheep for $97.12i; and some hogs for $60.87| ; I sold the whole for $510.10: what was my entire gain? 29. A dealer in coal bought 225.025 tons of coal : ho sold to A, 1.05 tons ; to B, 20.007 tons ; to C, 40.1255 tons -. and to D, 37.00056 tons : how much had he left ? 30. A man owes $2346.865: and has due him, from A, $1240.06 ; and from B, $1867.984 : how much will he have left after paying his debts? DECIMALS. 175 MULTIPLICATION. 200. Multiplication of decimals is the operation of taking one number as many times as there are units in another, when one or both of the factors contain decimals. 1. Multiply 8.03 by 6.102. Analysis. — If we change both factors to common frac- tions, tlie product of the nu- merators will be the same as that of the decimal numbers, and the number of decimal places will he equal to the OrERATION. 8.03 = StJij = m 6.102 = 6lWlcices in the quotient icill be equal to the greatest number of factors^ 2 or 5, ^V^ the divisor. 3. Can ^^ be exactly expressed decimally ? How many places ? Analysis. — 25 =5x5; hence, the fraction can be ex- actly expressed decimally, and by two decimals, because 5 is taken twice as a factor in the divisor. operation. 25)7.0(.28 50 200 200 DECIMALS. 193 Examples. Find the decimals and number of places in the following : 1. Express yf^ decimally. 2. Express -^^q decimally. 3. Express ^^ decunally. 4. Expicss ji^ decimally. 5. Express -^ decimally. 6. Express ^lt^ decunally. T. Express -^ decimally. 8. Express ^f^^ decunally. CASE II. 218. When the division does not terminate. 1. Let it be required to reduce J to its equivalent decimal. Analysis. — ^By annexing decimal ciphers to the operation. numerator 1, and making the division, we find the 3)1.0000 equivalent decimal to be .3333 + , «&c., giving 3's ^333 + as far we choose to continue the division. The further the division is continued^ the nearer the value of the decimal will approach to \, the exact value of the com- mon fraction. We express this approach to equality of value, by saymg, that if the division be continued without limits that is, to infinity, the value of the decimal will then become equal to that of the common fraction j thus, .3333 .. . , continued to infinity = J ; for, each succeeding 3 brings the value nearer to \. Also, .9999 . . . , continued to infinity = 1 ; for, each succeeding 9 brings the value nearer to 1, 2. Find the decimal corresponding to the common fraction §. Analysis. — Annexing decimal ciphers and operation. • dividing, wo find the decimal to ho .2222+,. 9)2.0000 in which we see that the figure 2 is continually .2222 repeated. 9 L 194- K'iPEATlNO Examples. 1. Express the fraction f decimally. 2. Change 1*3- into a decimal fraction. 3. Reduce j\ to a decimal fraction. 4. Reduce -j^ to a decimal fraction. 219. Definitions. 1. A Repeating Decimal is a decimal in which a sin(,le figure, or a set of figures, is constantly repeated. 2. A Repetend is a single figure, or a set of figures, which is constantly repeated. 3. A Single Repetend is one in which only a single figure Is repeated ; as j = .2222+, or f = .3333+. Such rcpetends are expressed by simply putting a mark over the first figure ; thus, .2222+, is denoted by .^2, and .3333+ by .^3. 4. A Compound Repetend has a set of figures repeated ; thus, If = .5t 5t+, and jj|3 ^ 5^33 5123 + are compound repetends, and are distinguished by marking the first and last figures of the set. Thus, 51 57+ is written .^57', and .5723 5723 + is written .^5723'. 5. A Pure Repetend is one which begins with the first decimal figure ; as, .^3, .^5, .^473', &c. 217. How do you determine wlien a common fraction can be ex actly expressed decimally? How many decimal places will there bo in the quotient ?— 218. Can one-third be exactly expressed decimally? Wliat is the form of tlie quotient? To what does the value of this quotient approach? Wlien does it become equal to one-third ?— 219. 1. What is a repeating decimal? 2. What is a repetend? 3. What is a single re^xitend? 4. What is a compniud repetend? DECIMALS. 19o 6. A Mixed Repetexd is one which has significant figures or ciplicrs between the decimal point and the repetend ; or which has whole numbers on the left hand of the decimal point ; such figures are called finite figures. Thus, .0^733', .4^73; .3^573', 6/5, are all mixed repetends ; .0, .4, .3, and 6, are the finite figures 7. Similar Repetends are such as begin at equal distance from the decimal points ; as .3^54', 2.7'534'. 8. Dissimilar Repetends are such as begin at different dis- tances from the decimal points ; as .'^253', .47'52'. 9. Conterminous Repetends are such as end at equal dis- tances from the decimal points ; as .1^25', /354'. 10. Similar and Conterminous Repetends are such as begin and end at the same distances from the decimal point ; thus, 53.2V753', 4.6^325', and .4^632', are similar and conterminous. REDUCTION OF REPETENDS TO COMMON FRACTIONb. CASE I. 220. To reduce a pure repetend to its equivalent common fraction. Analysis. — This proposition is to be analyzed hy examining the law of forming the repetends. let. ^=.llll+«&c. = .n; and f = 4444+ &c. = .H: 2d. ^V = . 010101+ &c. = .^01'; and f J = .2727+ &c. = .•»27': 3d. ^Jy =.001001 + &c.=.^001' ; and f ff = .324324+ &c. = .^324'; &c., &c., &c., &c. The above law for the formation of repetends does not depend on the multipliers 4, 27, and 324, but would be the same for any other figures. Rule. — Divide the number denoting the repetend by as many 9's as there are figures, and reduce the fraction to its loiuest terms. 196 KEPEATING Examples. 1. What is the equivalent common fraction of the repetecd 0.^3? We have, f = J = 0.33333+ = ^3. 2. What is the equivalent common fraction of the repeten .n62' ? We have, Jf J = j\\. Ans. 3. What are the simplest equivalent common fractions of the repetends /6, /162', 0;t69230', /945', and /09'? 4. What are the least equivalent common fractions of the repetends- /594405', .^36', and .^142857' ? CASE II. 221. To reduce a mixed repetend to its equivalent common fraction. Analysis. — A mixed repetend is composed of the finite figures which precede, and of the repetend itself; hence, its value must be equal to such finite figures plus the repetend. When the repetend begins at the decimal point, the unit of the first figure is .1. But if the repetend begins at any place at the right of the decimal point, the unit value of the first figure will be diminished ten times for each place at the right, and hence, O's must be annexed to the 9's which form the divisor. Rule. — To the finite figures, add the repetend divided by as many 9's as it contains places of figures, with as many O's annexed to them as there are places of decimal figures vreceding the repetend ; the sum reduced to its simplest form will be the equivalent fraction sought. 219. 5. What is a pure repetend ? 6. What is a mixed repetend 7. What are similar repetends? 8. Wliat are dissimilar repetends? 9. What are conterminous repetends? 10. What are similar and conterminous repetends ? — 220. How do you reduce a pure repetend to an equivalent common fraction? — 321. How do you find the value of a mixed repetend? DECIMALS. 197 Examples. 1. Rcqiured the least equivalent common fraction of the mixed repetend, 2.4^18'. Now, 2.4^18' = 2 + ^V + '18' = 2 + A + 9'd% = 2Ii. Ans. 2. Required tM) least equivalent common fraction of the mixed rcpetcud .5'925'. We have, .5^925' = tV + qWo = if ^l"^'- 3. What is the least equivalent common fraction of the repetend .008^497133'? We have, .008^497133' = ^^'oo + 99W9W00 = ms- 4. Required the least equivalent common fractions of the mixed repetends .13^8, 7.5^43', .04^354', 37.5^4, .6^75', and .7^54347'. 5. Required the least equivalent common fractions of the mixed repetends 0.7^5, 0.4^38', .09^3, 4.7^543', .009^87', and .4^5. CASE III. 222. To find the finite figures and the repetends correspond- ing to any common fraction. 1. Find the finite figures and the repetend corresponding to the fraction ^Iq. Analysis. — 1st. Reduce the fraction to its lowest terms, and tlien find all the factors 2 and 5 of the denominator. 2d. Add decimal ciphers to tlie numerator and make the division. 8d. The number of Jinite decimals preceding the first figure of tlie repetend will he equal to the greatest number of factors 2 or 5: in this example it is 3. OPERATION, 6 560 ~ 3 ' 280 3 3 280 ~2x2 X2x5x7 280)3.000+ (. 010^714285' lOS REPEATING 4th. When a remainder is found which is the same as a previous dividend, the second repetend begins. 5th. The number of figures in any repetend will never exceed the number, less 1, of the units in that factor of the denominator which does not contain 2 or 5. In the example, that number is 7, and the number of figures of the repetend. is 6. Rule. — Divide the numerator of the common fraction, r duced to its lowest terms, by the denominator, and poini o// in Ike quotient the finite decimals, if any, and the rep: te-iui. Examples. 1. Find whether the decimal, equivalent to the common fraction 29 f oTj ^^ finite or repeating : required the finite figures, if any, and the repetend. Analysis. — We first reduce the fraction to its lowest terms, giving gfffif* ^^^© then search for the factors 2 and 5 in the denominator, and find that 2 is a factor 3 times ; hence, we know that there are three finite decimals preceding the q*7A8\qq qq ■ ( 008^49*riSS' repetend. We next divide the numerator 83 by the denominator 9768, and note that the repetend begins at the fourth place. After the ninth division, we find the remainder 83; at this point the figures of the quotient begin to repeat ; hence, the repetend has 6 places. 2. Fmd the finite decimals, if any, and the repetend, if any, of the fraction yV&* 3. Find the finite decimals, if any, and the repetend, if any, of the fraction xieo- 4. Find the finite decimals, if any, and the repetend, if any, cf the fractions ^ft, tVj, tVs- OPERATION. 249 83 29304 9768 83 83 9768 2x ;2x2xl221 DECIMALS. 100 223. Properties of the Repetends. There are some properties of repetends which it is important to remark. 1. Any finite decimal may be considered as a repeatmg decimal by making ciphers recur ; thus, .35 = .35^0 = .35^00' =.35^000' = .35^0000', &c. 2. Any repeating decimal, whatever its number of figures, may be changed to one having twice or thrice that number of figures, or any multiple of that number. Thus, a repetend 2.3^57' having two figures, may be changed to one having 4, 6, 8, or 10 places of figures. For, 2.3^5r = 2.3^575r =2.3^575757' = 2.3^57575757', &c. ; so, the repetend 4.16'316' may be written 4.16^316' = 4.16^316316' = 4.16^316316316', &c. ; and the same ma^ be shown of any other. Hence, two or more repetends, having a different number of places in each, may be reduced to repetends having the same number of places, in the following manner : Find the least common multiple of the number of places in each repetend, and reduce each repetend to such number of places, 3. Any repeating decimal may be transformed into another having finite decimals and a repetend of the same number of figures as the first. Thus, .^57' = .5^5' = .57^57' = .575^75' = .5757^57' ; and 3.4^785' = 3.47^857' = 3.478^578' = 3.4785^785' ; and hence, any two repetends may be made similar. 222. How do you find the finite figures and the repetend corro- pponding to any common fraction? — 223. 1. How may a finite decimal be made a repeating decimal? S. Wlien a rejpetend has a given number of places, to what other form may it be reduced? How? 3. Into what form may any repeating decimal be transformed? L 200 REPEATING These properties may be proved by changing the repetends into their equivalent common fractions. 4. Having made two or more repetends similar by the last article, they may be rendered conterminous by the previous one ; thus, two or more repetends may always he made similar nd conterminous. 5 If two or more repeatmg decunals, having several repe- tends of equal places, be added together, their suu> will have a repetend of the same number of places ; for, every two sets of repetends will give the same sum. 6. If any repeatmg decimal be multiphed by any number, the product will be a repeating decimal having the same number of places in the repetend ; for, each repetend will be taken the same number of times, and consequently must pro- duce the same product. Examples. 1. Reduce .13^8, Y.5'43' .04^354', to repetends havmg the same number of places. Since the number of places are now 1, 2, and 3, the least common multiple is 6, and hence each new repetend will con- tain 6 places ; that is, .13^8 = .13^888888' ; T.5^43' = 7.5^434343' ; and .04^354' = .04^354354'. 2. Reduce 2.448', .6^925', .008^497133', to repetends having the same number of places. 3. Reduce the repeating decunals 165.464', .^04', .03^7 to such as are similar and conterminous. 4. Reduce the repeating decunals .5'3, .4^75', and 1.^757', o such as are similar and conterminous. 223. 4. To wliat form may two or more repetends be reduced? DECIMALS. 201 ADDITION. 224. To add repeating decimals. I, Make the repetends, in each number to be added, similar and conterminous : II. Write the places of the same unit value in the same 'olumn, and so many figures of the second repetend in each as shall indicate xdlh ceirtainty, how many are to be carried from one repetend to the other : then add as in whole numbers. Note. — If all the figures of a repetend are 9's, omit them and add 1 to the figure next at the left. Examples. 1. Add .12^5, 4.U63', 1.^7143', and 2.^54', together. DLSSIMILAR. SIMILAR. SIMILAR AND COXTERMINOUS. .12^5 = .12'5 = .12^555555555555' - - - 5555 4.n63' = 4.10^316' = 4.1G^31631G316316' - - - 31C3 1.^7143' = 1.7r4371' = 1.7^437143714371' - - - 4371 2.^54' = 2.54^54' = 2.54^545454545454' - - - 5454 The true sum = 8.54^854470131697' 1 to carry. 2. Add 67.3^45', 9.^G51', .^25', 17.4^, .\5, together. 3. Add .^475', 3.75^43', 64.^5', .^57', .r788', together. 4. Add .^5, 4.3^7, 49.4^57', .4^954', .^7345,' together. 5. Add .^175', 42.^57', .3^753', .4^954', 3.7^54', together. 6. Add 165, .464', 147.^04', 4.^95', 94.3^ 4.^12345'. SUBTRACTION. 225. To subtract one repeating decimal from another. I. 3Iake the repetends similar and conterminous : II. Subtract as in finite decimals, observing that when the repetend of the lower line is the larger, 1 must be carried to the first right-hand figure. 224. How do you add repeating decimals? — 225. How do you sub- tract repeating decimals. 9* 202 REPEATING Examples 1. From 11.4^75' take 3.45^35'. DISSIMILAR. SIMILAR. SIMILAR AND CONTERMINOUS. 11.4^75' = 11.47^5r = 11.4r575757' - - 3.45^35' = 3.45^735' = 3.45^35735' - - 575 735 The true difference = 8.0r840021' 1 to carry. 2. From 47.5^3 take i;757'. 3. From 17.^573' take 14.5^7. 4. From 17.4^3 take 12.34^3. 5. From 1.12^754' take .4^384'. 6. From 4.75 take .37^5. 7. From 4.794 take .1^744'. 8. From 1.45^7 take .3654. 9. From 1.4^937' take .1475. MULTIPLICATION. 226. To multiply one repeating decimal by another. Change Ihe repeating decimaU into their erjaivalent com moii fraetiom^, then multiply them together, and reduce tht product to its equivalent repealing decimal. Examples. 1. Multiply 4.25^3 by .257. OPKUATION. 4 '>ry3 — 4 4- -?5. _!_ __3 _ _ 4 I 22 5 I 3_ __ *•-•' "^ — ^ ' 100 "^aOO — ^ ^ 900 ^900 — 3_ __ 22 8 »00 3_8 2_8 900 L9J 4 4 50 957 2 2.5 Also. ^25 -^ 257 — -^^"^ ♦ hencf .-/.J i — 100 5 "^"t.1,, 1000 MiUl = 1.09310^0 ; and since 225000 = 5x5x5x5x5x2x2x2x9, there will be five places of finite decimals, and one figure in the repeteiid. Note. — Much labor will be saved in this and the next rule by keep- ing every fraction in its lowest terms ; and when two fractions are to be multiplied together, cancel all the factors common to both term before making the multiplication. 2. Multiply .375^4 by 14.75. 3. Multiply .4^253' by 2.57. 4. Multiply .437 by 3.7^5. 5. Multiply 4.573 by .3^75'. 6. Multiply 3.^5^6 by .42^5. 7. Multiply 1.H56' by 4.2^3. 8. Multiply 45.r3 by .^245'. 9. Multiply .4705^3 by 1.7^35'. DECIMALS. 203 DIVISION. 227. To divide one repeating decimal by another. Change the decimals into their equivalent common fractionb, and find the quotient of these fractions, Tlien change the quotient into its equivalent decimal. Examples. 1. Divide 5G.^6 by 13t. OPERATION. 56.^0 = 56 + f = ^^ = ip. Then, i^ ^ 137 = ip x yir = iiT = .'41362530'. 2. Divide 24.3^18' by 1.192. 3. Divide 8.59G8 by .2^45'. 4. Divide 2.295 by .^29r. 5. Divide 47.345 by 1.^6'. 6. Divide 13.5469533' by 4.^29t' 1. Divide .^45' by .418881'. 8. Divide .^475' by .3^753'. 9. Divide 3.^153' by .^24'. CONTINUED FRACTIONS. 228. A Continued Fraction has 1 for its numerator, and for its denominator a whole number plus a fraction, which also has a numerator of 1, and for a denominator, a whole number plus a similar fraction, and so on. .1. If we take any irreducible fraction, as ^J, and 'divide both terms by the numerator, it will take the form, ' ^ = 2? = 1 , 14 ^^y making the division. If, now, we divide both terms of ||^ by 14, we have, 14__1 15-1+tV 220. How do you multiply repeating decimals? — 237. How do you divide repeating decimals ?— 228. What arc continued fractions? What hi the rule for finding tha approximate value? 204: CONTINUED If. now, we replace ^f 3y its value, , - , we shall liave 1 + 14 15 1 29 1 + 1 1+A; hence, this is a '.ontinued fraction, 2. Reduce \^ to the form of a continued fraction. 15 19" 1 "1 + 15 19 4 1 15 - 3 + V 1 ~1+1 3 + 1 1 + 3 4 I. l + i ^p^ 1 hence. Analysis. — Let us analyze this example. If we neglect what comes after 1, the first term of the first denominator, we shall have, T = 1? which is called the first approximating fraction. If we neglect what comes after 3, the first term of the second denomi- nator, we shall have, 1 _ 3 1+1-4 the second approximating fraction. If we neglect what comes after 1, the first term of the third denominator, we shall have, 1 1+1 =-5 3 + 1 the third approximating fraction; and so on, for fractions which follow. If we stop at the first approximating fraction, the denominator 1 will he less than the true denominator; for, the true denominator is 1 plus a fraction; hence, the value of the first approximating fraction will he too great; that is, it will exceed the value of the given fraction. If we stop at the second, the denominator 3 will he less than the true denominator; hence, ^ will he greater than the number to ho ad'lcd to 1 ; therefore, 1 + /f is too large, and 1 -f- 1 + 5, wliioli FRACTIONS. 205 is J, is too small : that is, it is less than the value of the given frac- tion. Thus, every odd approximating fraction gives a value too large, and every even one, gives a value too small. Rule. — Write the given fraction in the form of a continued fraction, using several terms when a near approximation is desired ; then take a mean betiveen the last and the preceding approximating fractions. Examples. 1 Reduce 4^1 to the form of a continued fraction. 437 2 + 1 l + l 1 + 1 3 + tV. 2. Place f J under the form of a continued fraction, and find the value of each of the approximating fractions. 3. Place U under the form of a continued fraction, and find the value of each of the approximating fractions. 4. Place \\ under the form of a continued fraction, and find the value of each approximating fraction. 5. Place f J under the form of a continued fraction, and find the value of each approximating fraction. G. Place -J^ under the form of a continued fraction, and find the value of each approximating fraction. 7. The solar year contains 365 da. 5 hr. 48 m. 48 sec. Find what fractional part of a day the excess of the solar year is bove the common year, when the operation is carried to the fifth approximating fraction. 206 RATIO AND PKOPORTION. RATIO AND PROPORTION. 229. Two numbers, of the same kind, may be compared in two ways : 1st. By considering how much one is greater or less thai the other, which is shown by their difference ; and, 2d. By considering how mafiy times one number is greater or less than another, which is shown by their quotient. In comparing two numbers, by means of their difference, the less is always taken from the greater. In comparing two numbers by their quotient, one is regarded AS a standard which measures the other ; hence, to measure a number, is to find how many times it contains the standard. 230. A Ratio is the quotient arising from dividing one num- ber by another. The Terms of a ratio are the divisor and dividend ; hence, every ratio has two terms. The Divisor is called the Antecedent ; and the Dividend is called the Consequent. The antecedent and consequent, taken together, are called a Couplet. 231. The ratio of one number to another is expressed in two ways : 1st. By a colon ; thus, 4 : 16 ; and is read, 4 is to 16 ; or, 16 divided by 4. 2d. In a fractional form, as Y > <^^> 1^ divided by 4. 229. In how many ways may two numbers of the same kind be compared with each other? If you compare by their difference, wliat do you do ? If you compare by the quotient, how do you regard one of the numbers? How do you measure a number? 230. What is a ratio? What are its terms? How many terms lias every ratio? What is the divisor called ? What the dividend ? — 231. In how many ways is the ratio expressed? What are they? How is it read ? RATIO AND niOFORTION. 207 Since every ratio may be expressed under the form of a fraction, and since tlie numerator and denominator may be mul- tiplied or divided by the same number, without altering the value (Arts. 143 and 144), it follows that. If both terms of a ratio be multiplied or divided by the mme number, the ratio will not be changed. 232. A Simple Ratio is when both terms are simple num bers ; thus, t : 12, is a simple ratio. 233. A Compound Ratio is one which arises from the mul- tiplication of two simple ratios : thus, in the simple ratios 5 : 10, and 3 : 12, if we multiply the corresponding terms together, we have 5 X 3 : 10 X 12, whicli is compounded of the ratios of 5 to 10, and of 3 to 12, 234. The Elements of a term are its factors : thus, 5 and 3 are the elements of the first term, and 10 and 12 of the second. These elements are generally WTitten in a column, thus, !! [- : 1 .) f ; ^"^^ read, 5 multiplied by 3, to 10 multiplied by 12. Note. — A compound ratio may be reduced to a simple ratio, by multiplying the elements; thus the last ratio is that of 15 to 130. 235. To find the ratio of one number to another. AN'hcn the antecedent is less than the consequent, the ratio >li(jws how many times the consequent is greater than the • ntecedent. When the antecedent is greater than the consequent, the ratio shows what part the consequent Ls of the antecedent. The phrase, "what part," implies the quotient of a less num- ber divided by a greater. 233. What is a simple ratio?— 333. What is a compound ratio?— 23 i. What are the elements of a term? 208 KATIO AND PKOPORTION". Examples. 1. What is the ratio of 9 tons to 15 tons ? Analysis. — In this example the antecedent is 9 tons, and the consequent is 15 tons ; the ratio is therefore expressed by the frac- 2. What 3. What 4. What 5. What 6. What t. What 8. What 9. What 10. What 11. What 12. What is the ratio of 6 inches to 24 inches ? is the ratio of 1 feet to 35 feet? is the ratio of fifteen dollars to 6 dollars ? is the compound ratio of 5 : 6 and 4 : 10 ? is the compound ratio of 6 : 9 and 3:4? is the compound ratio of 4 : 5, 9 : 8, and 3:5? part of 6 is 4? part of 10 is 5? part of 34 is It ? part of 450 is 300? part of 96 is 16? 13. 8 is what part of 12 ? 14. 16 is what part of 48? 15. 18 is what part of 90 ? 16. 15 is what part of 165? It. 9 is what part of 11? 236. To find the antecedent or consequent, when the ratio and one of the terms are given. 1. The ratio of two numbers is 5 ; and the antecedent i? 4 dollars : what is the consequent ? Analysis. — Since the ratio is equal to the quotient of the consequent divided by the antecedent, it follows: 1st. That the consequent is equal to the antecedent multiplied hy the ratio : 2d. That the antecedent is equal to the consequent divided hy the ratio. OPERATIOr. Batio 5 = conset {uent. antecedent. 5 X ant. = cons. $4 X 5 = $20 = cons. Examples. 1. The ratio of two numbers is t, and the antecedent is 16cwt. : what is the consequent? 2. The consequent is 30 tons, and the ratio is 6 : what is the antecedent ? RATIO AND PROPORTION. 209 3. The antecedent is 15, and the ratio is 4 : what is the consequent ? 4. The ratio of two numbers is 1|, and the consequent is 7 : what is the antecedent ? 5. The ratio of two numbers is J, and the antecedent is | . what is the consequent ? 6. The ratio of the monthly wages of two men is 8 : the greater wages of one is $256: what is the wages of the other? 7. The ratio is 25, and the consequent is 14 X 5 x 10 : what is the antecedent? 8. The value of a horse is 2J times that of an ox : the yalue of the horse is $143 : what is the value of the ox ? SIMPLE PROPORTION. 237. A Simple Proportion is an expression of equality between two simple and equal ratios. Thus, the two couplets, 4 : 20 and 1 : 5, having the same ratio 6, form a proportion, and are written, 4 : 20 : : 1 : 5, by simply placing a double colon between the couplets. The terms are read, 4 is to 20 as 1 is to 5, and taken together, they are called a proportion. 238. The 1st and 4th terms of a proportion are called the extremes; the 2d and 3d terms, the means. Thus, in the pro- portion, 6 : 24 : : 8 : 32, 6 and 32 arc the extremes^ and 24 and 8 the means : _,. 24 32 Smce ^ = 8" WQ shall have, by reducing to a common denominator, 24 X 8 _ 32 X 6 6X8 ~ 6X8 210 RATIO AND PROPORTION. But since the fractions are equal, and have the same de- nominators, their numerators must be equal, viz. : 24 X 8 = 32 X 6 ; that is, In any proportion, the product of the extremes is equal to the product of the means. Thus, in the proportions, j 1 : 8 : : 2 : 16 ; we have 1 x 16 = 2x8. 4 : 12 : : 8 : 24 ; " " 4 X 24 = 12 X 8. 239. Since, in any proportion, the product of the extremes is equal to the product of the means, it follows that, 1st. If the product of the means he divided by one of the extremes, the quotient will he the other extreme. Thus, in the proportion, 4 : 16 : : 6 : 24, and 4 x 24 = 16 x 6 = 96 ; then, if 96, the product of the means, be divided by one of the extremes, 4, the quotient will be the other extreme, 24 ; or, if the product be divided by 24, the quotient will be 4. 2d. If the product of the extremes he divided hy either of the means, the quotient will he the other mean. Thus, if 4 X 24 = 16 X 6 = 96 be divided by 16, the quotient will be 6 ; or if it be divided by 6, the quotient will be 16. Note. — Wc shall denote the required term of a proportion by the letter x. 235. When the antecedent is less than the consequent, what does the ratio express? What does it express when the antecedent is greater than the consequent ? — 336. To what is the consequent e(iual, in any ratio? To what is the antecedent equal? — 237. What is a simple proportion? — 238. Which are the extremes of a proportion? Which the means? What is the product of the means equal to? — 239, If the product of the means be divided by one of the extremes, what is the quotient? If the product of the extremes be divided by one of the means, what is the quotient? KATIO AND FKOPORTIOX. 211 Examples. Find the required term in each of the following examples : 5 30 9 X 5 $45 i nV 10 12 X X X. 36. 27. ^^• 5, 10, ami lii The first three 'terras of a proportion are vhat is the fourth terra? G. The first three terms of a proportion are 6, 24, and 14 what is the fourth term? 7. The first, second, and fourth terms of a proportion are W, 12, and 16: what is the third term? 8. The first, third, and fourth terms of a proportion are 1 6, 8, and 20 : what is the second term ? DIRECT AND INVERSE PROPORTION. 240. It often happens, that two numbers which are cora- })ured toj^ethcr, may undergo certain changes of value, in whicii case they represent variable and not Jixed quantities. Thus, when we say that the amount of work done, in a single day, will be proportional to the number of men employed, we mean, that if we increase the number of men, the amount of work done will also be increased; or, if we diminish the number of men, the work done will also be diminished. This is called Dii-ect Proportion. If we say that a barrel of flour will serve 12 men a certain time, and ask how long it will serve 24 men, the time will be less : that is, the time will decrease as the number of mea is increased, and will increase as the number of men is d^- creased. This is called, Inverse Proportion; hence, 1. Two numbers are directly projoortional, ichen they in- crease or decrease together; in which case their ratio is always the same. k 212 RATIO AND PItOPORTlON. 2. Two numbers are inversely proportional, when one increases as the other decreases ; in which case their product is always the same. Note. — This is sometimes called, Reciprocal Proportion. First Illustration. If we refer to the numeration table of integral and decimal num oers (Art. 190), we see that* the unit of the first place, at the left of 1, is 1 ten; that is, a number ten times as great as 1. Tlie unit of the first decimal place at the right, is 1 tenth, a number on]v one-tenth of 1. The unit of the second place, at the left, is one hundred times as great as 1 ; while the unit of the second place, at the right, is only one hundredth of 1 ; and similarly for all other corresponding places. Hence, The units of place, taken at equal distances from the m4t 1, are inversely proportional. Second Illustration. The floor of a room is 20 feet long: what must be its bread'Ji in order that it may contain 360 square feet? Analysis. — The length of the floor, operation. multiplied by its breadth, will give the 360 _ area or contents; hence, the area, di- 20 ~ 18 it. Dreaclia. vided by the length, will give the breadth. If the contents remain the same, the length will incrodse as the breadth diminishes; and the reverse. Hence, when the con- tents are the same^ the length and breadth are inversely proportional. COMPOUND PEOPOETION. 241. A Compound Proportion is the comparison of the erras of two equal ratios, when one or both are compound. Thus, ^l : ^l :: 5 : 6; Or. :( ■■ i\ ■■ i\ ■■ :i RATIO AND PROPORTION. 213 Any compound proportion may be reduced to a simple one, by multiplying the elements of each term together ; thus, by multiplying together the elements of the last proportion, we have, 20 r 24 : : 15 : 18. Dcnce, in any compound proportion, ^Tlie product of the extremes is equal to the product of the means ; and the r-equired term may he found as in Art. 239. What are the required terms in the following proportions ? 1. 3 X 9 : 12 X 6 : : 15 : a:. 2. 5 X 9 : 10 X 9 : : 18 : a:. 242. If an element is unknown, denote it by x. Then, if all the parts arc known except one element, as in the follow- ing proportion, I = ^cl == ?| = l\ that element is equal to the product of the means divided by the product of all the elements of the first term and the known elemerds of the fourth term ; thus, 5x6x3xT 2X 21 X5 1. What is the required clement in the proportion, 3J 5J 1 2^ : 3> :: 3 5) 8) 2 'I X) 240. When are two numbers directly proportional? When are two numbers inversely proportional ? What is then said of their product ? Give the first illustration of inverse proportion.. Give the second. — 241. What is a compound proportion? What is the product of the extromea equal to? — 242. ITow do you find the unknown element? 214 SINGLE RULE OF THREE. SINGLE BULE OF THREE. 243. The Single Rule of Three is the process of finding from three given numbers, a fourth, to which one of them sliall have the same ratio as exists between the other two. 1. If 8 barrels of flour cost $56, what will 9 barrels cost, at the same rate. Note. — We shall denote the required term of the proportion by the letter x. Analysis. — The condition, "at the statement. same rate," requires that the quantity^ lar. tar. $ $ 8 barrels of flour, have the same ratio 8 : 9 : : 56 : a?, to the quantity, 9 barrels, as $56, the 56 X 9 cost of 8 barrels, to x dollars, the cost 3 ~ ^ of 9 barrels. ' Note. — It is plain that 8 barrels of flour will cost less than 9 bar- rels : hence, the 3d term is less than the 4th, and these terms are directly proportional. 2. If 36 men, in 12 days, can do a certain work, in what time will 48 men do the same work? Analysis. — Write the required operation. term, tc, in the 4th place, and the 48 : 36 : : 12 : a?, term 12, having the same unit value, 36 X 12 in the third place. ^ = 43 = ^ days. Then, analyzing the question, we see that 48 meu will do the work in a less time than 86 Ta SINGLE RULPJ OF THREE. 6. If 120 sheep yield 330 pounds of wool, bow many pounds will 36 sheep yield? I. If 80 yards of cloth cost $340, what will 650 yards cost ? 8. What is the value of 4 cwt. of sugar, at 5 cents a pound ? 9. If 6 gallons of molasses cost $1.95, what will 6 hogs- heads cost? 10. If 16 men consume 560 pounds of bread in a month haw much will 40 men consume? II. If a man travels at the rate of 630 miles in 12 days, how far will he travel in a leap year, Sundays excepted ? 12. If 2 yards of cloth cost 13.25, what will be the cost of 3 pieces, each containing 25 yards ? 13. If 3 yards of cloth cost 18s. New York currency, what will 36 yards cost ? 14. If it requires eight shillings and four pence to buy eight ounces of laudanum, how many ounces can be purchased for Is. 6d. ? 15. If 5 A. IR. 16 P. of land, cost $150.5, what will 125 A. 2R. 20 P. cost? 16. If 13 cwt. 2qr. of sugar cost $129.93, what will be the cost of 9 cwt. ? It. The clothing of a regiment of 750 men cost iE2834 5s.: what will it cost to clothe a regiment of 10500 men? 18. If 3 J yards of cloth will make a coat and vest, when the cloth is IJ yards wide, how much cloth will be needed when it is |- of a yard in width ? 19. If I have a piece of land 16|- rods long and 3 J rods wide, what is the length of another piece that is t rods wide and contains an equal area ? « 20. How many yards of carpeting that is three-fourths of yard wide, will carpet a room 36 feet long and 30 feet in breadth ? 21. If a man can perform a journey in 8 days, walking hours a day, how many days will it require if he walks 10 hours a day ? APPLICATIONS. 217 22. If a family of 15 persons liave provisions for 8 months, Ijy bow many must the family be diminished that the provisions may last 2 years? 23. A garrison of 4600 men has provisions for G months : to what number must the garrison be dimmished that the pro- visions may last 2 years and 6 months? 24. A certain amount of provisions will subsist an army o* 9000 men for 90 days : if the army be increased by GO 00, DOW long will the same provisions subsist it? 25. If 3 yd. 2qr. of cloth cost $15.t5, how much will 8 yd. 3qr. of the same cloth cost? 26. If .5 of a house cost $201.5, what will .95 cost? 21. What will 26.25 bushels of wheat cost, if 3.5 bushels cost 18.40? 28. If the transportation of 2.5 tons of goods 2.8 miles costs $1.80, what is that per cwt. ? 29. If f of a yard of cloth cost $2.16, what will be the cost of 5^ pieces, each containing 44 T yards ? 30. If f of an ounce cost $-}J, what will l|oz. cost? 31. What will be the cost of 16|lb. of sugar, if 14|ilb. cost $lf ? 32. If $19i- will buy 14 J yards of cloth, how much will 89f yards cost ? 33. If |- of a barrel of cider cost ^j of a dollar, what will ]^ of a baiTcl cost? 34. If y\ of a ship cost $2880, what will Jf of her cost? 35. What will 1161 yards of cloth cost, if 462 yards cost $150.66? 36. If 6 men and 3 boys can do a piece of work in 330 days, how long will it take 9 men and 4 boys to do the same work, under the supposition that each boy does half as much as a man? 37. If 4 men can do a piece of work in 80 days, how many days will 16 men require to do the same work? 10 21$ SINGLE nULE OF THREE. 38. If 21 sappers make a trench in 18 clays, how many days will 7 men require to make a similar trench ? 39. A certain piece of grass was to be mowed by 20 men in 6 days ; one-half the workmen being called away, it is required to find in what time the remainder will complete the work ? 40. If a field of gram be cut by 10 men in 12 days, in liow many days would 20 men have cut it ? 41. If 90 barrels of flour will subsist 100 men for 120 days, how long will they subsist 75 ? 42. If a traveller perform a journey in 35.5 days, when the days are 13.566 hours long, in how many days of 11.9 hours, will he perform the same journey ? 43. If 50 persons consume 600 bushels of wheat in a year, how long would they last 5 persons I 44. A certain work can be done in 12 days, by working 4 hours each day : how many days would it require to do the same work, by working 9 hours a day ? 45. If 1j\ barrels of fish cost $31J, what will 32J bar- rels cost ? 46. How much wheat can be bought for $96|-, if 2 bu. Ipk. cost $1.93-1? 47. If I of a yard of cloth cost 8 If, what will 7-J yards cost ? 48. What will be the cost of 37.05 square yards of pave- ment, if 47.5 yards cost $72.25 ? 49. If 3 paces or common steps be equal to 2 yards, how many yards will 160 paces make ? 60. If a person pays half a guinea a week for his board, how long can he board for ^£21? 51. If 12 dozen copies of a certain book cost $54.72 what will 297 copies cost at the same rate ? 52. If an army of 900 men require $3618 worth of pro- visions for 90 days, what will be the cost of subsistenoe. AITLICATIONS. 219 for the same time, when the aiiny is increased to 4500 men ? 53. A grocer bouglit a hogshead of rum for 80 cents a gallon, and after adding water sold it for 60 cents a gallon, when he found that the selling and buying prices were pro- portional to the original quantity and the mixture : how much water did he add? 54. A man failing in business, pays 60 cents for every dollar which he owes ; he owes A 13570, and B $1875 : how much docs he pay to each? 55. A bankrupt's effects amount to $2328.75, his debts amount to $3726 : what will his creditors receive on a dol- lar? 56. If a person drinks 80 bottles of wine in 3 months of 30 days each, how much does he drink in a week ? 57. If 4f yards of cloth cost 14s. 8d. New York currency, what will 40A yards cost? 58. If a grocer uses a false balance, giving only 14' oz. for a pound, how much will 154|-lb. of just weight give, when weighed by the false balance? 59. If a dealer in liquors uses a gallon measure which is too small by ^ of a pint, what will be the true measure of 100 of the false gallons ? 00. After A has travelled 96 miles on a journey, B sets out to overtake him, and travels 23 miles as often as A travels 19 miles : how far will B travel before he over- takes A ? 61. A person owning |- of a coal mine, sold | of his share for $9345 : what was the value of the whole mine ? 62. At what time, between 6 and 7 o'clock, will the our and minute hands of a clock be exactly together ? 63. If a staff, 5 feet long, casts a shadow of 7 feet, what is the height of a steeple, whose shadow is 196 feet, at the same time of day? 220 SINGLE RULE OF THREE. 64. A can do a piece of work in 3 days, B in 4 days, and C in 6 days : in what time will they do it, working together ? 65. A can build a wall in 15 days, but with the assistance of C, he can do it m 9 days : in what time can C do it alone ? 66. If 120 men can build J mile of wall in 151 days, how many men would it require to build the same wall in 40| days ? 61. If 3 horses, or 5 colts, eat a certain quantity of oats in 40 days, in what time will t horses and 3 colts consume the same quantity? 68. If a person can perform a journey in 24 days of 10 J hours each, in what time can he perform the same journey, when the days are 12 J hours long? 69. A piece of land, 40 rods long and 4 rods wide, is equiv- alent to an acre : what is the breadth of a piece 15 rods long that is equivalent to an acre? 70. If a person travelling 12 hours a day finishes one-half of a journey in ten days, in what time will he finish the remain- ing half, travelling 9 hours a day? 11. How many pounds weight can be carried 20 miles, for the same money that 4^ cwt. can be earned 36 miles ? 12.' If 12 liorses eat a certain quantity of hay in 1J weeks, how many horses will consume the same in 90 weeks ? 13. A watch, which is 10 minutes too fast at 12 o'clock, on Monday, gains 3 min. 10 sec. per day: what will be the time, by the watch, at a quarter-past ten in the morning of the fol- lowing Saturday? 74. Two persons, A and B, are on the opposite sides of a wood, which is 536 yards in circumference ; they begin to travel in the same direction at the same moment ; A goes at the rate of 11 yards per minute, and B at the rate of 34 yards in 3 minutes : how many times must A go round the wood before he is overtaken by B ? DOUBLE RULE OF THREE. 221 DOUBLE RULE OF THREE. 244. The Double Rule of Three is an application of tht principles of Compound Proportion. 1. If 8 men iu 12 days can build 80 rods of wall, how iiuch will 6 men build in 18 days? 8 12 STATEMENT. 6 18 } ■■ d\ ': 80 OPERATION. 18 X 6 X 80 12x8 Analysis. — We write the required term in the 4th place, and the 80 rods in the 3d. Then, since the wall built IS directly proportional to the number of men multiplied by x =: ^^ ,^^ ^ ^ ^^ _ qq I'ods. the number of days, 6 x 18 is written in the second place, and the remaining term in the first place. 2. If 20 men can perform a piece of work in 12 days, work- ing 9 hours a day (that is, in 108 hours), how many men will accomplish the same work in 6 days, working 10 hours a day (that is, in 60 hours)? 8TATKMENT. [ -'11 - 20 X. X = OPKRATION. 12 x 9 X 20 Tx 10 = 36 men. Analysis. — Write the required term, x, in the 4th place, and 20 men, having the same unit, in the 3d place. Since 20 men require 108 hours to do the work, more men will be required to do the same work in 60 hours ; therefore, tlie terms named in connection with each other, are inversely proportional: hence, 6 X 10 = 60, must be written in the first place. 3. If 24 men, in 6 days, working t hour? a day, can build a wall 115 feet long, 3 feet thick, and 4 feei ^igh, how long a wall can 36 men build in 12 days, working 14 hours a day, if the wall is 4 feet thick and 5 feet high. 244. What is the double rule of three? 222 DOUBLE RULE OF THREE. Analysis.— In this STATEMENT. example, an element, 24) 3C ) 115) x) viz., length of wall 6 - : 12 :: 3 : 4[ is required. This el- U) i) 5 ) ement, denoted by cc, OPERATION. is put in the 4th place with the other 36 X 12 X 14 X 115 X 3 X 4 Jb 24x6x1x4x5 elements composing the 4th term. = 414 Rule. I. Write the required term, or the term containing the re- quired element, in the Uh place, and the term having the same unit value in the third place : II. Then analyze the question, and see whether the terms named in connection with each other are directly or inversely proportional : when directly proportional, write the term named in connection with the 4th term, in the 2d place ; and when inversely, ivrite it in the first place: then fi)id the required term or clement (Art. 242). Examples. 1. If 2 men can dig 125 rods of ditch in 75 days, in how many days can 18 men dig 243 rods ? 2. If 400 soldiers consume 5 barrels of flour in 12 days, how many soldiers will consume 15 barrels in 2 days ? 8. If a person can travel 120 miles in 12 days of 8 hours each, how far will he travel in 15 days of 10 hours each ? 4. If a pasture of 16 acres will feed 6 horses for 4 months Low many acres will feed 12 horses for 9 months? 5. If 60 bu.^uels of oats will feed 24 horses 40 days, how long will 30 bushels feed 48 horses ? 6. If 82 men build a wall 36 feet long, 8 feet high, und 4 feet thick, in 4 days; in what time will 48 men build a wail 864 feet long, 6 feet high, and 3 feet wide ? APPLICATIONS. 223 7. If the freiglit of 80 tierces of sugar, each weighiug 3J hundredweight, for 150 miles, is $84, what must be paid for tlie freight of 30 hogsheads of sugar, each weighing 12 hundredweight, for 50 miles ? 8. A family consisting of G persons, usually drink 15.6 gal- ons of beer in a week : how much will they drink in 12.5 weeks, if the number be increased to 9 ? 9. If 12 tailors in 7 days can finish 14 suits of clothes, how many tailors in 19 days can finish the clothes of a regi- ment of 494 men ? 10. If a garrison of 3600 men eat a certam quantity of bread in 35 days, at 24 ounces per day to each man, how many men, at the rate of 14 ounces per day, will eat twice as much in 45 days ? 11. A company of 100 men drank iB20 worth of wine at 2s. 6d. per bottle : how many men, at the same rate, will £1 worth supply, when wine is worth Is. 9d. per bottle? 12. A garrison of 3600 men has just bread enough to allow 24 oz. a day to each man for 34 days ; but a siege coming on, the garrison was reinforced to the number of 4800 men : how many ounces of bread a day must each man be allowed, to hold out 45 days again«t the enemy? 13. Bought 5000 planks, 15 feet long and 2| inches thick ; how many planks are they equivalent to, of 12^ feet long and 1 f inches thick ? 14. If 12 pieces of cannon, eighteen-pounders, can batter down a castle in 3 hours, in what time would nine twenty-four- pounders batter down the same castle, both pieces of cannon bemg fired the same number of tunes, and their balls flying with the same Telocity ? 15. If the wages of 13 men for 7 J days, be $149.76, what will be the wages of 20 men for 15J days? 16. If a footman travel 264 miles m 6f days of 12 J hours 224 DOUBLE rulp: of three. each, in how many days of lOf hours each will he travel 129f miles ? It. If 120 men in 3 days, of 12 hours each, can dig a a trench of 30 yards long, 2 feet broad, and 4 feet deep, how many men would be required to dig a trench, 50 yards long, () feet deep, and IJ yards broad, in 9 days of 15 hours each? 18. If a stream of water running into a pond of 115 acres, raises it 10 inches in 15 hours, how much would a pond of 80 acres be raised by the same stream in 9 hours? 19. A person having a journey of 500 miles to perform, walks 200 miles in 8 days, walking 12 hours a day : in how many days, walking 10 hours a day, will he complete the re- mainder of the journey ? 20. If 1000 men, besieged in a town, with provisions for 28 days, at the rate of 18 ounces per day for each man, be rein- forced by 600 men, how many ounces a day must each man have that the provisions may last them 42 days ? 21. If a bar of iron 5 ft. long, 2 J in. wide, and If in. thick, weigh 451b., how much will a bar of the same metal weigh that is t ft. long, 3 in. wide, and 2^ in. thick ? 22. If 5 compositors in 16 days, working 14 hours a day, can compose 20 sheets of 24 pages each, 50 lines in a page, and 40 letters in a line, in how many days, working T hours a day, can 10 compositors compose 40 sheets of 16 pages in a sheet, 60 lines in a page, and 50 letters in a line? 23. Fifty thousand bricks are to be removed a given dis- tance in 10 days. Twelve horses can remove 18000 in 6 days : how many horses can remove the remainder in 4 days? 24. If 248 men, in 5 J days of 11 hours each, dig a trench of t degrees of hardness, 232 1 yards long, of wide, and 2 J deep, in how many days, of 9 hours long, will 24 men dig a trench of 4 degrees of hardness, 33tJ yarc's long, 5f wide, and 3i deep? PARTNERSHIP. 225 PARTNERSHIP. 245. A Partnership, or Firm, is an association of two or more persons, under an agreement to share the profits and 'osscs of business. Partners arc the persons thus associated. 246. Capitai, or Stock, is the amount of money or prop- erty contributed by the partners, and used in the business. Profit is the increase of capital between two given dates. Loss is the decrease of capital between two given dates. Dividend is the amount of profit ajiportioned to each partner. 247. Assets of a Firm, are its cash on hand, property, and all debts due to it. 248. Liabilities of a Firm, embrace all the debts wliich it owes, and all its indorsements. 249. Solvency is when the assets exceed the liabilities. 250. Insolvency is when the liabilities exceed the assets. 25 L An Assignment is a transfer of the assets of an in- solvent person or firm to others, for the benefit of creditors. 