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University of California • Berkeley 
 
 The Theodore P. Hill Collection 
 Early American Mathematics Books 
 
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NEW 
 
 UNIVERSITY ARITHMETIC, 
 
 EMBRACING THE 
 
 SCIEJ^CE OF I^^UMBERS, 
 
 AND TUEIR 
 
 APPLICATIONS ACCORDIXG TO TIIK MOST I3IPKOVED METHODS 
 
 OF 
 
 ANALYSIS AND CANCELLATION. 
 
 BY 
 
 CHARLES DA VIES, LL. D., 
 
 AUTHOR OF rnniAKT, INTELLECTUAL, AND SCHOOL ARITHMETICS ; ELEMENTART 
 ALQEBUA ; ELEMENTARY GEOMETKY ; PRACTICAL MATHEMATICS : ELEMENTS 
 OF SURVEYING ; ELEMENTS OF ANALYTICAL GEOMETRY ; DESCRIPTIVE 
 GEOMETRY ; f-UADES, SHADOWS, AND PERSPECTIVE ; DIFFER- 
 ENTIAL AND INTEGRAL CALCULUS ; AND LOGIC AND 
 UTILITY OF MATHEMATICS. 
 
 NEW YOrvK: 
 
 PUBLISHED BY A . S. BARNES & CO., 
 No. 51 & 53 .1 O I! \ - L^ T 11 E E T . 
 1858. 
 
Entered according to act of Congress, in the year eighteen hundred and fifty-six, 
 
 BY CnATtLES DAVIES, 
 In the Clerk's OfRce of the District Court of the United States, for the Southern 
 
 District of New York. 
 
 JONES & DENYSE, 
 STEREOTYPErtS AND ELECTROTYPERS, 
 1S3 AViUiam-Strect. 
 
PEEFACE. 
 
 Science, in its popular signification, means knoAA'ledga 
 reduced to order ; that is, knowledge so classified and arranged, 
 as to he easily remembered, readily referred to, and advanta- 
 geously applied. 
 
 Arithmetic is the science of numbers. It is the foundation 
 of the exact and mixed sciences and a knowledge of it is an 
 important element either of a liberal or practical education. 
 While Arithmetic is a science in all that concerns the properties 
 of numbers, it is an art in all that relates to their practical 
 application. 
 
 It is the first subject in a well-arranged course of instruction 
 to which the reasoning powers of the mind are applied, and i? 
 the guide-book of the mechanic and man of business. It is the 
 first fountain at which the young votary of knowledge drinkf 
 the pure waters of intellectual truth. 
 
 It has seemed, to the author, of the first importance that this 
 subject should be well 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, embracing luur 
 books enthled, Primary Arithmetic ; Intellectual Arithmetics 
 School Arithmetic; and University Arithmetic — the latter of 
 which is the present volume. 
 
 PuiMAny AraTKMETic. This first book is adapted to the 
 capacities and wants of young children. Sensible objects are 
 employed to illustrate and make familiar the simple combi- 
 nations and relations of numbers. Each lesson embraces one 
 combination of numbers, or one set of combinations. 
 
VI PEEFACE. 
 
 Intellectual Arithmetic. This work is designed to pro- 
 sent a thorough analysis of the science of numbers, and to form 
 a complete course of mental arithmetic. It is thought to be 
 accessible to young pupils by the simplicity and gradation of 
 its methods, and to be particularly adapted to the wants of 
 advanced students, as the attempt has been faithfully made to 
 give the subjects of which it treats a scientific arrangement 
 and logical connection in all the higher methods of arithmeti- 
 cal analysis. 
 
 School Arithmetic. Great pains has been taken in the 
 preparation of this book to combine theory and practice ; to 
 explain and illustrate principles, and to apply them to the 
 common business transactions of life — to make it einjyhatically 
 a 2y''(^ctical 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 upon the U7tit or 
 unity as the bane of all numbers, whether integral or frac- 
 tional — to reason with constant reference to this base, and thus 
 make it the key to the solution of all arithmetical questions. 
 It is thought, that the language used in the statement of princi- 
 ples, in the definitions of terms, and in the explanation of 
 methods, will be ibund to be clear, exact, brief and compre- 
 hensive. 
 
 Univeksity Auitiimetic. This work is designed to answer 
 another object. Here, the entire subject is treated as a 
 science. The scholar is supposed to be familiar with the simple 
 operations in the four ground rules, and with the first principles 
 of fractions, 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 carefully taught, and the different .significations of which the 
 figures are susceptible, depending on the manner in whicli they 
 are written, are fully explained. It is shown, for example, 
 that the simple numbers in which the value of the unit 
 
PREFACE. Vll 
 
 increases from right to left according to the scale of tens, and 
 the Denominate or Compound numbers in which it increases 
 according to a varying scale, belong to the same class of num- 
 bers, and that both may be treated under the same rules. 
 Hence, the rules for Notation, Addition, Subtraction, Multipli- 
 cation 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 Arithmetic. 
 
 Li developing the properties of numbers, from their elemen- 
 tary to their highest combinations, great labor has been 
 bestowed in classification and arrangement. It has been a 
 leading object to present the entire subject of arithmetic as 
 forming a series of ilepeiident and coyinccled joropositions : so 
 that the pupil, while acquiring useful and practical knowledge, 
 may at the same time be introduced to those beautiful methoda 
 of reasoning, v.'hich science alone teaches. 
 
 Great care has been taken to demonstrate every proposition — 
 to give a complete analysis of all the methods employed, from 
 the simplest to the most difficult, and to explain fully, the 
 reason of every rule. A full analysis of the science of Num- 
 bers has developed but one law ; viz , ihe law tchich connects 
 all the units of arithmetic with the rinit one, and which points 
 out the relations of these loiits to each other. 
 
 In the Appendix, which treats of Units, Weights and Mea- 
 sures, &c., the methods of determining the Arbitrary Unit, as 
 well as the general law which prevails in the. formation of 
 numbers, are fully explained. 1 cannot too earnestly recom- 
 mend this part of the work to the special attention of Teachers 
 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 scieniific and practical character. In the present 
 edition the matter is presented in a new form ; the arrange- 
 
Vlll PEEFACE. 
 
 mcnt of the subjects is more natural and scientific ; the 
 methods have been carefully considered ; the illustrations 
 abridged and simplified ; the definitions and rules thoroughly 
 revised and corrected ; and a very large number and variety of 
 practical examples have been added. The subjects of Frac- 
 tions, Proportion, Interest, Percentage, Alligation, Analysis, and 
 Weights and Measures, present many new and valuable fea- 
 tures, 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 school room is the tribunal, and the intelligent 
 and practical teacher the judge, before whom all text-books 
 must stand or fill. 
 
 The terms, " Cause and Effect," used in developing the sub- 
 ject of Pi'oportion, were supposed by the author, at tlie time of 
 their introduction into his works, to be common literary property. 
 But he has since been informed, that Prof H. N. Robinson, 
 the distinguished author of a full course of Mathematics, chiims 
 to have been the first who made a practical application of these 
 terms, in a tangible form, to Proportion, as published in liis 
 Aritlimetic of 185-i. I have great pleasure, tlierefore, in assifrn- 
 ing to their proper author, the invention of so ingenious and 
 practical a method, and of acknowledging the courtesy of beinw 
 permitted to continue its use in my books, with, however, as I 
 deem, important differences in the definitions and development 
 of the rule. 
 
 FiSHKiLL Landing. ") 
 
 My, 1858. 
 
CONTENTS. 
 
 FIRST FIVE El'LES. 
 
 PAGES. 
 
 Definitions , 13 
 
 Expressing N^umbers 1<, 
 
 Notation and Numeration 15-24 
 
 Formation and Nature of Number? 24—26 
 
 Scales 26 
 
 United States Money 2Y-29 
 
 Integral Units 29 
 
 Properties of the 9's 30 
 
 Reduction 31-35 
 
 Addition 35-44 
 
 Subtraction 44—53 
 
 Multiplication 63-67 
 
 Division 67-83 
 
 Practice .84*-85* 
 
 Longitude and Time 84-86 
 
 Applications in the Four Rules 86-95 
 
 Properties of Numbers 95-97 
 
 Divisibility of Numbers 97-99 
 
 Greatest Common Divisor 99-102 
 
 Least Common Multiple 102-104 
 
 Cancellation 104-108 
 
 COMMON FRACTIONS. 
 
 Definition of, and First Principles 108-1 11 
 
 The six Kinds of Fractions 111-113 
 
 Six Propositions 113-117 
 
 Reduction of Common Fractions 117-124 
 
 Addition of Common Fractions 124—129 
 
 Subtraction of Common Fractions 129-133 
 
 Multiplication of Common Fractions 133-137 
 
 1 
 
COINTENTS. 
 
 PAGES 
 
 Division of Common Fractions 137-141 
 
 Complex Fractions l-il-142 
 
 Applications in Fractions 142-141 
 
 Duodecimals 144-150 
 
 DECIMAL FRACTIONS. 
 
 Definition of Decimals, &c 150-151 
 
 Decimal Numeration Table — First Principles, &c 151-156 
 
 Addition of Decimals 156-15S 
 
 Subtraction of Decimals, 158-160 
 
 Multiplication of Decimals 160-162 
 
 Contractions in Multiplication 162-164 
 
 Division of Decimals , 164-168 
 
 Contractions in Division 168-170 
 
 Reduction of Common Fractions to Decimals 170-171 
 
 Reduction of Denominate Decimals 171-175 
 
 Circulating or Repeating Decimals — Definition of, &c 175-179 
 
 Reduction of Circulating Decimals 179-184 
 
 Addition of Circulating Decimals 184 
 
 Subtraction of Circulating Decimals 184-185 
 
 Multiplication of Circulating Decimals 185-186 
 
 Division of Circulating Decimals 186 
 
 CONTINUED FEACTIONS. 
 
 Definitions and Principles 186-189 
 
 RATIO AND PROPORTION. 
 
 Ratio Defined 189 
 
 Proportion Defined 190 
 
 Simple and Compound Ratio 102 
 
 Simple Proportion, or Rule of Three 193 
 
 Cause and Ellect 198 
 
 Inverse Proportion 199-205 
 
 Compound Proportion 205-209 
 
 Partnership 209-214 
 
OONTEKTS. XI 
 
 PERCENTAGE 
 
 PAGES 
 
 Percentage Defined and Illustrated 214-219 
 
 Simple Interest 219-232 
 
 Compound Interest 282-235 
 
 Discount 235-237 
 
 Banking 237-240 
 
 Bank Discount 240-242 
 
 Commission , 242-245 
 
 Stocks and Brokerage 245-250 
 
 Profit and Loss 250-255 
 
 Insurance 255-257 
 
 Life Insurance 257-259 
 
 APPLICATIONS. 
 
 Endo-wments 259-261 
 
 Annuities 261-263 
 
 Assessing Taxes 263-267 
 
 Custom House Business 267-272 
 
 Equation of Payments 272-275 
 
 Alligation 275 
 
 Alligation Medial 275-276 
 
 Alligation Alternate 276-282 
 
 Coins and Currencies 282-283 
 
 Exchange 283-295 
 
 General Average 295-298 
 
 Tonnage of Vessels 298-300 
 
 PO'WEES AND ROOTS. 
 
 Involution 300-301 
 
 Evolution 301-302 
 
 Extraction of Square Root 302-31 1 
 
 Cube Root 311-317 
 
 AEITHMETICAt PROGRESSION. 
 
 Definition of, <fec 317-319 
 
 Different Cases 31 9-322 
 
 General Example* 322-323 
 
sal CONTENTS. 
 
 GEOMETRICAL PROGEESSION. 
 
 PAGES 
 
 Definition of, <fcc 323 
 
 Cases 323-32G 
 
 Examples 326 
 
 AlfALTSIS. 
 
 Analysis and Promiscuous Examples S27-351 
 
 MENSrKATION. 
 
 Mensuration of Surfaces 351-357 
 
 Mensuration of Volumes 357-362 
 
 Gauging 362-366 
 
 Mecliaiiical Powei'S 366-374 
 
 Questions in Katural Philosophy 374-383 
 
 APPENDIX 
 
 Different Kinds of Units 3S4-3S8 
 
 United States Money 38S-390 
 
 English Money 39(1-393 
 
 Linear Measure 393-394 
 
 Cloth Measure 394-395 
 
 Square Measure 395-396 
 
 Sui-ve3-oi'8' Measure 396 
 
 Cul.ic Measure 396-393 
 
 ■\Viiie Mt-asure 398 
 
 Beer Measure 393 
 
 Dry Measure 399 
 
 Aviiirdu[)ois Weight 400 
 
 Troy Weiglit 401 
 
 Apotliecaries' Weiglit 401 
 
 Measure of Time 402-405 
 
 Circular Measure 406-406 
 
UNIVERSITY ARITHMETIC. 
 
 DEFINITIONS. 
 
 1. A single thing, is called one or a unit. A number is a 
 unit, or a collection of units. 
 
 2. A single thing of a collection, is called the unit or hase of 
 the collection. The primary base of every number is the unit 
 one. 
 
 3. SciKNCE treats of the properties and relations of things: 
 Art is the practical application of the principles of Science. 
 
 4. ARiTiniETic treats of numbers. It is a science when it 
 determines the properties and relations of numbers ; and an 
 art, when it applies principles of science 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, in Arithmetic, is the act of doing some- 
 thing with numbers. The number obtained by an operation ia 
 called a result, or answer. 
 
 1. What is a single thing called ? What is a number 1 
 
 2. What is a single thing of a collection called ? What is the primary 
 base of nvery uiunber 1 
 
 3 Of what does science treat 1 \A''hat is art f 
 
 4. Of what does Aritlimetic treat 1 When is it a science ' When an art' 
 
 5. What is a proposition 1 
 
 6. What is an analysis 1 
 
 7. What is an operation 1 What is the number obtained called ? 
 
14 NOTATIOIC. 
 
 8. A Rule is a direction for performing an operation, and 
 may either be inferi'ed from an analysis, or deduced from a 
 demonstration. 
 
 9. There are five fundamental processes of Arithmetic: 
 Notation and Numeration, Addition, Subtraction, Multiplication, 
 and Division. 
 
 EXPRESSING NUMBERS. 
 
 10. There are three methods of expressing numbers : 
 
 1st. By words, or common language ; 
 
 2d. By letters, called the Roman method ; 
 
 3d. By figures, called the Arabic method. 
 
 BY WORDS. 
 
 11. A single thing is called - - - - One. 
 One and one more - - - - Two. 
 Two and one more - - - _ Tliree. 
 Three and one more - - - - Four 
 Four and one more - - - - Five. 
 Five and one more - - - _ Siix. 
 Six and one more - - - - Seven. 
 Seven and one more - - - _ Fight. 
 
 Eight and one more - - . - Nine. 
 Nine and one more - _ - - Ten. 
 
 &c. &c. &;c. 
 
 Each of the words, or terms, one, ttco, three, four, k,Q.,(kQ\-\oics 
 how many units are taken. These terms are generally called 
 numbers ; though, in fact, they are but the names of num- 
 bers. 
 
 8. What is a rule? How may it, bp deduced 1 
 
 9. How many fundamental processes are there in Arithmetic 1 What 
 are they] 
 
 10. How many methods arc there of expressing numbers 1 What are 
 they] 
 
 11. What docs each of the words, one, iiru, three, &c., denote] What 
 are these words generally called ] What are they, in fact ] 
 
NOTATION. 
 
 15 
 
 NOTATION. 
 
 12. Notation is the method of expressing numbers cither 
 by letters or figures. The metliod by letters, is called Roman 
 Nutation ; the method by figures is called Arabic Notation, 
 
 ROMAN NOTATION. 
 
 13. In the Eoman Notation, seven capital letters are used, 
 viz. : I, stands for one ; V, for five ; X, for ten ; L, for fifty , 
 C, for one hundred; D, for five hundred ; and M, for 07ie thou- 
 sand. All other numbers are expressed by combining these 
 letters according to the following 
 
 ROMAN TABLE. 
 
 I. - - 
 
 - - One. 
 
 LXX. - 
 
 Seventy. 
 
 II. - - 
 
 - - Two. 
 
 LXXX. 
 
 Eighty. 
 
 III.- - 
 
 - - Three. 
 
 XC. - 
 
 Ninety. 
 
 IV.. - 
 
 - - Four. 
 
 C. - - 
 
 One hundred. 
 
 V. - - 
 
 . - Five. 
 
 CC- - 
 
 Two hundred. 
 
 VI.- - . 
 
 ■ - Six. 
 
 ccc. . 
 
 Three hundred. 
 
 VII. - - 
 
 • - Seven. 
 
 cccc- 
 
 Four hundred. 
 
 VIII. - ■ 
 
 ■ - Eight. 
 
 D. - - 
 
 Five hundred. 
 
 IX. - ■ 
 
 • - Nine. 
 
 DC. - 
 
 Six hundi-ed. 
 
 X. - - . 
 
 • . Ten. 
 
 DCC. - 
 
 Seven hundred. 
 
 XX. - - 
 
 - Twenty. 
 
 DCCC- 
 
 Eight hundred. 
 
 XXX. - 
 
 - Thirty. 
 
 DCCCC 
 
 Nine hundred. 
 
 XL. - . 
 
 - Forty. 
 
 M. - - 
 
 One thousand. 
 
 L. - - - 
 
 • - Fifty. 
 
 MD. - 
 
 Fifteen hundred. 
 
 LX. - - 
 
 • - Sixty. 
 
 MM. - 
 
 Two thousand. 
 
 Note. — This Notation was used by the Romans : hence, its name. 
 It is still used for dates, numbering of chapters, pages, &c. 
 
 The principles of this Notation are these : 
 
 1 . Every time a letter is repeated, the number which it denotes is 
 also repeated. 
 
 12. What is Notation ? What is the method by letters called 1 What 
 is the method by figures called 1 
 
 13. How many letters are used in the Roman Notation 1 What are 
 ♦■hey ■? What does each stand for 1 
 
 Note. — What are the three principles of this Notation 1 
 
16 NOTATION. 
 
 2. If a letter denoting a less number is written on the right of one 
 denoting a greater, their sum will express the number. 
 
 3. If a letter denoting a less number is written on the left of one 
 denoting a greater, their difference will express the number. 
 
 EXAMPLES IN ROMAN NOTATION. 
 
 Express the following numbers in the Roman Notation: 
 
 1. Sixteen. 
 
 2. Fourteen. 
 
 3. Eighteen. 
 
 4. Sixty-nine. 
 
 5. Seventy-eight. 
 
 6. One hundred and fifteen. 
 
 7. Four hundred and nine. 
 
 8. Seven hundred and fifty-one. 
 
 9. One thousand and sixty. 
 
 10. Two thousand and ninety-one. 
 
 11. Five hundred and sixty-nine. 
 
 12. Seven hundred and forty-five. 
 
 13. Nine hundred and sixty-one. 
 
 14. Six hundred and ninety-nine. 
 
 15. Nine hundred and fifty-seven. 
 
 16. One thousand two hundred and six. 
 
 17. Four hundred and ninety-five. 
 
 18. Seven hundred and fiity-five. 
 
 19. Eighteen hundred and forty-seven. 
 
 20. Two thousand five hundred and twenty. 
 
 ARABIC NOTATION. 
 14. Arabic Notation is the method of expressing numhera 
 by figures. Ten figures are used, and Ihoy form the alphabet 
 of the Arabic Notation. 
 
 14. What is Arabic Notation ! How many figurrs are used ? "VA'hat 
 do they form 1 Name the figures ' What does the express ! What 
 are the other figures called 1 
 
NOTATION. 17 
 
 They are, called cipher, or Naught. 
 
 .1 - - One. 
 
 2 - - - Two. 
 
 3 - - - Three. 
 
 4 - - - Four. 
 
 5 - - - Five. 
 
 6 - - - Six. 
 
 7 - - - Seven. 
 
 8 - - - Ei";ht. 
 
 o 
 
 9 - - Nine. 
 
 The cipher 0, expresses no value. It is used to denote the 
 ahscnce of a thing. The nine other figures are called 5iV/«e/?- 
 cant Jjgurcs, or Digits. 
 
 15. We have no sino;le 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, Avhicli is read, ten. 
 
 This 10 is equal to ten of tlie units expressed by 1. It is, 
 however, but a single ten, and may be regarded as a unit, the 
 value of which is ten times as great as the unit 1. It is called 
 a unit of the second order. 
 
 16. When two figures are written by the side of each other, 
 the one on the right is in the flace of units, and the other in 
 the place of tens, or of units of the second order. 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 meoTit. 
 
 15. Have we a separate character for ten 1 How do we express ten ' 
 To how many units 1 is 1 ten equal 1 May ten be regarded as a single 
 unit ! Of what order 1 
 
 IG. When two figures are written by the side of each other, what place 
 does the right hand figure occupy 1 The figure on the left 1 ^\' hen 
 units simply are named, what unita are meant ! 
 
Li NOTATION. 
 
 17. In order to express ten units of the second order, or one 
 hundred, we form a new combination : 
 
 It is done thus, ----- 100, 
 
 by writing two ciphers on the right of 1. This number is read, 
 cue hundred. 
 
 li^ow, this one hundred expresses 10 U7iits of the second order, 
 or 100 U7iits of the first order. The one hundred is but an 
 individual hundred, and, in this light, may be regarded as a 
 unit of the third order. 
 
 We can now express any number less than one thousand. 
 
 For example, in the number two hundred and fifty- 
 five, there are 5 unit?;. 5 tens, and 2 hundreds. Write, 
 therefore. 5 units of the first order, 5 units of the second § c 'S 
 
 order, and 2 of the third ; and read from the risht, units. „ "^ ^ 
 ' > -''255 
 
 tens^ hundreds. 
 
 In the number five hundred and ninety-five, there are " « 2 
 
 5 units of the first order, 9 of the second, and five of the ^ 5 a 
 
 third ,• and it is read from the right, units, tens, hundreds. 5 9 5 
 
 In the number six hundred and four, there are 4 units '^ x Z 
 
 of the first order, of the second^ and 6 of the third. ^ 5 § 
 
 TJie right hand figure ahvays expresses units of the 
 first order ; the second, units of the second order ; and the third, 
 units of the third order. 
 
 18. To express ten units of the third order, or one thousand, 
 we form a new combination by writing three ciphers on the 
 right of 1 ; thus, 1000. 
 
 Now, tliis is but one single thousand, and may be regarded as 
 a unit of the fourth order. 
 
 17. How do you write one hundred ? To how many units of the second 
 order is it equal ? To liow many of the first order ! How may it be 
 roiiardcd 1 Of what order 1 How many units of the third order in 200 ! 
 In 600 ! fii 900 ! 
 
 18 To what are ten units of the tiiird order equal 1 Hew do you write 
 it 1 How do you write a single unit of the first order 7 How do you writo 
 a unit of the second order 1 Of the third T Of the fourth \ 
 
NOTATION. 19 
 
 Thus, we may form as many orders of units as we please : 
 
 a single unit of ihe first order is expressed hy - - 1, 
 
 a unit of the second order by 1 and 0; tlius, - - 10, 
 
 a unit of tlie third order by 1 and two O's ; - - 100, 
 
 a unit of the fourth order by 1 and three O's ; - - 1000, 
 
 a unit of the fifth order by 1 and four O's ; - - 10000 ; 
 and so on, for units of higher orders : 
 
 19. Therefore, 
 
 1st. The same jrgure expresses different units according to the 
 place tc/n'ch it occupies : 
 
 2d. Units ofthejirst order occupy the place at the right ; units 
 of the second order, the second place ; units of the third order, the 
 third place, and so on for places still to the left: 
 
 3d. Ten units of the first order make one of the second ; ten of 
 the second, one of the third ; ten of the third, one of the fourth ; 
 and so on for the higher orders : 
 
 4tli. IVhen figures are written by the side of each other, ten 
 units in any one place male one unit of the place next at tlie 
 left. 
 
 EXAHPLKS IN WRITING THE ORDERS OP UNITS. 
 
 1. Write 7 units of the 1st order. 
 
 2. Write 8 units of tlie 2d order. 
 
 3. Write 9 units of the 4th order. 
 
 4. Write 3 units of the 1st order, with 9 of the 2d. 
 
 5. Write 9 units of the 3d order, with 6 of the 2d, and 1 of 
 the 1st. 
 
 G. Write units of the 2d order, 8 of the 1st, with 4 of the 
 3d, and 7 of the 4th. 
 
 19. On what does the unit of a figure depend 1 What is the unit of 
 the place on the right 1 What is the unit of the second place 1 What of 
 the third place 1 What of the fourth 1 &c. 
 
 How many units of the first order make one of the second 1 IIo';» 
 many of the second make one of the third ] How many of the third one 
 of the fourth 1 &c. When figures are written by the side of each oilier, 
 how many units of any place make one unit of the place next at the 
 \eft 1 
 
20 NOTATION. 
 
 7. "Write 8 units of the 6tli order, 7 of the 4th, 9 of the 5th, 
 of the od, 2 of the 2d, and 1 of the 1st. 
 
 8. 'Write 8 units of the 8th order, 6 of the 7th, of the 1st, 
 3 of tlie 2a, 4 of the 3d, 9 of the 4th, of the Gtli, and 2 of 
 the 5th. 
 
 9. AVrite 4 units of the 10th order, 8 of the 7th, 3 of the 9th, 
 2 of tlie 8th, of the 6th, 3 of the 1st, G of the 2d, of the 
 Sd, 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, G of the 8th, 7 of the 7th, of the 
 Gth, and 4 of the 5th. 
 
 11. Write 3 units of the 11th order, of the 10th, 8 of the 
 4th, of the 5th, 2 of the Gth, of the 7th, 3 of the 8th, 4 of 
 the 9tli, 1 of tlie 3d, % of the 2d, and 3 of the 1st. 
 
 12. Write 3 units of the 12th order, G of the 11th, 3 of the 
 8th, 7 of the Gth, 2 of the 4th, and 1 of the 2d. 
 
 13. Write 5 units of the 13th order, 8 of the 12th, of the 
 9th, G of tlie 7th, 8 of the 3d, and 12 of the 1st. 
 
 14. Write 7 units of the 14th order, 5 of the 13th, G of 
 the 12th, 5 of the 10th, 7 of the 8th, 9 of the Gth, 5 of the 
 4th, and 8 of the 1st. 
 
 15. Write 9 units of the 15th order, 4 of the 13th, 8 of 
 the 9 til, 2 of the Gth, 7 of the 3d, and 2 of the 2d. 
 
 IG. Write G units of the IGth order, 9 of the 12th, 7 of the 
 9th, 4 of the 7th, of the Gth, 8 of the 4th, 9 of the 5th, and 
 2 of the 2d. 
 
 17. Write 8 units of the 20th order, 5 of the 18th, G of 
 the 13th, 4 of the 11th, 9 of the 9th, 1 of the 17th, 4 of the 
 5th, and 9 of the 3d. 
 
 18. Write 6 units of the 10th order, 5 of the 8th, 9 of the 
 Gth, of the 4th, and 1 of the 1st. 
 
 19. Write 9 units of the 18th order, and then diminish the 
 figure of each order by 1 till you come to and include ; then 
 increase the figure of each order by 1, till you reach the first 
 order; and then read each order. 
 
 20. Write tJie number which has 20 units of the 17tli order, 
 
NUMEEATION. 
 
 21 
 
 ol the 14tli, 8 of the 16th, 4 of the 13th, of the 8th, of 
 the 9th, and one in each of the other places. 
 
 NUMERATION. 
 
 20. Numeration is the art of reading correctly any number 
 expressed by figures or letters. 
 
 The pupil has already been taught to read all numbers from 
 one to one thousand. The Numeration Table will te^ch him to 
 read any number whatever; that is, to express numbers in 
 words. 
 
 TABLE. 
 
 7th Period. 6th Period. 5th Period. 4th Period. 3d Period. 2d Period. 1st Period. 
 Quintiilioiis. Quadrillions. Trillions. Billions. Millions. Thousands. Units. 
 
 09 
 
 s - • 
 
 o 
 
 . 
 
 
 
 
 , 
 
 « 
 
 
 W 
 
 
 • 
 
 o 
 
 
 
 
 
 
 
 
 
 ^<-j 
 
 
 
 
 
 m 
 
 
 
 
 X/2 
 
 
 
 
 
 
 ^ DQ ' 
 
 ." <» 
 
 s ■ 
 
 
 W 
 
 
 r* 
 
 • 
 
 
 rt . • 
 
 
 I • 
 
 
 t- !=< 
 
 _o 
 
 
 f-; 
 
 
 o 
 
 
 
 W OT 
 
 
 
 s-i • 
 
 n3 O . 
 
 
 
 O 
 
 •i-H ^ 
 
 
 
 w 
 
 
 o a 
 
 
 • 
 
 o-S • 
 
 G"! : 
 
 H-2 
 
 
 ..— ( (—1 
 
 P3 3 
 
 
 ^ 
 
 o 
 
 
 E— 1 <« 
 
 
 
 "— -iS -A 
 
 tt- rt » 
 
 c*.^ , 1 
 
 
 V-. '.^ 
 
 
 t— 
 
 "~^ 
 
 
 u^ o 
 
 
 
 = 5 
 
 ^ § s 
 
 O'm 
 
 
 o 'z3 
 
 
 o 
 
 • — 1 
 
 
 =>- m 
 
 
 • 
 
 K C?" 3 
 
 ^a° 
 
 «=H 
 
 
 Kpq 
 
 
 w 
 
 
 
 cc H 'Cl 
 
 M 
 
 
 
 ^ O i^ 
 
 go 
 
 _o 
 
 1^ 
 
 
 o 
 
 Cm 
 O 
 
 a; 
 
 a 
 o 
 
 1^ 1 
 
 4) 
 
 • 
 
 
 S J^ TO 
 
 s J2 
 
 •3 
 
 T3 cc 
 
 O 
 
 r^ 
 
 CO 
 
 
 a s § 
 
 c J2 -2 
 
 D o '3 
 
 C3 a> ^ 
 
 s § 
 
 S-l 
 
 3 O 
 
 f^ 
 
 ^ 
 
 o 
 
 s 
 
 a O -3 
 
 p o '3 
 
 KHC^ 
 
 KHGf 
 
 ffiHH 
 
 KHpq 
 
 
 H 
 
 KHH 
 
 EH^ 
 
 3 7 
 
 8 9 4 
 
 2 1 
 
 6 
 
 6 3 
 
 6 
 
 8 
 
 
 
 6 
 
 3 4 
 
 6 2 
 
 5 
 
 Notes. — 1. Numbers expressed by more than three figures are 
 written 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. The 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, still to the left. 
 
 4. To Quintillions succeed Sextillions, Septillions, Octillions, Non- 
 lllions, Decillions, UiidecilIion.=5, Daodecillions, &c. 
 
 5. The pupil should be required to commit, thoroughly, the names 
 
 20. What is Numeration 1 "What is the unit of the first period 1 Wha} 
 is the unit of the second 1 Of the third 1 Of the fourth 1 Fifth ? Sixth T 
 Seventh 1 &c. Give the rule for reading numbers. ' Give the rule for 
 writing numbers. 
 
22 
 
 NUMEKATION. 
 
 of the periods, so as to repeat tlicm in their regular order from left 
 to right, as well as from right to lefc. 
 
 6. Formerly, in the English Notation, six places Avere given to 
 Millions, Trillions, Quadrillions. &:c. They were read, Millions, Tens 
 of Millions, Hundreds of Millions, Thousands of Millions, Tens of 
 Thousands of Millions, Hundreds of Thousands of Millions ; and the 
 same for Billions, Trillions, Quadrillions. Sec. This method produc- 
 ed 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 each of the others. The French method, which gives three 
 places to the unit of each period, is fully adopted in this country, and 
 must soon become universal. 
 
 RULE FOR READING NUMBERS. 
 
 I. Separate the number into periods of three ficjures each, 
 beginning at the riijht'hand. 
 
 II. Name the unit of each figure, beginning at the right hand. 
 
 III. T7ie7i, beginning at the left hand, read each period as if 
 it itood alone, naming its unit. 
 
 EXAMPLES IN READING NUMBERS. 
 
 Let the pupil point oflf and read the following numbers — then 
 write them in words. 
 
 1. 
 
 97 
 
 6. 
 
 32045607 
 
 
 11. 784236704 
 
 2. 
 
 326 
 
 7. 
 
 90464213 
 
 
 12. 7403026054 
 
 3. 
 
 3302 
 
 8. 
 
 47364291 
 
 
 13. 217040S0495 
 
 4. 
 
 65042 
 
 9. 
 
 4037902169 
 
 
 14. 21896720421 
 
 5. 
 
 742604 
 
 10. 
 
 91046302 
 
 
 15. 8140290308097 
 
 16. 
 
 8504680467023 
 
 
 
 19. 30467214302704 
 
 17. 
 
 90403040720156 
 
 
 
 20. 167320410341204 
 
 18. 
 
 1723047368 
 
 93210 
 
 
 
 2 
 
 1. 2164032189765421 
 
 Let each of the above examples, after being written on tliQ 
 \)lack board, be analyzed as a class exercise ; thus : 
 
 1. In how many ways may the number 97 be read? 
 1st. The common way, 97. 
 2d. Wo may read, 9 tens, and 7 units. 
 
NATUKE OF NUMBEKs, 23 
 
 2. Ill how many ways may 32 G be read ? 
 
 1st. By the common way, tliree hundred and twenty-six. 
 2d. Three hundreds, 2 tens, and 6 units. 
 3d. Tliirty-two tens, and six units. 
 
 3. In how many ways may the number 5302 be read? 
 1st. Five tliousand three hundred and two. 
 
 2d. Five tliousand, tliree Iiundreds, tens, and 2 units. 
 3d. Fifty-three hundreds, tens, and 2 units. 
 4th. Five hundred and tliirty tens, and 2 unita. 
 
 4. In 65042, liow many ten thousands ? How many thou- 
 sands ? How many hundreds ? IIow many tens ? How many 
 units ? 
 
 5. In 742604, how many hundred thousands? How many 
 ten tliousands ? How many thousands ? How many hundreds ? 
 How many tens ? IIow many units ? 
 
 RULE FOR WRITING NUMBERS, OR NOTATION. 
 
 I. Begin at the left hand and write each period in order, as if 
 it were a period of units. 
 
 II. When the nuynher, in any period except the left-hand period, 
 can he expressed hy less than three fgures, prefix one or two 
 ciphers ; and when a vacant period occurs, fill it with ciphers. 
 
 EXAMPLES IN NOTATION. 
 
 Express the following numbers in figures : 
 
 1. Six hundred and twenty-one. 
 
 2. Five thousand seven hundred and two. 
 
 3. Plight 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. 
 
 7. Fifty-nine millions tliree hundred and ten. 
 
 8. Fleven thousand eleven hundred and eleven. 
 
 9. Three hundred millions, one thousand and six. 
 
 10. Sixty-nine billions, three millions, two hundred and 
 eleven. 
 
24 DEKOMINATh NUMBERS. 
 
 11. Forty-seven quadrillions, sixty-nine billions, four bun- 
 dred and sixty-five thousands, two hundred and seven. 
 
 12. Eight hundred quintillions, four hundred and twenty-nine 
 millions, six thousand and nine. 
 
 13. Ninety-five sextiliions, eighty-nine millions, eighty-nine 
 thousands, three hundred and six. 
 
 14. Six quintillions, four hundred and fifty-one billions, sixty- 
 five millions, forty -seven thousands, and one hundred and four. 
 
 15. Write, in figures, nine hundred and ninety-nine billions, 
 sixty-five millions, eight hundred and forty-one thousands, four 
 hundred and eleven, 
 
 1 G. Four hundred and seventy nonillions, forty octillions, four 
 millions, six thousands, two hundred and four. 
 
 17. Sixty-five sextiliions, eight hundred quadrillions, seven 
 hundred and fifty billions, seven hundred and fifty-one millions, 
 nine hundred and seventy-five thousands, three hundred and 
 ten. 
 
 FORMATION AND NATURE OF NUMBERS. 
 
 21. The term, one, may refer to any single thing : it has no 
 reference to kind or quality : it is called an abstract unit. 
 
 22. The term, one foot, refers to a single foot, and is called 
 a concrete or denominate unit. 
 
 23. An abstract number is one whose unit is abstract : thus, 
 three, four, six, &c., are abstract numbers. 
 
 24. A concrete or denominate number, is one whose unit is 
 concrete or denominate : thus, three feet, four dollars, five 
 pounds, are denominate numbers. 
 
 25. A Simple Number is a single unit, or a single collection 
 of units, either abstract or denominate. 
 
 21. Docs the term, one, refer to the kind of thing to which it is applied 1 
 What is it called? 
 
 22. To what docs one foot refer 1 What is it called ' 
 
 23. What is an abstract number? 
 
 24. What is a concrete, or denominate number 1 
 
 25. What is a simple number' 
 
DENOMINATE NUMBERS. 25 
 
 26. Quantity is anything which can be measured. 
 
 27. Numbers which have the same unit are of the same 
 denomination : and numbers having different units are of differ- 
 ent denominations. Tims, 4 yards and 6 yards are of the same 
 denomination i but 4 yards and G feet are of different denomi- 
 nations. 
 
 28. If two or more denominate numbers, having different 
 units, are connected together, forming a single expression, this 
 is called, a compound denominate numbei*. Thus, 3 yards 2 
 feet and 6 inches, is a cotnpotind denominate number. 
 
 29. We have seen (Art. 19) that when figures are written 
 by the side of each other, thus, 
 
 678904, 
 the language implies that ten units of any place make one unit 
 of the place next to the left. 
 
 30. When figures are written to express English Currency, 
 thus: £ s. d. far. 
 
 4 17 10 3 
 the language implies, that four units of the lowest denomination 
 make one of the second ; twelve of the second, one of the third ; 
 and twenty of the third, one of the fourth. 
 
 31. "When figures are written to express Avoirdupois weight, 
 thus : T. cwt. qr. lb. oz. dr. 
 
 27 17 2 24 11 10 
 the language implies, that 16 units of the lowest denomination 
 
 26. What is quantity \ 
 
 27. When are numbers paid to be of the same denomination 1 When 
 of different denominations ! 
 
 28. What is a compound denominate number 1 
 
 29. When several figures are simply written by the side of each other, 
 what docs the language imply 1 
 
 33. In the English Currency, how many units of the lowest denomina- 
 tion make one of the second '\ How many of the second one of the third ! 
 How many of the third one of the fourth ! 
 
 31. In the Avoirdupois weight, how many of the lowest make ^ 
 the second 1 How many of the second one of the third ? 
 
 2 
 
26 DENOMINATE NUMBEKS. 
 
 make one of the second; 16 of the second, one of the third; 
 25 of the third, one of the fourth ; 4 of the fourth, one of the 
 6fth ; and 20 of the fifth, one of the sixth. 
 
 All the other compound denominate numbers are foi'med on 
 the same principle ; and in all of them, we pass from a lower 
 to the next higher denomination hy considering how many units 
 of the lower make one unit of the next Jiigher* 
 
 32. A Scale expresses the relations between the different 
 units of a number. There are two kinds of scales — uniform 
 and varying. In the common scale, the number of units which 
 make 1 of the next higher is 10. In English Currency, 4, 12, 
 and 20, make up the varying scale ; and 16, 16, 25, 4 and 20, 
 in Avoirdupois weight. 
 
 SCALE OF TENS. 
 
 33. Let us write a row of O's, thus ; 
 
 o 
 
 rag S .2 H 
 
 a 
 
 o 
 
 
 rt p 
 
 J^or:^ '-'^ud So,^ J^ort 
 GHPq KHS KHH SHP 
 
 0, 0, 0, 0, 
 
 The language of figures determines the unit of each place, and 
 also, the laip of change in passing from one place to another. 
 This is called the decimal system, in v»'hich the units change 
 according to the scale of tens. 
 
 If it be required to express a given number of units, of any 
 
 * For the Tables of Denominate Numbers, see Appendix, page 383. 
 
 32. What is a scale 1 How many kinds of scales are there'? "What 
 are they ? What is the scale in the common system of numbers ! ^^'llat 
 is tlie scale in English Currency 1 What in Avoirdupois weight ? 
 
 33. If a row of O's be written, what does the language of figures deter- 
 mine ? What is such a system called 1 How does the unit change' 
 How do you express a given number of units of any order ? 
 
INTEGRAL UNITS. 27 
 
 Older, we first select from tlie arithmetical alphabet the figure 
 
 which designates tlie number, and then write it in the place 
 
 corresponding to the order. Thus, to express three millions, 
 
 we write 
 
 3000000; 
 
 and similarly for all numbers. 
 
 UNITED STATES MONEY. 
 34. United States money atlbrds an example of a system of 
 denominate units, increasing according to the scale of tens: 
 thus, 
 
 ^ a (O ^ _r- 
 Of) ;z5 g S " 
 
 W Q P O S 
 
 11111 
 
 may be read 11 thousand 1 hundred and 11 tnills ; or, 1111 
 cents and 1 mill; or, 111 dimes, 1 cent, and 1 mill; or, 11 dol- 
 lars, 1 dime, 1 cent, and 1 mill ; or, 1 eagle, 1 dollar, 1 dime, 
 1 cent, and 1 mill. Thus, we may read the number with any 
 one of its units as a base, or we may name them all ; as 
 1 eagle, 1 dollar, 1 dime, 1 cent, 1 mill. Generally, in United 
 States money, we read in the denominations of dollars cents 
 and mills ; and say, 11 dollars 11 cents and 1 mill. 
 
 United States money is denoted by the character, $. The 
 figures expressing dollars are separated from those which denote 
 cents and mills by a comma ; thus, 
 
 $11,111 
 is read, 11 dollars 11 cents 1 mill; the figures on the left of 
 the comma always denote dollars ; the first two on the right 
 denote cents, and the third, mills. 
 
 ALIQUOT TARTS. 
 
 One number is said to be an aliquot part of another, when 
 
 84. Are the numbers used in United States nionej' abstract or denomi- 
 nate ? According to what scale do the units change 1 How are <^ 
 separated from cents and mills I A\ hat is an rdiquot parti 
 aliquot parts of a dollar ' 
 
28 
 
 VARYING SCALKS. 
 
 it is contained in that other an exact number of times. Thus ; 
 50 cents, 25 cents, &c., are aliquot parts of a dollar : so also, 
 2 months, 3 months, 4 months an.l 6 months are aliquot parts 
 of a year. The parts of a dollar are sometimes expressed frac- 
 tionally, as in the following 
 
 TABLE, 
 cents. 
 
 $1 
 
 i of a dollar = 
 
 100 
 50 cents. 
 1 of a dollar = 33 J cents. 
 
 cents. 
 20 cents. 
 
 1- of a dollar = 25 
 1 of a dollar = 
 
 l of a dollar 
 Jq of a dollar 
 
 121 cents. 
 10 cents. 
 J^ of a dollar = 61 cents. 
 5 cents. 
 5 mills. 
 
 o\j^ of a dollar 
 
 J of a cent 
 
 VAKYING SCALES. 
 
 35. If we write the well-known signs of the English currency, 
 and place 1 under each denomination, we shall have 
 
 £ s. d. far. 
 1111 
 
 Now, the signs £.s. d. and far. fix the value of the unit 1 in 
 each denomination ; and they also determine the relations 
 between 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 
 
 2c?. 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, make 
 up the scale. 
 
 36. If we take the denominate numbers of Avoirdupois 
 weight, we have 
 
 Ton act. qr. lb. oz. dr. 
 111111; 
 
 35 In En^lisli currency, is the scale uiiiforni or varying ? How does 
 it vary 1 
 
 30. Name the units of the scnle in AvoirJupois weii^Iit. 
 
INTEGRAL UNITS. 29 
 
 in which the units increase in the following manner ; viz. : 
 counting from the right, IG units of the lowest denomination 
 make 1 unit of the second; IG of the second, 1 of the third; 
 25 of tlie third, 1 of tlie 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. 
 