252. Assignees are the persons to whom such transfer is made. 253. When the capital of each partner is employed for tha same time. Since the profit arises from the use of the capital, each man's share of it should be proportional to his amount of stock. Hence, 245. What is a partnership or firm? Wliat are partners? — 246. Wliat Is canitiil or stock? What is profit? What is loss? What is a div- idend?— 247. What arc assets?— 248. What arc liabilities ?— 249. What is solvency? — 250. What is insolvency? — 2ol. What is an assignment? 252. What are assignees ?— 253. What is the rule, when each man's capital is employed for the same time? io* 226 PARTNERSHIP. Rule. — As the wHiole stock is to each man^s stock, so is the whole gain or loss to each mail's share of the gain or loss. Examples. 1. Mr. Jones and Mr. Wilson form a copartnership, the former putting in $1250, and the latter $750 : at the end oi the year there is a profit of $720: what is the share of each? 1250 750 STATKMENT. 2000 : 1250 : : 720 : x = Jones' share = $450. 2000 : 750 : : 720 : x = Wilson's share = $270. OPERATION. 25 18 lt^0 X '^'^0 ^000 ^0 = x= $450. 15 18 — ^00- = ^=^^^'^• ^0 2. A, B, and C, entered into partnership with a capital of $7500, of which A put in $2500, B put in $3000, and C put in the remainder ; at the end of the year their gain was $3000 : what was each one's share of it ? 3. A and B have a joint stock of $4200, of which A owns $3600, and B, $600 ; they gain, in one year, $2000 : what is each one's share of the profits ? 4. A, B, C, and D, have $40000 in trade, each an equal share ; at the end of six months their profits amount to $16000 : what is each one's share, allowing A to receive $50, and D, $30, out of the profits, for extra services ? 5. Three merchants loaded a vessel with flour ; A loaded 50i barrels, B, 700 barrels, and C, 1000 barrels ; in a storm a sea it became necessary to throw overboard 440 barrels : what was each one's share of the loss ? 6. A man bequeathed his estate to his four sons, in the fol- lowing manner, viz. : to his first, $5000, to his second, $4500, APPLICATIONS. 227 to bis third, $4500, aucl to his fourth, $4000. But on settling the estate, it was found that after paying the debts and ex- penses, only $12000 remained to be divided: how much should each receive ? 7. A widow and her two sons receive a legacy of $4500, of wliich the widow is to have ^, and the sons, each \. But the elder son dying, the whole is to be divided in the sani proportion between the mother and younger son : what will each receive? 8. Four persons engage jointly in a land speculation ; D puts in $5499 capital. They gain $15000, of which A takes -1320.50, B, $5245.75, and C, $3600.75 : how much capital did A, B, and C put in, and what is D's share of the gain? 9. A steam-mill, valued at $4300, was entirely destroyed by fire. A owned -J- oC it, B i, and C the remainder ; supposing it to have been insured for $2500, what was each one's share of the loss ? 10. A copartnership is formed with a joint capital of •^10970. A puts in $5 as often as B puts in $7, and as oi'ten as C puts in $8 ; their annual gain is equal to C's stock : what is each person's stock and gain ? 11. A man failing in business is indebted to A, $475.50, to B, $362.12^, to C, $250.87J, and to D, $140. He is worth only $014.25 : to how much is each entitled ? 12. Four persons. A, B, C, and D, agreed to do a piece of work for $270. They were to do the work in the pro- portions of I, -J, J, and J J : what should each receive for his work ? 13. A, B, and C, form a copartnership, with a capital Oi $50000, of which A puts in $18500, B, $24650, and C, the remainder. C, on account of his superior knowledge of the business, was to receive -^^ of all the profits, exclusive of his share. At the end of the year, tlie net profit is $7300 ; what should each receive ? 228 PARTNERSHIP. 14. Two merchants, A and B, form a copartnership. A contributes $10500, and B, $16500. At the end of the year, the assets are $29400, and the liabihties $4750. Now, sup- posing the partnership to continue, with what capital does each partner commence the new year ? 15. Three persons buy a. piece of land for $45G9, and the >arts for which they pay bear the following proportions to each other, viz. : the sum of the first and second, the sum of the first and third, and the sum of the second and third, are to each other as i |, and -^q : how much did each pay, and what part did each own? 254. When the capital is employed for unequal times. When the partners employ their capital for unequal times, the profits of each will depend on two circumstances : 1st, On the amount of capital he puts in ; and 2dly, On the length of time it is continued in business: Therefore, the profit of each will depend on the product of these two elements. The whole profit will be proportional to the sum of these products. Hence, the following 'Rule.—3Iitltiply each man^s cajntal by the time he con- tinued it in the firm ; then say, the sum of the 2Jroducts is to each product, so is the whole gain or loss to each man^s share. Examples. 1. A put in trade $500 for 4 months, and B $600 for 5 months. They gained $240 : what was the share of each ? OPERATION. A^s cap. 500 X 4 = 2000 B's cap. 600 X 5 = 3000 Sum of products = 5000 : 2000 : : 240 : ^ = $96, A's 5000 : 3000 : : 240 : :p = $144, B's. 2. Three men hire a pasture for $70.20 : A put in 7 horses APPLICATIONS 229 for 3 months ; B, 9 horses for 5 months ; and C, 4 horses for 6 months : what part of the rent should each pay ? 3. A commenced business with a capital of $10000. Four months afterwards B entered into partnership with him, and put in 1500 barrels of flour. At the close of the year their profits were $5100, of which B was entitled to $2100 : what was the value of the flour per barrel? 4 Ou the 1st of January, 1864, A commenced business with a capital of $23000 ; two months afterwards he drew out $1800 ; on the 1st of April, B entered into partnership with him, and put in $13500 ; four months afterwards he drew out $10000 ; at the end of the year their profits were $8400 : how much should each receive ? 5. Three persons divided theu* profits to the amount of $798. A put out $4000 for 12 months; B, $3000 for 15 mouths ; and C, $5000 for 8 months : to what part of the profits was each entitled? 6. Tliree persons, C, D, and E, form a copartnership ; C's stock is in trade 3 months, and he claims ^^ of the gain ; D's stock is in 9 months ; and E put in $756 for 4 months, and claims i of the profits : how much did C and D put in ? 7. Two persons form a partnership for one year and six months. A, at first, put in $3000 for 9 months, and then '$1000 more. B, at first, put in $4000, and at the end of the first year, $500 more j but at the end of 15 months, he drew out $2000. At the end of 12 months, C was admitted as a partner with $7333^. The gain was $7400 : how much should each man receive ? 8. Three men take an interest in a mining company. A ])ut in $480 for 6 months; B, a sum not named for 12 months ; and C, $320 for a time not named : when the accounts were settled, A received $600 for his stock and in-ofits ; B, $1200 for his ; and C, $520 for'his : what was B' stock, and C's time ? 254. What is the rule when the capital is employed for unequal time*. :30 PERCENTAGE. PERCENTAGE. 255. Per cent, means by the hundred. Thus, 1 per cent, of a number is one-hundredth of it ; 2 per cent, is two-hun- drcdths of it ; 3 per cent, three-hundredths, &c. 256. The Rate per cent, is the number of hundredths taken. . Thus, if i hundredth is taken, the rate is 1 per cent. ; if 2 hundredths are taken, the rate is 2 per cent.; if 3 hundredths, the rate is 3 per cent., &c. 257. The Base is the number whose part is taken. 258. The Percentage is the result of the operation, and is the part of the base taken. The rate per cent, is generally expressed decimally ; thus, i per cent, of a number, is j-Jq of it = .01 of it. 2 per cent, of a number, is j^^ of it = .02 of it. 25 per cent, of a number, is ^^^ of it = .25 of it. 50 per cent, of a number, is -^q^j of it = .50 of it. 100 per cent of a number, is ^^J of it = 1 time it. 200 per cent of a number, is f gj of it = 2 times it. -J per cent, of a number, is jg^ of it = .005 of it. f per cent, of a number, is j^q of it = .0015 of it. .75 per cent, of a number, is ^ of it — .0075 of it. .S^ per cent, of a number, is j^^'^ of it = .0085 of it. Note. — Per cent, is often expressed by the character %. Thus 5 per cent, is written 5 % ; 8 per cent., 8 %• Write, decimally, 5%; 8%; 151%; 100%; 204%; 3271-% 672.3 Vo 4 49%; and 507.5%. 255. What is the meaning of per cent.? What is 5 hundredths of a number? — 256. What is the rate per cent.? If four hundredtlis of a number is taken, what is the rate? — 257. What is the base? — 258. What is the percentage? How is the rate per cent, generally ex pressed ? FEKCKNTAGE. 231 259. To find the percentage, when the base and rate are known. 1. What is the percentage of $450, the rate being 6 per cent. ? Analysis. — The rate, expressed decimally, is operation 00. The percentage is, therefore, six hundredths 450 of the base, or the product of the base and .06 rate. Hence, to find the percentage of any ^27.00 Ans nmnber: Rule. — Mulfvply the base by the rate, and the product will he the percentage. Examples. Find the percentage of the 1. 4 per cent, of §1256. 2. 12% of $956.50. 3. I per cent, of 475 yards. of 324.5 cwt. JVo of 125.25 lbs. If per cent, of 750 bush. 4^Vo of $2000. 9 per cent, of 186 miles. 9. lOJ per cent, of 460 sheep. 10. 5j^Q per cent, of 540 tons. 11. 82- per cent, of $3465.75. following numbers : 12. 12^% of 126 cows. 13. 50 per cent, of 320 bales. 14. 37^ per cent, of 1275 yds. 15. 95 7o of $4573. 16. 105 per cent, of 2500 bar. 17. 112i% of $4573. 18. 250 per cent, of $5000. 19. 305% of $1267.871 20. 500 per cent, of $3000. 21. What is 3% of $765? 22. What is 4^ % of 960 bush. ? •:;; What is the difference between 4f "/« of $1000 and 7 J ,..'r cent, of $1500? 24. If I buy 895 gallons of molasses, and lose 17 per cent. by leakage, how much have I left? 25. A grocer purchased 250 boxes of oranges, and found (li;i' lie had lost in bad ones 18 per cent. : how many full boxes d good ones had he left ? 259. How do you find the percentage from tho base and raio? 232 PERCENTAGE. 260. Parts of percentage. There are three part? in percentage : 1st. The Base ; 2d. The Rate ; and 3d. The Percentage. 26 1 . To find the rate, when the base and percentage are known. I. What per cent, of $64 is 116? or, $16 is what part of $64? Analysis. — In this example, 16 is the opeuation. percentage, and 64 the base, and the rate is 1 6 — i — 25^ or required. Since the percentage is equal to 25 per cent, the base multiplied bj the rate, the rate is equal to the quotient of the percentage divided by the base. Rule. — Divide the percentage hy the base, and the first two decimal places ivill express the rate. Examples. 1. What per cent, of 10 dollars is 2 dollars ? 2. What per cent, of 32 dollars is 4 dollars ? 3. What per cent, of 40 pounds is 3 pounds ? 4. Seventeen bushels is what per cent, of 125 bushels ? 5. Thirty-six tons is what per cent, of 144 tons ? 6. What per cent, is $84 of $96 ? T. What per cent, is J of -I ? 8. What per cent, is 3 miles of 400 miles ? 9. Four and one-third is what per cent, of 9J ? 10. One hundred and four sheep is what per cent, of a drove of 312 sheep? II. A grocer has $325, and purchases sugar to the amount of $12 1.87 J : what per cent, of his money does he expend? 12. Out of a bin containing 450 bushels of oats, 56 J bushels were sold : what per cent, is this of the whole ? Note. — If the base be regarded as a single tiling, and denoted by 1, a fractional percentage expressed decimally will denote the rate. 2G0. How many parts are there in percentage? What are tliey? 2C1. How do you find the rate, ^om the base and percentage? PERCENTAGE. 233 13. J of a number is what per cent, of the number ? 14. J of a sliip is what per cent, of the ship ? 15. ^Q of 50 is what per cent, of 50? 16. f of a cargo is what per cent, of it? n. 1| times a number is what per cent, of the number? 262. To find the base, when the rate and percentage are known. 1. Of what number is $960, 16 per cent.? Analysis. — By Art. 259, the percentage , , ,.,.-,, r OPERATION. IS equal to tlie base multiplied by the -,^-, ,_ »„„_ rate ; hence, to find the base, Rule. — Divide the percentage hy the rate, expressed deci- molly. Examples. 2. The number 475 is 25% of what number? 3. The number 87J is 12^ 7o of what number? 4. Five hundred and sixty dollars is 140% of what number? 5. The number 75 is J°/o of what number? 6. One dollar and twenty-five cents is -J % of what number ? 7. The fraction | is 45% of what number? 8. The fraction } is f % of what number? 9 If a person receives $5850, and that sum is 75 Vo of what is due him, what is the debt ? 10. A bankrupt can pay only 37^ per cent, of his debts : what did he owe to that merchant to whom he paid $1647 ? 11. In an army, 15600 men are mustered after a battle, in Avhich 25 % were killed and wounded : what was the original number of men ? 263. Amount is the percentage plus the base. 264. Difference is the percentage minus the base. 262. How do you find the base, when the rate and percentage are known ?— 26a. What is the amount?— 264. ^^^lat is tlie difierence? 230 rEKCENTAGE. 2. Mr. Wilson lost 18 per cent, of Lis sheep by disease, and had a flock of 615 left : how many had he at first ? 3. Mr. Jones invests 46''/o of his capital in land, and has $0 1 3 left : what is his capital ? 4. An army fought two battles j in the first it lost 15 per cent., and in the second, 20 per cent, of the original number ; fter which it mustered 19500 men : what was its original strength ? 5. A grocer bought a quantity of provisions, but finding them damaged, sold them at a loss of 19 per cent., and re- ceived $10935 : what did they cost him ? 6. A son, who inherited a fortune, spent 37 1 per cent, of it, when he found that he had only 131250 remaining : what was the amount of his fortune ? t. A grocer purchased a lot of teas and sugar, on which he lost 16 per cent, by selling them for $4200 : what did he pay for the goods ? 8. A speculator invested in stocks, which, falling rapidly in price, he sold out at a loss of 13 per cent., and received $2262 : what was the amount of his purchase ? 267. Formulas of percentage. Nearly every practical question, in Arithmetic, is a partic- ular case of one or other of the five operations of Percentage : hence, we write the formulas : 1. Percentage = Base X Rate Art. 259. 2. Rate = Z^'ES! Art. 261. Base o -r. Percentaore , , ^^^ 3. Base = — ^ ^ ° Art. 262. Kate A -o Amount . .. o/»r 4 Base = =r— - Art. 265. 1 4- Bate e -o Difference . , ^^^ 6. Base = :. =i— - - - - - Art. 266. 1 — Kate PKOFIT AND LOSS. 237 PROFIT AND LOSS. 268. Profit and Loss are commercial terms, indicating gain or loss in busiiicss transactions. The gain or loss is always estimated on the cost price. The cost of an article is the amount paid for it. The selling price of an article is the amount received for it. Tlie cost is the base ; the gain or loss is the percentage ; the rate per cent, of gain or loss is the rate; the selUng price is the sum of the base and percentage, when there is a gain, and their difference when there is a loss. The following examples may all be wrought by the five for- umlas and rules of Percentage : Examples. 1. Bought 9 barrels of sugar, each weighing 250 pounds, at 7 cents a pound : how much profit would be made if it were sold at 8 1 cents per pound? 2. If 15 pieces of muslin, each containing 43 yards, cost 2t cents per yard, what would be the gain if sold at 31i cents per yard? 3. A farmer bought a flock of 360 sheep ; their keeping for 1 year cost $0.75 a head ; their wool was worth 1 dollar and 25 cents a head, and one-fourth of them had lambs, each of which was worth one-half as much as a fleece : what was the profit of the purchase at the end of the year? 4. A merchant bought 65 barrels of flour, at $5J per barrel, and sold them so as to gain $42.50 : what was the price per barrel ? 5: A person bought 500 bushels of potatoes, at 62^ cents 2G8 What do you understand by the terras profit and loss? What is cost? What is the selling price? What is the base? What is the gain or loss? 238 PERCENTAGE. per bushel, and sold them so as to gain $35 : at what price were they sold? 6. A house and lot were bought for $6450. The house was repaired at an expense of $5t5, painted for $796, and was then sold so as to gain $945 : what was the price of the house and lot? T. If in 3 hogsheads of molasses, which cost $68.04, ono third leaked out, what must the remainder be sold for per gallon to realize a profit of $2.52 on the whole? 8. If a merchant's profits are 22 per cent, on the cost of the goods sold, what is his profit on $4162.50 ? 9. A quantity of goods were bought for $3612 ; the charges on them were $54 : they were sold at an advance of 20 % ; what was the profit? 10. A merchant, on taking an inventory of stock, finds it worth $37649 : what would be his profit on this stock, if sold at an advance of 31|-Vo? 11. A merchant bought goods to the amount of $2965 ; but being damaged, he sold them at a loss of 15% : what was the amount of his loss? 12. A quantity of flour was bought for $8550 ; J^ of the flour was so damaged as to be sold at a loss of 12"/o ; -^ of it was sold at a profit of 19%; and the remainder at a profit of 30 % : what was the net profit on the flour ? 13. If sugar costs 10 cents per pound, and is sold at an advance of 12^%, what is the profit per lb.? 14. A bank, whose capital is $200000, after reserving $2860 for a surplus fund, declared a dividend of 8 % on the capital : what were the entire profits ? 15. The profits of a merchant averaged 25% of his capital ; and his expenses are 5 % of his profits : what part of tho capital were the expenses? 16. A farmer sells 375 bushels of corn for 75 cents a bushel j PliOFlT AND LOSS. 23& the piircliaser sells it at au advance of 20 per cent.: how much a bu^^iiel did he receive for the corn? n. A merchant buys a pipe of wine, for which he pays $322.56, and he wishes to sell it at an advance of 25 per cent. : what must he sell it for per gallon ? 18. A man bought 3215 bushels of wheat, for which he paid $3493.33J, but finding it damaged, is willing to lose 10 per cent. : what must he sell it for per bushel ? 19. If I purchase two lots of laud for $150.25 each, and sell one for 40 per cent, more than it cost, and the other for 28 per cent, less, what is the gain on the two lots ? 20. Bought a cask of molasses containing 144 gallons, at 45 cents a gallon, 36 gallons of which leaked out : at what price per gallon must I sell the remainder to gain 10 per cent, on the cost ? 21. A person in Chicago bought 3500 bushels of wheat, at $1.20 a bushel: allowing 5 per cent, on the cost for risk in transportation, 3 per cent, for freight, and 2 per cent, commis- sion for selling, what must it be sold for per bushel in New York to realize 40 per cent, net profit on the purchase ? 22. Bought a quantity of goods for 1348.50, and sold the same for 8425 : what per cent, did I make on the amount received ? 23. Bought a piece of cotton goods for 6 cents a yard, and sold it for 7 J cents a yard : what was my gain per cent.? 24. If I buy rye for 90 cents a bushel, and sell it for $1.20, and wheat for $1.1 2^ a bushel, and sell it for $1.50 a bushel, uj)on which do I make the most per cent.? 25. If paper that cost $2 a ream, be sold for 18 cents a quire, what is gained per cent. ? 26. How much per cent, would be made upon a hogshead of sugar weighing 13cwt. 3 qr. 141b., that cost $8 per cwt., if sold at 10 cents per pound ? 27. A hardware merchant bought 45 T. 16 cwt. 25 lb. of iron. 2tl:0 PERCENTAGE. at $75 per ton, and sold it for $78.50 per ton : what was his whole gain, and how much per cent, did he make ? 28. A merchant buys 67560 feet of lumber for $7000 : the expense of cartage and piling was |425, and the loss of material amounted to $216. If the lumber be sold at $97.50 per 1000 feet, Avhat will be the entire loss ? 29. A gentleman, having gold coin to the amount of $475, sold it for bank bills and obtained $593.75 : what was the rate per cent, of premium on gold, and what the rate per cent, of depreciation on the bills ? 30. In selHng a quantity of wheat, a merchant gained $500 when his rate of profit was 31%: what was the cost? 31. In the course of 6 months a merchant gained $3745 : what amount of goods must he have sold, allowing a gain of 25 Vo? 32. The net profits of a shoe-dealer were $2965, and his ex penses were $1260. If the rate of profit were 40%, what amount of goods were sold ? 33. What must be the annual sales of a merchant, that he may realize $4500, after paying $2500 expenses, when his rate of profit is 35 % ? 34. The surplus fund of an insurance company, amounting tc $32500, will pay 12^% on its capital: what is the capital? 35. The profits of a bank are 12% of its capital ; the ex- penses are 10% of the profits : what Vo of the capital are the expenses ? 36. A grocer sold a lot of sugars for $477.12, which was aa advance of 12% on the cost: what was the cost? 37. Mr. A. bought a lot of sugars, but finding them of an hiferior qaulity, sold them at a loss of 15%, and found that they brought $340 : what did they cost him ? 38. I sold a parcel of goods for $195.50, on which I made 15%: what did they cost me? PROFIT AND LOSS. 241 39. Sold 18 cwt, 3 qr. 14 lb. of sugar, at 8 cents a pound, and gained 15%: how much did the whole cost? 40. A dealer sold two horses for $412.50 each, and gained on one 35 '/o, but lost 10 Vo on the other: what was the cost of each, and what was his net gain ? 41. A merchant havmg a lot of flour, asked dS^Vo more than it cost him, but was obliged to sell it 12^% less than hi3 asking price : he received $1 per bbl. : what was the cost per bbl.? 42. If a merchant in selling a quantity of merchandise for $3850, loses 12% of the cost, what was the cost? 43. If 25 per cent, be gained on flour when sold at $10 a barrel, what per cent, would be gained when sold at $11.60 a barrel ? Note. — In this class of examples, first find the cost, as in Art. 267 ; then find the gain, or losg; and tlien divide bj the number on which the per cent, is reckoned. 44. A lumber-dealer sold 25650 feet of lumber at $19.20 a thousand, and gained 20 per cent. : how much would he have gained or lost had he sold it at $15 a thousand? 45. A man sold his farm for $3881.25, by which he gamed 12J per cent, on its cost : what was its cost, and what would he have gained or lost per cent, if he had sold it for $3211.60? 46. If a merchant sells tea at 66 cents a pound, and gains 20 per cent., how much would he gain per cent, if he sold it at ^1 cents a pound ? 41. Sold 5520 bushels of corn at 50 cents a bushel, and lost 8 per cent.: how much per cent, would have been gained had it been sold at 60 cents a bushel ? 48. A grocer bought 3 hogsheads of sugar, each weighing 11121 pounds ; he sold it at 11 cents a pound, and gained 31^ per cent. : what was its cost, and for how much should he have sold it to gain 50 per cent, on the cost ? 11 242 PERCENTAGE. COMMISSION. 269. Commission is an allowance made to an agent for a transaction in business, and is reckoned at a certain rate per cent, on tlie amount of money used. 270. A Commission Merchant is one who sells or buys goods for another. 271. A consignment is a quantity of goods sent to a merchant for sale. A consignor is the one who sends the goods. A consignee is the one to whom the goods are sent. Note. — The commission for the purchase or sale of goods, in the city of New York, varies from 2h to 12^ per cent. ; and, under some circumstances, even higher rates are paid. For the sale of real estate the rates are lower, varying from one-quarter to 2 per cent. All the cases of Commission come under the rules and for- mulas of Percentage. 1. A commission merchant sold a lot of goods, for which he received $1540 ; he charged 2^ per cent, commission : what was the amount of his commission, and how much must he pay over? 2. A commission merchant receives $1399.11 to be invested in groceries ; he is to receive 3 per cent, on the amount of the purchase : what amount is laid out in groceries, and what the commission ? 3. An auctioneer sold a house for $3125, and the furniture for $1520 : what was his commission at f per cent. ? 4. A flour merchant sold on commission 150 barrels of flour, at $9.15 a barrel : what was his commission at 2 J- per cent. ? 269. Wliat is commission ? How is it reckoned ? — 270. What is a commission merchant? — 271. What is a consignment? What is a con- Bignor ? ^^^lat is a consignee ? COMMISSION. 243 6. I sold at auction 96 hogsheads of sugar, each weighing 9 cwt. and 50 lb., at $6.50 per hundred : what was the auctioneer's commission at If/o, and to how much was I entitled ? 6. An agent purchased 2340 bushels of wheat at $1.75 a bushel, and charged 2-| per cent, for buying, IJ per cent, fof shipping, and the freiglit cost 2 per cent. : what was his com mission, and what did the wheat cost the owner ? t. A town collector received 4 J per cent, for collecting a tax of $2564.25 : what was the amount of his percentage ? 8. I paid an attorney 6f per cent, for collecting a debt of $7320.25 : how much did I receive ? 9. ^ly commission merchant sold goods to the amount of .4000, on which I allowed him 5 per cent. ; but as he paid over the money before it became due, I allowed him 1 J per cent, more ; how much am I to receive ? 10. A dairyman sent an agent 3476 pounds of cheese, and allowed him 3J per cent, for selling it : how much would he receive after deducting the commission, if it were sold for 12 J cents per pound ? 11. A person has $1500 in bills of the State Bank of In- diana, upon which there is a discount of 2J per cent., and $1000 of the bank of Maryland, upon which there is a discount of 3J- per cent. : what will be the loss in changing the amount into current money? 12. I am obliged to sell $2640 in bills on the bank of Dela- ware, upon which there is a discount of 2f per cent. : how much bankable money should I receive ? 13. A merchant in New York received a consignment Oi 75 bbl. of flour, which he sold at $4.75 per bbl. He charged I commission of 2%, 1% for storage, and f'/o for guarantee \Vhat were the charges, and what amount was transmitted to ibe consignor? Note, — 2 per cent, -f i per cent. + } per cent. = 8 per cent 244 PERCENTAGE. 14. A commission merchant in 'Naw York receives $12000 for tlie purchase of sugar. He charges 2% commission. What amount is laid out in purchasing sugar, and what is the com- mission ? 15. A factor receives 1108.75, and is directed to purchase iron at $45 a ton ; he is to receive 5 per cent, on the money paid : how much iron can he purchase ? 16. I forward $2608.625 to a commission merchant in Chicago, requesting him to purchase a quantity of corn ; he is to receive 2^ per cent, on the purchase : what does his commission amount to, and how much corn can he buy with the remainder, at 56 cents a bushel? n. My agent at Havana purchased for me a quantity of sugar at 6^ cents a pound, for which I allow him a com- mission of IJ per cent. His commission amounts to $42.66 : how many barrels of sugar of 240 pounds each did he purchase, and how much money must I send him to pay for it, including his commission? 18. A merchant in New Orleans received $187.50, to be laid out in the purchase of cotton. After allowing for commission at 2Vo, freight at i%, insurance at J%, and incidental expenses ■j^qVo, what amount was expended in the purchase of cotton, and what was the commission? 19. A commission merchant, in selling a quantity of mer- chandise for $2*185, received a commission of $60 : what was the rate of commission ? Note. — In this example, the base and percentage are given, and the rate is required (Art. 267). 20. A land agent received $175 for selling a house for $6795 : what was his rate of commission? 21. A collecting agent received $15 for collecting a debt of $175 : what was his rate of commission? 22. A miller received for his toll 5 bushels on every 45 bushels of grain that he ground : what was the rate ? i INTEREST. 245 INTEREST. 272. Interest is a percentage paid for the use of money Principal, or base, is the money on which interest is paid. Rate of interest is the per cent, paid per year. Amount is tlie sum of the principal and interest. Per annum means by the year. 273. In interest, by general custom, a year is reckoned at 12 niunths, each having 30 days. The Rate of Interest is generally lixed by law, and is called Legal Interest. Any rate above the legal rate is usury, and is generally forbidden by law. 274. In most of the States, the legal rate is 6 per cent. ; in New York, South Carolina and Georgia, it is 7 % ; and in some of the other States the rate is fixed as high as 10 per cent. 275. There are five parts in interest : 1st, principal ; 2d, rate ; 3d, time ; 4th, interest ; 5th, amount. CASE I. 276. To find the interest of any principal for one or more years 1. What is the interest of 13920 for 2 years, at 7 per cent. ? Analysis. — In tliis example, the operation. base is $3920, the rate 7%, and the |3920 interest for 1 year is the percentage : q>j xqXq, this product multiplied by 2, the num- . !,er of years, gives the interest for 2 ^^74.40 mt. for 1 year. years; hence, ? ^^' ^^ ^^ars. $548.80 interest. Rule. — Midtiply the principal by the rate, expressed decimally, and the product by the numr ber of years. 246 PERCENTAGE. Examples. 1. What is the interest of $6t5 for 1 year, at 6| per cent.? 2. What is the interest of |8tl.25, for 1 year, at r/o? 3 What is the interest of 1535.50, for 7 years, at 6%? I What is the interest of $1125.885, for 4 years, at 8%? 6. What is the interest of $789.U, for 12 years, at 5% 6. What is the interest of $2500, for 7 years, at 7|Vo? 7. What is the interest of $3153.82, for 2 years, at 4i%? 8. What is the amount of $199.48, for 16 years, at 7%? 9. What is the amount of $897.50, for 3 years, at 8%? 10. What is the interest of $982.35, for 4 years, at 6J%? 11. What is the amount of $1500, for 5 years, at 5i%? 12. Wiiat is the interest of $1914.10, for 6 years, at ^'/o? 13. What is the interest of $350, for 21 years, at 10%? 14. What is the amount of $628.50, for 5 years, at 12i%? 15. What is the amount of $75.50, for 10 years, at 6%? 16. What is the amount of $5040, for 2 years, at 7|%? Note. — Wlien there are years and months, and the months are an aliquot part of a year, multiply the interest for 1 year hy t7i4 years and the montJis, reduced to the fraxXion of a year. 17. What is the interest of $119.48, for 2yrs. 6 mo., at 7Vo? 18. What is the interest of $250.60, for 1 yr. 9 mo., at 6%? 19. What is the interest of $956, for 5yrs. 4 mo., at 9%? 20. What is the amount of $1575.20, for 3yrs. 8 mo., at 7%? 21. What is the amount of $5000, for'2yrs. 3 mo., at 5J%? 22. What is the interest of $1508.20, for 4 yrs. 2 mo., at 10%? 23. What is the interest of $75, for 6 yrs. 10 mo., at 12i% 24. What is the amount of $125, for 5 yrs. 6 mo., at 4f%? 272. What is interest? What is the principal or base? What is rate? What is amount? What is the meaning of per annum? — 273. How is a year reckcmed, in computing interest? How many days are reckoned in a month ? What is legal interest ? What is usury ? INTEREST. 247 CASE II. 277. To find the interest on a given principal for any rate and time. 1. What is the iuterest of $1752.95, at 6 per cent., for 2yrs. 4 mo. and 29da. ? Analysis. — The interest for 1 year is the product of the prin oipal and rate. If the interest for 1 year be divided by 12, the quotient will be the interest for 1 month ; if the interest for 1 month be divided by 30, the quotient will be the interest for 1 day. The interest for 2 years is two times the interest for 1 year ; the Interest for 4 months, 4. times the interest for 1 month; and the interest for 29 days, 29 times the Interest for 1 day. OPERATION. $1752.96 .06 12) 105,1776 int. for lyr. $105.1776 x2 =$210.3552 2a/r. 30) 877648 int. for Imo. 8.7648 X 4 = 35.0592 47no. .29216 int. for Ida. 0.29216 X 29 = 8.47264 29da, Total interest, $253.88704 Hence, we have the following, Rule. I. Find the interest for 1 year: II. Divide this interest by 12, and the quotient will be the interest for 1 month: III. Divide the interest for 1 month by 30, and the quo- tient will be the interest for 1 day : IV. Multiply the interest for 1 year by the member of years, the interest for 1 month by the numbei' of months, and the interest for 1 day by the number of days, and the sum f the products will be the required interest. 274. What are the general rates of legal interest ? — 275. How many parts arc there in interest? What are they? — 276. How do you find the interest of any principal for one or more years? — ^77. How do you find the interest for any rate and time? 248 PERGKKTAGE. Note. — Tliis method of computing interest for days, is the one \»» general use. It supposes the month to contain 30 days, or the year 360 days; whereas, it actually contains 865 days. To find the exact interest for 1 day, we must regard the month as containing ^2^ days = 30j^2 ^^J^ 5 ^^^ *^s is *^® number by which the interest for one month should be divided, in order to find the ''.met interest for one day. As the divisor, commonly used, is too small, lie interest found for 1 day, is a trifle too large. If entire accuracy required, the interest for the days must be diminished by its jIs part = ^3 part. 2d method. 278. There is another rule resulting from liie last analysis which is regarded as the best general method of computing interest. Rule. I. Find the interest for 1 year, and divide it &?/ 12 ; the quotient will he the interest for 1 month: II. Multiply the interest for 1 month by the time expressed in months and decimal parts of a month, and the product will be the required interest. Note. — Since a month is reckoned at 80 days, any number of days is redticed to decim349.998 Richmond, Va., May 1st, 1846 1. For value received, I promise to pay James Wilson, or order, three hundred and forty-nine dollars ninety-nine cents mtil eight mills, with interest, at 6 per cent. James Paywei.l. On tliis note were indorsed the following payments : Dec. 25th, I84C, received $49,998 July 10th, 184T, " 4.998 Sept. 1st, 1848, " 15.008 June 14th, 1849, " 99.999 What was due April 15th, 1850? Principal on interest from May 1st, 1846 - - - $349,998 Interest to Dec. 25th, 1846, time of first pay- ment, t months 24 days 13.649 -f Amount $363,647 + Payment Dec. 25th, exceeding interest then due - 49.998 Remainder for a new principal 8313.649 Interest of $313,649 from Dec. 25th, 1846, to June 14th, 1849, 2 years 5 months, 19 days - - 46.472-1- Amount $360,121 Payment, July 10th, 1847, less than in- , ^ , ^ •^ ' $4,998 terest then due ) Payment, Sept. 1st, 1848 15.008 Their sum, less than interest then due - $20,006 Payment, June 14th, 1849 99.999 Their sum exceeds the interest then due - - - - $120,005 Remainder for a new principal, June 14th, 1849 - 240.116 Interest of $240,116 from June 14th, 1849, to April 15th, 1850, 10 months 1 day - - - 12.045 Total due, April 15th, 1850- - - - $252.1614- 279. How do you find the interest when the principal \b in poiuids, Bhillings, and pence? INTEREST. 256 16478.84 New Haven, Feb. 6th, 1850. 2. For value received, I promise to pay William Jenks, or order, six thousand four hundred and seventy-eight dollars and «ighty-four cents, with interest from date, at 6 per cent. John Stewart. On this note wore indorsed the following payments : May 16th, 1853, received $ 545.76 May 16th, 1855, " 1276 Feb. 1st, 1856, " 2074.72. What remained due, August 11th, 1857? 3. A\s note of $7851.04 was dated Sept. 5th, 1851, on Trhich were indorsed the following payments : viz., Nov. 13th, 1853, $416.98 ; May 10th, 1854, $152 : what was due March Ist, 1855, the interest being 6 per cent. ? $8974.56 Nj,^ York, Jan. 3d, 1854. 4. For value received, I promise to pay to James Knowles, or order, eight thousand nine hundred and seventy-four dollars and fifty-six cents, with interest from date at the rate of 7 per cent. Stephen Jones. On this note arc indoi*sed the following payments : Feb. 16th, 1855, received $1875.40 Sept. 15tli, 1856, " 3841.26 Nov. 11th, 1857, " 1809.10 June 9th, 1858, " 2421.04. What will be due, July 1st, 1858? ^345.50 BuFFAiA Nov. Ist, 1852. 5. For value received, I promise to pay C. B. Morse, or order, three hundred and forty-five dollars and fifty cents, with interest from date, at 7 per cent. John Dob. i 250 PERCENTAGE. On this note are the following indorsements : June 20th, 1853, received $75 Jan. 12th, 1854, " 10 March 3d, 1855, " 15.50 Dec. 13th, 1856, " 52.75 Oct. 14th, 1857, " .... - 106.75 What will there be due, Feb. 4th, 1858? ^^^Q Mobile, Oct. 19th, 1850. 6. For value received, we jointly and severally promise to pay Jones, Mead & Co., or order, four hundred and fifty dol- lars on demand, with interest, at 8 per cent. Mannii^g & Bros. The following indorsements were made on this note : Sept. 25, 1851, received $85.60 ; July 10, 1852, received $20 ; June 6, 1853, received $150.45; Dec. 28, 1854, received $25.12J ; May 5, 1855, received $169 : what was due, Oct. 18, 1857? PROBLEMS IN SIMPLE INTEREST. 282. In every question of Interest, there are four parts : 1st, Principal ; 2d, Rate ; 3d, Time ; and 4th, Interest. If any three of these parts are known, the fourth can be found. The interest is found by multiplying the principal by the rate and time in years (Art. 276) ; therefore, the interest is the product of the three factors, principal, rate, and time. Any on© of these factors is found by dividing then* product by the other two : Hence, we have the following principles : 1st, The interest is equal to the product of the principalf ate, and time; 2d, The principal is equal to the interest divided by the product of rate and time; 3d, The rate is equal to the interest divided by the product of the principal and time; 4th, The time is equal to the interest divided hy the product of the principal and rate. PROBLEMS IN INTEREST. 267 283. Formulaa. Interest = I = PxRxT. ' 3. 11 = JU: 4. T= ' R X T ' P X T ' P X R Examples. 1. At what rate per cent, must 1325 be put at interest for 1 year and 6 months, to produce an interest of $34,125 ? Analysis. — The product of principal hj the time is 325 x 1| = 487^. By principle 3d, the rate equals $34.125 -r 487.5 = .07, or 7 per cent. 2. What principal, at 6 per cent., will in 9 months give an interest of 8178.9552? 3. The interest for 2 years and 6 months, at 7 per cent., is $76,965: what is the principal? 4. What sum must be invested, at 6 per cent., for 10 mouths and 15 days, to produce an interest of $327.3249 ? 5. If my salary is $1500 a year, what sum invested at 5 per cent, will pay it ? 6. What sum put at interest for 4 years and 3 months, at 7 per cent., will gain $283.3914? 7. The interest of $2100 for 3 years 1 month and 18 days is $460.60: what is the rate per cent.? 8. A person owning property valued at $2470.80, rents it for 1 year and 10 months for $452.98 : what per cent, does it pay? 9. At what rate per cent, must $3456 be loaned for 2 years 7 months and 24 days, to gain $503,712? 280. What is a partial payment ?— 281. What is the rule for partial payments? — 283. IIow many parts arc there in a problem of simple interest? What are thoy?— 283. Write on the blackboard the for mulaa for the problems of simple intorc>st. 258 PERCENTAGE. 10. If I build a hotel at a cost of $56000, and rent it for $tOOO a year, what per cent, do I receive for the invest- nent? 11. The interest on $1119.48, at 1 per cent., is $195,909 : vhat is the time? 12 How long will it take $500 to double itself, at 6 pe. cut., simple interest ? ? 13. Wishing to commence business, a friend loaned me $3120, at 6^ per cent., which I kept until it amounted to $5009.60 : Uow long did I retain it? U. I borrowed $700 of my neighbor, for 1 year and 8 months, at 6 per cent. ; at the end of the time be borrowed of me $750 : how long must he keep it to cancel the amount of interest I owed him ? 15. What amount of money must I invest at 6'/o, that I may receive annually an income of $450 ? COMPOUND INTEREST. 284, Compound Interest is interest computed on the amount^ which is the sum of interest and principal (Art. 272). It may be computed annually, semi-annually, quarterly, monthly, weekly, or daily. In savings banks, the interest is generally computed semi-annually. Rule. — Compute the interest for one year, unless some other time is named ; then add it to the principal, and com- •oute the interest on the amount as on a new principal ; add the interest again to the principal, and compute the interest as before ; do the same for all the times at which payments of interest become due; from the last result subtract the first principal, and the remainder ivill be the compound in- terest. 284. Wliat is compound interest? COMPOUND INTEREST. 259 Examples. 1. What will be the compound interest of $3250 for 4 jcars, at 7 per cent. ? OPERATION. $3150.000 principal for 1st year. $3750 X .07 = 262 500 interest for 1st year. 4012.500 principal for 2(1 " $4012.50 X .07 = 280.875 interest for 2(] " 4293.375 principal for 3(1 " ^ $4293.375 x .07 = 300.536 + interest for 3d " 4593.911 + principal for 4th " $4593.911 X .07 = 321.573 + interest for 4th " 4915.484 4- amount at 4 years. 1st principal 3750.000 Interest $1165.484 f 2. What will be the compound interest of $175 for 2 years, at 7 per cent.? 3. What will be the amount of $240 at compound interest, for 4 years, at 5 per cent.? 4. What will be the compound interest of $300, for 3 years, at 6 per cent. ? 5. What will be the compound interest of $590.74, at 6 ptT cent., for 2 years ? 6. What will be the compound interest of $500, for 2 years, at 8 per cent. ? 7. What will be the compound interest of $3758.56, for 3 years, at 7 per cent.? 8. What will be the compound interest of $95637.50, for 7 years, at 6 per cent. ? 9. What will be the compound interest of $75439.75, for 4 years, at 4 J per cent.? 260 PERCENTAGE. DISCOUNT. 285. Discount is an allowance made for the payment oi in one J before it is due. The Face of a note is the amount named in the note. The present value of a note, is such a sum as, being put at interest until the note becomes due, would increase to an amount equal to its face. The discount, on a note, is the difference between the face of tlm note and its present value. 286. Kno-wing the face of a note, due at a future time, and the rate of interest, to find its present value. 1. I give Mr. Wilson my note for $106, payable in 1 year : what is the present value of the note, if the interest is 6 per cent.? What is the discount? Analysis. — The present value operation. is the base, the rate ^ is 6 per 106 cent., and the face of the note ^^^^' "^^^^^ "^ 14. .06 ~ is the amount (Art. 2G7). Rule. — Divide the face of the note hy 1 dollar plus the interest of 1 dollar for the given time. Note. — When payments are to be made at different times, find tJie 'present value of the sums separately, and their sum will be the present value of the note. Examples. 1. What is the present value of a note of $615, due 1 year 4 months hence, at t per cent. ? 2. What is the present value of 1202.58, due in 1 year t months and 18 days, at 6 per cent.? 285. What is discount? What is the face of a note? Wliat is present value ? What is the discount on a note ? — 283. Knowing the face of a note and rate, how do you find the present value? DISCOUNT. 261 3. How much should I deduct for the present payment of a note of $721, due in 7 months and 6 days, at 5 per cent.? 4. If a note for $5100 is payable Feb. 4th, 18G4, what is its value Sept. lOtb, 1803, interest being reckoned at 8 per cent? 5. What sum of money will amount to $2500, in 2 years 7 months and 12 days, at 12 per cent.? G. What is the present value and discount of $3000, pay- a])le in 1 year 2 months and 20 days, at 7 per cent.? 7. A held a note of $1400 against B, payable Aug. 1st, 1856 ; B paid it May 15th, 1856 : what sum did he pay, the interest being 7 per cent.? 8. A flour merchant bought for cash 300 barrels of flour, for $10.50 per barrel ; he sold it the same day for $12 a barrel, and took a note at 3 months : what was the cash value of the sale, and what his gain, if the interest is reckoned at 7 per cent.? 9. A man purchased a house and lot for $10000, on the following terms : 5000 in cash, 2500 in 3 months, and the balance in six months : what was the cash value of the prop- erty, interest being reckoned at 6 per cent.? 10. Which is the more advantageous, to buy sugar at 7 J cents a pound, on 4 months, or at 8 cents a pound on 6 months, at 6 per cent, interest ? 11. Bought land at $10 an acre : what must I ask per acre if I abate 10 per cent., and still make 20 per cent, on the purchase money? 12. A merchant owed three notes, viz., $1000, payable Aug. 1st, 1855 ; $500, payable Oct. 10th, 1855, and $900, payable Nov. 1st, 1855 : what was the cash value of the three notes, "^aly 1st, 1855, reckoning interest at 6 per cent.; and what was the difference between that value and their amounts at the times when they fell due, if interest were reckoned from July Ist. 262 PERCENTAGB. BANKING. 287. A Corporation is a collection of persons authorized by law to do business together. The instrument which defines tlieir rights and powers is called a Charter, 288. Banks are Corporations for the purpose of receiving de^ posits, loaning money, and furnishing a paper circulation repre- sented by specie. Bank Notes are the notes made by a bank to circulate as money, and should be payable in specie, on presentation at the bank. A Promissory Note is the note of an individual, and is a positive engagement, in writing, to pay a given sum, either on demand or at a specified time. FORMS «F NOTES, No- ^' Negotiable Note. ^^^•^^ Providence, May 1, 1856. For value received, I promise to pay on demand, to Abel Bond, or order, twenty-five dollars and fifty cents. Reuben Holmes. No- 2- Note Payable to Bearer. ^•^^^•^^ St. Louis, May 1, 1855. For value received, I promise to pay, six months after date, to John Johns, or bearer, eight hundred and seventy-five dol- lars and thirty-nine cents. Pierce Penny. No- 3- 2^ate by two Persons. ?^55:?1 Buffalo, June 2, 185G. For value received, we jointly and severally promise to pa;y to Richard Ricks, or order, on demand, six hundred and fifty nine dollars and twenty-seven cents. Enos Allan. John Allan. BANKING. 263 No. 4, Note Payable at a Bank. ^'20.2b Chicago, May 7, 1856. Sixty days after date, I promise to pay John Anderson, or order, at the Bank of Commerce, in the city of New York, wcnty dollars and twenty-five cents, for value received. Jesse Stokes. Remarks Relating to Notes. 1. The person wlio signs a note is called the drawer or maker ot the note ; thus, Reuben Holmes is the drawer of note No. 1. 2. The person who has the rightful possession of a note is called the holder of the note. 3. A note is said to be negotiable when it is made payable to A. B., or order, who is called the payee (see No. 1). Now, if Abel Bond, to whom this note is made payable, writes his name on the back of it, he is said to indoi'se the note, and he is called the indorser; and when the note becomes due, the holder must first demand payment of the maker, Reuben Holmes ; and if he declines paying it, the holder may then require payment of Abel Bond, the indorser. 4. When a note is not paid at the time it becomes due, the in- dorser must be notified of the fact, and of the time it was due. This notice is generally given by an officer called a notary public, and is called a lirotest. 5. If the note is made payable to A. B., or bearer, then the drawer alone is responsible, and he must pay to any person who holds the note. 6. The time at which a note is to be paid should always be named ; but if no time is specified, the drawer must pay when required to do so, and the note will draw interest after the payment is demanded. 7. When a note, payable at a future day, becomes due, it will draw ntercst, though no mention is made of interest. 8. In each of the States there is a rate of interest established by aw, which is called the legal interest; and when no rate is specified, he note will always draw legal interest. If a rate Idgher than legal .ntercst is named in the note, or agreed upon, the drawer, in most ol the States, is not bound to pay the note. 287. What are corporations? What is a charter ?— 288. What ar© biuiks? What uro buuk-uotos? Wliat is a promissory note? 261 PERCENTAGE. 9. If two i^ersons jointly and severally give tlicir note (see No. 3), it may be collected of either of them. 10. The words, " For value received," should be expressed in every note. 11. When a note is given, payable on a fixed day, and in a specific article, as in wheat or rye, payment must be oflfered at the specified time ; and if it is noc, the holder can demand the value in money. 12. Days of grace are days allowed for the payment of a not after the expiration of the time named on its face. By mercantile usage, a note does not legally fall due vmtil 3 days after the expira- tion of the time named on its face, unless the note specifies "without grace.'" For example, No. 2 w^ould be due on the 4th of November, and the three additional days are called days of grace. When the last day of grace happens to be a Sunday, or a holiday, such as New Year's day, or the 4th of July, the note must be paid the day before ; that is, on the second day of grace. 13. There are two kinds of notes discounted at banks : 1st. Notes given by one individual to another for property actually sold ; these are called business notes, or business paper. 2d. Notes made for the purpose of borrowing money, which are called accommodation notes, or accommodation paper. The first class of paper is much preferred by the banks, as more likely to be paid when it falls due, or, in mer- cantile phrase, "when it comes to maturity." BANK DISCOUNT. 289. Bank Discount is the deduction made by a bank from the face of a note due at a future time. Bunk discount, by custom, is the interest of the face of the note, calculated from the time when it is discounted to the time when it falls due ; in which time three days of grace are always included (see remark 12). The interest on notes dis counted at bank is always paid in advance. The proceeds of a note is the difference between its face and the discount. 289. What is bank discount? How is interest calculated? When is it paid ? What are the proceeds of a note ? BANK DISCOUNT. 265 290. To find the bank discount. Rule. — Add 3 days to the time which the note has to rwi, and then calculate the interest for thai time at the given rate. Examples. 1. What is the bank discount on a note of $300, for 4 Qonths, at 6 per cent, per annum. 2. What is the bank discount on a note of $200, payable in five months, at 9 per cent.? 3. What is the bank discount and proceeds of a note of $500, at 6J per cent., payable in 8^ months ? 4. What is the cash value of a note, payable at bank, of $1255.38, and due in 4 months, at 7 per cent.? 5. What was the bank discount on a note of $500, due August 13th, 1855, and discounted July 1st, 1855, reckoning interest at 7 per cent.? 6. I bought 4368 bushels of wheat, at $1.25 a bushel, and sold it the same day for $1.30 a bushel on a note of 4 months. If I get this note discounted, at bank, at t per cent., what do I gain or lose ? t. What is the difference between the true and bank dis- count, of $1000, payable in 7 months, at 6 per cent.? 8. What is the difference between the true and bank dis- 20unt, of $10000, payable in 4 J months, at 8 per cent.? 9. January 1st, 1855, a note was given for $1000, at 5 J per cent., to be paid May 1st, next following : what was its cash value at bank ? 10. A holds a note against B for $1500, to run 6 months from Aug. 1st, without interest. Oct. 1st, he wishes to pay a debt at the bank of $1000, and turns in the note at a dis- count of 5 per cent, in payment : how much should he receive bnck from the bank? 290. How do you find the bank discount? 13 2G6 PERCENTAGE. 291. To draw a note due at a future time, whose proceeds shall be a given amount. 1. For what sum must a note be drawn at 4 months and 12 days, so that, when discounted at bank, at 6 per cent., the proceeds shall be $400. Analysis. — The face of the note must be such, that the interest for the given time, subtracted from the face, shall leave the re- quired proceeds. Hence, the proceeds correspond to the d/ifference; the rate of interest of $1 for the given time, to the rate ; and the face of the note to the base (Art. 267). Rule. — Divide the given proceeds hij 1 minus the rate of %\ for the given time, and the quotient will he the face of the note. OrERATION. Proceeds = $400. K %\ for 4 mo. Zda. = 0.0225 _, Proceeds Facc= ^ = $409.20 1 + 1 - .0225 = 0.97t5 2. For what sum must a note be drawn at T per cent., pay- able in 6 months, so that when discounted at a bank it shall produce $285.95. 3. How large a note must I make at a bank, at 6 per cent., payable in 6 months and 9 days, to produce $674.89? 4. For what sum must a note be drawn, at 5 per cent., payable in 9 months and 15 days after date, so that when discounted at bank, it shall produce $1000. 5. Marsh, Dean & Co. purchase of John Jones 380 ban'cls of flour, at $9.12|- a barrel, for which they give him a note at 90 days, for such sum, that if discounted at 6 per cent., he shall receive the above price for his flour : what was the face of the note? 291. How do you draw a note whose proceeds shall be a given amorint ? STOCKS. 267 STOCKS. 292. Capital or Stock is the amount of money paid in to carry on the business of a corporation. Stockholders are the owners of the stock. Certificates are the written evidences of ownership. 293. United States or State Stocks are the bonds of the United States, or of a State, bearmg a fixed interest. A Coupon is a due-bill for interest, attached to bonds or certificates of stock, and payable at specified times. 294. Par Value of stock is the number of dollars named in each share, generally 100; sometimes 50, and sometimes 25. Market Value of a stock, is what it brings per share, when sold for cash. 295. Premium is the rate per cent, which a stock sells for above its par value. Discount is the rate per cent, which a stock sells for below its par value. 296. Dividend is a profit divided among the stockholders, and is generally estimated at a certain rate per cent, on the par value of the stock. 297. Brokerage is a commission made to an agent for buying and selling stock, uncurrent money, or bills of exchange. Notes. — 1. The brokerage in the city of New York is generally ooo-fourth of one per cent, on the par value. 2. In questions of stocks, the par taZue is always the hase. 3. In the examples, the shares are |100 each, unless another amount is named. 298. To find the dividend on a given amount of stock. 1. What is the percentage on 25 shares, $100 each, of Kings County Insurance Company, the dividend being 25 per cent.? Analysis. — Here, the base and rate oxq given to find the per- centage (Art. 259). 268 PERCENTAGE. 2. The Atlantic Fire Insurance Co. declares a semi-annual dividend of 4-|% on the capital stock: what is the annual dividend of 43 shares at that rate? 3. The Atlantic Bank of Brooklyn has declared a serai^ annual dividend of 6%: what is the dividend on 18 shares? 4. A bankrupt is indebted to A, $5416, and to B, |6t95 what does each receive, when the dividend to the creditors i 47 J per cent.? 5. A mining company, shares $25 each, declared a dividend of n"/o: what was the dividend on 36 shares? 299. To find the value of stock which is above or below par. 1. What is the value of $5600 of stock, reckoned at par, when the stock is at a premium of 9 per cent. ? Analysis. — In this class of examples, the base and rate are given. When the stock is above par, the amount is required (Art. 265) ; when it is below par, the difference (Art. 2C6). 2. What is the cost of 56 shares of !N'ew York Central Railroad stock, at 5 J per cent, below par, and the brokerage J per cent. ? 3. I bought 36 shares in the Pennsylvania Coal Company, at a discount of 12 J per cent., and sold them at a premium of 7 per cent., paying j- per cent, brokerage in each case : how much did I make by the operation? 4. What is the market value of 216 shares of bank-stock, each share $15, and the premium t|"/o ? 292. Wliat is capital or stock ? Who are stockholders ? What are certificates? — 293. Wliat are United States stocks? What is a coupon? — 294. What is the par value of a stock? What is market value?— 295. What is premium? Wliat is discount ?— 296. What is dividend ? — 297. What is brokerage ? W^liat is the general rate in the city of New York? What is the base, in stocks? — 298. How do you find the dividend on stock ? — 299. How do you find the value of stock when it is above or below par? I STOCKS. 2C9 5. The par value of 257 sliares of bank-stock is $200 a l.arc : what is tlie present value of ail tlic shares, the stock ) irii^ at a premium of 15 per cent.? 0. What is the value of 120 shares of Exchange Bank !ock, it being at a premium of 18| per cent., and the par ;i]ue being $150 a share? 7. What will be the cost of 69 sliares of Panama Railroail lock, at a discount of 8"'o, the par value being |125, and in'okeragc } per cent.? 8. Gilbert & Co. buy for Mr. A, 200 shares of United States stock, at a premium of GJ per cent., and charge -J- per cent, brokerage : if the sliares are $1000 each, how much money does A pay for the stock? 9. Mr. B. bought 125 shares of stock in the American Guano Company, at par, the shares being $20 each. At the end of 4 months, he received a dividend of 5 per cent., and at the end of 10 months, a second dividend of 4 per cent. At the end of the year, he sold his stock at a premium of 10 per cent. : how much did he make by the operation, reckomng the interest of money at 7 per cent.? 300. To find how much stock, at par value, a given sum of money will purchase, when the stock is at a premium or discount. i . 1. What value of stock, at par, can be purchased by t$3045.38, if the stock is at a premium of 10 per cent., and } per cent, is charged for brokerage ? Analysis. — When the stock is above par, the amount and the rate are given to find the base (Art. 207); when below par, the difference and rate are given to find the base (Art. 267). 2. A person wishes to invest $3000 in bank-stock, which is at a discount of 15 per cent. : what amount at par can he purchase ? 300. Hovr do you find how much stock, at par, a given sum of money will buy when the stock is at a preini um ? How when it is at a discount ? 270 PERCENTAGE. 3. How many shares of Galena and Chicago Ra,iIroad stock can be bought for $6384, at 14% premium? 4. When bank-stock sells at a discount of t^%, what amount of stock, at par value, will $3700 buy? 5. A person has $7000, which he wishes to invest ; what will it purchase in 5 per cent, stocks, at a discount of 3J per cent., if he pays \ per cent, brokerage ? 6 How much 6 per cent, stock, at par, can be purchased for $8700, at SJ per cent, premium, i per cent being paid for brokerage ? 7. A person owning $12000 in government funds, desires to purcliase stock in the American Exchange Bank. The funds are at a discount of 3^ per cent., while the bank-stock is at a premium of 10^ per cent. : what amount of stock, at par value, can he purchase, allowing the broker's charges for the purchase to be f per cent.? 301. To find the rate of interest on an investment in stock, when the stock is above or below par. 1. What is the rate of interest on an investment in 6 per cent, stocks, when they are at a discount of 25 per cent.? Analysis. — The interest on the opkeation. stock is computed on its par value; .75 ) _ <^i ^ Ofi — ^ Ofi the interest on the investment is ic J ~ . — . computed on the market value, and the percentage in each case is the ^ — - _2 ::::: Qg same. Hence, 1 dollar of the stock ''^ multiplied by its rate of interest, will be equal to the market value ^'*^- ^ ^^ ^^^^• ©f $1 of the stock multiplied by its rate of interest. Rule. — 3IuUiply $1 o/* the stock by its rate of interest, and divide the product by the mai'ket value of %\ of the stock: the quotient will be the rate of interest on the investment. 2. If I buy 7 per cent, stock at 12J per cent discount, B»^at is the rate per cent, on the investment? s'i'ooKs. 271 3. If the stock of the Erie Raih-oad sells at 62J per cent., nud pays semi-annual dividends of 2^ per cent., what would be the rate of interest on an investment ? 4. The bonds of the Illinois Central Railroad Company, which bear interest of 1 per cent., are worth 87 per cent., and the charge for brokerage is J per cent. : what would be the interest on an investment in these funds ? 5. The stock of the Hartford and New Haven Railroad is at a premium of 20 per cent. : reckoning the interest on money at 6 per cent., what will be the interest on an investment? 302. To find how much a stock must bo above or below par, to produce a given rate of interest. I. At what rate must a 6 per cent, stock be bought, so that the investment shall yield 9 % interest ? Analysis. — Since the per- operatiox. centage is the same in both ^ I = Si X 06 = $ 06 cases, $1 of the stock multiplied .09 f by its rate of interest, is equal to the market value of $1 of x = - — = .66J the investment multiplied by * ^ ^^^- 1 - Mi = .331 dis. Rule. — I. Multiply %1 of the stock by its rate of interest, and divide the product by the rate of interest on the invest- ment: the quotient ivill be the per cent, of the market value of ^l of the stock: II. If the market value is greater than 1, subtract 1 from it, and the remainder will be the per cent, of premium ; if less than 1, subtract it from 1, and the remainder will be the per cent, of discount, 301. How do you find the rate of interest on an investment v/licn tlio Block is above or below par ? — 302. How do you find how miicli a stock must be above or below par, to produce a given rate of interest ? 272 PERCENTAGE. 2. At what rate of discount must I invest in 8 per cent, stock, in order to yield me 10 per cent.? 3. If the par value of a stock is $100, and the interest t per cent., what is the discount when an investment yields 13 per cent.? 4. At what rate must I invest in a 9% stock, that I maj receive 8 per cent, on my investment? 303. Which is the best investment? 1. I invest $1250 in State stocks bearing aj interest of 6 per cent., and a premium of 15 per cent. I invest an equal amount in State fives at 12 per cent, discount. Which will yield the larger interest ? Analysis. — Find the rate of interest of each investment, and then compare the two rates. That investment which produces the greater rate is the more advantageous. operation. 1st. 2d. ^^=.0521==5,V.V. .88 05 _ = .0568 = 5/o%°/<' The second investment is the more advantageous. 2. Which is the better investment, to buy sixes at par, or sevens at 107 ? 3. Which will yield the larger profit, 8 per cent, stock at a premium of 20 per cent., or 5 per cent, stock at 80 per cent. ? 4. If I invest $2000 in State stocks at 5 per cent., at par, and an equal amount at 6 per cent., at 90, what will be the difference of the proceeds of the investments at the tmd of 5 years? 303. How do you determine which is the best investment ? INSURANCE. 273 INSURANCE. 304. An Insurance Company is a company chartered to iiLsure against risks. Insurance is an indemnity for loss or injury. It is made by iiipauies or individuals, in consideration of a certain sum paid. Underwriters or Insurers are the companies or persons who insure. 305. Insurance is now limited, chiefly, to three classes of i.-asi'S : 1. Fire insurance, or insurance against loss by fire. 2. Marine insurance, or insm-ance against loss by water. 0. Life insurance, or insurance against loss by death. 306. A Mutual insurance is one in which the insured share in tlie profits. 307. A policy is the mutual agreement of the parties. 308. Premium is the percentage paid by him who owns the [)roperty to him who insures it, as a compensation for risk. 309. All the cases of insurance are simple applications of tlie principles of percentage. There are four : 1. To find the premium, when the base and rate are known (Art. 259). •2. To find the rate, when the base and premium arc known (Art. 201). 3. To find the base, when the rate and premium are known (Art. 2C2). 4. To find the percentage, when the premium is insured as well as the base. The base insured is then the premium plus the first base. 304. Wliat is an insurimce company? What is insurance? Who are luidtnvr iters? 274 PERCENTAGE. Examples. 1. What would be the premium for insuring a ship and cargo, valued at $147614, at 3^ per cent.? 2. What would be the insurance on a ship, valued at $47520, at J of 1 per cent. At ^ of 1 per cent.? 3. What would be the insurance on a house, valued at 116800, at IJ per cent.? At f of 1 per cent.? 4. A merchant owns | of f of a ship, valued at $24000, and insures his interest at 2 J per cent. : what does he pay for his policy? 5. What will it cost to insure a store, worth $5640, at | per cent., and the stock, worth $7560, at f per cent.? 6. A carriage-maker shipped 15 carriages, worth $425 each: what must he pay to obtain an insurance upon them at 75 cents on a hundred dollars ? 7. A merchant imported 150hhd. of molasses, at 35 cents a gallon; he gets it insured for 3i per cent, on the selUng price of 50 cents a gallon : if the whole should be destroyed, and he get the amount of insurance, how much would he gain ? 8. If I get my house and furniture, valued at $3640, insured at 4J per cent., what would be my actual loss if they were destroyed ? 9. The ship Astoria was valued at $20450, and her cargo at $25600, and was bound on a voyage from New York to Canton. The vessel was insured at the St. Nicholas office for $12000, at 2f per cent., and the cargo for $18500, at 3} per cent. The vessel foundered at sea : what was the entire loss of the owner? 10. Shipped from New York to the Crimea, 5000 barrels of 305. To how many classes is insurance limited? What are they? —306. What is mutual insurance ?— 307. What is tlie policy?— 308. What is premium ?— 309. How many cashes are there of insurance? What are they? 'our, worth $10.50 a barrel. The premuim paid was $2887.50 : vhat was the rate per cent, of the hisurance? 11. Paid $120 for insurance on my dwelling, valued at 17500 : what was the rate per cent.? 12. A merchant imported 225 pieces of broadcloth, each )iece containing 40 yards, at $3.50 a yard: he paid $1323 for nsurance : what was the rate per cent.? 13. A merchant paid $1320 insurance on his vessel and cargo, which was 5^- per cent, on the amount insured : how much did he insure? 14. A man pays $51 a year for insurance on his storehouse, at IJ per cent., and $126.45 on the contents, at 2J per cent. : what amount of property does he get insured ? 15. A person shipped 15 pianos, valued at $275 each. He insures them at 3 per cent., and also insures the premium at the same rate : what insurance must he pay ? IG. A store and its contents are valued at $16750. The owner insures them at IJ per cent., and then insures the premium at the same rate : what insurance must he pay ? LIFE INSURANCE. 310. Life Insurance is an agreement to pay, in consideration of a premium, a specified amount to parties named in the agreement, in case of the death of the party insured. 311. To enable the company to fix their premiums at such rates as shall be both fair to the insured and safe to the association, they must know the average duration of life from any given time to its probable close. This average is called the "Expectation of Life," and is determined by collecting 310. "What is life insurance? — 311. What is necessary to enable a company to fix their preniiuma? How is the expectation dt'terniinedlf ^'' ' ' ' : ' ' ' , •■ . xpccUUiim ol lilcV 276 PERCENTAGE. from many sources the most authentic information in regard to the average duration of life from any period named. If we take 100 infants, some will die in infancy, some in childhood, and some in old age. It has been found, from care- ful observation, that if the sum of their ages, after the last shall have died, be divided by 100, the quotient will be 38.'I2 very nearly : hence 38.t2 is said to be the " Expectation of Life" at infancy. The Carlisle Tables, which are used in this country and Eng- land, show the "Expectation of Life" from 1 to 100 years At 10 years old it is found to be 48.82 ; at 20, 41.46 ; at 30, it is 34.34 ; at 40, 2t.61 ; at 50, it is 21.11 ; at 60, 14 years ; at tO, 9.19 ; at 80, 5.51 ; at 90, 3.28 ; and at 100, it is 2.28 years. If we wish the expectation of life, between the periods named in the table, we can readily find it by the rules of pro- portion. Thus, if we wished the expectation of life at 16 years, we should observe that, at 10 years, it has been found to be 48.82 ; at 20 years, it has been found to be 41.46 ; hence, for 10 years it varies 48.82 — 41.46 = 7.36 years : Then, 10 : 6 : : t.36 : 4.416; which number being subtracted from 48.82, leaves 44.40, the expectation of life at 16 years of age. 312. From the above facts, and the value of money (which is shown by the rate* of interest), a company can calculate with great exactness the amount which they should receive annually, for an insurance on a life for any number of years, or during its entire continuance. Among the principal life insurance companies in the United States, are the New York Life Insurance and Trust Company the Girard Life Insurance, Annuity and Trust Company Oi Philadelphia, and the Massachusetts Hospital Life Insurance and Trust Company of Boston. The rates of insurance, in tliose companies, differ but little. INSURANCE 277 313. All companies have published tables which show the quarterly, semi-annual, and annual premiums that must be paid on each $100 or $1000 insured. Note. — Experience has demonstrated that the risks are about equal on all ages between 14 and 25 years. Persons under the ago of 25 years are charged for wJiole life policies, the rate at that ago; though dividends are based on the true age. An extra charge, on the above rates, of one-half per cent, on the amount insured, is made for insuring the lives of women under the age of 48 years. Examples. 1. A person, 20 years of age, finds that the premium, per annum, is $1.36 on 8100 : what must he pay to insure his life for 1 year for $8950 ? 2. A man, aged 40 years, wishes to insure his life for 5 years, and finds that the annual rate is $1.86 for $100 : how much premium must he pay per annum on $12500? 3. A person, 38 years of age, obtains an insurance on his life for 5 years, at the rate of $1.75 per annum on $100 : ho\f much is the annual premium on $15000 ? 4. A person going to Europe, expecting to retura in 2 year», effects an insurance on his life at J of J per cent, premium on $100 ; he insures for $5000 : what is the annual premium ? 5. What will be the annual premium for insuring a person's life, who is 60 years of age, for $2000, at the rate of $4.91 on $100? 6. A person, at the age of 50 years, obtained an insurance at 4J per cent, per annum on each $100 ; he insured for ^1500, and died at the age of 70. How much more was the nsurance than the payments, without reckoning interest ? 7. A gentleman, 47 years of age, going to China as ambas- sador, obtains an insurance on his life for $10000, by paying a premium of $2,71 per annum on every $100, and dies at tlie liiidiile of tlic third year : reckoning simple interest on liLs payments vit 7 per cent., what is gained by the insurance ? 27S PERCENTAGE. ENDOWMENTS. 314. An Endowment is a certain sum to be paid at the expiration of a given time, in case the person, on whose life it is taken shall live till the expiration of the time named. Tlie following table shows the value of an endowment pnr- liiised for $100, at the several periods mentioned in the, column of ages, the endowment to be paid if the person attains the age of 21 years. The table is calculated under the hypothesis that money is worth 6 per cent, interest. TABLE OF ENDOWMENTS, Showing the sum to be paid at 21 years, if alive. Age. Birth $376.84 3 months 344.28 6 " 331.40 9 " .... 318.90 1 year 30G.58 2 " 271.03 3 " 243. G9 4 " 225.42 Age. 5 years. . ..$210.53 6 " .. .. 198.83 7 " .. .. 188.83 8 " .. .. 179.97 9 " .. .. 171.91 10 " . . .. 164.46 11 " . . .. 157.43 12 " . . .. 150.64 Age. 13 years $144.12 86 .83 97 .31 137. 131 125. 120. 114. 109. 104. This table shows that if $100 be paid at the birth of a child, he will be entitled to receive $376.84, if he hves to attain the age of 21 years. If $100 be paid when he is ten years old, he will be entitled to receive $164.46, if he lives to attain the age of 21 years. And similarly for other ages. We can easily find by proportion, 1st. How much must be paid, at any age under 21, to pur- chase a given endowment at 21 ; and, 2d. What endowment a sum paid at any age under 21, will purchase. Examples. 1. Wliat endowment, at 21, can be purchased for $250, paid at the age of 10 years? 2. What endowment, at 21, can be purchased for $360, paid at the age of 5 ye:irs ? ANNUITIES. 279 3. If my child is T years old, and I purchase an endowineni for $650, wiiat will he receive if he attains the age of 21 years 't ANNUITIES. 315. An Annuity is a fixed sum of money to be paid at regular periods, generally, yearly, either for a limited tune, or forever, in consideration of a given sum paid in hand. The Present Value of an annuity is that sum which, being put at compound interest, would produce the sums necessary to pay the annuity. Tiie purchaser of an annuity should pay more than the com- pound interest ; for the seller cannot afford to take the money of the purchaser, invest it, reinvest the interest, and pay over the entire proceeds. Knowing the rate of interest on money, and the present value of an annuity, a close estimate may be made of the price it ought to sell for. Table, Skoiciiiff the pitESKNT VALUK OF AN ANNUITY OF $1, /roiH 1 to 30 ijcars, at dijferent rates of interest . Tears. 5 percent 6 per cent. Years. 5 per cent 6 per cent 1 0.052.*]81 0.943390 16 10.837770 10.105895 2 1.8r)JJ410 1.833393 17 11.274060 10.477260 3 2.72:3248 2.073012 18 11.689587 10.827603 4 3.r)45i)50 3.405106 19 12.085321 11.158116 5 4.329477 4.212304 20 12.462216 11.469921 6 5.075692 4.917324 21 12.821153 11.764077 7 5.78G373 5.582381 22 13.163003 12.041582 8 0.403213 0.209794 23 13.488574 12.303379 9 7.107822 6.801092 24 13.798642 12.550358 10 7.721735 7.360087 1 25 14.093945 12.783356 11 8.300414 7.886875 I 26 14.375185 13.003166 12 8.803252 8.388844 27 14.043034 13.210534 13 9.393573 8.852083 28 14.898127 13.400164 U 9.808(541 9.294984 29 15.141074 13.590721 15 10 379058 9.712249 30. 15.372451 13.764831 280 PERCENTAGE. To find the present value of an annuity for any rate, and for any time, we simply multiply the present value of an an- nuity of $1 for the same rate and time, by the annuity, and the jDroduct will be its present value. Thus, the present value of an annuity of $600 for 8 years, t 6 per cent., is $6,209^94 X 600 - $3725.8'764 ; that is, pres. val. of $1 x annuity = pres. val. ; hence, ., pres. val. ^. _ annuity = — - — - — -^-j- ; therefore, ^ pres. val. of $1 ' ' 316. To find what sum will produce a certain annuity at a given rate and for a given time. Rule. — Maltiphj the present value of an annuity of $1, at the given rate and for the given time, by the given annu- ity ; the product will he that sum. 317. To find what annuity a given sum will produce at a given rate, and for a given time. R/Ule. — Divide the given sum, or present valuer by the pres- ent value of $1, for the given rate and time, and the quotient will be the annuity. Examples. 1. What is the present value of an annuity of $550, at 5 per cent., for 21 years? 2. What would be the value of an annuity that should yield eight hundred and thirty-five dollars a year for sixteen years, the interest being compound, and at the rate of 5 per cent, per annum ? 3. What is the present value of an annuity of $1500 a year, for 30 years, the compound interest being reckoned at per cent. ? 314. What is an endowment ? Wliat does the table of endo^^^nent8 show ? What may be foui^d from the table ? — 315, What is an annu- ity? What is tlie present vWue of an annuity? — 316. How do you find the present value of an annuity *or a given rate and time? ASSESSING TAXES. 281 4. What annuity, for twenty-four years, could be purchased for the sura of twenty-seven thousand five hundred and sixty dollars, the compound interest being reckoned at 6 per cent. ? 5. Mr. Jones having a small fortune of $25000, and calcu- lating that he would live about 20 years, purchased an annuity ai C per cent., with an agreement that he would pay $20 a ) car to an invalid sister : what was his annual income from the investment after making that payment? ASSESSING TAXES. 318. A Tax is a certain sum required to be paid by the in- habitants of a town, county, or State, for the support of gov- ernment. It is generally collected from each individual, in proportion to the amount of his property. In some States, however, every white male citizen, over the age of twenty-one years, is required to pay a certain tax. This tax is called a poll-tax ; and each person so taxed is called a poll. 319. In assessing taxes, the first thing to be done is to make a complete inventory of all the property in the town, on which the tax is to be laid. If there is a poll-tax, make a full list of the polls, and multiply the number by the tax on each poll, and subtract the product from the whole tax to be raised by the town ; the remainder will be the amount to be raised on the property. This remainder is the percentage or tax to be raised. The value of the property taxed is the base ; hence this remainder, divided by the value of the property, 317. How do you find what annuity a given sum will produce, at a given rate and for a given time? — 318. What is tax? IIow is it generally collected? What is a poll-tax? — 319. What is the first tiling to be done in assessing a tax ? If there is a poll-tax, hov/ do you find tlie amount? IIow, then, do you find the per cent, of tax to be levied on a dollar? IIow do you find the tax to bo raisf^d on each individual? 2S2 rEROKNTAGE. gives the rate. Each man's property, multiplied by the rate, gives his tax or percentage. Examples. A certain town is to be taxed $4280 ; the property on which the tax is to be levied is valued at $1000000. Now there are 200 polls, each taxed $1.40. The property of A is valued at $2800, and he pays 4 polls. B's at $2400, pays 4 polls, C's at $2530, pays 2 " D's at $2250, pays 6 " E's at $7242, pays 4 polls, F's at $1651, pays 6 " G's at $1600.80, " 4 " What will be the tax on one dollar, and what will be A*s tax; and, also, that of each on the list? First, $1.40 X 200 = $280, amount of poll-tax. $4280 — $280 = $4 000, amount to be levied on property. Tlien, $4000 -^ $1000000 = .004 = j^U = 4 mills on $1. Now, to find the tax of each, as A's, for example : A's inventory, $2800 .004 $11.20 4 polls, at $1.40 each, 5.60 A's whole tax, $16.80 In the same manner, the tax of each person in the town- ship may be found. Examples. 1. In a county embracing 350 polls, the amount of property on the tax-list is $318200 ; the amount to be raised is as fol- lows : for State purposes, $1465.50 ; for county purposes, $350.25 ; and for town purposes, $200.25. By a vote of the county, a tax is levied on each poll of $1.50 : how much per c^nt. will be laid upon the property ? 2. In a county embracing a pojxilation of 98415 persons, a ASSESSING TAXES. 2S3 tax is levied for town, county, and State purposes, amounting to $100400. Of this sum, a part is raised by a tax of 25 cents on each poll, and the remainder by a tax of two mills on the dollar : what is the amount of property taxed ? 3. In a county, embracing a population of 56450 persons, a tax is levied for town, county, and State purposes, amount ing to |!874Gt ; the personal and real estate is valued a $4890300. Each poll is taxed 25 cents : what per cent, is the lax, and how much will a man's tax be, who pays for five polls, and whose property is valued at $5400 ? What is B's tax, who is assessed for 2 polls, and whose property is valued at $3760.50? 4.»A banking corporation, consisting of 40 persons, was taxed $957.50 ; their property was valued at $125000, and each poll was assessed 50 cents : what per cent, was their tax, and what was a man's tax, who paid for 1 poll, and whose share was assessed for $2000? 5. What sum must, be assessed to raise a net amount of $5674.50, allowing 2J per cent, commission on the money col- lected ? 6. Allowing 4 per cent, for collection, what sura must be assessed to raise $21346.75 net? 7. In a certain township, it becomes necessary to levy a tax of $4423.2475, to build a public hall. The taxable prop- erty is valued at $916210, and the town contains 150 polls, each of which is assessed 50 cents. What amount of tax must be raised to build the hall, and pay 5 per cent, for collection, ud what Ls the tax on a dollar? What is a person's tax who pays for 3 polls, and whose j>ersonal property is valued at $2100, and his real estate at «3000 ? What is G's tax, who is assessed for 1 poll, and $1275.50? What is n's tax, who is asscssoxl for 1 poll, and $2456 ? 284 EQUATION OF PAYMENTS. 8. The people of a school district wish to build a new school-liouse, which shall cost $2850. The taxable property of the district is valued at $190000 : what will be the tax on a dollar, and what will be a man's tax. whose property is valued at $7500? How much is Mr. Merchant's tax, whose personal and real estate are assessed for $1200? 9. In a school district, a school is supported by a rate-bill. A teacher is employed for 6 months, at $60 a month ; the fuel and other contingencies amount to $66. They drew $41.60 public money, and the whole number of days' attendance was 1688 : what was D's tax, who sent 148 days ? What was F's tax, who sent 184^ days? EQUATION OF PAYMENTS. 320. Equation of Payments is the process of finding the average time of payment of several sums due at dififerent times, so that no interest shall be gained or lost. The average or equated time, is the time that elapses from the time at which we begin to reckon interest to the time of payment of all the debts. The equated date is the date of payment of all the debts. 321. When the times of payment are reckoned from the same date. 1. B owes Mr. Jones $5t : $15 is to be paid in 6 months ; $18 in T months ; and $24 in $10 months : what is the average time of payment, so that no interest shall be gained or lost ? Analysis. — The interest of $15 for 6 opehation. months, is the same as the interest of $1 $15 X Q = 90 for 90 months; the interest of $18 for 7 $18 X 7 = 126 mouths, is the same as the interest of $1 &24 x 10 = 240 for 126 months; and the interest of $24 7^ ^l^dT^fR ibr 10 months, is the same as the interest 456 of $1 for 240 months; hence, the sura of EQUATION OF PAYMENTS. 285 these products, 456, is the number of months it would take $1 to produce the required interests. Now, the sum of the payments, $57, will produce the same interest in one fifty -seventh part of the time; tliat is, in 8 months: hence, to find the average time of payment: Rule. — Multiply each j^ayment by the time before it becomes due, and divide the sum of the 2)roducts by the sum of the vaymenfs: the quotient will be the average time. Examples. 1. A merchant owes $1200, of which $200 is to be paid in 4 months, $400 in 10 months, and the remainder m 16 months : if he pays the whole at once, at what time must he make the payment ? 2. A owes B $2400 ; one-third is to be paid in 6 months, one-fourth in 8 months, and the remainder in 12 months : what is tlic mean tunc of payment? 3. A merchant has due him $4500 ; one-sixth is to be paid in 4 months, one-third in 6 months, and the rest in 12 months: what is the equated time for the payment of the whole ? 4. A owes B $1200, of which $240 is to be paid in three months, $360 in five months, and the remainder in 10 months : wliat is the average time of payment ? 5. Mr. Swain bought goods to the amount of $3840, to be paid for as follows, viz. : one-fourth in cash, one-fourth in 6 months, one-fourth in 7 months, and the remainder in one year: what is the average time of payment? 6. A man bought a farm for $5000, for which he agreed to pay $1000 down, $1200 in 3 months, $800 in 8 months, $1500 in 10 months, and the remainder in one year: if he pa}s the whole at once, what would be the average time of pay ment ? 320. Wliat is equation of payment«? What is the average or equated time? — 321. How do you find equated time? 2S6 EQUATION OF FAYMEXTS. *l. A person owes three notes : the first is for $200, pay- able July 1st ; the second for $150, payable August 1st ; and the third for $250, payable August 15th : what is the average time, reckoned from July 1st? 322. When the times are reckoned from different dates. 1 E. Bond, Bought of Trust & Co. 1861. Aug. 1, 450 yds. muslin, at 10 cents - - - $45 00 Aug. 16, 800 yds. calico, at 12J cents - - - 100 00 Sept. 5, 720 yds. bombazine, at 80 cents - - 576 00 Oct. 1, 300 yds. cloth, at $3.50 - - - - 1050 00 On what day may the whole be supposed to have been pur- chased ; or, what is the equated date of purchase ? Analysis. — The owner parted with his goods, and therefore with their values, at the dates specified; and the question is, to find at what time he could have sold the whole at the same advantage. Keckoning from Aug. 1st, the earliest date, he had the use of $45, the amount of the first sale, for no time ; of $100 for 15 days, viz., from Aug. 1st to Aug. IGth; of $576 for 35 days, viz., from Aug. 1st to Sept. 5th; of $1050 for 61 days, from Aug. 1st to Oct. 1st: then, by the preceding Article, we have the following operation: OPERATION. 45 100 576 1050 X X X X 15 35 61 = 000 1500 20160 64050 1771 1771)85710(483-Wt^ 7084 = equated 1 !;ime. 14870 14168 48 days equated from Aug. date, Sept. 1st. 18th. 702 1771 322. From what date may the equated time be reckoned ? What is the multiplier of the date used as the point of departure? What do you do when the quotient contains a fraction? What is the rule when the times are reckoned for different dates? KQU. n'lON OF PAYMENTS. 287 InsteafiR0..^ft Note. — 1. When the items have the same or different times of credit allowed, find when the items are payable, and then proceed as before. 2. If the cash balance is required on a day previous to the latest date of the items, find the cash balance for this latest date ; then find the present value for the given date : this wiU be the cash balance. 3. Allowing a credit of six months on each item, what is the interest and cash balance, Feb. 1st, 1856? Dr. R. Sherman. Gr, 1855. July 1st, To merch., $750 " 17th, " " 600 " 25th, " " 800 $2150 Feb. 6th, By merch., Mar. 7th, " " Interest balance. Cash balance, 900 46.20 403.80 $2150.00 4. Allowing a credit of 3 months on each of the items of the following account, what would be the interest and cash balance on October 31st, 1856 ? Dr. E,. Rivers. 856. May 1, To merch., $500 " 20, " " 675 June 6th, To cash, 350 July 9th, " merch., 175 Cash balance, 620.70 $2320.70 Or. May 6th, By cash, " 25th, " mer.. $400 620 June 16th, " cash. 900 July 20th, " mer.. Interest balance, 400 .70 $2320. 70 i EQUATION OF PAYMENTS. 293 326. To find the equated time of settling an account con- taining debtor and creditor items. To equate an account, is to fix the time of payment of the ulurcliandise bahmce in such manner that the interest of each side shall be equal. The object of equating accounts is two- 1 ill : 1st. To find for what time interest must be charged ou file balance ; 2d. To find the date of a note, whose running time is fixed, and which is given in payment of the balance. Properly, the face of the note should be the sum, whose present value is the balance. If the note is given without interest, then its face is the balance ; and if the note becomes payable before the latest date, then interest must be charged for the remaining time. The process of equating accounts is similar to that of finding the cash balance : hence, we have the following Rule. — I. Find the merchandise balance: II. Find the number of days between the latest date of cither side, and the date of each item, and consider these numbers as InultipHers : III. MuUiiily each item by its multiplier ; then take the difference of the sums of these products and divide it by the merchandise balance: the quotient is the number of days, which, carried backward or forward from the latest date^ will give the equated date. Note. — When the greater sura of the items and the greater sum of the products fall on the same side of the account, the quotient is to he car- ried backward from the latest date : and forward, when these sums ar« found on different sides. 1. Equate the following account : Dr. James '. 1801. Jan. 16, To merch., $716.75 " 25, " " 000.00 Feb. 7, " " 2705.50 Mar. 19, " cash 701.25 N . Cr. 1. Jan. 19, By cash. $500.15 Feb. .1, " merch. , 1015.25 Mar. 7, " cash. 1200.00 April 3, " merch. , 712.00 294 EQUATION OF PAYMENTS. 2. What is the balance of the folio winff account — when due? Dr. Israel Jenkins. !835. May 6, To merch., $7150.00 " 16, " " 475.00 June 17, " " 3475.25 " 21, " " 1516.50 July 5, " " 279.00 Gr. $2450.00 1835. May 9, By cash, " 21, " " 915.00 June 12, " merch., 4165.50 " 19, " merch., 2915.50 3. What is the equated date for the payment of the bal- ance of the following account ? Dr. Jacob Parton. Cr, 1861. June 6, To merch., $8000.00 " 23, " " 1756.50 " 30, " cash, 2890.75 July 12, " note, 3000.15 1861. June 2, By merch., $7450.75 " 19, " " 2695.25 July 10, " " 1865.50 " 16, " " 970.00 827. Account of Sales. An Account of Sales is an account of the goods sold, with their prices, the charges thereon, and the net proceeds. Such an account a consignee transmits to the consignor. The net proceeds of such sale is nothing but the cash balance, due at the equated date. We will illustrate by the following example : ACCOUNT OF SALES OF FLOUll FOR A. MATr^E^YS, CHICAGO. Date. Purchaser. Description. Price. 1863. Nov. 5, Dec. 6, Dec. 19, Dec. 23, James Jackson, Robert Fisk, Francis Sutton, James Lyon, 75 bbls. superfine, 89 bbls. Excelsior, 120 bbls. fine, 66 bbls. ordinary, 350 bbls. $6.90 7.20 6.30 5.90 $517.50 640.80 756.00 389.40 $2303.70 CHARGES. Nov. 10th, cash paid transportation, Nov. 6th, insurance, Dec. 23d, storage, Commission on $230'\70, at 2^%, Total, ■ $16 9 10 57.58 $152.58 ALLIGATION. 295 ALLIiS-ATION. 328. Alligation is the process of mixing substances in such a manner that the value of the compound shall be equal to the sum of the values of the several ingredients. It is divided into two parts : Alligation Medial and Alligation Alternate. ALLIGATION MEDIAL. 329. Alligation Medial is the method of finding the price or quality of a mixture of several simple ingredients whose prices and quantities are known. I. A grocer would mix 200 pounds of lump sugar, worth 13 cents a pound, 400 pounds of Havana, worth 10 cents a pound, and 600 pounds New Orleans, worth T cents a pound : what should be the price of the mixture? Analysis.— The quantity, 2001b., opebation. at 13 cents a pound, costs $2G; 400 200 X 13 = 26.00 poimds, at 10 cents a pound, costs 400 X 10 = 40.00 $40; and GOO lb. at 7 cents a pound, 600 X 1 = 42.00 costs $42: hence, the entire mix- 1200 ) 108.00(9 cts. ture, consisting of 1200 lb., costs $108. Now, the price of the mixture will be as many cents as 1200 is contained times in 10800 cents, viz., 9 times: hence, to find the price of the mixture, Rule. — I. Find the cost of the mixture : II. Divide the cost of the mixture by the sum of the simples, and the quotient will be the price of the mixture. Examples. 1. If 1 gallon of molasses, at 75 cents, and 3 gallons, at 60 cents, be mixed with 2 gallons, at 31 J, what is the mixture worth a gallon? 2. If teas at 3TJ, 50, 62i, 80, and 100 cents per pound, be mixed together, what will be the value of a pound of the mixture ? 296 ALLIGATION. 3. If 5 gallons of alcohol, worth 60 cents a gallon, and 3 gallons, worth 96 cents a gallon, be diluted by 4 gallons of water, what will be the price of one gallon of the mixture ? 4. A farmer sold 50 bushels of wheat, at $2 a bushel ; 60 bushels of rye, at 90 cents ; 36 bushels of corn, at 62 J cents ; and 50 bushels of oats, at 39 cents a bushel : what was the average price per bushel of the whole? 5. During the seven days of the week, the thermometer stood as follows : 70°, 73°, 13^°, 17°, 70°, 80^°, and 81° : what was the average temperature for the week ? 6. If gold 18, 21, 17, 19, and 20 carats fine, be melted together, what will be the fineness of the compound? 7. A grocer bought 341b. of sugar at 5 cents a pound, 1021b., at 8 cents, 1361b. at 10 cents a pound, and 341b. at 12 cents a pound. He mixed it together, and sold the mix- ture so as to make 50 per cent, on the cost : what did he sell it for per pound ? 8. A merchant sold 81b. of tea, 111b. of coffee, and 251b. of sugar, at an average of 15 cents a pound. The tea was worth 30 cents a pound ; the coffee, 25 cents a pound ; and the sugar 7 cents a pound : did he gain or lose, and how much? ALLIGATION ALTERNATE. 330. Alligation Alternate is the method of finding what proportion of several simples, whose prices or qualities are known, must be taken to form a mixture of any required price or quality. It is the reverse of Alligation Medial, and may be proved by it. The process of Alligation Alternate is founded on an equality of gain and loss. In selling a mixture at an average price, there is a gain on each simple below that price, and a loss on each simple above that price. The gain must be exactly equal to the loss, otherwise the 7alue of the compound would not be an average price. ALTERNATE. 297 CASE I. 331. To find the proportional parts. 1. A miller would mix wheat, worth 12 shillings a bushel ; corn, worth 8 shillings ; and oats, worth 5 shillings, so as to nmke a mixture worth T shillings a bushel : what are the pioportional parts of each ? OPERATION. oats. 5s.- 7s. -jcorn, 8s.- ( wheat, 1 2s. A. B. c. D. i 4 5 1 1 2 i 2 B. 6 or 3 2 " 1 2 " 1 Analysis. — On every bushel put into the mixture, whose price is leKS than the mean price, there will be a gain; on every bushel whose price is greater than the mean price, there will be a loss ; and since the gains and losses must balance each oilier, we must connect an ingredient on which there is a gain with j aj^y number, the proportion of ili^i numbers in k 300 ALLIGATION". column E will be changed Thus, if we multiply column D by 12, we shall have 60 and 12, and the numbers in column E become Q6, 12 and 1, numbers which will fulfil the conditions of the question. Examples. 1. What quantity of teas at 12s. 10s. and 6s. mast be mixed with 20 pounds, at 4s. a pound, to make the mixture worth 8s. a pound? 2. How many pounds of sugar, at *l cents and 11 cents a pound, must be mixed with t5 pounds, at 12 cents a pound, so that the mixture may be worth 10 cents a pound? 3. How many gallons of oil, at ts., Is. 6d., and 9s. a gallon, must be mixed with 24 gallons of oil, at 9s. 6d. a gallon, so as to form a mixture worth 8s. a gallon ? 4. Bought 10 knives at $2 each : how many must be bought at Sf each, that the average price of the whole shall be $1^? 5. A grocer mixed 50 lb. of sugar worth 10 cents a pound, with sugars worth 9 J cents, tj cents, 1 cents, and 5 cents a pound, and found the mixture to be worth 8 cents a pound : how much did he take of each kind ? CASE III. 333. When the quantity of the mixture is given. 