 37. If we take any other class of denominate numbers, as 
 Troy weight, or any of the systems of measures, Ave shall 
 have different scales for the formation of the dilFerent numbers. 
 But in all the formations, we shall recognize the application of 
 the same general principles. 
 
 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 10, 
 we have the system of common numbers. 
 
 The second method consists in the application of known, 
 though varying laws of change in the units. These changes in 
 the units, produce difl'erent systems of denominate numbers, 
 each of which has its appropriate scale. 
 
 INTEGRAL UNITS OF ARITHMETIC. 
 
 38. The Integral units of Arithmetic may be divided into 
 ei2;ht classes : 
 
 1st. Abstract units : 2d. Units of currency : 3d. Units of 
 length : 4:ih. Units of surface : 5th. Cubic units, or units of 
 volume : (J/h. Units of weight : 7ih. Units of time : 8lh. Units 
 of circular measure. 
 
 First among the units of arithmetic stands the abstract unit 1. 
 This is the primary base of all abstract numbers, and becomes 
 the base, also, of all denominate numbers, by merely naming, 
 in succession, the particular thing to which it is applied. 
 
 37. How many general methods are there of forming numbers from tlie 
 unit one ] What is the first ? WTiat is the second 1 
 
 38. Into liow many general classes may the units of Arithmetic bo 
 arranged ^ What are they 1 
 
30 rr.oPERTiES of 9's. 
 
 OF THE SIGNS. 
 
 39. The sign ~, is called the sign o^ 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 p^us, which signifies jnore. 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 the 
 right is to be taken from the one on the left : thus, 6 — 2 = 4. 
 
 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- 
 plied by 3. 
 
 The parenthesis is used to indicate that the sum of two or 
 more numbers is to be considered as a single number : thus, 
 
 (2 4- 3 + 5) X G 
 shows, that the sum of 2, 3 and 5 is to be multiplied by 6. 
 
 The parenthesis is also used to denote that the difference be- 
 tween two numbers is to be considered as a single number ; thus, 
 
 (5 - 3) X 6, 
 denotes that the difference between 5 and 3 is to be multiplied 
 by 6. 
 
 The sign -r-, 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-^-5, denotes that 4 is 
 to be divided by 5. 
 
 PKOPERTIES OF THE 9's. 
 
 40. In any number, written with a single significant figure, 
 as 4, 40, 400, 4000, &;c., the excess over exact 9's is equal to 
 
 39. What is the sign of equaHty \ AVliat is the sign of adJilioii ? 
 Wiiat of siil)traction 1 What of niuhipHcation ! For wliat is tlie i)aien- 
 thesis used ! What is Ihe sign of division 1 
 
 40. What will be the excess over exact 9's in any number expressed by 
 a single significant figure 1 How may the excess over exact 9's be found 
 in any number whatever'! 
 
REDUCTION. 
 
 31 
 
 the number of units in the significant figure. For, any such 
 
 number may be written, 
 
 4 = 4. 
 
 Also, - - - - 40 = (9 -f 1) X 4, 
 
 « - - - - 400 = (99 + 1) X 4, 
 
 « - - - - 4000 r= (999 + 1) X 4, 
 &c., &c., &c. 
 
 Each of the numbers 9, 99, 999, &c., contains an exact number 
 of 9's ; hence, when muUipliecl by 4, the several products will 
 contain an exact number of 9's : therefore, 
 
 TJie excess over exact 9's, in each mmiber, is 4 ; and the same 
 may be shotvn for each of the other significant figures. 
 If we write any number, as 
 
 6253, 
 we may read it 6 thousands 2 hundreds 5 tens and 3. Now, 
 the excess of 9's in the thousands is 6 ; in 2 hundreds it is 2 ; 
 in 5 tens it is 5 ; and in 3 it is 3 : hence, in them all, it is 16, 
 which is one 9 and 7 over : therefore, 7 is the excess over 
 exact 9's in the number 6253. In like manner, 
 
 The excess over exact 9's in any number whatever, is found by 
 addi?ig together the sign'ficmit fgurcs 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 f^ay, 
 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 48701 ? In €7498 ? 
 
 2. What is the excess of 9"s in 9472021 ? In 2704962 ? 
 
 3. What is the excess of 9's in 87049612 ? In 4987051 ? 
 
 REDUCTION. 
 
 CHANGE OF UNITS. 
 
 41. Reductiox is the operation of changing the unit of a 
 number without altering its value. Thus, if we have 4 yards, 
 
 41. What is Reduction 1 How do you change yards to feet 1 How do 
 you change feet to inches 1 How do you change inches to feet 1 How do 
 you change feet to yards '' 
 
/ 
 
 32 RKDCCTION. 
 
 in whicli the unit is 1 yard, and wisli to change to feer, the 
 units of the scale will be 3, since 3 feet make 1 yard : there- 
 fore, the number of feet will be 
 
 4x3 = 12 feet. 
 If it were required to reduce 12 feet to inches, the units of the 
 scale would be 12, since 12 inches malie 1 foot. Hence, 
 
 4 yards =z 4 x 3 =z 12 feet = 12 x 12 = 144 inches. 
 
 H', on the contrary, we wash to change 144 inches to feet, and 
 then to yards, we should first divide by 12, the units of the 
 scale in passing from inches to feet ; and then by 3, the units 
 of the scale in passing from feet to yards. Hence, Reduction 
 is of two kinds : 
 
 1st. To reduce a number from a higher unit to a lower: 
 Multiply tlie units of the Idgliesl denomination by the number 
 of units ill the scale which connect-^ it with the next lower, and 
 then, add to t e product the un'ts <f that deno mi nation : Pro- 
 ceed in the same manner through all the denominations till the 
 unit is brought to the requird denomm'ttion. 
 
 Id. To reduce a number from a lower unit to a higher: 
 Divide the given number hy the number of units in the scale 
 which connects it with the next higher denomination ; and set 
 down the remainder, if there be one. Divide the quotient thus 
 obtained, and each succeeding quotient in the satne manner, till 
 the unit is reduced to the required denomination: the last quo- 
 tient with the several remainders annexed, will be the answer. 
 
 EXAMPLES. 
 
 1. Reduce £3 14s. 4c?. to pence. "We first multiply the £3 
 by 20, which gives GO shillings. We then add 14, making 74 
 shillings : we next multiply by 12, and the product is 888 pence : 
 to this we add 4rf. and we have 892 pence, which are of the 
 same value as £3 14s. Ad. 
 
 If, on the contrary, we wished to change 892 pence to pounds 
 Bhillings and pence, we should first divide by 12: the quotient 
 is 74 shillings, and 4c?. over. We next divide by 20, and the 
 
REDUCTION. 
 
 33 
 
 quofient is £3, and 14s. over: hence, the rebuilt is £3 I'ls. 4c?., 
 which is equal to 892 pence. 
 
 The reductions, in all the denominate numbers, are made in 
 the same manner. 
 
 2. In £5 5s., how many shil- 
 lings, pence, and farthings .'' 
 
 £ s. 
 5 5 
 
 20 
 
 105 5 shillings added. 
 12 
 
 12G0 
 4 
 
 5040 
 
 Here the reduction is from a 
 greater to a less unit. 
 
 4. In 342"., IQcivt., oqrs., 
 Idlb., how many pounds ? 
 
 34 
 
 20 
 
 3. In 5040 farthings, how many 
 pence, shillings, and pounds ? 
 
 4)5040 farthings. 
 12)1260 pence. 
 2i0 )IO|5~s hillings. 
 £5 OS. 
 
 In this example, the reduc 
 tion is from a less to a greater 
 unit. 
 
 IGcwt. added. 
 oqr. added. 
 19 lb. added. 
 
 5. In GOQOAlb., how many 
 tons, cwt., qr., and lb. 
 
 25)69G94 
 
 A71S. 
 
 AJ2787 q r. . 
 20)G06cwt. . 
 3342: . 
 34 r. IGcwt. 
 
 . . 19lb. 
 . . 3qr. 
 . . IGcwL 
 Sqr. 19/5. 
 
 696 
 
 4 
 
 2787 
 
 25 
 
 13954 
 
 5574 
 
 69694/^5. 
 
 6. In $426, how many cents ? How many mills ? 
 
 7. In 36 eagles 8 dollars and 6 dimes, how many cents? 
 
 8. In 8750 mills, hoAv many dollars and cents ? 
 
 9. In 43 eagles 3 dollars and 5 mills, how many mills ? 
 
 10. In £37 9s. 8c?., how many pence? 
 
 11. In 1569 farthings, how many pounds, shillings, pence, 
 and farthings ? 2* 
 
34 REDUCTION. 
 
 12. In IT. Wciot. Iqr. 2011/s., Avoirdupois, how many 
 pounds ? 
 
 13. In 15445^3., Avoidupois, how many tons, cwts., qrs., and 
 lbs. 
 
 14. How many grains of silver in 4Ib., Goz., 12dwt. and 
 7grs.? 
 
 15. How many pounds, ounces, pennyweights, and grains of 
 gold, in 704121 grains? 
 
 16. In 5lfe, 13,15,19, 2gr., Apothecaries' weight, how 
 niany grains? 
 
 17. In 174947 grains, how many pounds, ounces, drachms, 
 scruples and grains ? 
 
 18. In 6 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, rooda 
 and perches ? 
 
 23. In 456 square chains, how many acres ? 
 
 24. In BA., 21i., 8P., how many perches ? 
 
 25. In 14 tons of round timber, how many cubic mches ? 
 
 26. In 31 cords of- wood, how many cubic feet ? 
 
 27. In 56320 cubic feet, how many cords ? 
 
 28. In 157 yards of cloth, how many nails ? 
 
 29. In 192 Ells Flem., how many yards? 
 
 30. 07yd., Sqr., how many Ells English ? 
 
 31. In A/thd. Wine measure, how many quarts? 
 
 32. In 7560 pints, Wine measure, how many liogsheads? 
 
 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 365c?. 5h. ASm. 4.8sec., each, how 
 many seconds ? 
 
 38. How many months, weeks and days in 254 days, reckon- 
 ing the month at 30 days ? 
 
ADDITION. 35 
 
 ADDITION. 
 
 42. The sum of two or more numbers is a number containing 
 as many units as all the numbers taken together. 
 
 Addition is the operation of finding the sum of two or more 
 numbers. 
 
 1. What is the sum of 769 and 487 
 
 Analysis. — Write the numbers thus : 
 
 di-aw a line beneath them, - - . - 
 sum of the units, ----.. 
 sum of the tens, - 
 
 sum of the hundreds, . . - - - 
 Entire sura, - - . 
 
 The example may be done in another way. thus : 
 Set down the number as before : then say, 7 and 9 operation. 
 are 16 : set down 6 in the units place, and the 1 ten 769 
 
 under the 8 in the column of tens. Then say, 1 to 8 487 
 
 are 9, and 6 are 15. Set down the 5 in the column n 6 
 
 of tens, and the 1 hundred in the column of hundreds. 1256 
 
 We then add the hundreds, and find their sum to be 
 12 : hence, the entire sum of 1256. 
 
 Note 1 . — Observe, that units of the same value are always written 
 in the same column, since every collection must contain units of the 
 same kind. 
 
 2. Wlien the sum in any column, exceeds 9, it produces one or 
 more units of a higher order, which belong to the next column at the 
 left. la that case, write down the excess over tens, and add the tens 
 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. 
 
 JBeginners, however, should set down the numbers to be carried, 
 
 42. What is the sum of two or more numbers'! What is Addition ! 
 How are numbers written down for Addition ? What do you do in simple 
 numbers when the sum of E.ny column exceeds 9 1 What is this called '* 
 What is the general rule for the addition of numbers '^ 
 
3G ADDITION. 
 
 each under its proper columiij as in the examples below. 
 
 (2) (3) (4) 
 
 85468 672143 4783614 
 
 9104 79161 504126 
 
 379 8721 872804 
 
 94951 760025 6160544 
 
 1012 12110 211101 
 
 5. What is the sum of 35 dollars 4 dimes 6 cents 5 mills, 
 
 4 dollars 7 mills, and 97 cents 3 mills ? 
 
 Note. — Write the units of the same value in o?eratio:m. 
 
 the same column, separating the dollars from the $35,465 
 cents and mills by a comma (Art. 40) : then add the 4,007 
 
 columns as m simple numbers. ,973 
 
 6. Let it be required to find the sum ot 'm 
 £14 7s. Sd. 3far., and £6 18s. dd. 2far. 
 
 Analysis. — Write the numbers, as before, so that units of the samo 
 order shall fall in the same column. Beginning 
 
 ■with the lowest denomination, yve find the sum to be operation. 
 
 5 farthings. But since 4 farthings make a penny, we £ s.d.far. 
 set down 1 farthing, and carry one penny to the column 14 7 8 3 
 
 of pence. The sum of the pence then becomes 1 8, which 6 18 ^ 2 
 
 is 1 shilling and 6 pence over. Set down the 6 pence, 21 6 6 1 
 and carry the 1 shilling to the column of shillings, the 
 
 sum of which becomes 26 ; that is, 1 pound and 6 shillings. Setting 
 down the 6 shillings and carrying 1 to the column of pounds, we lind 
 the entire sum to be £21 6s. 6d. Ifar. 
 
 Hence, for the addition of all numbers, 
 
 I. Write the numbers so that units of the same value shall fall 
 in the same column. 
 
 II. Add the tmits of the lowest denomination, and divide their 
 sum hj so many as make one unit of the denomination next higher. 
 Set down the remninder and carry the quotient to the next higher 
 denomination ; proceed in the same manner through all the denom- 
 inations and set down the entire sum of the last column. 
 
ADDITION. 
 
 37 
 
 PROOF. 
 
 43. The proof of an operation, in Addition, consists in show- 
 ing that the answer contains as many units as there are in all 
 the numbers added. There are three methods of proof. 
 
 I. Begin at the top of the units column and add, in succession, 
 all the columns doicnwards. If the two residts agree, the work is 
 supposed to he right ; for, it is not likely that the same mistake will 
 have been made in both additions. 
 
 ir. Divide the given numbers into parts, and add the parts 
 separately : then add together the partial sums ; if the results 
 agree, the toork is supposed to he right ; for, a whole is equal to the 
 sum of all its parts. 
 
 III. Find the excess o/9's in each number, and place it at the 
 right (Art. 20). Add these nuinbers and note the excess ofd's 
 in their sum. This excess shoidd be equal to the excess of 9's in 
 the sum of the numbers. 
 
 1st. Method. 
 
 
 2d. 
 
 Method. 
 
 182796 
 
 
 182796 
 
 
 32160 
 
 143274 
 
 
 143274 
 
 
 47047 
 
 32160 
 
 
 Partial sums 326070 
 
 
 79207 
 
 47047 
 
 
 
 
 
 Sum 405277 
 
 
 326070 
 
 1st 
 
 partial sum. 
 
 
 
 79207 
 
 2d 
 
 a 
 
 
 
 Sum 405277 
 
 
 
 
 
 3d Method. 
 
 
 
 182' 
 
 796 
 
 6 excess 
 
 of 9's, 
 
 143: 
 
 274 
 
 3 " 
 
 
 a 
 
 32: 
 
 160 
 
 3 " 
 
 
 i( 
 
 47( 
 
 347 
 
 •7 
 
 4 « 
 
 
 u 
 
 Sura 405277 •• 
 
 16 - - 
 
 • 
 
 7 excess of 9'h 
 
 READING. 
 
 44. The pupil should be early taught to omit the intermediate 
 
 43. What is the proof of an operation in Addition 1 
 methods of proof are there I Explain each separately 1 
 
 44. What 18 Ihe reading process, in Addition"! 
 
 How many 
 
38 
 
 ADDiTION. 
 
 words in the addition of columns of figures. Tiius, in the 
 above examjjle, 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. 
 
 (!•) 
 
 (2.) 
 
 (3.) 
 
 (4.) 
 
 94201 
 
 80032 
 
 98800 
 
 10304 
 
 4G390 
 
 4291 
 
 10926 
 
 67491 
 
 37467 
 
 2376 
 
 321 
 
 1324 
 
 4572 
 
 840 
 
 479 
 
 46 
 
 5. What is the sum of 
 
 6. What is the sum of 
 169? 
 
 7. What is the sum of 42300, 6000, 347001, 525, 47 
 
 1376, 38940, 8471, 23607, 891 ? 
 3480902, 3271, 567321, 91243, 6001, 
 
 (8.) 
 
 days. 
 1276 
 3718 
 9024 
 6357 
 1028 
 9131 
 
 (13.) 
 miles. 
 1600 
 2588 
 9101 
 6793 
 8267 
 4572 
 
 (9.) 
 
 husliels. 
 
 47917 
 
 12031 
 
 5672 
 
 8321 
 
 728' 
 
 4^ 
 
 (14.) " 
 furlongs. 
 47468 
 59012 
 23419 
 15760 
 27900 
 12317 
 
 (10.) 
 
 rods. 
 9003 
 1881 
 6035 
 7810 
 3176 
 2004 
 
 (15.) 
 pounds. 
 76389 
 1036 
 2671 
 5132 
 6784 
 1672 
 
 (11.) 
 
 minutes. 
 
 67321 
 
 4702 
 
 1067 
 
 456 
 
 377 
 
 99_ 
 
 (16.) 
 dollars. 
 1602 
 9614 
 4732 
 5675 
 8211 
 4455 
 
 (12.) 
 
 gallons. 
 
 760324 
 
 18720 
 
 5762 
 
 359 
 
 1082 
 47269 
 
 (17.) 
 
 casks. 
 
 40506 
 
 37219 
 
 50170 
 
 32614 
 
 73462 
 
 10001 
 

 ADDITION. 
 
 •6Si 
 
 (18.) 
 
 (19.) (20 
 
 .) (21.) 
 
 (22.) 
 
 $175,3G5 $30,365 $180,000 $300,40 
 
 $4802,279 
 
 278,050 
 
 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 
 
 125,092 
 
 41,04 
 
 7,14 9,011 10,08 
 (24.) (25.) 
 
 42,75 
 
 (23.) 
 
 (26.) 
 
 £ s. d. far 
 
 lb. oz. dwt. 
 
 -ft 5 3 
 
 lb. oz. dr. 
 
 14 11 3 1 
 
 174 11 19 
 
 17 11 7 
 
 17 15 12 
 
 17 18 10 2 
 
 75 10 13 
 
 94 10 6 
 
 29 32 10 
 
 29 7 6 
 
 642 3 10 
 
 60 9 2 
 
 84 10 9 
 
 42 14 11 3 
 
 125 7 5 
 
 42 3 9 
 
 14 3 7 
 
 17 10 00 1 
 
 62 16 
 
 12 6 
 
 40 9 9 
 
 84 00 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.) 
 
 ciot. qr. lb. 
 
 yd. qr. na. 
 
 ^. E. qr. na. 
 
 L. mi. fur 
 
 174 2 20 
 
 74 3 3 
 
 14 4 3 
 
 n 2 1 
 
 320 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 
 
 36 3 1 
 
 29 3 2 
 
 40 1 
 
 (31.) 
 
 (32.) 
 
 33.) 
 
 (34.) 
 
 yrds. ft. in. 
 
 A. JR. F. 
 
 Tun. hlid. gal. 
 
 gal. qt. pt. 
 
 174 11 1 
 
 77 3 39 
 
 714 3 56 
 
 14 3 1 
 
 260 2 
 
 64 2 37 
 
 626 1 48 
 
 74 2 1 
 
 150 10 2 
 
 16 1 29 
 
 320 29 
 
 96 1 
 
 126 9 1 
 
 72 18 
 
 156 2 31 
 
 47 2 1 
 
 96 7 
 
 36 2 20 
 
 225 1 42 
 
 22 1 
 
 72 4 1 
 
 42 2 14 
 
 84 17 
 
 65 1 
 
 8 6 2 
 
 11 3 7 
 
 96 1 34 
 
 19 
 
40 
 
 
 
 
 additio:n. 
 
 
 
 
 
 (35) 
 
 t 
 
 
 (36.) 
 
 
 
 (37.) 
 
 
 (38.) 
 
 
 chal. hiL 
 
 qt. 
 
 yr. 
 
 mo. 
 
 . wic. 
 
 da. 
 
 hr. 
 
 min. 
 
 qr. lb. 
 
 oz. 
 
 14 31 
 
 G 
 
 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 
 
 n 
 
 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 tlie United States and territories, in 
 1850, was as follows : White population, 19553068; Free 
 Colored population, 434495 ; Slave population, 3204313 ; In- 
 dians, 400674 ; -what Avas the entire population ? 
 
 40. In the year 1850, the expenditures of the United States, 
 amounted to 43002168 dollars; 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 twodaughters, 5976 dollars ; to his 
 wife, the remainder of his property, which exceeded the amount 
 bequeathed to his children by twelve hundred dollars : find the 
 amount of his property ? 
 
 42. A stage goes in one day 27 miles 3 furlongs 36 rods; 
 the next, 32 miles 10 rods ; the next, 36 miles 2 furlongs ; the 
 next, 25 miles 6 furlongs 38 feet : how far did it go in four 
 days? 
 
 43. The population of Boston, in 1854, was 178000 ; Provi- 
 dence, 60,000 ; Buflalo, 75000 ; New Orleans, 139190; Louis- 
 ville, 55000 ; Sacramento, 12000': what was the entii'e popula- 
 tion of these cities ? 
 
 44. The population of New York city, in 1850, Avas 515547; 
 Brooklyn, 127618; Baltimore, 169054; Washington, 40000: 
 Cincinnati, 115436; Chicago, 29963; St. Louis, 77860; Mil- 
 waukie, 20061 ; Detroit, 21119 ; Indianapolis, 8091 : what wu." 
 the entire population of these cities ? 
 
ADDITION. 4:1 
 
 45. Bouglit 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 ? 
 
 46. A lady bought a bonnet for $5,375 ; some silk for $12,03 ; 
 some ribbon for $0,875 ; a shawl for $9,46 : what did the whole 
 amount to ? 
 
 47. A wine-merchant taking an invoice of his liquors, finds 
 that he has ohhds. 36 gah. 2qts. of wine ; 3hhds. \6fjals. Iqt. Ipt. 
 of rum ; Ihhd. 2qts. of gin ; 40^a/s. Ipf. of whisky : how much 
 liquor in all ? 
 
 48. Tea Avas imported into the United States in the year 
 1851, to the value of $4798005 ; in 1852, 17285817 ; in 1853, 
 i^8224853 : what was the value of the tea imported during 
 these three years ? 
 
 49. 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 ? 
 
 50. A man sold his house and lot for $25840, w^iich was 
 $3186 less than he gave for them ; how much did they cost 
 him ? 
 
 51. A speculator bought three city lots, for the first lie 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 ? 
 
 52. The population of England, in 1851, was 16921888; of 
 Ireland, 6515794; of Scotland, 2888742: what was the entire 
 population of the three ? 
 
 53. The churches of the United States and territories in 
 1850, were, Baptists, 9375 ; Congregationalists, 1706 ; Presby- 
 terians, 4824 ; Methodists, 13280 ; Universalists, 529 : what was 
 the whole number of churches belonging to these five denomi- 
 nations ? 
 
 54. In the same year, the value of the church property 
 
42 ADDITION. 
 
 owned by the Baptists in the United States and territories, 
 was $] 1020855 ; by the Congregationalists, §7970195; by the 
 Tresbyterians, $14543789; by the Methodists, $14822870; by 
 the Universalists, $1752310: what was the Avhole amount? 
 
 55. During the year 1853, there was coined in the United 
 States, $51888882 of gold; 87852571 of silver; and $07059 
 of copper : what was the whole amount of money coined in the 
 United States in 1853? 
 
 56. A farmer sends to market the following quantities of 
 butter; IScwt. 2qrs. IGlbs. ; lion 5mot. 21lbs. ; 2qrs, lAIbs. ; 
 how much did he send in all ? 
 
 57. A man having 84 acres 3 roods 20 rods of land, buys 
 120 acres 14 rods more : how much did he then have? 
 
 58. 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 ? 
 
 59. A farmer has three fields of grain, the first yields 1375 
 bushels; the second, 1810 bushels; the third, 1205 bushels; 
 he values his entire farm at $2975 more than the number of 
 bushels of grain raised from these three fields : what was the 
 value of his farm ? 
 
 60. Bought a silver tea-pot weighing lib. Goz. 12du't. ; a 
 cream-cup, weighing lOoz. IMict. 20gr. ; a porringer, weigliing 
 Woz. lOyr. ; a dozen large spoons, weighing l^i^. l-idwt. 12gr.: 
 what was the weight of the whole ? 
 
 61. The whole number of ailults in the United States and 
 territories, over twenty years of age, who could not read and 
 write in 1850, were as follows; of whites, males, 3896G4; 
 females, 573234 ; free colored, males, 40722 ; females, 49800 : 
 what was the whole number ? 
 
 62. The whole number attending school the same year, were, 
 of wliites, males, 2146432; females, 1916614; free colored, 
 males, 13864 ; females, 12597 : what was the whole number 
 attending school ? 
 
 63. A forwarding merchant had in his store-room, at one 
 
ADDITION. 43 
 
 dme, 7500 bushels of corn ; 12865 bushels of wheat ; 4G80 
 bushels of oats ; 329G bushels of barley, and had room enough 
 left to store 4000 bushels of oats : how many bushels of grain 
 would the storehouse hold ? 
 
 64. A man engagmg in trade had $5164,50, in cash; 
 $11810,25, in goods ; $3004, in notes. His nett profits aver- 
 aged $2384,16, a year, for 3 years : what was the total value 
 of the property at the end of the three years ? 
 
 65. A person paid two eagles for a coat ; four dollars and 
 six dimes for a hat ; two dollars and sixty-three cents for a vest; 
 ei^-ht dimes, seven cents and five mills for a knife : what was 
 the amount of his bill ? 
 
 66. From a piece of cloth, I2ijds. 2qrs. were cut at one time ; 
 16)jds. Iqr. 3na. at another, Avhen there were lOyds. Iqr. Ina. 
 remaining : how much was there in the whole piece ? 
 
 67. A farmer purchased a plough for $9^ ; a wagon, for 
 $45i- ; a horse, for $110f ; a load of hay, for $121; a harrow, 
 for $34- : what was the cost of the whole ? 
 
 68. If a certain warehouse be worth $12540,371, and one- 
 fourth the contents is valued at $5632,108 : what is the value 
 of the warehouse and the whole of its contents ? 
 
 60. The number of bales of cotton used in manufactures in 
 1850, by Massachusetts, were 223607 ; New Hampshire, 83026 ; 
 Ehode "^Island, 50713; Virginia, 17785; New York, 37778; 
 Maryland, 23525 ; Connecticut, 39483 ; Alabama, 5208 : what 
 was the entire amount consumed in those states ? 
 
 70. In 1850, the State of New York produced 13121498 
 bushels of wheat ; Pennsylvania, 15367691 bushels; Virginia, 
 11212616 bushels ; Ohio, 14487351 bushels ; Missouri, 2981652 
 bushels ; Illinois, 9414575 bushels: what was the whole num- 
 ber of bushels produced by those states in that year ? . 
 
 71. A farmer sold his wheat for $825,87^-; his barley for 
 67,121; liis pork for $80,10; his apples for $46: how much 
 did he receive for the whole ? 
 
 72. Three persons enter into copartnership, the first put in 
 7825 dollars capital ; the second put in 1250 dollars more than 
 
44 SUBTRACTION. 
 
 the first ; and the third put in as much as the other two : what 
 was the whole amount of capital invested ? 
 
 73. A farmer raised in one field 2A0bnsIi. 3ph. 2qts. of 
 wheat ; in another 97bush. Gqts. ; in another A2hush. IpJc. : how 
 much did he raise in the three fields ? 
 
 74. Add together three hundred dollars, ten eagles, forty 
 dimes, ninety-six cents, seven mills, nine dollars, forty-seven 
 cents, five mills, four eagles, three dollars, and nine dimes. 
 
 75. What is the sum of £17 10s. 6d.; £25 4s. lO^d. ; 18s. 
 Gd. 3far.; £11 Q^^d.; £1 18s.; 21s. O^d.? 
 
 76. The Deluge, according to Chronology, occurred 1656 
 years after the creation ; the call of Abraham 427 after the 
 Deluge ; the departure of the Israelites, 430 after the call of 
 Abraham ; the foundation of the temple, 479 after the de- 
 parture of the Israelites ; the end of the captivity, 476 after 
 the foundation of the temple ; and the birth of Christ, 536 years 
 after the end of the captivity : how many years from tlie crea- 
 tion to the present time, it being the year 1856 ? 
 
 SUBTRACTION. 
 
 45. The Difference hetween two numbers is such a number 
 as, added to the less, will give the greater. 
 
 If the numbers are unequal, the larger is called the minuend, 
 and the less the subtrahend. If they are equal, either is the 
 minuend and the other the subtrahend. Their difference, 
 whether they are equal or unequal, is called the remainder. 
 
 Subtraction is the operation of finding the difference he- 
 tween two numbers. 
 
 1. From 869 take 327 ; that is, from 8 lumdreds 6 tens and 
 9 units, take 3 hundreds 2 tens and 7 units. 
 
 45. "What is the ililTerPiico Ix-twcen two numbers ! When the minibera 
 are unequal, what is the larijcr nuiiiher callcil ! "VViiat is the h^ss eallcd 1 
 Wliat is their ditlcrnnce called 1 ^^'hat is Subtraction ^ Give tlie rule 
 for findini' the diflercnce between two numbers. 
 
OPERATION. 
 
 
 
 TT 
 C 
 
 2 
 
 GO 
 
 10 
 
 
 
 
 624 - 
 
 5 
 
 12 
 
 4 
 
 393 - 
 
 3 
 
 9 
 
 3 
 
 1 
 
 
 
 
 SUBTRACTION. 45 
 
 Analysis. — Beginning -wilh the right .land operation. 
 
 rgure, we take units from units ; then, tens 869 minuend, 
 
 from tens ; then, hundreds from hundreds, and 327 subtrahend, 
 
 find tlie remainder to be 542. 542 remainder. 
 
 2. From 624 take 393. 
 
 Analysis. — Having Avritten do\vn the numbers, Ave subtract 3 fron? 
 4, and find a remainder 1. At the next step 
 Ave meet a difTiculty, for we cannot subtract 
 9 tens from 2 tens. 
 
 Take 1 hundred = 10 tens, from 6 hun- 
 dreds and add it to 2 tens. Then 9 tens from 
 12 lens leaves 3 tens, and 3 liuiidreds from 5 
 leaves 2 hundreds, and the remainder is 231. 231-2 3 1 
 
 The remainder can be found by adding, 
 mentally, 10 to 2 tens, and then saying, 9 from 12 leaves 3 tens; 
 then adding 1 to 3 hundreds, and say, 4 from 6 leaves 2 hundreds. 
 
 The process of adding 10 to a figure of the minuend and returning 
 1 to the next figure of the subtrahend, at the left, is called borrowing. 
 
 3. From GT. licwt. 2qr. 20lb. Uoz., take 4T. ITcio.. \qr. 
 
 2\Ib. IOgz. 
 
 Analysis. — Taking \Ooz. from 1203., loz. 
 remain. At the next step we find a difllcuUy, 
 for 2\lb. cannot betaken for 20/6. "We then 
 take \qr. = 25/6. from iqr. and add it to 20/6., 
 making 45/6. ; then say, 21/6. from 45/6. leaves 
 24/6.; we then add 1 to the next left hand 
 figure of the subtrahend, and .«ay %qr. from 
 2qr. leaves 0: then llcict. from 'iicici. leaves 
 \1cict , and 5 from 6 leaves 1 ton. 
 
 Hence, to find the difference between two numbers, 
 
 I. Set down the less nimiber under the greater, so thai units of 
 the same value shall fall in the same column. 
 
 II. Begin with the units of the lowest denomination and sub- 
 tract each number f-om the one above it. 
 
 III. When the units of any denomination in the subtrahend 
 exceed those of the same denomination in the yninuend, suppose 
 
 T. 
 
 cui. 
 
 20 
 
 qr 
 
 lb. 
 
 oz. 
 
 6 
 
 14 
 
 2 
 
 SO 
 
 12 
 
 4 
 
 17 
 
 1 
 
 21 
 
 10 
 
 1 
 
 
 ] 
 
 
 
 1 
 
 17 
 
 
 
 24 
 
 2 
 
 5 
 
 34 
 
 1 
 
 45 
 
 12 
 
 4 
 
 17 
 
 ! 
 
 21 
 
 10 
 
, 46 SUBTRACTION. 
 
 SO mamj units to he added as make one unit of the next MgJier 
 denomination ; after which add 1 to the next denomination of 
 the subtrahend, and subtract as before. 
 
 FIRST PROOF. 
 
 46. The difference or remainder, is such a number as added 
 to the subtraliend, will give a sum equal to the minuend (-45) ; 
 hence, 
 
 Add the remainder to the subtrahend. If the worh is right, 
 the sum zvill be equal to the minuend. 
 
 SECOND PROOF. 
 
 Since the remainder added to the subtrahend is equal to the 
 minuend (Art. 45), it follows that the excess of 9's in these two 
 numbers is equal to the excess of 9's in the minuend ; hence, 
 
 Find the excess of 9's in the minuend, in the subtrahend and 
 in the remainder ; if the work is right, the excess of 9's tJi the 
 two last numbers will be equal to the excess ofd's in the first. 
 
 What is the difference between 874136 and 45302 ? 
 
 2d. Method. 
 
 - 2 excess of 9's in the first. 
 
 - 5 « « 2d. 
 _ -G+5 = ll: 2 excess. 
 
 READING. 
 
 47. What is the difference between 426 and 295 ? 
 
 By the common method, wliieh is spelling, operation. 
 wo say, 5 from 6 leaves 1; 9 from 12 leaves 426 
 
 3 ; 1 to carry to 2 are 3 ; 3 from 4 leaves 1. 295 
 
 By reading the words which express the 131 
 
 final result, we make the operations mentally, and say, o?Jc. three, 
 cnc. 
 
 4G. \Miat is the diflcrcnce or ri'maitider ! How do you i)iove Sulttrac- 
 tion '' How do you prove Subtraction by the second method ! 
 47. Explain the process of reading in Subtraction. 
 
 1st. Method. 
 
 
 874136 
 
 874136 
 
 45302-1 
 
 45302 
 
 828834 
 
 828834 
 
SUBTRACTION. 
 
 47 
 
 OPERATION. 
 
 
 yr. 
 
 mo. 
 
 da. 
 
 h.r. 
 
 1855 
 
 1 
 
 4 
 
 15 
 
 1801 
 
 3 
 
 4 
 
 12 
 
 TIME BETWEEN DATES. 
 
 48. "What time elapsed between the inauguration of INIr. 
 JefFerson, March 4th, 12 o'clock, M., 1801, and July 4tli, 3 
 P. M., 1855 ? 
 
 Analysis. — Place the earlier date under the 
 later, writing the number of tlie year, reckoned 
 from the beginning of the Christian Era, on the 
 left. Then, write in the same line the num- 
 ber of the month, reckoned from the first of 54 4 3 
 January, the number of tlie day, reckoned from tlie first of the month, 
 tlie number of the hour, reckoned from 12 at night, and write tliei 
 number of minutes and seconds, if there are any, still at the right. 
 Hence, to find the time between two dates, 
 
 Write down as above, and subtract the earlier date from the latter. 
 
 Note 1 . — In finding the difference between dates, as in casting 
 interest, the month is regarded as tlie twelfth part of the year and 
 as containing 30 days. 
 
 2. The civil day begins and ends at 12 o'clock at night. 
 
 
 
 EXAMPLES. 
 
 
 
 
 (1.) 
 
 (2.) 
 
 (3.) 
 
 (4.) 
 
 From 
 
 472567 
 
 103796 
 
 900372 
 
 1760134 
 
 Take 
 
 . 109271 
 
 47217 
 
 167301 
 
 48207 
 
 
 (5.) 
 
 (6.) 
 
 (7.) 
 
 (8.) 
 
 
 rods. 
 
 dollars. 
 
 mills. 
 
 barrels. 
 
 From 
 
 74623457 
 
 8600000 
 
 162347 
 
 8462 
 
 Take 
 
 32700169 
 
 761820 
 
 56321 
 
 4071 
 
 
 (10.) 
 
 
 (11.) 
 
 
 bushels. 
 
 inches. 
 
 
 minutes. 
 
 From 
 
 100000 
 
 200763194 
 
 3 
 
 601789412 
 
 Take 
 
 37214 
 
 2142079 
 
 — 
 
 10031761 
 
 48. How do you find the difference of time between two dates! In 
 this computation, what part of a year is a month \ How many days arc 
 rccko-ned to the month 1 
 
48 
 
 6 
 
 SUBTRACTION. 
 
 
 
 (12.) 
 
 (13.) 
 
 (14.) 
 
 
 cords. 
 
 gallons. 
 
 pounds. 
 
 From 
 
 4200000 
 
 ^^ii^in 
 
 100000000 
 
 Take 
 
 325 
 
 9999 
 (16.) 
 
 23 
 
 
 (15.) 
 
 (17.) 
 
 From 
 
 $8475,656 
 
 $1000,759 
 
 $4871036,008 
 
 Take 
 
 32,015 
 
 194,375 
 
 17362,25 
 
 
 (18.) 
 
 (19.) 
 
 (20.) 
 
 
 £ s. d. far 
 
 . ton. cwt. qrs. lb. 
 
 yd. qr. ncu 
 
 From 
 
 25 12 6 2 
 
 5 17 3 21 
 
 137 1 3 
 
 Take 
 
 10 14 3 1 
 
 2 9 1 14 
 
 19 3 2 
 
 
 (21.) 
 
 (22.) 
 
 (23.) 
 
 
 X. mi. fur. rd. 
 
 tun. hhd. gal. qt. pt. 
 
 A. R. P. 
 
 From 
 
 16 2 1 2,1 
 
 14 1 26 2 1 
 
 100 2 27 
 
 Take 
 
 16 1 4 9 
 
 5 3 35 3 1 
 
 10 3 30 
 
 
 24.) 
 
 (25.) 
 
 (26.) 
 
 
 hush. ph. qt. 
 
 cords, ft. in. 
 
 E.E. qr. na. 
 
 From 
 
 1000 3 4 
 
 225 42 1242 
 
 42 1 2 
 
 Take 
 
 25 1 6 
 
 100 112 720 
 
 16 4 3 
 
 (27.) (28.) (29.) (30.) 
 
 ft 3 3 3 3 9 E.E qr. na. E:F. qr. na. 
 
 144 10 5 27 4 1 174 3 1 171 1 3 
 
 64 11 7 14 7 2 49 4 2 74 3 2 
 
 (31.) 
 
 (32.) 
 
 
 (33.) 
 
 (34.) 
 
 2\ civt. qr. 
 
 cwt. qr. lb. 
 
 qr. 
 
 lb. oz. 
 
 lb. oz. dr. 
 
 14 12 2 
 
 17 1 21 
 
 143 
 
 22 12 
 
 174 11 10 
 
 1 14 3 
 
 14 2 24 
 
 74 
 
 19 14 
 
 39 12 13 
 
 (35.) 
 
 (36.) 
 
 
 (37.) 
 
 (38.) 
 
 A. R. P. 
 
 A. R. P. 
 
 da. 
 
 hr. min. 
 
 Jir. min. sec. 
 
 12 1 32 
 
 112 1 31 
 
 167 
 
 21 50 
 
 147 50 51 
 
 1 3 14 
 
 74 2 37 
 
 19 
 
 23 54 
 
 94 59 57 
 
SUBTRACTION. 49 
 
 39. From $10000 take $1240,37i-. 
 
 40. From 183701289 take 34627. 
 
 41. From 17^/-. 9mo. Iwk. \Ma. take lOy. llmo. 2wk. bda. 
 
 42. From 1441fe 7 3 5 3 19 take 56t5 6 ! 7 3 19. 
 
 43. From two eagles seven dimes, take twelve dollars and 
 fifty cents. 
 
 44. From forty dollars twelve and a half cents, take twenty- 
 five cents and seven mills. 
 
 45. From one eagle five dollars six dimes and ten cents, 
 take five dollars seven cents and four mills. 
 
 46. What sum added to £11 145. 'd\d. will make £133 11*. 
 ^dl 
 
 47. An apprentice, who is 14 years 11 months 3 weeks, 
 14 hours 58 minutes old, is to serve his master until he is 21 
 years of age. How long has he to serve ? 
 
 48. Tiie greater of two numbers is seven millions three 
 hundred and four thousand and ten ; the less is nine hundred 
 and fifty thousand one hundred and forty. What is their differ- 
 ence ? 
 
 49. Mont Blanc, the highest mountain in Europe, is 15680 
 feet high ; Chimborazo, the highest in America, is 21427 feet. 
 What is the difierence in their heights ? 
 
 50. A man sold his farm for seven thousand five hundred 
 and thirty dollars, which was fifteen hundred and ten dollars 
 more than he gave for it. How much did he give for it ? 
 
 51. The revenue collected at the port of New York for the 
 year ending 30th June, 1853, was $38289341,58 ; at Philadel- 
 phia, $4537046,16 ; at Boston, $7203048,52 ; at Baltimore, 
 $836437,99. How much more was collected at the port of 
 New York than at the other three ? 
 
 52. A man engaging in trade found at the end of five years 
 that he had increased his capital ten thousand three hundred 
 and ten dollars, and that his whole capital amounted to forty- 
 six thousand five hundred dollars. How much did he com- 
 mence with ? 
 
 3 
 
60 SUBTRACTION. 
 
 53. The minuend exceeds the remainder by 683021, and the 
 remainder is 902563. What is the subtrahend ? 
 
 54. The amount of tea consumed in the United States in the 
 year 1846, was 16891020 pounds; the amount of coffee. 
 124336054 pounds. How much more coffee than tea ^Yas 
 consumed ? 
 
 55. "What number is that to which, if you add 3726, the sum 
 will be ten thousand ? 
 
 56. From a stack of hay containing dtons Sgr. 20Ib., I sold 
 4:tons 17 cwt. 22lb. How much was then left ? 
 
 57. A owes B £25 ; after paying him £5 9ic?., how much 
 will he still owe him ? 
 
 58. If the distance from New York to Liverpool be 3100 
 miles, after a ship has sailed 800/rt. ofur. 36roc?s, what distance 
 remains ? 
 
 59. Bought a farm for three thousand five hundi'ed dollars 
 and fifty cents ; sold the same for three thousand three hundred 
 dollars and eighty-seven and a half cents : how much did ho 
 lose by the bargain ? 
 
 60. If a lot of goods are bought for $750, and sold for 
 $925,871 what will be gained ? 
 
 61. If I buy a bushel of wheat for $1,871 ; ten gallons of 
 molasses for $2,50 ; five yards of cloth for $12,371 : how much 
 chanjre must I receive back for two ten dollar bills ? 
 
 62. The population of the United States in the year 1850 
 was 23191876, of which 3204313 were slaves: what was the 
 white population ? 
 
 63. England contains 50922 square miles ; Scotland, 31324 
 square miles ; "Wales 7398 square miles ; the United States 
 contain 2988892 square miles. How many more square miles 
 does the United States contain than the whole of Great 
 Britain ? 
 
 64. A gentleman of fortune owning an estate of two hun- 
 dred thousand dollars, bequeathed thirty thousand dollars to 
 objects of chaj'ity ; twenty-five thousand two lunulrcd and fifty 
 dollars to each of his three sons; twenty thousand five hundred 
 
SUBTRACTION. 51 
 
 and seventy-five dollars to his daughter ; and the remainder to 
 his widow. How much did the widow receive ? 
 