1. A silversmith has four sorts of gold, viz., of 24 carats fine, of 22 carats fine, of 20 carats fine, and of 15 carats fine; he would make a mixture of 42 ounces of 11 carats fine : how much must he take of each sort ? OPERATION. n A. B. C. D. E F. a. H. '15~ 20-1 ^ i i i 1 5 3 15 30 i 2 2 4 22 J 1 i 2 2 4 .24- -J } 2 2 4 rP.OPORTIONAL TARTS. 15 + 2 + 2 + 2 = 21 ; 42 -f 21 == 2. ALTERNATE. 301 Rule. — I. Find the proportional parts as in Case I. : II. Divide the quantity of the mixture by the sum of the proportional parts, and the quotient will denote hoiv many times each part is to be taken. Multiply the parts separately by this quotient, and each product will denote the quantity of the corresponding simple. Examples. 1. A grocer has teas at 5s., 6s., 8s., and 9s. a pound, and wishes to make a compound of 881b., worth 7s. a pound: how much of each sort must be taken? 2. A liquor dealer wishes to fill a hogshead with water, and with two kinds of brandy at $2.50 and 3.00 per gallon, so that the mixture may be worth $2.25 a gallon : in what proportions must he mix them ? 3. A person sold a number of sheep, calves, and lambs, 40 in all, for $48 : how many did he sell of each, if he received for each calf $lf, each sheep $1J, and each lamb $J? 4. A merchant sold 20 stoves for $180 ; for the largest size he received $19 each, for the middle size, $7, and for the small size, $6 : how many did he sell of each kind ? 5. A vintner has wines at 4s., 6s., 8s., and 10s. per gallon ; he wishes to make a mixture of 120 gallons, worth 5s. per gallon : what quantity must he take of each ? 6. A tailor has 24 garments, worth $144. He has coats, pantaloons, and vests, worth $12, $5, and $2 each, respect- vely : how many has he of each ? 7. A merchant has 4 pieces of calico, each worth 24, 22, 20, and 15 cents a yard: how much must he cut from each piece to exchange for 42 yards of another piece, wortli 17 cents a yard ? 8. A man paid $70 to 3 men for 35 days' labor ; to the Gi-st he paid $5 a day, to the second, $1 a day, and to the third, $J a day : how many days did each labor ? 302 ' CUSTOM-HOUSE BUSINESS. CUSTOM HOUSE BUSINESS. 334. All merchandise imported into the United States, must be landed at certain ports, called Ports of Entry. On such merchandise the General Government has imposed a greater or f. lb. 112 is 1, 112 to 224 it is 2, 224 to 336 " 3, 336 to 1120 *' 4, 1120 to 2016 " 7, 2016 any weight " 9, consequently, 91b. is the greatest draft generally allowed. Tare is an allowance made, after draft has been deducted, for the weight of the boxes, barrels, or bags containing the commodity, and is of three kinds : 1st, Legal tare, or such as is established by law ; 2d, Customary tare, or such as is established by the custom among merchants ; and, 3d, Actual tare, or such as is found by removing the goods and actually weighing the casks and boxes in which they are contained. On liquors in casks, customary tare is sometimes allowed ob the supposition that the cask is not full, or what is called its actual wants ; and then an allowance of 5 per cent, for leakage. A tare of 10 per cent, is allowed on porter, ale, and beer, in bottles, on account of breakage, and 5 per cent, on all other liquors in bottles. At the custom-house, bottles of the com mon size ar§ estimated to contain 2| gallons the dozen. For tables of Tare and Duty, see Ogden on the Tariff of 1842. Examples. 1. What is the net weight of 25 hogsheads of sugar, the gross weight being 66 cwt. 3 qr, 141b,; tare, 111b. per hogs- head? 304 CUSTOM-HOUSE BUSINESS. cwt. qr. lb, 66 3 14 gross. 25 X 11 = 2751b. - - 2 3 tare. Ans. 64 14 net. 2. If the tare be 4 lb. per hundred, what will be the tare on 6 T. 2 cwt. 3 qr. 14 lb. ? Tare for 6 T. or 120 cwt. = 480 lb. 2 cwt. = 8 3qr. = 3 141b. = 0^ Tare - - 491^ lb. 3. What will be the cost of 3 hogsheads of tobacco at 89. 4 1 per cwt. net, the gross weight and tare being of civt. qr. lb. lb. N-o .. 1 - - 9 3 24 - - tare 146 (< 2 - - 10 2 12 - - " 150 tt 3 - - 11 1 24 - - " 158 4. At 21 cents per lb., what will be the cost of 5 hhd. of coffee, the tare and gross weight being as -follows : civf. qr. lb. lb. sTo. 1 - - - 6 2 14 - . - tare 94 " 2 - - - 9 1 20 - _ - " 100 " 8 - - - 6 2 22 - - _ " 88 '* 4 - - - T 2 24 _ . - " 89 " 5 - - - 8 13 . . . " 100 at is the net weight of 18 hhd. of tobacco, 5. What is the net weight of 18 hhd. of tobacco, each weighing gross 8 cwt. 3qr. 14 1b.; tare 161b. to the cwt.? 6. What is the net weight and yalue of 80 kegs of fig?, ross weight 7T. 11 cwt. 3qr., tare 121b. per cwt., at $2.3 J per cwt. ? 7. A merchant bought 19 cwt. 1 qr. 24 lb. gross of tobacco in leaf, at $24.28 per cwt. ; and 12 cwt. 3qr. 191b. gross in rolls, at 828.56 per cwt.; the tare of the former was 1491b., and of the latter, 491b. : what did the tobacco cost him, net? CUSTOM-HOUSE BUSINESS. 305 8. A grocer bought 17Jlilitl. of sugar, each lOcwt. Iqr. 14 lb., draft 7 lb. per cwt., tare 4 lb. per cwt. : what is the value at $7.50 per c^vt. net ? 9. A merchant bought 7 hogsheads of molasses, each weigh- ing 4 cwt. 3 qr. 141b. gross, draft 7 lb. per cwt, tare 8 lb. per hogshead, and damage in the whole 99 j lb. : what is the value t 18.45 per cwt. net ? 10. The net value of a hogshead of Barbadoes sugar was 'i^22.50 ; the custom and fees 812.49, freight $5.11, factorage $1.31; the gross weight was 11 cwt. Iqr. 151b., tare lljlb. per cwt. : what was the sugar rated at per cv/t. net, in the bill of parcels. 11. I have imported 87 jars of Lucca oil, each containing 47 gallons ; what did the freight come to at $1.19 per cwt. net, reckoning lib. in 11 lb. for tare, and 91b. of oil to the gallon? 12. A grocer bought 5 hhd. of sugar, each weighing 13 cwt Iqr. 121b., at 7^ cents a pound; the draft was IJlb. per cwt., and the tare 5J per cent. : what was the cost of the net weight? 13. A wholesale merchant receives 450 bags of coffee, each weighing 7Clbs. ; the tare was eight per cent., and the invoice price lOJ cents per pound. He sold it at an advance of 33^ per cent. : what was his whole gam, and what his selling price ? 14. A merchant imported 176 pieces of broadcloth, each piece measuring 46Jyd., at $3.25 a yard; what will be the duty at 30 per cent.? 15. What is the duty on 54 T. 13 cwt. 3 qr. 201b. of iron, invoiced at $45 a ton, and the duty 33 J per cent. ? 16. What will be the duty on 225 bags of coffee, each weighing gross 1601b., invoiced at 6 cents per pound; 2 per cent, being the legal rate of tare, and 20 per cent, the duty ? 17. What duty must be paid on 275 dozen bottles of claret, estimated to contain 2f gallons per dozen, 5 per cent, being allowed for breakage, and the duty being 35 cents per gallon f 306 TONNAGE OF VESSELS. 18. A merchant imports 175 cases of indigo, each case weighing 1961b. gross ; 15 per cent, is the customary rate of tare, and the duty 5 cents per pound ; what duty must he pay on the whole ? 19. What is the tare and duty on ^5 casks of Epsom salts, each weighing gross 2 cwt. 2 qr. 24 lb., and invoiced at IJ cents per pound, the customary tare being 11 per cent, and the rate of duty 20 per cent.? TONNAGE OF VESSELS. 343. There are certain custom-house charges on vessels, which are made according to their tonnage. The tonnage of a vessel is the number of tons weight she will carry, and this is determined by measurement. [From the "Digest," by Andrew A. Jones, of the N. Y. Custom-house.] CuMoni'/iouse cJiarges on all ships or vessels entering from any foreign port or place. Ships or vessels of tlie United States, having three-fourths of tlie crew and all the officers American citizens, J9er ton, - $0.06 Ships or vessels of nations entitled by treaty to enter at the same rate as American vessels, .06 Ships or vessels of the United States not having three-fourths of the crew as above, .50 On foreign ships or vessels other than those entitled by treaty, .50 Additional tonnage on foreign vessels, denominated light- money, .50 Licensed coasters are also liable once in each year to a duty of 50 cents per ton, being engaged in a trade from a port in one State to a port in another State, other than an adjoining State, unless tlie officers and three-fourths of the crew are American citizens; to ascer- tain which, the crews are always liable to an examination by an officer. A foreign vessel is not permitted to carry on the coasting trade; but having arrived from a foreign port with a cargo consigned to TONNAGE OF VESSELS. 307 more than one port of the United States, she may proceed coastwise with a certified manifest until her voyage is completed. 344. The government estimate the tonnage according to one I lie, while the ship-carpenter, who builds the vessel, uses an- other. G-overiunent Rule. — I. Measure, in feet, above the ujype deck the length of the vessel, from the forepart of the main stem to the after-part of the stern-post. Then measure the breadth taken at the widest part above the main wale on the outside, and the depth from, the under-side of the deck-plank fa the ceiling in the hold: II. Fj'om the length take three-fifths of the breadth, and niultiphj the remainder by the breadth and depth, and the product divided by 95 will give the tonnage of a si^igle- decker ; and the same for a double-decker, by merely making the depth equal to half the breadth. Cajpenters' Rule. — Mxdtiply together the length of the ki'el, the breadth of the main beam, and the depth of the hold, and the i^roduct divided by 95 will be the carpenters^ tonnage for a single-decker ; and for a double-decker, deduct from tlie depth of the hold half the distance between decks. Examples. 1. What is the government tonnage of a single-decker, whose length is 75 feet, breadth 20 feet, and depth IT feet? 2. What is the carpenters^ tonnage of a single-decker, the length of whose keel is 90 feet, breadth 22 feet T inches, and depth 20 feet 6 inches? 3. What is the carpenters' tonnage of a steamship, double decker, length 154 feet, breadth 30 feet 8 inches, and depth, ifter deducting aalf between decks, 14 feet 8 inches? 4. What is the carpenterti' tonnage of a double-decker, its length 125 feet, breadth 25 feet 6 inches, depth of hold 34 feet, and distance between decks 8 feet? 308 GENERAL AVERAGE. GENERAL AVERAGE. 345. Average is a term of commerce signifying a contribution \ty individuals, where the goods of a particular merchant are thrown overboard in a storm, to save the ship from sinking ; or where the masts, cables, anchors, or other furniture of the ship are cut away or destroyed, for the preservation of the vessel. In these and like cases, where any sacrifices are de- liberately made, or any expenses voluntarily incurred, to pre- vent a total loss, such sacrifice or expense is the proper subject of a general contribution, and ought to be ratably borne by the owners of the ship, the freight, and the cargo, so that the loss may fall proportionably on all. The amount sacrificed is called the Jettison. 343. Average is either general or particular ; that is, it is either chargeable to all the interests, viz., the ship, the freight, and the cargo, or only to some of them. As when losses occur from ordinary wear and tear, or from the perils incident to the voyage, without being voluntarily incurred ; or when any particular sacrifice is made for the sake of the ship only, or the cargo only, these losses must be borne by the parties immediately interested, and are consequently defrayed by a particular average. There are also some small charges, called petty or accustomed averages, one-third of which is usually charged to the ship, and two-thirds to the cargo. No general average ever takes place, except it can be shown that the danger was imminent, and that the sacrifice was made indispensable, or was supposed to he so, by the captain and officers, for the safety of the ship. 347. In different countries different modes are adopted ot valuhig the articles which are to constitute a general average. In general, however, the value of the freightage is held to be the clear sum which the ship has earned after seamen's wages, pilotage, and all such other charges as come under the name GENERAL AVERAGE- 300 of petty charges, are deducted ; one-third, and in some cases one-half, being deducted for the wages of the crew. The goods lost, as well as those saved, are valued at the price they would have brought, in ready money, at the place of deli very, on the ship's arriving there, freight, duties, and all other charges being deducted : indeed, they bear their pro- portions, the same as the goods saved. The ship is valued at the price she would bring, on her arrival at the port of de- livery. But when the loss of masts, cables, and other furni- ture of the sliip is compensated by general average, it is usual, as the new articles will be of greater value than the old, to deduct one-third, leaving two-thirds only to be charged to the amount to be contributed. Examples. 1. The vessel Good Intent, bound from New York to New Orleans, was lost on the Jersey beach the day after sailing. She cut away her cables and masts, and cast overboard a part of her cargo, by which another part was injured. The ship was finally got off, and brought back to New York. AMOUNT OF LOSS. Goods of A cast overboard, - - - - $500 Damage of the goods of B by the jettison, - 200 Freight of the goods cast overboard, - - 100 Cable, anchors, mast, &c., worth - 8300 ) Deduct one-third, - - - - 100 ) Expenses of getting the ship off the sands, - 66 Pilotage and port duties going in and out] of the harbor, commissions, &c., - j Expenses in port, ------ 25 Adjusting the average, - - - . - - 4 Postage, ------ - 1 Total loss, $1186 310 GENERAL AVERAGE. ARTICLES TO CONTRIBUTE. Goods of A cast overboard, 1500 Value of B's goods at IS". 0., deducting freight, &c., 1000 '* of C's " (I ti it 500 " of D^s " (( tt u 2000 " of E's " it It u 5000 Value of the ship, - - - 2000 Freight, after deducting one-third, • • 800 $11800 Then, Total value : total loss ; : 100 : per cent. of loss. 111800 : 1180 : : 100 : 10; hence, each loses 10 per cent, on the value of his interest in the cargo, ship, or freight. Therefore, A loses $50 ; B, $100 ; C, 50 ; D, $200 ; E, $500 ; the owners of the ship, $280— in all, $1180. Upon this calculation, the owners are to lose $280 ; but they are to receive their disbursements from the contribution : viz., freight on goods thrown overboard, $100 ; damages to ship, $200 ; various disbursements in expenses, $180 ; total, $480 ; and deducting the amount of contribution, they will actually receive $200. Hence, the account will stand : The owners are to receive $200 A loses $500, and is to contribute $50 ; hence, he receives B loses $200, and is to contribute $100 ; hence, he receives Total to be received, - - $150 450 100 f C, 50 C, D, and E, have lost nothing, and are to pay } D, 200 t E, 500 Total actually paid, - - - $750 COINS AND CURRENCIES. 811 COINS AND CURRENCIES. 348. Coins are pieces of metal, of gold, silver, or copper, of fixed values, and impressed with a public stamp prescribed by the country where they are made. These are called specie, and are generally declared to be a legal tender in payment of debts. The Constitution of the United States provides, that the value of gold and silver coins shall be fixed by act of Congress. The coins of a country, and those of foreign countries hav- mg a fixed value established by law, together with bank-notes redeemable in specie, make up what is called the Cun^ency. 349. A Foreign coin may be said to have four values : 1st. The intrinsic value, which is determined by the amount of pure metal which it contains : 2d. The Custom-house, or legal value, which is fixed by law : 3d. The mercantile value, which is the amount it will sell for in open market : 4th. The exchange value, which is the value assigned to it in buying and selling bills of exchange between one country and another. Let us take, as an example, the English pound sterling, which is represented by the gold sovereign. Its intrinsic value, as determined at the Mint in Philadelphia, compared with our gold eagle, is $4,861. Its legal or custom-house value is $4.84. Its commercial value, that is, what it will bring in Wall-street, New York, varies from 14.83 to $4.86, seldom reaching either the lowest or highest limit. The exchange value of the English pound, is $4.44^, and was the legal value before the change in our standard. This change raised the cgal value of the pound to $4.84 ; but merchants, and dealers in exchange, preferred to retain the old value, which became nominal, and to add the diflference in the form of a premium on exchange^ which is explained m Art. 365. For the values of the various coins, see Table, page 406 312 EXCHANGE. EXCHANG-E. 350. Exchange is a term whicli denotes the payment of money by a person residing in one place to a person residing in another. The payment is generally made by means of a bill of exchange. 351. A Bill of Exchange is an open letter of request from one person to another, desiring the payment to a third party named therein, of a certain sum of money to be paid at a specified time and place. Of a bill of exchange three copies are made, and are called a set of exchange. They are sent by different ways to the drawee, so that in case one is lost, another may reach hhn. There are always three parties to a bill of exchange, and generally four : 1. He who writes the open letter of request, is called the drawer or maker of the bill ; 2. The person to whom it is directed, is called the drawee ; 3. The person to whom the money is ordered to be paid is called the payee ; and 4. Any person who purchases a bill of exchange is called the h^^yer or r^emitter. 352. Bills of exchange are the proper money of commerce. Suppose Mr. Isaac Wilson, of the city of New York, ships 1000 bags of cotton, worth £6000, to Samuel Johns & Co., of Liverpool ; and at about the same time William James, of New York, orders goods from Liverpool, of Ambrose Spooner, to the amount of six thousand pounds sterling. Now, Mr. Wilson draws a bill of exchange on Messrs. Johns & Co., in the following form, viz. : Exchange for £G000. New York, July 80th, 1846. Sixty days after sight of this my first Bill of Exchange (second and third of the same date and tenor unpaid), pay to EXCUANGK. 313 David C. Jones, or order, six thousand pounds sterling, with or without further notice. Isaac Wilsox. Messrs. Samuel Johna & Co.,) Merchants, Liverpool. ) Let us now suppose that Mr. James purchases this bill of David C. Jones, for the purpose of sending it to Ambrose Spooner, of Liverpool, whom he owes. We shall then have all the parties to a bill of exchange ; viz., Isaac Wilson, the maker or drawer ; Messrs. Johns & Co., the drawees; David C. Jones, the payee; and William James, the buyer or re- mitter. 353. A bill of exchange is called an inland bill, when the drawer and drawee both reside in the same country ; and when they reside in different countries, it is called a foreign bill. Thus, all bills in which the drawer and drawee reside m the United States, are inland bills ; but if one of them resides in England or France, the bill is a foreign bill. 354. The time at which a bill is made payable varies, and is a matter of agreement between the drawer and buyer. They may either be drawn at si^ht, or at a certain number of days after sight, or at a certain number of days after date. 355. Days of Grace are a certain number of days granted to the person who pays the bill, after the time named in the bill has expired. In the UnHed States and Great IBritaiu three days are allowed. 356. In ascertaining the time when a bill, payable so many days after sight, or after date, actually falls due, the day of presentment, or the day of the date, is not reckoned. When the time is expressed in months, calendar moiUhs are alway understood. If the month in which a bill falls due is shorter than the one in which it is dated, it is a rule not to go^on into the next month. Thus, a bill drawn on the 28th, 29th, 30th, or 31st of December, payable two months after date, falls due 14 >14 E,XCHANGE. on Hie last of February, except for the daj^s of grace, and would be actually due on the third of March. INDORSING BILLS. 357. In examining the bill of exchange drawn by Isaac Wilson, it will be seen that Messrs. Johns & Co. are requested to pay the amount to David C. Jones, or order ; that is, either to Jones or to any other person named by him. If Mr. Jones simply writes his name on the back of the bill, he is said to indorse it in hlanky and the drawees must pay it to any rightful owner who presents it. Such rightful owner is called the holder, and Mr. Jones is called the indorser. If Mr. Jones writes on the back of the bill, over his signa- ture, "Pay to the order of William James," this is called a special indorsement, and William James is the indorsee, and he may either indorse in blank, or write over his signature, " Pay to the order of Ambrose Spooner," and the drawees, Messrs. Johns & Co., will then be bound to pay the amount to Mr. Spooner. A bill drawn payable to bearer, may be transferred by mere delivery. ACCEITANCE. 358. When the bill drawn on Messrs. Johns & Co. is pre- sented to them, they must inform the holder whether or not they will pay it at the expiration of the time named. Their agreement to pay it is signified by writing across the face of the bill, and over their signature, the word "accepted," and they are then called the acceptors. LIABILITIES OF THE PARTIES. 359. The drawee of a bill does not become responsible for its payment until after he has accepted. On the presentation of the bill, if the drawee does not accept, the holder should immediately take means to have the drawer and all the in- KMCHANQE. 315 dorsers notified. Such notice is called a protest, and is giren by a public officer called a notary, or notary public. If the indorsers are not notified in a reasonable time, they are not responsible for the amount of the bill. If the drawee accepts the bill, and fails to make the pay. ment when it becomes due, the parties must be notified as before, and this is called protesting the hill for non-payment. If the indorsers are not notified in a reasonable time, they are not responsible for the amount of the bill. PAR OF EXCHANGE — COURSE OF EXCHANGE. 360. The intrinsic par of exchange, is a term used to com- pare the coins of different countries with each other, with re- spect to their intrinsic values ; that is, with reference to the amount of pure metal in each. Thus, the English sovereign, which represents the pound sterling, is intrinsically worth $4. 8 CI in our gold, taken as a standard, as determined at the Mint in Philadelphia. This, therefore, is the value at which the sovereign should be reckoned, in estimating the par of ex- change. 361. The commercial par of exchange is a comparison of the coins of different countries according to their market value. Thus, as the market value of the English sovereign varies from $4.83 to $4.85 (Art. 349), the commercial par of exchange will fluctuate. It is, however, always determined when we know the value at which the foreign coin sells in open market. 362. The course of exchange is the variable price which is paid at one place for bills of exchange drawn on another The course of exchange differs from the intrinsic par of ex hange, and also from the commercial par, in the same way that the market price of an article differs from its natural price. The commercial par of exchange would at all times de- termine the course of exchange, if there were no fluctuations in trade. 316 EXCHANGE. 363. When the market price of a foreign bill is above the commercial par, the exchange is said to be at a premium, or in favor of the foreign place, because it indicates that the foreign place has sold more than it has bought, and that specie must be shipped to make up the difference. When the market price is below this par, exchange is said to be beloiv vaVf or in favor of the place where the bill is drawn. Such place will then be a creditor, and the debt must be paid in specie or other property. It should be observed, that a favor- able state of exchange is advantageous to the buyer, but not to the seller, whose interest, as a dealer in exchange, is ident'- fied with that of the place on which the bill is drawn. INLAND BILLS. 364. We have seen that inland bills are those in which the drawer and drawee both reside in the same country (Art. 853). Examples. 1. A merchant at New Orleans wishes to remit to New York $8465, and exchange is IJ per cent, premium : how much must he pay for such a bill ? 2. A merchant in Boston wishes to pay in Philadelphia $8746.50 ; exchange between Boston and Philadelphia is IJ per cent, below par : what must he pay for a bill ? 3. A merchant in Philadelphia wishes to pay $98*16.40 in Baltimore, and finds exchange to be 1 per cent, below par : what must he pay for the bill ? 4. What must be paid for a draft of $10000, payable 60 days after sight, on St. Louis, exchange being at a premium of |"/o, interest bemg charged at 6%? 5. What amount of exchange on New Orleans can be bought for $14815, the discount being IVo? 6. For what amount must a bill of exchange, at 30 days, bo drawn, for which I paid $9650, discount 1%, and the interest being 6%? EXCHANGE. 817 ENGLAND. 365. It has been stated that exchanges between the United States and England are made in pounds, shillings and pence, and that the exchange value of the pound sterling is reckoned at 84.44|- = 4.4444 + ; that is, this value is the base in which the bills of exchange are drawn. Now this value *)eing below both the commercial and intrinsic value, the drawers of bills increase the course of exchange so as to make up this deficiency. For example, if we add to the exchange value of the pound, 9 per cent., we shall have its commercial value, very nearly. Thus, exchange value, - - - = $4.4444 + Nine per cent., = .3999 + which gives, - - - - $4.8443 and this is the average of the commercial value, very nearly. Therefore, when the course of exchange is at a premium of 9 per cent., it is at the commercial par; and as between England and this country, it would stand near this point but for the fluctuations of trade and other accidental circumstances. Examples. 1. A merchant in New York wishes to remit to Liverpool £\IQ1 10s. 6d., exchange being at 8 J per cent, premium: how much must he pay for the bill in United States money? First, ieiI67 10s. 6d. - - - = £1U1.525 Multiply by 8J per cent., - - .085 The product is the premium - - = 99.239625 This product added, gives - - iE1266.764625 which, reduced to dollars and cents, at the rate of $4.44| (o the pound, gives $5630.008 +, the amount which must be paid for the bill in dollars and cents. 2. A merchant has to remit £36794 8«. 9d. to London: 318 EXCHANGE. how much must he pay for a bill in dollars and cents, ex- change being Yf per cent, premium ? 3. A merchant in New York wishes to remit to London $67894.25, exchange being at a premium of 9 per cent. : what will be the amount of his bill in pounds shillings and pence? Note. — Add the amount of tlie premium to tlie exchange value of tlie pound; viz., $4.44^, which, in this case, gives $4.84444; and then divide the amount in dollars by this sum, and the quotient will be the amount of the bill in pounds and the decimals of a pound. 4. A merchant in New York owes ^1256 18s. 9d. in Lon- don ; exchange at a nominal premium of 7 J per cent. : how much money, in United States currency, will be necessary to purchase the bill? 5. I have $947.86, and wish to remit to London £364 18s. 8d., exchange being at S\ per cent. : how much additional money will be necessary? 6. Received, on consignment from London, an invoice of English cloths amounting to £1569 10s. The duties thereon amounted to |416 ; storage, cartage, and insurance, amounted to $85. The cloths were sold at an advance of 26 per cent. on the invoice. Supposing the commission 2J per cent., and the premium of exchange 12 per cent., what would be the face of the bill of exchange that would cover the net proceeds? FRANCE. 366. Accounts in France, and the exchange between France and other countries, are all kept in francs and centimes, which are hundredths of the franc. We see, from the table, that the value of the franc is 18.6 cents, which gives, very nearly, 5 francs and 38 centimes to the dollar. The rate of exchange is computed on the value 18.6 cents, but is often quoted by stat- ing the value of the dollar in francs. Thus, exchange on Paris is said to be 5 francs 40 centimes ; that is, one dollar will buy a bill on Paris of 5 francs and 40 hundredths of a franc. EXCHANGE. 310 Examples. 1. A merchant ia New York wishes to remit 161556 francs to Paris, exchange being at a premium of 1 J per cent. : what will be the cost of his bill in dollars and cents ? Commercial value of the franc, - - 18.6 cents, Add IJ per cent., - - - . .279 Gives value for remitting, - - - 18.819 cents ; then, 161556 X 18.879 = 131632.89724, which is the amount to be paid for the bill. 2. What amount, in dollars and cents, will purchase a bill on Paris for 86978 francs, exchange being at the rate of 5 francs and 2 centimes to the dollar? First, 86978 ~- 5.02 = $17326.29, the amount, nearly. Is this bill above or below par? What per cent.? 3. How much money must be paid to purchase a bill of exchange on Paris for 68097 francs, exchange being 3 per cent, below par? 4. A merchant in New York wishes to remit $16785.25 to Paris ; exchange gives 6 francs 4 centimes to the dollar : how much can he remit in the currency of Paris? HAMBURG. 367. Accounts and exchanges with Hamburg, are generally made in the marc banco, valued, as we see in the table, at 35 cents. Examples. 1. What amount, in dollars and cents, will purchase a bill of exchange on Hamburg for 18649 marcs banco, exchange being at 2 per cent, premium? 2. What amount will purchase a bill for 3678 marcs banco, reckoning the exchange value of the marc banco at 34 cents? Will this be above or below the par of exchange ? 320 EXCHANGE, ARBITRATION OF EXCHANGE 368. Arbitration of Exchange is the method by which the currency of one country is changed into that of another, through the medium of one or more intervening currencies, with which th« first and last are compared. 369. When there is but one intervening currency, it is called Simple Arbitration ; and when there is more than one, it is called Compound Arbitration. The method of performmg this is called the Chain Rule. 370. The principle involved in arbitration of exchange is simply this : To pass from one system of values through several others, and find the true proportion between the first and last. I. Let it be required to remit $65t0 to London, by the way of Paris, exchange on Paris being 5 francs 15 centimes for $1, and the exchange from Paris to London 25 francs and 80 centimes for iEl : what will be the value of the remittance to London? £1 Analysis. — $1 =: 5.15 francs : and 1 franc = • 25.80 If %\ were remitted to Paris, it would produce there 5.15 francs; an6 if 1 frano were remitted from Paris to London, it would pro' duce there • 25.80 But $6570 are remitted to Paris; hence, they produce there 6570 X 5.15 francs; and this amount is remitted to London; hence, it produces there, . 6570 X 5.15 X -^ = £1311 9s. Old. 25.80 Hule.—I. Find the value of a single unit of each of the moneys named, in the money of the place next named : II. Multi2Dly the sum to be remitted by these values in succession, and the product will be the equivalent in the money of the place to ivhich the remittance is to be made. EXCHANGE 321 Examples. 1. A merchant wishes to remit $4888.40 from New York to London, and the exchange is at a premium of 10 per cent. He finds that he can remit to Paris at 5 francs 15 centimes to the dollar, and to Hamburg at 35 cents per marc banco. Sow, the exchange between Paris and London is 25 francs 80 i-entiines for £1 sterling, and between Hamburg and London 13J marcs banco for £1 sterling : how had he better remit ? OPERATION. Ist. To London direct. $4888.40 X j.-sh-i = ieiOOO. 1.03 2d. Through Paris. 4888.40 X ^ X ^ = j£975.7852 = £d1o 15s. 8}d. 1 '^$M 6.16 3d. Through Hambiirg. $4888.40 X jV X T!>?75 = ^1015.771 = .£1015 15s. 5d. Hence, the best way to remit is through Haml3Hrg, then direct ; and the least advantageous, through Paris. 2. A merchant in New York wishes to transmit $1500 tc Vienna, through London and Hamburg : what will be tlui value when received, if £1 = $4.86, iBl = 14 marcs banco, and 6 marcs banco = 8 florms ? 3. A merchant at Natchez wishes to pay $10000 in Boston He transmits through New Orleans and New York. From Natchez to New Orleans exchange is |Vo premium, from New Orleans to New York f/o discount, and from New York to Boston i'/o discount : by this exchange, what amount at NaUoiea will pay the debt? 322 INVOLUTION. 4. A, of London, draws a bill of £862 10s. ou B, of Cadiz, and remits the same to C, of Havre, who, in turn, remits to D, of Amsterdam, and D remits to B, of Cadiz : how much will pay the bill, if 1 Spanish dollar = 2 florins 15 stivers, 12 florins = 26 francs, and 24 f. 15 c. = iEl? INVOLUTION. 371. A POWER OF A NUMBER is any product which arises from multiplying the number contmually by itself. The root, or simple factor, is called the first power : The second power is the product of the root by itself: The third power is the product, when the root is taken 3 times as a factor : The fourth power is the product, when it is taken 4 times : The fifth power is the product, when it is taken 5 times. 372. The number denoting how many times the root is taken as a factor, is called the exponent of the power. It is written a little at the right and over the root : thus, if the equal factor or root is 3, 3^= 3, the 1st power, root, or base. 32= 3 X 3 = 9, the 2d power of 3. 33= 3 X 3 X 3 = 27, the 3d power of 3. 3* = 3 X 3 X 3 X 3 = 81, the fourth power of three. 373. Involution is the operation of finding the powers ol numbers. Note. — 1. There are three things connected with every power: 1st, Tlie root ; 2d, The exponent ; and 3d, The power or result of the multiplication. 2. In finding any power, one multiplication gives the 2d power: hence, the number of multiplications is 1 less Mian the exponent. Rule. — Multiply the number into itself as many times less 1 as there are units in the exponent, and the last product will be the poioer. EVOLUTION. 32 Find the power 1. The square 2. The square 3. The square 4. The square 5. The square 6. The square 7. The square 8. The square 9. The square 10. The square 11. The square 12. The square 13. The square 14. The square 15. The square 16. The square 17. The square Examples, of the followhi;]^ numbers : of 4? 18. The cube of 6? of 15? 19. The cube of 24 ? of 142? 20. The cube of 125? of 463 ? 21. The cube of 136 ? of 1340? 22. The 4th power of 12 ? of 24.6 ? 23. The 5th power of 9 ? of .526? 24. The value of (4.25)' ? of 3.125? 25. The value of (1.8)^ ? of .0524? 26. The value of (.45)^^ ? off? 27. The value of (}f)^? off? 28. The cube of (|) ? of 1 ? 29. The 4th power of f ? off J? 30. The value of (21)" ? ofifl? 31. The value of {^y ? of tf? 32. The value of (24f)'? of 15x\? 33. The value of (.25)^ ? of 225/5 ? 34. The value of (142.5)^? EVOLUTION. 374. Evolution is the operation of finding the root of a number ; that is, of finding one of its equal factors. 375. The Square Root of a number is tlie factor which, multiplied by itself once, will produce tlie number. Thus, 8 is the square root of 64, because 8 X 8 = 64. The sign ^ is called the radical sign. When placed before a number, it denotes that its square root is to be extracted : Thus, y'36 = 6. . 376. The Cube Root of a number is the factor whicli, mul- tiplied by itself iicnce, will produce the number. 824 EXTRACTION OF THE SQUARE ROOT. Thus, 3 is the cube root of 21, because 3 x 3 x P» = 21. We denote the cube root by the sign -y/ , with 3 written over it : thus, -^^27, denotes the cube root of 21, which is equal to 3. The small figure 3, placed over the radical, is called the index of the root. The terms Power and Root, are dependent on each other : thus, the power is the product of equal factors ; and the root is one of the equal factors. EXTRACTION OF THE SQUARE ROOT. 377. The Square Root of a number is one of its two equal factors. To extract the square root is to find this factor The first ten numbers and their squares are : 1, 2, 3, 4, 5, 6, 1, 8, 9, 10. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The numbers in the first line are the square roots of those in the second. The numbers 1, 4, 9, 16, 25, 36, &c., haying two exact equal factors, are called perfect squares. A PERFECT SQUARE is a uumbcr which has two exact equal factors. Note. — The square root of a number less than 100 %\'lll be loss than 10 ; while the square root of a number greater than 100 will be greater than 10: hence, the square root of a number expressed by ono or two figures, is a number expressed by one figure. 378. To find the law of the square of a number. Any number expressed by two or more figures may be ro garded as composed of tens and units. i. What is the square of 36 ~ 3 tens -f 6 units? OPKRATION. 3 + 6 3 + 6 3 3' + 3 X 6 + 6^ X 6 EXTKACTION OF THE SQUARE ROOT. 325 Analysis. — The square of 36 is found by taking 36, thirty-six times. This is done by first taking it 6 units times, and then 3 tens times, and adding the products. 36 taken 6 units times, gives 6^ + 3 X 3; and taken 3 tens times, 3' + 2 (3 X 6) + 6' gives 3 X 6 + 3^; and their sum is, 8^ + 2 (3 X 6) + 6=: that is, Rule. — The square of a number is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units. 379. To find the square root of any number. 1. Let it now be required to extract the square root of 2025. Analysis. — Since the number contains more than two places of figures, its root "will contain tens and units. But as the square of one ten is one Imndred, it follows that the square of the tens of the required root must be found in the figures on the left of 25. Hence, beginning at the right, we point off the number into periods of two figures each. We then find the root contained in 20 bun- operation. (h-eds, which is 4 tens or 40. We then square 20 25(45 4 tens, which gives 16 hundred, and then place jg 16 under the first period, and subtract; this 85^)42 5 takes away the square of the tens, and leaves 42 5 425, w7iich is twice the product of the tens by the units plus the square of the units. If, now, we double the tens, and then divide the remainder, ex- clusive of the right-hand figure (since that figure cannot enter into the product of the tens by the units), by it, the quotient will be the units figure of the root. If we annex this figure to the root and to the augmented divisor, and then multiply the whole divisor thus increased by it, the product will be twice the tens by the units, plus the square of the imits* and hence, we have found both figures of the root. 326 EXTRACTION OF THE SQUARE KOOT. Rule. — I. Separate the given number into periods of tivo Jlgur'es each, by writing a dot over the place of units, a second over the place of hundreds, and so on for each alternate figure to the left: II. Note the greatest square contained in the period on the left, and place its root on the right, after the manner of a quotient in division. Subtract the square of this root from the first period, and to the remainder bring down the second period for a dividend: III. Double the root thus found for a trial divisor, and, place it on the left of the dividend. Find how many times (he trial divisor is contained in the dividend, exclusive of its right-hand figure, and place the quotient in the root, and also annex it to the divisor: TV. Multiply the divisor thus increased, by the last figure of the root y subtract the product from the dividend, and to the remainder bring down the next period for a new divi- dend: Y. Doid)le the ichole root thus found, for a new trial di- visor^ and continue the operation as before, until all the periods are brought down. Examples, I. What is the square root of 425104 ? Analysis. — We. first place a dot over operation. the 4, making the right-hand period 04. ^2 5i 04(652 We then put a dot over the 1, and also 35 over the 2, making three periods. 195'ifi'Sl The greatest perfect square in 42 is g25 Sa, the root of which is 6. Placing 6 in ^ ,^ the root, subtracting its square from 42, ^ "" ofOi ^d bringing down the next period 51, we have 651 for a dividend; and by doubling the root, we liave 12 for a trial divisor. Now, 13 is contained in 65, 5 times. Place 5 both in the root and in the divisor; then multiply 125 by 5; sub- tract the product, and bring down the next period. EXTKACTION OF THE SQUARE ROOT. 327 We must now double the whole root G5 for a new trial divisor; or we may take the first divisor, after having doubled the last figure 6; then dividing, we obtain 2, the third figure of the root. Notes. — 1. The left-hand peri(xl may contain but one figure; each of the others will contain two. 2. If any trial divisor is greater than its dividend, the correspond- ing root figure will be a cipher. 3. If the product of the divisor by any figure of the root exceed the corresponding dividend, the root figure is too large, and must be diminished. 4. There will be as many figures in the root as there are i)eriod8 in the given number. 5. If the given number is not a perfect square, there will be a remainder after all the periods are brought down. In this case, lx3riods of ciphers may be annexed, forming new periods, each of which will give one decimal place in the root. 2. What is the square root of 758692 ? OPERATION. 75 86 92(871.029 +. 64 Analysis. — After using all the periods of the given num- ber, we annex periods of deci- mal ciphers, each of which gives one decimal place in the quo- tient. What are the square roots of 3. Square root of 49? 4. Square root of 144? 5. Square root of 225 ? 6. Square root of 2304 ? 7. Square root of 7994? 8. Square root of 6275025 ? 167)11 86 11 69 1741)17 92 17 41 174202)510000 348404 1742049)16159600 15678441 48ll59 Rem. the following numbers : 9. -v/10000 = what No. ? 10. ^2768456 = what No. ? 11. i/3«T54 =: what No. ? 12. VH>00000 = what No.? 13. -/y 6Y2^ = what No.?. 14. -v/30225'' = what No. ? 328 EXTRACTION OF THE SQUARE ROOT. 380. To extract the square root of a fraction. 1. What is the square root of .6? Analysis. — "We first annex one cipher, operation. to make even decimal places; for, one .60(.*It4 + decimal multiplied by itself will give 49 wo places in the product. We then extract the root of the first period, and to the remainder annex a decimal period; and so on, till we have found a sufficient number of decimal places. 2. What is the square root of if' Analysis. — The square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator. 3. What is the square root of f ? Analysis. — When the terms are not perfect squares, reduce the common fraction to a decimal, and then extract the square root of the decimal. 141)1100 1029 1544)7100 6176 924 rem. OPERATION. — -s/Ie — 4 ~ -^25 5' OPERATION. J - .75 ; -/!= V'?15 = .8545 + Rule. — I. If the fraction is a decimal, point off the periods from the decimal point to the right, annexing ciphers if necessary, so that each period shall contain two places, and then extract the root as in integral numbers: II. If the fraction is a common fraction, and its terms verfect squares, extract the square root of the numerator and denominator separately : III. If, after being reduced to their lowest terms, the numerator and denominator are not perfect squares, reduce the fraction to a decimal, and then extract the square root of the result. EXTRACTION OF THE SQUARE ROOT. 329 Examples. What are the square roots of the following numbers? 4. Square root of f f ? 5. Square root of ^^^^ ? 6. Square root of .0196? 7. Square root of 6.25 ? 8. Square root of 278.89 ? 9. Square root of .205209 ? 10. Square root of J? 11. Square root of |J ? 12. Square root of J^ ? 13. Square root of 5 J- ? 14. Square root of .7994? 15. Value of -yJ^Mi ? 16. Square root of .60794? 17. Value of -v/.02220i ? 18. Value of -^25.1001 ? 19. Value of -v/l9G.425 ? 20. Value of yTs"? 21. Value of n/lfls ? '' 6 2 4 1 * 22. Value of >/T ? 23. Value of ^X ? 24. Value of ^l35 ? 25. Value of -v/."784 ? 26. Square root of 5647.5225? 27. Square root of 160048.0036? Applications in Square Root. 381. A TRIANGLE Is a plain figure which has three sides and three angles. If a straight line meets another straight line, making the adjacent angles equal, each is called a right angle ; and the lines are said to be perpen- dicular to each other. 382. A RIGHT-ANGLED triangle is one which has one right angle. In the right-angled triangle ABC, the side AC, opposite the right angle B, is called the hypothenuse ; "^^ the side AB, the base; and the side BC, a the pei^endicular. Base. 383. A SQUARE is a figure bounded by. four equal sides, at rij^ht an"cles to each other. 384. In a right-angled triangle the square described on the 330 EXTRACTION OF THE SQUARE HOOT.' D _. hypothenuse is equal to the sum of the squares described on the other two sides. Thus, if ACB be a right-angled triangle, right-angled at C, then will the large square, D, described in the hypothenuse AB, be equal to the sum of the squares F and E, described on the sides AC and CB. This is called the carpenter's theorem. By counting the small squares in the large square D, you will find their number equal to that contained in the small squares F and E. In this triangle the hypothenuse AB = 5, AC = 4, and CB = 3. Any num- bers having the same ratio, as 5, 4, and 3, such as 10, 8, and 6 ; 20, 16, and 12, &c., will represent the sides of a right- angled triangle. 385. When the base and perpendicular are known, to find the hypothenuse. Analysis. — Wishing to know the distance from A to the top of a tower, I measured the height of the tower, and found it to bo 40 feet ; also the distance from A to B, and found it 30 feet: what was the distance from A to 02 AB = 30; AB2 = 30^ = 900 BO = 40; BO- = 40= = 1600 AC2=AB2-f B02 = 900 + 1600 AC = v/2500 = 50 feet. Rule. — Square the base and square the perpendicular, add the results, and then extract the square root of their sum. 386. To find one side, when we know the hypothenuse and the other side. 1. The length of a ladder wliich will reach from the middle EXTRACTION OF THE SQUARE ROOT. 331 of a street 80 feet wide to the eaves of a house, is 50 feet : what is the height of the house? Analysis. — Since the square of the length of the ladder is equal to the sum of the squares of half the width of the street and the height of the house, the square of the length of tlie ladder dimin- ished by the square of half the width of tlie street, will he equa to the square of the height of the house : hence. Rule. — Square the hypothenuse and the known side, and take the difference; the square root of the difference will be the other side. Examples. 1. A general having an army of 111649 men, wished to form them into a square : how many should he place on each front ? 2. In a square piece of pavement there are 48841 stones, of equal size, one foot square : what is the length of one side of the pavement? 3. In the center of a square garden, there is an artificial cu-cular pond, covering an area of 810 square feet, which is -|^ of the whole garden : how many rods of fence will inclose the garden ? 4. Let it be required to lay out 6t A. 2 R. of land in the form of a rectangle, the longer side of which is to be three times as great as the less : what is its length and width ? 5. A farmer wishes to set out an orchard of 8200 dwarf pear-trees. He has a field twice as long as it is wide, which he appropriates to this purpose. He sets the trees 12 feet apart, and in rows that are Ukewise 12 feet apart : how many rows will there be, how many trees in a row, and how much land will they occupy ? G. There is a wall 45 feet high, built upon the bank of a -tream 60 feet wide: how long must a ladder be that will reach from the oae €ide of the stream to the top of the wall on the other? 632 EXTRACTION OF THE SQUARE ROOT. 7. A boy having lodged his kite in the top of a tree, finds that by letting out the whole length of his line, which he knows to be 225 feet, it will reach the ground 180 feet from the foot of the tree : what is the height of the tree ? 8. There are two buildings standing on opposite sides of the street, one 39 feet, and the other 49 feet from the ground to the eaves. The foot of a ladder 65 feet long rests upon the ground at a point between them, from which it will touch the eaves of either building : what is the width of the street? 9. A tree 120 feet high was broken off in a storm, the top striking 40 feet from the roots, and the broken end resting upon the stump : allowing the ground to be a horizontal plane, what was the height of the part standing? 10. What will be the distance from corner to corner, through the center of a cube, whose dimensions are 5 feet on a side? 11. Two vessels start from the same point, one sails due north at the rate of 10 miles an hour, the other due west at the rate of 14 miles an hour : how far apart will they be at the end of 2 days, supposing the surface of the earth to be a plane ? 12. How much more will it cost to fence 10 acres of land, in the form of a rectangle, the length of which is four times its breadth, than if it were in the form of a square, the cost of the fence being 12.50 a rod? 13. What is the diameter of a cylindrical reservoir contain- ing 9 times as much water as one 25 feet in diameter, the height being the same ? Note. — If two volumes have the same altitude, their contents will be to each other in the same proportion as their bases; and if the bases are similar figures (that is, of like form), they will be to each other as the squares of their diameters, or other like dimensions. 14. If a cylindrical cistern eight feet in diameter will hold 120 barrels, what must be the diameter of a cistern of the same depth to hold 1500 barrels ? CUBF. ROOT. 333 15. If a pipe 3 inches in diameter will discharge 400 gallons in 3 minutes, what must be the diameter of a pipe that will discharge IGOO gallons in the same time? 16. What length of rope must be attached to a halter 4 feet long, that a horse may feed over 2 J acres of ground? 17. Three men bought a grindstone, which was 4 feet in liameter : how much of the radius must each grind off to usd up his share of the stone? CUBE ROOT. 387. The Cube Root of a number is one of its three equal factors. Thus, 2 is the cube root of 8 ; for, 2 x 2 x 2 = 8 : and 3 is the cube root of 2t ; for, 3x3x3 = 27. To extract the cube root of a number, is to find one of its three equal factors. 1, 2, 3, 4, 6, 6, 7, 8, 9, 1 8 27 64 125 216 343 512 729 The numbers in the first line are the cube roots of the cor- responding numbers of the second. The numbers of the second line are called perfect cubes. ,A Perfect Cube is a number which has three exact equal factors. By examining the numbers in the two lines, we see, 1st. That the cube of units cannot give a higher order than hundreds : 2d. That since the cube of one ten (10) is 1000, and the cube of 9 tens (90), 729,000, the cube of tens wilt not give a lower denomination than thousands, nor a higher denomina- tion than hundreds of thousands. Hence, if a number contains more than three figures, its cube root will contain more than one ; if it contains more than six, its root will contain more than two, and so on ; every additional three figures giving one additional figure in the root, U-. = 10 + 6 10 + 6 60 + 36 100 + 60 100 + 120 + 36 10 + 6 83-1 CUBE MOOT. and the figures whicli remain at the left hand, although less than three, will also give a figure in the root. This law ex- plains the reason for pointing off into periods of three figures each. 388. Let us see how the cube of any number, as 16, is ^ormed. Sixteen is composed of 1 ten and 6 units, and may le written, 10 + 6. To find the cube of 16 = 10 + 6, we must multiply the number by itself twice. To do this we place the number thus. Product by the units, - - - - Product by the tens, - - - - Square of 16, Multiply again by 16, - Product by the units, - - - - 600 + 720 + 216 Product by the tens, - - - 10 00 + 1200 + 360 Cube of 16, - - - - 1000 + 1800 + 1080 + 216 1. By examining the parts of this number, it is seen that the first part 1000 is the cube of the tens; that is, 10 X 10 X 10 = 1000 : 2. The second part 1800 is three times the square of the tens multiplied by the units; that is, 3 X (10)= X6 = 3xl00x6= 1800 : 3. The third part 1080 is three times the square of the units multiplied by the tens; that is, 3 X 6'^ X 10 = 3 X 36 X 10 = 1080 : 4. The fourth part is the cube of the units; that is, 6» = 6 X 6 X 6= 216. 1. What is the cube root of the number 4096 ? Analysis. — Since the num- operation. ber contains more than three ; aoA/i^ - , - , 4 uyD(lo figures, we know that the root j will contain at least units and tens. 12 X 3 = 3)3 (9-8-t-6 Separating tlie three right- ^^ — 4 096 ci'ni.: HOOT. 335 hand figures from tlie 4, we know that the cube of the tens will be found in the 4; and 1 is the greatest cube in 4. Hence, wc place the root 1 on the right, and this is the tens of the required root. We then cube 1, and subtract the result from 4, and to the remainder we bring down the first figure of the next period. We have seen that the second part of the cube of 16, viz., ^800, is three times the square of the tens multiplied by the units; and hence, it can have no significant figure of a less denomination than hundreds. It must, therefore, make up a part of the 30 liun- dreds above. But this 30 hundreds also contains all the hundreds which come from the 3d and 4th parts of the cube of 16. If it were not so, the 30 hundreds, divided by three times the square of the tons, would give the unit figure exactly. Forming a divisor of three times the square of the tens, we find the quotient to be ten; but this we know to be too large. Placing 9 in the root, and cubing 19, we find the result to be 0859. Then trying 8, we find the cube of 18 still too large; but when we take 6, we find the exact number. Hence, tlie cube root of 4096 is 16. 389. Hence, to find the cube root of a number: Rule. — I. Separate the given mimher into periods of three figures each^ beginning at the rights by placing a dot over the place ofunits^ a second over the place of thousands^ and so on over each third figure to the left: the left-hand period icill often contain less than three places of figures : II. Note the greatest perfect cube in the first period^ a?id set its root on the rights aftei' the manner of a quotient i7i division. Subtract the cube of this nvmber from the first 2:>eriod, and to the remainder bring down the first figure of tiie next period for a dividend: III. Take three times the square of the root just found for a t) ial divisor, and see how often it is contained in the dividend^ and place the quotient for a second figure of the root. Then cube the figures of the root thus founds and if tlieir cube b< greater than the first two x>eriods of th^ 336 CUBE ROOT. given number ^ diminish the last figure j hut if it be less^ subtract it from the first two periods^ and to the remainder bring down the first figure of the next 'period for a new dividend: IV. Take three times the square of the whole root for a second trial divisor, and find a third figure of the root as before. Cube the whole root thus found, and subtract the result from tJie first three periods of the given number loheu it is less than that number ^ but if it is greater , diminish the last figure of the root: proceed in a similar way for all the periods. Examples. I. What is the cube root of 20t968t5? OPERATION. 20 196 815(215 2^= 8 2=' X 3 = 12)121 Fh-st two periods, - - - - 20 196 (2iy = 21 X 21 X 21 = 1 9 683 3 X (2iy = 2181)li 138 First three periods, ... - 20 196 815 (215)^= 215 X 215 X 215 = 20 196 815 Find the cube roots of the followmff numbers : 1. Cube root of 1128? 2. Cube root of 111649? 3. Cube root of 46656 ? 4. Cube root of 15069223? 6. Cube root of 5135339 ? 6. Cube root of 48228544 ? 1. Cube root of 84604519? 8. Cube root of 28991029248 ? 390. To extract the cube root of a decimal fraction. Rule. — Annex ciphers to the decimal, if necessary, so that it shall consist of 3, 6, 9, &c., decimal places. Then put the first point over the place of thousandths, the second over the CUBE ROOT. 337 place of millionths^ and so on over every third place to the right; after which, extract the root as in whole numbers. Notes. — 1. There "will be as many decimal places in the root as there are jxjriods of decimals in the given number. 2. If, in extracting the root of a number, there is a remainder after all the periods have been brought down, periods of ciphers may be annexed by considering them as decimals. Examples. Fnid the cube roots of the followin": numbers : 1. Cube root of 8.343 ? 2. Cube root of 1T28.729 ? 3. Cube root of .0125 ? 4. Cube root of 19683.46656? 5. Cube root of .38T420489? 6. Cube root of .000003375 ? T. Cube root of .0066592? 8. Value of -^81.729? 391. To extract the cube root of a common fraction. Rule. — I. Heduce compound fractions to simple 09ies, mixed numbers to improper fractions^ and then reduce the fraction to its loicest terms: II. Extract the cube root of the numerator and denom- inator separately^ if t/iey have exact roots ; but if either of them has not an exact root, reduce the fraction to a deciwMl, and extract the root as in the last case. Examples. Find the cube roots of the following fractions 1. Cube root of ^*^ ? 2. Cube root of ^ ? 3. Cube root of 31^? 4. Cube root of 91 J ? 5. Cube root of ffj ? 6. Cube root of yJlf^ ? 7. Cube root of ^HjW ? 8. Cube root of i|ff| ? 9. Cube root of 7f ? 10. Cube root of 66f ? 15 338 Cl^BE ROOT. Applications. 1. What must be the dimensions of a cubical bin, that its volume of capacity may be 19683 feet? 2. If a cubical body contains 6859 cubic feet, what is the length of one side ? what the area of its surface ? 3. The volume of a globe is 46656 cubic inches ; what would be the side of a cube of equal solidity ? 4. A person wishes to make a cubical cistern, which ^hall hold 150 barrels of water: what must be its depth? 5. A farmer constructed a bin that would contain 1500 bushels of grain ; its length and breadth were equal, and each half the height : what were its dimensions ? 6. What is the difference between half a cubic yard, and a % = = xf (J, what A can do m 1 day ; 1- ■^72 0' = 40da. A'o- tVo = " T2^» H J? 11 It 1 ^ih = 30 da. T%- AV = " if (J» li Q tl tl 1- ^ih = 24 da. T%- - ITSf (< T\ tl It 1 -Xio = n^da. Hence, the share of each will be : $312 X A = U9.2Qj%, A's share. *312 X tV = $65.68iV, B's share. $312 X A = $82.10fo, C^s share. $312 X tV = |114.94}J, D's share. $312.00, amount paid to A, B, C, and D 22. A person owning § of a vessel, sold f of his share for $1736 : what was the value of the whole vessel? 23. If a man performs a journey in tj- days, traveling 14| hours a day, in how many days will he perform the same ourney by traveling 10-J hours a day? 24. If 1^ of a pole stands in the mud, 2 feet in the water, and f above the water, what is the length of the pole ? 25. After spending -J- of my money, and J of the remainder, I had $1062 left : how much had I at first ? 360 ANALYSIS AND 26. Suppose a cistern has two pipes, and that one can fill it in T-i hours, and the other in 41- hours : in what time can both fill it, running together? 21, If 54 yards of ribbon cost $9, what will 26 yards cost? 28. If 2 acres of land cost J of f of | of $300, what wil' I of -J of 10 J acres cost ? 29. A regiment of soldiers, consisting of 1000 men, is to be clothed ; each suit is to contain 3^ yards of cloth If yards wide : how much shalloon that is ^ yards wide is necessary for lining ? 30. How much tea, at Ts. 6d. a pound, must be given for 234 bushels of oats, at 3s. 9d. a bushel, New York currency? 31. What will 3 pipes of wine cost, at 2s. 9d. per quart, New England currency ? 32. A gives B 165 yards of cotton cloth, at 2s. 6d. per yard, Missouri currency, for 625 pounds of lump sugar : how much was the sugar worth a pound? 33. If the expense of keeping 1 horse 1 day is 3s. 4d., Canada currency, what will be the expense of keeping 4 horses 3 weeks, at the same rate? 34. Bought 10 bales of cloth, each bale containing 14 pieces, and each piece 22^ yaids, at 10s. 8d. per yard,"- lUinois currency : what was the cost of the cloth ? 35. A has TJcwt. of sugar, worth 12 cents a pound, for which B gave him 12Jcwt. of flour: what was the flour worth a pound? 36. Bought 120 yards of cloth, at 6s. 8d. a yard. New York currency, and gave in payment 16 bushels of rye, at 4s. 6d. a bushel, New England currency, and the balance in money : how many dollars will pay the balance ? 31. A merchant bought 21 pieces of cloth, each piece con- taining 41 yards, for which he paid $1260 ; he sold the cloth at $1.15 per yard : did he gain or lose, and how much ? PKOMISCL'UUS EXAMPLES. 361 38. The hour and mmute hands of a watch are together at 12: at what moment will they be together between 5 and G? 39. How many yards of carpeting J of a yard wide will cover the floor of a room 18 feet long and 15 feet wide? 40. If 9 men can build a house in 5 months, by working 12 hours a day, how many hours a day must the same mei ork to do it in 6 months? 41. B and C can do a piece of work in 12 days ; with the assistance of A they can do it in 9 days : in what time can A do it alone ? 42. A can mow a certain field of grass in 3 days, B can do it in 4 days, and C can do it in 5 days : in what time can they do it, workmg together? 43. Divide the number 480 into 4 such parts that they shall be to each other as the numbers 3, 5, T, and 9 ? 44. What length of a board that is 8j- inches broad, will make a square foot? 45. The provisions in a garrison were sufficient for 1800 men, for 12 months ; but at the end of 3 months, it was re- inforced by 600 men, and 4 months afterward, a second rein- forcement of 400 men was sent in : how long would the provisions last after the last reinforcement arrived? 46. A merchant bought a quantity of broadcloth and baize for $488.80 ; there was 11 7 J yards of broadcloth, at 13^ per yard ; for every 5 yards of broadcloth he had 1 J yards of baize : how many yards of baize did he buy, and what did it cost him per yard? 4t. If the freight of 40 tierces of sugar, each weighmg 3 J cwt., for 150 miles, costs |42, what must be paid for the freight of 10 hhd., each weighing 12 cwt., for 50 miles? 48. If 1 pound of tea be equal in value to 50 oranges, and 70 oranges be worth 84 lemons, what is the value of a pound 9f tea, when a lemon is worth 2 cents? 16 362 ANALYSIS AND 49. What amount must be discounted, at t per cent., to make a present payment of a note of $500, due 2 years 8 months hence ? 50. If the interest on $225 for 4-J- years is $91.12^, whafc would be the interest on |640, at the same rate, for 2} years? 51. A farmer having 1000 bushels of wheat to sell, can have 11.15 a bushel cash, or $1.80 in nmety days : which would be most advantageous to him, money being worth 1 per cent. ? 52. A merchant bought goods to the amount of $1515 on 9 months' credit ; he sells the same for $1800 in cash : money being worth 6 per cent., what did he gain? 53. Three persons in partnership gain $482.G2 ; A put in f as much capital as B, and B put in f as much as C : what was each one's share of the gain? 54. A father divided his estate, worth $9268.60, among his 4 children, giving A -J- of it, B i and C $5 as often as he gave D $6 : how much did each receive ? 55. A tax of $415.50 was laid upon 4 villages. A, B, C, and D ; it was so distributed, that as often as A and B each paid $5, C paid $1, and D $8 ; what part of the whole tax did each village pay ? 56. There are 1000 men besieged in a town, with provisions for 5 weeks, allowing each man 16 ounces a day. If they are reinforced by 400 men, and no relief can be afforded till the end of 8 weeks, what must be the daily allowance to each man? 51. A reservoir has 3 pipes ; the first can fill it in 10 days, the second, in 16 days, and the third can empty it in 20 days : in what tune will the cistern be filled if they are all allowed to run at the same time? 58. Two persons, A and B, are on opposite sides of a wood, which is 536 yards in circumference ; they begin to PROMISCUOUS EXAMPLES. 3C3 travel in the same direction at the same time ; A goes at the rate of 11 yards a minute, and B, at the rate of 34 yards in 3 minutes : how many times will B go round the wood before he overtakes A ? 59. Two men and a boy were engaged to do a piece of work. One of the men could do it in 10 days, the other in 1 6 days, and the boy could do it in 20 days : how long would t take them together to do the work ? 60. A owes B $500, of which $150 is to be paid in 3 months, $175 in 6 months, and the remainder in 8 months : what would be the equated time for the payment of the whole ? 61. If 42 men, in 270 days, working 8 J hours a day, can build a wall 98f feet long, 7J feet high, and 2^ feet thick ; in how many days can 63 men build a wall 45J feet long, 6/^ feet high, and 3|- feet thick, working 11 J hours a day? 62. After one-third part of a cask of wine had leaked away, 21 gallons were drawn, when it was found to be half full : how much did the cask hold ? 63. A man had a bond and mortgage for $2500, dated July 1st, 1854. Not satisfied with 7Vo interest, he sold the mortgage for its nominal value, and on Sept. 1st, 1854, pur- chased 10 shares of railroad stock, par $100, at 115. On Nov. 1st, he bought 8 shares more of the same stock, at 98 ; and on April 1st, 1855, he bought 5 shares more at the same rate. On the first days of August and February, in each year, he received a regular semi-annual dividend of 4 per cent., and at the end of the year (January 1st, 1856) sold his whole stock at 99 : did he lose or gain by the investment in stocks, and how much ? 64. A landlord being asked how much he received for the cut of his property, answered, that after deducting 9 cents from each dollar, for taxes and repairs, there remained ^3014.30 : what was the amount of his rents ? 65. If 165 pounds of soap cost $16.50, for liow much will 3(14: ANALYSIS AND it be necessary to sell 390 pounds, in order to gain the cost of 36 pounds ? 66. What is the height of a wall which is 14 J yards in length, and ^q of a yard in thickness, and which cost $406, it having been paid for at the rate of $10 per cubic yard? 67. A thief escaping from an oflScer, has 40 miles the start and travels at the rate of 5 miles an hour ; the officer in pur suit travels at the rate of 1 miles an hour : how far must he travel before he overtakes the thief? 68. Two families bought a barrel of flour together, for which they paid $8, and agreed that each child should count half as much as a grown person. In one family there were 3 grown persons and 3 children, and in the other, 4 grown per- sons and 10 children ; the first family used from the flour 2 weeks, and the second 3 weeks : how much ought each to pay? 69. At 842 a thousand, how many thousand feet of lumber should be given for a farm containing 33 A. 2R. 16 P., valued at 1125 an acre? 10. A person paid $150 for an insurance on goods, at 3|- per cent., and finds that in case the goods are lost, he will receive the value of the goods, the premium of insurance, and $25 besides : what was the value of the goods ? n. A distiller purchased 5000 bushels of rye, which he could have at 96 cents a bushel, cash, or at $1, 2 months' credit ; which would be the more advantageous, to buy on credit, or to borrow the money at 7 per cent., and pay the cash? 12. A stockholder bought | of the capital of a company at par ; he sold i of his purchase at par, and the remainder for $25000, and by the latter sale made $5000: what was the value of the whole capital? 13. How many bushels of grain will a bin contain, that is 3ft. 5 in. wide, 2ft. 6 in. long, and 6ft. deep? PROMISCUOUS EXAMPLES. 365 74. Three travelers have each to make the same journey of 2160 miles ; the first travels 30 miles a day, the second 27, and the third 24 : how many days should one set out after the other, that they may all arrive together? 75. A house which was resold for $7180, would have given profit of $420, if the second proprietor had purchased it ^130 cheaper than he did: at what price did he purchase it? 76. A piece of land of 188 acres was cleared by two com- panies of men, working together ; the first numbered 25 men, and the second 22 ; the first company received $84 more than the second : how many acres did each company clear, and what did the clearing cost per acre ? 77. I have three notes payable as follow : one for $100, due Feb. 12th; the second for $400, due March 12th; and the third for $300, due April 1st : what is the average time of payment from January 1st? 78. How many marble slabs, 15 in. square, will it take to pave a floor 32 feet long, and 25 feet wide? What will be tlie cost at $3 a square yard for the marble, and 40 cents a square yard for labor? 79. A man, in his will, bequeathed $500 to A, $425 to B, $300 to C, $250 to D, and $175 to E ; but after settling up tlie estate and paying expenses, there was but $1155 left: what is each one's share ? 80. If 31b. of tea are worth 71b. of coflfee, and 1411). of coffee are worth 481b. of sugar, and 181b. of sugar are worth 27 lb. of soap ; how many pounds of soap are 6 lb. of tea worth ? 81. What is the hour, when the time past noon is | the time to midnight ? 82. If f of a yard of cloth cost $f , being J of a yard wide, what is the value of |- of a yard IJ yards wide, of the same quality ? 83. A farmer sold 60 fowls, a part turkeys, and a part S66 ANALYSi;^ AND chickens ; for the turkeys he received $1.10 apiece, and for the chickens 50 cents apiece, and for the whole he received $51 60 : how many were there of each? 84. A person hired a man and two boys ; to the man he gave 6 shiUings a day, to one boy 4 shilhngs, and to the other 3 shiUings a day, and at the end of the time he paid them 104 shillings : how long did they work? 85. Divide $6471 among three persons, so that as often as the first gets |5, the second will get $6, and the third $7. 86. Two partners have invested in trade 11600, by which they have gained 1300 ; the gain and stock of the second amount to $1140 : what is the stock and the gain of each? 87. What is the height of a tower that casts a shadow 75.75 yards long, at the same time that a perpendicular staff 3 feet high, gives a shade of 4.55 feet in length? 88. A can do a certain piece of work in 3 weeks ; B can do 3 times as much in 8 weeks; and C can do 5 times as much in 12 weeks : in what time can they all together do the first piece of work? 89. Two persons pass a certain point, at an interval of 4 hours ; the first traveUng at the rate of 1 1 J, and the second 17 J miles an hour : how long, after passmg the fixed point, and how far, will the first travel before he is overtaken by the second ? 90. Three persons engage in trade, and the sum of their stock is $1600. A^s stock was in trade 6 months, B's 12 months, and C's 15 months ; at the time of settlement, A re- ceives $120 of the gain, B $400, and C $100 : what was each person's stock ? 91. A, B, and C, start at the same time, from the same point, and travel in the same direction, around an island 73 miles in circumference. A goes at the rate of 6 miles, B 10 miles, and C 16 miles per day : in what time will they all be together again? PliOMISCUOUS EXAMPLES. 367 92. What length of wire, J of an inch in diameter, can be drawn from a cube of copper, of 2 feet on a side, allowing 10 per cent, for Waste? 93. A person having $10000 invested in 6 per cent, stocks, sells out at 65, and invests the proceeds in 5 per cents at 82 J : what will be the difference in his annual income ? 94. In order to take a boat through a lock from a certain river into a canal, as well as to descend from the canal into the river, a volume of water is necessary 46 J yards long, 8 yards wide, and 2f yards deep : how many cubic yards of water will this canal throw into the river in a common year, if 40 boats ascend and 40 descend each day, except Sundays and eight holidays ? 95. A company numbering sixty-six shareholders have coa- Btructed a bridge which cost $200000 : what will be the gain of each partner at the end of 22 years, supposing that G400 persons pass each day, and that each pays one cent toll, the expense for repairs, &c., being $5 per year for each share- holder? 96. Five merchants were in partnership for four years, the first put in $60, then, 5 months after, $800 ; the second put in first $600, and 6 months after $1800 ; the third put in $400, and every six months after, he added $500 ; the fourth did not contribute till 8 months after the commencement of the partnership ; he then put in $900, and repeated this sum every 6 months; the fifth put in no capital, but kept the ac- counts, for which the others agreed to allow him $800 a year, to be paid in advance, and put in as capital. What is each one's share of the gain, which was $20,000 ? 9t. A general, arranging his army in the form of a square, found that he had 44 men remaining : but by increasing each side by another man, he wanted 49 to fill up the square : how many men had he? 98. A, B, and C, are to share $987 in the proportion of 368 ANALYSIS AND i i and J resi3ectively ; but by the death of C, it is required to divide the whole sum proportionally between the other two : what will each have? 99. A lady going out shopping, spent at the first place she stopped, one-half her money, and half a dollar more ; at the icxt place, half the remainder, and half a dollar more ; and at he next place, half the remainder, and half a dollar more, Allien she found that she had but three dollars left : how much had she when she started ? 100. If a pipe of 6 inches discharges a certain quantity of fluid in 4 hours, in what time will 4 pipes, each of 3 inches bore, discharge twice that quantity? 101. A man bought 12 horses, agreeing to pay $40 for the first, and in an increasing arithmetical progression for the rest, paying $310 for the last : what was the difference in the cost, and what did he pay for them all ? 102. A bill for goods, amounting to $15000, is to be paid for in three equal payments without interest ; the first in 4 months, the second in 6 months, and the third in 9 months, money being worth t per cent. : how much ready money ought to pay the debt ? 103. If an iron bar 5 feet long, 2 J inches broad, and If inches thick, weigh 45 pounds, how much will a bar of the same metal weigh, that is Y feet long, 3 inches broad, and 2^ inches thick? 104. A market woman bought a certain number of eggs at the rate of 4 for 3 cents, and sold them at the rate of 5 for 4 cents, by which she made 4 cents : what did she pay apiece for the eggs? What did she make on each egg sold? How many did she sell to gain 4 cents? 105. A person passed i of his life in childhood, ^^ of it in youth, 5 years more than -i of it in matrunony ; he then had a son, whom he survived 4 years, and who reached only i the age of his father : at what age did he die ? PROMISCUOUS EXAMPLES. 369 106. A well is to be stoned, of which the diameter is 6 feet 6 inches, the thickness of the wall is to be 1 foot 6 inches, leaving the diameter of the well within the wall 3 feet 6 inches ; if the well is 40 feet deep, how many cubic feet of stone will be required? 101. A surveyor measured a piece of ground in the form of rectangle, and found one side to be 37 chains, and the other ■12 chains 16 links : how many acres did it contain ? 108. A farmer bought a piece of land for $1500, and agreed to pay principal and interest in 5 equal annual instal- ments : if the interest was 7 per cent., how much was the annual payment ? 109. A fountain has 4 receiving pipes, A, B, C, and D ; A, B, and C will fill it in 6 hours ; B, C, and D in 8 hours ; C, D, and A in 10 hours ; and D, A, and B in 12 hours : it has also 4 discharging pipes, E, F, G, and II ; E, F, and Q will empty it in 6 hours ; F, Or, and H in 5 hours ; G, H, and E in 4 hours ; H, E, and F in 3 hours. Suppose the fountain full of water, and all the pipes open, in wliat time would it be emptied? 110. If a ball 2 inches in diameter weighs 5 pounds, what will be the diameter of another ball of the same material that weighs 78.12p pounds ? 111. A gives B his bond for $5000, dated April 1st, 1861, payable in ten equal annual instalments, the first payment of $500 to be made April 1st, 1802. Afterward, A agreed to take up his bond on the 1st day of April, 1863. He was to pay, on that day, the instalment due on the 1st of April, 1862, with interest at 7 per cent., the instalment due April 1st, 1863, and to be allowed compound interest, at 7 per cent., to be computed half-yearly, on each of the subsequent payments: what sum, on the first day of April, 1863, will cancel the bond? 370 MENSURATION. MENSURATION. 405. Mensuration is the art of measuring, and embraces all the methods of determining the contents of geometrical figures. It is divided into two parts, the Mensuration of Surfaces, and the Mensuration of Volumes. o 1—1 MENSUKATION OF SURFACFS. 406. Surfaces have length and breadth. They are measured by means of a square, which is called the unit of surface. A SQUARE is the space included between four i foot, equal lines, drawn perpendicular to each other. Each line is called a side of the square. If each side be one foot, the figure is called a square foot. The number of small squares that is con- tained in any large square, is always equal to the product of two of the sides of the large square. As in the figure, 3x3 = 9 square feet. The number of square inches contained in a square foot is equal to 12 X 12 = 144. If the sides of a square be each four feet, the square will contain sixteen square feet. For, in the large square there are sixteen small squares, the sides of which are each one foot. Therefore, the square whose side is four feet, contains sixteen square feet. Triangle. 407. A TRIANGLE is a figure bounded by three straight lines. Thus, ACB is a triangle. The lines BA, AC, BC, are called sides; and the corners, B, A, and C, are called a7igles. The side AB is the base. When a line Hke CD is drawn, making the angle CD A equal to the angle CDB, then CD is said to be at right angles to OF SURFACES. 37] AB, and CD is called the allitade of the triangle. Each tri- angle CAD or CDB is called a right-angled triangle. The side BC, or the side AC, opposite the right angle, is called the hypothenuse. The area or contents of a triangle is equal to half the product of its base by its aUitude (Bk. lY., Pro2?. YL). Note. — All the references are to Davies' Legendre. Examples. 1. The base, AB, of a triangle is 50 yards, and the per- pendicular, CD, 30 yards : what is the area? operation. 60 Analysis. — Wo first multiply tho 30 base by the altitude, and the product 2)1500 is square yards, which we divide by ^^^^^ "^^ ^^^^ 2 for the area. 2. In a triangular field the base is 60 chams, and the per- pendicular 12 chains : how mjich does it contain ? 3. There is a triangular field, of which the base is 45 rods, and the perpendicular 38 rods : what are its contents ? 4. What are the contents of a triangle whose base is 15 chains, and perpendicular 36 chains? Rectangle and Parallelogram. 408. A RECTANGLE is a four-sided figure, or quadrilateral, like a square, in which the sides are perpendicular to each other, but the adjacent sides are not equal. 409. A PARALLELOGRAM is a quadrilat- eral which has its opposite sides equal and parallel, but its angles not right angles. The line DE, perpendicular to the base, is called the altitude. 372 MENSURATION Tlie area of a square, rectangle, or iJarallelogram, is equal to the x^roduct of the base and altitude. Examples. 1. What is the area of a square field, of which the sides are each 66.16 chains? 2. What is the area of a square piece of land, of which the ides are 54 chains ? 3. What is the area of a square piece of land, of which tlie sides are 75 rods each ? 4. What are the contents of a rectangular field, the length of which is 80 rods, and the breadth 40 rods ? 5. What are the contents of a field 80 rods square? 6. What are the contents of a rectangular field, 30 chains long and 5 chains broad? T.. What are the contents of a field, 54 chains long and 18 rods broad? 8. The base of a parallelogram is 542 yards, and the per- pendicular height 120 feet : what 'is the area ? 9. The measure of a rectangular field is 24000 square feet, and its length is 200 feet : what is its breadth ? Trapezoid. 410. A TRAPEZOID is a quadrilateral, D ABCD, having two of its opposite sides, / AB, DO, parallel. The pei-pendicular, / EF, is called the altitude. A F B The area of a trapezoid is equal to half the product of the sum of the two parallel sides by the altitude {Bk. lY., Prop. TIL). Examples. 1. Required the area or contents of the trapezoid ABCD, having given AB = 643.02 feet, DC = 428.48 feet, and EF = 342.32 feet. OF SURFACES. 373 Analysis. — We first find operation. tho sum of the paraUel' 643.02 + 428.48 = 1071.50 = sum sides, and then multiply it of parallel sides. Then, 1071.50 x by the altitude ; after which 342.32 = 3GG795.88 • and ^^-ilii-^J =: we divide the product by 183397.94 = the area. 2, for the area. 2. What is tlie area of a trapezoid, the parallel sides of which are 24.82 and 16.44 chains, and the perpendicular dis- tance between them 10.30 chains? 3. Requu-ed the area of a trapezoid, whose parallel sides are 51 feet and 37 feet 6 inches, and the perpendicular dis- tance between them 20 feet and 10 inches. 4. Required the area of a trapezoid, whose parallel sides are 41 and 24.5, and the perpendicular distance between them 21.5 yards. 5. What is the area of a trapezoid, whose parallel sides are 15 chains, and 24.5 chains, and the perpendicular height 30.80 chains ? C. What are the contents of a trapezoid, when the parallel sides are 40 and 64 chains, and the perpendicular distance be- tween them 52 chains? Circle. 411. A Circle is a portion of a plane bounded by a curved li-ne, every point of which is equally dis- tant from a certain point within, called the center. The curved line AEBD is called the circumference ; the point C, the center ; the line AB, passing through the center, a diameter ; and CB, a radius. The circumference, AEBD, is 3.1416 times as great as the diameter AB. Hence, if the diameter is 1, the circumference will be 3.1416. Therefore, if the diameter is known, the circumference is found by multiplying 3.1416 by the diameter {Bk. V., Prop, XVI.). 374 MENSURATION Examples. 1. The diameter of a circle is 8 : what is the circumference? OPERATION. Analysis. — The circumference is found 3.1416 by simply multiplying 3.1416 by the di- 8 ^'^^t^^- Ans. 25.1328 i 2. The diameter of a circle is 186 : what is the circum- ference ? 3. The diameter of a circle is 40 : what is the circum- ference ? 4. What is the circumference of a circle whose diameter is 5t? 412. Since the circumference of a circle is 3.1416 times as great as the diameter, it follows, that if the circumference is knoivn, we may find the diameter by dividing it by 3.1416. Examples. 1. What is the diameter of a circle whose circumference is 157.08? 2. What is the diameter of a circle whose circumference is 23304.3888? 3. What is the diameter of a circle whose circumference is 13700? 413. To find the area or contents of a circle. Hule.— Multiply the square of the radius by 3.1416 {Bk. Y., Prop. XY.). Examples. 1. What is the area of a circle whose diameter is 12 ? 2. What is the area of circle whose diameter is 5 ? 3. What is the area of a circle whose diameter is 14? OF VOLUMES. 375 4. now many square yards iu a circle whose diameter is 3J feet ? 5. What is the area of a circle whose dianiotcr is -J mile ? Sphere. . 414. A SPHERE is a portion of space bounded by a curved surface, all the l)oints of which are equally distant from a certain point within, called the center. The line AD, passing through its center C, is called the diameter of the sphere, and AC its radius. 415. To find the surface of a sphere. Rule. — 3TuUipIy (he square of the diameter by 3.1416 {Bh VIII., Prop. X., Cor. 1.). Examples. 1. What is the surface of a sphere whose diameter is 6? 2. What is the surface of a sphere whose diameter is 14 ? 3. Required the number of square inches in the surface of a sphere whose diameter is 3 feet or 36 inches. 4. Required the area of the surface of the earth, its mean diameter being 7918.7 miles? MENSUKATION OF VOLUMES. 416. A SOLID or volume is a portion of space having three dimensions : length, breadth, and thickness. It is measured by a cube, called the cubic unit, or unit of volume. A CUBE is a volume having six equal faces, which are squares. If the sides of the cube be each o ,- i. i i lect = 1 yard. one foot long, the figure is called a cubic foot But when the sides of the cube arc one yard, 1 ^-y / A in ':':P ■r-< Pi II ■■'1 ■-^ : ! '¥ : I 1 f 376 MENSURATION as in the figure, it is called a cubic yard. The base of the cube, which is the face on which it stands, contains 3x3 = 9 square feet. Therefore, 9 cubes, of one foot each, can be placed on the base. If the figure were one foot high, it would contain 9 cubic feet ; if it were 2 feet high, it would contain two tiers of cubes, or 18 cubic feet ; hence, the contents are equal to the product of the length, breadth, and height. 417. To find the volume or contents of a sphere. Rule. — Multiply the surface by the diameter, and divide the product by 6 ; the quotient will be the contents {Bk VIII., Prop. XIY., Sch. 3), Examples. 1. What are the contents of a sphere whose diameter is 12 ? Analysis. — We find the sur- face by multiplying the square of the diameter by 3.1416. We then multiply the surface by the diameter, and divide the product by 6. OPERATION. 12' = 144 multiply by 3.1416 surface 452.3904 diameter 12 6)5428.6848 solidity 904.7808 2. What are the contents of a sphere whose diameter is 8 ? 3. Find the contents of a sphere whose diameter is 16 inches. 4. What are the contents of the earth, its mean diameter being 7918.7 miles? 5. Find the contents of a sphere whose diameter is 1.2 feet. Prism. 418. A Prism is a volume whose ends or bases are equal plane figures, and whose faces are par- allelograms. The sum of the sides which bound the base, is called the perimeter of the base ; and the sum of the parallelograms which bound the prism, is called the eonvcx surface. (CD OF VOLUMES. 377 419. To find the convex surface of a right prism. Rule. — Multiply the perimeter of the base by the perpen- dicular height, and the product will be the convex surface (Bk. YIT., Prop). I.). Examples. 1. What is the convex surface of a prism whose base is bounded by five equal sides, each of which is 35 feet, the alti- tude being 52 feet? 2. What is the convex surface when there are eight equal sides, each 15 feet in length, and the altitude is 12 feet? 420. To find the volume or contents of a prism. Rule. — Multiply the area of the base by the altitude, and the product ivill be the contents {Bk. YII., Prop. XIV.). Examples. 1. What are the contents of a square prism, each side of the square which forms the base being 16, and the altitude of the prism 30 feet? OPERATION. Analysis. — We first find the area of the Tp2 _ or/* square which forms the base, and then multi- ~ on ply by the altitude. . 1680 2. What are the contents of a cube, each side of which is 48 inches? 3. IIow many cubic feet in a block of marble, of whicl the length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 5 feet ? 4. How many gallons of water will a cistern contain, whose dimensions are the same as in the last example? 5. Required the measure of a triangular prism, whose height is 20 feet, and area of the base 691. 378 MENSURATI02J' Cylinder. 421. A Cylinder is a volume generated by the revolution of a rectangle, AF, about BF. The line EF is called the axis, or altitude ; the circular surface, the convex surface of the cylinder ; and the circular eads, the hasea. 422. To find the convex surface of a cylinder. Rule. — Multiply the circumference of the base by the altitude, and the product will be the convex surface {Book Yin., Prop, I.). Examples. 1. What is the convex surface of a cylinder, the diameter of whose base is 20, and the altitude 40 ? 2. What is the convex surface of a cylinder whose altitude is 28 feet, and the circumference of its base 8 feet 4 inches ? 3. What is the convex surface of a cylinder, the diameter of whose base is 15 inches, and altitude 5 feet ? 4. What is the convex surface of a cylinder, the diameter of whose base is 40, and altitude 50 feet ? 423. To find the volume or contents of a cylinder. Rule. — 3fuUiply the area of the base by the altitude: the product will be the contents or volume (Book YIII., Prop. II.). Examples. 1. Required the contents of a cylinder of which the altitude is 1 1 feet, and the diameter of tlie base 16 feet. Analysis. — We first find the area of the base, and then multiply by the altitude: the product is the so- lidity. OPERATION. 16^=256 .1854 area base, 201.0624 11 2211.6864 OF VOLUMES. 379 2. What are the contents of a cylinder, the diameter of whose base is 40, and the altitude 29 ? 3. What are the contents of a cylinder, the diameter of whose base is 24, and the altitude 30 ? 4. What are the contents of a cylinder, the diameter of whose base is 32, and altitude 12 ? 5. What are the contents of a cylinder, the diameter o whose base is 25 feet, and altitude 15 ? Pyramid. 424. A Pyramid is a volume bounded by several triangular planes united at the same point, S, called the vertex, and by a plane figure or base, ABODE, in which they terminate. The altitude of tlie pyramid is the line SO, drawn per- pendicular to the base. A 425. To find the volume or contents of a pyramid. Rule. — Multiply the area of the base by the altitude^ and divide the product by 3 {Bk, VII., Prop. XVII.). Examples. 1. Required the contents of a pyramid, operation. the area of whose base is 86, and the alti- 86 tude 24. ^^ Analysis. — We simply multiply the area of ll the base 86, by the altitude 24, and then di- ^ns. 6SS vide the product by 3. 2. What are the contents of a pyramid, the area of whose ase is 365, and the altitude 36? 3. What are the contents of a pyramid, the area of whoso base is 207, and altitude 36? 4. What are the contents of a pyramid, the area of whose biise is 562, and altitude 30 ? 380 MENSUKATION OF VOLUMES. 5. What are the contents of a pyramid, the area of whose base is 540, and altitude 32 ? 6. A pyramid has a rectangular base, the sides of which are 50 and 24 ; the altitude of the pyramid is 36 : what are its contents ? 7. A pyramid with a square base, of which each side is 15 has an altitude of 24 : what are its contents ? Cone. 426. A Cone is a volume generated by the revolution of a right-angled triangle ASC, about the side CB. The point C is the vertex, and the line CB is called the axis, or altitude. 427. To find the volume or contents of a cone. Rule. — 3Iultiply the area of the base by the altitude, and divide the product by 3 ; or, multiply the area of the base by one-third of the altitude {Bk. YIII., Prop. Y.). Examples. 1. Required the contents of a cone, the diameter of whose base is 6, and the altitude 11. Analysis. — We first square the diameter, and multiply it by .7854, which gives the area of the base. We next multiply by the altitude, and then divide the product by 3. OPERATION. 6^^ = 36 36 X .1854 = 28.2744 11 3 )311.0184 Ans. 103.6728 2. What are the contents of a cone, the diameter of whose base is 36, and the altitude 27 ? 3. What are the contents of a cone, the diameter of whose base is 35, and the altitude 27 ? 4. What are the contents of a cone, whose altitude is 27 feet, and the diameter of the base 20 feet? GAUGING. 381 aAUG-ING. 428. Cask-Gauging is the method of finding the number of gallons which a cask contains, by measuring the external di- mensions of the cask. 429. Casks are divided into four varieties, according to the L'urvature of their sides. To which of the varieties any cask belongs, must be judged of by inspection. 1st Variety — least curvature. 2d Variety — least mean curvature. 3d Variety — greatest mean curvature. 4th Variety — greatest curvature. 430. The first thing to be done is to find the mean di- ameter. To do this, Rule. — Divide the head diameter by the bung diameter, and find the quotient in the first column of the following table, marked Qu. Then if the bung diameter be multijjlied by the number on the same line with it, and in the column answering to the proper variety, the product will be the true mean diameter, or the diameter of a cylinder having the same altilude and the same contents with the cask proposed. oSl GAUGING. 50 l8t Var. 2(1 Var. 3d Vmr. 4th Var. Qu. Ist Var. 2d Var. 3d Var. 4tli Var 8660 8465 7905 7637 76 9370 9337 8881 8827 51 8680 8493 7937 7681 77 9396 9358 8944 8874 52 8700 8520 7970 7725 78 9334 9290 8967 8922 53 8720 8548 8002 7769 79 9353 9320 9011 8970 54 8740 8576 8036 7813 80 9380 9353 9055 9018 55 8700 8605 8070 7858 81 9409 9383 9100 9066 5G 8781 8633 8104 7903 83 9438 9415 9144 9114 57 8803 8662 8140 7947 83 9467 9446 9189 9163 58 8834 8690 8174 7993 84 9496 9478 9334 9211 59 8846 8720 8210 8037 85 9536 9510 9280 9260 60 8869 8748 8246 8083 86 9556 9543 9336 9308 61 8893 8777 8283 8138 87 9586 9574 9372 9357 63 8915 8806 8330 8173 88 9616 9606 9419 9406 63 8938 8835 8357 8330 89 9647 9638 9466 9455 64 8963 8865 8395 8365 90 9678 9671 9513 9504 65 8986 8894 8433 8311 91 9710 9703 9560 9553 66 9010 8924 8473 8357 93 9740 9736 9608 9602 67 9034 8954 8511 8404 93 9773 9768 9656 9652 68 9060 8983 8551 8450 94 9804 9801 9704 9701 69 9084 9013 8590 8497 95 9836 9834 9753 9751 70 9110 9044 8631 8544 96 9868 9867 9802 9800 71 9136 9074 8673 8590 97 9901 9900 9851 9850 73 9163 9104 8713 8637 98 9933 9933 9900 9900 73 9188 9135 8754 8685 99 9966 9966 9950 9950 74 9215 9166 8796 8733 100 10000 10000 10000 10000 75 9243 9196 8838 8780 Examples. 1. Supposing the diameters to be 32 and 24, it is required to find the mean diameter for each variety. Dividing 24 by 32, we obtain .75 ; which being found in the column of quotients, opposite thereto stand the numbers, .9242 .9196 which being each mul- tiplied by 32, produce I 8780 J ^^sP^^^t^^'^ly, 29.5U4 29.4272 28.2816 28.0960 for the correspond- ing mean dianietca required. 2. The head diameter of a cask is 26 inches, and the bung diameter 3 feet 2 inches : what is the mean diameter, the cask being of the third variety ? 3. The head diameter is 22 inches, the bung diameter 34 GAl'GINQ. S83 iiiclies : what is the mean diameter of a cask of the fourth variety ? 431. Having found the mean diameter, we multiply the square of the mean diameter by the decunal .7854, and tlie product by the length ; this will give the contents in cubic inches. Then, if we divide by 231, we have the contents iii wine gallons (see Art. 4*75) ; or if we divide by 282, we have the contents in beer gallons (Art. 4^6). Analysis. — For wine measure, we operation. multij)ly the length by the square of I X d* X ^-^^ = the mean diameter, then by the deci- I x (P X 0034 mal .7854, and divide by 231. If, then, we divide the decimal .7854 by 231, the quotient car- ried to four places of decimals is .0034 ; and this decimal, multiplied by the square of the mean diameter and by the length of the cask, will give the contents in wine gallons. For similar reasons, the content is operation. found in beer gallons by multiplying ; 72 ^ 7854 together the length, the square of the mean diameter, and the decimal .0028. I X d^ X .0028. Hence, for gauging or measuring casks, Tlule.— Multiply the length by the square of the mean di ameter ; then multiply by 34 for ivine, and by 28 for beer measure, and point off in the product four decimal places. The jyroduct will then express gallons, and the decimals of a gallon. 1. How many wine gallons in a cask, whose bmig diameter is 30 inches, head diameter 30 inches, and length 50 inches ; the cask being of the first variety? 