 65. The population of New Orleans in 1850 was 116375; 
 in 1854 it was 139190 : what was the increase in four years ? 
 
 66. Having deposited $1500 in a bank, I drew out at one 
 time $475,121; at another time $300; at another $526,25: 
 how much remained ? 
 
 67. If the Declaration of Independence was made at pre- 
 cisely 12 o'clock, on the 4th day of July, 1776 ; how much time 
 will have passed to the 4th day of March, 1857, at 30 minutes 
 past 3 o'clock, P. M. ? 
 
 68. If I borrow $1576 of a friend, and afterwards pay him 
 $920,871, how much would I still owe him ? 
 
 69. The first settlement made in the United States was at 
 Jamestown, in Virginia, May 23, 1607 : how many years from 
 that time to the 4th of July, 1856 ? 
 
 70. The sum of two numbers is 36804, and the less number 
 is eighteen thousand nine hundred and twenty-seven : what is 
 the sri'eater number ? 
 
 71. The revenue of the United States in the year 1853 was 
 $61337574 ; the expenditures $54026818 : how much did the 
 revenue exceed the expenditures ? 
 
 72. Tlie exports of the State of New York in the year 1853 
 were valued at $66030355 ; those of the State of Virginia, for 
 the same year, $3302561 : how much did the exports of New 
 York exceed those of Virginia ? 
 
 73. A ship-builder sold a vessel for $50376, which cost him 
 $42978 : how much did he gain ? 
 
 7 I. A farmer sold his farm for six thousand three hundred 
 and seventy-five dollars ; after paying his debts, he has four 
 thousand and fifteen dollars left : Avhat was the amount of his 
 debts ? 
 
 75. Gunpowder was invented in the year 1330 : how many 
 years from that time to the year 1856 ? 
 
 76. What number is that to which if you add 3726 the sum 
 will be ten thousand ? 
 
SUBTKACTION. 
 
 77. A speculator bought a quantity of flour for $2084,50 ; 
 of bacon for $760,871 ; of hops for $1836,25. He sold the 
 flour for §2375,60; the bacon for $912,375; the hops for 
 $1750 : what did he gain or lose on the \Yhole ? 
 
 78. A farmer has two pastures, one containing 9A. 3Ii. 32P. ; 
 the other 12^. 29P. He has also two meadows, one contain- 
 ing 10^. 2Ii. ; the other 15^, IB. 20P. : how much more 
 meadow than pasture has he ? 
 
 79. From a pile of wood containing 76 cords and 6 cord 
 feet, was taken at one time 20 cords and 48 cubic feet ; at 
 another time 14 cords 1 cord foot and 80 cubic feet : how 
 much remained in the pile ? 
 
 80. A gentleman purchased a house wortli $9430 ; a cai*- 
 riage for $475,50 ; a span of horses for $840,40. He paid at 
 onetime $5260; at another $1275,371; at another $936,42: 
 how mucli remained unpaid ? 
 
 81. If a ship and cargo are valued at $47568,487, and the cargo 
 alone at $3406,50, what is the value of the ship without the cargo? 
 
 82. A gentleman dying left an estate of $50000 ; after pay- 
 ing his debts, which amounted to $5647,50, he desired that 
 each of his two sons should receive $15000, and his widow the 
 remainder : how much did the widow receive ? 
 
 83. A note on interest, dated July 1st, 1853, was to be paid 
 March 20th, 1856 : how long was it on interest? 
 
 84. Bouglit a hogshead of wine, from Avhich was drawn 
 S2ffals. Iqt. 'ipt. ; how much i-emained in the cask ? 
 
 85. The population of Chicago in 1850 was 29963 ; in 1855 
 it was 80025 : what was the increase in three years ? 
 
 86. A land speculator owning twenty-five thousand acres of 
 land, sells at one time fifteen hundred acres ; at anotlier four 
 thousand seven hundred ; at another twenty-iive hundred acres ; 
 at anotlier seven hundred and fifty acres : what number of acres 
 has he left ? 
 
 87. The latitude of New Orleans is 29° 57' 30"; tiiat of 
 Boston, 42° 21' 23": what is the diflerence in the latitude of 
 these two places ? » 
 
MULTIPLICATION. 53 
 
 88. A person bought a span of horses for three hundred 
 dollars ; a carriage for $410,50 ; a harness for $50,G7o ; he 
 sold the wliole for six hundred dollars : did he gain or lose, and 
 how much ? 
 
 89. The population of Great Britain and its adjacent islands 
 in the year 1841, was 18GG47G1 ; in 1851 it was 209364G8 : 
 Avhat was the increase of population in ten years ? 
 
 90. From a piece of cloth containing 47 yards, a tailor cut 
 I'^yds. oqrs. 2na. : how much was left ? 
 
 91. A ti'adesman failing in business, was indebted to A 
 £105 19s. lid.; to B, £127 10s. 9i(7. ; to C, £34 I85. lOd. ; 
 to D, £500 19s. ; to E, £700 14s. Gif/. When this took place, 
 he had in cash £50, in goods to the amount of £350 14s. ^d., 
 his household furniture was worth £24 lis., his book accounts 
 amounted to £94 14s. 8c?. If all these were given np to the 
 creditors, how much would they lose ? 
 
 MULTIPLICATION. 
 
 48. Multiplication is the operation of talcing one number 
 as many times as there are units in another. 
 
 The number to be taken is called the midtiplicand. 
 
 The number denoting how many times the multiplicand is 
 taken, is called the midtiplier. 
 
 The result of the operation is called the product. 
 
 The multiplicand and multiplier are called factors, or pro- 
 ducers of the product. 
 
 Note. — Since, when the multiplier is an integral number, the 
 product may be obtained by adding the multiplicand to itself as many 
 times less 1 as there are unils in the multiplier, Multiplication is 
 sometimes called a short method of addition. 
 
 48. What is Multiplication ' What is the numl>er to be taken called! 
 What docs the multiplier denote 1 What is the result called ! What are 
 the multiplier and multiplicand called 1 Why is Multiplication called a 
 short method of Addition 1 
 
54 MULTIPLICATION. 
 
 49. Thei'e are three parts in every operation of multiplica* 
 tion. First, the multiplicand: second, the muUiplier : and third, 
 the product. 
 
 From the definition of Multiplication, we see that, 
 
 Is;;. If the multiplier is 1, the product Avill be equal to the 
 multiplicand. 
 
 2c?. If the multiplier is greater than 1, the product will be as 
 many times greater than the multiplicand, as the multiplier is 
 greater than 1. 
 
 3d. If the multiplier is less than 1, that is, if it is a proper 
 fraction, then the product will be such a part of the multipli- 
 cand as the multiplier is of 1. 
 
 50. Let it be required to multiply any two numbers together, 
 say 6 to 4. 
 
 Analysis. — If we write, in a horizontal line, fi 
 
 as many stars as there are units in the nmlti- , '^ \ 
 
 plicand, and write as many such lines as there 
 
 are units in the multiplier, it is evident that, 
 
 4 ■ 
 all the stars will represent the number of units 
 
 which arise from taking the multiplicand as 
 
 many times as there are units in the multiplier. 
 
 Change now the multiplier into the multiplicand : that is, multi- 
 ply 4 by 6. 
 
 Make in a vertical line, as many stars as there are units in the 
 new multiplicand, (4), and as many vertical lines are there are units 
 in the new multiplier, (6), when it is again evident that, all the stars 
 will represent the number of units in the product. Hence, 
 
 T7ie product of two factors is the same whichever factor is 
 used as the midtipUer. 
 
 3x7 = 7x3 = 21: also, Gx3 = 3x6 = 18. 
 9x5 = 5x9 = 45: also, 8x6 = 6x8 = 48. 
 
 and, 8x7 = 7x8 = 56: also, 5x7 = 7x5 = 35. 
 
 49. How many parts . are there in every operation of Multiplication 1 
 Wliat are they ? How many principles follow from the definition of Mul- 
 tiplication ■? What arc they ! 
 
 50. In how many way.s mny 6 and 4 be multiplied together I How do 
 the two produf.ts compare with each other 1 \\'lial docs this proved 
 
 vf> ^ 9f- ^ 9f ^ 
 •ff- ^ ^ ^ ^ ^ 
 ^ ^ *• ^ ^ ^ 
 
MULTIPLICATION. 55 
 
 52. A Composite Number is one that may be produced by 
 the multiplication of two or more numbers, called factors^ 
 Thus, 2 X 3 rz: 6, in which G is the composite number, and 
 2 and 3 the factors. Also, 16 r=: 8 X 2, in which IG is a com- 
 posite number, and 8 and 2 the factors ; and since 4 X 4 = 16, 
 we may also regard 4 and 4 as factors of 1 G. 
 
 A Prime Number is one which cannot be produced by mul- 
 tiplication, and is divisible only by itself and 1. 
 
 53. Let it be required to multiply 7 by the composite number 
 6, of which the factors are 2 and 3. 
 
 7 
 
 _3 
 
 21 
 
 42 
 
 C£> 
 
 
 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 « 
 * 
 * 
 
 * 
 * 
 * 
 
 * 
 * 
 * 
 
 * 
 * 
 * 
 
 * 
 * 
 * 
 
 * 
 * 
 * 
 
 * 1 
 
 * J 
 '* 1 
 
 1-2 X 7= 14 
 
 » 3 
 
 » 
 
 * 
 
 * 
 
 * 
 
 * 
 
 * 
 
 * 
 
 * J 
 
 ;-2 42 
 
 * 
 
 * 
 
 * 
 
 * 
 
 * 
 
 « 
 
 * 1 
 
 *2 
 
 * 
 
 * 
 
 * 
 
 * 
 
 * 
 
 * 
 
 * J 
 
 1 
 
 If we write 6 horizontal lines with 7 units in each, it is evi- 
 dent that the product of 7 X 6 = 42, will express the number 
 of units in all the lines. 
 
 Let us first connect the lines in sets of two each, as at the 
 right ; the number of units in each set will then be expressed 
 by 7 X 2 = 14. But there are 3 sets; hence, the number of 
 units in all the sets, is 14 x 3 = 42. 
 
 Again, if we divide the lines into sets of 3 each, as at the 
 left, the number of units in each set will be equal to 7 x 3 = 21, 
 and since there are two sets, the whole number of units will be 
 expressed by 21 X 2 = 42. 
 
 53. What is a composite number 1 Give an example of a composite 
 number 1 What are its factors 1 What are the factors of 161 What is 
 a prime number 1 
 
 53. If several factors are multiplied together, will the product be altered 
 jy changing their order 1 How do you multiply by a composite number ' 
 
56 irULTIPLI'JATlON. 
 
 Since the product of either two of the three factors 7, 3 and 
 2, will be the same whichever be taken for the multiplier (Art. 
 50), and since the same principle will apply to that product and 
 lo the other foctor, as well as to any additional factor, if intro- 
 iuced, it follows that, 
 
 The product of any number of factors will he the same in 
 vhatever order they are multiplied : 
 
 Hence, to multiply by a composite number, 
 
 I. Separate the composite number into its factors : 
 
 II. Midtiply the midtiplicand and the partial products hy the 
 factors, in succession, and the last product will be the entire pro- 
 duct sought. 
 
 Note. — Any number whatever, as 440, ending with 0. is a com- 
 posite number of which 10 is a factor: for, 440 = 44 x 10. If there 
 are two O's on the right of the significant figures, then 100 is a fac- 
 tor, and so on for a greater number of ciphers. Hence, when there 
 are ciphers on the right of significant figures, either in the multipli- 
 cand or multiplier, or both, 
 
 Multiply the significant figures together, and then annex the ciphers 
 to the product. 
 
 54. 1. Multiply G27 by 214. 
 
 Analysis. — The multiplicand 627 is to be ta- operation. 
 
 ken 214 times : that is, 4 units times, 1 ten times, 627 
 
 and 2 hundred times. Taking it 4 units times, 214 
 
 gives 2508 ; taking it 1 ten times gives 627, of 2508 
 
 which the lowest unit is 1 ten ; hence. 7 is written 627 
 
 in the tens place : taking it 2 hundred times, gives 1 254 
 
 1254. the lowest unit of which is 1 hundred. 134178 
 Adding, we have 134178 for the product. 
 
 Note. — What is one factor of a number ending in ' "What is one 
 factor of a number ending in two O's ! In throe 0"s ! &c. How du vou 
 multiply by such a number when there are cipliers in one or both factors ? 
 
 54. Explain the operation of niulliplying 6'-i7 by 214. Explnin the five 
 principles which come from this analysis. What is a partial produ -t ! 
 Give the general rule for multiplication. What must be observed in tli« 
 multiplication of United States money? 
 
MULTIPLICATION. 57 
 
 It is seen, from the preceding analysis, that 
 
 1. If units be multiplied hy units, the unit of the product will 
 he 1. 
 
 2. If tens be multiplied by units, the unit of the product will 
 he 1 ten. 
 
 3. If hundreds be multiplied by imits, the unit of the product 
 will he 1 hundred; and so on: 
 
 And since the product of the factors is the same whichever 
 is taken for the multiplier (Art. 50), it follows that, 
 
 4. If units of the first order he multiplied hy units of a higher 
 order, the units of the product will be the same as that of the 
 higher order. 
 
 5. If units of any order he multiplied hy units of any other 
 order, the iinit of the product luill he of an order one less than 
 the sum of the units denoting the two orders. 
 
 Note. — When the multiplier contains more than one figure, the 
 product obtained by multiplying the multiplicand by a single figure, 
 is called a •partial product. In the last example there are three 
 partial products, 2508, 627, and 1254. The sum of the partial pro- 
 ducts is equal to the product sought : 
 
 2. Multiply £3 8s. M. 3/ar. by 6. 
 
 Analysis. — Multiplying 3 farthings by 6, we operation. 
 
 have 1 8 farthings, equal to Ad. and 2far. : set £ s. d. far. 
 down the 2/ar. : then, 6 times Qd. are 36rf., and 3 8 6 3 
 
 4 pence to carry are 40fZ., equal to 3 shillings 6 
 
 and Ad.: then, 6 times 85. are 485. and 3s. to 20 11 4 2 
 carry are 51 shillings, equal to £2 and 11 shil- 
 lings: then, 6 times jC3 are jClS and jC2 to carry are jC20, which 
 set down. 
 
 Note. — The vmit of each product will be the same as the unit of 
 the multiplicand. Hence, for the multiplication of all numbers, we 
 have the following 
 
 Rule. — Multiply every order of units in the multiplicand, in 
 succession, beginning with the lowest, hy each figure in the mul- 
 tiplier, and divide each product so formed hy so many units as 
 
 naJce one unit of the next higher denomination : write down each 
 
 3* 
 
5S MULTIPLICATION. 
 
 remainder under the units of its oivn order, and carry the quo* 
 tient to the next product. 
 
 Note. — In multiplj-ing United States money, care must be taken 
 to point off as many places for cents and mills as there are in the 
 multiplicand. 
 
 1- Multiply 14 dollars IG cents and 8 mills, by 5, 6, and 7. 
 
 $14,168 $14,168 $14,168 
 
 5 6 7 
 
 (2.) (3.) (4.) 
 
 $870,40 $894,120 $2141,096 
 
 9 14=i7 X 2 36 = 6 X 6. 
 
 PROOFS OF MULTIPLICATION. 
 
 55. There are three methods of proof for multiplication : 
 
 I. Write the multiplier in the place of the multiplicand, and 
 h'.d the product as before : if the two products are the same, 
 t>6 ycvk ia supposed to be right (Art. 50). 
 
 II, 7>:vide the product by one of the factors, and the quotient 
 "wijl be ths^- ol\\2x factor. 
 
 HI. L'y iSa method of casting out the 9's. 
 
 FIRST METHOD. 
 
 MuLiply • - 80432 506 
 
 by - - 506 80432 
 482592 4048 
 402 160 2024 
 
 40o98592 1518 
 
 1012 
 40698592 
 
 Note 1. — Althougii wc generally begin the multiplication by tho 
 figure of the lowcsi unit, yet we may multiply in any order, if we 
 only preserve the places of the different orders of units. In the ex- 
 ample at the right, we began with the order of tens of tlicusands (r 
 5tl: order. 
 
 55. How many proofs are there for multiplication'! What is the first 1 
 What is the second 1 Wliat is the third 1 
 
OPERATION. 
 
 
 641 = 639 + 
 
 2 
 
 232 = 225 + 
 
 7 
 
 4473 + 14 
 
 450 
 
 
 3195 
 
 
 1278 
 
 
 1278 
 
 
 148698 + 14 
 
 MULTIPLICATION. 59 
 
 2. Although either factor may be used as tlie multiplier (Art. 50), 
 Etill it is best to use that one which contains the fewest figures. 
 For, if \vc change the process and use the multiplicand as the mul- 
 tiplier, there -vvill be more multiplications, as shown in the last 
 example. 
 
 PROOF BY THE O'S. 
 
 56. Let it be required to multiply any two numbers together, 
 as 641 and 232. 
 
 Analysis. — We first find the excess over 
 exact 9's in both factors, and then separate 
 each factor into two parts, one of which 
 shall contain exact 9's, and the other the 
 excess, and unite the two by the sign plus. 
 It is now required to take 639 + 2 = 641, 
 as many times as there are units in 
 225 + 7 = 232. 
 
 Every partial product, in this multiplica- 
 tion, contains exact 9's, except 14, which 
 
 contains one 9 and 5 over; and as the same may be shown for any 
 two numbers, we see that, 
 
 If we find the excess of 9's in each of two factors, and then 
 multi'ply them together, the excess of 9's in their product will be 
 equal to the excess of 9's in the product of the factors. 
 
 (1.) Ex. (2.) Ex. 
 
 Multiply 87G03 - - 6 818327 - - 2 
 
 by 98G5 ^ 9874 1 
 
 Prod. 864203595 - - 6 80S01G0798 - - 2 
 
 3. By multiplication we have 
 
 Ex.4. Ex.8. Ex.4. Ex. of product, 2. 
 7285 X 143 X 976 = 1016752880. 
 
 Note. — Is it necessary to commence the multiplication with the lowest 
 unit 1 Which factor is it most convenient to use as a multiplier 1 
 
 56. How do you find the excess of 9's in the product of two factors'! 
 If the excess of 9's in any factor is 0, what is the excess of 9's in the 
 product 1 
 
60 
 
 MULTIPLICATION. 
 
 Ex.5. Ex.4. Ex.0. Ex.0. 
 4. We also have 869 x 49 X 36 = 1532916. 
 
 Note. — When the excess of 9's in any factor is 0, the excess of 9's 
 ill the product is always 0. 
 
 EXAIIPLES. 
 
 (1.) (2.) (3.) (4.) 
 847046 9807602 570409 216987 
 8 7 6 9 
 
 (5.) 
 
 (6.) 
 
 (7.) 
 
 103672 
 
 8163021 
 
 90031746 
 
 42 - 
 
 126 
 
 274 
 
 (8.) 
 
 (9.) 
 
 (10.) 
 
 $14,168 
 
 $894,126 
 
 $20034,645 
 
 5 
 
 14 
 
 48 
 
 (11.) 
 
 (12.) 
 
 (13.) 
 
 47321809 
 
 1237506 
 
 437024 
 
 4261 
 
 3460 
 
 400 
 
 (14.) 
 
 (15.) 
 
 (16.) 
 
 8703600 
 
 107030 
 
 30671200 
 
 34600 
 
 5700 
 
 482 
 
 (17.) 
 
 (18.) 
 
 (19.) 
 
 £ s. d. 
 
 T. qr. lb. oz. 
 
 yd. ft. in. 
 
 20 6 8 
 
 3 3 21 14 
 
 16 2 9 
 
 4 
 
 8 
 
 7 
 
 (20.) 
 
 (21.) 
 
 (22.) 
 
 
 hlid. gal. qt. pt. 
 
 E.F. qrs. na. 
 
 12° 42' 55" 
 
 4 42 2 1 
 
 24 2 3 
 
 9 
 
 12 
 
 24 
 
 23. Mulfiply IStons 2qrs. IGlbs. Ooz. by 48. 
 
 24. JMultiply Oyr. 8ino. 2ivL Ma. 42m. by 56. 
 
 25. Multiply 68 by the factors 9 and 8 of the composit > 
 number 72. 
 
MULTIPLICATION. 61 
 
 26. Multiply 3657 by the factors of 64. 
 
 27. Multiply 37046 by the factors of 121. 
 
 28. Multiply 2187406 by the factors of 144 
 
 29. Multiply 430714934 by 743. 
 
 30. Multiply 37157437 by 14972. 
 
 31. Muhiply 47157149 by 37049. 
 
 32. Multiply 57104937 by 40709. 
 
 33. Multiply 79861207 by 890416. 
 
 34. Multiply 9084076 by 9908807. 
 
 35. Multiply 2748 by 200. 
 
 36. Multiply 67046 by 10 : also by 100. 
 
 37. Multiply 57049 by 100 : also by 1000. 
 
 38. Multiply 4980496 by 1000 : also by 10000. 
 
 39. Multiply 90720400 by 100 : also by 10000. 
 
 40. Multiply 74040900 by 1 : also by 10. 
 
 41. Multiply 674936 by 100 : also by 100000. 
 
 42. Multiply 478400 by 270400. 
 
 43. Multiply 367000 by 37409000. 
 
 44. Multiply 7849000 by 84694000. 
 
 45. Multiply 89999000 by 97770400. 
 
 46. Muhiply 9187416300 by 274987650000. 
 
 47. Multiply 86543291213456 by 12637482965. 
 
 48. Multiply 76729835645873 by 217834569. 
 
 49. If it costs 2479 dollars to build one mile of plank road, 
 bow much would it cost to build 25 miles ? 
 
 50. How far would a vessel sail in 9 days, of 24 hours each, 
 at the rate of 15 miles an hour ? 
 
 51. A man bought two farms, one of 125 acres, at 26 dollars 
 an acre ; another of 96 acres, at 32 dollars an acre ; he paid at 
 one time 2500 dollars ; at another time 1725 dollars : what 
 remained to be paid ? 
 
 52. In 9 pieces of kersey, each containing lAyd. 3qr. 2na., 
 how many yards ?* 
 
 * NorE. — When the multiplicand is a compound denominate number, 
 and the multiplier a composite number, it is best always to multiply by the 
 factois of the composite number. 
 
62 MULTIPLICATION. 
 
 53. Wliat will 15 gallons of wine cost at 5s. S^d. per gallon ? 
 
 54. What will be the value of 416 sheep at $2,48 a head ? 
 
 55. Bought 40 barrels of flour at $8,75 a barrel, and sold 
 them for $9,121 a barrel : what was the whole gain ? 
 
 56. What is the weight of 11 hogsheads of sugar, each 
 weighing 7cwL 2qr. 18lb., and what would be its value at 6 
 cents a pound ? 
 
 57. A merchant bought 36 pieces of broadcloth, each piece 
 containing 44 yards, at 4 dollars a yard : what did the whole 
 cost? 
 
 58. A gentleman whose annual income is $3479, expends for 
 pleasure and travelling $600 ; for books and clothing $570 ; for 
 board and other expenses $1200 : how much will he save in 
 5 years ? 
 
 59. The number of milch cows in the state of New York in 
 1850 was 931324 : Avhat would be their value at 18 dollars 
 each ? 
 
 60. If a man travel 20/?u*. ofur. IQrd. in one day, how far 
 will he travel in 24 days ? 
 
 61. How long will it take a man to mow 14 acres of grass, 
 allowing 10 working hours a day, if he mow one acre in 4Jir. 
 4omt. SOsec. ? 
 
 62. If a man spend six cents a day for segars, how mucli 
 will he spend in thirty years, allowing three hundred and sixty- 
 five days to the year ? 
 
 63. A farmer sold 118 bushels of barley for 62^ cents a 
 bushel, and receives 5 barrels of flour at $9,87^ a barrel, and 
 the remainder in cash : how much cash did he receive ? 
 
 64. Two persons start at the same point and travel in oppo- 
 site directions, one at the rate of 34 miles a day, the other at 
 the rate of 28 miles a day : how far apart will they be at the 
 end of 14 days? 
 
 65. An apothecary sold 8 bottles of laudanum, each con- 
 taining 10 T 6 5 2 9 14^7*. : what was the weight of the 
 whole ? 
 
 GO. A farmer took 7 loads of oats to market, each load 
 
MULTIPLICATION. 63 
 
 having 20 bags, and each bag containing 2hush. opk. Qqi. : how 
 many bushels of oats did he take to market ? 
 
 67. If in a woollen factory 4G8 yards, of cloth are made in 
 one day, how many yards will be made in 313 days ? 
 
 G8. The greatest number of whales ever captured in the 
 northern seas, in one season, was 2018. Estimating the oil 
 produced from each to have been 212 barrels, what was the 
 amount of oil produced ? 
 
 69. What will be the value of an ox weighing 7cwf. 2qr. 
 IQlh., at 11 cents a pound ? 
 
 70. What will be the cost of 245 hogsheads of sugar, each 
 weighing 984 pounds, at 7 cents a pound ? 
 
 71. Bought 6 loads of hay, each weighing 18cw^ Sqis. 21/6. ; 
 after letting a neighbor have 2tons Ibcwt. Iqr. bib,, how much 
 will there be Left ? 
 
 72. In an orchard there are 136 apple trees, each tree yield- 
 ing 17 bushels of apples : how many bushels did the whole 
 orchard yield, and what would they be worth at 42 cents a bushel? 
 
 73. A flour mei'chant bought 1845 barrels of flour at 7 dol- 
 lars per barrel. He sold at one time 528 barrels, at 9 dollars a 
 barrel ; at another time 856 barrels at 8 dollars a barrel ; how 
 many barrels had he left, and at what cost ? 
 
 74. What are 25 hogsheads of sugar worth, each weighing 
 872 pounds, at 6^ cents a pound ? 
 
 75. It is estimated that the whole amount of land appropriat- 
 ed by the General Government for educational purposes, to 
 the 1st of January, 1854, was 52770231 acres. What would 
 be the value of this land at the Government price of one dollar 
 and twenty-five cents an acre ? 
 
 76. If 30 men can do a piece of work in 25 days, how long 
 will it take one man to do it ? 
 
 77. A man desired that his property should be equally divided 
 omono- his 5 children, giving each twenty-seven hundred dol- 
 lars : what was the amount of his property ? 
 
 78. Bought 9 chests of tea, each containing 72 pounds, at 
 37:^ cents a pound : wha/ was the cost of the whole ? 
 
64: MULTIPLICATION. 
 
 79. A farm consisting of 127 acres, was sold at auction foi 
 S37,565 an acre : what sum of money did it bring ? 
 
 80. A drover bought 127 head of beef cattle at an average 
 of 39 dollars per head ; he sold 86 of them for 43 dollars per 
 head; for how much per head must he sell the remainder, to 
 clear on the first cost 1246 dollai's ? 
 
 81. What will 75 firkins of butter cost, each firkin weighing 
 56 pounds, at 16 cents a pound? 
 
 82. A merchant bought a box of goods containing 37 pieces, 
 each piece containing 46 yards, worth 7 dollars a yard : what 
 did the box of goods cost ? 
 
 .. 83. A bond was given April 20th, 1850, and was paid Sept. 
 4th, 1856 : what will be the product, if the time which elapsed 
 from the date of the bond to the time it was paid be multiplied 
 by 45 ? 
 
 84. "What distance will a Wheel 16 feet 8 inches in circum- 
 ference measure on the ground, if rolled over 84 times ? 
 
 85. What is the difference between twice eight and fifty, and 
 twice fifty-eight ? 
 
 86. How much wood in 4 piles, each containing 5 cords, 
 6 cord feet and 32 cubic feet ? 
 
 87. A man bought 56 acres of land for $25 an acre, and 94 
 acres for $32 an acre ; if he sells the whole at $30 an acre, 
 will he gain or lose, and how much ? 
 
 88. If 12 men can build a wall in 16 days, how many men 
 will be required to build a wall nine times as long m half the 
 time ? 
 
 89. A former sold 4 cows for $25,50 each; 12 sheep for 
 $2,12-1^ each ; and 3 calves for $7,25 each ; what was the 
 amount of the sale ? 
 
 90. If it requires 116 tons of iron to construct one mile of 
 railroad, how much would it require to construct a railroad 
 from Albany to Buff;ilo, it being 326 miles? 
 
 91. A merchant bought 960 pounds of cheese at 9 cents a 
 pound ; 148 pounds of butter at 12^ cents a pound. He gave 
 in payment, 12 yards of cloth, at $4,75 a yard ; 186 pounds of 
 
MTTLTIPLICATION, 
 
 65 
 
 sugar at 7 cents a pound, and the remainder in cash : how much 
 cash did he pay ? 
 
 92. If a family consume 12^'/. 25-1;. \pt. of ale in a week, 
 how much will they consume in 14 weeks? 
 
 93. How much brandy will supply an army of 25,000 men 
 for one month, if each man requires Igal. 2qf. Ijyt. 2gi. 
 
 94. It is estimated that the French, during the years 1854 
 and 1855, transported to the Crimea 80000 horses, and that 
 70000 of them were lost in the same time. Supposing the 
 first cost of each horse to be $100, and the cost of transporta- 
 tion $95 per head, what was the value of the horses lost ? 
 
 95. A man purchased a piece of woodland containing 27 
 acres, at 39 dollars per acre ; each acre produced on an average 
 70 corda of wood, which, being sold, yielded a nett profit of 45 
 cents a cord : how much did the profit on the wood fall short of 
 paying for the land ? 
 
 BILLS OF PAKCELS. 
 
 9G. Chicago, June 10, 1856. 
 
 J/r. John C. Smith, Bought of David Toombs. 
 
 14 pounds of tea, at 75 cents, - - $ 
 
 9 " 
 
 " coffee. 
 
 14 " 
 
 42 « 
 
 " sugar, 
 
 11 " 
 
 3 " 
 
 " pepper, 
 
 12^ " 
 
 5 " 
 
 " chocolate. 
 
 56 " 
 
 12 " 
 
 " candles. 
 
 16 " 
 
 Received payment, 
 
 
 David Toombs. 
 
 97 New York, March 20th, 1857. 
 
 Mr. Jacob Johns, Bought of George Bliss Sf Co 
 48 pounds of sugar at 9i cents a pound, - - - $ 
 6 hogs, of molasses, each containing 63 gallons, 
 at 27 cents a gallon, 
 
 8 casks of rice, 285 lbs. each, at 5 cts. a pound, 
 
 9 chests of tea, 86 lbs. each, at 87^ cts. a pound, 
 
 4 bags of coffee, each 67 lbs., at 11 cts. a poiuid, 
 
 Received payment, ^ 
 
 4 Geo. Bliss &'. C-o- 
 
66 MULTIPLICATION. 
 
 98. Hartford, Novembet 21s<, 1856. 
 
 Gideon Jones, Bought of Jacoh Thrifty. 
 
 78 chests of tea, at $55,G5 per chest, - - % 
 251 bags of coffee, 100 pounds each, at) 
 12^ cts. per pound, - - ) 
 
 317 boxes of raisins, at $2,75 per box, 
 1049 barrels of shad, at $7,50 per barrel, - - - 
 76 barrels of oil, 32 gallons each, at $1,08 per gal., 
 
 Amount, $ 
 Received the above in full, Jacob Thrifty. 
 
 99. Baltimore, Jan. \st, 1855. 
 Mr. Abel Wirt, Bought of Timothy Stout. 
 
 10 yards of broadcloth, at $4,37i, - - - $ 
 75 " " sheeting, " ,09 - 
 42 " " plaid prints '' ,45 - 
 
 5 barrels of Genesee flour, at $7,87^, - 
 
 7 pairs of boots, at $1,60 per pair, 
 18 bushels of corn, at 72 cts. per bushel, 
 
 "$ 
 
 100. Montreal, Oct. 16th, 1855. 
 3fr. Chas. Snow, Bought of Vose, Duncan Sf Co. 
 
 45 yai-ds of broadcloth at 9s. M. - - £ s. d. 
 
 56 " " " 12s. ^d. - 
 
 16 " vesting?, " 6s. 8^d. - - - 
 
 24 lbs. colored thread, " os. Ad. - - - 
 
 72 pairs silk hose, " 7s. 5|(/. - 
 
 108 yards carpeting, " 14s. 10c?. - - - 
 
 Received payment, £ 
 
 VosE, Duncan & Co. 
 
DIVISION. 
 
 67 
 
 DIVISION 
 
 57. Division is the operation of finding from two munhers a 
 ihird, which midtiplied by the first, to ill produce the second. 
 
 The first number, or number by which we divide, is called 
 the divisor. 
 
 The second number, or number to be divided, is called the 
 divide7id. 
 
 The third number, or result, is called the quotient. 
 
 The quotient shows how many times the dividend contains 
 the divisor. 
 
 When the quotient is expressed by an integral number, the 
 division is said to be exact. When it cannot be so expressed, 
 the part of the dividend that is undivided, is called the remainder. 
 
 58. There are always three numbers in every division, and 
 sometimes four : 1st. the dividend ; 2d. the divisor ; od. the 
 quotient ; and 4th, the remainder. 
 
 There are three methods of denoting division ; they are the 
 following : 
 
 12 -r 3 expresses that 12 is to be divided by 3. 
 -L2 expresses that 12 is to be divided by 3. 
 
 3)12 expresses that 12 is to be divided by 3. 
 When the last method is used, if the divisor does not exceed 
 12, we draw a line beneath the dividend and set the quotient 
 under it. If the divisor exceeds 12, we draw a curved line on 
 the right of the dividend, and set the quotient at the right. 
 
 59. Short Division is the operation of dividing when the 
 work is performed mentally, and the results only written down. 
 It is limited to the cases in which the divisors do not exceed 12. 
 
 57. What is division l What is the number to be divided called 1 What 
 is the number called by which we divide 1 What is the answer called 1 
 What is the number called which is left 1 
 
 58. How many parts are there in division 1 Name them. How many 
 eigns arc there in division 1 Mako and name them. 
 
 59. What is short division ? How is it generally performed 1 Where is 
 the quotient written ] To what cases is it limited 1 
 
68 DIYISTON. 
 
 1. Divide 45G by 4. 
 
 Analysis. — Tlie number 456 is made up of 4 hundreds, 5 tens, 
 ?nd 6 units, each of which is to be divided by 4. 
 Dividing 4 hundreds by 4, we have the quotient, operation. 
 1 hundred : 5 tens divided by 4, gives 1 ten and 4)456 
 
 1 ten over : reducing this to units and adding in 114 
 
 die 6, we have 16 units, which contains 4, 
 4 times: hence, the quotient is 114: that is, the dividend contains 
 the divisor 114 times. 
 
 2. Divide £11 8s. Id. Zfar. by 5. 
 
 Analysis. — Dividing £11 by 5, the quofient is £2 and £1 re- 
 maining. Reducing this to shillings and adding 
 in the 8, we have 28s., which divided by 5, gives operation. 
 5s. and 35. over. This being reduced to pence £ s. d. far. 
 and Id. added, gives 43J. Dividing by 5, we 5)11 8 7 3 
 have ^d. and 3c?. remainder. Pteducing 2d. io 2 5 8 3 
 
 farthings, adding 3 farthings, and again dividing 
 by 5, gives the last quotient figure 3/ar. 
 
 3. Divide £6 85. M. by 8. operation, 
 
 £ s d 
 Here we have to pass to shillings before 
 
 making the first division. 
 
 S)6 8 8 
 
 16 1 
 
 4. Divide 11772 by 327. 
 
 Analysis. — Flaving set down the divisor on the left of the dividend, 
 it is seen that 327 is not contained in the first 
 three iigvues on the left, which are 117 hundreds. opekation. 
 
 But by observing that 3 is contained in 11, 327)11772(36 
 3 times and something over, we conclude that the 981 
 
 divisor is contained af least 3 times in the first 1962 
 
 four figures, 1177 tens, which is a pnrtialdividcnd. 1962 
 
 Set down the quotient figure 3, and multiply the 
 divisor by it : we thus get 981 tens, which being less than 1 177. the 
 quotient figure is not too great: we subtract the 981 tens from the 
 first four figures of the dividend, and find a remainder 196 tens, 
 which being less than the divisor, the quotient figure is not too small. 
 Reduce this remainder to units and add in the 2, and we have 1962. 
 
 As 3 is contained in 19, 6 times, we conclude that the divisor is 
 contained in 1962 as many as 6 times. Setting down 6 in the quo- 
 
DIVISION. 69 
 
 tient and mnltipJymg the divisor by it, we find the product to be 
 19C2. Therefore the entire quotient is 36. or the divisor is contained 
 36 times in the dividend. 
 
 60. From the above analysis, we have the following rule for 
 the division of numbers. 
 
 1. Begin icith the highest order of miits of the dividend, and 
 pass on to the lower orders until the fewest nmnher of figures he 
 found that will contain the divisor : divide these figures by it for 
 the first figure of the quotient : the unit of this figure will be the 
 same as that of the lowest order in the partial dividend. 
 
 II. Mnhiplg the divisor by the quotient figure so found, and 
 subtract the product from the partial dividend. 
 
 III. Reduce the remainder to units of the next loiver order, 
 and add in the units of tJiat order found in the dividend: this 
 gives a new partial dividend. Proceed in a similar manner until 
 units of every order shdl have been divided. 
 
 DIRECTIONS FOR THE OPERATIONS. 
 
 Notes. — There are tivc operations in Long Division. 1st. To write 
 down the numbers : 2d. To divide, or find how many times : 3d. Tc 
 multiply: 4th. To subtract: 5th. To bring down, to Ibrm the partia 
 dividends. 
 
 2. The product of a quotient figure by the divisor must never b 
 larger than the corresponding partial dividend : if it is, the quotien 
 figure is too large and must be diminished. 
 
 3. When any one of the remainder.s is greater than the divisor, the 
 quotient figure is too Fmall and must be increased. 
 
 4. Tlie unit of any quotient figure is the same as that of the partial 
 dividend from which it is obtained. The pupil should always name 
 the unit of every quotient figure. 
 
 60. Give the rule for the division of numbers. 
 
 Notes. — 1. How many operations are there in division 1 Name them. 
 
 2. If a partial product is greater than the partial dividend, what does it 
 indicate '^ What then do you do ? 
 
 3. What do you do when an}' one of the remainders is greater than the 
 divisor '\ 
 
 4. What is the unit of any figure of the quotient \ \^'hen the divisoi 
 18 contained in simple units, what will be the unit of tlic quotient figure 
 
70 DIVISION. 
 
 0. If any partial dividend is less than the divisor, the correspond 
 ing quotient figure is 0. 
 
 6. When there is a remainder, after division, write it at the right 
 of tiie quotient, and place the divisor under it. 
 
 PRINCIPLES RESULTING FROM DIVISION. 
 
 1. When the divisor is equal to the dividend, the quotient will he 1. 
 
 2. When the divisor is less than the dividend, the quotient will ha 
 greater than 1. The quotient will be as many times greater than 1 
 as the dividend is times greater than the divisor. 
 
 3. Wlien the divisor is greater than the dividend, the quotient will 
 be less than 1. The quotient will be such a part of 1, as the divi- 
 dend is of the divisor. 
 
 4. When the divisor is 1, the quotient will be equal to the divi- 
 dend. 
 
 PROOF. 
 
 61. There are two methods of proof for division : 1st. By 
 multiplication ; 2d. By the excess of 9's. 
 
 FIRST METHOD. 
 
 By the definition of division, the quotient is such a number 
 as multiplied by the divisor will produce the dividend (Ai-t. 57). 
 
 In example 4, each product of the divisor by a figure of the quo- 
 tient is a partial product, and the sum of these products is the product 
 of the divisor and quotient (page 57, Note). Each product is taken, 
 
 M'hcn it is contained in tens, what will be the unit of the quotient figure 1 
 Wlien it is contained in hundreds'! In thousands] 
 
 5. If any partial dividend is less than the divisor what is the correspond- 
 ing figure of the quotient ? 
 
 6. When there is a remainder after division, what do you do witli it? 
 Note. — 1. When the divisor is equal to the dividend, what will the quo- 
 tient be 1 
 
 2. When the divisor is less than the dividend, how will the quotient 
 compare with 1 1 How many times will it be greater than 1 ? 
 
 3. When tlie divisor is greater than the dividend, how will the quotient 
 compare with 1 ? What part will the quotient be of 1 ? 
 
 4. When the divisor is 1, what will the dividend be ! 
 
 61. How many methods of proof are there for division ! What are they ! 
 What is the proof by multiplication ! What is the proof by the 9's ? 
 
DIVISION. 71 
 
 geparately, from the dividend, and nothing operation 
 
 remains. But, taking each product away, in 327)11772(36 
 
 succession, leaves the same remainder as would 1981 
 
 be left if their sum were taken away at once. 1962 
 
 Hence, the number 36, when multiplied by 1962 
 the divisor 327, gives a product equal to the 
 
 dividend 11772; therefore, 36 is the quotient (Art. 57): hence, to 
 prove division, 
 
 Blidtiply tlie divisor hy the quotient and add in the remainder, 
 if any. If the work is right, the result will be the same as the 
 dividend. 
 
 Note. — Divide 325 by 19. The quotient is 17, and 16 remainder: 
 the true quotient is 17^| , for. this being multiplied by the divisor 19, 
 will give the dividend. If the pupil knew how to dispose of the frac- 
 tional part, we should simply say, " Multiply the divisor by the 
 quoticntj^' which is exactly what we do under the rule. 
 
 PROOF BY 9's. 
 
 Since the dividend is equal to the product of the divisor and 
 quotient, it follows that if the excess of 9's in the divisor be 
 multiplied by the excess of the 9's in the quotient, the excess 
 of 9's in the product will be equal to the excess of 9's in the 
 dividend (Art. 56). Hence, 
 
 Find the excess of 9's in the divisor and in the quotient : mul- 
 tiply them together, and note the excess of 9's in the product : if 
 this is equal to the excess of 9's in the dividend, the work may he 
 regarded as Hght. 
 
 Divisor, 327, excess of 9's - - 3 ] 
 
 Quotient, 36, « - - . o\ ^''^^^^^> ^' 
 
 Dividend, 11772, " - - - 
 
 EXAMPLES. 
 
 (1.) (2.) (3.) (4.) 
 
 3)19737 4)147308 5)1346840 6)1650930 
 
 (5.) (6.) (7.) 
 
 6)47689872 9)10324683 7)506321494 
 
72 
 
 DIVISION. 
 
 £ 
 
 3)47 
 
 (8.) 
 
 s. 
 19 
 
 A. 
 9)37 
 
 (9.) 
 i?. 
 3 
 
 P. 
 
 17 
 
 (11.) 
 
 $ cts. 
 8)634 75 
 
 6 
 
 (10.) 
 
 yd. qr. na. 
 5)47 3 1 
 
 (12.) 
 
 $ cts. m. 
 7)1468 96 
 
 (13.) 
 
 $ 
 12)802346 
 
 16 
 
 Divide $29,25 by 26. 
 Divide $10,125 bv 27. 
 Divide $347,49 by 429 
 Divide $751,50 by 150. 
 Divide $571 1,04 by 108. 
 Divide $315 by $35. 
 Divide $50065 by $527. 
 Divide $432 by 54. 
 