2. How many wine, and how many beer gallons in a cask \\'hose length is 36 inches, bung diameter 35 inches, ^nd head diameter 30 inches, it being of the first variety? 3. How many wine gallons in a cask of which the head diameter is 24 inches, bung diameter 36 inches, and length 3 feet 6 inches, the cask being of the second variety? 384: MECHANICAL POWERS. OF THE MECHANICAL POWERS. 432. There are six simple machines, which are called Me- chanical powers. They are, the Lever, the Pulley, the Wheel and Axle, the Inclined Plane, the Wedge, and the Screw. 433. To understand the nature of a machine, four things must be considered. 1st. The power or force which acts. This consists in the efforts of men or horses, of weights, springs, steam, &c. : 2d. The resistance which is to be overcome by the power. This generally is a weight to be moved : 3d. The center of motion, called a fulcrum or prop. The prop or fulcrum is the point about which all the parts of the machine move : 4th. The respective velocities of the power and resistance. 434. A machine is said to be in equilibrium when the re- sistance exactly balances the power ; in which case all the parts of the machine are at rest, or in uniform motion, and in the same direction. Lever. 435. The Lever is a bar of wood or metal, which moves around the fulcrum. There are three kinds of levers. 1st. When the fulcrum is between the weight and the ower : 2d. When the weight is between the power and the fulcrum : MECHANICAL POWERS. 385 ; 3d. When the power is between the fulcrum and tlie weight : t==== The perpendicular distance from the fulcrum to the di- rections of the weight and power, are called the arms of the lever. , 436. An equilibrium is produced in all the levers, when the weight, multiplied by its distance from the fulcrum, is equal to the power multiplied by its distance from the fulcrum. That is, Rule. — The weight is to the power, as the distance from Jie power to the fulcrum^ is to the distance from the weight to the fulcrum. Examples. 1. In a lever of the first kind, the fulcrum is placed at the middle point : what power will be necessary to balance a weight of 40 pounds ? 2. In a lever of the second kind, the weight is placed at the middle point : what power will be necessary to sustain a weight of 50 lb. ? 3. In a lever of the third kind, the power is placed at the middle point : what power will be necessary to sustain a weight of 25 lb. ? 4. A lever of the first kind is 8 feet long, and a weight of CO lb. is at a distance of 2 feet from the fulcrum : what power will be necessary to balance it ? 5. In a lever of the first kind, that is 6 feet long, a weight of 200 lb. is placed at 1 foot from the fulcrmn : what power will balance it ? 6. In a lever of the first kind, hke the common steelyard, the distance from the weight to the fulcrum is one inch ; at what distance from the fulcrum must the poise of 1 lb. bo 17 SS6 MECHANICAL POWEkS. placed, to balance a weight of lib? A weight of l|Ib. ? Of 2 lb. ? Of 4 lb. ? 7. In a lever of the third kind, the distance from the ful- crum to the power is 5 feet, and from the fulcrum to the weight 8 feet : what power is necessary to sustain a weight of 40 lb. ? 8. In a lever of the third kind, the distance from the ful- crum to the weight is 12 feet, and to the power 8 feet : what power will be necessary to sustain a weight of 1001b.? 437. Remarks. — In determining the equilibrium of the lever, we have not considered its weight. In levers of the first kind, the weight of the lever generally adds to the power, but in the second and third kinds, the weight goes to diminish the effect of the power. In the previous examples, we have stated the circumstances under which the power will exactly sustain the weight. In order that the power may overcome the resistance, it must of course be somewhat increased. The lever is a very important mechanical power, being much used, and entering, indeed, into most other machines. Of the Pulley. 438. The pulley is a wheel, having a t , ■ i groove cut in its circumference, for the /^ ^v purpose of receiving a cord which passes over it. When motion is imparted to the cord, the pulley turns around its axis, which is generally supported by being attached ^^ ^J to a beam above. 439. Pulleys are divided into two kinds, fixed pulleys and movable pulleys. When the pulley is fixed, it does not increase the power which is applied to raise the weight, but merely changes the direction 4n which it acts. MECHANICAL rOWKRS. 3ST advantage. Thus, r 440. A movable pulley gives a mechanical iu the movable pulley, the hand which sus- tains the cask actually supports but one- half of the weight of it ; the other half is supported by the hook to which the other end of the cord is attached. 441. If we have several movable pul- leys, the advantage gained is still greater, and a very heavy weight may be raised by a small power. A longer time, however, will be required, than with the single pulley. It is, indeed, a general principle in ma- cliines, that what is gained in power, is lost in time; and this is true for all ma- chines. There is also an actual loss of power, viz., the resistance of the machine to motion, arising from the rubbing of the parts against each other, which is called the friction of the machine. This varies in the different machines, but must always be allowed for, in calculating the power neces- sary to do a given work. It would be wrong, however, to suppose that the loss was equivalent to the gain, and that no advantage is derived from the mechanical powers. We are unable to augment our strength, but by the aid of science we so divide the resistance, that by a continued exertion of power, we accomplish that which it would be impossible to effect by a single effort. If, in attaining this result, we sacrifice time, we cannot but see that it is most advantageously exchanged for power. 442. It is plain, that in the movable pulley, all the parts of the cord will be equally stretched ; and hence, each cord run- 388 MECHANICAL PuWERS. ning from pulley to pulley, will bear an equal part of the weight ; consequently, Rule. — The power will always he equal to the weight divided by the number of cords which reach from pulley to pulley. Examples. 1. In a single immovable pulley, what power will support weight of 601b.? 2. In a single movable pulley, what power will support a weight of 80 lb. ? 3. In two movable pulleys, with 4 cords (see last fig.), what power will support a weight of 1001b.? 443. This machine is com- posed of a wheel or crank, firmly attached to a cylin- drical axle. The axle is supported at its ends by two pivots, which are of less diameter than the axle around which the rope is coiled, and which turn freely about the points of support. In order to balance the weight, we must have, Rule. — The power to the weight, as the radius of the axle, to the length of the crank, or radius of the tvheel. Examples. 1. What must be the length of a crank or radius of a wheel, in order that a power of 401b. may balance a weight of 600 lb. suspended from an axle of 6 inches radius ? MECHANICAL POWERS. 389 2. What must be the diameter of an axle, that a power of 1001b., applied at the circumference of a wheel of 6 feet diameter, may balance 400 lb. ? Inclined Plane. 444. The inclined plane is nothing more than a slope or l(.'clivity, which is used for the purpose of raising weiglits. It is not difficult to see that a weight can be forced up an in- clined plane, more easily than it can be raised in a vertical line. But in this, as in the other machines, the advantage is obtained by a partial loss of power. Thus, if a weight W, be supported on the in- clined plane ABC, by a cord passing over a pulley at F, and the cord from the pulley to the weight be parallel to the length of the plane AB, the power P will balance the weight W., when P : W : : height BC : length AB. It is evident that the power ought to be less than the weight, since a part of the weight is supported by the plane : hence, Rule. — The power is to the weight , as the height of the plane is to its length. Examples. 1. The length of a plane is 30 feet, 'and its height 6 feet what power will be necessary to balance a weight of 2001b.? 2. The height of a plane is 10 feet, and the length 20 feet what weight will a power of 501b. support? 3. The height of a plane is 15 feet, and length 45 feet: what power will sustain a weight of 1801b.? 390 MECHANICAL POWERS. The Wedge. 445. The wedge is composed of two inclined planes, united together along their bases, and form- ing a solid ACB. It is used to cleaye masses of wood or stone. The resistance which it overcomes is the attraction of cohesion of the body which it is employed to separate. The wedge acts, principally, by being struck with a hammer, or mallet, on its head, and very Uttle effect can be produced with it, by mere pressure. All cutting instruments are constructed on the principle of the inclined plane or wedge. Such as have but one sloping edge, like the chisel, may be referred to the inclined plane ; and such as have two, like the ax and the knife, to the wedge. Rule. — Half the thickness of the head of the wedge, is to the length of one of its sides, as the power which acts against its head to the effect produced at its side. Examples. 1. If the head of a wedge is 4 inches thick, and the length of one of its sides 12 inches, what will measure the effect of a force denoted by 96 pounds ? 2. If the head of a wedge is 6 inches thick, the length of the side 2t inches, and the force applied measures 250 pounds, what will be the measure of the effect? ♦ 3. If the head of a wedge is 9 inches, and the length of the side 2 feet, what will be the effect of a blow denoted by 200 pounds? 4. If the head of a wedge is 10 inches, and the length of the side 30 inches, what will measure the effect of a blow denoted by 600 ? MECHANICAL POWERS. 391 . The Screw. 446. The screw is composed of two parts — the screw, S, and the nut, N. The screw, S, is a cylinder with a spiral projection winding around it. The nut, N, is per- forated to admit the screw, and within it is a grove into which the thread of the screw fits closely. The handle, D, which projects from the nut, is a lever which works the nut upon the screw. The power of the screw depends on the distance between the threads. The closer the threads of the screw, the greater will be the power ; but then the number of revolutions made by the handle, D, will also be proportionably increased : so that we return to the general principle — what is gained in power is lost in time. The power of the screw may also be increased by lengthening the lever, D, attached to the nut. The screw is used for compression, and to raise heavy weights. It is used in cider and wine presses, in coining, and for a variety of other purposes. Riile. — As the distance between the threads of a screw, is to the circumference of the circle described by the power, so is the poioer employed to the weight raised. Examples. 1. If the distance between the threads of a screw is half an inch, and the circumference described by the handle 15 feet, what weight can be raised by a power denoted by 120 pounds ? 2. If the threads of a screw are one-third of an inch apart, and the handle is 12 feet long, what power must be appUed to sustain 2 tons? 392 QUESTIONS IN PHILOSOPHY. 3. What force applied to the handle of a screw 10 feet long, with threads one inch apart, working on a wedge whose head is 5 inches, and length of side 30 inches, will produce an effect measured by 100001b.? 4. If a power of 300 pounds applied at the end of a lever 15 feet long will sustain a weight of 282^44 lb., what is the listance between the threads of the screw ? QUESTIONS IN NATUEAL PHILOSOPHY. UNIFORM MOTION. 447. If a moving body passes over equal spaces in equal portions of time, it is said to move with uniform motion, or uniformly. 448. The velocity of a moving body is measured by the space passed over in a second of time. 449. The space passed over in any time is equal to the pro- duct of the velocity multiplied by the number of seconds in the time. If we denote the velocity by Y, the space passed over by S, and the time by T, we have S = Y X T. LAWS OF FALLING BODIES. 450. A body falling vertically downward in a vacuum, falls through IGj'jft. during the first second after leaving its place of rest, 48ift. during the second second, 80^ ft. the third second, and so on : the spaces forming an arithmetical pro- gression of which the common difference is 321 ft., or double the space fallen through during the first second. This numljer is called the measure of the force of gravity, and is denoted by g. 451. It is seen from the above, that the velocity of a body QUESTIONS IN PHILOSOPHY 393 is continually increasing. If H denote the height fallen through, T, the time, V, the velocity acquired, and g, the force of gravity, the following formulas have been found to express the relations between these quantities : Y = g XT . . . (1). Y' = 2g XU . . . (2). H = |Y X T . . . (3). R =:^g XT . . . (4). From which we see, 1st. That the velocity acquired at the end of any time, is < qual to the force of gravity (32 Jr) multiplied by the time. 2d. That the square of the velocity is equal to twice the force of gravity multijjlied by the height; or, the velocity is equal to the square root of that quantity. od. That the space fallen through is equal to one-half the velocity multiplied by the time. 4th, That the sjmce fallen through is equal to one-half the force of gravity multiplied by the square of the time. 452. If a body is thrown vertically upward in a vacuum, its motion will be continually retarded by the action of gravi- tation. It will finally reach the highest point of its asceht, and then begin to descend. The height to which it will rise may be found by the second formula in the preceding para- graph, when the velocity with which it is projected upward is known ; for the times of ascent and descent will be equal. 453. The above laws are only approximately true for bodies falling through the air, in consequence of its resistance. We may measure the depths of wells or mines, and the heights of elevated objects approximately, by using dense bodies, as leaden bullets or stones, which present small surfaces to the air. Examples. 1. A body has been falling 12 seconds : what space did it describe m the last secoid, and what in the whole time? 17* 39i QUESTIONS IN PHILOSOPHY. 2. A body has been falling 15 seconds : find the space described and the velocity acquired. 3. How far must a body fall to acquire a velocity of 120 feet? 4. How many seconds will it take a body to fall throuo^h a space of 100 feet? 5. Find the space through which a heavy body falls in lO seconds, and the velocity acquired. 6. How far must a body fall to acquire a velocity of 1000 feet? 7. A stone is dropped into a well, and strikes the water in 3.2 seconds : what is the depth of the well ? 8. A stone is dropped from the top of a bridge, and strikes the water in 2.5 seconds : what is the height of the bridge ? 9. A body is thrown vertically upward with a velocity of 160 feet : what height will it reach, and what will be the time of ascent? 10. An arrow shot perpendicularly upward, returned again m 10 seconds. Required the velocitv with which it was shot, and the height to which it rose. 11. A ball is let fall from the top of a steeple, and reaches the ground in three seconds and a half: what is the height of the steeple ? 12. What time will be necessary for a body falling freely, to acquire a velocity of 2500 feet per second? 13. If a ball be thrown vertically upward with a velocity of 35v) feet per second, how far will it ascend, and what will be the time of ascent and descent? 14. How long must a body fall freely to acquire a velocity of 3040 feet per second ? 15. If a body falls freely in a vacuum, what will be its velocity afte* 45 seconds of fall? QUESTIONS IN rJIILOSOPIIY. 304 16. During how many seconds must a body fall in a vacuum to acquire a velocity of 1970 feet, which is that of a camion- ball? n. What time is required for a body to fall in a vacuum, from an elevation of 3280 feet? 18. From what height must a body fall to acquire a vo- ocity of 984 feet ? 19. A rocket is projected vertically upward with a velocity of 380 feet : after what time will it begin to fall, and to what height will it rise ? SPECIFIC GRAVITY. 454. The specific gravity of a body is the weight of a unit of volume compared with the weight of a unit of the tandard. Distilled rain-water is the standard for measuring the i>ccific gravity of bodies. Thus, 1 cubic foot of distilled rain- water weighs 1000 ounces avoirdupois. If a piece of stone, of Uic same volume, weighs 2500 ounces, its specific gravity is 2.5 ; that is, the stone is 2.5 times as heavy as water. If, then, we denote the standard by 1, the specific gravity of all other bodies will be expressed in terms of this standard; and if we multiply the number denoting the specific gravity of any body by 1000, the product will be the weight in ounces of 1 cubic foot of that body. If any body be weighed in air and then in water, it will weigh less in water than in air. The difference of the weights will be equal to the sustaining force of the water, which is found to be equal to the weight of an equal volume of water: hence. Rule. — If we know or can find the weight of a body in air and in water, the difference of these lueights will he equal to that of an equal volume of water; and the weight of the body in air divided by this difference will be the measure of the specific gravity of the body, compared with woier as a standard. . 396 QUESTIONS IN PHILOSOPHY. Table OF SPECIFIC GRAVITIES— WATER, 1. NAMES OF BODIES. SPEC. GRAV. NAMES OF BODIES. SPKG GBAV. METALS. 21.000 19.500 13.500 11.350 10.51 8.800 8.758 8.000 7.800 7.500 7.291 7.215 3.10 3.10 3.00 3.00 2.93 2.90 2.75 2.72 2.69 2.66 2.62 2.60 PorDlivrv 2.60 2.50 1.86 1.049 0.9859 0.9822 0.9476 0.9452 0.9206 0.9121 0.9036 0.9036 0.9012 0.8941 0.8614 1.480 2.050 2.150 2.542 0.926 1.842 0.942 0.969 Platinum Brick Quicksilver WOODS. Oak, fresh felled White Willow Box Lead Silver Bronze Elm Steel Horbeam Larch Tin Pine Zinc Maple BUILDING STONES. Hornblende Ash Birch Fir Horse-chestnut SOLID BODIES. . Common earth Moist sand Basalt Alabaster Syenite Dolerite Gneiss Quartz Clay Limestone Flint Phonolite Ice Granite Lime Stone for building . . . Trachyte Tallow Wax By inspecting this Table, we see the weight of each body compared with an equal volume of water. Thus, platinum is 21 times as heavy as water ; gold, 19 times as heavy ; iron, t^ times as heavy, &c. Examples illustrating Specific G-ravity. 1. A piece of copper weighs 93 grains in air, and 82 J grains in water : what is its specific gravity ? 2. How many cubic feet are there in 2240 pounds of dry oak, of which the specific gravity is .925, a cubic foot of standard water weighing 1000 ounces ? QUESTIONS IN PHILOSOPHY. 397 3. A piece of pumice-stone weighs in air 50 ounces, and when it is connected with a piece of copper which weighs 390 ounces in air, and 345 ounces in water, the compound weighs 344 ounces in water ; what is the specific gravity of the stone ? 4. A right prism of ice, the length of whose base is 20.45 yards, breadth 15.75 yards, and height 10.5 yards, floats on the sea; the specific gravity of the ice is .930, and that of the sea-water 1.026 : what is the height of the prism above the surface of the water? 5. A vessel in a dock was found to displace 6043 cubic feet of water : what was the weight of the vessel, each cubic foot of the water weighing 03 pounds ? 0. A piece of glass was found to weigh in the air 33 ounces, and in the water 21 ounces : what was its specifiti gravity ? 7. A piece of zinc weighed in the air IT pounds, and lost when weighed in water 2.35 pounds : what was its specific gravity ? 8. If a piece of glass weighed in water loses 318 ounces of its weight, and weighed in alcohol loses 250 ounces, what is tlie specific gravity of the alcohol? 9. A flask filled with distilled water weighed 14 ounces ; filled with brandy, it weighed 13.25 ounces ; the flask itself weighed 8 ounces : what was the specific gravity of the brandy ? 10. What is the weight of a cubic foot of statuary marble, of which the specific gravity is 2.83T, the cubic foot of water weighing 1000 ounces ? 11. A jar containing air weighed 24 ounces 33 grains ; the air was then excluded, and the jar weighed 24 ounces ; the jar being then filled with oxygen gas, weighed 24 ounces 36.4 grains : what was the specific gravity of the oxygen the air being taken as the standard? g98 QUESTIONS IN PHILOSOPHY. mariotte's law. 455. This law, which relates to air and all other gases, steam i ad all other vapors, was discovered by the Abbe Mariottc, a French philosopher, who died in 1684. It will be easily understood from a particular example. Suppose an upright cylindrical vessel in a vacuum contains gas which is confined in the vessel by a piston at the upper. r,nd. Suppose the gas or vapor fills the whole vessel, and the |)lston is loaded with a weight of 5 pounds. If now, the piston be loaded with a weight of 10 pounds, the gas will be compressed and occupy only half its former space. If the weight be increased to 15 pounds, the gas will have only one- third of its original volume, and so on. At the same time, the density of the gas or vapor will be doubled, made three times as great, and so on. The law, therefore, may be thus stated : Rule. — The temperature remaining the same, the volume of a gas or vapor is inversely proportional to the pressure which it sustains. Also, the density of a gas or vapor is di- rectly proportional to the pressure. Examples. 1. A vase contains 4.3 quarts of air, the pressure being 10 pounds : what will be the volume of the air when the pressure is 12.3 pounds, the temperature remaining the same? 2. Under a pressure of 15 pounds to the square inch, a certain quantity of gas occupies a volume of 20 quarts : what pressure must be applied to reduce the volume to 8 quarts? 3. A quart of air weighs 2.6 grains under a pressure of 15 pounds : what will be the weight of a quart, if the pressure be reduced to 14.2 pounds ? 4. The pressure upon the steam contained in a cylinder is increased from 25 pounds upon the square inch to 4t pounds : what part of the original volume will be occupied ? APPENDIX. DIFFERENT KINDS OF UNITS. I. ABSTRACT UNITS. 456. The abstract unit I is the base of all numbers, and is tailed a unit of the first order. The unit 1 ten, is a unit of the second order ; the unit 1 hundred, is a unit of the third order ; and so for units of the higher orders. These are ab- stract numbers formed from the unit 1, according to the scale of tens. All abstract integral numbers are collections of the unit one. II. UNITS OF CURRENCY. 457. Ill all civilized and commercial countries, great care is taken to fix a standard value for money, which standard is culled the Unit of Currency. In the United States, the unit of currency is 1 dollar ; in Great Britain it is I pound sterling, equal to $4.84 ; in France it is 1 franc, equal to I8j% cents nearly. All sums of money are expressed in the unit of currency, or in units derived from the unit of currency, and having fixed proportions to it. III. UNITS OF LENGTH. 458. One of the most important units of measure is that for distances, or for the measurement of length. A practical want has ever been felt of some fixed and invariable stand- ard, with which all distances may be compared : such fixed standard has been sought for in nature. There are two natural standards, either of which affords this desired natural element. Upon one of them, the English have founded their system of measures, from which ours is taken ; and upon the other, the French have based their system. 400 APPENDIX. These two systems, being the only ones of importance, will be alone considered. First. — The English system of measures, to which ours con- forms, is based upon the law of nature, that the force of gravity is constant at the same point of the earth's surface, and consequently, that the length of a pendulum which oscil ates a certain number of times in a given period, is also con stant. Had this unit been known before the adoption and use of a system of measures, it would have formed the natural unit for divisior and been the natural base of the system of linear measure. But the foot and inch had long been used as units of linear measure ; and hence, the length of the pendu- lum, the new and invariable standard, was expressed in terms of the known units, and found to be equal to 39.1393 inches. The new unit was therefore declared invariable — to contain 39.1393 equal parts, each of which was called an inch; 12 of these parts were declared by act of Parliament to be a stand- ard foot, and 36 of them an Imperial yard. The Imperial yard and the standard foot are marked upon a brass bar, at the temperature of 62^°, and these are the linear measures from which ours are taken. The comparison has been made by means of a brass scale 82 inches long, manufectured by Troughton, in London, and now in the possession of the Treasury Department. Second. — The French system of measures is founded upon the principle of the invariability of the length of an arc of the same meridian between two fixed points. By a very minute survey of the length of an arc of the meridian from Dunkirk to Barcelona, the length of a quadrant of the me- ridian was computed, and it has been decreed by the French aw that the ten-millionth part of this length shall be regarded as a standard French metre, and from this, by multiplicatioa and division, the entire system of linear measures has been established. On comparing the two scales very accurately, it has been UNl'Ib OJ^ VOLUME. 401 found that the French m^tre is equal to 39.31079 English inches — differing somewhat from the Englisli yard. This rela- tion enables us to convert all measures in either system into the corresponding measures of the other. IV. UNITS OF SURFACK 459. The linear unit having been established, the most con- venient UNIT OF SURFACE is the area of a square, one of whose sides is the unit of length. Thus, the units of surface in com- mon use, are — • A square inch = a square on I inch. A square foot = a square on 1 foot. A square yard = a square on I yard. A square rod = a square on 1 rod. V. UNITS OF VOLUME. 460. The unit of volume, for the measurement of solids, is taken equal to the volume of a cube one of whose edges is equal to the linear unit. The units of volume in common use are — A cubic inch = a cube whose edge is 1 inch ; A cubic foot = a cube whose edge is 1 foot = 1728 cubic in. A cubic yard = a cube whose edge is 1 yard = 27 cubic feet. A perch of stone = 24j cubic feet ; or a block of stone 1 ro^d long, 1 foot thick, and IJft. wide. The standard unit of volume for the measurement of liquids is the wine gallon, which contains 231 cubic inches. The standard unit of dry measure is the Winchester bushel, which contams 2150.4 cubic inches, nearly. VI. UNITS OF WEIGHT. 461. Having fixed an invaiiable unit of length, we passed easily to an invariable unit of surface, and then, to an invari- able unit of volume. We wish now to define an invariable unit of weight. 402 APPENDIX. It has been found that distilled rain-water is the most in- variable substance ; hence, this has been adopted as the standard. We have two units of weight, the avoirdupois pound, and the pound troy. The standard avoirdupois pound is the weight of 2 1. 701554 cubic inches of distilled water. The standard Troy pound is the weight of 22.194422 cubic inches. This standard is at present kept in the United States Mint at Philadelphia, and is the standard unit of weight. VII. UNITS OF TIME. 462. Time can only be measured by motion. The diurnal revolution of the earth affords the only invariable motion ; hence, the time in which it revolves once on its axis, is the natural unit, and is called a day. From the day, by multipli- cation, we form the weeks, months, and years ; and by division, the hours, minutes, and seconds. VIII. UNITS OF CIRCULAR OR ANGULAR MEASURE. 463. This measure is used for the measurement of angles, and the natural unit is the right angle. But this is not the most convenient unit. The unit chiefly used is the 360 part of the circumference of a circle, called a degree, which is di- vided into 60 equal parts, called minutes, and these again into 60 equal parts, called seco7ids. Hemarks. 464. It is seen that all the units, determined by the pen- dulum, depend on time as the ultimate base ; that is, the length of a pendulum which will vibrate seconds determines all ^he units of 'tneasure and weight. Now, time is measured by motion, and the motion of the earth on its axis is the only invariable motion. Hence, we refer to this to fix the unit of time, on which the unit of length depends, and from which all the other units are derived. ABSTRACT UNITS. 4.03 No class of pupils can rightly and clearly apprehend the nature of numbers and the operations performed upon them, without distinct and fixed notions of the units ; hence, every teacher should labor to point out their absolute and relative values : this can only be done by means of sensible objects. Every school-room, therefore, should be provided with complete set of all the denominate units. The inch, the foot the yard, the rod, should be accurately marked off on a con- spicuous part of the room, together with the principal units of surface, the squre inch, square foot, square yard, &c. The units of volume should also be exhibited. The cubic inch and the cubic foot will serve as illustrations for one class of the units of volume ; and the pint, quart, gallon, and bushel, should be exhibited to illustrate the others. The unit of weight should also be seen and handled. A child even can apprehend what is meant by an ounce or a pound, when it takes one of these weights in its hand ; and mature years can acquire the idea in no other way. Let, therefore, every school-room be furnished with a com- plete set of models to illustrate and explain the absolute and rdatiijc values of the different units. I. ABSTRACT NUMBERS. 465. An Abstract Number is one whose unit is not named. Table. 10 Umts make 1 Ten. 10 Tens 1 Hundred. 10 Hundred 1 Thousand. 10 Thousand 1 Ten-thousand. &c., &c. Table Reversed. Ten. TJnIta. nuncL 1 = 10. Thous. 1 = 10 = 100. Ton-thous. 1 = 10 = 100 = 1000. 1 = 10 = 100 = 1000 = 10000. Scale. — Uniform — units, 10. 4.01 APPENDIX. II. CURRENCY. I. UNITED STATES MOXEY. 466. United States Money is the currency established by Congress, a. d. 1786. The names or denominations of its nits are, Eagles, Dollars, Dimes, Cents, and Mills. The coins of the United States are of gold, silver, copper, nd nickel, and are of the following denommations : 1. Gold : Double-eagle, eagle, half-eagle, three-dollars, quar- ter-eagle, dollar. 2. Silver : Dollar, half-dollar, quarter-dollar, dime, half-dime, and three-cent piece. 3. Copper : Cent. 4. Nickel: Cent. * Table. 10 Mills .... make 1 Cent, .... marked ct. 10 Cents 1 Dime, d. 10 Dimes 1 Dollar, |. 10 Dollars 1 Eagle, B. 20 Dollars 1 Double Eagle. . . 2 E. Table Reversed. ct m. d. 1 = 10. $ 1 = 10 = 100. E. 1 : = 10 = 100 = 1000. 1 = 10 100 = 1000 = 10000. Scale. — Uniform — units, 10. In all the States the shilling is reckoned at 12 pence, the ^ound at 20 shillings, and the dollar a^ 100 cents. The following table shows the number of shillings in a dol lar, the value of iEl in dollars, and the value of $1 in the fraction of a pound : ENGLISH MONEY. 4:05 Til English currency, 4s. Gd. - ^21 = $4.84, and |1 re £^\^. In New York, Ohio, ) „ ^, ^^i , *, ,,. , . ' ^ 8s. - £1= $21, and $1 = M. Michigan, ) ^ * North Carolina, 10s. r„KJ„Pa.. DeI.,J ,,«,.. ^i^^2^_ aud|l = ^. In S. CaroUna & Ga., 4s. 8d. - £1 = $4|, and $1 = £^, In Canada and Nova,) ^ ^, *. , <,^, ^, „ ,. 7 5s. - iei = $4, and $1 = £i, Scotia, 3 * Notes. — The present standard or degree of purity of the coins was fixed by Act of Congress in 1837. It is this: 1. Nine hundred equal parts of pure gold, are mixed with 100 parts of alloy, of copper, and silver, (of wliich not more than one-half must be silver) thus forming 1000 parts, equal to each other in weight. The silver coins contain 900 parts of pure silver, and 100 parts of pure copper. The copper coins are of pure copper. The nickel cent is 88 parts copper and 12 nickel. 2. The eagle contains 258 grains of standard gold, and the other gold coins in the same proportion. The dollar contains 412i grains of standard silver, and the others in the same proportion. The cent, 168 grains of pure copper. 3. If a given quantity of gold or silver be divided into 24 equal parts, each part is called a carat. If any number of carats be mixed with so many equal carats of a less valuable metal, that there be 24 carats in the lAixture, then the comx)ound is said to be as many carats fine as it contains carats of the more precious metal, and to contain as much aUoy as it contains carats of the baser. 4. Although the currency of the United States is in dollars, cents, and mills, yet in some of the States the old currency of pounds shillings, and pence, is still nominally preserved. II. ENGLISH MONEY. 467. The units or denominations of English money are ejuineas, ponnds, shillings, pence, and farthings. •.too ArricNDix. Table. 4 fartliings, marked far., make 1 penny, marked d. 12 pence, 1 sliilling, " ». 20 shillings, 1 pound, or sovereign, £. 21 shillings, 1 guinea. Table Reversed. d. fax, 1 = 4. £ 1 = 12 = 48. 1 = 20 = 240 = 960. Notes. — 1. The primary unit in English money is 1 farthing. The units of the scale, in passing from farthings to pence, are 4 ; in passing from pence to shillings, the units of the scale are 12 ; in pass- ing from shillings to pounds, they are 20. 2, Farthings are generally expressed in fractions of a penny. Thus, Ifar. = id. ; 2 far. = id. ; 3 far. = |d. 3. The standard of the gold coin is 22 parts of pure gold and 2 parts of copper. The standard of silver coin is 37 parts of pure silver, and 3 parts of copper. A poimd of gold is worth 14.2878 times as much as a pound of silver. In copper coin, 24 pence make 1 pound avoirdupois. By reading the second table from left to right, we can see the value of any unit expressed in each of the lower denominations. Thus, Id. = 4 far. ; Is. = 12d. = 48 far. £1 = 20s. = 240d. = 960 far. TABLE OF LEGAL VALUES OF FOREIGN COINS. Franc of France and Belgium.. Florin of the Netherlands Guilder of do Livre Tournois of France Milrea of Portugal Mih-ea of Madeira , Milrea of the Azores Marc Banco of Hamburg , Pound Sterling of Great liritain. Pagoda of India , Real Vellon of Spain Real Plate of do Rupee Company $ ct. 18^ 40 40 18-i 1 12 1 00 831 35 4 84 1 84 05 10 44| VALLK OF COINS. 407 1^111)60 of Britisli Jiulia Jiix Dollar of Denmark Jiix Dollar of Prussia Kix Dollar of Bremen Kouble, silver, of Russia Tale of China Dollar of Sweden and Norway Si>ecie Dollar of Denmark Dollar of Prussia and Northern States of Germany... Florin of Southern States of Germany Florin of Austria and City of Augsburg Lira of the Lombardo-Venetian Kingdom Lira of Tuscany Lira of Sardinia Ducat of Naples Ounce of Sicily Pound of Nova Scotia, New Brunswick, Newfound land, and Canada 44i 1 00 i)8.\ T8| 75 1 48 1 OG 1 05 69 40 . 481 16 16 18 ,-•'3 80 2 40 4 04 TABLE OF FOREIGN COINS OF USAGE VALUES. Berlin Rix Dollar Current marc Crown of Tuscany . . . Elberfeldt Rix Dollar Florin of Saxony " Bohemia Elberfddt. . . " Prussia Trieste " Nuremburg. Frankfort.., " Basil St. Gaul . . . . Creveld Livre do do l^lorence Genoa Geneva Jamaica Pound Leghorn Dollar Leghorn Livre (GJ to the doUar) Livre of Catalonia Neulchatel Livre Pezza of Leghorn Rhenish Rix Dollar Swiss Li\Te Scuda of Malta Turkish Piastre ct 69i 28 05 69| 48 48 40 221 48 40 40 41 40 15 18| 21 00 90 1511 53i 26i 90 60} 27 40 05 [Tlie above Tables arc taken from a work en tho Tariff, by E. D. Ogden, Esq., of Misf ^e\v York Custom-house]. 408 APPENDIX. III. LINEAR MEASURE. I. LONG MEASURE. 468. This measure is used to measure distances, lengths, breadths, heights, and depths. Table. 12 inches make 1 foot, marked ft. 3 feet 1 yard, yd. 51 yards, or IGi feet .... 1 rod, rd. 40 rods 1 furlong, fur. 8 furlongs, or 320 rods ... 1 mile, mi. 3 miles 1 league, L. 69^ statute miles, nearly, or . ) 1 degree on the equator,; 60 geographical miles ... ) or any great circle, 360 degrees a circumference of the earth. Table Reversed. [ deg., a in. ' yd. 1 12. rd. 1 = 3 36. far. 1 = H = 161 198. mi. 1 = 40 = 220 = 660 1920. 1 = = 8 = 320 = neo = 5280 63360. Notes. — 1. "A fathom is a length of six feet, and is generally used to measure the depth of water. 2. A hand is 4 inches, and is used to measure the height of horses ; a common pace is 3 Teet ; a military pace, 2} feet ; a geo- graphical mile equals a minute of a great circle; a knot (used by sailors) is a geographical mile. 3. The units of the scale, in passing from inches to feet, are 12, in passing from feet to yards, 3; from yards to rods, 5i; from rods to furlongs, 40; and from furlongs to miles, 8. ENGLISH SYSTEM. 469. The Imperial yard of Great Britain is the one from which ours is taken. Hence, the units of measure are identical. CLOTH MEASURE. 409 FRENCH SYSTEM. 470. The base of the new French system of measures is the measure of the meridian of the earth, a quadrant of which is 10,000,000 mitres, measured at the temperature of 32° Fahr. The multiples and divisions of it are decimals, viz. : 1 metre = 10 decunetres = 100 centimetres = 1000 millhnetres — 3.280899 United States feet, or 39.3t0t9 inches. This relation enables us to convert all measures in either system into the corresponding measures of the other. Austrian, 1 foot = 12.448 U. S. inches = 1.03t3T foot. Prussian, } ^ ^^^^ ^ j^.sci " " = 1.0800 " Rhineland, ) Swedish, 1 foot =11.690 " " = 0.9t4145 " f 1 foot = 11.034 " " = 0.9195 " Spanish, -J league (royal) = 25000 Span. ft. = 4 J miles ) ^ ( " (common) = 19800 " = 3i " ) I II. CLOTH MEASURE. 471. Cloth measure is used for measuring all kinds of cloth, ribbons, and other things sold by the yard. Table. 2k inches, (in.) . . make 1 nail, .... marked na. 4 nails 1 quarter of a yard, . . . qr. 3 quarters 1 Ell Flemish, . . . . E. Fl. 4 quarters 1 yard, yd. 5 quarters 1 EU EngUsh, E.R Table Reversed. □a. in. qr. 1 = 2i. KFL 1=4=9. ya. 1 = 3 = 12 = 2t. E.E. 1 = li = 4 = 16 = 36. 1 = IJ = If = 5 = 20 = 45. Note. — The units of the scale, in this measure, are 2i, 4, 3, \ and f. 18 410 APPENDIX. IV. UNITS OF SUEFACE. I. SQUARE MEASURE. 472. Square measure is used in measuring land, or any thiKg in which length and breadth are both considered. Table. i 44 square inches {sq. in.) make 1 square foot, .... 9 square feet 1 square yard, . . . 80i square yards 1 square rod, or 1 perch, 40 square rods, or 40 perches, 1 rood, 4 roods, or 160 perches, . . 1 acre, ... 640 acres 1 square mile JSq.ft Sq.yd. P. B. A. M. A. 1 1 = 4 p. 1 40 160 Table Reversed. 6q. ft sq. in. 1 = 144. 9 = 1296. 2121 = 39204. 1210 = 10890 = 1568160. 4840 = 43560 = 6272640. Bq. yd. 1 = 30J = Note. — The units of the scale in this measure are 144, 9, 80}, 40, and 4. II. surveyors' measure. 473. The Surveyor's, or Gunter's chain, is generally used in surveying land. It is 4 poles, or 66 feet in length, and is divided into 100 links. Table. 7^0% inches .... make 1 link, marked I. 4 rods = 66 ft. = 100 links . 1 chata, c. 80 chains 1 mile, mi 1 square chain 16 square rods, or perches, . . P 10 square chains 1 acre, A Notes. — 1. Land is generally estimated in square miles, acres roods, and square rods, or perches. 2. The units of the scale, in this measure, are Ty'/o, 4, 80, 1, and 10. UNITS OF VOLUME. 411 V. UNITS OF VOLUJME, OR CAPACITY. I. CUBIC MEASURE. 474. Cubic measure is used for measuring stone, timber, earth, and sucli other things as have the three dimensions, cngtli, breadth, and thickness. Table. 1728 cubic inches {cu. in.) make 1 cubic foot, . marked cu. ft. 27 cubic feet 1 cubic yard, cu. yd. 40 feet of roimd, or ; ^ ^ _, ^„ , , - ' . , h .1 ton, T. 50 feet of liewn timber,) 42 solid feet 1 ton of shipping, .... T. 8 cord feet, or) 1 /i n 128 cubic feet, f • • • • ^' ' 24i cubic feet of stone . , . 1 perch, P. Notes. — 1. A cord of wood is a pile 4 feet wide, 4 feet bigh, and 8 feet long. 2. A cord foot is 1 foot in length of the pile which makes a cord. 3. A CUBE is a solid or volxmie bounded by six equal squares, called faces; the sides of the squares are called edges. 4. A cubic foot is a cube, each of whose faces is a square foot; its edges are each 1 foot. 5. A cubic yard is a cube, each of whose edges is 1 yard 6. A ton of round timber, when square, is supposed to produce 40 cubic feet: hence, one-fifth is lost by squoHng. II. LIQUID MEASURE. 475. Liquid, or wine measure, is used for measuring all liquids except ale, beer, and milk. 4:12 APPENDIX. Table. 4 giiiS (gi.) .... make 1 pint, marked pt 2 pints 1 quart, qi 4 quarts .1 gallon, gal. 31i gallons 1 barrel, bar. or bbl. 42 gallons 1 tierce, tie?' 63 gallons 1 hogsliead, Jihd 2 hogsheads 1 pipe, ^)^ 3 pipes, or 4 hogsheads, . 1 tun, tun. Table Reversed. pt gi. qt 1 = 4. gal 1 = 2 = 8. bar. 1 = 4 = 8 = 32. tier. 1 = 311 = 126 = 252 = 1008. hbd. 1 =11 42 = 168 = 336 = 1344. pi. 1 = 11 = 2 63 = 252 = 504 = 2016. tun. 1 = 2 = 3 =4 126 = 504 = 1008 = 4032. 1 = 2 = 4 = 6 =8 252 = 1008 = 2016 = 8064. Notes. — 1. The standard unit, or gallon of wine measure, in the United States, contains t31 cubic inches, and hence, is equal to the weight, avoirdupois, of 8.839 «ubie '4nche8 of distilled water, very nearly. _ * 2. The English Imperial wine gallon contains 277.274 cubic inches, and hence, is equal to 1.2 times the wine gallon of the United States, nearly. III. BEER MEASURE. 476. Beer measi^e was formerly used for measuring ale, beer, and milk. They are now generally measured by wine measure. Table. 2 pints {pt.) . . . make 1 quart, marked gf. 4 quarts 1 gallon, gal 36 gallons 1 barrel, bar 54 gallons - 1 hogshead, 7i7id DRY MEASURE. 413 Table Reversed. gal. bar. 1 = qt 1 4 = pL 2. 8. hhd. 1 = 3G = 144 = 288. 1 = IJ = 54 = 216 = 432. Notes.— 1. The standard gallon, beer measure, contains 282 cable mches, and hence, is equal to the weight of 10.1799 cubic inches o» distilled rain-water. 2. Milk is generally bought and sold by wine measure. III. DRY MEASURE. 477. Dry measure is used iu measuring all dry articles, such as grain, fruit, salt, coal, &c. Table. 2 pints (pL) .... make 1 quart, marked qt. 8 quarts 1 peck, pk. 4 pecks 1 bushel, Im, 36 bushels 1 chaldron, cTi, Table Reversed. qt pt pk. 1 = 2. bn. 1 = 8 = 16. Ob. 1 = 4 = 32 = 64. 1 = = 36 = 144 = 1152 = 2304. Notes. — 1. The standard hushd of the United States is the Win diester bushel of England. It is a circular measure 18i inches it diameter, and 8 inches deep, and contains 2150.4 cubic inches, nearly It contains 77.627413 pounds avoirdupois of distilled water. 2. A gallon, dry measure, contains 268.8 cubic inches. 3. Wine measure, beer measure, and dry measure, and all meas- ures of volume, differ from the cubic measure only in the unit which is used as a standard. 4:14 APPENDIX. VI. UNITS OF WEIGHT. I. AVOIRDUPOIS WEIGHT. 478. By this weight all coarse articles are weighed, such as hay, grain, chandlers' wares, and all metals except gold and silver. Table. 16 drams {di\) .... make 1 ounce, .... marked oz. 16 ounces 1 pound, lb. 25 pounds 1 quarter, qr- 4 quarters 1 hundred weight cwt. 20 hundred weight 1 ton, T. Table Reversed. ot dr. lb. 1 = 16. qr. 1 ^zz 16 =z 256. 1 = 25 =: 400 = 6400. 4 = 100 = 1600 = 25600. ewt T. 1 = 1 = 20 = 80 = 2000 = 32000 = 512000. Notes. — ^1. The standard avoirdupois pound is the weight of 27.7015 cubic inches of distilled water; and hence, 1 cubic foot weighs 1000 ounces, very nearly. 2. By the old method of weighing, adopted from the English sys- tem, 112 pounds were reckoned for a hundred weight. But now, the laws of most of the States, as well as general usage, fix the hun- dred weight at 100 pounds. 3. The units of the scale, in passing from drams to ounces, are 16; from ounces to pounds, 16; from pounds to quarters, 25; from quarters to hundreds, 4; and from hundreds to tons, 20. II. TROY WEIGHT. 479. Gold, silver, jewels, and liquors, are weighed by Troy velght. Table. •24 grains ( 5'/'.) . . . make 1 pennyweight, . . marked ^wt. 20 pennyweights 1 ounce, oz, 12 ounces 1 pound, ..•»••*«• lb. apothecaries' weight. Table Reversed. pwt. gf- oz. 1 = 24. lb. 1 z= 20 = 480. 1 = 12 = 240 = 5T60. Notes.— 1. The standard Troy pound is tlio weight of 22.794377 cubic inches of distilled water. Hence, it is less than the pound avoirdupois. 2. 7000 troy grains = 1 pound avoirdupois. 175 troy pounds = 144 pounds " 175 troy ounces = 192 ounces " 437i troy grains = 1 ounce " 3. The Troy pound being the one deposited in the Mint at Phila- delphia, is generally regarded as the standard of weight. 4. The units of the scale are 24, 20, and 12. III. APOTHECARIES^ WEIGHT. 480. This weight is used by apothecaries and physicians iu mixing their medicines. But medicines are generally sold, in the quantity, by avoirdupois weight. Table. 20 grains (gr.) .... make 1 scruple, .... marked 3. 3 scruples 1 dram, 3. 8 drams 1 ounce, f . 12 ounces 1 pound, lb Table Reversed. 3 gr. 3 1 = 20 1 3 = 60 8 = 24 r= 480 ib 1 = 1 = 12 . = 96 = 288 = 5760 Notes. — 1. The pound, ounce, and grain, are the same as the pound, \mce, and grain, in Troy weight. 2. The units of the scale, in passing from grains to scruples, are 20 ; in passing from scruples to drams, 3 ; from drams to ounces, 8 ; and from ounces to pounds, 12. 416 APPENDIX. IV. FRENCH SYSTEM. 481. The basis of tMs system of weights is the weight in vacuo of a cubic decimetre of distilled water. This weight is called a kilogramme, and is the unit of the French system. P is equal to 2.20413t pounds avoirdupois. The other denom nations are as follows : 10 milligrammes = 1 centigramme ; 10 centigronmes = 1 decigramme ; 10 decigrammes = 1 gramme ; 10 grammes = 1 decagramme ; 10 decagrammes = 1 hectogramme ; 10 hecto- grammes = 1 kilogramme ; 10 kilogrammes = 1 quintal ; 10 quintals = 1 ton of sea-water. COMPARISON OF WEIGHTS. English^ 1 pound = 1.000936 pounds avou'dupoia. French, 1 kilogramme = 2.204131 It tt Spanish^ 1 pound = 1.0152 U it Swedish, 1 pound = 0.9316 tt tt Austrian, 1 pound = 1.2351 tt tt Prussian, 1 pound = 1.0333 tt It VII. UNITS OF TIME. 482. Time is a part of duration. The time in which the earth revolves on its axis is called a day. The time in which it goes round the sun is called a solar year. Time is divided into parts accordmg to the following Table. 60 seconds, see. make 1 minute, marked m. 60 minutes 1 hour, , , 7ir. 24 hours 1 day, ....... da. 7 days 1 week, . , wlc.^' 52 weeks, nearly 1 year, yr.^ 12 calendar months = 365 da. 1 Julian or common year, . . yr. 366 days make 1 leap-year 100 years 1 century, cem. DATES. 417 Table Reversed. wk. 1 = 521 = da. 1 = 1 = 365 = hr. 1 1 = GO 24 = 1440 1G8 = 10080 StGO = 525600 BCC 60. 3600. 86400. 604800. 31536000. The year is divided into 12 calendar months No. No. days. 1st. January, . . . . 31 2d. February, .... 28 Sd. March, 31 4th. April, 30 5th. May, 31 6th. June, 30 No. 7th. July, . . 8th. August, 9 th. September, 10th. October, . 11th. November, 12th. December, . No. days. 31 31 30 31 30 31 The number of days in each month may be remembered by the following : Thirty days hath September, April, June, and November ; All the rest have thirty-one, Excepting February, twenty-eight alone. Notes. — 1. Days are numbered in each, month from the first day of the month. 2. Months are numbered from January to December. 3. The centuries are numbered from the beginning of the Cliristian Era. The year 30, for example, at its commencement, was called the 30th year of the first century, though neither the century nor the year had elapsed. Thus, Juno 2d, ISoG, was the 6th month of the 5Gth year of the 19th century. 4. The civil day begins and ends at 12 o'clock at night. Li tlie civil day, the hours are reckoned from that time. Dates. 1. The length of the solar year is 365 da. 5 hr. 48 m. 48 sec, very nearly. It is desirable to have the periods and dates of the civil year corresiwnd to those of the solar year ; else, the summer months of tho 18* 4:18 APrENDlX. one would in time become tlie winter months of the other, thereby producing great confusion in dates and history. 2. The common civil year is reckoned at 365 da., and the solar year at 365 da. 6hr. The 6 hours accumulate for 4 years before they are counted, when they amount to 1 day, and are added to February; and the year is called a bissextile, or Imp-year. 3. The odd 6 hours have been so added, that the leap-years occur in those numbers, which are divisible by 4. Thus, 1856, 1860, 1864 &c., are leap-years ; and when any number is not divisible by 4, the remainder denotes how many years have passed since a leap-year. 4. This method of disposing of the fractional part of the year would be without error, if the solar year were exactly 365 da. 6 hr. in length; but it is not; it is only 365 da. 5hr. 48m. 48 sec. long: hence, the leap-year is reckoned at too much, and to correct this error, every centennial year is reckoned as a common year. But this makes an error again, on the other side, and every fourth centennial year the day is retained. Thus, 1800 was not, and 1900 will not be, reckoned a leap-year: the error will then be on the other side, and" 2000 will be a leap-year. This disposition of the fractional part of the year causes the civil and solar years to correspond very nearly, and indicates the following rule for finding the leap-years: Rule. — Every year ivhich is divisible by 4 is a leap year, unless it is a centennial year, and then it is not a leap-year unless the number of the century is also divisible by 4. 5. The registration of the days, by reckoning the civil year at 365 da., was established by the Eoman emperor, Julius Caesar, and hence this period is sometimes called the Julian year. The error, arising from the fractional part, continued to increase until 1582, when it amounted to 10 days; that is, as the year had been reckoned too long, the number of days had been too few, and the count, in the civil year, was behind the count in the solar year. In this year (1582), Pope Gregory decreed the 4th day of October to be called the 14th, and this brought the civil and the solar years together. The new calendar is sometimes called the Gregorian Caleridar. 6 The method of dating by the old count, is called Old Style; and by the new, New Style. The difference is now 12 days In Russia, they still use the old style; hence, their dates are 12 days behind ours. Their 4th of January is our 16th. UNITS OF CIRCULAR MEASURE. 410 VUI. UNITS OF CmCULAK MEASURE. 483. Angular, or circular measure, is used iu latitude and longitude, iu measuring the motions of the heavenly bodies, and also in measuring angles. The circumference of every circle is supposed to be divide into 360 equal parts, called degrees* Each degree is divided into GO minutes, and each minute into 60 seconds. Table. GO seconds ("). . . , make 1 minute, marked '. GO minutes 1 degree, °. 30 degrees 1 sign, 8. 13 signs, or 300°, 1 circle, o. Table Reversed. / . // o 1 ZiZ 60 1 = 60 = 3600 a 1 = 30 = 1800 = 108000 1 = = 12 = 3G0 = 21600 = 1296000 Miscellaneons Table. 12 units, or things, . . make 1 dozen. 12 dozen 1 gross. 12 gross or 144 dozen, .... 1 great gross. 20 things 1 score. 100 pounds 1 quintal of fish. 19G pounds 1 barrel of flour. 200 pounds 1 barrel of pork. 18 inches 1 cubit. 22 inches, nearly, 1 sacred cubit. 14 pounds of iron or lead ... 1 stone, 21i stones 1 pig- b i>igs 1 fother. 420 APPENDIX. BOOKS AND PAPER. The terms, folio^ quarto, octavo, duodecimo, &c., indicates the number of leaves into which a sheet of paper is folded. A sheet folded in 2 leaves A sheet folded in 4 leaves A sheet folded in 8 leaves, A sheet folded in 13 leaves A sheet folded in 16 leaves A sheet folded in 18 leaves, A sheet folded in 24 leaves A sheet folded in 32 leaves 24 sheets of paper, 20 quires, . . . 6 biindleo. is called, a folio. a quarto, or 4to. an octavo, or 8vo. a 12mo. a 16mo. an ISmo. a 24mo. a 82mo. make 1 quire. ... 1 ream. 1 bundle. 1 UMe. METRIC SYSTEM OF WEIGHTS AND MEASURES. The primary base, in this system, for all denominations of weights and measures, is the one-ten-millionth part of the dis- tance from the equator to the pole, measured on the earth's surface. It is called a Meter, and is equal to 39.3t inches, very nearly. The change from the base, in all the denominations, is accord- ing to the decimal scale of tens : that is, the units increase ten times, at each step, in the ascending scale, and decrease ten times, at each step, in the descending scale. MEASUKES OF LENGTH. Base, 1 meter = 39.37 inches, nearly. Table. Ascending Scale • 1 Descending Scale. Myriameter. Kilometer. -I o o w c 1 Centimeter. Millimeter. J 1 1 1 1 1 1 1 1 The names, in the ascending scale, are formed by prefixing to the base, Meter, the words, Deca (ten), Hecto (one hundred), Kilo (one thousand), Myria (ten thousand), from the Greek nu- merals ; and in the descending scale, by prefixing Deci (tenth), Centi (hundredth), Milli (thousandth), from the Latin numerals. 4:23 METIUC SYSTEM. Hence, the name of a unit indicates whether it is greater or less than the standard ; and, also, how many times. The table is thus read : 10 millimeters make 1 centimeter. 10 centimeters make 1 decimeter. 10 decimeters make 1 meter. 10 METERS make 1 decameter. 10 decameters make 1 hectometer. 10 hectometers make 1 kilometer. 10 kilometers make 1 myriameter. Table of Equivalents. i .-a|i§ a I a &flH(Sl ft I I 1= 10 1= 10= 100 1= 10= 100= 1,000 1= 10= 100= 1,000= 10,000 1= 10= 100= 1,000= 10,000= 100,000 1= 10= 100= 1,000= 10,000= 100,000= 1,000,000 1 = 10=100 = 1,000 = 10,000 = 100,000 = 1,000,000=10,000,000 Table of Equivalents in English Measure. 1 Millimeter = 0.0394 inches, nearly. 1 Centimeter = 0.393t " 1 Decimeter = 3.9310 " 1 Meter = 39.31 in. = 3.280833 ft. 1 Decameter = 32 ft. 9.1 in. 1 Hectometer = 19 rd. 14 ft. 1 in. 1 Kilometer = 4 fur. 38 rd. 13 ft. 10 in. 1 Myriameter = 6 mi. 1 fur. 28 rd. 6 ft. 4 in. Besides a clear ajDprehension of the length of the base, 1 meter, it is well to consider the length of the largest unit, tho MEASURES OF LENGTH. 423 myriameter, equal to nearly 6 and one-fourth miles ; and also tho length of the smallest unit, the millimeter, about four-hundredths of an inch. Compare, also, each of the smaller measures, the decimeter and centimeter, with the inch. When, in the metric S3'stem, the value of any single unit is fixed in the mind, the values of all the others may be readily apprehended, since they always arise from multiplying or dividing by 10. Note. — In all the tables, tlie unit is in small cax)itals, and should bo constantly referred to. Methods of Reading. The number 25365.891 meters, is read, in English, Twenty-five thousand three hundred and sixty-five meters, and 897 thousandths of a meter. But in the language of the metric system, it may be read, Two myriamcters, 5 kilometers, 3 hectometers, 6 decameters, 5 meters, 8 decimeters, 9 centimeters, and 7 millimeters. It may also be read, beginning with the lowest denomination, 7 milli- meters, 9 centimeters, &c., &c. In reading, remember that the unit of any place is ten thnes as great as the unit of the place next at the right, and one- tenth of the unit of the place next at the left. Hence, the change from one unit to another, and the methods of reduction and reading, are identical with those in the system of decimal currency. 1. Write, numerate, and read, five hundred and ninety-six hectometers. 2. Write, numerate, and read, eighty-nine thousand and forty- one centimeters. Questions. — VVHiat is tlio primary base of the metric system? To ■what portion of the earth's surface is it equal? What is its length? What is the ascending scale from the meter? What is the descending scale ? What is the length of a myriameter ? According to what law do the different units increase and decrease ? 424: METEIC SYSTEM. MEASURES OF SURFACES, OR SQUARE MEASURE. Base, 1 Are = the square whose side is 10 meters. = 119.6 square yards, nearly. = 4 perches or square rods, nearly. The unit of surface is a square whose side is 10 meters. It is called an Are, and is equal to 100 square meters. Table. 1 1 !2; < 1 1 6 1 The table is thus read : 100 centares make 1 ARE. 100 ares make 1 hectare. Table of Equivalents. Hectare. Akb. Centare. , 1 = 100 1 = 100 = 10,000 Equivalents in acres, roods, and perches. 1 Centare = 1.195985 sq. yards, nearly. 1 Are = 3.9536t perches. 1 Hectare = 2A. IR. 35.367P. MEASURES OF VOLUMES. Base, 1 liter = the cube of- the decimeter. = 61.023378 cubic inches. = a little more than a wine quart. Questions. — ^What is the primary base of the measure for surfaces ? To what is it equal, in square yards ? What are the denominations, be- ginning with the least ? To what is the centare equal ? To what is the hectare equal ? MEASURES OF VOLUMES. 4:2o The unit for the measure of vohime is the cube whose edge is one-tenth of the meter — that is, a cube whose edge is 3.93t inciies. This cube is called a Liter, and is one-thousandth part of the cube constructed on the meter, as an edge. The Table. Ascending Scale. Descending Scale. ' ' ' 2 3 OQ B 55 55 o '3 Decaliter. Liter. U 1 u S 1 1 1 1 1 1 1 1 1 :>\q is thus read : 10 milliliters make centiliter. 10 centiliters make deciliter. 10 deciliters make liter 10 liters make decaliter. 10 decaliters make hectoliter. 10 hectoliters make kilolitcr, or stere, Table of Equivalents I 1 a 1 1 = 10 = = 10 = 100 = I 1 = 10 = 100 = 1 = 1 = 10 = 10 = 100 = 100 = 1,000 = 10,000 = 1,000 = 10,000 = lOOjOOO = 1,000 = 10 100 1,000 10,000 100,000 1,000,000 NoTK— The kiloliter, or stere, is the cube constructed on the meter, uf; an edge. Honce, the liter is one-thousandth part of the kiloliter. 426 METRIC SYSTEM. Equivalents in Cubic Measure. 1 milliliter = .061023 cubic inches. 1 centiliter = .610234 cubic inches. 1 deciliter = 6.102338 cubic inches. 1 LITER = 61.023378 cubic inches. 1 decaliter = 610.233179 cubic inches. 1 hectoliter = 6102.337795 cu. in. = 3.5314454 cu. ft. 1 kiloliter, or stere = 61023.377953 cu. in. = 35.314454 cu. ft. Note. — Law of change in the units, and methods of reading, are the same as in linear measure. DRY MEASURE. EQUIVALENTS IN THE WINCHESTER BUSHEL. Since 1 bushel = 2150.4 cu. in. ; 1 pk. = 537.6 cu. in. ; 1 qt.= 67.2 cu. in ; 1 pt. = 33.6 cu. in. ; therefore, = .001816 pints. = .018161 pints. = .181611 pints. = 1.816112 pints. = 1 pk. 1.08056 qt. = 2bu. 3pk. 2qt. 1.6112 pt. = 28 bu. Ipk. 4qt. 0.112pt. aid, is a little less than 1 quart, and the stere, nearly 30 Winchester bushels. LIQUID MEASURE. EQUIVALENTS IN THE WINE GALLOIT. Since 1 wine gallon contains 231 cubic inches, 1 quart will contain 57.75 cubic inches; 1 pint, 28.875 cubic inches; and 1 gill, 7.21875 cubic inches ; we have, Questions. — What is tbe unit for the measure of volumes ? To wbat is it equal in cubic inches ? What part is it of the cube on the meter? Name all the denominations of volume. What is the unit of Dry Meas- ure ? To what is it equal ? To what is the stere or kiloliter equal? milliliter centiliter deciliter LITER decaliter hectoliter kiloliter, or stere Note.— -The liter, or stand milliliter centiliter deciliter LITER decaliter hectoliter WEIGHTS. = 0.008453 gills. = .084534 gills. = .845345 gills. = Iqt. .11336 pt. = 2 gal. 2qt. 1 pt. .1330 pt. = 26 gal. Iqt. 1 pt. 1.344 gills. 427 1 kiloliter, or stere = 1 tun, 12 gal. qt. 1 pt. 1.44 gilis. WEIGHTS. Base, 1 gram = weight of a cubic centimeter of rain-water. = 15.432 grains, Troy, nearly. = .0352746 ounces, Avoirdupois, nearly. The unit of weight is also equal to the one-millionth part of the weight of a cubic meter of pure rain-water, weighed in va- cuum. It is called a Gram, and is equal to 15.432 grains, Troy, which is equal to .0352746 ounces, Avoirdupois, very nearly. Table. Ascending Scale. Descending Scale. 'o &i [Ulier, tonn uintal. yriagram. ilogram. ectogram. ecagram. ! 1 1 1 ^ C ^ M W P C!5 ft 6 S 1111 1 1 ; 1 1 1 The table is thus read : 10 milligrams make I centigram. 10 centigrams make 1 decigram. 10 decigrams make 1 gram. 10 grams make 1 decagram. 10 decagrams make 1 hectogram. 10 hectograms make 1 kilogram. 10 kilograms make 1 myriagram. 10 myriagrams make 1 quintal. 10 quintals make 1 millier, or touHcau. 428 METRIC SYSTEM. i Table of Equivalents. 1 1 Myriagram. Kilogram. g 2 Decagram. Gram. Decigram. Centigram. Milligram. , 1= 10 . 1= 10= 100 . 1= 10= 100= 1,000 . 1= 10= 100= 1,000= 10,000 1= 10= 100= 1,000= 10,000= 100,000 1 10= 100= 1,000= 10,000= 100,000= 1,000,000 1= 10= 100= 1,000= 10,000= 100,000= 1,000,000= 10,000,000 . 1= 10= 100= 1,000= 10,000= 100,000= 1,000,000= 10,000,000= 100,000,000 1=10= 100=1,000=10,000= 100,000=1,000,000=10,000,000=100,000,000=1,000,000,000 Equivalents in Avoirdupois and Troy Weights. 1 Milligram = 0.0154 grains, Troy. 1 Centigram = 0.1543 grains, It 1 Decigram = 1.5432 grains, It 1 Gram = 15.4327 grains, It 1 Decagram = 0.352t ounces, Avoirdupois, 1 Hectogram = 3.5274 ounces, (( 1 Kilogram = 2.2046 pounds, (( 1 Myriagram = 22.046 pounds, (( 1 Quintal = 220.46 pounds. ti 1 Millier, or ton. =2204.6 pounds, *' Note. — Law of change in the units, and methods of reading, the same as in linear Measure. NATURE OF THE METRIC SYSTEM. The Metric system is based on the meter. From the meter, three other units are derived ; and the four constitute the primary units of the system. They are : Questions. — What is the unit of weight ? To what is it equal in Troy weight ? To what is it equal in Avoirdupois ? Name aU the units of the weight, from the lowest to the highest. To what is the millier, or ton, equal ? GENERAL PRINCIPLES. 429 Meter = 39.3Y inches, nearly : unit of length. Are = a square on 10 meters : unit of surface. Liter = a cube whose edge is a dociiiieter : unit of volume. Gram = the weight of a cube of rain-water, each edge of which is a centimeter : unit of weight. From these four units all others are derived, according to the decimal scale. Every system of Weights and Measures must have an inva- viable unit for its base ; and every other unit of the entire system should be derived from it, according to a fixed law. Tlie French Government, in order to obtain an invariable unit, measured a degree of the arc of a meridian on the earth^s surface ; and from this computed the length of the meridional arc from the equator to the pole. This length they divided into ten million equal parts, and then took one of these parts for the unit of length, and called it a Meter. The length of this meter is equal to 1 yard, 3 inches, and 37 hundredths of an inch, very nearly. Thus they obtained the length of the unit which is the base of the Metric System of Weights and Measures. The next step was to fix the law^ by which the other units should be obtained from the base. The scale of tens was adopted. rRONUNCIATION. Me'teb. Mil'li-me-ter. Cen'ti-me-ter. Dcc'i-me-ter. Dee'a-me-ter. Hec'to-me-ter. Kiro-me-ter. Myr'i-a-me-ter. Abe. Cen'tare. Hee'tare. Li'tee. MQ'li-li-ter. Cen'ti-li-ter. Dec'i-li-ter. Dee'a-li-ter. Hee'to-li-ter. Kiro-li-ter. Myrl-a-li-ter. Geam. Milli-gram. Cen'ti-gram. Dec'i-gram. Dee'a-gram. Hee'to-gram. Kil'o-gram. Myr'i-a-gram. 430 METRIC SYSTEM. TO CHANaE FKOM ONE SYSTEM TO THE OTHER. To change, in Linear Measure, from the Metric to the Common system. Rule. Multiply the meters and decimals of a meter by 3.280833 (the value of a meter), and the product will he the result in feet. To change from the Common to the Metric system. Rule. Beduce the linear measure to feet and decimals of a foot^ and then divide by 3.280833 ; the quotient will be the result in meters and decimals of a meter. Examples. 1. In 5961.814 meters, how many feet and inches? 2. In 814163 meters, and 31 hectometers, how many feet and inches ? 3. Express 320 rods, 5 yards and 6 inches in the Metric Measures. 4. Express I mile, 3 furlongs, 39 rods and 5 yards in the Metric Measures. To change, in Square Measure, from the Metric to the Common system. Rule. Beduce the number to ares and decimals of the are; then multiply by 3.95361, and the product will be the residt in perches. To change, from the Common system, to the Metric system. Rule. Find the value of the number in perches and decimals of a perch : then divide by 3.95361, and the quotient tvill he the result in ares and decimals of the are. REDUCTION. 431 Examples. 1. In 6127 ares, 4 liectares and 3 centares, bow many acres, roods and perclies? 2. In 32t ares, 15 hectares and 89 centares, how many > square feet? ^ 3. In 4 acres, 3 perches and 200 square feet, how many hectares, ares and centares ? 4. In 1375 square yards and 250 square feet, how many hectares, ares and centares? To change, in measures of volume, from the Metric to the Com- mon system. Rule. Reduce the number to liters and decimals of the liter : then multiply by Gl. 023378, and the product will be the result in cubic inches. To change, in measures of volume, from the Common to the Metric system. Rule. Reduce the number to cubic inches : then divide by 61.023378, and the quotient will be the result in litebs and decimals of the liter. Examples. 1. In 6 kiloliters, 9 hectoliters, 6 decaliters, 8 liters and 4 centiliters, how many cubic feet and inches ? 2. In 8 hectoliters, 9 decaliters, 27 liters and 15 milliers, how many cubic yards, feet and inches ? 3. Change 27 cubic yards, 16 cubic feet and 16 cubic inches, to the Metric measures. 4. Change 40 cubic yards, 25 cubic feet and 1167 inches, to the Metric measures. 432 METEIC SYSTEM. To change, in weights, from the Metric to the Common system. Rule. Beduce the number to grams and decimals of a gram : then multiply by 15.423, and the product will be the result in grains Troy; or, multiply by .0352146, and the product will be ounces in Avoirdupois. To change, in weights, from the Common to the Metric system. Rule. Beduce the number to Troy grains, or to Avoirdupois ounces: then divide by 15.423, or by .0352746, and the quotient mil be GRAMS and decimals of the gram. Examples. 1. Change 4 quintals, 6 kilograms, 4 decagrams, T grams and 6 centigrams, to Avoirdupois and Troy weights. 2. Change 2 milliers, 6 myriagrams, 9 grams, 4 decagrams and 9 milligrams, to Troy and Avoirdupois. 3. Change 1 T. 3 cwt. 3 qr. 20 lb. 6 oz., to the Metric weights. 4. Change 161b. 11 oz. 4 pwt. 19gr,, Troy, to the Metric weights. Ques. — In linear measure, how do you change from the Metric to the Common system ? How do you change from the Common to the Metric system ? In square measure, how do you change from the Metric to the Common system ? How do you change from the Common to the Metric system ? In measures of volume, how do you change from the Metric to the Common system ? How do you change from the Common to the Metric system ? In weights, how do you change from the Metric to the Common sys- tem ? How do you change from the Common to the Metric system ? ANSWERS. PACK. EX. AN3. EX. ANS. EX. ANS. 16. 16. 1 I J«. II 2 I XIY. II 3 I XYI. II 4 I XYII. || 5 | XIX 6 I XXII. II t I XXYIII. II 8 I XXIX. II 9 | XXXIII. 16. 10 I XXXYIL II 11 I XXXYIIL || 12 | XLIII. || 13 | XLYII. II 14 I XLIX. II 15 I LYL|| 16 | LYIII. || 17 | LIX. li). I 16. 16 16. 16 16 18 I LXY. II 19 I LXIX. II 20 | LXYII. || 21 | LXXY 52"7 LXXYI. II 23 I LXXXI. || 24 | LXXXYII. || 25 | LXXXIX. II 26 I XCIY. || 27 | XCY. || 28 | XCYIL 29 I XCIX. I 30 I CXY. || 31 | DCCL. || 32 | MLX. 33 I MMXL. II 34 | DLX. || 35 | DCCCCLX. || 36 | DCXC. a7 I ML. li 38 I MMMMIY. || 39 | YMIX. || 40 | IXIX. 16. 41 I DCCCYI. II 42 I DCYIII. j 43 | YMMMYI. || 44 | 16. MMI. II 19. II 1 I 7 II 2 I 80 II 3 I 9000 || 4 | 93 W. 5 I 961 II 6 I 7408 || 7 | 897021 || 8 | 86029430 || 9 | 4328- 20. I 021063 II 10 I 967040932 i| 11 | 30430208123 || 12 | 360- 20. 030702010 II 13 I 5800006000812 || 14 | 75605070905008 20, I 15 I 904000800200720 || 16 | 6000900704098020 || 17 | 20. 80510006040900040900 || 18 | 6050900001 || 21. || 19 21 987054321012345678 || 22. || 1 | 621 || 2 | 5702 || 3 | 8001 22. 4 I 10406 II 5 I 65029 || 6 | 40000241 || 7 | 59000310 22. I 8 I 12111.11 9 I 300001006 ||.10 | 69003000200 23. 32 I 47000069000465207 || 33 | 800000000000429006009 434 ANSWERS. •23. II 34 I 95000000000000089089306 || 35 | 6000000451065- 23. II 047104 11 36 1 999065841411 H 30. 1| 1 | 2 ; 1 || 2 | t ; 3 30. II 3 I 1 ; t II 32. 1| 6 | 42600 ; 426000 || 1 \ 36860 |1 8 | $8.75 32. II 9 I 433005 \\ 10 | 8996 i 11 1 £1 12s. 8d. 1 far. H 12 | 154451b. 32. II 13 I IT. Ucwt. Iqr. 201b || 33. |1 14 | 26215 grs 33. li 15 I 1221b. 2oz. 18pwt. 9gr. |! 16 | 29362gr. || 17 | 33. II 301b. 4 § 3 3 2^ 7gr. |1 18 [ 249 in.|l 19 | 1600rd. 8800yd.; 33. II 26400 ft. 316800in. || 20 j 75yd. 2ft. Gin. || 21 | 33. II 6 sq. yd. 2 sq. ft. || 22 | 2 A. OR. 35 P. || 23 | 45 A. 6sq. Ch. 33. II 24 1 568 P. || 25 | 967680 cu. in. |1 26 1 3968 ou. ft. 33. II 27 I 440 cords || 28 1 2512 na. U 29 | 144 yd. 1| 30 | 33. II 78 E. E. 1 qr. || 31 1 1008 qt. || 32 1 15 hlid. || 33 1 3024 pt. 33. II 34 1 129 bar. ( 35 | 1984 pt. | 36 | 32 bu. 3pk. 7 qt. 33. II 37 I 63113856 sec. || 38 | 8mo. '2wk. || 37. || 1 | 182630 37. II 2 I 87539 || 3 | 110526 || 4 | 79165 1 5 | 73285 || 6 | 4148- 37. II 907 II 7 I 395873 || 8 j 24177 || 9 | 66395 |j 10 | 22099 37. II 11 I 73566 || 12 | 833157 || 38. -|| 13 | 32921 || 14 | 185876, 38. II 15 I 93684 || 16 | 34289 |1 17 | 243972 1 18 | $991,546 38. II 19 I $85,465 || 20 I $770,560 1| 21 1 525.892 ( 22 | $9638.495 38. II 23 I ie223 2s. 5d. Ifar. || 24 | 12961b. 10 oz. 2pwt. 38. II 25 1 453 ft 9 5 3 3 || 26 | 2 cwt. 3 qr. 8 lb. 8 oz. 5 dr. as. II 27 1 43 T. 2 cwt. Oqr. 71b. || 28 | 312 yd. Oqr. 2iia. 38. II 29 I 251 E. E. Iqr. 3 na. || 30 | 143 L. 2 mi. 6 fur. 39. II 31 I 4 fur. Ird. 4 yd. Oft. 7 in. || 32 .| 322 A. IK 4 P. 39. II 33 I 2224 Tun Ohhd. 5 gal. || 34 | 339 gal. 3 qt. ANSWERS. 435 39. |35 1 230 chal. 25 bu. 3 pk. 4 qt. || 36 | 820 yr. 4 mo. 5 da. 39. |37 1 904 da. 18 hr. Imi. || 38 | 2T. 14cwt. Iqr. 201b. 15oz. 39. 1 39 1 23592550 || 40 | $137915940 J 41 | 88056 39. |42 1 121 mi. 4 fur. 8rd. 5 ft. || 40. 1 43 | $22,009 10. 1 |44 1 $27,740 1 45 1 2 Tun 2hhd. 29 gal. 2qt. Opt 40. |46 1 $20308675 [ 47 | $30569853 J 48 | $29026 40. |49 1 $8209.75 II 50 | $150106 || 51 | 29714 |1 41. || 52 1 $50- 41. 1 110025 1 53 1 59808512 || 54 | 2T. 4cwt. 2 qr. lib. 41. |55 1 205 acres. || 56 | $75002.295 | 57 | $7425 41. 1 58 1 41b. 5oz. 6pwt. II 59 1 1053420 1 42. || 60 | 1842yrs. 42. |61| 32341 II 62 1 $27131.23 || 63 | $28,105 || 64 | 39yd. Iqr. 42. 1 G5 1 $180,825 II 66 1 $35068.807 | 67 | £59 2s. 3d. 2 far. 42. 1 68 1 66585383 || 43. || 69 | $1019.10 || 70 | $33800 43. |71 380 bu. 1 pk. II 72 1 $458,342 || 73 | £51 14s. 2d. 3 far. 43. 1 174 1 $6235 II 75 1 66° 50' || 76 | 10 cents. | 77 | 5860 47. 1 |1 1 363296 1 2 | 56579 || 3 | 733071 1 4 | 1711927 47. 1 5 1 41923288 || 6 | 7838180 || 7 | 106026 || 8 | 4391 47. 1 9 1 62786 II 10 1 198621115 || 11 | 3591757651 48. 1 12 4199675 II 13 | 8878778 || 14 | 99999977 || 15 | 48. 1 1 88443.641 5 16 | $806,384 | It | $4853673.758 48. 1 18 1 £U 18s. 3d. Ifar. || 19 | 3T. 8cwt. 2qr. 71b. 48. 1 1 20 1 117yd. 2qr. Ina. || 21 | 59 L. Imi. 3 fur. 28 rd. 48. 1 22 1 8 Tun Ihhd. 53gal. 3qt. || 23 | 89 A. 2 R. 37 P. 48.1 24 1 975 bu. Ipk. 6qt. || 25 | 124 cords 58 ft. 522 in. 48. 1 126 1 25 E. E. Iqr. 3na. [ 27 [ 79ft> 10 1 6 3 43G ANSWERS. 28 1 123 43 23 II 29 | 124 E. E. 3qr. 3 na 30 I 96 E. F Iqr. 1 ua. \\ 31 j 12 T. lUwt. 3qr 32 I 2cwt. 2qr. 221b. || 33 j 69 qr. 21b. 14 oz. 34 I 1341b. 14 oz. 13 dr. || 49. || 35 j 10 A. 2R. 18 P. 36 I 3tA. 2R. 34P. || 3T | 14t da. 21 hr. 66 mm 38 I 52 hr. 50mm. 54sec. || 39 | $8759.625 I 40 | 183666662 41 I 6yr. 9mb. 3wk. Ida. || 42 | 88ft) 0| 63 43 I $8.20 II 44 I $39,868 || 45 | $10,626 || 46 | £121 17s. Od. Ifar. | 47 | 6yr. Omo. Owk. 6da. 9hr. 2mm. 48 I 6353870 || 49 | 5747 || 50 | $6020 || 51 | 25712808.91 52 I 36190 II 53 | 683021 || 54 | 107445034 || 55 | 6274 56 I 4T. 3cwt. 2qr. 231b. || 57 | £19 19s. 2d. 3 far. 58 I 2299 mi. 2 fur. 4 rd. || 59 | $199,625 || 60 | $175,875 61 I $3.25 II 61. II 62 | 19987563 || 63 | 2899248 64 I $73675 || 65 | 22815 || 66 | $198,625 || 67 | 80 yr. 8 mo. Oda. 3hr. 30 min. || 68 | 655.125 69 I 249yr. Imo. llda. || 70 | 17877 || 71 | $7310756 72 I 4cwt. Iqr. 181b. || 73 | 7398 || 74 | 2360 || 75 | $526 76 I 6274 II 77 | $356.35 gain. || 78 | 3 A. 2K 39 P. 79 I 41 cords 5 cord ft. || 80 | $3280.105 || 81 | $44161.987 82 I 2yr. 8 mo. 19 da. || 53. || 83 | $14352.50 || 84 | 30 gal. 2 qt. 1 pt. || 85 | 50062 || 86 j 15550 || 87 1 12° 23' 53' 88 I $161,175 loss. II 89 | 2271707 || 90 | 32 yd. Oqr. 2ua 91 I £950 2s. 8d. || 60. || 1 | 6776368 || 2 | 68653214 3 I 3422454 || 4 | 1952883 || 5 | 4354224 || 6 j 1028540646 ANSWERS. 437 7 I 246G8698404 || 8 | 3329480 || 9 | 4036084764 10 i 129844534245 || 61. || U | 810444 || 12 | 23()13 13 I 72127422 || 14 | 5403312 || 15 | 12440'.)7 16 I 1990170000 II 17 | 3165172200 1| 18 | 582400000000 19 I $104448.48 | 20 | $2501.136 || 21 | $23121.312 22 I $71997.312 |j 23 | $7019.168 | 24 | $30780.960 25 I $21597.440 || 26 | $38824.056 || 27 " | $278879.3(i4 28 I $379255.968 || 29 | $9282001.666 || 30 j £Sl 6s. 8d. 31 I 24 T. 7c\vt. 3qr. || 32 | 118 yd. 1 ft. 3 in. 33 I 114° 26' 15" II 34 | 561ihd. 7 gal. 2qt. Opt. 35 I 698 E.F. || 62. || 1 | 865T. llcwt. 3qr. 201b. 2 I 320 JT. 2 mo. Owk. Ida. 15 hr. 12 rain. || 3 | 4896 4 I 670460 ; 6704600 || 5 | 5704900 ; 57049000 6 I 4980496000 ; 49804960000 || 7 | 9072040000 ; 907204000000 || 8 | 74040900 ; 740409000 || 9 | 67493600 ; 67493600000 || 10 | 129359360000 || 11 | 13729103000000 12 I 664763206000000 || 13 | 8799238229600000 14 I 2526426017908695000000 || 15 | 1093689368445084- 378777040 || 16 [ 16714410677359581583737 1 17 | $61975 18 I 3240 I 19 I 2097 || 20 | 133 yd. 3qr. 2na. 21 I £^ 19s. 4d. 2 far. || 22 | $1031.68 | 63. || 23 | $15 24 I $506.88 II 25 | $6336 || 26 | $5545 || 27 | $16763832 28 I 496 mi. 1 fur. 24 rd. || 29 | $657 || 30 | $24,375 31 I 868 miles | 32 | 7lb 2 3 7.3 0^ 12gr. j 33 | 411 l)u. Ipk. Oqt. II 34 I 427816 || 64. || 35 | $84.26 4ZS ANSWERS. 64. II 36 I $168t5.60 (| 3T | 2T. 18 cwt. 1 qr. 21 lb. J 38 | $971.04 64. II 39 I 461 left ; $1315 price. i| 40 | $1417 || 41 j $65962788.15 64. II 42 I 750 II 43 j 13500 || 44 [ $243.00 || 45 j 11914 65. II 46 I $4770.755 || 47 | $61 || 48 | 1672 | 49 | 286 yr. 9 mo. 65. II 50 I 84 rd. 14 ft. || 51 | 50 || 52 | 24 cords. || 53 | $92 gain. 65. II 54 I 216 II 55 | $149.25 || 56 j 37816 || 57 | $34.88 66. H 58 I 669 hhd. 40 gal. 2qt. \\ 59 | 13650000 || 60 | $202.50 66. II 61 I $21,475 J 62 | $927.35 || 67. || 63 | $18844.01 67. II 64 I $132,935 || 65 | £115 18s. 6d. || 72. |1 1 1 6579 72. ||2 136842 || 3 | 269368 ||. 4 | 275155 || 5 | 7948312 72. II 6 I 1147187 || 7 j 72331642 || 8 | £lb 19s. 9d. 72. II 9 I 4A. OR. 33 P. || 10 | 9 yd. 2qr. 1 na. 1| 11 | $79.3445 72. II 12 |, $209,728 || 13 | $66862.18 || 73. || 14 | 15311409^1 73.11^1237132 || 16 | 177242 || 17 | 68 || 18 | 44670 73. U 19 I 27i| 1 20 I $17.4512 || 21 | $3.842j-«^% || 22 | $1,125 73. II 23 I $0,375 || 24 | $0.81 || 25 | $5.01 || 26 | $52.88 || 27 | 9 73. II 28 I 95 J 29 I $8 || 30 | 763521 || 31 | 407294|-?-f J 73. J 32 j 13195133if|f j 33 | 125139204||if 73. II 34 I 269577255882T-Y4V3 II 35 | 14243757 748fffJ:i 73. 1 36 I 15395919iffiJ || 37 | 30001000/y\V3 II ^^ | 73. II 131809655J^|J^ | 39 | 300335575Jf?i-Jf || 40 | 9948157- 73. 1 977/-,VWt II 41 I 59085714tVt II 42 | 1258127JfIM 73.1143 I 119191753j%V4V6- II 44 | 17A. 311. 7P. 73. 11 45 I Ida. 12 hr. 31min. 30 sec. || 46 | 35 mi. Ofur. 29 rd. 73. II 47 I 49 gal. 3,^^ q*- II 48 1 2 bu. pk. 7 qt. || 74. || 49 | $25.25 ANSWERS. 439 74. 1 50 I 2s. 4d. J 51 | 22 mi. 1 fur. 8rd. J 52 | 316A. IR 35P. 74. l 53 I $2*1.397+ J 54 | 98765 || 55 | $11250 || 5(j 1 148018fA| 74. II 57 I $4.75 | 58 j $12.50 \\ 59 | 757l88yV4 74. 1 60 I $1,625 1 61 | 365 days. | 62 | 800008 || 63 | 47 75. II 64 I IT. 13cwt. 3qr. || 65 | 45 cu. ft. 995}| cubic inches. 75. II 66 I 301^1 tons. | 67 | 4424^?3 J 68 | 59' lO'Hf 75. li 69 I 5doz. J 70 | $4.50 || 71 1 ^£273 7s. 6d. ] 72 | 41684xVt 75. J 73 I 9 11 74 I $56 || 75 | 666-J-§| U 76 | 200000 76. } I I 7175 II 2 I 4600 \\ 3 | 168525 || 4 1 76850 || 1 | 2725 76. 1 2 1 387321 J 3 | 4413^40 || 4 | 15423 |1 5 1 2674584 76. II 6 I 280082 |1 77. || 1 | 4800 U 2 | 5950 || 3 | 185000 77. 1 4 I 8380225 || 1 | 55975066f || 2 ] 493574Gfif 77. 1 3 I 355850400 || 4 | 148072400 j 1 | 7408000 77. 1 2 I 2199176000 | 3 | 242601500 || 4 1 17573500 79. II 1 I $142 II 2 I $17 II 3 I $14 || 4 1 835 ] 5 | $864 1| 6 | $172 79. 11 7 I $120 II 8 I $90 || 80. || 1 | $121,615 J 2 1 $67.50 80. J 3 I $737.88 || 4 | $496,875 I 5 | $118.9145 \\ 81. |1 1 1 $3,024 81. II 2 I $12.8915 II 3 1 $5,922 ; $6.4575 ; $9,198 || 4 | $18.22765 81. 1 5 I $736.68468f ] 6 1 $876,434 || 7 1 $2423.09925 81. 1 8 I $339286.5375 J 82. || 1 | 254 || 2 | 26251^2^0^ 82. 1 3 I 291147 ] 4 | 2U4:SSj% \\ 5 | 978 J 6 1 954 J 7 I 140848 82. II 8 1 2025 y 9 | 39252 ] 10 | 475542 \\ 11 | 242172 82. 1 12 I 484344 || 13 | 951084 || 14 | 2250 ] 15 | 48120 82. II 16 1 16215 II 17 1 4S645 || 18 | 144378 y 84. || 1 I 387 84. 1 2 I 1548 1 3 I 532 II 4 I 804 I 5 I 15911 1 6 | 1935 440 ANSWERS. 84. II 1 I 1809 II 8 I 3216 !| 85. || 1 | 1322Hf || 2 | n40j|§ 85. II 3 I 218|ff§ II 4 I 83253^%% || 5 | 2459^1 || 6 | md^\%% 85. II 1 I 950HfH II 86. II 1 I imUi II 2 I 146 II 3 I 9lUim S6. II 4 I 158t32ff|gg II 5 | 2b^%\%% \\ 6 | 224tffM-§ "6. II 1 I 196ff II 2 I 3inft II 3 I 61096ff || 4 | mUi 86. II 5 I 909511 II 6 | 6992A| || 1 \ 6150^^0 || 8 j 40^9,-3^ 89. II 1 I ^631 17s. 6d. n 2 I ^2 9s. 5fd. || 3 f iSl 16s. lO^d. 89. II 4 I ^594.50 || 5 | ^£469 5s. || 6 | iE931 || 1 | ^58t 5s. 89. II 8 I ie82 10s. II 9 | $2.t0 || 10 | $555 || 11 | $547.50 89. II 12 I 13.00 II 13 I $812.25 || 14 | $24,375 || 15 | $63.4375 89. li 16 I $315.40 II 17|$469.03||18|^615s. II 91. || 3 | 10° 34' 0" 91. II 4 I 35° 11' 0" II 5 I 13° 23' 0" || 92. || 1 | Ihr. 2mm. 8sec. p.m. 92. II 2 I 2 hr. 55 min. 24 sec. p. M. || 3 | 8 lir. 12 min. a. m. 92. II 4 I Ihr. 2mm. 20 sec. Fast. || 93. || 1 | 33° 55' W. 93. II 2 I 95° 48' W.; lOhr. 17mm. 48 sec. p.m. || 3 | 23° 45' 22" W. 93. II 4 I 120° W. II 5 I 156° 59' E. || 94. || 1 | $128 || 2 | 2 bu. 1 pk. 94. II 3 I 32 II 4 I 463684 || 5 | 416664§ || 6 | 57979}Jf 94. H 7| 7mo.lwk.4Jd.||8|12yr.||9|6mo.0wk.5d.l41ir.40mm. 94. II 10 I 765 II 11 I $72 || 12 | $5 || 13 | $812.25 || 14 | $147.9375 95. II 15 I £14 14s. II 16 I iE166 2s. 8d. || 17 | 6d. || 18 | $6.95175 95. II 19 I $8.64 II 20 | $93 || 21 | 36 || 22 | 451b. 6oz. Upwt. 95. II 23 I 50 II 24 | $2480 gain ; $19 per acre. || 25 | 6780 cu. ft 95. II 26 I $773,395 || 27 | $4.2408 || 28 1 $16.7025 || 96. || 29 | 768C 96. II 30 I lib. 7oz. 12pwt. 11 gr. || 31 | $10 || 32 j 2bu. Ipk. 7qt. 96. II 33 I $0.75 II 34 | 104 1| 35 | 16 || 36 | 52 gal. 1 qt. || 37 | 96 ANSWERS. 441 9B. II 38 I $598281 || 39 | 31680 || 97- || 40 | 130 || 41 | n93%V-, 97. II 42 I 11 hr. 4min. 32 sec. a.m. || 43 | 127° 30' 97. i 44 I 67° 35' A's long.; 9hr. 19mm. p. m. B's time. : r. II 45 I 10 cords 7 C. ft. 15cii. ft. || 46 | Icwt. 3qr. 9lb. lOoz, I. jl 47" 1 $164,475 ||. 48 | 282}t. 6 mo. 8da. || 49 \u. II 6 gal. 2qt. Opt. 2 gi. || 50 | 6° 13 mi. Ifur. 34 rd. 2 yd. 97. II 51 I 1000000 II 98. || 52 | 13824 || 53 | 36100 98.11 54 I 14 mi. 5 fur. 21 rd. 8 ft. || 55 | 10 || 56 | 3' 98. II 57 I 3yd. 1 qr. 3na. || 58 | 33 || 59 | 13209i?^/, 98. II 60 I 111.88 II 61 I lyr. 205da. 17 hr. 15min. 98. II 62 I $10591021.60 || 99. || 63 | 25 yr. 6 mo. 16 da. 9 hr. 99. II 64 I $2478, Widow's share ; $1239, Child's share. || i^o \ 99. II 13068 II 66 | 107° 47'; 1 hr. llmin. 8 sec. p. m. 99. II 67 I 4hr. 56min. p.m.; 26° east of New York. 99. II 68 I 48 hr. || 69 | 4333fg-S || 100. || 70 | $2 || 71 | 46ilbs. 100. II 72 I 14 days. || 73 | 28 bar. 6 gal. || 74 | 24 bar. 19 gal. 100. II 75 1 $85.33J || 76 | llf^ rolls. || 77 | 7 mi. 6fKr. 20 rd. 100. 1 78 I 87501b. || 79 | $18,025 || 80 | 2500 bbl. || 101. || 81 | 101. II 482bu. Ipk. 2qt.= 1st; 160 bu. 3pk. Oqt. IJpt. = 2d ; 101. II 321 bu. 2pk. Iqt. Of pt. = 3d. || 82 | 40° 50' East ; 101. II 35^o°y II 83 I $2400 = Captain's ; 81000 = LieutenantV: ; 101. II $600 = Midshipman's ; and $200 = Sailor's. || 84 | 87° ao 101. 11 85 I 9hr. 33min. 14scc. a.m.|| 86 | lOhr. 54min. IOscc.a.m 101. II 87 I 19° II 88 I 4800 yd. || 89 | $7410 || 102. i| 90 | 514 102. II 91 I 2011bu. II 92 I 1 yr. 338 da. 22 hr. || 93 | 72 = greater; 442 ANswjr,KS. 102. II 26 = less. || 94 | $5T = less ; ^$133 = greater || 95 | HOda. 102. II 96 1 $1.1875 1 97 1 $8383^ = A' s ; $85201 = = B's; 102. II $7708i = C's. II 98 1 $1 1651.25 :rr 1st ; $11576.25 = 2d; 102. II $11496.25 = 3d ; $11401.25 = 4th . II 104. II 1 1 3 X 3 ; 104. 1 2 X 5 ; 2 X 2 X 3 ; 2x7; 2 X 2 X 2 X 2; 104. i|2x3x3;2x2x2x3;3x3x3;2x2x7 104. II 2 1 2 X 3 X 5 ; 2 X 11; 2 X 2x2 X 2 X 2; 104. 12 X 2 X 3 X 3, 2 X 19 ; 2 X 2 X 2x5; 3 X 3 X 5 ; 104. V X 7 II 105. II 3 1 2 X 5 X 5 ; 2 X 2 X 2 X 1; 2 X 29; 105. II 2 X 2 X 3 X 5 ; 2 X 2 X 2 X 2 X J 2X2; 2 X 3 X 11 ; 105. ||2x2xl7; 2x5x7; 2x2x2x3x3 105. M 1 2 X 2 X 19 ; 2 X 3 X 13; 2 X 2 X 2 X 2 X 5 ; 105. II 2 X 41 ; 2 X 2 X 3 X ^ 2 X 43; 2 X 2 X 2 x 11 ; 105. II 2 X 3 X 3 X 5 II 5|2 X 2 X 5 >: 5 ; 2 X 3 X 17 ; 105. II 2 x2 X 2 X 13; 5x5x 11 ; 5x 2x2 X 2x2 X 2 X 2 X 3 ; 105. ||2X2X2X59; 2x2x2x2x2x5; 2x2x11x19 105. 1 6 1 5X3X7 ; 2 X 53 ; 2X2X3X 3x3; 2x5x H ; 105. J 5 X 23 ; 2 X 2 X 29; 2X2X2X3X5; lOo. II 5 X 5 X 5 ; 5X5X5X3X3; 2x2x2x5x3x3 105. M 1 2, 5, 3 II 2 1 2, 3, 7 1 3 1 5, 7, 3 II 4 1 2, 3, 7 II 5 1 2 107. t 1 1 32 II 2 1 3f II 3 1 14 II 4 1 48 II 5 1 8f II 6 1 4f II 7 1 8 107. II 8 1 A II 9 1 6| II 10 1 27 II 11 1 9 II 12 1 36 II 108. II 13 1 46 108. 11 U 1 4 II 15 16J II 16 1 8 II 17 1 4711 II 18 1 15 II 19 16210 108. II 20 1 6f II 21 1 Hi II 22 illj||23|4i|| 110. II 111260 110 II 2 1 7200 B 3 1 1260 II 4 1 1008 II 5 1 10500 II 6 1 10800 ANSWKllS. 44;:^ 110. I 7 I 540 J 8 I 420 || 9 | 336 || 10 | 1176 || 11 | 110. II 144 rods. 16 days = A's time ; 12 days = B's time 110. II 9 days = C's time. || 111. || 12 | $1680. 112 at Uo 111. II 105 at $16 ; 80 at $21; 70 at $24)1 13 | 210 bu. 105 bags 1 1 1 . II 70 bbls. ; 30 boxes ; 14 hhds. || 14 | 60 days. A = 3 times 111. IB = 4 times C = 5 times ; D = 6 times. | 112. 2 1 18 112. 13| 12 1 *\ 5 II 5 1 6 H 6 1 10 n t 1 28 II 8 1 14 114. II 1| 16 1 211 11 3 1 22 1 4 1 124 ||.5 | 62 1 6 1 81 II 1 |45 114. f8| 25 1 9| 12 1 10 1 3 i U 1 122 per head. 13 = A's 114. INo . ; 21 = B's No. ; 29 = C's No. 11 121. II 1 1 V- ;¥ 121. II 2 1^; ¥-1 SIM;!? 1 i\U; II II 5|¥f; -Yf 121. l|6| n-, W l|f IW-;-V/-l|8|fJ; Wlil¥-; -f ; 121. II V- 11 2 1^ f ; « ; f ; S II 3 1 J ; J;J;i IMI ii; 121. wa ;H ; V ;¥-IIM^;lif;f l|6|J;in If; 121. II f; f ; A; 5"! II sin; V-; ¥; ¥; ¥ II 122. 1 1 A; 122. II tV ;tV ; I's II 2|t't;iV; tV II 3 1 i§ ; r^» ; tV ; A 122. II 4 1 H; «; A; A; AIIsIt'f; A; i'»; A II e 1 8 , 51 » 122. IU\ ; A ; A II 'JUV; A; A- II 8 1 /s ; /? ; A ;?'s 122. Ml 3 . 57 » A; 3=5 11 2|:i«,; A; A II 3|H;M; AV 122. IIM 5% ; »¥a;/T"Tll5|T¥ff;ili HI 6|t^; A*6; 122. 1 tV ; II 7 1 H;t¥j;5¥s 1 8 1 AV;H; -fVo- 123. II M M; «; H 1 3 1 -JJ ; 4f ; H ; T¥Tll3|HJ;H§i 123. 1 in II 4 h^'^ -;Mf;TVj;M115. 1 5 J Ys'y 10% J i^i 124.11 II; i 1 2|J||3|||; A;§; J;f || 4 ||-< ; ^ ; 444 ANSWERS. 124. 1 f ; f il5|-!f;M;M;il;lll e 1 -51; if ; A; A; 124. II i II 125. 11 1 1 if^ II 2 1 -W- I 3 1 if Ml 4 1 -?/ 11 5 1 H^ 125. II 6 1 IJJIA 11 1 1 5404 11 8 1 IJJii 11 9 1 If Aiii 11 10 1 li«563 125.11 11 1 H¥-^ 11 1 1 HMl 2 1 m^~ 11 126. il 3 1 !-%« 126. II 4 1 HV-' 11 5 1 af 1 Ji 11 6 1 WiyJ 11 t 1 4i>ic„,.»jJ 126.8 1 "^AV-' 1 9 1 ^flF II 10 1 H« II 11 1 '-Vs^ 126. II 12 1 '^\W^' 11 13 1 ^ff" 1 14 1 iijas 1 15 1 241. 126. II 16 1 '-¥-' II n 1 H¥-' II 18 1 HU^ II 19 1 '-"tW-' 126. 11 20 1 iill^ II 21 1 -LH-F II 22 1 "1 II 23 1 3381 127. II 1 1 li H 2 1 12 II 3 1 5i^-A II 4 1 241S§ II 5 1 9 II 6 1 66j% 127. ni 11214 1 8 1 225''Jr 1 9 1 040^1, 1 10 | 5f ^ II H I HyVA 127. II 12 1 225 II 13 1 lOl^i 11 14 1 14 II 15 |, 376|ii 127. 11 16 1 1073l3\V 1 128. II 1 1 + il 2 1 i 1 3 1 J II 4 II * 128. II 5 1 # 1 6 KH 1 1 1 A II 8 1 ill 9 1 i 11 10 1 V' = ^ 128. II 11 It II 12 1ffi 11 13 IH 1! 14 1AV II 15 lit! 128. II 16 1 A¥t II " 1 a II 18 1 A II 19 i il 1 20 1 /J, 128. 1 21 1 A II 22 1 rh 11 129. 1 1 1 A 1 2 1 3', II 3 1 T», 129. 11 4 1 A 11 5 1 A 11 6 1 lli in 1 1 11 8 1 35f 1 9 1 141 129. i 10 1 8i 1 11 1 :i¥ff II 12 1 sth II 13 1 ISfi II 130. Ill 1 e 130. 1 Vt' '. u 11 2 1 m. in. ^¥0-, m ii 3 1 'ivv, m m, m 130. i 4 1 if. Ii. If. il « II 5 1 VaV, Mg. us. Ill II 6 i ftf 130. 11 m \ iiiitVt,ij-hi 8 1 n, fi, !i. ¥«' II 9 1 ¥o^ ii, w ISO. 11 Vj" 11 10 1 %\ V IPl 1 %', w II 12 1 ¥/. 3\. W, ANSVVfcKS. 445 130. II Ji II 1 I A. tV. a, }§ II 3 I if, A, it II 3 I U. il Vif 130. II 4 I W, ii W II 5 I '#, Fo, Vif II 6 I fg, »-», ^«, Vo° liol ^ I 4J. M. 'tV. ft II 8 1 M. A. fl. IS II 9 I H. H. U-. 130. II H 11 '0 I VV. Vo°. ih \V II 132. II 1 1 m. m. h% ■32. II 2 I H, ^1, a I 3 I fl, i^, A II 4 I W. W. A 132- II 5 I w. H. A II 6 I %%'. A. il I -t I A'. W, A '32. II 8 I w, M. H. mi 9 I i°A. #?. AV II 10 I ¥/■ W. 132. II |g I 11 I W. Vo', M II 12 I U, '^', H II 13 I V-/. 132. IIW. ij. A II 134. II 1 I 120 far. | 2 | 1606?- lb. 134. 1 3 I 7840 min. || 4 j 3240 grs. 1 5 | ^-jtH. j 6 j Ayl- 134. i 1 I ,AW° II 8 I ^h '=»'-d- i 9 I 8«- ^d. ; 9s. 4d. 134. 1 10|6fur. 8rd. 4yd. 2ft. Sin. ; 6fur. 34 rd. 1yd. 1ft. Siin 134. II 11 I 33 rd. 1yd. 2 ft. 6 in. || 12 | jhe»'"- II 13 I tAt, 134.11 14 I ^'^ II 1 I if"i- II 2 I jE4H II 3 I jUl 135. II 4 I j-gg II 5 I ^f [| 6 I till T I iflll 8 I tVA I 9 I A 135. II 10 I A°„- II 136. II 2 I f, A. H. U I 3 I A'o. aV», il il 137. II 1 I 2igf II 2 I 3 /A II 3 I 2§i I 4 1 1A"A i 5 | 4H || 6 | 4jg 137. 1 T I 2A^ 1 8 I 9|i I 9 I 16|3J || 10 | llA 11 H I 2CU 137. 1 12 I 10/A II 13 I 4|f 1 14 I 6,*A 1 15 1*611 || 16 | 133,\ 137. II n I 18A I 18 I 89A n 19 | 212A || 20 1 161| || 21 1 65A'5 137. II 22 I 10013 II 23 | 891f || 24 | 341li bushels ; UllMl 139. 1 1 I 14Ain. I 2 I 2da. 14 hr. 30min. || 3 | Icwt. 1 qr. _ _ 446 ANSWEKS. 139. I 191b. 4}oz. II 4 I 2oz. 10 pwt. 12 gr. || 5 | 9 cwt. 1 qr. 139. II 51b. 8f oz. II 6 | 20 bu. 1 pk. 5f qt. || 7 | 3 hhd. 37 gal. 139. II 3 qt. Opt. Ifgil. || 8 | 55 da. 2 hr. 4tmin. 30 sec. m II 9 I 2R. 20 P. Usq. ft. SSg^-sq. in. |] 10 | 7 in. || 11 | 139. H 13s. lOfd. ]| 12 ] 7 fur. 2 ft. 9in. || 13 | 222 da. 1 hr. 24min 1 J 39. II 14 1 7oz. 7 pwt. 23 gr. || 15 | 5 s. 16° 16' iO^%%-" 139. H 16 I 1 yd. qr. 2^ na. i| 17 | 1 C. ft lieu. ft. 466f cu. in. 139.1118 1 2C. 4C. ft. 2cu. ft. J) 19 | 3 yd, 2qr. Ofna. 140. II 20 I 3A. 2R. 33}?. || 21 | llcwt. 3qr. 211b. lloz. l^dr. 140. II 22 I 3 fur. 0»rd. 2 ft. 6 in. || 141. || 1 | f || 2 | j\ \\ 3 | ^\ 141. a H If II 5 I A I 6 I A in Tih II 8 I 35-11 II 9 I X 141. II 10 I ff II 11 I 24^,- II 12 I Ijh II 13 I 1 II 14 I 3A 141. 1 15 I 7f II 16 I 14f I 17 I A II 18 I H II 19 I ^ II 20 | 6f^ 141. i 21 I 81 II 22 I H II 23 | $72 || 142. || 24 | $f || 25 | 18H 142. II 26 I 1811 11.27 | 33/^ || 28 | 22^% || 29 | 1^ j 30 | $18f i"42. II 2 I tIj II 3 I ^i^ II 4 I 3!^ II 5 Uf II 143. || 2 | 2^^ 143. i 3 I 761 II 4 1 73fi || 5 | 6ff || 6 | 182^^^ 144.1 1 I 9oz. 7 pwt. 12 gr. 1| 2 | 7 cwt. 1 qr. 241b. 8 oz. 144. II 3 I 29 gal. 3f qt. || 4 | 1 mi. 1 fur. 16rd. || 5 | Is. 3d. 144. II 6 I 38' 34f' || 7 | 563 A. OR. 35|-P. J 8 | 10 cwt. Iqr. 144. II 221b. 91? oz. || 9 | 1 lb. 8oz. 16 pwt. 16 gr. || 10 j '144. II 2 cords 2 C. ft. 4 cu. ft. || 11 | 5iin. || 12 | 4 ^ 3 3 23 4gr 144. II 13 I lA. IR. 17P. 21isq. yd. || 14 | Ipwt. ISJgr 1464 1 I 3f II 2 I 1/^ II 3 I 71 II 4 I llA II 5 I 16 || 6 | 70 146. II 7 I 44 II 8 1 1584 |1 9 | 6O8/2 || 10 | 5987f || 11 | 4536 ANSWKUS. 447 146. 1 12 I 6405 II 13 | 6975 | U | 11725 || 15 | 3| | 10 | 12? 146. a n I 63 II 18 I 1781 || 19 | 14J-,4 | 20 | 19^ || 21 | f i 146. II 22 I Jj II 23 I 5'i II 24 | ^% || 25 j 14 1 26 j 18 | 27 | 130^ 146. II 28 I A a 29 I 14 || 30 j 6316^ || 31 | j^ \\ 32 | 6|| 146. II 33 I I a 34 I A a 35 I 2f a 36 I 20 a 37 I fl II 38 I jl 146. a 89 I 2^V 1 40 I 51 || 41 | 14^ || 42 j If a 147. I 43 | llj 147. 1 44 I 22| II 45 | 3,*^ || 46 j 14f a ^^ I ^ 11 ^8 I »r\ 147. II 49 I 55 cents, a 50 | 34J a 5' I *H II 52 | 325 a 53 | i 147. a 54 I ^% a 55 I H a 56 I H I 57 | 20^ | 58 | I2II59 | 5-} 148. a 60 I i a 61 I 120 A. = A's share ; 80 A. = B's share ; m. a 20 A. = C'8 share. || 149. || 1 | j a 2 | A I 3 j ,V^ 149. a 4 Ufj 1 5 U'A 1 6 I 7^ a ^ I 36 I 8 I ^ i 9 I 2^ 149. a 10 1 iH a HI m ii 12 1 It^ a 13 m im cit', 149. a 15 I 1662^ a 16 I 1383 i " || a 18 I M I 19 I Hi 149. \\ 20 I 2if II 21 I 9tV II 22 | 48g'^ || 23 | ^'^ I 24 | l^^J- 149. a 25 I rh II 26 I tV II 27 j ^\ I 28|tV a 29 j A II 30 I A 149. a 31 I 40 II 32 I 1120 || 33 | 1} a 34 j ^ I 35 | f^^- 149. a 36 I HI- II 37 I Hi a 38 | Ij || 39 | 825^ a ^0 | 4193tV 149. a 41 I 16046^ a 42 I ? IM3 I ^ a 44 I ^ a 45 I 22f 149. a 40 I 68f|f i 47 j 3^ |1 48 | 1} j 49 | 9-| 1 50 | 72^';, 150. i 51 I 51 lbs, a 52 I l^^yds. a 53 | 1^1 54 | 4 B 55 | ^ 150. a 56 I 3A- II 57 I ItV i 58 I 6 | 59 | A || 60 | 21 a 61 | 27^- 150. a 62 I U^V a 63 I ^% i 64 | j'y a 65 | A l 151- 11 66 | ^^ 151. a 67 I'lOfa a 68 I 14j II 69 | H || 70 | $H a 71 I '3^ 151. a 72 I 108/j. II 73 I ^ a 74 I 4 a 75 I 24i i 70 | l-J | 77 | lOJ 448 ANSWERS. 151. II 78 I 41 |n9 I 6 II 80 I ^ II 81 I i II 82 | 6096 151. II 83 I HtV II 152. II 1 I Ij^ I 2 I Hf || 3 | 2|f || 4 j 100 152. II 5 I if II 6 I f n I Ij II 8 I 35 II 9 I ffg- || 10 | 2A | 11 | 152. II 531 f 12 I i, I 153. || 1 1 15 || 2 | Hf || 3 | ttf || 4 | 42i^ 53. II 5 I ^V^ II '6 I 26tV n I 15 II 8 I 16^ || 9 | 8bu. lipk. 153. II 10 i 1 mi. 2 fur. 16 rd. || 11 | 4mi. Uur. 19 rd. 3yd. 02Jft. 153. II 12 I 20i II 13 I 14 II 14 | 20Ji || 15 | 2700 = A's share-, 153. II 2800 = B's share ; 800 = C's share. || 154. || 16 | 40 154. II n I £n 17s. 5d Oi-far. || 18 | 24 = John's ; 32 r= James' 154. II 19 I 285f- II 20 | A, 80 ; B, 24 ; C, 30 ; D, 40 ; 66 rem. 154. II 21 I 467| II 22 | $2j\ sellmg price ; $^%\ = 1st one's gain ; 154. II $^^8 = 2d one's gain. || 23 | 257J| || 24 | 7^ || 25 | 154. II 1724i = A's ; 1231f = B's || 26 | 165 || 156. || 1 | 7ft. 2' 156. II 2 I 5 ft. 2' 6" II 3 | 21 ft. 4' 11" 4'" || 4 | 5 ft. T' 156. II 5 I 3' 3" 2'" II 6 | 2ft. 7' 3" || 7 | 15 ft. 4' 10" 4'" 156. II 8 I 3ft. 6' 5" 5'" || 9 | 87 ft. 10' 7" 4'" || 10 | 183ft. 5' 6" 2"' 156. II 11 I 223 ft. 8' 4" 9"' || 12 | 87 ft. 2' 7" 9"' 6"" 156. II 13 I 317 ft. 11' 0" 4'" II 14 | 543 ft. 6' 3" 2"' sum ; 156. II |107 ft. 8' 9'' 2"'diff. || 160. || 1 | 41eu. ft. 3' 10" 160. II 2 I 43 sq. ft. 6' 6" || 3 | 82 sq.ft. 9' 4" || 4 | 347 sq. ft. 10' 3" 160. 1 5 I 554 sq.ft. 7' 8" 8"' 3"" || 6 | 2917 sq. ft. 0' 0'^ V' 4."' 160. II 7 I 194 sq.ft. 4' 3" ^"' \\ 8 | 39 sq.ft. 11' 2'' 3" 160. II 9 I 296 sq.ft. 10' ^" \\ 10 | 96sq. yd. 2 sq.ft. 8' 3' 160. II 11 I 3150 sq.ft. || 12 | 327Jsq.yd. || 13 | 21 sq. ft 161. II 14 I $26.40 II 15 I 10 A. IR. 25 P. || 16 | 3119 sq.ft. 6' 9" ANSWERS. 449 161. II 11 I 99 II 18 I $208,011,11 19 | 89cu. ft. 3' || 20 | I18.49J 1 , 161. II 21 I 504CU. ft. II 22 | 11 jj cords || 23 | 24124f? 161. II 24 I 41958 || 25 | 19419cu. ft. 9' || 26 | 849cu. ft. 8' 8" 161. II 27 I $15,403+ 1 28 | 2t5^*^cu. yd. || 162. || 29 | $19,803 163. II 1 I 4ft. 7' II 2 I 5ft. 3' 3" || 3 | 48ft.' 6' || 4 | 8ft. 7' 163. II 5 I 12ft. 6' II 6 I 37ft. 3' || 7 | 1ft. 7' || 8 | 8ft. 163. II 9 I 6ft. 6' ^jziW II 167. II 1 I .06 II 2 I 1.7 jj 3 | .005 167. II 4 i .27 II 5 I .047 || 6 | 6.41 || 7 | 7.008 || 8 j 9.05 || 9 1 11.50 167. II 10 I 44.7 II 1 I 27.4 || 2 | 36.015 |1 3 | 99.0027 || 4 | .320 167. II 5 I 200.000320 || 6 | .3600 || 7 | 5.000003 || 8 | 40.0000009 167. II 9 I .4900 II 10 I 59.0067 || 11 | .0469 || 12 | 79.000415 167. II 13 I 67.0227 || 14 | 105.0000095 || 15 | 40.204000 168. II 1 I $37,265 || 2 | $17,005 || 3 | $215.08 || 4 | $275,005 168. II 5 I $9,008 II 6 | $15,069 || 7 | $27,182 || 8 | $3,059 171. II 1 I 130G.1805 II 2 | 528.697893 || 3 | 159.37 || 4 | 1.5415 171. II 5 I 446.0924 || 6 | 27.2087 || 7 | 88.76257 || 8 | 71.01 171. II 9 I 1835.599 || 10 | 397.547 || 11 | 31.02464 || 12 | 90.210129 171. II 13 I 204.0278277 || 14 | 400.33269960 || 15 | .1008879 171. II 16 I $85,463 || 172. || 17 | $1065.19 || 18 | 3.8896 172. II 19 1 $427,835 || 20 | $19,215 || 21 | $670,975 || 22 1 $30,286 172. II 23 I $328,202 || 24 | $248,011 || 25 | $134,634 173. II 1 I 875.0033 || 2 | 368.5631 !| 3 | 7141.51354 || 4 1 51.722 173. II 5 I 2.7696 || 6 | 1571.85 || 7 | .6946 || 8 | .89575 173. II 9 I 603.925 || 10 | 1379.25922 || 174. || H | 99.706 174. II 12 I 17.949 || 13 | .699993 || 14 | 328.9992 || 15 | .999 450 ANSWERS. 174. II 16 I 6314.9 || It | 365,007495 || 18 | 20.9942 174. II 19 I 260.3608?)53 | 20 j 10.030181 || 21 | 2.0294 174. II 22 I 999.999 || 23 | 2499.75 || 24 | 103.015 || 25 | .4232 174. II 26 I 171.925 jj 27 | $82,625 || 28 | $26.60 || 29 | 126.84194 174. I 30 I $76hl8 II 175. || 1 | .796875 || 2 | 2.6387 ^ 175. 11 3 I .0000500 || 4 | 1..50050 || 5 1 26.99178 || 6 | 10376.283913 175. II 7 I 165235.5195 || 176. || 8 | .0206211250 || 9 | 28033.797- 176. II 099 II 10 I 175.26788356 || 11 | .000432045770 176. II 12 I 216.94165850 || 13 | .000000000294 || 14 | 18616.74 176. II 15 I 933.8253150762 || 16 |.00715248 || 17 | .608785264 176. II 18 I .02860992 || 19 | 2.435141056 || 20 | 1296 176. D 21 I 312.5 II 22 | .375 || 23 | .0036 || 24 | 148.28125 176. I 25 I 12.13035 || 26 | $24.0625 || 27 j $3192.005625 176. II 28 I $210.03125 || 29 | $708.901875 || 30 | $2.06525 gain. 177. 1 1 I 4796.4 ; 47964 || 2 | 69472.9 ; 694.729 J 3 | 415300. ; 177. 1 4153. II 4 I 2704 ; 27040. || 5 | 129072. ; 1290.72 H 6 | 177. II 871000. ; 8710, | 7 | 140100. ; 1401. || 179. || 2 | 258.13007 179. 1 3 I 162.525 || 4| 2757.89785 U 5 | 3566163 || ISO. || 1 1 2.22 180. I 2 I 8.522 II 3 | 33.331 || 4 | 1.0001 || 5 | 12420.5 || 6 | .005 180. 1 7 I 4.25 II 8 I .007 || 9 | .075 ]| 10 j 1.27 || 11 | .015 180. II 12 I 17.008 U 13 I 25.05068 ; 250.5068 ; 2505.068 ; 180. II 25050.68 ; 250506.8 || 14 | 48.65961 ; 4865.961 180. 1 48659.61 ; 486596.1 ; 4865961. || 15 | 41.622 ; 416.22 180. II 4162.2 ; 41622. ; 416220. ; 4162200. || 16 | 254.7347748 180. II 25473.47748; 254734.7748; 2547347.748; 25473477.48 ANS\r£K(4. • 451 180. II 2547347U.8 || U | .1395646+ || 18 | 1918.515 + 181. II 19 I .004735 || 20 | 174.412 1 21 1 6^.7125 || 22 | 1.36832 + 181. I 23 I 12976.816+ || 24 j .004958+ || 25 | 6.165 181. II 26 I $9,875 || 27 | $2.15 || 28 | $.62 || 29 j 18 I 30 | 8 181. 1 31 I 14 II 32 I 65.5 || 33 | 269 acres ; $13573.204 cost 181. II $50,458 average price. || 34 | $7631.8855 share of eldest ; 181. II $5723.914125 share of others. || 182. [ 2 | 10970 182. II 3 I 60200 II 4 | 1000 jj 5 | 100 || 6 | 10 ; 100 ; 1000 ; 182. II 30 ; 20 ; 2000 ; 12 ; 1200 ; 500000 || 183. ] 3 | 8.311 + 183.14 I 1.563+ II 5 I 1.16049+ || 6 j 16.11902+ 184. I 1 I 31.69274 ; 3.169274 jj 2 | 57.13562 ; 571.3562 184. II 5713.562 || 3 | .675 ; .0675 ; .0000675 | 4 | .049 ; .0049 184. 11-00049 II 5 I .030467 ; .0030467 ; .00030467 || 6 | .004741 184. II .0004741 ; .00004741 | 7 | .497 ; .0497 ; .00497 186. II 1 I 79.1188 II 2 | 35.2843 || 3 | 11.5834036 1| 4 | 3202.8870 187. 1 1 I .25 ; .5 ; .75 || 2 | .8 ; .875 ; .3125 | 3 | .375 ; .04 187. II 4 I .015625; .2666f || 6| .125; .003 || 6| .25714 + ; 187-11.44117+ 1 7 I .23903+ || 8 | .07157+ || 9 | .4375; 187. S .078125 li 10 I .00448 || 11 | .536 ; .372 || 12 | .9 187. II 13 I .73333J || 14 |. 48375 || 15 | .51282+ || 16 | .5375 ; 187. S .005606+ II 17 | .16666+ J 18 | 1.5555-| || 19 | .15909^, 187. II 20 I $100.80 II 21 | $17.85 || 22 | 30.61111 || 23 | 2.9166? 187.1 24 I 2.8412+ || 188. || 1 | j; f B 2 | j ; | j 3 ^^ ; Tii> 1«8- II 4 I ilU ; nih II 5 I MS- II 6 I tI-o i- ^ I tVo^ II 8 I H 188. I 9 I J II 10 I i II 189. II 1 I .0546875 || 2 | .325 | 3 | 3.9375 452 • ANSWERS. 189.14 1 .375 il 5 | 71.15113+ || 6 |. .(3625 || 7 | . 15375 189. II 8 [3225 II 9' | .26175 |1 10 | 100511+ I 11 | .04 189. II 12"|. 91111+ II 13 I .875 || 14 1 .01587+ || 15 | .712 9 i) + 189rj| 16 I .2325 || 17 |. 