 14. Divide 734947644 by 48. 22. 
 
 15. Divide 8536752 by 36. 23. 
 
 16. Divide 3367598 by 19. 24. 
 
 17. Divide 49300 by 725. 25. 
 
 18. Divide 6477150 by 145. 26. 
 
 19. Divide 770 by 28. 27. 
 
 20. Divide $87,256 by 5. 28. 
 
 21. Divide $495,704 by 129. 29. 
 
 30. Divide 334422198 by 438. 
 
 31. Divide 714394756 by 1754. 
 
 32. Divide 47159407184 by 3574. 
 
 33. Divide 5719487194715 by 45705. 
 
 34. Divide 4715714937149387 by 17493. 
 
 35. Divide 671493471549375 by 47143. 
 
 36. Divide 571943007145 by 37149. 
 
 37. Divide 1714347149347 by 57143. 
 
 38. Divide 49371547149375 by 374567. 
 
 39. Divide 171493715947143 by 571007. 
 
 40. Divide 6754371495671594 by 678957. 
 
 41. Divide 7149371478 by 121. 
 
 42. Divide 71900715708 by 57149. 
 
 43. Divide 14714937148475 by 123456. 
 
 44. Divide 729^. 2R. IP. by 41. 
 
 45. Divide oQMa. Qhr. by 240. 
 
 46. Divide 1298mi. 2fur. 33rd. by 37. 
 
 47. Divide 95Md. Ggal by 120. 
 
 48. Divide 232bush. 3ph. Igt. by 105. 
 
 49. Divide $18306,25 by 725. 
 
 50. Bonp;ht 7 yds. of clDtli for 1 6.«. Ad. : wlint did it cost per yd. ? 
 
DIVISION. 73 
 
 51. A man travelled 2Gomi. Gfur. IQrd. in 12 clays : how far 
 did he travel in one day? 
 
 52. If 569^. 2Ii. 23P. be equally divided between 9 per- 
 sons, how much will 5 of them have ? 
 
 53. The annual income of a gentleman is $10000: how 
 much is that per day, counting 365 days to the year ? 
 
 54. What number multiplied by 9999 will produce 987551235 ? 
 
 55. A gentleman owning an estate of $75000, gave one-fourtL 
 of it to his wife, and the remainder was divided equally among 
 his five children : liow much did each receive ? 
 
 56. The expenditure of the United States for 1853 was 
 $54026818 : how much Avould that be per day, allowing 365 
 days to^the year ? 
 
 57. If 28 yai'ds of cloth cost $133, what will one yard cost? 
 
 58. If I pay $637,50 for 51 yards of cloth, what is the price 
 per yard ? 
 
 59. The city of New York, in 1850, had 104 periodical i)ub- 
 lications, with an aggregate circulation of 78747600 copies: 
 what would be the average circulation of each ? 
 
 60. Bought 19. bushels of wheat for $30,875 : what was the 
 cost of one bushel ? 
 
 61. How long will 9125 loaves of bread last 5 families, if 
 each family consume 5 loaves a day ? 
 
 62. The product of two numbers is 7207272072, and the 
 multiplier 9009 : what is the multiplicand ? 
 
 63. How many rings, each weighing Adwt. 12g7\, can be 
 made from lOor. 11 dwt. 12gr. of gold ? 
 
 64. If iron is worth 2 cents a pound, how much can be 
 bought for $67,50 ? 
 
 65. If 14 sticks of hewn timber measure 12 2^. 38/;!. 118m., 
 how much does each stick contain ? 
 
 66. In 1850, Pennsylvania manufactured 285702 tons of pig 
 iron, and employed 9285 hands : what was the average product 
 of each hand ? 
 
 67. The number of collesre libraries in the United States in 
 1850, was 213, containing 942321 volumes : what would be the 
 
 average number of volumes in each ? 
 
 ■1 
 
74 OONTEAOTIONS 
 
 CONTRACTIONS AND APPLICATIONS. 
 
 CONTRACTIONS IN MULTIPLICATION. 
 
 62. Contractions in Multiplication are short methods of find- 
 ing products when the multipliers are particular numbex'S. 
 
 63. To multiphj hy 25. 
 
 1. Multiply 356 by 25. 
 
 Analysis. — If we annex two ciphers to the niul- operation. 
 tiplicand, we multiply it by 100 (Art. 5a): this 4)35600 
 product is 4 times too great; for the multiplier is 8900 
 
 but one-fourth of 100 ; hence, to multiply by 25, 
 
 Annex two ciphers to the multiplicand and divide the result 
 hy 4:. 
 
 EXAMPLES. 
 
 2. IMultiply 287 by 25. 
 
 3. Multiply 184 by 25. 
 
 4. IMultiply 6741 by 25. 
 
 5. Multiply 3074 by 25. 
 
 OPERATION 
 
 15 
 
 3^ 
 
 3 
 
 45 
 
 48 Ans 
 
 64. When the multiplier contains a fraction. 
 
 AVhat is the jjroduct of 15 multiplied by 31 ? 
 
 Analysis. — The multiplicand is to be taken 
 3 and one-fifth times : taking it one-fifth times, 
 gives 3, wliich we write in the units place : 
 then, taking it 3 times, gives 45, and the sum 
 48 IS the product ; hence 
 
 Rule. — Take such a pari of the multi- 
 plicand as the fraction is of 1 ; then multiply hy the integral 
 number, and the sum of the products will be the required 
 product. 
 
 2. Multiply 327 by 8.}. 5. Multiply 1272 by 12|. 
 
 3. Multiply 23474 by 16^. I 6. Multiply 9824 by 272^. 
 
 4. Multiply 34700 by 127"^. | 7. Multiply 3828 by 731. 
 
 62. Wliat are contractions in multiplication 1 
 
 G3 How do you multiply by 251 
 
 C4. How do you multiply when the multiplier contains a fraction! 
 
IN MULTIPLICATION. 
 
 76 
 
 65. To multiplij hj 12?,. 
 1. Multiply 286 by 12J. 
 Analysis. — Since 121 is one-eighth of 100, 
 Annex two ciphers to the multiplicand and divide 
 the result by 8. 
 
 EXAMPLES. 
 
 OPERATION. 
 
 8)28600 
 
 1. Multiply 384 by 121. 
 
 2. Multiply 476 by 12^. 
 
 3. 
 
 Multiply 14800 by 12^ 
 4. Multiply G70418 by 12-1. 
 
 G6. To multiply hy 331. 
 1. Multiply 975 by 33 J. 
 
 Analysis. — Aunoxiiig Uxo cipliers to the 
 multiplicand, multiplies it by 100: but the 
 multiplier is one-third of 100 : hence, 
 
 Annex two ciphers and divide the result by 3. 
 
 operation. 
 3)97500 
 32500 
 
 EXAMPLES. 
 
 1. Multiply 1670252 by 33i 
 
 2. Multiply 1480724 by 33i 
 
 67. 2"o multiply hy 125. 
 ]. Multiply 1125 by 125. 
 
 Analysis. — Annexing three ciphers to the 
 multiplicand, multiplies it by 1000 : but 125 
 is but one-eighth of one thousand : hence, 
 
 Annex three ciphers and divide the result by 8 
 
 EXAMPLES. 
 
 3. Multiply 10675512 by 331. 
 
 4. Multiply 4442172 by 33i. 
 
 OPERATION. 
 
 8)1125000 
 140625 
 
 1. Multiply 59264 by 125. 
 
 2. Multiply 17593408 by 125. 
 
 3. Multiply 1940812 by 125. 
 
 4. Multiply 140588 by 125. 
 
 COXTRACTIO^VS IN DIVISION. 
 68. Contractions in Division are short methods of finding 
 the quotient, when the divisors are particular numbers. 
 
 65. How do you multiply by 12.| 1 
 
 66. How do you multiply by 33^1 
 
 67. How do you multiply by 125 ] 
 
 68. What are Contractions in Division? 
 
76 CONTKACTIONS 
 
 69. By revei'siiig the last four processes, we have the four 
 following rules : 
 
 1. To divide any number by 25 : 
 
 Multiply the number by 4, and divide the product. by 100. 
 
 2. To divide any number by 12i : 
 
 Multiply the number by 8, and divide the product by 100. 
 
 3. To divide any number by 33 J : 
 
 Multiply the number by 3, and divide the product by 100. 
 
 4. To divide any number by 125 : 
 Multiply by 8, and divide the product by 1000. 
 
 EXAMPLES. 
 
 1. Divide 6350 by 25. 
 
 2. Divide 21345 by 25. 
 
 3. Divide G5G280 by 25. 
 
 4. Divide 7278675 by 25. 
 
 5. Divide 5287215 by 25. 
 
 6. Divide 12225 by 12^. 
 
 7. Divide 10650 by 12i. 
 
 8. Divide 11925 by 121. 
 
 9. Divide 1760000 by 12^. 
 
 10. Divide 67500 by 331. 
 
 11. Divide 1308400 by 331. 
 
 12. Divide 15851400 by 331. 
 
 13. Divide 8072400 by 33J. 
 
 14. Divide 281250 by 125. 
 
 15. Divide 6015750 by 125. 
 
 16. Divide 2026875 by 125. 
 
 70. Wlien the divisor is a composite number. 
 
 1. How many feet and yards are there in 288 inches ? 
 
 Analysis. — Since there are 12 inches in 1 foot, there will be as 
 many feet in 288 inches as 12 is contained times 
 in 288 ; viz., 24 feet, in which the unit is 1 foot. operation. 
 
 Since 3 feet make 1 yard, there will be as many 12)288 
 
 yards in 24 feet as 3 is contained times in 24 ; 3)24 
 
 viz., 8 yards : in which the unit is 1 yard. We 8 
 
 have thus passed, by division, from the unit 1 
 inch to the unit 1 foot, and then to the unit 1 yard ; that is, in each 
 
 69. What rules do you get by reversing the four previous rules ? Gito 
 them. 
 
 70 What is a composite number 1 Under ".<o\v ninny points of view 
 may division be regarded 1 What arc they ! A'liat is the rule for division 
 when the divi.sor i~ ;> romposile nunibor ' When there are remainders 
 after division, how do -ov And tli remainder in units of the dividenrl ■" 
 
IN DIVISION. 77 
 
 operation, we have increased the unit as many times as there are units in 
 the divisor. 
 
 Let us now use the same numbers, ii an entirely difFerent question.' 
 
 2. If 288 dollars be equally divided among 3G men, what 
 will be the share of each ? 
 
 Analysis. — Since 288 dollars is to 36 = 12 x 3 operation. 
 be equally divided among 36 men, each 12)288 
 
 ■will liavo as many dollars as 36 is con- 3)24 
 
 tained times in 288. Dividing 288 into 8 
 
 12 equal parts, we find that each part 
 
 is 24 dollars. If each of these parts be now divided into 3 equal 
 parts, there will then be 36 parts in all, each equal to 8 dollars : 
 here, the unit of the result is the same as that of the dividend. Hence, 
 we may regard division under two points of view : 
 
 1. As a process of reduction, in which the unit of each suc- 
 ceeding dividend is increased as many times as there arc units in 
 the divisor : 
 
 2. As a process of separating a number into equal parts ; in 
 %vhich case the unit of a part will he the same as that of the 
 dividend. 
 
 Hence, the follow^ing rule when the divisox* is a composite 
 number : 
 
 RuLii;. — Divide the dividend hy one of the factors of the divi- 
 sor ; then divide the quotient, thus arising, hy a second factor, 
 and so on, till every factor has been used as a divisor: the last 
 quotient will be the answer. 
 
 EXAMPLES. 
 
 Divide the following numbers by the factors : 
 1. 2322 by 6 = 2 x 3. 5. 1145592 by 72 = 8 X 9 
 
 2. 37152 by 24 = 4 x G. 
 
 3. 19152 by 36 z= G X G. 
 
 4. 38592 by 48 = 4 X 12. 
 
 G. 185760 by 96 = 8 X 12 
 
 7. 115776 by 64= 8 x 8. 
 
 8. 4G3104 by 144 = 12 x 12 
 
 Note. When there are remainders, after division, the operation la 
 to be treated as one of Reduction. 
 
78 CONTE ACTIONS 
 
 9. Divide the number 3671 by 3C = 2 X 3 x 5. 
 
 Analysis. — Dividing 3671 by 2, yve have a quotient 1S35, and a 
 remainder, 1. Alter the second 
 
 division, we have a quotient 611, operation. 
 
 and a remainder, 2 ; and after the 2)3671 
 
 third division, the quotient 122, 3)1835 . . 1. 
 
 and the remainder, 1. Now. it is 5)611 . . 2. 
 
 plain, from the first analysis, that, 122 .. 1. 
 
 1. The unit of the first quotient 1X3 + 2= 3 + 2= 5; 
 
 is as many times greater than the 5X2 + 1 = 10 + 1 = 11 rem. 
 unit of the dividend, as the divi- Ans. 122^^. 
 
 sor 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 imits of the given dividend, is 
 simply a case of reduction in which the divisors denote the 
 units of the scale : therefore, 
 
 I. Multiphj the last remainder hy the last divisor hut one, and 
 add in the preceding remainder. 
 
 II. Multiphj this residt hy the next preceding divisor, and add 
 in the remainder, and so on, till you reach the unit of the divi- 
 dend. 
 
 Divide the following numbers by the factors, and find the 
 remainders : 
 
 1. 41G705 by 315 - 7x9x5. 
 
 2. 80410G by 462 = S x 2 x 7 x 11. 
 
 3. 756807 by 3456 = 4 X 8 x 9 x 12. 
 
 4. 8741659 by 105 =: 3 x 5 x 7. 
 
 5. 917043 by 385 = 5 x 7 x 11. 
 
 6. 4704967 by 1155 = 11x7x5x3. 
 
 7. 71874607 bv 7560 = 8x7x9x5x3. 
 
IN DIVISION. 79 
 
 71. When the divisor is 10, 100, 1000, S^c. 
 
 1. Divide 3278 by 1000 = 10 x 10 x 10. 
 
 Analysis. — We divide 3278 by 10, by operation. 
 
 simply cutting off 8, giving 327 tens and 10)327|8 
 
 8 units remainder. We again divide by 10)32|7 . 8 rem. 
 
 10, by cutting ofT the 7, giving 32 hun- 10)3|2 . . 7 rem. 
 
 dreds and 7 tens remainder. We again 3 . . 2 rem 
 
 divide by !0 by cutting ofF tlie 2, giving ^^-^^ Ans. 
 a quotient of 3 tliousands and 2 Imndreds 
 
 remainder. The quotient then is 3, and a remainder of 2 hundreds 
 7 tens and 8 units, or 278. 
 
 Rule. — Cut off from the right of the dividend as many 
 fgurcs as there are ciphers in the divisor, considering the figures 
 at the left the quotient, and those at the right the remainder. 
 
 72. When any divisor contains significant figures with one or 
 more ciphers at the right hand. 
 
 2. Divide 87589G by 32000. 
 
 Analysis. — The divisor 32000 = 32 x 1000. operation. 
 
 Dividing by 1000 gives a quotient 875, and 896 32l000)875l896(27 
 remainder. Then dividing by 32 gives a quo- 64 
 
 tient 27, and 11 remainder, Vi^hicli gives the 235 
 
 result 271^11}. Hence, 224 
 
 Rule. — Cut off, by a line, the ciphers from ^ , ^ „ „ 
 
 **^ ^115. 27 A — . 
 
 the right of the divisor, and an equal num- ' 3 2000- 
 
 ber of fgurcs from the right of the dividend : divide the remain' 
 
 ing figures of the dividend by ilie remaining figures of the divisoVy 
 
 and the remainder^ if any, with the figures cut off from the divi' 
 
 dend annexed, will form the true remainder. 
 
 EXAMPLES. 
 
 Divide the followinjr numbers : 
 
 1. 1972G54 by 420000. 
 
 2. 1752000 by 12000. 
 
 3. 73199006 by 801400. 
 
 4. 11428729800 by 72000. 
 
 5. 3G981400 by 14G000. 
 
 6. 141614398 by 63000. 
 
 71. How do you divide when the divisor is 1, with ciphers annexed 1 
 
 72. How do you divide when the divisor contains significant figures, 
 with ciohers annexed^ How do vou find the true remainder 1 
 
80 APFLICATiO.N.S 
 
 78. When the divisor contains a fraction. 
 
 1. Divide 856 by 4i operation. 
 
 -Analysis. — There are 5 filthy in 1 ; hence, 21=3x7 3)4280 
 in 4 there are 20 fifths; therefore, 4^ = 21 7)142(i -2 
 
 fifths. In the dividend 856, there are 5 times 203 -5 
 
 as many fifths as units 1 ; that is, 4280 fifths ; Ans. 203t4. 
 
 therefore, the quotient is 4280 divided by 21, 
 equal 203-^|^. Hence, when the divisor contains a fractional part, 
 
 Reduce the divisor and dividend to the fractional unit of the 
 divisor, and then divide as in integral numbers. 
 
 Find the quotients in the following examples : 
 
 1. 
 
 3245-^ 
 
 -16^. 
 
 5. 
 
 87317^ 9|. 
 
 2. 
 
 47804-^ 
 
 -15}. 
 
 6. 
 
 87906 ^12f 
 
 3. 
 
 870631 -f- 
 
 -14i. 
 
 7. 
 
 95675 ^ 15f. 
 
 4. 
 
 37214-^ 
 
 -511. 
 
 8. 
 
 71096 -M7f. 
 
 APPLICATIONS IN MULTIPLICATION. 
 
 74. The analysis of a practical question, in Multiplication, 
 requires that the multiplier be an abstract number ; and then 
 the unit of the product will be the same as the unit of the mul- 
 tiplicand. 
 
 75. To find the cost of several things, lohen we kiwio the 
 price of one and the numher of things : 
 
 1. What will six yards of cloth cost at 8 dollars a yard ? 
 
 Analysis. — Six yards of cloth will cost 6 times as much as 1 yard. 
 Since 1 yard of cloth cost 8 dollars, 6 yards Avill cost 6 times 8 dol- 
 lars, which are 48 dollars ; therefore. 6 yards of cloth at 8 dollars a 
 yard, will cost 48 dollars : hence, 
 
 TJie cost of any numher of things is equal to the price of a 
 single thing midtiplied by the nu7nber of things. 
 
 But we have seen that the product of two numbers will be 
 the same, (that is, will contain the same number of units) which- 
 
 73. How do you divide when the divisor contains a fraction ! 
 
 74. What docs the analysis of a practical question lequire! 
 
 75. How do you find the cost of a single thing 1 How is it done in 
 practice 1 
 
ATPLICATIONS. 81 
 
 ever be talcen for the multiplicand (Art, 50). Hence, in prac- 
 tice, we may multiply the two factoi-s together, taking either for 
 the muhi2)lier, and then assign the 2>^'oper unit to the product. 
 We generally take the less number for the multiplier. 
 
 76. To find the cost when the price is an aliquot part of a dollar : 
 1. Find the cost of 45 bushels of apples, at 25 cents a bushel. 
 
 Analysis. — If the price were 1 dollar a bushel, operation. 
 the cost would be as many dollars as there aie 4)45,00 
 
 bushels. But the price is 25 cents = ^ of a dol- $11,25 
 
 ]ar; hence, the cost will be one-fourth as many 
 dollars as there are bushels ; that is, as many dollars as 4 is contained 
 times in 45 = 11 dollars and 1 dollar over. This is reduced to cents 
 by adding two ciphers j then dividing again by 4, we have the entire 
 cost : hence. 
 
 Tithe such a part of the number which denotes the amount (f 
 the commodity, as the price is of 1 dollar : the result ivill be the 
 cost in dollars. 
 
 EXAMPLES. 
 
 1. What would be the cost of 284 bushels of potatoes, at 50 
 cents a bushel ? 
 
 2. At 331 cents a gallon, what will 51 gallons of molasses cost? 
 
 3. What cost 112 yards of calico, at 12^ cents a yard ? 
 
 4. If a ponnd of butter cost 20 cts., what will 175 pounds cost ? 
 
 5. What will 576 bushels of 2)S576 cost at 1 dollar a bushel. 
 
 wheat cost, at $1,50 a bushel ? J^ '' j^^e^its 
 
 i?>64 " Sfl,50 " 
 
 6. What will it cost to dig a ditch 129 rods long, at §1,331 
 a rod ? 
 
 7. At $1,25 a barrel, what will 96 barrels of apples cost? 
 
 8. What will 5 pieces of cloth cost, each piece containing 25 
 yards, at $1,20 a yard ? 
 
 77. To find the cost of articles sold by the 100 or 1000. 
 
 1. What will 544 feet of lumber cost at 2 dollars per 100 ? 
 
 76. How do you find the cost, when the price is an aliquot part of a dollar 1 
 
 77. How do you find the cost of articles sold by the hufidred or thou- 
 sand 1 
 
 4* 
 
82 APPLICATIONS^ S. 
 
 Analysis. — At 2 dollars a foot, the cost would be 544 x 2 = 1088 
 dollars : but as 2 dollars is the price of 100 feet, it foJlows that 1088 
 dollars is 100 times the cost of the lumber; therefore, if ^ye divide 
 1088 dollars by 100 (which is done by cutting off two of the right 
 hand figures, Art. 71), we obtain the cost. 
 
 Note. — Had the price been so much per 1000, we should have 
 divided by 1000 : hence, 
 
 Multiply the quantity hy the number denoting the price ; if the 
 price he by the 100, cut of two figures on the right hand of the 
 product ; if by the 1000, cut off three, and the remaining fig- 
 ures will be the answer in the same denomination as the price, 
 tvhich, if cents or mills, may be reduced to dollars. 
 
 EXAMPLES. 
 
 1. What will be the cost of 3742 feet of timber at $3,25 
 per 100 ? 
 
 2. At $12,50 per 1000, what will 5400 feet of boards cost? 
 
 3. Richard Ames, Bought of John Maple. 
 1275 feet of boards at $9,00 per 1000 
 
 3720 
 
 
 u 
 
 
 15,25 
 
 it 
 
 715 
 
 
 scantling 
 
 
 8,75 
 
 u 
 
 1200 
 
 
 timber 
 
 
 12,0G 
 
 u 
 
 2550 
 
 
 lathing 
 
 
 75 
 
 lOi 
 
 965 
 
 
 plank 
 
 
 1,121 
 
 (< 
 
 Received payment, John Maple. 
 
 *76. To find the cost of articles sold by the ton. 
 What is the cost of 640 pounds of hay at $11,50 per ton? 
 
 Analysis. — Since there arc 2000 operation. 
 
 pounds in a ton, the cost of 1000 2)$1].50 
 
 l)ouiids will be half as much a.s of 5.75 price of 1000 lbs 
 
 1 ton : viz., $5,75. Mulliply this 640 
 
 by the number of pounds (640). and 23000 
 
 cut off three places from the right 3450 
 
 (Art. 71), in addition to the two $3^68000 
 
 places cut off for cents ; hence, 
 
 *76. How do you find the cost of articles sold by the toni 
 
APPLICATIONS. 83 
 
 MxdiqAy one-half the price of a ion by the number of pounds, 
 and cut ojf three figures from the right hand of the product. The 
 remaining figures will be the ansiver in the same denomination as 
 the price of a ton. 
 
 EXAMPLES. 
 
 1. What will be the cost of 1575 pounds of plaster at $3,84 
 per ton ? 
 
 2. At $7,37-1- a ton, what will 3496 pounds of coal cost ? 
 
 3. What will 1260 pounds of hay cost at $9,40 per ton ? at 
 $10,25 ? at $14,60 ? 
 
 4. What will be the cost of transportation of 5482 pounds 
 of iron from Pittsburgh to New York at $6,65 per ton ? 
 
 APPLICATIONS IN DIVISION. 
 
 *77. Abstractly, the object of division is to find from two given 
 numbers a third, which, multiplied by the first, will produce 
 the second. Practically, it has three objects : 
 
 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 ot 
 equal parts. 
 
 Analysis. — Consider the number denoting cost or price as abstract; 
 then make the division and assign the proper unit to the quotient. 
 Hence, we have the following 
 
 RULES 
 
 I. Divide the number denoting the cost ly the number of 
 things : the quotient will be the price of one. 
 
 II. Divide the number denoting the cost hy the price of one : 
 the quotient will be the number of things. 
 
 III. Divide the whole number of things by the number of 
 2)arls into which they are to be divided: the quotient will be the 
 number in each part. 
 
 ^77. What is the object of Division abstractly? How many objects haa 
 it practically ^ Name the objects. Give the rules for the three cases. 
 
84* 
 
 PRACTICE. 
 
 PRACTICE. 
 
 77*. Practice is an easy method of applying the rules oS 
 Anthraetic to questions which occur in trade and business. 
 
 An aliquot part of a number is an exact part : hence, if 
 a number be divided by an aliquot part, the quotient will be an 
 integral number. 
 
 TABLE OF ALIQUOT PARTS. 
 
 Cts. 
 
 Parts 
 of$l. 
 
 Parts of£l. 
 
 Parts of 1 
 f<hilling. 
 
 Mo. 
 
 Parts of 
 a year. 
 
 Days. 
 
 Parts of 
 1 month. 
 
 50 = 
 
 h 
 
 10s. =^ 
 
 6 d.=i 
 
 6 = 
 
 i 
 
 15 =- 
 
 i 
 
 33^= 
 
 i 
 
 Gs. 8(1.=^ 
 
 4 J.=l 
 
 4 = 
 
 i 
 
 10 = 
 
 i 
 
 25 =- 
 
 i 
 
 5s. =1 
 
 3 d.=\ 
 
 3 = 
 
 i 
 
 H = 
 
 i 
 
 20 = 
 
 i 
 
 4s. =1 
 
 2 d.=l 
 
 2 = 
 
 i 
 
 6 = 
 
 i 
 
 12i = 
 
 i 
 
 3s. 4rf.=i 
 
 lirf.=i 
 
 1 = 
 
 tV 
 
 5 = 
 
 1 
 
 6 
 
 6i = 
 
 iV 
 
 2s. 6d.=\ 
 
 1 d. = ^^ 
 
 
 or ^ of 
 
 3 = 
 
 iV 
 
 5 == 
 
 ^ 
 
 ls.8d.=^\ 
 
 
 
 3mo. 
 
 
 
 EXAMPLES. 
 
 1. What is the cost of 37G yards of cloth at $1,75 = 1^ 
 dollars per yai'd? 
 
 Analysis. — At $1 a yard it would cost $376 ; At Si, — $376 
 
 at 50 cents = S^; a yard, it would cost $188; At 50cts. = Si = 188 
 
 at 25 cents = $|- a yard, it would cost $94 ; At 25cts.=t^= 94 
 
 hence, at $1,75 = $1|, it will cost $658. total cost, $658 
 
 2. What is the cost of 196 yards of cotton at 9 c?. = |s. per 
 yard? Ans. £7 7s. 
 
 3. What is the cost of 425 yards of tape at lie?, per yard ? 
 
 Jiis. £2 13s. lirf. 
 
 4. What is the cost of 475 yards of tape at l^c?. per yard ? 
 
 Ans. £2 9s. o^d, 
 
PEAOTICE. 85* 
 
 5. "What is the cost of 354 yards of cord at l^d. per yard ? 
 
 Ans. £1 16s. lO^d. 
 
 6. At 121 cents = || a yard, Avhat will be the cost of 4756 
 yards of bleached shirting ? A7is. $594,50. 
 
 7. At 2s. 6d. = £J per pair, what will be the cost of 3754 
 pairs of gloves ? - Ans. £469 5s. 
 
 8. If wlieat is 3s. Gd. a bushel, what will be the cost 5320 
 bushels? Aiis. £931. 
 
 9. If bi-oadcloth costs £1 7s. a yard, what will be tlie cost 
 of 435 yards? Ans. £587 5s. 
 
 10. If linen is 2^. Gd. = 2^s. a yard, what will be the cost 
 of 660 yards ? Ans. £82 10s. 
 
 11. What will be the cost of AOlbs. of soap, if 1 pound costs 
 6f cents? ylns. $2,70. 
 
 12. If a yard of twisted cord costs 21 cents, what will be 
 the cost of 140 yards ? Ans. $3,15. 
 
 13. If one bushel of apples cost 62^ cents, what will be the 
 cost of 876 bushels ? Ans. $547,50. 
 
 14. What will be the cost of 1000 quills, if every 5 quills 
 cost 11 cents ? Ans. $3,00. 
 
 15. If 1 yard of extra-superfine cloth costs |91, what will 
 be the cost of 851 yards ? ^,i5. f 812,25. 
 
 16. What will be the cost of 1848 yards of linen cambric 
 at 87^ cents a yard? Ans. $1617. 
 
 17. If one yard of broadcloth cost $4,871 cents = $41, 
 what will be the cost of 696 yards ? Ans. $3393. 
 
 18. If the price of 1 yard of cloth is $4,75 z=: $4f, what 
 will be the cost of 281 yards ? Ans. $135,375. 
 
 19. If 1 quart of oil costs 14^ cents, what will be the cost 
 of Ihhd. 2gaL 3qts. ? Ans. $38,135. 
 
 20. What will be the cost of 350 bushels of potatoes, at 3s. 
 Qd. = 3^s. a bushel ? Ans. £61 5s. 
 
84 LONGITUDE AND TIME. 
 
 LONGITUDE AND TIME. 
 
 78. The equator of the earth, like other circles, is divided 
 into 360°, which are called degrees of longitude. 
 
 79. The sun apparently goes round the earth once in 24 
 hours. This time is called a day. 
 
 Hence, in 24 hours, the sun apparently passes over 3G0^ of 
 longitude; and in 1 hour over 360'^-i-24 = 15''. 
 
 80. Since the sun, in passing over 15° of longitude, requires 
 1 hour or 60 minutes of time, 1° will require 60 minutes 
 -r-lo = 4 minutes of time; and 1' of longitude will require 
 one-sixtieth of 4 minutes, which is 4 seconds of time : hence, 
 
 15° of longitude require 1 hour. 
 1° of longitude requires 4 minutes. 
 1' of longitude requires 4 seconds. 
 Hence, we see that, 
 
 1. If the degrees of longitude be multiplied by 4, the 2>'>'oduct 
 will be the corresponding time in minutes. 
 
 2. If the minutes in longitude be multiplied by 4, the jyroduct 
 will be the corresp)onding time in seconds. 
 
 1. What is the time in hours, minutes and seconds in 56° 47' ? 
 
 Analysis. — First reduce the de- operation. 
 
 grees to hours and minutes ; then m. hr. m. sec. 
 
 reduce the minutes to minutes and 56° x 4 = 224 =3 44 
 seconds, and take the .svim. 47' X 4 = 18S sec. = 3 8 
 
 3 47 8 
 
 81. When the sun is on the meridian of any place, it is 12 
 o'clock, or noon, at that place. 
 
 78. How is the equator of the earth supposed to be divided i 
 
 79. How does the sun appear to move ' "What is a day 1 How far 
 docs the sun appear to move in 1 hour ! 
 
 80. How do you reduce degrees of longitude to time 1 How do you 
 reduce minutes of longitude to time ? 
 
 8L What is the hour when the sun is on tlie meridian 1 When the 
 eun is on the meridian of any place, how will the time be for all places 
 east 1 How for all places west 1 If you have the difference of time, how 
 do you find the time 1 
 
LONGITUDK AND TIME. 85 
 
 Now, as the sun apparently goes from east to west, at the 
 instant of noon, at one place, it will be past noon for all places 
 at the east, and before noon for all places at the west. 
 
 If then, Ave 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 their difference, if 
 that other be east, or by subtracting \i,\i west. 
 
 82. 2^0 reduce time to degrees and minutes of longitude. 
 
 1. The difference of time between Boston and New Orleans 
 is 1 hour 11 minutes and 48 seconds : what is the difference of 
 lonfritude ? 
 
 Analysis. — Since 1 hour corresponds operation. 
 
 to 15° of longitude, there will be as 15° x 1 = 15° 
 
 many times 15° as thei-e are hours: 11 -i- 4= 2° 45' 
 
 Since 1° corresponds to 4 minutes of 48 -i- 4= 12' 
 
 time, there will be as many degrees as Diff. 17° 57' 
 
 4 is contained times in the minutes : 
 
 Since 1' corresponds to 4 seconds of time, there will be as many 
 minutes as 4 is contained times in the seconds : hence, 
 
 1. Multiply 15° by the number of hours, and the 'product will 
 he degrees of longitude : 
 
 2. Divide the minutes by 4, and the quotient will be degrees 
 and minutes of longitude : 
 
 3. Divide the seconds by 4, and the quotient ivill be minutes 
 and secunds of longitude. The sum of these results will be the 
 difference of longitude. 
 
 EXAIMPLES. 
 
 1. The longitude of Albany is 73° 42' west, and that of 
 Buffalo 78° 55' west : what is the difference of longitude and 
 what the difference of time ? 
 
 2. 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 ? 
 
 82. How do you reduce time to degrees and minutes of longitude '^ 
 
86 APPLICATIONS. 
 
 3. When it is 12 M. at New York, it is 11 o'clock 6 minutea 
 and 28 seconds at Cincinnati : wliat is their difference of lon- 
 gitude ? 
 
 4. The longitude of Philadelphia is 75° 10' Avest, and that 
 of New York 74° 1' west: what is the difference of time be- 
 tween these two places ? 
 
 5. Washington is in longitude 77° 2' west, New Orleans in 
 89° 2' west. When it is 9 o'clock A. M. at Washing-ton, what 
 is the time at New Orleans ? 
 
 6. If the difference of time between two places be 42/».z. 
 IGiCc, what is the difference of longitude? 
 
 7. What is the difference of longitude between two places if 
 the difference of time is 2//. 20»i/. 44sec, ? 
 
 8. The longitude of St, Louis is 90° 15' west; a person at 
 that place observed an eclipse of the moon at lOA. 40;/i?. P. M. ; 
 another person, in a neighboring state, observed the same eclipse 
 22/m/. 125fr. earlier: what was the longitude of the latterplace, 
 and the time of observation ? 
 
 9. If the difference of time between London and Ore"-on 
 Citj is 8 hours, what is the difference in longitude ? 
 
 10. The difference of longitude between St. Louis and New 
 l^ork 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 ? 
 
 APPLICATIONS OF THE PRECEDING RULES. 
 
 1. What will it cost to build a wall 9G rods long, at $1,33} a 
 rod ? 
 
 2. A farmer wishes to put lOGGiws/^ Iph. of potatoes into 
 474 barrels, how much nuist he put into each barrel ? 
 
 3. At $4,32 a yard, what will 121 yards of cloth cost ? 
 
 4. How many barrels of apples, each containing 1\ bushels, 
 can I buy for $3G, at 45 cents a bushel ? 
 
 5. The quotient arising from a certain division is 123G ; the 
 divisor is 375, and the remainder 184 : what is the dividend 1 
 
APPLICATIONS. 87 
 
 G. The Croton Water Works of New York are capable of 
 discharging 60000000 gallons of water every 24 hours : what 
 would be the average amount per minute ? 
 
 7. The population of the United States, in 1850, was 23191876. 
 It has been estimated that 1 person in every 400 dies from 
 intemperance : how many deaths then may be attributed to this 
 cause, in the United States, during that year ? 
 
 8. At the rate of 45 miles an hour, how long would it take a 
 railroad car to pass around the globe, a distance of 25000 miles ? 
 
 9. If a quantity of provisions will last 25 men 2mo. owl:. Gda., 
 how long will it last 10 men ? 
 
 10. If a man's salary is $1200 a year, and his expenses are 
 $640 annually, how many years will it be before he will save 
 $6720 ? 
 
 11. How long will it take to count 20 millions at the rate of 
 80 per minute ? 
 
 12. If 3160 barrels of pork cost $47400, how many barrels 
 can be bought for $11475? 
 
 13. What will be the cost of 6 firkins of butter, each con- 
 taining 96 pounds, at 12^ cents a pound ? 
 
 14. What will 1000 quills cost, at 1 cent a piece ? 
 
 15. What will be the cost of 851- yards of cloth, at $9i a 
 yard? 
 
 16. What will be the cost of l/thJ. 2gcd. oqt. of brandy, at 
 56^ cents a quart ? 
 
 17. What will be the cost of 196 yards of cotton goods, at 
 Is. 6d. per yard ? 
 
 18. At 2s. 8d. per bushel, what will 1246 bushels of oats cost ? 
 
 19. If 112Z5. of cheese cost £2 16s., what is that per 
 
 pound ? 
 
 20. What will be the cost of 1426 pounds of hay, at $9,75 
 
 per ton ? 
 
 21. How much must I pay for the transportation of 3840 
 pounds of iron, from Albany to Buflxilo, at $4,50 per ton ? 
 
 22. Bought 124 barrels of potatoes, each containing 2\ bush 
 els, at 33i cents a bushel : what is the whole cost ? 
 
88 APl'I.ICATIONS. 
 
 23. If fifteen hundred tons of coal cost $11812,50, what will 
 one ton cost ? 
 
 24. If 789 pounds of leatlier cost $142,02, what is that per lb. ? 
 2a. There are three numbers, whose continued product is 
 
 10)200; one of the numbers is 25; another 18: what is the 
 third number? 
 
 2G. If Idwt. of gold is worth 92 cents, what would be the 
 weight of $10059,28 in gold? 
 
 27. 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 ? 
 
 28. 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 dollai's, and for the remainder he received 
 84560. What was his entire gain, and what did he receive 
 per acre on the last sale ? 
 
 29. A piece of ground 60 feet long and 48 feet wide is en- 
 closed by a wall 12 feet high and 2^ feet thick : how many 
 cubic feet in the wall ? 
 
 30. What will be the cost of transportation, from Montreal 
 to Boston, of 325640 feet of lumber at $2,37i per thousand ? 
 
 31. Bought 684 pounds of hay, at $12.40 a ton : what will it 
 cost me ? 
 
 32. At $2,121 a hundred, what will 786 feet of lumber cost? 
 
 33. How many shingles will it require to cover the roof of a 
 building 40 feet long and 26 feet wide, witli rafters 16 feet long, 
 allowing one shingle to cover 24 square inches ? 
 
 34. If 14/3. %oz. 12dwt. 3ffr. of silver be made into 9 tea-pots 
 of equal weight, what will be the weight of each? 
 
 35. A man bought 320 barrels of flour for $2688: at what 
 rate must he sell it to gain $1,60 on each barrel ? 
 
 36. A farmer has a granary containing Ai[)bi(s/i. Ipk. 2qt. of 
 wheat ; he wishes to put it into 182 bags: how much must he 
 put in each bag ? 
 
 37. A trader bought 750 barrels of flour for which he paid 
 $4875 ; ho sold the same for $7,25 a barrel : what was his 
 profit on each banei ? 
 
AI'l'LICATIONS. 89 
 
 38. How many sheep, at $1,62J a head, can be bought for $1 G9 ? 
 
 39. If a person save $6,87-l- a day, how long will it take hira 
 to save $267,75 ? 
 
 40. How many canisters, each holding 3lb. lOoz., can be filled 
 from a chest of tea containing 58lb. 
 
 41. In 26 hogsheads the leakage has reduced the whole 
 amount to l35Sff(il. 2qt. ; if the same quantity has leaked out of 
 each hogshead, how much still remains in each ? 
 
 42. The number of college libraries in the United States in 
 1850 was 213, containing 942312 volumes : what would be the 
 average number of volumes to each ? 
 
 43. A man bought a piece of land for $3475,25, and sold the 
 same for $3801,65, by which transaction he made $3,40 an 
 acre : how many acres were there ? 
 
 44. 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 in 
 California was $57715000 : how much more was produced in 
 1855 than in 1854 ? 
 
 45. If the forward wheels of a carriage are 12 feet in cir- 
 cumference, and the hind wheels 16 feet 6 inches, how many 
 more times will the forward wheels turn round than the hind 
 wheels, in runninfj a distance of 264 miles ? 
 
 46. If a certain township is 9 miles long, 41 miles wide, how 
 many farms of 192 acres each does it contain ? 
 
 47. The total number of land warrants issued during the 
 year ending Sept. 30th, 1855, was 34337, embracing 4093850 
 acres of land : what was the average number of acres to each 
 warrant ? 
 
 48. The amount of foreign imports brought into the Uniled 
 States during the fiscal year of 1855 Avas $261382960; during 
 the year 1854 it was $305780253 : how much was the decrease ? 
 
 49. The longitude of Philadelphia is 75" 10', and that of 
 New Orleans 89° 2' ; when it is 12 M. at Philadelphia, what is 
 the time at New Orleans .'' 
 
90 APPLICATIONS. 
 
 50. The sun passes the meridian at 12 M., the moon at 
 %hr. 30m. P. M. : what is the difference in longitude between 
 the sun and moon ? 
 
 51. Two persons, A and B, observed an eclipse of the moon ; 
 A observed its commencement at Qhr. A2mi. P. INI. ; B was in 
 longitude 73° 20', and observed its commencement 23 minutes 
 eai-lier than A: what was A's longitude, and B's time of 
 observation ? 
 
 52. If in 11 piles of wood there are 120 cords, 7 cord feet, 
 5 cubic feet, how much is there in each pile ? 
 
 53. If IQavL 2q)\ lllb. lOoz. of flour be put into 9 barrels, 
 how much will each barrel contain ? 
 
 54. A miller bought a quantity of wheat for $G25,40, which 
 he floured and put into barrels at an expense of Si 10,12^: 
 what profit did he make by selling it for $900 ? 
 
 55. America was discovered Oct. 11th, 1492 : liow long to 
 the commencement of the Revolution, April 19th, 1775 ? 
 
 56. From a hogshead of wine a merchant draws 18 bottles, 
 each containing Ijyt. Zgills ; he then fills three 6 gallon demi- 
 jons, and 4 dozen bottles each containing 2qt. Iq^t. Skills: how 
 much remained in the cask ? 
 
 57. In 753 G89 yards, how many degrees and statute miles ? 
 
 58. In 189mz. 3fur. 6rd. \ft. how many feet ? 
 
 59. If 24 men can build 768 rods of wall in 1 day, how 
 many rods can 48 men build in 9 days ? 
 
 60. A certain number increased by 1764, and the sum mul- 
 tiplied by 209, gives the product of 7913576 : what is the 
 number ? 
 
 61. If a man travel 146mi. Ifur. Ih-d. 14/L in 5 days, how 
 much is that for each day ? 
 
 62. If 325 acres of land cost $17712,50, how many acres 
 can be bought for $545 ? 
 
 63. A merchant having $324 wishes to purchase an equal 
 number of yards of two kinds of cloth ; one kind was worth 
 4 dollars a yard, the other was worth 5 dollars a yard : how 
 many yards of each can he buy? 
 
APPLICATIONS. 91 
 
 64. From one-fourth of a piece of cloth containing QSyd. 3qr. 
 a tailo" cut 5 suits of clothes : how much did each suit contain ? 
 
 Go. A manufacturer having £5 lO.v,, distributed it among his 
 laborers, giving every man 18d., every woman 12c/., and every 
 boy lOd. ; the number of men, women and boys, were equal : 
 what was the number of each ? 
 
 GG. 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 2093G4G8 : how many deaf and dumb 
 persons were there in the entire population ? 
 
 67. 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 e2:2;s ? 
 
 68. How much time will a man save in 50 years, beginning 
 witli a leap year, l)y rising 45 minutes earlier each day ? 
 
 GD. liichard Roe was born at 6 o'clock, A. M., June 24th, 
 18-32 : what will be his age at 3 o'clock, P. M., on the 10th day 
 of January, 1858? 
 
 70. During the year 1855, there were shipped to Great 
 Britain, from the United States, 408434 barrels of tloiir ; 2550092 
 bushels of wheat; 1048540 bushels of corn. Supposing the 
 flour to have sold for $10,25 a barrel, the wheat for $2,121 a 
 bushel, and the corn for $0,94 a bushel, what was the value 
 of the whole ? 
 
 71. A man dying without making a will, left a vridow and 
 4 children. The law provides, in such cases, that the Avidow 
 shall receive one-third of the personal propei'ty, and thai the re- 
 mainder shall be equally divided among the children. The 
 estate was valued as follows : a farm, at $5000 ; 5 horses, at 
 $85 each; a yoke of oxen, for $110; 25 cows, at $22 each; 
 150 sheep, at $2 each ; some lumber, at $45 ; forming utensils, 
 at $174; household furniture, at $450 ; grain and hay, at $380 : 
 what was the share of the widow and each child ? 
 