972916+ || 18 | .48125 || 19 | 55 189. II 20 I .001617+ II 21 | .25625 || 22 | .063 || 23 | .10416 + 189. II 24 1.00994318+ || 25 | .791666+ || 26 | .3375 I 27 | .3125 189. II 28 I .040909 || 29 | .01875 \\ 30 | .020265 + 189^ II 31 I .19672+ || 32 | .34895+ || 33 | .0153t+ || 34 | .005 190. Jl I 2qr. 171b. 4oz. || 2 | Ihhd. 13gal. 3.44 qts. 190. II 3 I 16s. 7d. 2.99 far. || 4 | 2 gal. 1 qt. || 5 | 190. II Iwk. 4 da. 23 hr. 59 min. 56.54+ sec. || 6 | 8 P. 190. II 7 I 6cwt. 3qr. || 8 | 1 lihd. 47 gal. 1 qt. || 9 | 20 gal. 1 qt. 191. II 10 I lOoz. 18pwt. 15.99+ gr. || 11 | 3qrs. 1.5 iia. 191. II 12 I 1yd. 2 ft. 11.9+ in. || 18 | 24P. 23sq. jd. 5sq. ft. 191. II 82.4832 sq. in. || 14 | 32 mi. 1 fur. 14 rd. 4 yd. 2 ft. 9.408 in. 191. II 15 I 2ft. 7.5 in. || 16 | 43 13 13 9.6+gr. 191. 1 17 I 3R. IP. 13.31 sq. yd. || 18 | 9 sheets. || 19 | 111b. 191. II 20 I 7d. 2ftir. || 21 | 1 R. 14 P. || 22 | 286da. Hhr 191. II 18min. 36scc. || 193. || 1 | .06 || 2 | .08125 || 3 | .034375 193. II 4 I .01328125 || 5 | .0171875 |] 6 | .034 || 7 | .028 193. II 8 I .024219375 || 194. 1 1 | .71428571+ || 2 | .2666 + 194.1 3 I .4545+ || 4 | .3888+ || 196. || 3 | j ; /^ ; jg ; jf ; ,\ ^96. II 4 I t¥3 ; t\; I II 197. II 4 I A; 7 j-g| ; -jj^; 37fg; 22 3. 7^4 3 4 II n I 3.4 . 217 . 7_ . 412 5^6. _J6 3_. AL 3 3 0? 99399 II ^ I 45')495>T5>^]G"6 5JI6500>90 198. II 2 I .1875^0' II 3 | .0^0344827586206896551724137931' ANSWERS. 453 198. II 4 I .'09t56^;..^592'; .5^3' 1 200. || 2 | 2.4481818'; 200. II .5^925925'; .008^497133' || 3 1 165.16^416416'; .04^040404' 200.11.03^777777' || 4 | .5^333333'; .4^57575'; 1.7^5775tr 201. II 2 I 95.2^829647' || 3 1 69.74^203112' || 4 | 55.6^209780437503' 201. II 5 I 47.3^763490' || 6 j 416.2^542876' || 202. || 2 | 45.7^755' 202. II 3 I 2.9^957' || 4 | 1.64ir7' || 5 | .65^370016280907' 202. II 6 1 4.37^4' || 7 | 4.619^525' || 8 | 1.0923^' || 9 | 1.3462^937' 202. II 2 I 5.53780^5' || 3 1 1.093^086'"|| 4 1 1.64ir7' || 5 1 1.7183^39' 202. II 6 I 1.4710^037' || 7 | 6.r656' || 8 | ll.'^068735402' 202. II 9 I .81654468350' || 203. || 2 | 13.570413^961038' 203. II 3 I 35.024^0' || 4 | 7.719^54' || 5 | 26.7837^428571' 203. II 6 I 3.r45' II 7 | 3.^8235294117647058' || 8 | 1.2^6' 203. II 9 I 15.48^423' 205. II 205. II " 21 1 . II 4| 17 1 l39~l + l~ , 27~1 + 1~^ , 205. II l>;-~' 1 + 1"' 2 205. 1 205. II ^ 1 47 1 ._ 67 1 1 + 1 ^ ^ 1 65~1 + 1 "2^ '85 1 + 1 ^ g2 + i~"'^ 205. II 2+l~^ 3 3+1-* ^ 205. II 205. II 1 + 1~*_5 H-1 ^ _,, 1 + 1 ^ „ 2 + 1 ^ ,3 205. II 1 + 1-^^ ,, 1 + 1-- , 205. II 1 + i '' i+r" 205. II 205. II ^ 205. II 37 1 , 109 1 , 1 87 2 + 1 * " ' 1 450 4 + 1 * ^ 2 + 1-* r ^ 7.+ 1 ^ ^ 205. II 1 + 1 « ,, 1 + 1 ^^ ,, 205.11 5 + i~^ 3 + l~^'%o 205. II GVf + t¥t) - 2 = i^SW Arts. 1 + i '^^ 454 ANSWERS. 208.112 i V = 4 II 3 I V = 5 II 4 \.j\ = i || 5 | 60 -f- m.j20Z^ 6 I 1^^ = 2 II 7 I |$|^ = f gf ^ Ijl m JS II- = f II 9 I T^O = 1 II 10 I XX ^ 1 II 11 I 30g ^ 2 208. II 12 Uf = i l: 13 I f II IM ^ II 15 I i II 16 I tV II H | /, 208. II 1 I 112 cwt. II 2 I 5 tons. || 209. || 3 | 60 || 4 | 5 || 5 | ,? 209. II 6 f32 II 7 I 28 II 8 I $65 || 211. || 1 | a; = 60 || 2 | ar = 2T 209. II 3 I ^ = 9 II 4 I :r z::: i II 5 I 38 II 6 I 56 |i 7 | 12 || 8 | 40 213. II 1 I or = 21 II 215. II 1 I 330 II 2 I 90 II 3 I 504 || 4 | 2.08 215. II 5 I 875 II 216. || 6 | 99 || 7 | 27621 || 8 | 20 || 9 | 122.85 216. II 10 I 1400 11 11 I 164«5 || 12 | 121.875 || 13 | $27 216. II 14 I 710Z. II 15 I 3533.936 || 16 | 86.62 || 17 j £39679 10s. 216. II 18 I 9||19|8Jrd.|| 20 | 160 yds. || 21 | 7-1 || 217- || 22 | 10 217. II 23 I 920 II 24 | 54 || 25 | 39.375 || 26 | 382.85 || 27 | 63 217. II 28 I $.036 II 29 | $7080.48 [ 30 | $1,925 || 31 | 2.10 217. II 32 I 52.50 || 33 | $f| || 34 | 7200 || 35 | $37,909 + 217. II 36 I 225 || 37 | 20 || 218. || 38 | 54 || 39 | 12 1| 40 | 6 218. II 41 1 160 II 42 I 40.47 || 43 | 10 yr. || 44 | 51 || 45 1 132.589-f 218. II 46 I 1121 II 47 | 18.66| || 48 | 66.355 || 49 | 106f || 50 [ 40 218. II 51 I 112.86 II 52 | 18090 || 219. || 53 | 21 gal. || 54 I 219. II 2142 to A ; 1125 to B || 55 | .625 || 56 | 6f || 57 | $15.86| 219. II 58 I 168 lbs. || 59 | 93f || 60 | 552 || 61 | 17444 219. II 62 I 6hr. 32mm. 43j-\sec. || 63 | 140 || 220. || 64 | l^da 220. II 65 I 221 da. || 66 | 45 || 67 | Uj\ \\ 68 | 20i- || 69 | 10^ 220. II 70 I 131 II 71 I 810 II 72 I 6 II 73 | lOlir. 40 min. Se^^^sec 220. II 74 I 16^ times. || 222. || 1 | I6I || 2 | 7200 || 3 | 1871 ANSWERS. 455 t 222. II 4 I 72 II 5 I 10 i 6 I 92J II 223. t | 36 | 8 | 292.5 || 9 | 156 223. II 10 I 9600 II 11 I 50 || 12 | 13f || 13 | 8571f J 14 | 3 hr. 223. 1 15 I $471.04 || 16 | 3if || 224. || 17 | 180 J 18 | 13^ in. 224. I 19 I 14 j II 20 I 7^ || 21 | 97i || 22 | 32 j 23 | 32 || 24 1 132 226. II 2 I $1000, A's; 81200, B's ; $800, C's || 3 1 1714.28f, A's 226. II 285.7 If, B's || 4 | $4030, A's ; $3980, B's j $3980, C's , 226. II $4010, D's II 5 | 100, A's ; 140, B's ; 200, C's || 6 | 226. II $3333i 1st ; $3000, 2d ; $3000, 3d ; $2666|, 4th 227. II 7 I $3000, widow's ; $1500, son's | 8 | $12961.50, A's; 227. II $15737.25, B's ;' $10802.25, C's ; $1833, D's gain. 227. II 9 I $450, A's ; $600, B's ; $750, C's || 10 | 4242.50, A s 227. II stock ; 1697, A's gain: 5939.50, B's stock; 2375.80 B's 227. II gain : 6788, C's stock ; 2715.20, C's gain. || 11 | 237.75, 227. II A's ; 181.0625, B's ; 125.4375, C's ; 70, D's. jj 12~j 227. II 87.831+ A's; 65.06+ B's; 48.795, C's; 68.313, 227. II D's. II 13 I 2553, A's ; 3401.70, B's ; 1405.30, C's. 228. II 14 I 15063f, B's ; 9586J, A's. || 15 | 1015.331 the first ; 228. 11 1523.00, the second ; 2030.66f , the third. || 2 | 16.38, A's ; 228. II 35.10, B's; 18.72, C's. || 229. || 3 | $7 || 4 | 6577.23fJ5, 229. 1 A's ; 1822.76111, B's. || 5 | 288, A's ; 270, B's ; 240, C's. 229. II 6 I 280, D's ; 168, C's. || 7 | 2648.86^^, A's ; 2901.13/^. 229. II B's ; 1850, C's. | 8 | $800, B's stock ; 15mo., C's time. 231. I 1 I 50.24 II 2 I 114.78 || 3 | 1.1875 || 4 | 2.839375 231. II 5 I 1.002 II 6 I 12 II 7 I 90 1 8 I 16.74 || 9 | 47.725 231. I 10 I 27.54 II 11 I 300.365 || 12 | 15.75 || 13 | 160 456 ANSWERS. 231. II 14 I 478.125 || 15 | 4344.35 || 16 | 2625 || 11 | 5144.625 231. II 18 I 12500 II 19 | 3867.018t5 || 20 | 15000 || 21 | 22.95 231. II 22 I 43.20 || 23 | 65 || 24 | U2.85 || 25 | 205 i^l I 20 II 2 I 121 II 3 I H IM I 13f II 5 I 25 II 6 1 87i 232. II T I 571 II 8 I f "A || 9 [ 47/g || 10 | 331- || 11 j 3ti || 12 | 12^ 233. II 13 I 8tJ II 14 I 80 |1 15 | 70 || 16 j 66f |1 It | 160 233. II 2 I 1900 II 3 1100 [j 4 | 400 || 5 | 15000 || 6 | 142f 233. II t I 1.9^ II 8 I 90 II 9 I 1800 || 10 | 4392 || 11 | 20800 234. II 1 I 388.1188 || 2 | 9000 || 3 | 550 || 4 | 156 || 5 | 30.123-f 235. II 6 I 140 II 1 I 3ni1.77J || 8 | 5400 || 9 | 4 || 10 | 5425 235. II 1 I 160 II 236. i| 2 | 150 |1 3 | 950 \\ 4 | 30000 \\ 5 | 13500 236. II 6 I 50000 || 1 | 5000 || 8 | 2600 || 237. || 1 | 33.15 237. II 2 I 21.411 II 3 | 236.25 || 4 j 6.15/^ || 5 | $.695 per bushel 238. II 6 I 8166 || 1 | $.56 || 8 j 915.15 || 9 | 133.20 238. II 10 I 11165.311 II 11 I 444.15 || 12 | 1910.115 || 13 | 2cts. 238. II 14 I 18860 || 15 j 11% || 16 | 90 cents. || 239. || 11 | $3.20 239. II 18 I $.96 II 19 | $18.03 || 20 | $.66 || 21 | $1.80 || 22 | 18% 239. II 23 I 25% || 24 | Neither. || 25 | 8'0% || 26 | 25 % 239. II 21 I 160.34315 gain ; 4f % |1 240. || 28 | $1041.15908i 240. II 29 I 25% on gold; 20% on paper. || 30 | 1612.90ii^ 240. II 31 I 14980 || 32 | 10562.50 || 33 | 20000 || 34 | 260000 240. II 35 I 11% II 36 I $426 || 31 j $400 || 38 | 110 241. II 39 I 548.80 || 40 | 350, 1st; 525, 2d ; 10, gain. || 41 | 6 241. 11 42 I 4315 || 43 | 45% || 44 | 25.65 lost. || 45 | 3450, cost; 241. II 5% loss. I 46 I 40 % || 41 | lOf % || 48 j 339, cost ; 508.50 ANgWERS. 457 242. II 1 I 188.50, com. ; ^1351.50 paid over. || 2 | 40.n, com. ; 242. 1 1359 laid out. jj 3 j 34.8375 jj 4 | 164.53125 243. II 5 I 96.33; 5831.67 || 6 | 163.80, com. ; 4340.70, cost. 243.11 7 I 115.391 || 8 | 6835.283 || 9 | 935 || 10 | 420.922 243. II 11 I $70 II 12 I 2571.36 || 13 | 39.1875, charges ; 1267.062£ • — — — 1 243. II trans. |i 244. || 14 | 11764.705+ ; 235.295+ || 15 | 15 tons 244. II 16 I 63.625, com. ; 4544.642+ bu. J 17 | 158bbls. , 244. II $2412.66 || 18 | 183.0607+ ; 3.6612+ || 19|2^8_t''/» 244. II 20 I 2-^YU \\ 21. 1 8f % || 22 | llJVo || 246. || 1 | 43.875 246. II 2 I 60.9875 || 3 | 224.91 || 4 | 360.2832 || 5 | 473.844 246. II 6 I 1312.50 || 7 | 283.8438 || 8 | 422.8976 || 9 | 1112.90 246. II 10 I 265.2345 || 11 | 1893.75 || 12 | 373.2495 || 13 | 735 246. II 14 I 1016.075 || 15 | 120.80 || 16 | 5796 || 17 | 20.909 246. II 18 I 26.313 || 19 | 458.88 || 20 | 1979.5013J || 21 | 5618.75 246. 1 22 I 628.416J || 23 | 64.0625 || 24 | 157.65625 249. II 2 I 42.24325 || 3 | 420.253125 || 4 j 213 || 5 | 181.25 249. II 6 I 11.0415 II 7 | 132.7707+ || 8 | 26.9586+ || 9 | $49. II 416.1673+ || 10 | |334.2187+ || 11 | 120.0693 + 249. II 12 I 40.0968 || 13 | 81.'6778+ || 14 | 162 || 15 | 221.266 249. B 16 I 389.2406 || 17 | 135.3714 \\ 18 | 42.9404 + 249. II 19 I 84.6855 \\ 20 | 55.6685+ j 21 | 32.666? 2.50. II 22 I 8590.8333J || 23 | 36 || 24 | 93.7843+ || 25 250. II 160.4408+ || 26 | 12.963+ || 27 | 82.036+ | 28 | 70.964 250. II 29 I 879.46703+ || 30 | 801.769 || 31 | 933.1573 + 250. II 32 I 499.339+ || 33 | 140.6444+ || 34 | 5085 458 ANSWEKS. 251. II 35 I 403.858 \\ 36 | 9337.50 || 1 | 394.325625 || 2 | 697.986 251. II 3 I 3339.613 || 252. |i 4 | 823.902+ || 5 | 4640.532.-{- 252. II 6 1 1976.6305+ |i 2|ie45 8s. Ifcl. 1| 253. || 3 | ^£45 12s. 4d.+ 253. (I 4 I iei54 7s. Od. 2far. || 5 | £1133 10s. 9id. || 6| 253. II £199 6s. 3fd. || 7 | £6 16s. 5d. j| 255. || 2 ] 5359.3664 + 255. II 3 I 8925.5443: || 4 | 1127.041 || 5 | 190.758 \\ 256. || 6 | 256. II 156.20+ II 257. |1 2 | 3976.782| || 3 | 439.80 || 4 | 6234.76 257. II 5 I 30000 || 6 | 952.576+ || 7 | 7% || 8 | 10% || 9 | 51 % 258. II 10 I 121% II 11 I 2yr. 6mo. |] 12 | 16yr. 8mo. 258. II 13 I 5yr. 4mo. || 14 | 1 yr. 6'mo. 20 da. || 15 | 7500 259. II 2 I 25.3575 || 3 | 291.7215 1| 4 ] 57.3048 [ 5 | 73.0154 + 259. II 6 I 83.20 || 7 j $845.8376+ || 8 | $48165.9388+ || 9 | 259. II $14523.55509+ || 260. || 1 | 562.50 || 2 | 184.499 + 261. II 3 I 21 II 4 I 5000 || 5 | 1902.587+ || 6 | 236.438 = dis..; 261. II 2763.562 pres. value. || 7 j 1379.6123+ || 8 | 3538.0835 + 261. II cash value ; 388.0835+ gain. || 9 1 9890.23864 + 261. II 10 I .00414+ at7J cts. || 11 | 13.33J || 12 | 2369.2617, 261. II cash value ; 61.9883, diff. || 265. || 1 [ 6.15 || 2 | 7.65 265. II 3 I 23.2913 dis.; 476.708 pres. value, jj 4 | 1225.3555 265. II 5 I 4.375 |i 6 | 82.5916 gain. || 7 | 11.785 || 8 | 15.4044 265. il 9 I 981.21 || 10 ] 474.375 || 266. || 2 | 296.50 || 3 | 697.20 266. II 4 I 1041. 666f || 5 | 3522.092 || 268. || 2 | 387. || 3 | 90 268. 11 4 1 2559.06, A's ; 3210.6375, B's. || 5 | 153 i| 2 | 5320 268. il 3 I 666 II 4 1 17455.50 || 269. || 5 | 59110 || 6 j 21375 269. II 7 I 7999.6875 || 8 | 213500 |1 9 | 307 || 2 | 3529.411 + ANSWERS. 459 270.13 I 5G II 4 I 4000 || 5 j 7235.142+ || 6 | 8000 270. II 7 I 10432.432+ || 2 | 8»/« || 271. || 3 18V. || 4 | 8"/. 271. II 5 15% II 272. || 2 | 20% || 3 | 41 J % || 4 | 12i% premium. 272. II 2 I 7 "/» best. 1 3 | 8% best., || 4 | 166.66f || 274. || 1 •274. II 5168.59 || 2 | 158.40 ; 237.60 || 3 | 252 ; 126 || 4 | 300 274. 1 5 I 89.55 || 6 | 47.811 || 7 | 1252.12^ || 8 | 163.80 274- II 9 I 16481.25 || 10 | 5^% || 275. || 11 | l|°/o || 12 | 44% 275. II 13 I 24000 f 14 | 9020 || 15 | 127.4625 || 16 | 298.2546 277. II 1 I 121.72 II 2 | 232.50 || 3 | 262.50 || 4 | 20 || 5 | 98.20 277-1 6 I 120 II 7 I 9101.635 || 278. || 1 \ 411.15 || 2 | 757.908 279. II 3 I 1227.395 || 280. || 1 | 7051.63415 ] 2 | 9049.53795 280. II 3 I 23058.6765 || 281. || 4 | 2195.95 || 5 | 2159.613 + i82. II 1 t H % II 2 I 37901125 || 283. 1 3 | 1^% ; 82.25, A's tax ; 283. 156.9075, B's tax. j 4 | f % ; 15.50 || 6 | 5820 i 6 | 283. II 22236.197 || 7 | 4656.05, whole tax ; 5 mills on $1 ; $27 ; 283. 1 6.8775, G's tax ; 12.78, H's tax. 1 284. || 8 m cts. on $1 ; 284. II 112.50 ; 18. || 9 | 7.40 ; 9.225 || 285. || 1 | 12 mo. || 2 | 9 285. I 3 I 8|mo. || 4 | 7mo. 3da. || 5 | 6Jmo. || 6 | 6mo. 6da. 286. II 7 1 26 J da., or July 28th. || 287. || 2 | 28^^. or ^pril 287. II 29th, Eq. time of purchase ; Dec. 29th, Eq. time of paym*t, 287. II 3 I 78^ da., or Oct. 18th. || 288. 1 4 | 76yYTV/H ^a., or 288. II July 13th. || 5 | 21^1^, or 21 days. || 2 | 9-Jmo. 2«9. II 3 I 5^ mo. ] 2 | 25 mo. J 3 | 5f mo. | 4 | 421 da., from 289. II Jan. 1st ; or on Feb. 26d, next year. || 5 | July 8th, 1857. 291. II 2 I $.58 hit. balance ; $700.58 cash balance. 1| 292. 1 3 | 460 ANSWERS. 292. II '!>46.20 int. balance ; 1403.80 cash balance. || 4 j $.10 int. 292. II balance ; $620.70 cash balance. || 293. || 1 | 109 da. from 293. II April 2d, or Dec. 14, 1860. || 294. || 2 | 2449.15 balance ; 294. II April 9th. || 3 | July 13th. || 295. || 1 | 8.50 || 2 | $ 06 296. II 3 I $.49 II 4 I $1.00 || 5 | 75° || 6 | 19 || 7|^^.ir,i 296 II 8 I $.30 loss. || 298. || 1 j 1 lb. at 8 cts. ; 1 lb. at lOcts. ; 298. II 31b. at 14 cts. || 2 | 1 lb. of each. || 299. || 3 | 1 calf, 299. II 2 cows, 1 ox, 1 colt. || 4 | 3 gallons. || 300. || 1 | 20 lb. of 300. II each || 2 | 75 lb. of each. || 3 | 3G gal. at 7s. ; 24 gal. at 300. II 7s. Cd. and 9s. Gd. ; 12 gal. at 9s. || 4 | 10 at $2, and 15 at 300. II $f II 5 I 25 lb. at 5 and 7 ; 100 lb. at 1^ • 37^ at 9J ; and SCO. II 50 at 10 I 301. || 1 | 221b. of each. || 2 | 9 gal. water, 301. II 401 gal. at $2.50, and IS^gal. at $3.00 || 3 | 12 sheep, 16 301. II lambs, 12 calves. || 4 | 8 at $6, 8 at 17, 4 at $19 301. II 5 I 90 gal. at 4s., and 10 gal. each at 6s., 8s., and 10s. 301. II 6 I 6 vests, 12 pants, 6 coats. || 7 | 30 at 15, 4 each of 20, 301. II 22, and 24 || 8 | 10 at $^, 15 at $1, 10 at $5 304. II 3 I 1260.9932 || 4 | 713.37 || 5 | 6T. 14cwt. 1 qr. 16.681b. 304. II 6 I 6T. 13cwt. 2qr. 4 1b. ; $308.4774 || 7 | 792.612 305. II 8 I 1196.343+ || 9 | 255.835+ || 10 | 4.09+ || 11] 305. II 398.1199 || 12 | 466.27875 || 13 1 1101.24 gain ; 14 cts., price 306. II 14 I 7936.50 i| 15 | 820.4625 || 16 | 423.36 || 17 | 251.453-+ 306. II 18 I 1457.75 || 19 | 22.605 cwt., tare ; $68.5856 dutv 307. II 1 I 225 j^ tons. || 2 | 4384| tons. || 3 | 729gV5 tons. 307. II 4 I 1006.57+ tons. || 316. || 1 | 8591.975 || 2 | 8637.16875 ANSWERS. 461 316. II 3 I 9177.036 || 4 1 9970 || 5 | $15000.305 || 6 | 9801.9299-f 317. 11 2 I 176204.4729+ || 318. || 3 | jeUOU 17s 7id. 318. II 4 I 0005.368+ || 5 | 807.874+ || 6 | 9096.806 + 319. II 2 I 7^'o premium. || 3 | 12280.06 || 4 | 84597 francs 66 319. I centimes. || 1 j 6057.693 || 2 | 1250.52 ; 3% nearly belovf 319. II par. || 321. || 2 | 5761.31 + florins. || 3 | 9962.219+ 322. II 4 I 3495.839+ Sp. doll. || 323. [ 1 | 16 | 2 | 225 323. II 3 I 20164 || 4 | 214369 || 5 11795000 | 6 | 605.16 323. II 7 I .276676 ] 8 | 9.765626 || 9 j .00274576 || 10 | j% 323. I 11 I H II 12 U? II 13 I i gf I II 14 I mU II 15 | 58.140625 323. II 16 I 250^2^^ II 17 | 51030.81 || 18 | 216 || 19 | 13824 323. II 20 I 1953125 || 21 | 2515456 || 22 | 20736 || 23 | 59049 323. II 24 I 76.765625 || 25 | 10.4976 || 26 | .0184528125 323. II 27 I iUi II 28 | j^- \\ 29 | ^U^ || 30 | 57^^, 323. II 31 I II^Ht II 32 | 14880.930 || 33 | .000244140025 323. II 34 I 2893640.625 || 327. || 3 | 7 || 4 | 12 || 5 | 15 || 6 | 48 327. II 7 I 89.409+ || 8 | 2505 || 9 | 137.84+ || 10 | 1003.8677 + 327. II 11 I 191.713+ II 12 I 1000 || 13 | 311.011+ || 14 | 173.853 + 329. II 4 i f II 5 I if II 6 I .14 || 7 | 2.5 || 8 | 16.7 || 9 | .453 || 10| 329. II .93+ II 11 1.9682+ || 12|.1581+ || 13 | •>/5l = 2J. Arts, 329. II 14| ^."7994 = . 89409 +^ns. II lb\YJ22l = A1U+ Aus. 329. II 16 I .779+ || 17 | .149+ || 18 | 5.01 | 19 | 14.015 329. II 20 I 1.2247+ || 21 | fj || 22 | f || 23 | .2828+ || 24 329. II 11.618+ II 25 | .885+ || 26 |- 75.15 || 27 | 400.06 331. 1 1 I 343 II 2 I 221 j 3 I 21 ^P II 4 I 00 rd. wide ; 180 rd. long. 462 ANSWERS. 331. II 5 I 40 rows ; 80 trees in a row ; 10 A. R. 29 P. 168f sq. 33?. II ft. area. || 6 | 75 ft. || 332. || 7 | 135 || 8 j 94.708 + 332. II 9 I 53.33^- ft. || XO | 8.66+ ft. || 11 | 825.8 mi. || 12 | ilOO 332^1113 175 11 14 I 28.28+ ft. || 333. || 15 | 6 in. || 16 | 333. i 11.041+ rd. || 17 | 4.405+ in., 1st man's share: 333. 1 5.739+ in., 2d man's share ; 13.856+ in., 3d man's share. 336. II 1 I 12 II 2 I 49 II 3 I 36 II 4 I 247 || 5 | 179 || 6 | 364 336. I 7 I 439 II 8 I 3072' } 337. || 1 1 2.028+ I 2 | 12.0016+ 337. 1 3 I .232+ 1 4 | 27.0002+ || 5 | .729+ || 6 | .015 337. II 7 I .188+ II 8 | 4.339+ || 1 j j || 2 J |- i| 3 | 3||| 4 [Tj 337. II 5 I I II 6 I ^^ II 7 I fi II 8 | jj II 9 I 1 987+ || 10 | 3.83 + 338. i 1 I 27 ft. II 2 I 19 ft. long ; 2166 sq. ft. area. || 3 | 36 ft., 338. i leng-th of side. || 4 | 8.57+ it. || 5 I 9.77+ ft., length ; 338. II 19.54+ ft., height. || 6 | 10.125 cu. ft. || 7 | 45cts. per yd. ; 338. II 2025 yd. j 9 | 641b. || 10 | 8 ft., length of side. || 11 | 8 338. II 12 I $1331 II 13 | 12 in. long; 6 in. wide; 1 in. thick. 339. II 14 I 24 ft. long ; 20 ft. wide ; 9 ft. deep. i| 15 | 20 ft. 339. II 16 I .54+ in., 1st woman's share; .69+ in., 2d woman's 339. II share; .99+ in., 3d woman's share; 3.77+ in., 4th 339. II woman's share. || 340. || 1 | 89 || 2 | $80 || 341. || 3 | $396 341. 1 4 I $1550 II 5 I 171 rd. || 6 | 201ft. || 342. || 1 | 5 miles. 342. II 2 I $2 II 3 I I inches. || 4 | 15, 18, 21, 24, 27, 30, S3 343. II 1 I 2730 II 2 | 226 last term ; $64.96 whole. || 3 | 791imi 343. II 4 I 10 mi. 7 fur. 27 rd. 1| yd. || 344. i| 1 | 5551 344. i 2 I 13 da. ; 312 m. i| 3 | 6 || 346. || 1 | 1 || 2 | 3125000 ANSWERS. 4(53 346. S 3 I ^V I 4 I $100000000000 || 5 | $3200 || 6 | $54000 346. 1 1 I 8327.68 || 8 | $595,508 1| 347- || 1 | 118081 1 2 1 2044 347. B 3 I 11184810 II 4 | $42949072.95 || 348. || 5 | 938249- 348. i 9224- ships. || 355. || 16 | $4.4954 || 359. || 22 | 4166.40 359. I 23 I 9f II 24 | 36 ft. | 25 | 1770 || 360. || 26 | 2if 360. j 27 1 4.33^ | 28 | 100 | 29 j 5500 yd. || 30 | 117 || 31 |^ 360. 1 32 I Ucts. II 33 | $56 || 34 | $5600 J 35 | 7^ cts. || 36 | 360. II 843 I 37 I $246.75 gain. || 361. |1 38 1 5hr. 27min. 16i\sec. 361. 11 39 I 40 II 40 I 10 II 41 I 36 days. || 42 | l-J-f 361. II 43 I 60 = 1st part ; 100 = 2d part ; 140 = 3d part ; 361. I 180 = 4th part. || 44 j 16| in. || 45 | 25\rao. || 46 | $2.20 361. J 47 I $12 II 48 | $1.20 || 362. || 49 | 78.652 -f- discount. 362. 1 50 I $129.60 J 51 | $19,375 most advantageous on time» 362. 1 52 I 292.823 gain || 53 | 122.70 = A's share ; 163.00 - 362. I B's share ; 196.32 = C's share. || 64 j $2317.15 = A's ; 362. II $1853.72 = B's ; $2317.15 = C's ; $2780.58 = D's. 362.' II 55 I $95.10 = A's ; $95.10 = B's ; $133.14 = C's ; 362. B 152.16 = D's. J 56 | 7ioz. 1 57 | 8| da. || 58 | 17 times. 363. I 59 I 4}f da. 1 60 | 5 mo. 24 da. || 61 | 68 da. J 02 | 126 gal. 363. i 63 I $172.78 loss on stocks. J 64 | $3312.417+ 1 65 | $42.60 364. I 66 I 4 yd. || 67 | 140 miles. | 68 | $2, the 1st ; $6, the 364. J second. J 69 | 100 || 70 ] $3825 || 71 | $144 gain by bor- 364. II rowing. J 72 | $36000 || 73 | 41.183-f B 36.5. | 74 | Tiie 365. I second, 10 days after the 3d ; this 1st, 8 days after the 365. J 2d, or 18 days after the 3d. j] 75 | $6890 || 76 | 100 A., 484 ANSWE-HS. 1st Co. ; 88A., 2dCo. ; $7 per acre. || 11 \ Uda., or March 16th. || IS | 512 slabs ; $302.22| |1 19 \ 350, A's ; 210, C's ; 291.50, B's ; 175, D's ; 122.50, E's. || 80 | 72 81 I 5hr. 20min. p.m. || 82 | S0.66f | 83 | 24 chickens and 36 turkeys. || 366. || 84 | 8 days. |] 85 | 1797.50, 1st, 2157, 2d ; 2516.50, 3d. || 86 | |640 stock, and $120 gain 2d. ; $960 stock, and $180 gain, 1st. || 87 | 49.945+ ft. f wk. II 89 I llf hr. ; 1341- miles. || 90 | 533^-, A's ; 888f, B^s ; 177J, C's. || 91 | 36^ days. 367. 92 84485.006+ ft. || 93 | $206.06+ in favor of 1st inyest. 94 I 23599680 cu. yd. || 95 | 4646.363 + 96 555.017+, 1st; 4354.717+, 2d; 4304.663 + , 3d; 5781.263+, 4th; 4004.338 + , 5th. 97 I 2160 98 I $564, A's ; $423, B's. || 368. || 99 | $31 || 100 | 8hr. 101 I $30 com. diff. ; $246f cost. || 102 | $14461.50^ 103 I 97Jlb. II 104 I f ct.,cost; ^ct. sold for; -^oCt., gain on each ; 80 eggs sold. 105 I 84 years old. 106 I 942.48+ cubic feet. 107 I 155A. 311. 38.72 P. 108 I $365,837+ || 109 | 6/g-bours. || 110 | 5 inches 111 I $4006.54+ II 371. i 2|36A.||3|5A. IK 15P. || 4 371. II 135 A. II 372. II 1|437A. 2R. 34.32+P. II 2 1 291A. 2R. 16P 372. B3 35 A. OR. 25 P. II 4 1 20A. 11 5 1 40A. || 6 1 15A. 372. n 24A. IR. 8 P. II 8 1 26 A. 3R. 20 P. 5 sq. yd. ANSWERS. 465 372. II 9 i 120 feet. || 373.. || 2 | 21 A. OR. 39.824P. || 3 j 373. II 921.875 sq.ft. || 4 | 704.125 sq. yd. || 5 | GOA. 3R. 12.8P. 373. II 6 I 270 A. IR. 24 P. || 374. || 2 | 584.3376 || 3 | 125.664 374. il 4 I 179.0712 || 1 | 50 || 2 | 7418 || 3 | 4300.8354- 71. II 1 I 113.0976 II 2 I 19.035 || 3 j 153.9384 || 375. J 4 j ;75: II 1.069+ II 5 I 20 A. OR. 16.9984 P. || 1 | 113.076 375. II 2 I 615.7536 11 3 | 4071.5136 || 4 | 196996571.722104 sq. mi. 376. II 2 I 268.0832 || 3 | 2144.6656 cu. in. || 4 | 259992792- 376. II 082.6374908 || 5 | .9047808 cu. ft. || 377. || 1 | 9100 sq. ft. 377. II 2 I 1440 sq. ft. || 2| 110592 cu. in. || 3 | 42f cu. ft. 377. II 4 I 315f|gal. || 5 | 13820cu. ft. || 378. || 1 | 2513.28 378. II 2 I 233.33Jsq. ft. || 3 | 2827.44 sq. in. || 4 | 6283.2 sq. ft. 379. II 2 I 36442.56 || 3 | 13571.712 || 4 j 9650.9952 379. II 5 I 7363.125 || 2 | 4380 || 3 | 2484 || 4 | 5620 380. II 5 I 5760 || 6 | 14400 || 7 | 1800 || 2 | 9160.9056 380. II 3 I 8659.035 || 4 | 2827.44 || 382. || 2 | 32.4938 inches. 382. II 3 I 28.2574 in. || 383. || 1 | 197.459+ gal. wine. || 2| 383. II 136.9209+ gallons wine; 112.7583+ gallons beer. 383. II 3 I 148.3772+ gal. wine. || 385. jj 1 | 401b. || 2 | 251b. 38.5. II 3 I 501b. II 4 j 201b. || 5 | 401b. || 6 | lin.; l^in.; 2 in.; 4 in. 386. II 7 I 641b. || 8 | 1501b. || 388. | 1 | 60 lb. || 2 | 401b 388.11 3 I 251b. II 1 I 7ift. || 389. || 2 | IJft. || 1 | 40 Ih. 389. II 2 I 1001b. II 3 | 601b. || 390. || 1 j 576 ] 2 | 2250 390. II 3 I lOGGf II 4 | 3000 || 391. || 1 | 2592001b. || 2 | 1.47+ lb. 392. 1 3 I 1.15+ lb. II 4 | 1.2 in. || 393. || 1 | 3G9fi ft. ; 2316ft. 466 ANSVVEUS. 394. Il-^i 3G18fft. ; 482ift. II 3 |223f|lft.i| 4 1 2^ sec. nearly. 394. II H 1608i-ft. space ; 32 If velocity. || 6 1 2 mi. 4984i|3ft. 394. ni 164.69 J II 8 1 100.52iLft. II 9 1 397151 ft; 4{|| sec. :?94. II 10 1 160| ft. = velocity ; 402^V¥fi ft- II 11 1 197J-^n m. II 12 1 77;-f|see. II 13 1 1904i2^«3 ft- = height; 101511 :: 394. II time of ascent. || 14 | 94jV3sec. II 15 1 1447.5 395 II 16 1 61.24 sec. II 17 1 14.28+ sec. || 18 1 15050-ii|rt. 395. II 19 1 12 sec; 2316 ft. II 396. 1 r| 8.857 II 2|38l|fcu. fi. 397. II 3 1 .980 II 4 1 2 ft. 11.388 in. II 5 | 190 T '. 709 11). II () i 2.75 397. II 7 1 7.234 + II 8 1 .786+11 9 1 .875 || 10 1 177 11). 5(»z 397. II 11 1 1.103 II 398. i 1 1 3.49 qt. || 2 | 37.5 II 3 1 2.4r) oT 398. 1 4 1 JA = .5319. ^lie Rational ^erics of ^taadard ^chool-^ooks, PUBLISHED BY A. S. BABNES & COMPANY, ^ 111 & 113 WILLI A:fl STREET, KEW TOBK. This Seuies embraces nbout Three Hundred Volnmes of Standard Ediicationnl Works, composing the most complete and uniformly meritorious collection of text- books ever published by a single firm. TuK Sekies is complsU, covering every variety and grade of science and literature, from the Primer which guides the lisping tongue of the infant, to the abstruse and difficult *' West Point Course.'' The Seeiks is uniformlj excellent. Each volume, among so many, maintains its own standard of merit, and assists, in its place, to round the perfect whole. The Sbeies is know)x and popularly used in every section of Via United States, and b'j evyry cluas of citizenJt, representing all shades of political opinion and religious be- lief. In proof of this^ it 1^ only necessary to name the folloxring popular works, with which every ons is familiar, and which fairly represent the whole : PARKER & WATSON'S Readers. &c, DAVIES' Course of Mathematics- WILLARD'S Course of History. PECK'S GANOT'S Natural Philosophy. STEELE'S 14- Weeks in each Science. JARVIS' Physiology and Health. WOOD'S Text-Books in Botany. SMITH'S Orthography and Etymology. BOYD'S Course in English Literature. MONTEITH'3 & McNALLY'S Geog's. CLARK'S Diagram English Grammar. " P., D., & S.'s" System of Penmanship. ANDREWS & STODDARD'S Latin. CROSBY'S Greek Series. WORMAN'S German Series. PUJOL'S French Class-Book. ROOT'S (GEO F.) Sch. Music- Books. MANSFIELD'S Political Manual. THE SCHOOL-TEACHER'S LIBRARY. Twenty-five Volumes. Who would knov/ mors of this unrivaled Series should consult for details, 1. THE nESCJtJrriri: C at JLLOGUE-frco to Tcnchcm; others, 5 ccnt.s. S. THE ILL U ST It AT ED EDUCATIONAL Ji U L LETI N— Vcriodical organ of the Publishers. Full of instruction for Teachers. Subscription, 20 cciits. Sample free. TERMS OF EXAMlNATiOK. — We propose to supply any teacher who deBircs to examine tcvt-booka, with a vinw to introduciion, if a2>2)rvvcd, with sample copies, on receipt of onk-ii.vlf the ])rico annexed (in Catalogue), and Iho books will be sent by mail or express without expense to the purchaser. Books marked thus (*) are ex- cepted from this offer. TERMS OF INTRODUCTION.— The Publishers are prepared to make special and vory favorable terms for first introduction of any of llie Natioxal Seuies, and will furnish the reduced introductory price-list to teachers whose application presents evi- dence of good faith. Teachers desiring to avail themselves of any of the prirrilefjcif of the prof ession^ il not known to the Publishers, should mention the name of one or more of their Trus- tees or Patrons, as pledge of good faith. For farther information, address the ruhlishers. The JVatio7ial Se^-ies of Standard Sc?iool-^ooks, ORTHOGRAPHY AND READING. JSTATIOFAL SERIES OP READERS AND SPELLERS, BY PAEKEE & WATSOK The National Primer $25 National First Reader 38 National Second Reader 63 National Third Reader 95 National Fourth Reader 1 50 National Fifth Reader 1 88 National Elementary Speller 25 National Pronouncing Speller 45 Tliis unrivaled series has acquired for itself during a very few years of publication, a reputation and circulation never before attained by a series of school readers in the Bame space of time. No contemporary books can be at all compared -with them. Tho average annual increase, in circulation exceeds 100,000 volumes. We cliallengo rival publishers to show such a record. The salient features of these works which have combined to render them so popular may be briefly i-ecapitulated as follows ; 1. THE WORD METHOD SYSTEM— Tins famons progressive method for young children originatad and was copyrighted with these books. It constitutes a process by which the beginner with toords of one letter is gradually introduced to additional lists formed by prefixing or affixing single letters, and is thus led almost insensibly to tho mastery of tlio more difficult constructions. This is justly regarded as one of tluj most striking modern improvements in methods of teaching. 2. TREATMENT OF PEONUKOIATION.— The wants of the youngest scholars in this department are not overlooked. It may be said that from the first lesson the student by this method need never be at a loss for a prompt and accurate render- ing of every word encountered. 3. ARTICULATION AND ORTEO^T are recognized as of primary in*. Jtortanca ^t * ^ 3 (0»«>C) TAe JVatioiiut Series of Standard Sc?ioot-2)Ooks, ORTHOGRAPHY AND READING-Continued. 4. PUNCTUATION is inculcated by a scries of interesting reading lessong. the simple perusal of which suffices to fix its principles indelibly upon tke mind. 5. ELOCUTION. Each of the higher Readers (3d, 4th and 5th) contains elaborate, Echolarly, and thoroughly practical treatises on elocution. This feature alone has secured for the series many of its warmest friends. 6. THE SELECTIONS are the crowning glory of the Eeries. Without exception it may be said that no volumes of the same size and character contain a collection bo diversified, judicious, and artistic as tliis. It embraces the choicest gems of English literature, so arranged as to afford the reader ample exercise in every department of style. So acceptable has the taste of the authors in this department proved, not only to the educational public but to tlio reading community at large, that thousands of copies of the Fourth and Fifth Headers have found their way into public and private libraries throughout the country, where they are in constant use as manuals of liter- ature, for reference as well as perusal. 7. ARRANGEMENT. Tlic exercises arc so arranged as to present constantly al- ternating practice in the different styles of composition, while observing a definite plan of progression or gradation throughout the whole. In the higher books the ar- ticles are placed in formal sections and classified topically, thus concentrating the in- terest and inculcating a principle of association likely to prove valuable in subsequent general reading. 8. NOTES AND BIOGRAPHICAL SKETCHES. Tliese are full and adequate to every want. The biograpliical sketches present in pleasing style the history of every autlior laid under contribution. 9. ILLUSTRATIONS. These are plentiful, almost profuse, and of the highest character of art They are found in every volume of the scries as far as and including the Third Reader. 10. THE GRADATION is perfect E.ich volume overlaps its companion pre- ceding or following in tlie series, so that the scholar, in passing from one to another, is barely conscious, save by the presence of the new book, of the transition. 11. THE PRICE is reasonable. The books wore not trimmed to the minimum of size in order tliat the publishers might be able to denoniinate them " the cheapest in the market," but were made large enoxcgh to cover and suffice for the grade indi- cated by the respective numbers. Thus the child is not compelled to go over his First Reader twice, or be driven into the Second before he is prepared for it The compe- tent teachers who compiled the series made each volume just what it should be, leav- ing it for their brethren who should use the books to decide what constitutes true cheapness. A glance over the books will satisfy any one that the same amount of matter is nowhen. furnished at a price more reasonable. Besides which another con- sideration enters into the question of relative economy, namely, the 12. BINDING. By the use of a material and process known only to themselves, in common with .ill the publications of this house, the National Readers are warranted to out-last any with which they may be compared—the ratio of relative durability be- ing in their favor as two to one. ^ The JVatto7iat Seizes of Standm'd Sc?ioot-!Books, SCHOOL-KOOM CARDS, To Accompany the National headers. ^-•-♦^••^- Eureka Alphabet Tablet *i 50 Presents tha alphabet upon tlie Word Method System, by which the child will learn the alphabet in nine days, and make no small progress in reading ana spelling in the same time. National School Tablets, lo Nos *7 60 Embrace reading and conversation;'.! exercises, object and moral les- sons, form, color, &c. A complete set of these large and elegantly illus- trated Oards will embellish the school-room more than any other article of furniture. READING Fowle's Bible Reader $l 00 The narrative portions of the Bible,, chronologically and topically ar- ranged, judiciously combined with selections from the Psalms, Proverbs, and other portions which inculcate important moral lessons or the great truths of Christianity. The embarrassment and difficulty of reading the Piide itself, by course, as a class exercise, are obviated, and its use made feasible, by this means. North Carolina First Reader 50 North Carolina Second Reader 75 North Carolina Third Reader i oo Prepared expressly for the schools of this State, by C. 11. Wiley, Super- intendent of Common Schools, and F. M. Hubbard, Professor of Litera- ature in the State University. Parker's Rhetorical Reader i oo Designed to familiarize Readers with the pauses and other marks in general use, and lead them to the practice of modulation and inflectien of the voice. Introductory Lessons in Reading and Elo- cution 75 Of similar character to the foregoing, for less advanced classes. School Literature l 50 High Admirable selections from a long list of the world's best writers, for ex- ercise in reading, oratory, and composition. Speeches, dialogues, and model letters represent the latter department. 5 90 77ie JVational Series of Sta7idard Schoot-^ooks, ORT HOGRAP HY. SMITH'S SERIES BuppHcs a Bpeller for every class in graded schools, and comprises the most coin> " pleto and excellent treatise on English Orthography and its companion branches extant. t. Smith's Little Speller $20 Finfit Itonnd in the Ladder of Learning. 2. Smith's Juvenile Definer 45 Lessons composed of familiar words grouped with reference to similar Bis are to be found on the page opposite the map itself, and each book is complete in one volume. The mechanical execution is unrivalled. Paper and printing are everything that could be desired, and the bind- kig is— A. S. Barnes and Company's. Ripley's Map Drawing $1 25 This system adopts the circle as its basis, abandoning the processes by triangulation, the square, parallels, and meridians, &c., which have been proved not feasible or natural in the development of this science. Suc- cess seems to indicate that the circle '• has it." National Outline Maps Foi thfi school-room walls. In preparation. 9 The National Seines of Standard Schooi-^ooks* MATHEMATICS. AKITHMETIC. 1. Davics' Primary Arithmetic . . .$ 2o 2. Davies' Intellectual Arithmetic 40 3. Davies' Elements of Written Arithmetic 50 4. Davies' Practical Arithmetic 1 00 Key to Practical Arithmetic *1 00 5. Davies' University Arithmetic 1 50 Key to University Arithmetic *1 50 ALGEBRA. 1. Davies' New Elementary Algebra 1 26 Key to Elementary Algebra *1 25 2. Davies' University Algebra 1 60 Key to University Algebra *1 GO 3. Davies' Bourdon's Algebra 2 25 Key to Bourdon's Algebra *2 25 GEOMETRY. 1. Davies' Elementary Geometry and Trigoncrtietry . 1 40 2. Davies' Legendre's Geometry 2 25 3. Davies' Analytical Geometry and Calculus .... 2 50 4. Davies' Descriptive Geometry 2 75 MENSURATION. 1. Davies' Practical Mathematics and Mensuration . . . 1 40 2. Davies' Surveying and Navigation 2 50 3. Davies' Shades, Shadows, and Perspective . . . . 3 75 MATHEMATICAL SCIENCE. Davies' Grammar of Arithmetic * 50 Davies' Outlines of Mathematical Science *1 00 Davies' Logic and Utility of Mathematics *1 50 Davies & Peck's Dictionary of Mathematics *3 75 10 7%e JVa^onat Series of Sta7idard Sc?iool'-^ooks, DAYIES' NATIONAL COUESE of MATHEMATICS. ITS RECORD. In claiming for this series the first place among American text-books, of whatever class, the Publishers appeal lo the magnificent record which its volumes have earned during the thirty-five years of Dr. Charles Davies' mathematical labors. The unre- mitting exertions of a life-time have placed the modern series on the same proud emi- nence among competitors that each of its predecessors has successively enjoyed in a course of constantly improved editions, now rounded to their perfect fruition — for it Bccms indeed that this science is susceptible of no further demonstration. During the period alluded to, many authors and editors in this department have started into public notice, and by borrowing ideas and processes original with Dr. Davies, have enjoyed a brief popularity, but are now almost unknown. Many of tho series of to-day, built upon a similar basis, and described as "modem books," are destined to a similar fate ; while the most far-seeing eye will find it difficult to fix tho time, on the basis of any data afforded by their past liistory, when these books will cease to increase and prosper, and fix a still firmer hold oa the affection of every educated American. One cause of this unparalleled popularity is found in the fact that the enterprise of the author did not cease with the original completion of his books. Always a practi- cal teacher, he has incorporated in his text-books from time to time tho advantages of every improvement in methods of teaching, and every advance in scioncc. During nil the years in which he has been laboring, he constantly submitted his own theorica and those of others to the practical test of the class-room — approving, rejecting, or modifying them as the experience thus obtained might suggest. In this way he has been able to produce an almost perfect scries of class-books, in which every depart- ment of mathematics has received minute and exhaustive attention. Nor has he yet retired from the field. Still in the prime of life, and enjoying a rlpo experience which no other living mathematician or teacher can emulate, his pen is ever ready to carry on the good work, as the progress of science may demand. AVit- ness his recent exposition of the " Jletric System," wTiich received the ofiicial cu- dorsemeat of Congress, by its Committee on Uniform Weights and Measures. Davies' System is tiii! ackxowledqed National Standaeo fo3 the United States, for the following reasons : — 1st. It is the basis of instruction in the great national Gchools at West Point aad Annapolis. 2d. It has received tho quasi endorsement of the National Congress. 8d. It is exclusively used in the public schools of the National Capital. 4th. The officials of the Government use it as authority in all cases involving mathe- matical questions. 5th. Our great soldiers and sailors commanding the national armies and navies were educated in this system. So have been a majority of eminent scientists in this country. All these refer to " Davies" as authority. 6th. A larger number of American citizens have received their education from thia than from any other series. 7th. The series has a larger circulation throughout the wholo country than any othei", being extensively imd in every State in the Union, The JVatio7ial Series of Standat'd SchoolSooks, MATHEMATICS-ContinuBd. ARITHMETICAL EXAMPLES. Reuck's Examples in Denominate Numbers % 50 Reuck's Examples in Arithmetic .... l 00 These volumes differ from the ordinary arithractie in their peculiarly practical character. They are composed mainly of examples, and afford the most severe and thorough discipline for the mind. While a book which should contain a complete treatise of theory and practice would ba too cumbersome for every-day use, the insuftlcicucy of j»aciicaZ examples has been a source of complaint HIGHER MATHEMATICS. Church's Elements of Calculus 2 50 Church's Analytical Geometry 2 so Church's Descriptive Geometry, with Shades, Shadows, and Perspective 4 50 These volumes constitute the " West Point Course" in their several departments. Courtenay's Elements of Calculus .... 3 25 A work especially popular at the South. Hackley's Trigonometry 3 oo With applications to navigation and surveying, nautical and practical geometry and geodesy, and logarithmic, trigonometrical, and nautical tables. THE METRIC SYSTEM. The International System of Uniform Weights and Measures must hereafter be taught in all common-schools. Professor Charles Davies is the official exponent of the system, as indiaated by the following resolutions, adopted by the Committee of the House of Representatives, ona " Uniform System of Coinage, Weights, and Measures^" February 2, 18G7 :— Resolved, That this committee has observed with gratification the efforts made by the editors and publishers of several mathematical works, designed for the use of com- mon-schools and other institutions of learning, to introduce the Metric System of VVeights and Measures, as authorized by Congress, into the system of instruction of the youth of the United States, in its various departments ; and, in order to extend further the knowledge of its advantages, alike in public education an* in general use by the people, Le it further resolved, That Professor Charles Davies, LL.D., of the State of New York, be requested to confer with superintendents of public instruction, and teachers of schools, and others interested in a reform of the present incongruous system, and, by lectures and addresses, to promote its general introduction and use. The official version of the Metric System, as prepared by Dr. Davies, may be found in the Written, Practical, and University Arithmetics of the Mathematical Scries, and ta also published separately, price postpaid, ylre cent3. 12 The JVational Se?^les o/ Standa7*d SchoolSooks, HI ST OB Y. Monteith's Youth's History, $75 A History of the United States for begiuners. It is arranged upon the catcchetic.il plan, with illustrative maps and engravings, review questions, dates in parentheses (tliat their study may be optional with the younger class of learners), and interesting liiographical Sketches of all persons who have been prominently identified with the history of our country, Willard's United States, School edition, ... i 25 Do. do. University edition, . 2 25 The plan of tliis standard work is chronologically exhibited in front of the title-page ; the Maps and Sketches are found useful assistants to the memory, and dates, usually so difficult to remember, are bo systematically arranged as in a great degree to ol)viate the dilhculty. Candor, impai'- tiality, and accuracy, arc the distinguishing features of the narrative portion. Willard's Universal History, 2 25 The most valuable features of the " United States" arc reproduced in this. The peculiarities of \\\z work arc its great conciseness and the prominence given to the chronological order of events. The margin marks each successive era with great distinctness, so that the pupil re- tains not only the event but its time, and thus fixes the order of history firmly and usefully in his mind. Mrs. AVi Hard's books are constantly revised, and at all times written up to embrace important historical events of recent date. Berard's History of England, i ^^ By an aulhuress well known for the success of her TTistory of the United States. The social life of the English people is felicitously intei-woven, as iu fact, with the civil and military transactions of the realm. Ricord's History of Rome, i 25 Possesses the charm of an attractive romance. The Tables with which this history abounds are introduced in such a way as not to deceive the inexperieaced, while adding materially to the value of the work as a reli- able index to the character and institutions, as well as the history of the Soman people. Banna's Bible History, l 25 The only compendium of Bible narrative which affords a connected and chronological view of the important events there recorded, divested of all superfluous detail. Alison's History of Europe 2 50 An abridgment f )r Schools, by Gould, of this great standard work, covering the eventful period from A. U. 17S9 to 18i5, being mainly a his- tory of the career of Napoleon. Marsh's Ecclesiastical History, i gb Questions to ditto, 73 Affording the History of the Church in all ages, with accounts of the pagan world during Biblical periods, and the character, rise, and progress of all Religions, as well as the v:irious sects of the worshipers of Christ. The work is entirely non-sectarian, though sh-ictly catholic. 13 The A^utioiial Sadies of Standard Sc?ioot-7iooks, PENMANSH l"p^ 1 * ^ Beers' System of Progressive Penmanship. Per do?eA .$2 50 This "round hand" system of penmanship in twelve nnrabcre com> mends itself by its simplicity and thoroughness. The first fojir numbers are primary books. Nos. 5 to 7, adyancod books for boys. Nos. 8 to 10 advanced books for girls. Nos. 11 and 12, ornumenfcil penmanship. These books are printcil from steel plates (engraved by McLccs), and are unexcelled fh mechanical execution. Large quantities are aunually sold. Beers' Slated Copy Slips, per set *50 All beginners should practice, for a fww weeks, slate exercises, familiar* izing them with the form of the letters, the motions of the hand and arm, &c., &c. These copy slips, 32 in number, supply all th« copies found in a complete series of writing-books, at a trifling cost Fulton & Eastman's Copy Books, per dozen l 50 A series for the economical, — complete in three nunihnrs. (1) Elemen- tary Exercises : (2) Gentlemen's Hand : (3) Ladies' Hand. Fulton k Eastman's Chirographic Charts, 2 Nos., per set . . . *5 00 To enibcllisli the school-room walls, and furnish class exercise in the elements of Penmanship. DRAWING. Clark's Elements of Drawing 1 00 Containing full instructions, with appropriate designs and copies for a complete course in this graceful art, from the first rudiments of outline to the finished sketches of landscape and scenery. Fowle's Linear and Perspective Drawing 60 For the cultivation of the eye and hand, with copious illustrations and directions which- will enable the unskilled teacher to learn the art himself while instructing his pupils. Monk's Drawing Books— Six Numbers, each, .* 40 A series t)l proirressive Drawing Books, presenting copy and blank on opposite pages. Tlie copies are f ic-simiics of the best imported litho- graphs, the oHginals of which cost from 50 cents to $1..50 each in the print-stores, llach book contains eleven largo patterns. No. 1. — Ele- mentary stmlics ; No. 2.— Studies of FoliHge; No. 3. — landscapes; Na 4.— Animals, I. ; No. 5. — Animals, IL ; No. U.— Marine Views, &c Ripley's Map Drawing 1 25 One of the most efTicient aids to the acquirement of a knowledge of geography is the practice of map drawing. It is useful for the same rea- son that th(! best exercise in orthography is the wnting of difficult words. Sight comes to the aid of hearing, and a double impression is produced upon the memory. Knowledge bocomos less mechanical and more intui- tive. The student who has sketched the outlines of a country, and dotted the important places, is little likely to forget either. The impression pro- duced may be compared to that of a traveler who has been over the ground — while more comprehensive and accurate la detail 14 The JVati07ial Series of Standard Schoot-Sooks, ELOCUTION. Northend's Little Orator *60 Contains simple and attractive pieces in prose and poetry, adapted to the capacity of ciiildren under twelve years of age. Northend's National Orator *l lo About one hundred and seventy choice pieces happily arranged. Tho design of the author in making the selection has been to cultivate versa- tility of expression. Northend's Entertaining Dialogues . • . .*l lo Extracts eminently adapted to cultivate the dramatic faculties, as well as entertain an audience. Zachos' Analytic Elocution 1 25 All departments of elocution — such as the analysis of the voice and the sentence, phonology, rhythm, expression, gesture, «&c. — are hero arranged for instruction in classes, illustrated by copious examples. Sherwood's Self Culture l 25 Self culture in reading, speaking, and conversation — a very valuable treatise to those who would perfect themselves in these accomplishments. BOOK-KEEPING. Smith & Martin's Book-keeping l lo Blanks to ditto *60 This work is by a practical teacher fcnd a practical book-keeper. It is of a thoroughly popular class, and wCl be welcomed by every one who loves to see theory and practice oombined in an easy, concise, and methodical form. The Single Entry portion is well adapted to supply a want felt in nearly all other treatises, which seem to be prepared mainly for the use of wholesale merchants, leaving retailers, ^ mechanics, farmers, &c., who transact the greater portion of the business of the country, without a guide. The work is also commended on this account for general use in Young Ladies' seminaries, where a thorough grounding in the simpler form of accounts will be invaluable to the future housekeepers of the nation. The treatise on Double Entry Book-keeping combines all the advan- tages of the most recent methods, with the utmost simplicity of applica- tion, thus affording the pupil all the advantages of actual experience in the counting-house, and giving a clear comprehension of the entire sub- ject through a judicious course of mercantile transactions. The shape of the book is such that the transactions can be presented as in actual practice ; and the simplified form of Blanks, three in number, adds greatly to the ease experienced in acquiring the science. 15 t.' i% '