 72. The amount of gold coin in the United States in 1855 
 was estimated at about $241200000. Adopting the same ratio 
 of increase as from 1850 to 1855, the population of the United 
 
92 
 
 APPLICATIONS. 
 
 States in 1855 would be about 26800000. In an equal distri- 
 biition of the gold, how much would eacli person receive ? 
 
 73. How many sliingles will it take to cover the two sides of 
 the roof of a building, 55 feet long, with rafters IGi feet in 
 length, allowing each shingle to be 15 inches long and" 4 inches 
 wide, and to lay one-third to the weather ? 
 
 74. If the longitude of St. Petersburgh is 30° 45' east, and 
 that of Washington 77° 2' west, what is the difference of lon- 
 gitude between the two places, and the difference of time ? 
 
 75. When it is 6 o'clock, A. M., at Washington, what is the 
 time at St. Petersbursrh ? 
 
 76. 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 \hr. 4:4-m. from the time at the place of observation. If 
 his chronometer shows the time to be Shi: 12 mi. P. M., what is 
 the correct time, at the place of observation, and how far is he 
 east from New York ? 
 
 77. A cistern containing 960 gallons, has two pipes; 45 
 gallons ]-un in every hour by one pipe, and 25 gallons run out 
 by the other: how long a time will be required to fdl the 
 cistern ? 
 
 78. A speculator sold 840 bushels of wheat for $2180, which 
 was S500 more than he gave for it : what did it cost him a 
 bushel ? 
 
 79. The wliole number of gallons of rum manufactured in 
 the United States in 1850, was 6500500 gallons ; if it be 
 valued at 50 cents a gallon, how many schoolhouses could be 
 built, worth $750 each, with the proceeds ? 
 
 80. A farmer sold a grocer 30 bushels of potatoes at 37-i cents 
 a bushel, for which he received 6 gallons of molasses at 45 
 cents a gallon ; GO pounds of mackerel at 6^- cents a pound, and 
 tlie remainder in sugar at 10 cents a pound : how many pounds 
 of sugar did he receive ? 
 
 81. If a man travel 12;///. Sfu?: 20a/. in one day, how long 
 will it take him to travel 174////. \far. at the same rate ? 
 
APPLICATIONS. 93 
 
 82. If a man sell '2har. 12^/al. 2qi. of beer in one week, Low 
 much will he sell in 12 weeks ? 
 
 83. A hquoi- merchant had ooO pint bottles, 400 quart bottles, 
 350 two quart bottles, 375 three quart bottles, and 150 jugs, 
 liolding a gallon each : hew many barrels of wine will fill them ? 
 
 84. How many yards of carpeting, one yard wide, will it 
 take to cover the floors cf two parlors, each 18 feet long, and 
 IG feet wide, and what will it cost at $l,o3i a yard? 
 
 85. 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 and 1 6 feet wide and 9 feet high ? 
 
 8G. Two persons ai'e Imi. Afiir. 20/ d. 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 over- 
 take the foremost ? . 
 
 87. A man sold 500 bushels of wheat at $1,75 a bushel, and 
 took his pay in sugar at 5 cents a pound. He afterwards sold 
 one-half of it : what quantity of sugar had he left ? 
 
 88. A man bought 7 barrels of sugar at Sl2,87-l- 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 ? 
 
 89. A flour merchant bought a quantity of flour for $18750, 
 and sold the same for $26250, by which he gained $3 a barrel 
 how many barrels were there ? 
 
 90. Three men rented a farm and raised d64:biish. 2jjI: AqL 
 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 had what was left : how 
 much did each have ? 
 
 91. A vessel in longhude 70^ 25' east, sails 105° 30' 56'- 
 west, then 40° 50' east, then 10' 5' 40" west, then 39° 11' 36" 
 east ; in what longitude is she then, and how many days will it 
 take her to sail to longitude 77° west, if she sail 3° 20' each 
 da,}' ? 
 
 92. A privateer took a prize worth $25000, which was 
 divided into 125 shares, of which the captain took 12 shares ; 
 
94 APPLICATIONS. 
 
 2 lieutenants, eacli 5 shares ; 6 midshipmen, each 3 shares ; 
 ajid the remainder was divided equally among 85 seamen : how 
 much did each receive ? 
 
 93. If the longitude of Boston is 71° 4', and a gentleman in 
 travelling from Boston to Chicao;o iinds that his watch is Ihr. 
 5m. 44sec. too fast by the time of the latter place, what is 
 the longitude of Chicago, provided his watch has kept time 
 accurately ? 
 
 94. What time would it be in Boston when it was 8/ir. 21 mi. 
 ZOsec, A. M., in Chicago ? 
 
 95. What time would it be at Chicago when it was 12 M. at 
 Boston ? 
 
 96. 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: 
 bow far apart are they in degrees and minutes of longitude ? 
 
 97. In 12 bales of cloth, each bale containing 16 pieces, and 
 each piece containing 20 ell English, how many yards ? 
 
 98. How many eagles can be made from 2Ub. 4oz. (5pwt. 
 ISgr. of gold, making no allowance for waste, if each eagle 
 weighs llpwls. 9(/r. ? 
 
 99. A man paid $3284,82 for some wheat. He sold 740 
 bushels at 2 dollars a bushel ; the remainder stood him in $1,42 
 a bushel : how many bushels did he purchase ? 
 
 100. A speculator gave $8968 for a certain number of barrels 
 of flour, and sold a part of it for $2618, at $7 a barrel, and by 
 so doing lost $2^ on each barrel ; for how much must he sell 
 the remainder to gain $1060 on the whole? 
 
 101. A man sold 105/1. 2R. 2QF. of land ibr as many dollars 
 as there were perches of land, payable in instalments, at the rate 
 of 1 dolhu- an hour. If the contract was closed at 12 o'clock, 
 M., April 1st, 1856, what length of time will be allowed the 
 purchaser to pay the debt, reckoning 365 days 6 hours to the 
 year ? 
 
PKOPERTIKS OF NUMBERS. 95 
 
 PROPERTIES OF NUMBERS. 
 
 PRIME AND COMPOSITE NUMBERS. 
 
 83. An Integral Number is the unit 1, or a collection of 
 Buch units. 
 
 84. One number is said to be divisible by another when the 
 quotient is an integral number. The division is then said to be 
 exact. 
 
 85. A Composite Number is one that may be produced 
 by the multiplication of two or more numbers, called factors ; 
 thus, 30 = 2 X 3 X 5, is a composite number, in which the 
 factors are 2, 3 and 5. 
 
 Note 1. — A composite number is exactly divisible by any one of 
 its factors. 
 
 86. A Prime Number is a number that is divisible only 
 by itself and by 1 ; thus, 1, 2, 3, 5, 7, 11, 13, &c., are prime 
 numbers. 
 
 87. Two numbers are said to be prime to each other when 
 they have no common factor ; thus, 4 and 9 are 2^r//?ie to each 
 other, though both are composite numbers. 
 
 88. Any number (prime or composite), as 26, may be put 
 under the form, 1 X 26; hence, every number is divisible by 
 itself and by 1, and therefore, these are not reckoned among 
 i\iQ factors or divi'sos either of prime or composite numbers. 
 
 83. What is an integral number 1 
 
 84. When is one number said to be divisible by another] How is the 
 division then said to be 1 
 
 85. What is a composite niamber 1 Bv what is a composite numbej 
 always divisible 1 
 
 86. What is a prime number '; 
 
 87. When are two numbers prime to each other 1 
 
 88. What numbers are not reckoned among the divisors of prime or 
 composite numbers 1 
 
96 PKOPEKTIES OF NrHIBERS. 
 
 89. Every factor of a composite number is a divisor, and 
 is either prime or composite ; and, since every composite factor 
 may be again divided, it follows that 
 
 Every number is equal to the product of all its prime factors. 
 
 For example, 24 = 3 X 8 ; but 8 is a composite number of 
 which the factors are 2 and 4 ; and 4 is a composite number of 
 which the factors are 2 and 2 ; hence, 
 
 24 = 3x 8 = 3x2x4=33x2x2x2; and 
 60 = 5x12 = 5x3x4 = 5x3x2x2. 
 
 Hence, to find the prime factors of any number : 
 Divide the number by any prime member that will exactly 
 divide it: then divide the quotient by any prime number that 
 will exactly divide it^ and so on, till a quotient is found which 
 is prime ; the several divisors and the last quotient will be the 
 prime factors of the given number. 
 
 Note. — It is most convenient, in practice, to use at each division 
 the least prime number that is a divisor. 
 
 1. What are the prime factors of 105 ? 
 
 Analysis. — Three being the least divisor that operation. 
 is a prime number, Ave divide by it, giving the 3)105 
 
 quotient 35 ; then 5 is the least prime divisor 5)35 
 
 of this quotient : hence, 3, 5 and 7 are the 7 
 
 prime factors of 105. 
 
 EXAMPLES. 
 
 1. What are the prime factors of 9? 10? 12? 14? IC? 1«? 
 24? 27? 28? 
 
 2. What are the prime factors of 30? 22? 32? BG? 38? 40? 
 45? 49? 
 
 3. What are the prime factors of 50 ? 56? 58? 60? 64 66? 
 68? 70? 72? 
 
 4. What are the prime foctors of 7G ? 78? 80? 82? 84? 
 86? 88? 90? 
 
 89. To what product is every number equal ] How do you find the 
 prime factors of any number 1 
 
PROPERTIES OF NUMBERS. 97 
 
 5. What are the prime factors of 92? 94? 96? 98? 99? 
 100? 102? 104? 
 
 G. What are the prime fxctors of 105? lOG ? 108? 110? 
 115? 116? 120? 125? 
 
 7. What are the prime foctors of 302 ? 305 ? 604 ? 875 ? 
 975 ? 055 ? 
 
 Note. — The prime factors, "when the numbers are small, may 
 generally be seen by inspection. The teacher can easily increase 
 the number of examples. 
 
 90. When there are several numbers, and it is required to 
 find the prime factors common to all of them : 
 
 Find the in'une factors of carh, and then select thoae factors 
 ivhich are common to all the members. 
 
 8. What are the puime factors common to 150, 210 and 270 ? 
 
 9. What are the prime factors common to 42, 126, and 168? 
 
 10. What are the prime factors common to 105, 315 and 525 ? 
 
 11. Wliat are the prime factors common to 84, 126 and 210? 
 
 12. What are the prime factors common to 168, 25.6, 410, 
 and 820 ? 
 
 13. What are the prime factors common to 420, 630, 1050, 
 and 2100 ? 
 
 91. DIVISIBILITY OF NUMBERS. 
 
 1. Two is the only even number which is prime. 
 
 2. Two divides every even number, and no odd number. 
 
 3. Three divides every number the sum of Avhose figures is 
 divisible by 3. 
 
 4. Four divides every number when the two right hand 
 figures are divisible by 4. 
 
 5. Five divides every number which ends in or 5. 
 
 6. Six divides every even number that is divisible by 3. 
 
 7. Ten divides every number ending in 0. 
 
 90. How do you find the prime factors common to several numbers ? 
 
 91. 1. How many even numbers arc prims'! 2. What numliers will 2 
 divide ! 3 What numbers will 3 divide 1 4. What numbers will 4 divide 1 
 5. What numbers will ^ divide 1 6. ^^'hat numbers will 6 divide 1 7. What 
 
98 PKOPERTIES OF NUitBERS. 
 
 8. When the divisor is a composite number, and when fhe 
 iividend and partial quotients arc successively divisible by i/i 
 factors, the division will be exact (Art. 57)- 
 
 For, dividing by the factors separately, gives the same quo 
 tient as dividing by their product (Art. 70). 
 
 9. Any number u-hich will divide one factor of a product loil 
 divide the product. 
 
 Thus, take any number, as 30 = 5x6; any number which 
 will divide 5 or 6 will divide 30. 
 
 10. Any number which will exactly divide each of ta-o num- 
 bers will divide their sum : and any number which will divide 
 their sum and one of the numbers, toill divide the other. 
 
 For, take any two numbers, an 9 and 12 ; then, 
 
 9 + 12 = 21. 
 Now, any divisor that will divide two of these numbers will 
 divide the other ; else, we should have a whole number equal to 
 a fraction, which is impoi>sible. 
 
 11. Any number which unll exactly divide each of two num- 
 bers will divide their difference : and any number ivhich will 
 divide their diffirence and one of the numbers, will divide the 
 other. 
 
 For, let 24 and 8 be any two numbers ; then, 
 
 24-8 = 16. 
 Now, any divisor that will divide two of these numbers will 
 divide the other ; else, we shoidd have a whole number equal to 
 u fraction, which is impossible. 
 
 12. Jf there is a remainder after division, any number which 
 wilt exactly divide the dividend and divisor will also divide the 
 remainder. 
 
 numbers will 10 divide 1 8. "When will llif divi-or exactly divide the divi- 
 dend ! 9. When will any minihor divide a p rod net ! 10. \\ hen u ill a 
 number divide the sum of two numbers ! When will it divide either .)f 
 them separately ! 11. When will a number exactly divide the dilVereiico 
 of two numbers ! 12. If a nmnher divides the dividend and divisor whil 
 kthcr number will it alwavs divide'' 
 
GREATEST COMMON DIVISOR. 99 
 
 For, we always have 
 
 Dividend = Divisor x Quotient -|- Eem. 
 or Dividend — Divisor X Quotient = Rem. ; 
 
 hence, by principle (11) any number which will divide the divi- 
 dend and divisor will also divide the remainder, after division. 
 
 GREATEST COMMON DIVISOR. 
 
 92. A Common Divisor of two or more numbers is any 
 number that will divide each of them without a remainder; 
 hence, it is merely a common factor of the numbers. 
 
 93. The Greatest Common Divisor of two or more 
 numbers is the greatest number that will divide each of them 
 without a remainder; hence, it is their greatest coxanon factor. 
 
 For example, 2 and 3 are common divisors of 12 and 18; 
 but G is their greatest common divisors, since there is no num- 
 ber greater than 6 that will exactly divide both of them ; hence, 
 it is their greatest common factor. 
 
 Note. — Since 1 and the number itself will divide every number, 
 they are not reckoned among the common divisors. 
 
 Hence, to find the greatest common divisor of two or more 
 numbers, 
 
 I. Resolve each number into its 2>i'ime factors : 
 
 n. Tlie product of all the factors common to each result will be 
 the greatest common divisor. 
 
 EXAMPLES. 
 
 1. "What is the greatest common divisor of 12 and 20? 
 
 Analysis. — There are three prime fac- 
 tors in 12 ; viz., 2, 2 and 3 : there are three operation. 
 prime factors in 20; viz., 2, 2 and 5 : the 12 = 2X2x3 
 factors 2 and 2 are common; hence, 2x2 ==4 20 = 2x2x5 
 is the greatest common divisor. 
 
 92. Wliat is a common divisor of two or more numbers ? 
 
 93. Wbat is the sreatest common divisor of two or more numbers ? Ho^v 
 
 o 
 
 4o you find the greatest common divisor of two or more numbers I 
 
100 GREATEST COMMON DIVISOR. 
 
 2. What is the greatest common divisor of 18 and 36. 
 
 3. What is the greatest common divisor of 12, 24 and 60 
 
 4. What is the greatest common divisor of 15, 50 and 40 ? 
 
 5. What is tlie greatest common divisor of 24, 18 and 144? 
 
 6. Wliat is the greatest common divisor of 50, 100 and 80 ? 
 
 7. What is the greatest common divisor of 56, 84 and 140 ? 
 
 8. What is tlie greatest common divisor of 84, 154 and 210 ? 
 
 SECOND METHOD, 
 
 94. When the numbers are large, another method is used for 
 finding their greatest common divisor. 
 
 1. Let it be required to find the greatest common divisor of 
 the numbers 216 and 408. 
 
 Analysis. — The greatest common divisor opkration. 
 
 cannot be greater than the least number 216. 216)408(1 
 Now, as 216 will divide itself, let us see if it 216 
 
 will divide 408; for if it will, it is the great- 192)216(1 
 
 est common divisor sought. Making the 192 
 
 division, we find a quotient 1 and a remain- 24)192(8 
 
 der 192; hence, 216 is not a common di- 192 
 
 visor. 
 
 The greatest common divisor of 216 and 408 will divide the 
 remainder 192 (Art. 91-12); and if 192 will exactly divide 216, it 
 will be the greatest common divisor. Wc find that 192 is contained 
 in 216 once, aud a remainder 24. The greatest comtnon divi.^^or of 
 192 and 216 will divide the remainder 24; and if 24 will exactly 
 divide 192, it will also divide 216, and consequently 408 ; now 24 
 exactly divides 192, and hence is the, greatest common divisor sought. 
 
 Hence, to find the greatest common divisor, 
 
 Divide the greater number hy the leas, and then, divide the 
 preceding divisor by the remainder, and so on, till nothing 
 remains : the last divisor will be the greatest coinmon divisor. 
 
 No FES. — 1. If the last remainder is 1, the numbers haA^c no com 
 mon divisor ; that is, they are prime with respect to each other 
 (Art. 87). 
 
 94. ^"^'l^at is tlic rule when the numbers are large ? 
 
GREATEST COMMON DIVISOK. 101 
 
 2. If, in the course of tne opcralion, any one of the remainders is 
 a prime number^ and will not exactly divide the ■preceding divisor, it 
 is certain that no common divisor exists, aud it is unnecessary to 
 divide further. 
 
 EXAMPLES. 
 
 1. "What is the greatest common divisor of 3328 and 4592 ? 
 
 2. What is the greatest common divisor of 2205 and 4501 ? 
 
 3. What is the greatest number that -will divide 1G082 and 
 25740 ? 
 
 4. What is the greatest number that will divide 620, 1116 
 and 1488 ? 
 
 5. What is the greatest common divisor of 5270, 5952, 5394 
 and 3038 ? 
 
 6. What is the greatest common divisor of 4617, 7695, 6642 
 and 8424 ? 
 
 7. A farmer has 315 bushels of corn, and 810 bushels of 
 wheat ; he wishes to draw the corn and wdieat to market 
 separately in the fewest number of equal loads : how many- 
 bushels must he draw at a load ? 
 
 8. The Illinois Central Railroad Company have 15750 acres 
 of land in one location, and 21725 acres in another. They 
 wish to divide the whole into lots of equal extent, containing 
 the greatest number of acres that will give an exact division : 
 how many acres will there be in each lot ? 
 
 9. A man has a corner lot of land 1044 feet long, and 744 
 feet wide. The adjacent sides are bounded by the highway 
 and he wishes to build a boai'd fence with the fewest panels of 
 equal length : what must be the length of the panels ? 
 
 10. A farmer has 231 bushels of barley, 369 bushels of oats, 
 and 393 bushels of wheat,' all of which he wishes to put into 
 the smallest number of bags of equal size, without mixing : 
 how many bushels must each bag contain ? 
 
 11. Three persons. A, B, and C, agree to purchase a lot 
 of 63 cows at the same price per head, provided each man can 
 thus invest his whole money. A has $286, B .^462, and C 8638 ; 
 how many cows could each man purchase ? 
 
102 LEAST COMMON MULTIPLE. 
 
 LEAST COMMON MULTIPLE. 
 
 95. A Multiple of a number is any product in -which th« 
 number enters as a factor ; hence, a muhiple of any number is 
 exactly divisible by the number. 
 
 96. A Common Multiple of two or more numbers is any 
 number which each -will divide -without a remainder. 
 
 97. The Least Common Multiple of two or more num- 
 bers is the least number -svhich they will -separately divide with 
 out a remainder. 
 
 Notes. — 1. Since the least common multiple is exactly divisible 
 by a divisor, it can be resolved into two factors, one of which is the 
 divisor and tlie other the quotient. 
 
 2. !f the divisor be resolved into its prime factors, the equal factoi 
 of the least common multiple maybe resolved into the same factors 
 hence, the least common multiple will contain every prime factor of it. 
 divisor. 
 
 3. The question of finding the least common multiple of several 
 numbers, is therefore reduced to finding a number which shall con- 
 tain all their prime factors and none others. 
 
 1. What is the least common multiple of 6, 12 and 18 ? 
 
 A.NALTSIS. — Havmg placed the given num- 
 bers in a line, if we divide by 2, we find the operation. 
 quotients 3, 6 and 9 ; hence, 2 is a prime fac- 2 )6 . . 12 . . 18 
 tor of all the numbers. Dividing by 3, -we 3)3 . . 6 . . 9 
 find that 3 is a prime factor of the quoti- 1 . . 2 . . 3 
 ents 3, 6, and 9 ; and hence, the quotients 2X3X2x3 = 36 
 2 and 3 are prime factors of 12 and 18; 
 
 hence, the prime factors of all the numbers are 2, 3, 2 and 3, and 
 their product 36 is the least common multiple. 
 
 98. Therefore, to find the least common multiple of several 
 numbers : 
 
 95. Wliat is a multiple of a number \ 
 
 96. What is a common multiple of two or more numbers 1 
 
 97. What is the least common multiple of two or more numbers ? 
 9S. How do you find the least common multiple ol several numbers' 
 
COMMON MULTIPLIi. 103 
 
 I. Place the numbers on the same line, and divide by any prime 
 number that tuill exactly divide tiuo or more of them, and set 
 doivn in a line below the quotients and the undivided members. 
 
 II. Then divide as before, until there is no prime number 
 greater than 1 that will exactly divide any two of the numbers. 
 
 III. Then multiply together the divisors and the numbers of 
 the lower line, and their product tvill be the least common 
 multiple. 
 
 Note. — If the numbers have no common prime factor, their pro- 
 duct will be their least common midtiple. 
 
 EXAMPLES. 
 
 1. What is the least common multiple of 4, 9, 10, 15, 18, 
 20, 21 ? 
 
 2. What is the least common multiple of 8, 9, 10, 12, 25, 
 32, 75, 80 ? 
 
 3. What is the least common multiple of 1, 2, 3, 4, 5, 6, 
 7,9? 
 
 4. What is the least common multiple of 9, 16, 42, 63, 21 
 14, 72 ? 
 
 5. What is the least common multiple of 7, 15, 21, 28, 35, 
 100, 125 ? 
 
 6. What is the least common multiple of 15, 16, 18, 20, 24, 
 25, 27, 30 ? 
 
 7. What is the least common multiple of 9, 18, 27, 36, 45^ 
 54? 
 
 8. What is the least common multiple of 4, 10, 14, 15, 21 ? 
 
 9. What is the least common multiple of 7, 14, 16, 21, 24? 
 
 10. What is the least common multiple of 49, 14, 84, 108, 
 98? 
 
 11. A can dig 9 rods of ditch in a day; B 12 rods in a day; 
 and C 1 6 rods in a day : what is the smallest number of rods 
 that would afford exact days of labor to each, working alone ? 
 In what time would each do the whole work ? 
 
 12. A blacksmith employed 4 classes of workmen, at $15, 
 $16, 121 and $24 per month, for eacli man respectively, paying 
 to each class the same amount of wages. Required the least 
 
104 CANCELLATION. 
 
 amount that will j^ay either class for 1 month ; also, the num- 
 ber of men in each class ? 
 
 13. A farmer has a number of ba"js containinsr 2 bushels 
 each ; of barrels, containing 3 bushels each ; of boxes, containing 
 7 bushels each ; and of hogsheads, containing 15 bushels each : 
 what is the smallest quantity of wheat that would fill each an 
 exact number of times, and hoio many times would that quan- 
 tity fill each ? 
 
 14. Four persons start from the same point to travel round 
 a circuit of 300 miles in circumfei'cnce. A goes 15 miles a 
 day, B 20 miles, C 25 miles, and D 30 miles a day. How 
 many days must they travel before they will all come together 
 again at the same point, and how many times will each have 
 gone round ? 
 
 Note. — First find the number of days that it will take each to 
 travel round the circuit. 
 
 CANCELLATION. 
 
 99. Cancellation is a method of shortening Arithmetical 
 operations by omitting or cancelling common factors. 
 
 1. Divide 36 by 18. First, 36 = 9 x 4 ; and 18 = 9 x 2 
 
 2. 
 
 Analysis. — Thirty-six divided by 18 is operation. 
 
 equal to 9 X 4 divided by 9 x 2 : by can- 36 x 4 _ 
 
 celling, or striking out the 9's, we have 18 ~ 0X2 
 4 divided by 2, which is equal to 2. 
 
 Note. — The figures cancelled arc slightly crossed. 
 
 The operations, in cancellation, depend on two principles : 
 
 1 . JVie cancelling of a factor, in any number, is equivalent to 
 dividing the number by that factor 
 
 2. If the dividend and divisor he both divided by the same 
 number, the quotient will not be changed. 
 
 99. What is cancellation 1 On what principles do the operations of 
 cancellation depend ? 
 
CANCELLATION. 105 
 
 PRINCIPLES AND EXAMPLES. 
 
 1. Divide 5G by 32. 
 
 Analysis. — Resolve the dividend and ' operation. 
 
 divisor into factors, and tlien cancel those 56 S x 7 7 
 
 which are common. 32 ;8 x 4 4 
 
 2. In 72 times 25, how many times 45 ? 
 
 Analysis. — We see that 9 is a factor of 72 and 45. Divide by 9, 
 and write the q^uotient 8 over 72, and the 
 quotient 5 below 45. Again, 5 is a fac- operation. 
 
 tor of 25 and 5. Divide 25 by 5, and 8 5 
 
 write the quotient 5 over 25. Dividing "^^ ^ ^-■^ _ .„ 
 
 5 by 5, reduces the divisor to 1, which A? 
 
 need not be set down : hencC; the true 
 quotient is 40. 
 
 Note. — The operation may be performed in another way, by wri- 
 ting the divisor on the left of a vertical 
 
 line, and the dividend on the right: in operation. 
 
 which case, the quotients, in cancelling, ^ 
 
 are written above, and at the side of the 
 numbers, as 5, 8 and 5. If we conceive 
 
 5 
 
 the horizontal line, first used, to be turn- a .^ 
 
 ed up from left to right, the dividend, 
 
 which was above the line, will fall at the right, and the divisor, 
 
 which was below it, at the left. 
 
 100. Hence, to perform the operations of cancellation : 
 
 I. Resolve the dividend and divisor into such factors as shall 
 give all the factors common to both. 
 
 II. Cancel the common factors and then divide the product of 
 the remaining factors of the dividend hy the product of the re- 
 maining factors of the divisor. 
 
 Notes. — 1. Since every factor is cancelled by division^ the quotient 
 1 always takes the place of the cancelled factor, but is omitted when 
 it is a multiplier of other factors. 
 
 2 If one of the numbers contains a factor equal to the product of 
 two or more factors of the other, all such factors may be cancelled. 
 
 100. How do you perform the operations of cancellation T 
 
106 OA]S-CELLATION. 
 
 3. If the product of two or more factors of the dividend is equal to 
 the product of two or more factors of the divisor, such factors may be 
 cancelled. 
 
 4. It is generally more convenient to set the dividend on the right 
 of a vertical line and the divisor on the left. 
 
 EXAMPLES. 
 
 1. What is the quotient of 2x4x8x13x7x16 divid- 
 ed bj 26 X 14 X 8 ? 
 
 2. What is the quotient of 42 X 3 X 25 x 12 divided hj 
 28 X 4 X 15 X 6? 
 
 3. What is the quotient of 125 x 60 X 24 X 42 divided by 
 25 X 120 X 36 X 5? 
 
 4. How many times isllx39x7x2 contained in 44 x 
 18 X 26 X 14? 
 
 5. What is the quotient of 8 times 240 multiplied by 5 times 
 114, divided by 24 time.s 57 multiplied by 6 times 15 ? 
 
 6. What is the value of (22 + 8 + 16) X (18 + 10 + 21) 
 divided by (9 + 5 + 7) x (15 + 8) ? 
 
 7. Divide (140 + 86 - 34) x (107 - 19) by (237 - 141) 
 X (17 + 20 - 15) ? 
 
 8. Divide [12 X 5 - 2 X 9] x (42 + 30) by (5 x 8) 
 X (2 X 9) X (10 + 17) ? 
 
 9. What is the quotient of 240 x 441 X 16 divided by 
 175 X 56 X 27? 
 
 10. What is the quotient of 64 times 840 multiplied by 
 9 times 124, divided by 32 times 560 multiplied by 4 times 31 ? 
 
 11. How many dozens of eggs, worth 14 cents a dozen, must 
 be given for 18 pounds of sugar, woi'th 7 cents a pound ? 
 
 12. A dairyman sold 5 cheeses, each weigliing 40 pounds, at 
 9 cents a pound : how many pounds of tea, worth 50 cents a 
 pound, must he receive for the cheeses ? 
 
 13. Bought 12 yards of cloth at Sl,84 a yard, and paid for 
 it in potatoes at 48 cents a bushel : how many bushels of pota- 
 toes will pay for the cloth ? 
 
 14. How many firkins of butter, each containing 56 pounds, 
 
CANCELLATION. 107 
 
 at 25 cents a pound, will pay for 4 barrels of sugar, each 
 weighing 175 pounds, at 8 cents a pound ? 
 
 15. A man bought 10 cords of wood, at 20 shillings a cord, 
 and paid in labor at 12 shillings a day : how many days must 
 he labor ? 
 
 16. How many pieces of cloth, each containing 36 yards, av 
 $3,50 a yard, must be given for 96 barrels of flour, at $10,50 a 
 barrel ? 
 
 17. A farmer exchanged 492 bushels of wheat, worth §1,84 
 a bushel, for an equal number of bushels of barley, at 87 cents 
 a bushel, of corn at GO cents a bushel, and of oats at 45 cents a 
 bushel : how many bushels of each did he receive ? 
 
 IS. How many barrels of flour, worth $7 a barrel, must be 
 given for 250 bushels of oats, at 42 cents a bushel ? 
 
 19. If 48 acres of land produce 2484 bushels of corn, how 
 many bushels will 120 acres produce? 
 
 20. A man works 12 days at 9 shillings a day, and receives 
 in pay wheat at two dollars a bushel : how many bushels did 
 he receive ? 
 
 21. A grocer sold 6 hams, each weighing 14 pounds, at 10 
 cents a pound, and took his pay in apples at 48 cents a bushel : 
 how many bushels of apples did he receive? 
 
 22. How long will it take a man, travelling 36 miles a day, 
 to go the same distance that another man has travelled in 15 
 days at the rate of 27 miles a day ? 
 
 23. A man took 4 loads of apples to market, each load con- 
 tainins: 12 barrels, and each barrel 3 bushels. He sells them 
 at 45 cents a bushel, and receives in payment, a number of 
 boxes of tea, each box containing 20 pounds, Avorth 72 cents a 
 pound : how' many boxes of tea did he receive ? 
 
108 COMMON FRACTIONS. 
 
 COMMON FRACTIONS. 
 
 101. The unit 1 denotes an entire thing, as 1 apple, 1 chair, 
 1 pound of tea. 
 
 If the unit 1 be divided into two equal parts, each part is 
 called one-half. 
 
 If the unit 1 be divided into three equal parts, each part is 
 called one-third. 
 
 If the unit I be divided into four equal parts, each part is 
 called one-fourth. 
 
 If the unit 1 be divided into twelve equal parts, each part is 
 called one-twelfth ; and if it be divided into any number of equal 
 parts, we have a like expression for each part. 
 
 The parts are thus written : 
 
 1 is read, one-half. 
 1 - - one-third. 
 J- - - one-fourth, 
 
 4 
 
 i - - one-liflh. 
 i - - one-sixth. 
 
 1 is read, one-seventh. 
 •|- - - one-eighth. 
 
 one-tenth. 
 
 one-fifteenth. 
 
 one-fiftieth. 
 
 1 
 
 15 
 
 JL 
 
 50 
 
 The ^, is an entire half ; the 1, an entire third ; the 1, an 
 entire fourth ; and the same for each of the other equal parts ; 
 hence, each equal part is an entire thing, and is called a frac- 
 tional unit. 
 
 The unit, or whole thing which is divided, is called the unit 
 of the fraction. 
 
 NoTK. — In every fraction let the pupil distinguish carefully, be- 
 tween the unit of the fraction and the fractional unit. The first is 
 the whole thing from which the fraction is derived; the second, orie 
 of the equal parts into which that thing is divided. 
 
 101. Wha» is a unit 1 What is each part called when the unit 1 is divid- 
 ed into two equal parts ! When it is divided into 3] Into il Into 51 
 
 Into lai 
 
COMMON FRACTIONS. 109 
 
 102. Each fractional unit may, like the unit 1, become the 
 base of a collection : thus, suppose it were required to express 
 2 of each of the fractional units : we should then write 
 
 2 
 2 
 
 which is read 
 
 2 halves = i x 2. 
 
 2 
 3 
 
 _ . . . 
 
 2 thirds = ^ x 2. 
 
 2 
 4 
 
 _ - - _ 
 
 2 fourths = 1x2. 
 
 * 
 
 - - . _ 
 
 2 fifths = i X 2. 
 
 
 &;c., &;c., 
 
 &c., &c. 
 
 If it were required to express 3 of each of the fractional 
 units, we should write 
 
 |- which is read 3 halves —1x3. 
 f - - - - 3 thirds =1x3. 
 f - - - - 3 fourths = 1x3. 
 3 . . . . 3 fifths =1x3. 
 &c., &c., &c., &c. ; hence, 
 
 A Fraction is one of the equal parts of a unit, or a collec- 
 tion of such equal parts. 
 
 Eractions are expressed by two numbers, one written above 
 the other, with a line between them. The lower number is 
 called the denominator, and the upper number the numerator. 
 
 The denominator denotes the number of equal parts into 
 which the unit is divided ; and hence, determines the value of 
 the fractional unit. Thus, if the denominator is 2, the fractional 
 unit is one-half; if it is 3, the fractional unit is one-third ; if it 
 is 4, the fractional unit is one-fourth, &cc., Sec. 
 
 The numerator denotes the number of fractional units taken. 
 Thus, f denotes that the fractional unit is i, and that 3 such 
 units are taken; and similarly for other fractions. 
 
 How may the one-half be regarded 1 The one-third ? The one fourth ? 
 What is each part called ! "What is the unit of a fraction 1 What is a 
 fractional unit ! How do you distinguish between the one and the other ! 
 
 102. May a fractional unit become the base of a collection 1 What is a 
 fraction ? How are fractions expressed ? What is the lower number 
 called ? What is the uppemumber called \ What does the denominator 
 denote 7 What does trhe numerator denote ! In the fraction .'} fifths, what 
 
110 COMMON FRACTIONS. 
 
 In the fraction f , the base of the collection of fractional units 
 is ^, but this is not the primary base. For, ^ is one-Jrfth of the 
 unit 1 ; hence, the primary base of every fraction is the u?iit 1. 
 
 103. If we suppose a second unit to be divided into the 
 eame number of equal parts, such parts may be expi'essed in 
 the same collection with the parts of the first : thus, 
 
 I is read 3 halves. 
 
 •^ - - - 7 fourths. 
 
 i_6 . . . 16 fifths. 
 
 i_8 . . . 18 sixths. 
 
 ^-f - - - 25 sevenths. 
 
 104. A %\hole number may be expressed fractionally by 
 writing 1 below it for a denominator. Thus, 
 
 3 may be written y and is read, 3 ones. 
 
 5 --.- A ---5 ones. 
 
 6 ---- |. -.. 6 ones. 
 8 ---- ^ _._ 8 ones. 
 
 But 3 ones are equal to 3, 5 ones to 5, 6 ones to 6, and 
 8 ones to 8 ; hence, the value of a number is not changed by 
 placing 1 under it for a denominator. 
 
 105. If the numerator of a fraction be divided by its denomi- 
 nator, the integral part of the quotient will express the number 
 of entire units used in forming the fraction ; and the remainder 
 will show how many fractional units are over. Thus, y are 
 equal to 3 and 2 thirds, and is written y = 3|- : hence, 
 
 A fraction has the same form as an unexecuted division. 
 
 is the fractional base 1 What is the primary base 1 What is the primary 
 base of every fraction 1 
 
 103. If a second unit be divided into the same number of equal parts, 
 may the parts be expressed with those of the first I How many unita 
 have been divided to obtain G thirds 1 To obtain 9 halves 1 12 fourths 1 
 
 104. How may a whole number bo expressed fractionally] Does this 
 chanrre the value of the number ! 
 
 105. If the numerator be divided by the denominator, what does the 
 quotient show 1 What does the remainder show 1 What form has a frai> 
 lion 1 Wluit arc the seven i)rinciplc8 which foiliiw ? 
 
COMMON FRACTIONS. Ill 
 
 From what has been said, we conclude that, 
 
 1st. A fraction is one of the equal parts of a unit, or a col- 
 lectioji of such equal parts : 
 
 2d. Tlie denominator shous into hoxo many equal parts the unit 
 is divided, and hence indicates the vcdue of the fractiomd unit : 
 
 3d. 27ie numerator shows how inany fractional units are taken . 
 
 4th. 'llie value of every fraction is equal to the quotient arising 
 from dividing the numerator hy the denominator : 
 
 5th. When the numerator is less than the denominator, the value 
 of the fraction is less than 1 : 
 
 6th. When the numerator is equal to the denominator, the value 
 of the fraction is equal to 1 : 
 
 7th. When the numerator is greater than the denominator, the 
 value of the fraction is greater than 1. 
 
 EXA3IPLES IN WRITING AND READING FRACTIONS. 
 
 1. Read the following fractions : 
 
 S 7 5 _6 2J ]_6 j_a_ 
 
 9' 12' 3' 15J 9 ' 7 ' 104' 
 
 What is the unit of the fraction, and what the fractional unit in 
 each example ? How many fractional units are taken in each ? 
 
 2. Write 15 of the 19 equal parts of 1. Also, 37 of the 49 
 equal parts of 1. 
 
 3. If the unit of the fraction is 1, and the fractional unit 
 one-fortieth, express 27 fractional units. Also, 95. Also, 106. 
 Also, 87. Also, 41. 
 
 4. If the unit of the fraction is 1, and the fractional unit 
 one 68th, express 45 fractional units. Also, 56. Also, 85. 
 Also, 95. Also, 37, 
 
 5. If the unit of the fraction is 1, and the fractional unit one 
 90th, express 9 fractional units. Also, 87. Also, 75. Also, 65. 
 Also, 85. Also, 90. Also, 100. 
 
 DEFINITIONS. 
 
 106. A Proper Fraction is one whose numerator is less 
 than the denominator. 
 
 100. What is a proper fraction ? Give examples. 
 
112 COMMOX FK ACTIONS. 
 
 The following are proper fractions : 
 
 1 JL 1 3. 3^ S JL 8. 5 
 
 2' 3' 4' 4' 7' 8' 10' 9' 6' 
 
 107. Ax Improper Fraction is one whose numerator is 
 equal to, or exceeds the denominator. 
 
 Note. — Such a fraction is called improper because its value equals 
 or exceeds 1. 
 
 The following are improper fractions : 
 
 3 A 6. 8^ 1 12. JLt ±9_ 
 
 2' 3' 5' 7' 8' 6 ' 7 ' 7 * 
 
 108. A Simple Fraction is one whose numerator and de- 
 nominator are both whole numbers. 
 
 Note. — A simple fraction may be either proper or improper. 
 The following are simple fractions : 
 
 J.3^5_8.98..6X 
 4' 2' 6' 7' 2' 3' 3' 5* 
 
 109. A Compound Fraction is a fraction of a fraction, or 
 eeveral fractions connected by the word of. 
 
 The following are compound fractions : 
 
 1 of 1 1 of 1 of 1 i of 3 J- of 1 of 4 
 
 2^4' 3 "^ 2 3' 6 ' 7 Ul g Ui "i. 
 
 110. A Mixed Number is made up of a whole number and 
 a fraction. 
 
 The following are mixed numbers : 
 
 31, 41, 6f, 5f, Gf, 31. 
 
 111. A Complex Fraction is one whose numerator or de- 
 nominator is fractional ; or, in which both are fractional. 
 
 The following are complex fractions : 
 
 {}) J_ (t) ^ 
 
 5' 191' (!') 691- 
 
 107. Wliat is an improper fraction? Wliy improper 1 Give examples 
 
 108. Wliat is a simple fraction 1 Give examples. May it be proper or 
 improper 1 
 
 109. M'liat is a compound fraction 1 Give examples, 
 
 110. ^\■hat is a mixed numberl Give examples, 
 
 111. What is a complex fraction 1 Give examples. 
 
COMMON FJRACTIONS. 113 
 
 112. The numerator and denominator of a fraction, taken 
 together, are called the terms of the fraction : hence, every 
 fraction has two terms. 
 
 FUNDAMENTAL PROPOSITIONS. 
 
 113. By multiplying the unit 1, we form all the whole 
 numbers, 
 
 2, 3, 4, 5, G, 7, 8, 9, 10, &c.; 
 
 and by dividing the unit 1 by these numbers we form all the 
 fractional units, 
 
 2' 3' 4' 5' 6' 7' 8' 9' TO' "^^* 
 
 Now, since in 1 unit there are 2 halves, 3 thirds, 4 fourths, 
 5 fifths, 6 sixths, &;c., it follows that the fractional unit becomes 
 less as the denominators are increased: hence, 
 
 Any fractional imit is such apaj-i ofl, as 1 is of the denominator. 
 
 Thus, ^ is one-half of 1, since 1 is one-half of the denomi- 
 nator 2 ; i is one-third of 1, since 1 is one-third of 3 ; i is one 
 fourth of 1 ; |-, one-fifth of 1, &c. &c. 
 
 114. Let it be required to multiply ^ by 4. 
 
 Analysis. — In f there arc 3 fractional units, operation. 
 
 each of which is i, and these are to be taken -1x4=: ^^ = J^. 
 4 times. But .3 things taken 4 times, gives 12 
 
 things of the same kind ; that is, 12 eighths ; hence, the product is 4 
 times as great as the multiplicand : therefore, we have 
 
 Proposition I. — If the numerator of a fraction he multiijlicd 
 by any number^ the fraction will he muIti'pUed as many times as 
 there are units in that number. 
 
 112. How many terms has every fraction 1 "What are they 1 
 
 113. How may all the whole numbers be formed ? How may the frac- 
 tional units be formed ? What part of one, is one-half] What part of 1 
 is any fractional unit ] 
 
 114. What is proved in proposition 11 
 
 6 
 
114 
 
 PRINCIPLES OP 
 
 EXAMPLES. 
 
 1. Multiply f by 6, by 7. 
 
 2. Multiply I by 4, by 9. 
 
 3. Multiply g?,- by 11, by 12. 
 
 4. Multiply^^by 12, by 14, 
 115. Let it be required to multiply -^ by 4. 
 Analysis. — In ^ there are 5 fractional 
 lits, each of which is -^^- ^^ ''^'® '^'"^"ide operation. 
 
 the denominator by 4, the quotient is 3, and -^ X 4 r= -pr^ 
 
 5. Multiply fj- by 3, by 4. 
 
 G. Multiply If by 7, by 9. 
 
 7. Multiply A|. by 5, by 10. 
 
 8. Multiply -1^- by 3, by 11. 
 
 the fractional unit becomes ^, which is 4 
 times as great as -^j because, if ^ be divided into 4 equal parts each 
 part will be j\j. If wc take this fractional unit 5 times, the result 
 ^ will be 4 times as great as ^ ; therefore, we have 
 
 Proposition II. — If the denominator of a fraction be divided 
 by any number^ the value of the fraction will be increased as 
 many times as there are units in that number. 
 
 EXAMPLES. 
 
 1. Multiply I by 2, by 4. 
 
 2. Multiply 1^ by 8, by 4, 2. 
 
 3. Multiply ^\ by 2, 3, 4, 
 6,8. 
 
 4. Multiply ^ by 6, by 5, 
 10, 15. 
 
 5. Multiply -}| by 2, 3, 4, G, 
 8, 12, 16, and 24. 
 
 116. Let it be required to divide -^j by 3 
 
 G. Multiply fo by 2, 4, 5, 
 10, 20. 
 
 7. Multiply 3^3- by 7, and 
 by 5. 
 
 8. Multiply ^y by 21, 6, 7, 
 3, and 2. 
 
 9. Multiply i;i- by 2, 3, 4, % 
 9, and 12. 
 
 Analysis. — In -,"1. there are 9 fractional 
 
 1 1 
 
 OPERATION. 
 
 9 ^3 _ 8 
 
 units, each of whic'.i is -jlj, and these are to ^^ -7- 3 
 
 be divided by 3. But 9 things, divided by 
 
 3, gives 3 things of the same kind lor a quotient : hence, the quotient 
 
 is 3 elevenths, a number Avhich is one-third of ^^ ; hence, we have 
 
 Proposition III. — If the nuuxrator of a fraction be divided 
 Inj any number, the fraction will be diminished as many times as 
 there are units in that number. 
 
 11. "j. Wiiat is proved in projiosition 11 ! 
 116. What in piovcd in proposition III ! 
 
COMMON FKACTIONS. 
 
 115 
 
 EXAMPLES. 
 
 1. DivideJf by2,4,8, 16. 
 
 2. Divide jf by 2, 7 and 
 
 14. 
 
 3. Divide ^ by 2, 5, 4 and 
 
 10. 
 
 4. Divide fl by 5, G, 10, 
 15 and 20. 
 
 5. Divide l| by 2, 3, 6, and 
 G. Divide If by 3, 6, 8, and 
 
 12. 
 
 7. Divide f | by 3, 9 and 27 
 
 8. Divide fA by G, 9, 27 
 and 54. 
 
 117. Let it be required to divide -^ by 3. 
 
 Analysis. — In -^, there are 9 fractional operation. 
 
 units, each of which is ^V- Now, if we J--:_3 = -? — = A. 
 
 ; 11 ' 11- 11X833 
 
 multiply the denominator by 3, it becomes 
 
 33. and the fraciioual unit becomes -^j which is one-third part of 
 ^. If, then, we take this fraciioiial unit 9 times, the result -^ is 
 just one-tliird part of ^j : hence, we have divided the fraction y\ 
 by 3 : therefore, we have, 
 
 Proposition IV. — Jf the denominator of a fraction he mul- 
 tiplied hy any niimhrr, the fraction will be diminished as many 
 times as there are units in that number. 
 
 EXAMPLES. 
 
 1. Divide f by G, 7 and 8. 
 
 2. Divide | by 5, 4 and 9. 
 
 3. Divide If by 3, 4 and 12. 
 
 4. Divideff byG, 8andll. 
 
 5. Divide |f by 7, 5 and 3. 
 
 G. Divide \\ by 7, 8 and 6. 
 
 7. Dividers by 3, 7 and 11. 
 
 8. Divide -fi by 8, 4 and 10. 
 
 118. Let it be required to multiply both terms of the frac- 
 tion 1^ by 4. 
 
 Analysis. — In f . the fractional unit is -^, and 
 it is taken 3 times. By multiplying the denomi- 
 nator by 4. the fractional unit becomes -^. the 
 value of which is is one-fourth of \. By multiplying the numerator 
 by 4. we increase the number of fraclional units taken. 4 times ; that 
 
 operation. 
 
 3X4 12^ 
 
 5X4 ~ '2 
 
 117. If the denominator of a fraction be multiplied by any number, how 
 will the value of the fraction be effected ? 
 
 118. If both terms of a fraction be multiplied by any number, how will 
 the value of the fraction be effected ^ 
 
116 PRINCIPLES OF 
 
 is, we increase the number of parts taken just as many times as we 
 decrease the value of the fractional unit ; hence the value of the trac- 
 tion is not changed : therefore, we have 
 
 Pkoe^sition V. — //■ both terms of a fraction he multl^^lied 
 by the same number, the value of the fraction will not be 
 changed. 
 
 EXAMPLES. 
 
 1. Multiply both terms of the fraction |- by 4, by 6, and by 5. 
 
 2. Multiply both terms of -fj by 5, by 8, by 9, and 11. 
 
 3. Multiply both terms of if by 7, by 8, and 9. 
 
 4. Multiply both terms i^- by 5, 8, 6, and 12. 
 
 5. Multiply both terms of |-| by 2, 3, 4, and 5. 
 
 119. Let it be required to divide the numerator and denomi. 
 nator of -^ by 3. 
 
 Analysis. — In ^^, the fractional unit is ■^, operation. 
 and it is taken 6 times. By dividing the dcnonii- 6 h-3^2 
 
 nator by 3. the fractional unit becomes \, the 15^3 5 
 
 value of which is 3 times as great as ■^^. By ; 
 
 dividing the numerator by 3, we diminish the number of fractional 
 luiits taken 3 times ; that is. we diniinish the number of parts taken 
 just as many ti77ies as we increase the value of the fractional tinit : 
 hence, the value of the fraction is not changed ; therefore, we have 
 
 Proposition VI. — If both terms of a fraction be divided by 
 the same number, the value of the fraction will not be changed. 
 
 EXAMPLES. 
 
 1. Divide both terms of |- by 2 and by 4. 
 
 2. Divide botli terms of \ by 3. 
 
 8. Divide both terras of || by 2, 3, 4, 6, and 12. 
 
 4. Divide both terms of A| by 2, 4, 8, and 16. 
 
 5. Divide both terms of ^ by 2, 3, 4, G, and 12. 
 G. Divide both terms of -f-^ by 2, 3, 4, G, and 36. 
 
 119. If botli trrins of a fraction be divided by any number, how will the 
 ■ iuo of the fraction be ctlectcd \ 
 
KEDUOTION OF FJi ACTIONS. 117 
 
 REDUCTION OF FRACTIONS. 
 
 120. Reduction of Fkactions is the operation of cliang- 
 in-T the fractional unit without alterinjjr the value of the fraction. 
 
 A fraction is in its loioest terms, when the numerator and 
 denominator have no common factor. 
 
 CASE I. 
 
 121. To reduce a fraction to its lowest terms. 
 
 1. Reduce y'/'g- to its lowest terms. 
 
 Analysis. — By inspection, it is seen that 5 is 1st. operation. 
 a common factor of the numerator and denomi- ^)i-^ — W' 
 nator. Dividing by it, we have ^. We then see 
 that 7 is a common factor of 14 and 35 : divid- 7)|^ — ^. 
 
 iug by it, we have -|. Now, there is no factor 
 common to 2 and 5 : therefore, f is in its lowest terms. 
 
 2d. The greatest common divisor of 70 and 175 is 35. (Art. 93) ; if we 
 divide both terms of the fraction by it, we ob- ^ 
 
 tain, -S-. The value of Ihe fraction is not chang- 2d operation. 
 ed in either operation, since the numerator and ^5)^-ts ~ !• 
 denominator are both divided by the same num- 
 ber (Art. 119): hence, the following 
 
 ]^ULE. — Divide the numerator and denominator hy their com' 
 mon factors, until they become prime toith respect to each other. 
 
 Or : 2d. Divide the numerator and denominator hy their greatest 
 common divisor. 
 
 exa:\iples. 
 
 Reduce the following fractions to their lowest terms : 
 
 6. Reduce y|-i. 
 
 7. Reduce jVy"^- 
 
 1. Reduce -^-^. 
 
 2. Reduce j^o- 
 
 3. Reduce ^^. 
 
 4. Reduce i^4.|. 
 
 5. Reduce \y^. 
 
 8 Reduce ff-hy 2d. method. 
 9. Reduce f^f " " 
 
 10. Reduce V^^ " 
 
 120. What is reduction of fractions 1 When is a fraction in its lowest 
 terms 1 
 
 121. How do you reduce a fraction to its lowest terms 1 
 
118 
 
 REDUCTION OF FRACTIONS. 
 
 1 1. Reduce y^g^^ by 2(i. metL 
 
 12. Reduce yVaV 
 
 13. Reduce ||{}. 
 li. Reduce j^.%. 
 15. Reduce f|A|. 
 
 IG. Reduce yYA- 
 
 17. Reduce f ^. 
 
 18. Reduce jViV 
 
 19. Reduce iQff§. 
 
 20. Reduce g'^Z/aV 
 
 21. Reduce sffo^- 
 
 22. Reduce J^V'sV 
 
 CASE II. 
 
 122, To reduce an improper fractio)i to an equivalent whole or 
 
 mixed number. 
 
 1. In ^l^ how many entire units ? 
 
 Analysis. — Since there are 5 fifths in 1 unit, operation. 
 there will be in 278 fifths as many units 1 as 5)278 
 
 5 is contained times in 278, viz.. 55 and -f times. 55|-. 
 
 Hence, the following 
 
 Rule. — Divide the numerator hy the denominator, and the qm 
 tient will be the equivalent whole or mixed number. 
 
 EXAMPLES. 
 
 Reduce the following fractions to whole, or mixed numbers 
 
 1. 
 
 Reduce \^^. 
 
 9. 
 
 Reduce ^VrV^ acres 
 
 2. 
 
 Reduce ^-^. 
 
 10. 
 
 Reduce \\\\ 
 
 3. 
 
 Reduce \^*. 
 
 11. 
 
 Reduce Y-,Y6^- 
 
 4. 
 
 Reduce ^^T- 
 
 12. 
 
 Reduce ^|^?|". 
 
 5. 
 
 Reduce Y/ pounds. 
 
 13. 
 
 Reduce ''ff ". 
 
 6.' 
 
 Reduce 2||8 ^ays. 
 
 14. 
 
 Reduce Vgr . 
 
 7. 
 
 Reduce ^|f * yards. 
 
 15. 
 
 Reduce ='VW- 
 
 8. 
 
 Reduce "gV/. 
 
 16. 
 
 Reduce ^'H^''*. 
 
 CASE in. 
 123. To reduce a mixed manber to an equivalent imprope 
 fraction. 
 
 1. Reduce 12|- to its equivalent improper fraction. 
 
 122. What is an improper fraction 1 How do you reduce an irapropev 
 fraction to its equivalent whole, or mixed number ! 
 
 123 What iii a mixed number! How do you reduce a mixed numbei 
 to an improper fraction 1 How do you reduce a whole number to a frac- 
 tion having a given denominator! 
 
REDUCTION OF FE ACTIONS. 119 
 
 Analysis. — Since in any number 
 
 there are 7 times as many 7tlis as operation. 
 
 units 1, there will be 84 sevenths in 12 x 7 = 84 sevenths. 
 
 12: To these add 5 sevenths, and add 5 sevenths, 
 
 the equivalent fraetion becomes 89 gives 12|=89 sevenths, 
 
 sevenths, hence, the following Aiis. = %p. 
 
 Rule. — Midiipli/ the ivhole number by the denominator : to the 
 2)roducl add the numerator, and place tlie sum over the given 
 denominator. 
 
 EXAIUPLES. 
 
 1. Reduce 39|- to its equivalent improper fraction. 
 
 2. Reduce 112j?q to its equivalent improper fraction. 
 
 3. Reduce 427ii to its equivalent improper fraction. 
 
 4. Reduce G7G|y to an improper fraction. 
 
 5. Reduce 367^^^ to an improper fraction. 
 
 6. Reduce 847 y^-^ to an improper fraction. 
 
 7. Reduce G7426 |^yf to an improper fraction. 
 
 8. How many 200ths in 6751^ J ? 
 
 9. How many 151 ths in 187yVr? 
 
 10. Reduce 149g- to an improper fraction. 
 
 11. Reduce 375^^ to an improper fraction. 
 
 12. Reduce 17494^|-|f3- to an improper fraction. 
 
 13. Reduce 4834|^|- to an improper fraction. 
 
 14. Reduce 1789|- to an improper fraction. 
 
 15. In 125|- yards, how many sevenths of a yard ? 
 IG. In 375|- feet, how many fourths of a foot ? 
 
 17. In 4641^ hogsheads, how many sixty-thirds of a hogs- 
 head ? 
 
 18. In 96Jj'j5- aci'es, how many 640ths of an acre ? 
 
 19. In 984y'y'2- pounds, how many 112ths of a pound ? 
 
 20. In 353?^ years, how many 366ths of a year? 
 
 21. How many one hundred and thirty-fifths are there in the 
 mixed number 87j'*Jy ? 
 
 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. 
 
120 REDUCTION OF FKACTI0N8. 
 
 CASE IV. 
 
 123.* To reduce a wliole number to a fraction having a given 
 denominator : 
 
 1. Reduce 17 to a fraction of whicli tlie denominator shall 
 be 5. 
 
 Analysis. — There are 17 times as many operation. 
 
 fifths in 17 as there are in 1. In 1, there 17 X 5 = 85 
 
 are 5 fifths ; therefore, in 17 there are 17 17 = ^ 
 times 5 fifths or 85 fifths ; hence, 
 
 Rule. — Multiply the whole number by the denominator, and 
 write the product over the required denominator. 
 
 EXA3IPLES. 
 
 1. Change 18 to a fraction whose denominator shall be 7. 
 
 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 haviner 9 for its denominator. 
 
 7. Reduce 450 to a fraction having 12 for its denominator. 
 
 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 a fraction whose denominator shall be 89. 
 
 11. Reduce 325 to a fraction whose denominator shall be 75. 
 
 CASE V. 
 
 124. To reduce a compound fraction to a simple fraction. 
 
 1. What is the equivalent fraction of f of |-? 
 
 Analysis. — Three-fifths of \ is three times 
 \oi i^: 1 fifth of -^ is -gij (Art. 117) : and 3 opkration. 
 
 times ^\ is H (Art. 1 14) ; hence, | of ^ - ^ ^ ^ = ^ 
 
 hence, the following 
 
 123.* How Jo you reduce a vvliole number to a fraction having a given 
 denominator ! 
 
 124. \\'hat is a compound fraction 1 How do you reduce a compound 
 fraction to a simple fraction 1 
 
KEDCrCTION OF FKACTIONS. 121 
 
 Rule. — Multiply the numerators together for a new nurnera^ 
 tor, and the denominators tog e titer fur a new denominator. 
 
 Note. — 1. If there are mixed numbers, reduce them to their 
 equivalent improper fractions. 
 
 2. Cancel every factor common to the numerator and denominatpr 
 before multiplying. 
 
 EXAMPLES. 
 
 1. Reduce |- of -| of |- to a simple fraction. 
 
 2. Reduce -| of | of |- to a simple fraction. 
 
 3. Reduce |- of f of 2^ to a simple fraction. 
 
 4. Change -| of f of f of 3^- to a simple fraction. 
 
 5. Change -f^ of f of |- of -f-^ to a simple fraction. 
 
 6. What is the value of i of i of | of 12i ? 
 
 7. What is the value of f of | of 4| ? 
 
 8. What is the value of ,% of 7^ of o-V ? 
 
 9. Reduce -^ of 9|- of 6f of 2| to a whole or mixed number- 
 
 10. Reduce j\ of -^^ of 21|^ to a whole or mixed number. 
 
 11. Reduce |- of f of f of y^^^ ^^ T3 ^^ ^ simple fraction. 
 
 12. Reduce yY^ of j^g- of j^^- of f to a simple fraction. 
 
 13. Reduce 3f off of ^^^-j of 49 to a simple fraction. 
 
 CASE VI. 
 
 125. To reduce fractions of different denominators to equiva- 
 lent fractions that shall have a common denominator. 
 1. Reduce -|, |- and |- to a common denominator. 
 
 Analysis. — Multiplying both terms 
 
 of the first fraction by 20, the product operation. 
 
 of the olher denominators, gives |5.. 2X5x4 = 40 1st num. 
 
 Multiplying both terms of the second 4 x 3 x 4 = 48 2d num. 
 
 fraction by 12, the product of the other 3 X .5 X 3 = 45 3d num. 
 
 denominators, gives |§. Multiplying 3 x 5 X 4 = 60 denom. 
 both terms of the third by 15. the pro- 
 duct of the other denominators, gives ||-. In each case both terms 
 
 125. How do you reduce fractions of different denominators to equiva- 
 lent fractions having a common denominator 1 Note 1. — What reductions 
 are first made T 2. When the numbers are small, how may the work be 
 done'' 3. How may the work often be shortened"! 
 
122 REDUCTION OF FRACTIONS. 
 
 of the fraction have been multiplied by the same number ; there- 
 fore, the value is not changed (Art. 118) : hence, the following 
 
 ]^ULE. — Multiply the numerator of each fraction by all the 
 denominators except its own, for the new numerators, and all the 
 denominators together for a common denominator. 
 
 Nq-j-j. — 1 Before multiplying, reduce to simple fractions when 
 necessary. 
 
 2. When the numbers are small, the work may be performed 
 mentally ; thus, 
 
 1, 1 I become, |^, if, ^ ; and f , 1, f become, if, |f, ff . 
 
 EXAMPLES. 
 
 Eeduce the following fractions to common denominators : 
 
 1. Reduce f , 51 and f . 
 
 2. Reduce f , f , \ and i of 5. 
 
 3. Reduce 9i ' 4i, 2| and f 
 
 4. Reduce f , l, f , ^ and 2i. 
 
 5. Reduce 21 of 3, f , f and f . 
 
 6. Reduce 2i of 31 of f , and 
 
 G3 of |. 
 
 7. Reduce | of f of -| and f 
 of f of |. 
 
 8. Reduce 4§, 21 5iand 6. 
 
 9. Reduce 51 f, 3i and 3f . 
 
 10. Reduce f"'of 5i i of 3i 
 and J 2 of 8^. 
 
 11. Reduce 61 of 2, f , ^ and i. 
 
 IS^OTE. — 3. We may often shorten tlie work by multiplying the 
 numerator 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 by 
 this method : 
 
 1. Reduce ^, -j^, |- and f to a common denominator. 
 
 2. Reduce f, ^y and f to a common denominator. 
 
 3. Reduce 41, -f^ and 71 to a common denominator. 
 
 4. Reduce 10|, f and 71 to a common denominator. 
 
 5. Reduce Gl. # and 71 to a common diMiominator. 
 
 G. Reduce ■^-, ^, 141 and 3|- 1o a common denominator. 
 
 7. Reduce -^, |, 2f and If to a common denominator. 
 
 8. Reduce f, i, ^% and ^ to a common denominator. 
 
 9. Reduce W* f ' ^t ''^"'^ I *° ^ common denominator. 
 
 ''^ Reduce 24,, 51, -^^^r 'i"^^ "^tV *" -"^ common denominator. 
 
EEDUCTION OF FRACTIONS. 123 
 
 CASE VII. 
 
 125*. To reduce fractions io their hast common denominator. 
 
 The least common denominator of two or more fractions is 
 the number which contains only the prime foctors of their 
 denominators. Hence, before beginning the operation, reduce 
 every fraction to a simple fraction and to its lowest terms. 
 
 1. Reduce |-, |- and | to their least common denominator. 
 
 OPERATION. 
 
 (36^4) X 3 = 27 1st numerator. 2)4 . 6 . 9 
 
 (36 -=-6) X 5 = 30 2d numerator. 3)2 . 3 . 9 
 
 (36 -^ 9) X 4 = 16 3d numerator. 2.1.3 
 
 2X3X2X3 = 36, least common denominator : 
 
 therefore, the fractions, reduced to their least common de- 
 nominator, are 
 
 3 6' 3 6' '*"^ 3 6' 
 
 Hence, the following 
 
 Rule. — I. Find the least common multiple of the denomina-' 
 tors (Art. 98) : this ivill be the least common denominator of 
 the fractions. 
 
 H. Divide the least common denominator hy the denominator 
 of each fraction, sejwratehj ; multiply the quotient hy the nume- 
 rator and place the product over the least common denominator : 
 the results will be the new and equivalent fractions. 
 
 EXAMPLES. 
 
 1. Reduce I, ^ and -^-^ to their least common denominator. 
 
 2. Reduce fj, |- and iy to their least common denominator. 
 -3. Reduce 2|, yV and -^-^ to their least common denominator. 
 
 4. Reduce 5|, 4:-f^ and -^^ to their least common denominator. 
 
 5. Reduce 8y^, f and 3^ to their least common denominator. 
 
 6. Reduce 9^, -^^^ and r^^ to their least common denominator. 
 
 125*. What is the least common denominator of two or more fractions 1 
 How do you find the least common denominator of two or more fractions J 
 
124 ADDITION OF FRACTIONS. 
 
 7. Reduce 2-|, 3-j^y and y^^ to their least common denominator. 
 
 8. Reduce o^, |-, f and j-^ to their least common denominator. 
 9. Reduce |, -jy and -jg to their least common denominator. 
 
 10. Reduce Aj-, 7-^^ and ~ to their least common denominator. 
 
 11. Reduce 3^, 6j^ and 1 Jg- to their least common denominator. 
 
 12. Reduce 6|, Sy^- and 2-^ to their least common denominator. 
 
 13. Reduce Oyy, G^^ and 3-3 to their least common denominator. 
 
 14. Reduce j^y, 2^ and 1^''^- to their least common denominator. 
 
 15. Reduce 5|^, Gj-^, -^ and -^ to their least common de- 
 nominator. 
 
 ADDITION OF COMMON FRACTIONS. 
 
 126. The Sum of two or more fractions is a number which 
 contains the unit 1 as many times as it is contained in the frac- 
 tions taken together. 
 
 Addition op Fractions is the operation of finding the sum 
 of two or more fractions. There are two cases : 
 
 Is/. When the fractions have the same unit. 
 2d. When they have different units. 
 
 CASE I. 
 
 127. When the fractions have the same rmit. 
 
 1. What is the sum of i, f , | and ^ ? 
 
 Analysis. — In this example, the unit operation. 
 
 of the fraction is 1, and the fractional 1 + 3 + 6+3 = 13 
 unit ^. There is 1 half in the first. 3 hence, ^^ = 6-J sum. 
 halves in the second. 6 in the third, and 
 3 in the fourth : hcnce^there are 13 halves in all, equal to 6 J. 
 
 2. "What is the sum of 1£ and g£ ? 
 
 Analysis. — The unit of both fractions is operation. 
 
 l£. In the first, the fractional unit is ^£, ^.£ = f £ 
 
 and in the second, ^£. These fractional |-jG = ^£ 
 
 units, being dilTerent, cannot be expressed. |£ + ^£ = £^ = £\^. 
 
 126. What is the sum of two or more frartions ? What is idditijn of 
 fractions 1 How many rases are there ? M'hat arc they 1 
 
 127. How do you add fractions which have the same unit ? 
 
COMMON FRACTIONS. 
 
 125 
 
 in one collection. But ^£ = f jC and fjC = ^£, in each of which 
 the fractional unii is -JjG : hence, their sum is |-jC = £1-^. 
 
 Note. — Only ur.its of the same value, ickether fractional or integral^ 
 can be expressed in the same collection. 
 
 From the above analysis, we have the following 
 
 Rule. — I. When the /radians have the same denominator, 
 add their niunerators^ and -place the sum over the common de- 
 nominator : 
 
 II. When they have not the same denominator, reduce them 
 to a common denominator, and then add as before. 
 
 JVoTE. — 1. 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. 
 
 2. It often abridges the operations in fractions to reduce them to 
 f^ieir least common denominators, before adding (Art. 125*.) 
 
 1. Add f , f, f and f 
 
 2. Add f £, f £, f £ and ii-£. 
 
 3. Add $ i-L , %fj, $if and $/y 
 4- Add 1^- i^ i^ and -^- 
 
 5. Add f , f , f, -If and Jf . 
 
 6. Add ^2, ^2, 3?2 ' T2 '^"t^ f 1 
 
 7. Add i f and ^^. 
 
 8. Add f , I, I and ,\. 
 
 EXAMPLES. 
 
 9. Add f , f , f and ^\. 
 10. Add I, j^, ^ and fl. 
 
 1 1 Add 7 7 13 J 1 „„,] 1 9 
 
 19 Arid 3 5 9 5 „,-,/i ] 5 
 J. 4. j^\.aa -J, -g-, jg, g^ ana g^. 
 
 13. Add j\, f, f and |. 
 
 14. Add 1 41 and |. 
 
 15. Add y\, j5-^, -2^ and |. 
 
 16. Add j% fV I and l.* 
 
 Note. — Reduce each fraction to its least common denominator be- 
 fore adding. 
 
 17. What is the sum of } and } ? 
 
 Analysis. — Reducing to a common 
 denominator, we find the fractions to 
 be ^ and /^, and their sum to be 
 H- That is, 
 
 128. The sum of two fractions whose numerators are each 1, is 
 equal to the sum of their denominators divided by their product. 
 
 OPERATION. 
 
 128. What is the sura of two fractions equal to when each numerator 
 is equal to 1 1 
 
i2G ADDITION uF 
 
 18. What is the sum of i and i? of i and i? of 1 and i? 
 of 1 and yV ? 
 
 19. What is the sum of yV and ^\ ? of yV and jL? of i and 
 1? of i. audi? 
 
 20. What is the sum of 12|, llf and lof ? 
 
 OPERATION. 
 
 IVi^o/e Numbers. Fractions. 
 
 12 + 11 + 15 = 38 14-1^-1-4— -63_ 4- _7_Q_ 1 _7 5 208_iJL03 . 
 
 then, 38 + 11^3 = 391 o|. ^,,5 
 
 Note. — When there arc mixed numbers, add the whole numbers 
 and the fractions separately, and then add their sums. 
 
 Find the sums of the following fractions : 
 
 27. Add f , j% of j\ of 8 and 21. 
 
 28. Add 4f, ,»j- of ^ of 151. 
 29.Add3f, 4f andl6yV 
 SO.AddSf, 4| and ^ of 16. 
 
 31. Add 6|, 13f, 181 and 1321. 
 
 32. Add 124, 201!-, and 40 
 
 I 8' 
 
 21. Add If, 31 and 1 of 7. 
 
 22.Add3ff,7f,iland2ii. 
 
 23. Add 2f , 41 and f of 5^\. 
 
 24. Add 12f , 9f, f of 61 
 
 25. Add j\ of 61 and f of 71. 
 
 26. Add 1 of 9f and | of 4f. 
 
 33. Bought a cord of wood for 2|- dollars ; a barrel of flour 
 for $9^^ ; and some pork for $5^ : what was the entire cost ? 
 
 34. A person travelled in one day 351 miles; the next, 28^ 
 miles ; and the next, 25^^^ miles : how many miles did he ti-avel 
 in the three days ? 
 
 35. A grocer bought 4 firkin.s of butter, weighing re.'pectively 
 54|, 55f, 51 j'^^ and 50|4 pounds : what w.as their entire weight ? 
 
 36. I paid for groceries at one time j^ of a dollar; at ano- 
 ther, 31 dollars ; at another, 7f dollars ; and at another, 5^ 
 dollars : what was the whole amount paid ? 
 
 37. A merchant had three pieces of Irish linen ; the first 
 piece contained 22^- yards ; the second 201 yards ; and tlie 
 third 21^- yards : how many yards in the three pieces ? 
 
 38. A man sold 5 loads of hay ; the first weighed 18j^C'r^. ; 
 the second 19^1c(6'A ; tlie third 19-^f;w/.; the fourth 21 J-ltvt'/. ; and 
 the fifth 2{^\^cw(. : what was the weight of the whole ? 
 
CX)MMON FKACTIONS. 127 
 
 39. A farmer has three fields ; the first contains 17f acres ; 
 the second 25^j acres; and the third 46 j^^ acres: how many- 
 acres in the three fields ? 
 
 40. A man sold 112f bushels of wheat for 2504 dollars ; O/g 
 bushels of corn for 62| dollars; 225^^ bushels of oats for 104^ 
 dollars : how many bushels of grain did he sell, and how much 
 did he receive for the whole ? 
 
 CASE II. 
 
 129. When the fractions have different units. 
 
 1. What is the sum of ^Ib. and ^oz. ? 
 
 Analysis. — In ^!b. there are operations. 
 
 ^z. (Art. 41). Then, the units ^Ib. = f x IGoz. = ^^oz. 
 
 of the fractions being the same, ^^ oz. + -^oz. = -^^oz. -\- ^oz. 
 viz., \oz.. we reduce to a com- = -^-^oz. = IZ^oz 
 
 mon denominator and add, and 
 obtain IS^oz. 
 
 Second Method. — Three- ^oz.=\x^Jh.=-^^lh. 
 
 fourths of an ounce is equal to f''j- + ^V^-=l~2f^^-+F57'^-=lli' 
 ^Ib. (Art. 41). Then, by add- 
 ing, we iind the sum to be ^^Jb.= \2^'^oz.= \Zoz. %\dr . 
 
 Third Method.— Find the {lb.=^xi&oz.=^oz. = Vloz.\2\dr. 
 value of each fractional part ^oz. = \x\Gdr. — ^dr.=z 12 
 
 in terms of integers of the Sum - - - - 13 8^ 
 
 lower denominations, and then add. 
 
 Rule. — Reduce the given fractions to the same unit, and then 
 add as in Case 1. 
 
 Or: Reduce the fractions sejjaratehj to integers of lower de- 
 nominations, and then add the denominate numbers. 
 
 EXAMPLES. 
 
 1 . Add f of a yard to -| of an inch. 
 
 2. Add together ^ of a week, \ of a day, and i of an hour. 
 
 3. Add ^cwt., ^Ib., 13os'., ^cwt. and 6/i. together. 
 
 4. Add -g^ of a pound troy to i of an ounce. 
 
 129. How do you add fractions when they have different units 1 
 
128 ADDITION OF FRACTIONS. 
 
 5. Add ^ of a ton to -j^^ of a hundred weight. 
 
 6. Add ^ of a chaldron to ^ of a busheL 
 
 . 7. What is the sum of f of a tun, and f of a hogshead of 
 wine ? 
 
 8. Add 1- of f of a common year, f of | of a day, and I of 
 I of I- of 191 hours, together. 
 
 9. Add I" of an acre, f of 19 square feet, and |- of a square 
 inch, together. 
 
 1 0. What is the sum of i of a yard, i of a foot, and \ of an 
 inch ? 
 
 11. What is the sum of f of a £, and f of a shilling ? 
 
 12. What is the sum of i of a week, i of a day, i of an 
 hour, and f of a minute ? 
 
 13. Add together |- of a mile, f of a yard, and ' of a foot. 
 
 14. What is the sum of f of a leap year, ^ of a week, and 
 i of a day ? 
 
 15. Add -i of a ton to |- of a hundred weiofht. 
 
 16. Add ^Ib. troy, J02. and ^pivt. 
 
 17. Add together -^^ of a circle, 3|- signs, |^ of a degree, and 
 ^ of 5i minutes. 
 
 18. What is the sum of ^ijcL, | of ^q?: and S^na. ? 
 
 19. Add j^g of a cord, f cubic feet, and |- of 1 of 24|- cubic 
 feet. 
 
 20. What is the sum of f of i of 4 cords, ^^ of -^^ of 15 cord 
 feet, and ^ of 31^ cubic feet ? 
 
 21. Add I of 3 ell English to j^^ o^ a yard.' 
 
 22. Add together A of 3.-J. IE. 20P., | of an acre, and | of 
 3E. 15P 
 
 23. What is the sum of ^^ of a ton, -^ of a cwt., and -^ of 
 an ounce ? 
 
 24. Wliat is the sum of •?,- of ^ of a mile, ? of a furlon"-, -'*- 
 of a rod and -^ of a foot ? 
 
 25. AVliaL is ilie sum of -^'^ of a common year, ^^ of a week, 
 3 of a day, and f of an hour ? 
 
8UETRACT10N OF FKACTI0N3. 129 
 
 SUBTRACTION. 
 
 130. The diiference between two fractions is such a number 
 as added to the less will give the greater. 
 
 SuBTRA-CTiON of Common Fractions is the operation of finding 
 the ditl'erence between two fractional numbers. Thei'e are two 
 cases : 
 
 1. When the fractions have the same unit: 2d. When the 
 fractio7is have different units. 
 
 CASE I. 
 
 131. When the fractions have the same unit.. 
 1. What is the difference between ^ and \ ? 
 
 4 4 
 
 Analysis. — The unit of both fractions is the operation. 
 same, being the abstract unit 1. The fractional f — ^ = f . 
 luiit is also the same, being ^ in each ; hence, 
 the diiference of tlie fractions is equal to the difference of the frac- 
 
 2. 
 
 4' 
 
 tional units, which is 
 
 2. "What is the difference between ^Ib. and f of a pound ? 
 
 Analysis. — The unit in both fractions is operation. 
 
 lib. The fractional unit of the first is \1h. |— |=i|_l-^=^y6. 
 and of the second ^Ib. Reducing to the 
 
 Bame fractional unit, we have ^fZ6. and ^^/6., the difference of 
 which is -fjlb. ; hence, 
 
 Rule I. — If the fractional unit is the same in both, subtract 
 the less numerator from the greater, and place the difference over 
 the common denominator. 
 
 II. 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 denominator. 
 
 130. What is the difference between two fractions 1 What is Subtrac- 
 tion of Common Fractions 1 How many cases are there ! What are they ? 
 
 131. How do 3'ou make the subtraction when the fractions have the 
 same unit 1 
 
130 SUBTRACTION OF 
 
 Note. — Reduce each fraction to a simple form and to its lowest 
 terms before reducing to a common denominator. 
 
 EXAMPLES. 
 
 1. From 1^ take 1. 
 
 2. From f f take {J-. 
 
 3. From if take if. 
 
 4. From ^^i take i^f 
 
 3 0a 305 
 
 5. From | take |-. 
 
 6. P>om ji take J-f 
 
 7. From || take ||. 
 
 8. From 37|i take \ of 5^. 
 
 9. From | take ^. 
 
 10. From | take j\. 
 
 11. From 25 take \}. 
 
 12. From ^"^ of 3 take 1 of |. 
 
 13. From i of f of 7 take ^. 
 
 14. From 3f take f of ^. 
 
 15. From f of 15 take | of 3. 
 
 16. From 7^ of 2 take -} of -|. 
 
 17. To ^vliat fraction must I add f that the sum may be -I ? 
 
 18. AVhat number added to 1 1 will make 5 ? 
 
 19. What number is that to which if 1^ be added the sum 
 will be 17|-? 
 
 20. From the sum of 3f and 10| take the difference of 25i 
 and 17^. 
 
 21. What number is that from which if you subtract i of j 
 of a unit, and to the remainder add -| of -g- of a unit, the sum 
 will be 9 ? 
 
 22. If I buy f of 1^ of a vessel, and sell ^ of ^ of my share, 
 how much of the whole vessel have I left ? 
 
 23. A man bought a horse for i of 4 of j^ ^^ $500, and sold 
 him again for | of I- of ^ of $1680 : what did he gain by the 
 bargain ? 
 
 24. Bought wheat at 1|- dollars a bushel, and sold it ibr 2^ 
 dollars a bushel : what did I gain on a bushel ? 
 
 25. From a barrel of cider containing 31^ gallons, 12f gal- 
 lons were drawn : how much Avas there left ? 
 
 26. Bought 10| cords of wood at one time, and 24|- cords at 
 another; after using IGJy cords, how much remained? 
 
 27. A merchant bought two firkins of butter, one containing 
 ^^To pounds, and the other 56}^- pounds ; he sold 43|| poundd 
 at one time, and 34^ pounds at another : how much had he left? 
 
COMMON FKACTIONS. ]31 
 
 28. A man having $501 expended $15^^^ for dry goods, and 
 $12|- for groceries : liow much had he left ? 
 
 29. A boy having ^ of a dollar, gave -i of a dollar for an 
 inkstand, and } of it for a slate : how much had he left ? 
 
 30. Bought two pieces of cloth, one containing 27f yards, 
 the other 321- yards, from which I sold 4lO\^ yards : how much 
 had I left ? 
 
 132. 1. ^yhat is the difference between i and i ? 
 
 Analysis. — Reducing both fractions to operation. 
 
 a coinmou denominator and subtracting, i — 1 = -^ — -j^^ = -^, 
 we find the difference to be -^ ; that is, 
 
 The difference behveen two fractions, each of ivhose numerators 
 is 1, is -equal to the difference of the denominators divided by 
 their product. 
 
 2. From i take ^. 
 
 3. From jL take -jL.. 
 
 4. From -jL take ^. 
 
 5. From Jy take Jg. 
 
 133. 1. What is the difterence between IQ>\ and 3i ? 
 
 Analysis. — Since we cannot take -^-^ from 
 ■j^. wc borrow 1 -.= \^ from the wliole nuin- operation. 
 
 her of the minuend, wliich added to j^^-, gives 16^^= 16A- 
 
 W : then -j^ from ^| leaves if. We must 3|- = 3y\ 
 
 now carry 1 to the next figure of the subtra- 12W 
 
 hend, and say 4 from 16 leaves 12. Hence, 
 to subtract one mixed number from another, 
 
 Subtract the fractional part from the fractional part, and the 
 integral 'part from the integral part. 
 
 1. What is the diifei-ence between 144 and 12y^jy? 
 2 What is the difference between 115| and 391- ? 
 
 3. What is the difference between 78-i^ and A^ ? 
 
 i h 3 2 
 
 4. What is the difference between 48y^j^ and 41l|- ? 
 
 5. What is the difference between 287^V and 104 -M^. 
 
 132. What is the difference between two fractions whose numerators 
 are each 1 \ 
 
 133. How do you subtract one mixed number from another? 
 
132 BUBTKACTION OF 
 
 CASE II. 
 
 134. When the fractions have different iniits. 
 
 1. What is tlie difference between i of a £ and ^ of a shil- 
 ling ? 
 
 OPERATION. 
 
 Analysis. — Reducing to the com- ■|£ = | x 205. = ^s. 
 mon unit \s.^ we find the difference "^-^s. — ^s. = ^^s— ^s. = ^s. 
 to be ^s. = 9s. 8d. = On. 8d. 
 
 Second Method. — Reducing to the ^■^- — 3 ^ it^ ~ ^eV- 
 common unit IjG, we find the differ- 2^~ eV-^ ~ %q^ ~ i^^- 
 ence to be |f £ = 95. 8^. = |f £ = 95. 8d. 
 
 Third Method. — Reduce the frac- ^£ = IO5. 
 tions to integral units, and then sub- ^5. = Ad. 
 
 tract as in denominate numbers. 95. 8c?. 
 
 Rule. — Reduce the fractions to the same unit, and then sub- 
 tract as in Case 1. 
 
 Or : Find the value of each fraction in units of loiuer da- 
 nominatiom, arid then suht7-act as in denominate numbers. 
 
 EXAMPLES. 
 
 1. From f of a pound troy take ^ of an ounce. 
 
 2. From | of a ton take |- of | of a pound. 
 
 3. Fi'om |- of y of a hogshead of wine take ^ of | of a 
 quart. 
 
 4. From >- of a lea2;ne take f of a mile. 
 
 5. What is the difference between Ifs. and f of l\d. ? 
 
 G. What is the difference between ^1 of a degree and ~ of 
 \ of a degree ? 
 
 7. From f|- of a squai-c mile take 3GJ acres. 
 
 8. From ^ of a ton take | of 12cwt. 
 
 9. From l^lb. troy take \ of an ounce. 
 
 10. From 2g- cords take f of a cord foot. 
 
 11. From -^ of a yard take | of an inch. 
 
 131. How do vou subtract w\ien the fractions have different units 1 
 
COMMON FE ACTIONS. 133 
 
 12. From i of f of a pound take | of ^ of a dram, apothe- 
 caries' weiglit. 
 
 13. A pound avoirdupois is equal to 14:0Z. Il2nvt. IGf/r. troy, 
 what is the difference, in troy weight, between the ounce avoir- 
 dupois and the ounce troy ? 
 
 MULTIPLICATION OF FRACTIONS. 
 
 135. 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. 
 
 1. If 1 pound of tea cost | of a dollar, what will f of a 
 pound cost ? 
 
 Analtsis. — The cost »,"ill be equal to operation. 
 
 the price of unity taken as many times as $| X y = ^^^ := $3^' 
 there are luiits in the quantity (Art. 75). 
 
 One-seventh of a pound of tea will cost one-scvcnth as much as 1/6. 
 Since lib. cost S|. \ of 1/6. will cost | of Sf^SgV (Art. 117). 
 But 3 sevenths of 1/6. will cost 3 times as much as \\ that is, 
 Sj^g X 3 = $|f (Art. 114). Hence, to multiply one fraction by 
 another, 
 
 Rule. — Multiply the numerators together for a new numera- 
 tor^ and the denominators tor/ether for a new denominator. 
 
 Notes. — 1. When the multiplier is less than 1, we do not take the 
 whole of the multiplicand, but only such a part of it as the multi- 
 plier is of 1. 
 
 2. When the multiplier is a proper fraction, multiplication does 
 not imply increase., as in the multiplication of whole numbers. The 
 product is the same part of the multiplicand which the multiplier 
 is of 1. 
 
 135. What is multiplication of fractions ] How do you multiply ono 
 fraction by another ? When the multiplier is less than 1, what part of the 
 multiplicand is taken 1 If the fraction is proper, does multiplication imply 
 increase ? What part is the product of the multiplicand 1 What do you 
 do when either factor is a whole number 1 
 
134: 
 
 MULTIPLICATION OF 
 
 3. If either of the factors is a whole number. ^ATite 1 under it for 
 a denominator. 
 
 4. ^Yhen either of the factors is a mixed number, it maybe reduced 
 to an improper traction, or we may multiply the parts separately and 
 take their sum. 
 
 1. Multiply I by 8. 
 
 2. Multiply fg by 12. 
 
 3. Multiply ll by 9. 
 
 4. Multiply 1^ by 15. 
 
 5. Multiply ^^1 by 12. 
 
 6. Multiply I of i- by 35. 
 
 7. Multiply 3i off by 14. 
 
 8. Multiply If of 2^ by 16. 
 
 9. ]\ruhii)ly 2i of f by 70. 
 10. Multiply 4| of 8 by 36. 
 
 11. Multiply 36 by 4i. 
 
 Analysis. — The number 3(5 is to be taken 
 4 J times; that is, 4 times and i times. One- 
 ninth of 36 is 4, which is wTitten in the units 
 place : then. 4 times 36 is 144; and the sum 
 148 is the product. 
 
 OPERATION. 
 
 36 
 
 4 
 144 
 
 148 Ans. 
 
 12. Multiply 67 by 9^^. 
 
 13. Multiply 842 by 71-. 
 
 14. Multiply 360 by 12f. 
 
 15. Multiply 460 by llf. 
 
 16. Multiply 620 by lOf. 
 
 17. Multiply 1340 by 8f. 
 
 1. Multiply ^ by 8. 
 
 2. Multiply 15 by f 
 
 3. Multiply 11 hyj%. 
 
 4. Multiply 7| by 8. 
 
 5. Multiply 9^ by 18f. 
 
 6. INIultiply 32- by 411. 
 
 7. Multiply Vr- by 9. 
 
 8. Multiply f by I-. 
 
 9. Multiply 1 by f . 
 
 10. Multiplyloff by ^. 
 
 11. Multiply -iV by /o of A- 
 
 EXAJrrLES. 
 
 12. 
 13. 
 14. 
 15. 
 16. 
 17. 
 18. 
 19. 
 20. 
 21. 
 22. 
 
 Multiply i of 1 by A of T^. 
 
 INIultiply I- by 16. 
 
 iNIultiply 28 by -j^. 
 
 :\rultiply ll- by 18. 
 
 IMultiply 8^0 by 15. 
 
 Multiply 
 
 IMultiply 
 
 Multi])ly 8421 by 7^. 
 
 ISrultiply 
 
 Multiply 
 
 Multiply 
 
 r. 
 TT 
 
 of 2- by ^J. 
 
 51 by A of 31. 
 
 f\ bv ^-. 
 
 9 '•V 7 
 
 TTJ 
 
 by 7^ 
 
 TT* 
 
 ^? by -J of ^g. 
 
COMMON FRACTIONS. ' 135 
 
 23. Multii)ly -j\, |f and || together. 
 
 24. Multiply if, ^^, -^j and f{i together. 
 
 25. What is the product of |^ by | of 17 ? 
 2G. What is the product of 6 by f of 5 ? 
 
 27. AVhat is the product of i of i of 3 by loj ? 
 
 28. Require the product of f of |- by f of 3|. 
 
 29. Require the product of 5, ^, | of |, and 41-. 
 
 30. Require the product of 14, f , ^ of 9, and 6f . 
 
 31. What A\ill 7 yards of cloth cost at $f a yard ? 
 
 32. Wliat will 12|- bushels of apples cost at $| a bushel ? 
 
 33. If one bushel of wheat cost $1^, what will f of a bushel 
 cost ? 
 
 34. If one horse eat f of a ton of hay in one month, how 
 much will 18 horses eat in the same time ? 
 
 35. If a man earn $i| in one day, how much can he earn in 
 24 days ? 
 
 36. What will 31 yards of cloth cost at |- of a dollar a yard ? 
 
 37. At $16 a ton, what will |i of a ton of hay cost ? 
 
 38. If one pound of tea cost $1\, what will 61 pounds cost? 
 
 39. What will 3| boxes of raisins cost at $2^ a box ? 
 
 40. At 75 cents a bushel, what will ii of a bushel of corn 
 cost ? 
 
 41. If a lot of land be worth $75y^3-, what will ^x of it be 
 worth ? 
 
 42. If a man earn $56 in one month, how much can he earn 
 in y-j of a month ? 
 
 43. What will 171 yards of cambric cost at 21 shillings a 
 yard ? 
 
 44. Bought 15|- barrels of sugar at $20| a barrel, what did 
 the whole cost ? 
 
 45. If one bushel of corn is worth f of a dollar, what is f of 
 a bushel worth ? 
 
 46. If I own -^j of a farm and sell -^^ of my share, what 
 pai-t of the whole farm do I sell ? 
 
 47. Bought a book for -j^ of a dollar and a knife for j^ as 
 much; how mucli did I pay for the knife ? 
 
136 MDLTIPLICATION 
 
 48. At f of |4 of a dollar a pound, what will f of ||- of a 
 pound of tea cost ? 
 
 40. If ha J is Avorth $9|- a ton wliat is ^ of Si ton worth. 
 
 50. If a man can dig a cellar in 22^ days, how many days 
 will it take him to dig ^ of it ? 
 
 51. If a railroad train run 1 mile in -^L. of an hour, how 
 long will it be in running 106j miles ? 
 
 52. What will be the cost of 20|- cords of wood at $3J a 
 cord ? 
 
 53. If a man walk 3^ miles an hour, how far will he walk in 
 Of hours ? 
 
 54. "What will 14| bushels of potatoes cost at 311 cents a 
 bushel ? 
 
 55. What will 12J dozens of eggs bring at 18|- cents a 
 dozen ? 
 
 56. At "I of a dollar a bushel, what will 1021 bushels of ry8 
 cost? 
 
 57. What will |- of a firkin of butter cost at $18Jg- a 
 firkin ? 
 
 58. A man, at his death, left his wife $15000; she at her 
 death left |- of her share to her daughter : what part of the 
 father's estate did the daughter receive ? 
 
 59. A person owning ^ of a cotton factory sold f of his paii; 
 to A, and the rest to B : what jiart of the whole did each 
 buy? 
 
 GO. A owTied I- of a farm and sold 4 of his share to B, who 
 sold f of what he bought to C, who sold £ of what he bought 
 to D : what part of the whole did D have ? 
 
 61. A owned | of 200 acres of land, and sold ^ of his sliare 
 to B, who sold i of what he bought to C : how many acres had 
 each? 
 
DIVISION OF FRACTIONS. 137 
 
 DIVISION. 
 
 136. Division of Fractioxs is tlie operation of finding 
 a number wliicli multiplied by the divisor will produce the 
 dividend, when one or both are fractional. 
 
 1. What is the quotient of |- divided by -J^*- ? 
 
 Analysis. — How many lanes is ^^ con- operation. 
 
 tallied in |-? If ^ be divided by 14, the l-^XA^l-^J^ 
 
 quotient will be gx. . (Art. 117). Since ~ iVa' ~ 1^6 -^^'* 
 
 the true divLsor is but ^ of 14, the divisor 
 used is 5 times too large ; hence, the par- 
 tial quotient r^^^ is 5 times too small. 
 Multiplying this by 5, we have the true 
 quotient, ,3_5^ = ■^\. 2 ^ ! 
 
 The operation may be made by means X4 
 
 of the vertical line, by .simply placing the 
 
 dividends on the right and the divisors on 16 | — -^^ A. 
 
 the left. 
 
 Since the same process is applicable to any two fractions, 
 we have the following rule : 
 
 I. Invert the terms of the divisor : 
 
 II. Multiply the numerators together for the numerator of the 
 quotient, and the denominators together for the denominator of 
 the quotient. 
 
 Notes. — 1. If either the dividend or divisor is a whole number. 
 
 make it fractional, by writing 1 under it for a denominator. 
 
 2. If the vertical line is used, the denominator of the dividend and 
 the numerator of the divisor fall on the left, and the other terms on 
 the right. . 
 
 3. Cancel all common factors. 
 
 136. What is division of fractions ^ AMiat is the rule for the division 
 effractions'! What do you do when cither the dividend or divisor is a 
 whole number 1 Where do the parts fall when you use the vertical line '^ 
 What do you do when either term of the fraction is a mixed number or 
 a compound fraction 1 If the terms of the dividend are exactly divisible 
 l)y the corresponding terms of the divisor, how do you find the quotient ? 
 
 7 
 
L38 
 
 DIVISION OF 
 
 4. If the dividend and divisor have a common denominator, it 
 will cancel, and the quotient of the numerators will he the 
 answer. 
 
 5. When either term of the fraction is a mixed number, or a com- 
 pound fraction, reduce to the form of a simple fraction before 
 dividing. 
 
 6. If the numerator of the dividend is exactly divisible by the 
 numerator of the divisor, and the denominator by the denominator, 
 the division may be made without inverting the terras of the divisor. 
 
 EXAMPLES. 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 12. 
 13. 
 14. 
 lo. 
 16. 
 17. 
 18. 
 10. 
 20. 
 21. 
 22. 
 23. 
 24. 
 25. 
 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 
 H by 7. 
 t\ by 6. 
 
 If by 9. 
 m by 40. 
 M by 13. 
 
 5by tV- 
 27 by I 
 
 ^byf 
 
 fVby 
 ^by 
 off by f off. 
 
 5 
 
 TT* 
 
 I off by. 
 
 * of ^ 
 
 I Of f by f off. 
 56 by 11. 
 1000 by T^. 
 725 by f f 
 4f by 5. 
 9^5^ by 12. 
 of 16iby 41. 
 by 1 of 7. ' 
 ^ of 50" by 41. 
 300^'V by 61. 
 1 of"3ii by l>f 71 
 91 by 81. 
 
 1 
 
 h 
 
 » OI yy 
 
 by 6^. 
 
 26. 
 27. 
 28. 
 29. 
 30. 
 31. 
 32. 
 33. 
 34. 
 35. 
 36. 
 37. 
 
 oo 
 OO. 
 
 39. 
 40. 
 41. 
 42. 
 43. 
 44. 
 45. 
 46. 
 47. 
 48. 
 49. 
 50. 
 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 Divide 
 
 by 4. 
 
 _9 
 
 ro- 
 
 il 
 
 il by 5. 
 ffby8. 
 m by 48. 
 tS's by 21. 
 36 by 
 420 by f 
 
 3 
 
 ¥• 
 
 7 
 15" 
 
 byM- 
 I- by if. 
 
 foffbyfoff. 
 lofioffbylofl 
 650 by J-^f . 
 1273 by li. 
 4324 by 
 
 2'o by 
 
 25 
 
 2 of 2 7. 
 
 3 "^ 5 
 
 by 
 
 ] 28 
 475' 
 
 6f by 8. 
 
 121 
 
 by 42 
 r7 by 91. 
 100 by 43 
 
 44^ by 
 lll-Vby 
 
 by 
 
 by I of U. 
 
 by I of 
 
 191J- 
 5205} 
 
 '8' 
 2 13 
 3'3 3* 
 
 33i. 
 1591. 
 
 90 
 
COMMON FRACTIONS. 139 
 
 51. At J- of a dollar a pound, how much butter can be bought 
 for I k of a dollar ? 
 
 52. At f of a dollar a yard, how muck cloth can be bought 
 for |- of a dollar ? 
 
 5-), If a bushel of potatoes cost ^ of a doljar, how many can 
 be bought for -^^ of a dollar ? 
 
 54. If ^ of a ton of hay will feed 1 horse one week, how 
 many hoi^^es will -f^j of a ton feed, the same time ? 
 
 55. If ^^ of a bushel of apples cost | of a dollar, what will a 
 bushel cost ? 
 
 5G. What will a barrel of flour cost, if j^^ of a barrel cost 
 I of a dollar ? 
 
 4 
 
 57. If |- of a bushel of apples cost -| of a dollar, what will 
 1 bushel cost ? 
 
 58. How much molasses at f of a dollar a gallon, can be 
 bought for ly dollars ? 
 
 59. A man sold |-|- of a mill, which was ^ of his share : 
 what part of the mill did he own ? 
 
 60. Yf hat number multiplied by f, will give lof for the pro- 
 duct ? 
 
 61. What number multiplied by 51, will give 146 for the pro- 
 duct? 
 
 62. The dividend is 520^, and the quotient 36y% : what is 
 the divisor? 
 
 63. What number is that which if multiplied bv I- of ^ of 
 151, will produce -I? 
 
 64. If lib. of sugar cost ^j of a dollar, what will 1 pound 
 cost? 
 
 65. If lQ\Ih. of nails cost y of a dollar, what is the price per 
 pound ? 
 
 66. If y of a yard of cloth cost $3, what is the cost of a 
 yard ? 
 
 67. A fiuiiily consumes 165| pounds of butter in 8^ weeks: 
 how much do they consume in 1 week ? 
 
 68. At $9-1 a barrel, how much flour can be bought for 
 
 $138f ? 
 
140 REDUCTION OF 
 
 69. If a man divides $3| equally among 8 beggars, how 
 much will he give them apiece ? 
 
 70. If 8 pounds of tea cost 7f dollars, what is the price per 
 pound ? 
 
 71. If I of a ton of hay sell for $10f, what should 1 ton sell 
 for ? 
 
 72. If ^ of an acre of ground produce 84y^ bushels of pota- 
 toes, how many bushels will 1 acre produce? 
 
 73. What quantity of cloth may be purchased for $5Jg-, at 
 the rate of $6f a yard ? 
 
 74. How long would a person be in travelling 125^ miles, if 
 he travelled oljj miles per day ? 
 
 75. How many bottles, each holding 1^ gallons, can be filled 
 from a barrel of wine, containing 311 gallons ? 
 
 7G. How long will it take 11 men to do a piece of work that 
 I man can do in 15| days ? 
 
 77. If |- of a barrel of flour cost G 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 3- of a gallon, 
 be filled from i of a barrel of 311 gallons ? 
 
 81. If 5}Jb. of tea cost $41-, what is the price of 1 pound ? 
 
 82. If f of -^ of a ship is worth $2540, what is the whole 
 vessel worth? 
 
 83. If |- of an acre of land cost S361, Avhat will be the value 
 of an acre ? 
 
 84. If I of -J of a bai'rel of flour will last a family 1 week, 
 how lonn; will 9fV barrels last them? 
 
COMPLKX FRACTIONS. 141 
 
 REDUCTION OF COMPLEX FRACTIONS. 
 137. A Complex Fraction is only another form of expres- 
 Eion for the division of fractious: thus, -3, is the same as | 
 
 "6 
 
 divided by f ; and may be written, i^^=is 
 
 
 5 
 
 o3 
 
 X 
 
 "$ 
 
 7 
 
 138. To reduce a complex fraction to a simple fraction : 
 
 1. Reduce —y- to a simple fraction. 
 
 Analysis. — Reducing the divisor operation. 
 
 and dividend each to a simple frac- G|- = ^j) and 1| = f-. 
 
 tion, we have %» and f . Then 2_o 20 _:. 8 ^ 2j) x 1 = 83 - 54 
 divided byf is equal to 5/ xX^Jj^a ^ "^ '' ' ^ ^' 
 
 _3_5 _ t5 
 
 — 6 — "^a- 
 
 Rule. — Divide the numerator g | 35 
 
 of the compilcx fraction hy its de- Ans. ^^- = 5|. 
 
 nomi/iator. 
 
 Or: Midtiply the numerator of the upper fraction into the de- 
 nominator of the lower, for a new numerator ; arid the denominator 
 of the upper fraction into the numerator of the lower, for a new 
 denominator. 
 
 ]v;oTES. — 1. When either of the terms of a complex fraction is a 
 mixed number, or a compound fraction, it must first be reduced to the 
 form of a simple fraction. 
 
 2. ^Yhen the vertical line is used, the numerator of the upper and 
 Ihe denominator of the lower numbers fall on the right of the vertical 
 
 line, and the other terms on the left. 
 
 137. What is a complex fraction 1 
 
 138. How do you reduce a complex fraction to a simi)le fraction ^ 
 
142 
 
 APPLICATIONS m 
 
 EXAMPLES. 
 
 Reduce the following to simple fractions 
 
 1. Reduce f- 
 
 4 
 5 
 
 2. Reduce ^' 
 
 T6 
 
 3. Reduce -^• 
 
 87i 
 
 4. Reduce -y^- 
 
 8 
 
 5. Reduce -^• 
 
 84 
 
 6. Reduce r^- 
 
 7. Reduce ~-^^' 
 
 8. Reduce 
 
 20 
 
 7 
 
 ^ of 7 3- 
 9. Reduce ^~J~^^- 
 
 26.8 
 10. Reduce 
 
 '3 5 
 
 11. Reduce 
 
 I of 17 
 554 
 
 H 
 
 12. Reduce f off^ of -| 
 
 APPLICATIONS IX FRACTIONS. 
 
 1. What -will 51 cords of wood cost at L of | of | of $50 a 
 cord ? 
 
 2. A farmer sold f of a ton of hay for $6| : what would be 
 the price of a ton at the same rate ? 
 
 3. A person walks 77 1 miles in 10^ hours : at what rate 
 is that per hour ? 
 
 4. From the product of ^ and llJ,take y^, and multiply the 
 remainder by 20'. 
 
 5. How mucli greater is ^ of the sum of ^, ^, i and J , than 
 the sum of -^, i and ^ ? 
 
 6. If •} of a ton of hay is worth $7i, what is 2| tons worth? 
 
 7. If - of a dollar will pay for |- of a yard of cloth, how 
 piany yards can be bouglit for $ll-i?- ? 
 
 8. AVhat is the value of 3J- cords of wood at S-l| a cord ? 
 
 4 3- A 
 
 9. "What is the continued product of 144, -— , v, ^^^ ^? 
 
 <■ '' oi' f 9 
 
 49''- 34?- 
 
 10. "Wiiat is the sum and (lidi'rencc of --3 and vr,rV ? 
 
 9/ 14"TT 
 
COMPLEX FRACTIONS. 143 
 
 11. At } of a dollar a peck, how many busliels of apples can 
 be bought for $Gf ? 
 
 12. What is the difference between f of a league and -^ of 
 a mile ? 
 
 13. Subtract 8|Z6. from |- of a cwf. 
 
 14. What is the sura of A^^ miles, ^ of a furlong, and f of 
 11 yards ? 
 
 15. At $lf per day, how many days labor can be obtained 
 for$36f? 
 
 16. Sold 7i bushels of apples for $3| : what should I receive 
 for 24f bushels ? 
 
 17. A has G34 sheep, which are 124 more than f of 21 times 
 B's number : how many sheep had B ? 
 
 18. At 3^ of a dollar a yard, how many yards of ribbon 
 can be bought for ^ of a dollar ? 
 
 19. Paid $56 1 for 94 yards of muslin : how much was that 
 per yard ? 
 
 20. Bought 51 yards of cloth at $41 a yard, and paid for it 
 in wheat at $11 a bushel : how many bushels were required ? 
 
 21. What number must be taken from 27|, and the remainder 
 multiplied by 14f , that the product shall be 100 ? 
 
 22. Three persons. A, B, and C, purchase a piece of pro- 
 perty for $6300; A pays f of it, B, land C the remainder: 
 what is the value of each one's share ? 
 
 23. What number is that which being diminished by the dif- 
 ference between f and | of itself leaves a remainder equal to 34 ? 
 
 24. Add together 1 of a week, 1 of a day, and 1 of an hour. 
 
 25. What is the sum of f of £15, £3f, i of f of | of £1, 
 and f of f of a shilling ? 
 
 26. If 1 of John's marbles are equal to i of James', and to- 
 gether they have 56 : how many has each ? 
 
 27. A person owning f of 2000 acres of land, sold f of his 
 share : how many acres did he retain ? 
 
 28. A boy having 240 marbles, divided them in the following 
 manner : he gave to^A, l, to B, j\, to C, 1, and to D, i, keeping 
 the remainder himself : what number of marbles had each ? 
 
144 DUODECIMALS. , 
 
 29. A man engaging in trade with $3740, found at the end 
 of 3 years that he had gained $156^ more than ^ of his capital : 
 what was his average annual gain ? 
 
 30. Two boys having bought a sled, one paying |- of a dollar, 
 and the other |- of a dollar, sold it for y^g- of a dollar more than 
 they gave for it : what did they sell it for, tind what was each 
 one's share of the gain ? 
 
 31. A farmer having 126y bushels of wheat, sold -I of it for 
 $21 a bushel, and the remainder for $1|- a bushel : how much 
 did he receive for his wheat ? 
 
 32. A man having $19-1-, expended it for wheat and corn, of 
 each an equal quantity ; for the wheat he paid $14 a bushel, 
 and for the corn $|- a bushel : how much of each did he buy ? 
 
 33. Two persons engage in trade : A furnished -^^^ of the 
 capital, and B, -^ ; if B had furnished $49 2|- more, their shares 
 would have been equal : how much did each furnish ? 
 
 34. A man being asked how many sheep he had, said he 
 had them in 3 fields : in the first he had G3, which was ^ of. 
 what he had in the second, and that f of what he had in the 
 second was just 4 times what he had in the third : how many 
 sheep had he in all ? 
 
 DUODECIMALS. 
 
 139. Duodecimals are a system of numbers which arise from 
 dividing the unit 1 according to the uniform scale of 12 ; tlms, 
 
 If the unit 1 foot be divided into 12 equal parts, each part 
 is called an inch or prime, and marked \ If an inch be divi- 
 ded 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. 
 
 139. What arc duodecimals 1 If the unit 1 foot be divided into 12 equal 
 parts, what is each part called? If 1 inch be divided into 12 equal parts, 
 what is each part called ] If the second be divided in like manner, what 
 is each part called 1 What are indices 1 
 
DUODECIMALS. 145 
 
 This division of the foot gives 
 
 1' inch or prime — ^- of a foot. 
 
 1" second is Jj of jL - - - - = J-^ of a foot. 
 V" third is tV of yV of t'2 - - = W^ of a foot. 
 
 Note.— The marks ', ", '", &c., which denote the fractional units, 
 are called indices. 
 
 
 TABLE. 
 
 
 
 12'" 
 
 make 
 
 1" 
 
 second. 
 
 12" 
 
 a 
 
 r 
 
 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 it, at 
 every point, being 12. 
 
 Note. — Duodecimals are chiefly used in measuring surfaces and 
 solids. 
 
 ADDITION AND SUBTRACTION. 
 
 140. The units of duodecimals are reduced, added, subtracted, 
 and muhiplied like those of other denominate numbers. The 
 units of the scale are 12, at every change of the unit. 
 
 EXAMPLES. 
 
 1. In 86' how many feet? 
 
 2. In 750" how many feet? 
 
 3. In 37000'" how many ft. ? 
 
 4. In G7' how many feet? 
 
 5. In 470'" how many feet? 
 
 6. In 375" how manv feet ? 
 
 7. What is the sum of 8/?. 9' 7" and G/>. 7' 3" 4'"? 
 
 8. What is the difference between 32/?. 6' G" and 20ft. 7'"? 
 
 9. Add together 9/^. G' 4" 3"', 12//. 2' 9" 10"', 26ft. 0' 5", 
 and 40/2. 1' 0" 3'". 
 
 10. Wliat is the sum of 126ft. 0' G", 45/?. 11' 0" 2'" and 
 12ft. G' ? 
 
 140. By what rules do you operate on duodecimal units I What are tha 
 unils of the scale ! 
 
1 4:0 DUODECIMALS. 
 
 11. What is the sum of 84//. 7', 96//. 0' 11", 42//. 6' 9'' 
 10"' and 5' 7" 11'"? 
 
 12. From 127//. 3' 6" 4'" 11"" take 40//. 0' 10" 7"' 5"". 
 
 13. What is the difference between 425/7. 9' 10" and 107//. 
 10' 9" 8'"? 
 
 14. What is the sum and difference of 325//. 7' 6" 2'" and 
 217/. 10' 9"? 
 
 15. What is the sum and difference of 1001//. 0' 0" 10'" 
 and 720//. 10' 9" 1'" ? 
 
 MULTIPLICATION. 
 
 141. Multiplication of duodecimals is the operation of 
 finding the superficial contents, or the contents of volume, when 
 the linear dimensions are known. 
 
 To do this we begin with the highest unit of the multiplier 
 and the loiuest of the multiplicand, and recollect, 
 
 1st. That 1 linear foot x 1 linear foot = 1 square foot, (Art. 
 411), or, that a part of a foot x a part of a foot = some part 
 of a square foot. 
 
 2d. That a square foot X by a foot in length =: a cubic foot. 
 
 Note. — Ob.serve that in the first muUiplication the unit is changed, 
 from a linear to n superficial unit; in the second muUiplication, from 
 a supevficial unit to a unit of volume. 
 
 1. Multiply 6//. 7' 8" by 2//. 9'. 
 
 Analysis. — Since a prime is ^ of a 
 foot, and a second y^, 2X8"= ^^ of 
 a square foot ; which reduced to 12ths, 
 is r and A"; that is, 1 twelfth, and 4 
 twelfths of twelfths of a square foot. 
 2 X 7' = 14 twelfths = 1ft. 2' - - - 
 
 2 X 6 = 12 square feet 2X6=12 
 
 9' X 8"= ylfg- of a square foot ---- 6" - 9'x 8"= 6" 
 
 9'x T = ^^ = 5' 3" 9'X 7' = 5' 3" 
 
 9 X G' =14 = 4 6' 9'X 6 = 4 6' 
 
 Prod. 18 3' 1" 
 
 
 
 OPERATION, 
 
 
 
 
 
 ft- 
 
 
 
 
 
 
 6 
 
 7' 
 
 8" 
 
 
 X 
 
 8" = 
 
 o 
 
 9' 
 
 
 2 
 
 
 1' 
 
 4" 
 
 2 
 
 X 
 
 7' = 
 
 1 
 
 2' 
 
 
 HI. Wliat is multiplication of duodecimals 1 
 
DUODECIMALS. 147 
 
 Rule. — I. Write the midtijylier under the mtdtij^licand, so 
 that units of the same order shall fall in the same column. 
 
 II. Begin with the highest unit of the multiplier and the 
 lowest of the multiplicand, and make the index of each product 
 equal to the sum of the indices of the factors. 
 
 III. Reduce each product of the first midtijylication to square 
 feet and 12ths of a square foot, and when there ore three factors 
 reduce the second products to units of volume, 
 
 NoTK. — The index of the unit of any product is equal to the sum 
 of the indices of the factors. 
 
 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 ? 
 
 Analysis. — Beginning with the 1 foot, operation. 
 
 we say 1 time 4' is 4' = ^^ of a square ft. 
 
 foot : then, 1 time 2 is 2 square feet. 2 4' 
 
 Next, 5 times 4' are 20" = 1' and 8" : 1 5^ 
 
 then, 5 times 2 feet = 10', and the V to 
 carry, makes 11' 8". Then multiplying 
 by the length 12 feet 6', we find the 
 contents to be 41 3' 10" cubic feet. 
 
 2 
 
 4' 
 
 
 
 11 
 
 8" 
 
 3 
 
 3' 
 
 8" 
 
 12 
 
 6' 
 
 
 39 
 
 8' 
 
 
 
 1 
 
 7' 
 
 10" 
 
 41 3' 10" 
 
 2. Multiply 9/7. 6' by 4/if. 7'. 
 
 3. Multiply 12/^. 5' by Qft. 8'. 
 
 4. MuUiply 35/^. 4' 6" by 9/?. 10'. 
 
 5. What is the product of 45//. 4' 3" by 12^ 2' 9" ? 
 
 6. What is the product of 140//. 0' 2" 4'" by 20//. 10' ? 
 
 7. What is the product of 279//. 10' 6" by 8' 4" ? 
 
 8. What are the contents of a board 14//. 6' 3" long and 
 2//. 9' wide ? 
 
 9. How many square feet in a floor 18//. 9' long, and 15/f. 
 10' wide ? 
 
 10. How many square yards in a ceiling 70//. 9' long, and 
 12//. 3' wide ? 
 
148 DnasioN OF 
 
 11. What will be the cost of paving a yard 64/);. 6^ square, 
 at 5 cents a square foot ? 
 
 12. What are the cubic contents of a block of marble, 6/?. 9' 
 long, 4/'/. 8' wide, and 2//!. 10' thick ? 
 
 13. There is a room 97/7. 4' around it; it is 9//. G' high: 
 what will it cost to paint the walls, at 18 cents a square yard ? 
 
 14. How many cubic feet of wood in a pile 36/^. 5' long, 
 6//. 8' high, and 3//. 6' wide ? 
 
 15. What will a pile of wood 26/if. 8' long, 6//'. 6' high, and 
 Sft. 3' wide, cost, at $3,50 a cord ? 
 
 16. How many cubic yards of earth were dug from a cellar 
 which measured 38/V. 10' long, 20ft. 6' wide, and 9/V. 4' deep? 
 
 17. At 16 cents a yard, what will it cost to plaster a room 
 22/;. 8' long, 18/^ 9' wide, and lift. 6' high? There are to 
 be deducted 8 windows, Gft. 4' high and 2/7. 9' wide ; 2 doors, 
 1ft. 6' high and ^ft. 2' wide, and the base moulding, which is 
 1 foot wide. 
 
 DIVISION OF DUODECIMALS. 
 
 142. Division of Duodecimals is the operation of finding 
 from two duodecimal numbers a third, which multiplied by the 
 first, will give the second. 
 
 1. A hall contains lOSsq.ft. 4' 5" 8'" 4^ and is Gft. IV S" 
 wide : what is its length ? 
 
 AnATA'SIS. The OPF.RATION. 
 
 units of the dividend ft. s(]. ft. ft. 
 
 are square feet and 6 11' 8") 103 4' 5" 8'" 4-(14 9' 11" 
 
 fractions of a square 
 foot. The units of 
 the divisor arc linear 
 feet and fractions of 
 a linear foot. 
 
 First, consider how 
 often the first two parts of the divisor are contained in the first part 
 of the dividend. The first two parts of the divisor arc nearly equa^ 
 
 97 
 
 7' 
 
 4" 
 
 
 5 
 
 9' 
 
 1" 
 
 8'" 
 
 5 
 
 2' 
 
 9" 
 
 0'" 
 
 
 6' 
 
 4" 
 
 8"' i' 
 
 
 fi' 
 
 4" 
 
 8'" 4 
 
 142. What is Division of Duodecimals 1 How is it performed ] 
 
DIVISION OF DUODECIMALS. 149 
 
 to 7 feet, and this is contained in I02sg. ft. 14 times and something 
 over. Multiplying the divisor by this term of the quotient and sub- 
 tracting, we find the remainder 5ft. 9' 1", to which bring down 8"'. 
 
 Next, consider how many times the first two parts of tlie divisor, 
 (equal to 7 feet, nearly.) are contained in the first two parts of the 
 remainder, reduced to the next lower unit ; that is, 5ft. 9' = 69'. 
 Multiplying the divisor by the quotient figure 9', and making the 
 subtraction, we have, 6' 4" 8", to which bring down 4'"'. 
 
 Consider, again, how often, nearly 7 feet is contained in 6' 4"=76" 
 Multiplying the divi-sor by the quotient 11", we find a product equal 
 to the last remainder. Hence, the process of division is the same as 
 that of other denominate numbers^ except in the manner of selecting the 
 quotient figure. 
 
 Notes. — 1. If the integral unit of the dividend and divisor is the 
 same, the unit of the quotient unll he 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 linear. 
 
 3. If the unit of the dividend is a unit of volume, and the unit of 
 the divisor linear, the unit of the quotient loill be superficial. 
 
 4. If the unit of the dividend is a unit of volume, and the unit of 
 the divisor superficial, the unit of the quotient will be linear. 
 
 EXAMPLES. 
 
 1. Divide 2^sq.fL 0' 4" by Qft. 4'. 
 
 2. Divide bOsq.ft. 0^ 10" 6'" by 9/;. 6'. 
 
 3. What is the length of a floor whose area is 1176sg'./i;. 1' 
 6", and breadth 24/;. 3'? 
 
 4. A load of wood, containing ll^cv.fL 2' 6" 8'", is Zft. 4' 
 high, and 4ft. 2' wide : what is its length ? 
 
 5. In a granite pillar there are IQocu.ft.^ 1" 6'"; it is 
 ?>ft. 9' wide, and 1ft. 3' thick : what is its length ? 
 
 6. There are 394sg./i!. 2'9"in the floor of a hall that is \^ft. 
 7' wide : what is its length ? 
 
 7. A board Ylft. 6' long, contains Tl sq.ft. 8' 6" : what is its 
 width ? 
 
 8. From a cellar 42/V. 10' long, 12/. C wide, were thrown 
 158c2«. yds. lieu, ft 4' of earth : how deep was it? 
 
150 DECIMAL FKACTIOrJS. 
 
 DECIMAL FRACTIONS. 
 
 143. There are two kinds of Fractions : Common Fractions 
 and Decimal Fractions. 
 
 A Common Fraction is one in which the unit is divided into 
 any number of equal parts. 
 
 A Decimal Fraction is one in which the unit is divided 
 according to the scale of tens, 
 
 144. 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, each 
 part is called one-lmndrcdth. 
 
 If the unit 1 be divided into one thousand equal parts, the 
 parts are called thousandths, and we have like expressions for 
 the parts, when the unit is further divided according to the 
 scale of tens. 
 
 These fractions may be written thus : 
 
 Three-tenths, ------ ^. 
 
 Seventh-tenths, ----- ^^. 
 
 Sixty-five hundredths, - - - - _f^. 
 
 215 thousandths, - - . - - 
 1275 ten thousandths, - - - - 
 
 2 1 5 
 1000' 
 12-5 
 
 loooo- 
 
 From which we see, that in each case the denominator indi- 
 cates the fractional unit ; that is, determines whether the parts 
 are tenths, hundredths, thousandths, &c. 
 
 143. How many kinds of fractions are tli-ere ? What are thoyi What 
 is a common fractionl What is a flrcimal fraction 1 
 
 144. When the unit 1 is divided into 10 equal parts, what is each part 
 called 1 What is each pari called when it is divided into 100 equal parts ? 
 When into 1000 1 Into 10,000, cVc. 1 How are decimal fractions formed? 
 What gives denomination to the fraction 1 
 
DECIMAL FKACTIONS. 151 
 
 145. The denominators of decimal fractions are s(!ldom set 
 down. The fractions are usually exj^ressed by means of a 
 pei'iod, placed at the left of the numerator. 
 
 Thus, y^ - • is written - - .3 
 
 Too - - - - .UO 
 
 __2JL5_ .215 
 
 1000 \ .i--!^" 
 
 10000 .x-i^/ 
 
 Tliis method of writing decimal fractions is a form of lan- 
 guage, employed to avoid writing the denominatoi's. The denomi- 
 nator, however, of every decimal fraction is always understood : 
 
 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. 
 
 w 
 
 .^3 2 
 
 • tn S O "S 
 
 '" ^ '^ ^ t^ 
 
 _rf T^ ™ '^ • S 
 
 •£ rg tn -^ ui o 
 
 m i, g ^ iH C a 
 
 c = o c - = c 
 
 hKhhKSh 
 
 .4 is read 4 tenths. 
 
 .5 4 - - 54 hundredths. 
 
 .0 6 4 - - 64 thousandths. 
 
 .6754 - - 6754 ten thousandths. 
 
 .01234 - - 1234 hundred thousandths. 
 
 .007654 - - 7654 millionths. 
 
 .0043604 - - 43604 ten millionths. 
 
 Note. — Decimal fractions are numerated from left to right ; thus, 
 tenths, hundredths, thousandths, &c. 
 
152 DECIMAL FRACTIONS. 
 
 146. Write and numerate the following decimals : 
 Six-tenths, - - - - .6 
 Six hundredths, - - - .0 G 
 
 Six thousandths, - - - .0 6 
 Six ten thousandths, - - - .0006 
 Six hundred thousandths, - - .00006 
 Six miUionths, - - - - .000006 
 Six ten millionths, - - - .0000006 
 Here we see that the same figure denotes different decimal 
 units, according to the place which it occupies ; therefore, 
 
 The value of the unit, in the different places, in passing from 
 the left to the right, diminishes according to the scale of tens. 
 
 Hence, ten of the units in any place, are equal to one unit in 
 the place next to the left ; that is, ten thousandths make one 
 hundredth, ten hundredths make one-tenth, and ten-tenths make 
 the unit 1. 
 
 This scale of increase, from the right hand towards the left, is 
 the same as that in whole numbers ; therefore, 
 
 Whole numbers and decimal fractions may be united by 
 placing the decimal point between them ; thus. 
 
 Whole numbers. Decimals. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 to 
 
 ■5 
 
 
 
 
 
 CO 
 
 5 
 
 w 
 
 TS 
 
 
 
 
 
 
 
 
 
 en 
 
 3 
 O 
 
 
 
 w 
 C 
 
 .2 
 
 
 o 
 
 ■B 
 
 3 
 
 
 
 
 
 
 
 aj 
 
 C 
 
 
 to 
 
 .a 
 
 a 
 
 
 o 
 
 o 
 
 •5 
 
 "13 
 
 0) 
 
 
 
 
 
 •5 
 
 ■.-> 
 
 t/3 
 
 a 
 o 
 
 
 • 
 
 •n 
 
 
 c 
 
 --a 
 
 
 
 
 -a 
 
 c 
 
 3 
 
 ^ 
 
 ,J^ 
 
 't^ 
 
 o 
 
 CA 
 
 
 o 
 
 D 
 O 
 
 H 
 
 (a 
 1^ 
 
 
 
 wi 
 
 
 a 
 
 O 
 j2 
 
 
 c 
 
 *a 
 
 
 .2 
 
 
 C 
 
 c 
 
 V3 
 
 H 
 
 
 c 
 
 c 
 
 3 
 
 3 
 O 
 
 H 
 
 "3 
 C 
 3 
 
 o 
 
 a 
 
 3 
 u 
 H 
 
 4 
 
 2 
 
 
 
 3 
 
 1 
 
 4 
 
 5 
 
 1 
 
 .2 
 
 
 
 4 
 
 3 
 
 
 
 7 
 
 8. 
 
 145. Arc the denominators of decimal fractions generally set down 1 
 How are Ihc fractions expressed \ Is the denominator understood ! M'hat 
 is it ? What is the place ne.Yl the decimal point called ! What is its 
 unit! What is the next place called ? What is its unitT What is the 
 third place called ! What is its unit 1 Which way are decimals numerated ^ 
 
DECIMAL FRACTIONS. 153 
 
 A number composed pai-tly of a whole number, and partly 
 of a decimal, is called a mixed number. 
 
 RULE FOR WRITING DECIMALS. 
 
 Write the decimal as if it were a ichole number, prefixing as 
 many ciphers as are necessary to make it of the required denomi- 
 nation. 
 
 RULE FOR READING DECIMALS. 
 
 Read the decimal as though it were a whole number^ adding 
 the denomination indicated by the lowest decimal unit. 
 
 EXAMPLES. 
 
 Write the following numbers decimally : 
 
 (1.) (2.) (3.) (4.) (5.) 
 
 6 11 5 _2_7_ i±_ 
 
 Too 10 1000 100 Tooo* 
 
 (6.) (7.) (8.) (9.) (10.) 
 "loo 'Tooo *^ioo ■^'^loo -^''lO' 
 
 "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. 
 
 7. Five, and three millionths. 
 
 8. Forty, and nine ten millionths. 
 
 146. On what does the unit of a figure depend 1 How does the value 
 change from the left towards the right 1 What do ten units of any ono 
 place make 1 How do the units of the places increase from the right to- 
 wards the left ^ How may whole numbers be joined with decimals ! What 
 is such a number called \ Give the rule for writing decimal fractions. 
 Give the rule for reading decimal fractions. 
 
154 DECIMAL FRACTIONS. 
 
 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 twenty-seven ten 
 thousandths. 
 
 14. One hundred and five, and ninety-five ten millionths. 
 
 UNITED STATES MONEY. 
 
 147. The denominations of United States Money correspond 
 to the decimal division, if we regard 1 dollar as the unit. 
 
 For, the dimes are tenths of the dollar, the cents are hun- 
 dredths of the dollar, and the mills, being tenths of the cent, are 
 thousandths of the dollar. 
 
 EXAMPLES. 
 
 1. Express $37 and 26 cents and 5 mills, decimally. 
 
 2. Express $17 and 5 mills, decimally. 
 
 3. Express $215 and 8 cents, decimally. 
 
 4. Express $275 5 mills, decimally. 
 
 5. Express $9 8 mills, decimally. 
 
 6. Express $15 6 cents 9 mills, decimally. 
 
 7. Express $27 18 cents 2 mills, decimally. 
 
 ANNEXING AND PREFIXING CIPHERS. 
 
 148. Annexing a cipher 15 placing it on the right of a number. 
 If a cipher is annexed to a decimal it makes one 7nore decimal 
 
 place, and, therefore, a cipher must also be added to the denomi- 
 nator (Art. 145). 
 
 The numerator and denominator will therefore have been 
 multiplied by the same number, and consequently the value of 
 the fraction will not be changed (Art. 118) : hence, 
 
 147. If the denominations of Federal Money he expressed decimally, 
 ■what is the unit 1 What part of a dollar is 1 dime ? What part of a dime 
 is a centl What part of a cent ia a milH What part of a dollar is I 
 cent 1 1 mill 1 
 
DECIMAL FRA.CTIOi!fS. 155 
 
 Annexing ciphers to a decimal fraction does not alter its value. 
 We may take as an example, .5 = -f^. 
 
 If we annex a cipher to the numerator, we must, at the same 
 time, annex one to the denominator, which gives, 
 
 .5 =: Y^ = .50 by annexing one cipher. 
 .5 = tVoV — '^0*^ ^y annexing two ciphers, 
 .5 = yo%%'V = .5000 by annexing three ciphers. 
 
 Also, .■* _ y^j _ .t\J _ j-^Q _ .'iW — ^000* 
 
 Also, .7 = .70 = .700 = .7000 = .70000. 
 
 149. Prefixing a cipher is placing it on the left of a number. 
 
 If cij)hers are prefixed to the numerator of a decimal frac- 
 tion, the same number of ciphers must be annexed to the de- 
 nominator. Now, the numerator will remain unchanged while 
 the denominator will be increased ten times for every cipher 
 annexed ; and hence, the value of the fraction will be dimin- 
 ished ten times for every cipher prefixed to the numerator 
 (Art. 117). 
 
 Prefixing ciphers to a decimal fraction diminishes its value 
 ten times for every cipher prefixed. 
 
 Take, for example, the fraction .3 = ^^. 
 
 .3 becomes -^^ = .03 by prefixing one cipher. 
 
 .3 becomes jWo = .003 by prefixing two ciphers. 
 
 .3 becomes yHqq — .0003 by prefixing three ciphers: 
 in which the fraction is diminished ten times for every cipher 
 prefixed. 
 
 148. When is a cipher annexed to a number? Does the annexiinr of 
 ciphers to a decimal alter its value 1 Why not ? What does five-tenths 
 become by annexing a cipher 1 What by annexing two ciphers ! Three 
 ciphers \ AVhat does 7 tenths become by annexing a cipher ? By annex- 
 ing two ciphers 1 By annexing three ciphers 1 
 
 149. When is a cipher prefixed to a number 1 Whe» prefixed lo a decimal, 
 does it increase the numerator \ Does it increase the denominator '\ What 
 effect, then, has it on the value ef the fraction ? 
 
156 
 
 ADDITION OF 
 
 ADDITION OF DECIMALS. 
 
 150. Addition of Decimals is the operation of finding the 
 sum of twe or more decimal 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 expressmg the same unit must be 
 placed in the same column. 
 
 The addition of decimals is then made in the same manner 
 as that of whole numbers. 
 
 I. Find the sum of 87.06, 327.3 and .0567. 
 
 OPERATION. 
 
 Place the decimal points in the same column : 87.06 
 
 this brings units of the same value in the same 327.3 
 
 column : then add as in whole numbers j hence, .0567 
 
 414.4167 
 
 Rule. — I. Set doion the numbers to be 
 added so that figures of the same unit value shall stand in the 
 same column. 
 
 II. Add as in simple numbers, and 2'>oint off in the sum, from 
 the right hand, a number of places 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 6.035, 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. 
 
 ITjO. What is Atldition 1 ^Vhat parts of unity may be added together 1 
 How do you set down the numbers for addition ! How will the decimal 
 points fain How do you then add! How many decimal places do j-ou 
 point off in the sum 1 
 
DECIMAL FEACTIONS. 157 
 
 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 hundred 
 twenty six-millionths, one hundred fifty, and seventeen ten-mil- 
 lionths. 
 
 14. Find the sum of three hundred twenty-seven thousandths, 
 fifty-six ten-thousandths, four hundred, eighty-four milHonths, 
 and one thousand five hundred sixty hundred-millionths. 
 
 15. What is the sum of 5 hundredths, 27 thousandths, 476 
 hundred-thousandths, 190 ten-thousandths, and 1279 ten-mil- 
 lionths ? 
 
 16. "What is the sum of 25 dollars 12 cents 6 mills, 9 dollars 
 8 cents, 12 dollars 7 dimes 4 cents, 18 dollars 5 dimes 8 mills, 
 and 20 dollars 9 mills ? 
 
 17. What is the sum of 126 dollars 9 dimes, 420 dollars 
 75 cents 6 mills, 317 dollars 6 cents 1 mill, and 200 dollars 
 4 dimes 7 cents 3 mills ? 
 
 18. A man bought 4 loads of hay, the first contained 1 ton 
 25 thousandths ; the second, 997 thousandths of a ton ; the third, 
 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? 
 S127,055, and for harness and robes, $75,28 : what was the 
 entire cost ? 
 
 20. Bou'o-ht a barrel of flour for $9,375 ; a cord of wood for 
 $2,121 ; a barrel of apples for $1,621 ; 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 ; 
 
158 SDBTK ACTION OF 
 
 corn, for $12G,12i; oats, for $97,371; rye, for $100,10 ; and 
 barley, for §oO,G2i : 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^ : 
 what was the amount of his bill ? 
 
 SUBTRACTION OF DECIMALS. 
 151. Subtraction of Dechials is the operation of finding 
 the difference between two decimal numbers. 
 
 I. From 6.304 to take .0563. 
 
 Note. — In this example a cipher is annexed to operation. 
 the minuend to make the number of decimal places 6.3040 
 
 equal to the number in the subtrahend. This docs -05(33 
 
 not alter the value of the mmuend (Art. 148) : 6.2477 
 
 hence, 
 
 KuLE. — 1. Write the less number under the greater, so that 
 figures of the same unit value shall fall in the same column. 
 
 II. Subtract as in simple numbers, and point off the decimal 
 places in the remainder, as in addition. 
 
 Proof. — Same as in simi^le numbers. 
 
 EXAMPLES. 
 
 1. From 3278 take .0879. 
 
 2. From 291.10001 take 41.496. 
 
 3. From 10.00001 take .111111. 
 
 4. Required the difference between 57.49 and 5.768. 
 
 5. What is the difference between .3054 and 3.075 ? 
 
 6. Required the difference between 1745.3 and 173.45. 
 
 7. AVhat is the diflerence between seven-tenths and 54 ten- 
 thousandths ? 
 
 151. Wliat is subtraction of decimal fractions ! How do you set down 
 the numbers for subtraction 1 How do you then subtract ' How many 
 Jcciinal placcfl do you point off \n the remainder 1 
 
DECIMAL FRACTIONS. 159 
 
 8. What is the difference between .105 and 1.00075 ? 
 
 9. What is the difference between 150.43 and 754.355 ? 
 
 10. From 1754.754 take 375.49478. 
 
 11. Take 75.304 from 175.01. 
 
 12. Required the difference between 17.541 and 35.49. 
 
 13. Ecquired the difference between 7 tenths and 7 mil- 
 lionths. 
 
 14. From 396 take 67 and 8 ten-thousandths. 
 
 15. From 1 take one-thousandth. 
 
 16. From 6374 take fifty-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 hundred 
 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 $617.37^, and sold it 
 for $700 : how much did I gain by the sale ? 
 
 28. Having bought some cattle for §325.50 ; some sheep for 
 $97.12^; and some hogs for $60.87i ; I sell the whole for 
 $510.10 : what was my entire gain ? 
 
 29. A dealer in coal bought 225.025 tons of coal ; he sold to 
 A, 1.05 tons, to B, 20.007 tons, to G, 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 Ay 
 $1240.06, and from B, $1867.981 : how much will he have left 
 after paying his debts ? 
 
160 MULTIPLICATION OF 
 
 31. Bought of eacli of two persons, 1284.0-'5 pounds of wool, 
 from which I sell to three persons, each 262.125 pounds : how 
 much will I still have on hand ? 
 
 MULTIPLICATION OF DECIMAL FRACTIONS. 
 
 152. Multiplication of decimal fractions is the operation 
 of taking one number as many times as there are units in ano- 
 ther, when one of the factors contains a decimal, or when they 
 both contain decimals. 
 
 1. Multiply 8.03 by G.102. 
 
 OPERATION. 
 
 Analysis. — If we change both factors 
 to common fractions, the product of the 
 mimerators will be the same as that of 
 the decimal numbers, and the number of 
 decimal places loill he equal to the number 
 of ciphers in the two denominators ; hence, 
 
 Rule. — Multiply as in simjyle man- 48.99906 
 
 hers, andiwint off in the product, from 
 
 the right hand, as many firjures for decimals as there are decimal 
 places in both factors ; and if there be not so many in the pro- 
 duct, supply the deficiency by prefixing ci2}hc7-s. 
 
 examples. 
 
 1. Muhiply 2.125 by 375 thousandths. 
 
 2. Muhiply .4712 by 5 and G tenths. 
 
 3. INIuhiply .0125 by 4 thousandths. 
 
 4. Muhiply G.002 by 25 hundredths. 
 
 5. Muhiply 473.54 by 57 thousandths. 
 
 G. IMultiply 137.549 by 75 and 437 ^hou-sandths. 
 7. ]\Iultiply 3 and .7495 by 73487. 
 
 152. After multiplying, how many decimal places will you point oil' :n 
 the product 1 When tlicrc are not so many in tlie product, whjit do you 
 do 1 Give the rule for the multiplication of decimals. 
 
 803 
 100 
 
 = 8.03 
 
 6102 
 1000 
 
 = 6.102 
 1606 
 
 
 803 
 
 
 4818 
 
DECIMAL FRACTIONS. 161 
 
 8. Multiply ,04375 by 4713 i liundi-ed thousandths. 
 
 9. Multiply .371343 by seventy-five thousand 493. 
 
 10. Multiply 49.0754 by 3 and 5714 ten thousandths. 
 
 11. Multiply .573005 by 754 millionths. 
 
 12. Multiply .375494 by 574 and 375 hundredths. 
 
 13. Multiply two hundred and ninety-four millionths, by ono 
 millionth. 
 
 14. Multiply three hundred, and twenty-seven hundredths 
 by G2. 
 
 15. Multiply 93.01401 by 10.03962. 
 IG. Multiply 59G.04 by 0.000012. 
 
 17. Multiply 38049.079 by 0.000016. 
 
 18. Multiply 1192.08 by 0.000024. 
 
 19. Muhiply 7G098.158 by 0.000032. 
 
 20. Multiply thirty-six thousand, by thirty-six thousandths. 
 
 21. Multiply 125 thousand, by 25 ten thousandths. 
 
 22. What is the product of 50 thousand, by 75 ten millionths ? 
 
 23. What is the product of 48 hundredths, by 75 ten thou- 
 sandths ? 
 
 24. What are the contents of a lot of land 16.25 i-ods long, 
 and 9.125 rods wide ? 
 
 25. What are the contents of a board 12.07 feet long, and 
 1.005 feet wide? 
 
 26. What will 27.5 yai'ds of cloth cost, at .875 dollars per yard ? 
 
 27. At $25,125 an acre, what will 127.045 acres cost? 
 
 28. Bought 17.875 tons of hay, at $11.75 a ton: what was 
 the cost of the whole ? 
 
 29. A gentleman purchased a farm of 420.25 acres, for 
 $35.08 an acre ; he afterwards sold 196.175 acres to one man 
 for $37.50 an acre, and the remainder to another person, for 
 $36,125 an acre : what did he gain on the first cost ? 
 
 30. A merchant bought two pieces of cloth, one containing 
 
 87.5 yards, at $2.75 a yard, and the other, containing 27.35 
 
 yards, at $3,125 a yard ; he sold the whole at an average price 
 
 of $2.94 a yard : did he gain or lose by the bargain, and how 
 
 much ? 
 
 8 
 
162 CONTKACTIOXS IN 
 
 CONTRACTIONS IN MULTIPLICATION. 
 
 153. Contraction, in the multiplication of decimals, is a 
 short method of finding the product of two decimal numbers in 
 such a manner, that it shall contain but a given number of 
 decimal places. 
 
 1. Let it be required to find the product of 2,38645 multi- 
 plied by 38.2175, in such a manner that it shall contain but four 
 decimal places. 
 
 Analysis. — It is proposed, in this example, to operation. 
 
 take the multiplicand 2.38645, 38 times, then 2.38645 
 
 2 tenths times, then 1 hundredth times, then 5712.83 
 
 7 thousandths times, then 5 ten-thousandths 715935 
 
 times, and the sum of these several products 190916 
 
 will be the product sought. 4773 
 
 Write the unit figure of the multiplier directly 239 
 
 under that place of the multiplicand which is 167 
 
 to be retained in the product, and the remaining 1 2 
 
 places of integer figures, if any, at the right, 91.2042 
 and then write the decimal places at the left in 
 their order, tenths, hundredths, &c. 
 
 When the numbers are so written, the product of any figure 
 in the multiplier hij the figure of the multiplicand directly over 
 it, will be of the same order of value as the last figure to be re- 
 tained in the product. 
 
 Therefore, the first figure of each product is always to be 
 arranged directly under the last retained figure of the multlpli- 
 cand. But it is the whole of the multiplicand which should be 
 multiplied by each figure of the multiplier, and not a j^art of it 
 only. Hence, to compensate for the part omitted, we begin 
 
 153. What is contraction in the multiplication of decimals ? What is 
 proposed in the example ! liow arc the numbers written down for multi- 
 (jlication 1 When the numbers are so written, what will be the oriicr of 
 value of the product of any figure of the multiiilier by the figure directly 
 over it 1 Where then is the first figure of each product to be written ! 
 liow do you compciibutc for tlie j»art ouiiUvd ! 
 
 \ 
 
MULTIPLICATION. 163 
 
 with tlie figure at tlie riglit of the one directly over any 
 muUiplier, and carry one when the product is greater than 
 5 and less than 15, 2 when it falls between 15 and 25, 3 
 when it falls between 25 and 35, and so on for the higher 
 numbers. 
 
 For example, when we multiply by the 8, instead of saying 
 8 times 4 are 32, and writing down the 2, we say first, 8 times 
 5 are 40, and then carry 4 to the product 32, which gives 36. 
 So, when Ave multiply by the last figure 5, we first say, 5 times 
 3 are 15, then 5 times 2 are 10 and 2 to carry make 12, which 
 is written down 
 
 EXAMPLES. 
 
 1. Multiply 36.74637 by 127.0463, retaining three decimal 
 places in the product. 
 
 Contraction. Oonunon ivay. 
 
 36.74637 36.74637 
 
 3640.721 127.0463 
 
 3674637 11023911 
 
 734927 22047822 
 
 257225 14698548 
 
 1470 25722459 
 
 220 7349274 
 
 11 3674637 
 
 4668.490 4668.490346931 
 
 2. Multiply 54.7494367 by 4.714753, reserving five places 
 of decimals in the product. 
 
 3. Multiply 475.710564 by .3416494, retaining three decimal 
 places in the product. 
 
 4. Multiply 3754.4078 by .734576, retaining five decimal 
 places in the product. 
 
 5. Multiply 4745.679 by 751.4549, and reserve only whole 
 numbers in the product. 
 
16-i 
 
 DIVISION OF 
 
 154. Note. — When a decimal num'ber is to be multiplied by 10, 
 100, 1000, kc, the multiplication may be made by removing the 
 decimal point as many places to the right hand as there are ciphers 
 in the multiplier ; and if there be not so many figures on the right 
 of the decimal point, supply the deficiency by annexing ciphers. 
 
 Thus, 4.27 multiplied by 
 
 Also, 59G.027 multiplied by 
 
 10 
 
 
 42.7 
 
 100 
 
 
 427 
 
 > 1000 
 
 > = < 
 
 4270 
 
 10000 
 
 
 42700 
 
 100000 
 
 
 ^ 427000 
 
 10 ' 
 
 
 ' 5960.27 
 
 100 
 
 
 59G02.7 
 
 > 1000 
 
 - = ^ 
 
 596027 
 
 10000 
 
 
 5960270 
 
 100000 
 
 
 59602700 
 
 I 
 
 i 
 
 DIVISIOxN OF DECIMAL FRACTIONS. 
 1.5.5. Division of Decim.^l Fractions i.s the operation 
 of divi.sion when eitlier the divisor or dividend is a decimal, or 
 when both are decimals. 
 
 Analysis. — Since the dividend must be 
 equal to the product of the divisor and quo- 
 tient, it must contain as many decimal 
 places as both of them (Art. 152) : there- 
 fore. 
 
 There must be as many decimal places in the 
 quotient as the number of decimal places in the 
 dividend exceeds thnt in the divisor : hence, 
 
 OPERATION. 
 
 2.043). 71505(35 
 ^129 
 10215 
 10215 
 
 Ans. 0.35. 
 
 Rule. — Divide as in simple numhers, and point off in the 
 
 154. How do you multiply a decimal number by 10, 100. 1000, &c. ? 
 If there are not as many decimal figures as there are ciphers in the nnilti- 
 pUer, what do you do ? 
 
 155. If one decimal fraction be multiplied by anothor, how many deci- 
 mal places will there be in the product ' How docs the number of deci- 
 mnl jilaces in the dividend compare with those in the divisor and quotient' 
 How do you determine the number of decimal places in the quotient 1 
 Clivc the rule for the division of decimals. 
 
DECIMAL FE ACTIONS. 165 
 
 quotie7it,from the right hand, as mamj -places for decimals as the 
 numhcr of decimal 'places in the dividend exceeds that in the divi- 
 sor ; and if there are not so many, mpphj the deficiency hy pre- 
 fixing ciphers. 
 
 EXAMPLES. 
 
 1. Divide 4.6842 by 2.11. 
 
 2. Divide 12.825G1 by 1.505. 
 
 3. Divide 33.6G431 by 1.01. 
 
 4. Divide .010001 by .01. 
 
 5. Divide 24.8410 by .002. 
 
 6. Divide .0125 by 2.5. 
 
 7. Divide .051 by .012. 
 
 8. Divide .0G3 by 9. 
 
 9. Divide 1.05 by 14. 
 
 10. Divide 5.1435 by 4.05. 
 
 11. Divide .46575 by 31.05. 
 
 12. Divide 2.46616 by .145. 
 
 13. Wliat is the quotient of 75.15204, divided by 3 ? By .3 ? 
 By .03 ? By .003 ? By .0003 ? 
 
 14. What is the quotient of 389.27688, divided by 8? 
 By .08 ? By .008 ? By .0008 ? By .00008 ? 
 
 15. What is the quotient of 374.598, divided by 9 ? By .9 ? 
 By .09 ? By .009 ? By .0009 ? By .00009 ? 
 
 16. What is the quotient of 1528.4086488, divided by 6? 
 By .06 ? By .006 ? By .0006 ? By .00006 .? By .000006 ? 
 
 17. Divide 17.543275 by 125.7. 
 
 18. Divide 1437.5435 by .7493. 
 
 19. Divide .000177089 by .0374. 
 
 20. Divide 1674.35520 by 9.60?, 
 
 21. Divide 120463.2000 by 1728. 
 
 22. Divide 47.54936 by 34.75. 
 
 23. Divide 74.35716 by .00573. 
 
 24. Divide .37545987 by 75.714. 
 
 25. If 25 men remove 154,125 cubic yards of eartli in a day, 
 how much does each man remove ? 
 
 26. If 167 dollars 8 dimes 7 cents and 5 mills be equally 
 divided among 17 men, how much will each receive ? 
 
 27. Bought 45.22 yards of cloth for ^97.223 : how much was 
 it a yard ? 
 
 28. If 375.25 bushels of salt cost $232,655, wliat is the price 
 per bushel ? 
 
L66 DIVISION OF 
 
 29. At $0,125 per pound, how much sugar can be bought for 
 $2.25 ? 
 
 30. How many suits of clothes can be made from 34 yards 
 of cloth, allowing 4.25 yards for each suit ? 
 
 31. If a man travel 26.18 miles a day, how long will it take 
 him to travel 366.52 miles? 
 
 32. A miller wishes to purchase an equal quantity of wheat, 
 corn, and rye ; he pays for the wheat, $2,225 a bushel ; for the 
 corn, $0,985 a bushel ; and for the rye, SI. 168 a bushel: how 
 many bushels of each can he buy for S242.979 ? 
 
 33. A farmer purchased a farm containing 56 acres of wood- 
 land, for which he paid $46,347 per acre ; 176 acres of meadow 
 land, at the rate of $59,465 per acre ; besides which there was 
 a swamj) on the farm that covered 37 acres, for which he was 
 charged $13,836 per acre. What was the area of the land ; 
 what its cost ; and what the average price per acre ? 
 
 34. A person dying has $8345 in cash, and 6 houses, valued 
 at $4379.837 each ; he ordered his debts to be i)aid, amounting 
 to §3976.480, and $120 to be expended at his funeral ; the 
 residue was to be divided among his five sons in the following 
 manner : the eldest was to have a fourth part, and each of the 
 other sons to have equal shares. What was the share of each 
 son ? 
 
 ( 
 
 PARTICULAR CASES. 
 
 156. Note. — 1 . When any decimal number is to be divided by 10, 
 100, 1000, &c., the division is made by removing the decimal point 
 as many places to the left as tlieie are O's in the divisor : and if 
 there be not so many figures on the left of the decimal point, the 
 deficiency must be supplied by prefixing ciphers. 
 
 l.'jG. How do you divide a decimal number by 10. 100, 1000, &c. 1 II 
 thorn be not as many figures at tlie left of tlic deci.nal point as there cipher* 
 in the divisor, what do you do ? 
 
DECIMAL FRACTIONS. 
 
 167 
 
 10 ] 
 
 
 [4.987 
 
 100 
 
 
 .4'J87 
 
 1000 
 
 > = -^ 
 
 .01987 
 
 10000 J 
 
 
 .004987 
 
 10 1 
 
 
 r 32.756 
 
 100 
 
 
 3.2756 
 
 1000 
 
 y = 
 
 [ .32756 
 
 10000 
 
 
 .032756 
 
 100000 
 
 
 .0032756 
 
 49.87 divided by 
 
 327.56 divided by 
 
 157. Note. — 2. When there are more decimal places in the divi- 
 sor than in the dividend, annex as many ciphers to the dividend aa 
 are necessary to make its decimal places equal to those of the dii^i 
 sor; all the figures of the quotient will then he lohole numbers. Always 
 bear in mind that the quotient is as many times greater than 1, as 
 the dividend is times greater than the divisor. 
 
 EXAMPLES. 
 
 1. Divide 4397.4 by 3.49. 
 
 We annex one to the dividend. Had 
 it contained no decimal place, we should 
 have annexed two. 
 
 OPERATION. 
 
 3.49)4397.40(1260 
 349 
 
 907 
 698 
 2094 
 2094 
 
 2. Divide 1097.01097 by .100001. Ans. 1260 
 
 3. Divide 9811.0047 by .1629735. 
 
 4. Divide .1 by one ten-thousandths. 
 
 5. Divide 10 by one-tenth. 
 
 6. Divide 6 by .6. By .06. By .006. By .2. By .3. By 
 .003. By .5. By .005. By .000012. 
 
 158« Note. — 3. When it is necessary to continue the division 
 farther than the figures of the dividend will allow, we may annex 
 ciphers to it, and consider them as decimal places. 
 
 157. If there are more decimal places in the divisor than in the dividend, 
 what do you do 1 AVhat will the figures of the quotient then be 1 
 
 158. How do you continue the division after you have brought down all 
 the figures of the dividend ] When the division does not terminate, what 
 sign do you place after the quotient 1 What does it show 1 
 
1G8 
 
 CONTRACTIONS IN 
 
 EXAMPLES. 
 
 1. Divide 4.25 by 1.25. 
 
 In this example, after having exhausted 
 the decimals of the dividend, we annex an 
 0, and then the decimal places vised in the 
 dividend will exceed those in the divisor 
 by 1. 
 
 OPERATION. 
 
 1.25)4.25(3.4 
 3.75 
 500 
 500 
 
 Ans. 3.4 
 
 2. Divide .2 by .06. 
 
 We see, in this example, that the division 
 will never terminate. In such cases the di- 
 vision should be carried to the third or fourth 
 place, which will give the answer true 
 enough for all practical purposes, and the 
 sign + should then be written, to show that 
 the division may still be continued. 
 
 OPERATION. 
 
 .06). 20(3. 333 + 
 18 
 
 18 
 20 
 
 18 
 20 
 
 Ans. 3.333 + 
 
 3. Divide 37.4 by 4.5. 
 
 4. Divide 58G.4 by 375. 
 
 5. Divide 94.0360 by 81.032. 
 
 6. Divide 36.2678 by 2 25. 
 
 159. Note. — 4. If we regard 1 dollar as the unit of United States 
 Currency, all the lower denominations, dimes, cents, and mills, are 
 decimals of tiie dollar. Hence, all the operations uponUiiitid States 
 Money arc the same as the correspondins operations on decimal 
 fractious. 
 
 CONTR.ICTIONS IN DIVISION. 
 
 160. Contractions in division of decimals, are short 
 methods of finding a quotient Avhich shall contain a given 
 number of decimal places. 
 
 159. M'liat is tlie unit of the currnicy of the United States ? '^^'llat 
 parts of lhi.s unit arc tlie inferior dcnoniinalions. dimes, cnits, and mills 1 
 
 IGO. "What are the contractions in division? E.\plaiii the process of 
 making tho ilivision T 
 
DIVISION. 
 
 EXAMPLES. 
 
 169 
 
 1. Divide 754.347385 by G1.34775, and find a quotient 
 which shall contain three places of decimals. 
 
 Common Method. 
 01.34775)754.34738500(12.296 Contracted Method. 
 
 75 61.34775)754.347385(12.290 
 988 61348 
 
 61347 
 
 14086 
 12260 
 
 1817 
 1226 
 
 ~590 
 552 
 
 "38 
 
 550 14086 
 
 438^ 12269 
 
 9550 1817 
 
 48350 12^1 
 
 12975 590 
 
 353750 ^ 
 
 361808650 38 
 
 37 
 
 1545100 
 
 1 
 
 In the operation, by the common method, the figures at the 
 right of the vertical line, do not aflect the quotient figures : 
 
 1. Note the unit of the first quotient figure and then note the 
 number of figures which the quotient is to contain. 
 
 2. Select as many figures of the divisor as you wish places of 
 figures in the quotient, and midtiphj the figures so selected by the 
 first quotient figure, observing to carry for the figures cast off as 
 in the contraction of multiplication. 
 
 3. Use each remainder as a new dividend, and in each follow 
 ing division omit one figure at the right of the divisor. 
 
 ]S^OTE. — In the example above, the order of the first quotient figure 
 M'ns obviously tens : hence there were two places of whole numbers ; 
 and as there were three decimal places required in the quotient. /I'c 
 fip:u)es of the divisor must be used. 
 
 2. Divide 59 by .74571345, and let the quotient contain four 
 pl.'u't's of decimals. 
 
 160 What fifrures may be omitted in the contracted method'! 
 
ITO KEDUCTION OF 
 
 3. Divide 17493.407704962 by 495.783269, and let the 
 quotient contain four places of decimals. 
 
 4. Divide 98.187437 by 8.4765G18, and let the quotient 
 contain seven places of decimals. 
 
 0. Divide 47194.379457 by 14.73495, and let the quotient 
 contain as many decimal places as there will be integers in it. 
 
 REDUCTION OF COMMON AND DECIMAL FRACTIONS. 
 
 161. To change a common to a decimal fraction. 
 
 The value of a fraction is the quotient of the numerator di 
 vided by the denominator (Art. 105.) 
 
 1. Reduce |- to a decimal. 
 
 Analysis. — If we place a decimal point after the 7, operation. 
 and then write any number of O's, alter it, the value 8)7.000 
 of the numerator will not be changed (Art. 148). .875 
 
 If then, we divide by the denominator, the quo 
 tient will be the decimal number : hence, 
 
 Rule. — Annex decimal ciphers to the numerator and then 
 divide by the denominator, pointing off as in divif^ion of decimal' 
 
 EXAMPLES. 
 
 Reduce the following common fractions to decimals : 
 1. Reduce \, l, and f. 9. Reduce j"^ and ^^. 
 
 i 
 
 2. Reduce |^, ^, and -f^. 
 
 3. Reduce f and -^^. 
 
 4. Reduce j^-^ and y*j. 
 
 5. Reduce l and tttoTT' 
 
 6. Reduce -^^ and \^. 
 
 7. Express |i5^| decimally. 
 
 8. Express -^^^.j decimally. 
 
 10. Express 3^} ^5- deciraullj 
 
 11. Reduce y^j and 2^/j." 
 
 12. Reduce -^- of j of 6. 
 
 13. Reduce | of }\. 
 
 14. Reduce y\ of ||. 
 
 15. Reduce | of r!-- 
 
 16. Reduce ?°hind ^*\. 
 
 20 7 5 8 
 
 17. What is the decimal value off of |- multiplied by yg. 
 
 161, How do you cliaiigcaconimon toadccimal fraction'! 
 
 SfZ. How ilo yoii rh.iiii;'^ a dcciiiial to llic loriii of a comiDoii fraction? 
 
COMMON FK ACTIONS. 171 
 
 18. What is the value, in decimals, of -} of f of I divided 
 Djfoff? 
 
 19 A man owns |- of a ship ; he sells 2% of his share : what 
 part is that of the whole, expressed in decimals ? 
 
 20. Bought 11 of 87fV bushels of wheat for ^^^ of 7 dollars 
 a bushel : how much did it come to, expressed in decimals ? 
 
 21. If a man receives f of a dollar atone time, 71 at another, 
 and 8|- at a third : how much in all, expressed in decimals ? 
 
 22. What mixed decimal is equal to f of 18, and /^ of 11 
 plus 74, added together ? 
 
 23. What decimal is equal to f of 31 taken from f of 8f ? 
 
 24. What decimal is equal to 4fj V^j h added together? 
 
 162. To change a decimal to the form of a common fraction. 
 A decimal fraction may be changed to the form of a common 
 fraction by simj^ly writing its denominator (Art. 145). 
 
 EXAMPLES. 
 
 Express the following decimals in vulgar fractions. 
 
 1. Reduce .25 and .75. 
 
 2. Reduce .125 and .625. 
 
 3. Reduce .105 and .0025. 
 
 4. Reduce .8015 and .6042. 
 
 5. Reduce .68375. 
 
 6. Reduce .01875. 
 
 7. Reduce .22575. 
 
 8. Reduce .265625. 
 
 DENOMINATE DECIMALS. 
 
 163. A Denominate Decimal is one in which the unit of 
 the fraction is a denominate number. Thus, .3 of a dollar, .7 of 
 a shilling, .8 of a yard, &;c., are denominate decimals, in which 
 the units are, 1 dollar, 1 shilling, 1 yard. 
 
 CASE I. «• 
 
 164. To find the value of a denominate number in decimals of 
 a liigher tmit. 
 
 163. What is a denominate {lecimaH 
 
 164. How do you find the value of a denominate number in decimals of 
 a higher unit I 
 
OPEUATION. 
 
 ■|f?. = .75d. ; hence, 
 
 9f(/. =9.75d. 
 
 12)9.75(7. 
 
 .8125s., and 
 
 20)4.8125.5. 
 
 172 REDUCTION OF 
 
 I. Reduce £1 4s. 9^d. to the decimal of a £. 
 
 Analysis. — Wc first reduce 3 
 fartliings to the decimal of a penny, 
 by dividing by 4. ^Ye then annex 
 the quotient .756?. to the 9 pence. 
 We next divide by 12, giving .81 25, 
 which is the decimal of a shilling. 
 
 This we annex to the shillings, and £.240625 : therefore, 
 
 then divide by 20. ^1 4^ 93J. ='^^1.240625. 
 
 Rule. — I. Divide the lowest denomination hy the units of the 
 scale which connect it toith the next higher, annexing ciphers, if 
 nccessa?y. 
 
 II. Annex the quotient to the next higher dcnominntion and 
 divide hy the units of the scale ; and proceed in the same manner 
 through all the denominations, to the required unit. 
 
 Note. — When any denomination, between the highest and the 
 lowest is wanting, the number to be prefixed to the corresponding 
 quotient, is 0. 
 
 EXAMPLES. 
 
 1. Reduce 14 drams to the decimal of a Ih., Avoirdupois. 
 
 2. Reduce 78(f. to the decimal of a £. 
 
 o. Reduce 03 pints to the decimal of a peck. 
 
 4. Reduce 9 hours to the decimal of a day. 
 
 5. Reduce 375G78 feet to the decimal of a mile. 
 
 6. Reduce 7oz. Iddict. of silver to the decimal of a pound, 
 
 7. Reduce Sctct. lib. 8oz. to the decimal of a ton. 
 
 8. Reduce 2.45 sliillings to the decimal of a £. , 
 
 9. Reduce 1.047 roods to the decimal of an acre. 
 
 10. Reduce 176.9 yards to the decimal of a mile. 
 
 11. Reduce 2qr. lilb. to the decimal of a C7rt. 
 
 12. Reduce IO02;. ISdivt. ^('■•gr. to the decimal of a lb. 
 
 13. Reduce oqr. 2na. to the decimal of a yard. 
 
 14. Reduce ^gal. to the decimal of a hog.-^head. 
 
 15. Reduce 17//. Gm. 43ser. to the decimal of a day. 
 
 16. Reduce 4ctct. '^^qr. to the decimal of a ton. 
 
 17. Reduce 19s. 5d. 2fa7\ to the decimal of a poiuid. 
 
DKNOMINATE DECIMALS. 173 
 
 18. Reduce 1^. ."7/'. to the decimal of an acre. 
 
 19. Reduce 2qr. ona. to the decimal of an Eng. Ell. 
 
 20. Reduce 2ijd. 2ft. G^««. to the decimal of a mile. 
 
 21. Reduce 15' 22^." to the decimal of a degree. 
 
 22. Reduce 290 cubic inches to the decimal of a ton of round 
 timber. 
 
 23. Reduce obusJi. oph. to the decimal of a chaldron. 
 
 24. Reduce 17yc?. 1ft. Qin. to the decimal of a mile. 
 
 25. "What decimal part of a year is 9^ months ? 
 
 2G. What decimal part of a lb. is lOoz. 18dwt. 16gr.? 
 
 27. What decimal part of an acre is 1^. 14P. ? 
 
 28. AYhat decimal part of a chaldron is Aojyk. ? 
 
 29. What decimal part of a mile is 72 yards ? 
 
 30. What part of a ream of paper is 9 sheets ? 
 
 31. What part of a rod in length is 4.0125 inches? 
 
 32. Reduce IQwk. 2£?a. to the decimal of a leap year. 
 
 33. Reduce 4 5 13 19 lO^-r. to the decimal of a ife. 
 
 34. Reduce 3qt. 1.7 opt. to the decimal of a /ihd. 
 
 35. Reduce 24:sq. yd. l.^sq.ft. to the decimal of an acre. 
 
 36. Reduce 2qr. Ina. 0.3 6m. to the decimal of a yard. 
 
 37. Reduce 3ft. 4' 8" 3'" to feet and decimals of a foot. 
 
 CASE II. 
 
 165. To find the value of a decimal in integers of less denomi- 
 nations. 
 
 1. What is the value of .832296 of a £ ? 
 
 Analysis. — Fu-st multiply the decimal operation. 
 
 by 20, which brings it to the denomination .832296 
 
 of shillings, and after cutting off from the 20 
 
 right as many places lor decimals as there 16.645920 
 
 are in the given nunlber, we have 1 65. and 1 2 
 
 the decimal .645920 oter. This is re- 7.751040 
 
 duced to pence by multiplying by 12, and 4 
 
 then to farthings by multiplying by 4. 3.004160 
 
 Ans. 165. "id. 3far. 
 
 165. How do you find the value of a decimal in integers of less denomi 
 nations 1 
 
174r DKNOMINATE DECIMALS. 
 
 Rule. — I. Mulliply (he decimal by the uiiis of the scaCe 
 which connect it with the next less denomination, pointing off as 
 in the multijjli cation of decimals. 
 
 II. Mullijily the decimal part of the product as before, and 
 tontinue so to do until the decimal is reduced to the required deno- 
 minations. The integers cut off at the left form the ansuer. 
 
 EXAMPLES. 
 
 1. "What is the value of .6725 of a hundred weight ? 
 
 2. What is the value of .61 of a pipe of wine ? 
 
 3. What is the value of .83229 of a £ ? 
 
 4. Required the value of .0625 of a barrel of beer. 
 
 5. Requii'ed the value of .42857 of a month. 
 
 6. Required the value of .05 of an acre. 
 
 7. Required the value of .3375 of a ton. 
 
 8. Required the value of .875 of a pipe of wine. 
 
 9. What is the value of .375 of a hogshead of beer ? 
 
 10. What is the value of .911111 of a pound troy? 
 
 11. What is the value of .675 of an English ell ? 
 
 12. What is the value of .001136 of a mile in length ? 
 
 13. What is tire value of .000242 of a square mile ? 
 
 14. Required the value of .4629 degrees. 
 
 15. Required the value of .875 of a yard. 
 
 16. Required the value of .3489 of a pound apothecaries. 
 
 17. Required the value of .759 of an acre. 
 
 18. Required the value of .01875 of a ream of paper. 
 
 19. Requii'ed the value of .0055 of a ton. 
 
 20. Required the value of .625 of a sliilling. 
 
 21. Required the value of .3375 of an acre. 
 
 22. Required the value of .785 of a year, of 3651 daya 
 
REPEATING DECIMALS. 175 
 
 CIRCULATING OR REPEATING DECIMALS. 
 
 166. In changing a common to a decimal fraction, there are 
 two general cases : 
 
 1st. When the division terminates ; and 
 2d. When it does not terminate. 
 
 In the first case, the quotient will contain a limited number 
 of decimal places, and the value of the common fraction will be 
 exactly expressed decimally. 
 
 In the second case, the quotient will contain an infinite num- 
 ber of decimal places, and the value of the common fraction can- 
 not be exactly expressed decimally. 
 
 CASE I. 
 
 167. When the division terminates : 
 
 When a common fraction is reduced to its lowest terms (which 
 we suppose to be done in all the cases that follow), there wuU 
 be no factor common to its numerator and denominator (Art. 
 120). 
 
 1. Reduce ^ to its equivalent decimal. 
 
 Analysis. — Annexing one decimal to the operation. 
 
 numerator multiplies it by 10, or by 2 and 5; 50)17. 00(. 34 
 
 hence, 2 and 5 become prime factors of the nu- 15 
 
 merator every time that an is annexed. But 2 00 
 
 if the division be exact, these prime factors, and 2 GO 
 none others, must also he found iii the denomi- 
 nator (Art. 91). 
 
 166. How many cases are there in chanfring a vulgar to a decimal frac- 
 tion 1 What are they ? . What distinguishes one of these cases from tho 
 
 other 1 
 
 167. How do you determine when a vulgar fraction can be exactly ex- 
 pressed decimally ^ How many decimal places will there be in the 
 ciuotient ' 
 
176 CIRCULATING OR 
 
 OPERATION. 
 
 2. Reduce -^ to its equivalent decimal. 35) 50 (.1388 + 
 
 Analysis. — 36 = 18x2 = 9X2x2= ?^ 
 
 3 X 3 X 2 X 2 : in which we see that the de- ^'^^ 
 
 nominator contains other factors than 2 and o. ^^^ 
 aud liencc, the fraction cannot be exactly ex- "^-^ 
 
 pressed bij decimals (An. 91). Hence, to deter- ^^^ 
 
 mine whether a common fraction can be exactly ^^^ 
 
 expressed decimally : 288 
 
 I. Decompose the denominator into its prime factors; and if 
 there are nn factors other than 2 and 5, (he exact division 
 can be made : 
 
 II. If there are other prime factors, the exact division cannot 
 
 he made. 
 
 Note.— Every decimal annexed to the numerator, introduces the 
 two factors 2 and 5 ; and these factors must be introduced until we 
 have as many of each as there are in the denominator after it shall 
 have been decomposed into its prime factors 2 and 5. But the quo- 
 tient will contain as many decimal places as there arc decimal O's in 
 the dividend (Art. 155); hence, 
 
 The number of decimal jylaces in the quotient will be equal to 
 ike greatest number of equal factors 2 or 5, in the divisor. 
 
 EXAJrPLES. 
 
 1. Can ^^3- be exactly expressed decimally ? opkration. 
 how many jdaces ? 25) 70 (.28 
 
 50 
 
 25 = 5 X 5 ; hence, the fraction can be ex- 200 
 
 actly expressed decimally. 200 
 
 Find the decimals and number of places in the following: 
 
 1. Express ^?^ decimally. 5. Express ^Vo decimally. 
 
 2. Express ^-/^ decimally. G. Express -^^ decimally. 
 
 3. Express ^y^y decimally. 7. Express ^f^ decimally. 
 
 4. Express yi^^ decimally. J 8. Express /^j decimally. 
 
REPEATING DECIMALS. 177 
 
 CASE II. 
 168. When (he division does not terminate. 
 
 1. Let it be required to reduce i to its equivalent decimal. 
 
 Analysis. — By annexing decimal ciphers to 
 the numerator 1, and making the division, wc operation. 
 
 find the equivalent decimal to be .3333 +, &c., 3)1 0000 
 
 giving 3's as far as avc choose to continue the .3333 + 
 
 divifcion. 
 
 The further the division is continued, the nearer the value of 
 the decimal will approach to ^, the true vaZ;/eof the common 
 fraction. We express this approach to equality of value, by 
 saying, that if the division be continued without limit, that is, to 
 infill ity, the value of the docimal will then become equal to that 
 of the common fraction ; thu s, 
 
 .33.33 +, continued to infinity 
 
 1 . 
 
 3 ' 
 
 for, every annexation of a 3 brings the value nearer to ^. 
 
 Also, .9999 +, continued to infinity = 1 ; 
 
 for, every annexation of a 9 brings the value nearer to 1. 
 
 2. Find the decimal corresponding to the common fraction |^. 
 
 Analysis. — Annexing decimal ciphers and operation. 
 dividing, we find the decimal to be .2222 +, in 9)2.0000 
 
 which we see that the figure 2 is continually .2222 + 
 
 repeated. 
 
 169. A Circulating Decijial is a decimal fraction in 
 which a single figure, or a set of fiigures, is constantly repeated. 
 
 170. A Repetend is a single figure or a set of figures, which 
 is constantly repeated. 
 
 168. Can one-third be exactly expressed decimally'! What is the foni) 
 of the quotient 1 To what docs the value of this quotient approach ? 
 When does it become equal to one-third 1 
 
 169. What is a circulatinjr decimall 
 
 170. What is a repetcnd 1 
 
178 CIKCr LATINO OK 
 
 171. A Single Repetend is one in -which only a single 
 figure is repeated ; as 
 
 I = .2222+, or | = .3333 +. 
 
 Such repetends are expressed by simply putting a mark over 
 the first figure ; thus, 
 
 .2222 +, is denoted by .% and .3333 + by .'3. 
 
 172. A Compound Repetend has the same set of figures 
 circulating alternately ; thus. 
 
 If = .57 57 +, and -|4|| = .5723 5723 +, 
 are compound repetends, and are distinguished by marking the 
 first and last figures of the circulating period. Thus, .57 57 + 
 is written .'57', and .5723 5723 + is written .'5723'. 
 
 173. A Pure Repetend is one which begins with the first 
 decimal figure ; as 
 
 .'3, .'5, .'473', &c. 
 
 174. A Mixed Repetend is one which has significant figures 
 or ciphers between the repetend and the decimal point ; or 
 which has whole numbers at the left hand of the decimal point ; 
 such figures are caWo.^ finite figures. Thus, 
 
 .0733', .473', .3'573', G.'o, 
 
 are all mixed repetends ; .0, .4, .3, and 6, are ih^ finite figures. 
 
 175. Similar Repetends are such as begin at equal dis- 
 tances from the decimal points ; as .3'54', 2.7'534;.'. 
 
 176. Dissimilar Repetends are such as begin at different 
 ^.'stances from the decimal point ; as .'253', .47'52'. 
 
 177» Conterminous Repetends are such as end af equal 
 distances from the decimal points ; as .1'25', .'354'. 
 
 171. ^Vhat is a single repetend ? 
 
 372. What is a compound repetend T 
 
 173. What is a pure repetend 1 
 
 174. What is a mixed rcj)etcnd 1 
 
 175. What are similar repetends t 
 17(). What are dissimilar repctoudal 
 
REPEATING .DECIMALS. 179 
 
 178. Similar and Conterminous Repetends are sucli as 
 bei^in and end at the same distances from the decimal point ; 
 thus, 53.2753', 4.6'325', and .4^632', are similar and conter- 
 minous repetends. 
 
 REDUCTION OF REPETENDS TO COMMON FRACTIONS. 
 
 CASE I. 
 
 179. To reduce apure repetendto its equivalent common fraction. 
 
 Analysis, — This proposition is to be analyzed by examining the 
 law of forming the repetends. 
 
 1st. .| =.llll+&c. = .M; and f =.4444 + &c. = .M : 
 
 2d. ^=.010101+&:c =>01'; and f|- =.2727+&c. = .'27' : 
 3d. ^1^=^.001001 +Scc. = .^001'; and fff = .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; hence, 
 
 The value of any jmre repetend is equal to the 7iiimher denot- 
 ing the repetend, divided hy as many 9's as there are figures. 
 
 EXAMPLES. 
 
 1. What is the equivalent common fraction of the repetend 
 0.3?^ 
 
 Now, 9 = i = 0.33333 +.= 0.^3. 
 
 2. What is the equivalent common fraction of the repetend 
 :i62' ? 
 
 We have, gf I = t'tt -^"*' 
 
 3. What are the simplest equivalent common tractions of the 
 repetends .^6,.^162', 0.769230', .^945', and .^09'? 
 
 4. AVhat are theleastequivalentcommonfractionsof the repe- 
 tends .^594405', .^36', and .^142857' ? 
 
 177. What are conterminous repetends \ 
 
 178. What are similar and conterminous repetends 1 
 
 17'J. How do you find a common fraction equivalent to a puie repetend ! 
 
180 CIRCULATING OR 
 
 CASE II. 
 
 180. To reduce a mixed repelend to its cquivalen t common fraction. 
 
 Analysis. — A mixed repetend is composed of the finite figures 
 wliich precede, and of tlie 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 limes for each place at the right, and hence, O's must be annexed 
 to the 9"s Avhich form the divisor ; therefore, 
 
 To the finite figures, add the repetend divided hy as many 9'* 
 as it contains places of figures, ivith as many O's annexed to 
 them as there are places of decimal figures preceding the repe- 
 tend ; the sum reduced to its simplest form icill be the equivalent 
 fraction sought. 
 
 EXAMPLES. 
 
 1. Required theleast equivalent common fractioQ of the mixed 
 repetend, 2.4:'18'. 
 
 Now, 
 
 2.4^18' = 2 + T-V +^18' = 2 + T^ + J4 = 2f 3. Ans. 
 
 2. Required theleast equivalent common fraction of the mixed 
 repetend .o'925'. 
 
 We have, .5^925' = -^ + ^^ = If- ^««- 
 
 3. What is the least equivalent common fraction of the repe- 
 tend .008^497133'? 
 
 Wo havf 008M971?!i' a i 4971 33 S3_, 
 
 nCIKHt, .UUOiJ/IOO — Yoo^ i- 99-9999-000- — TTTaS' 
 
 4. Required the least equivalent common fractions of the mixed 
 repetends .13^8, 7.5^43', .04'3o4', 37.5^4, .G7o', and .7^:i4347'. 
 
 5. Required the least equivalent common fractions of tlie mixed 
 rcpetends 0.7^5, 0.4^38', .09^3, 4.7^543', .009'87^ and .4'o. 
 
 180. How do you find the value of a mixed repetend ' 
 
REPEATING DECIMALS.. 181 
 
 CASE III. 
 
 181. To find the finite figures and the repetend corresponding 
 to Q,ny common fraction. 
 
 1. Find the finite figures and the repetend corresponding to 
 
 the fraction ^^^q- 
 
 Analysis. — 1st. Reduce the frac- 
 tion to its lowest terms, and then 
 
 OPKRATION. 
 fi 3 
 
 560 280 
 3 
 
 280 2X2X2X5X7 
 280)3.000 +(.010^714285' 
 
 find all the factors 2 and 5 of the 
 denominator. 
 
 2d. Add decimal ciphers to the 
 numerator and make the division. 
 
 3d. The number of finite decimals preceding the first figure of tho 
 repetend will be equal to the greatest number of factors 2 or 5 
 (Art. 167). In this example it is 3. 
 
 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 tha 
 number, less l , of the units in that factor of tlie denominator which 
 does not contain 2 or 5. Tn the example, that number is 7, and the 
 number of figures of the repetend, is 6. Hence, 
 
 Divide the numerator of the common fraction, reduced to its 
 lowest terms, hj the denominator, and point off in the quotient 
 the finite decimals, if any, and the repetend. 
 
 EXAMPLES. 
 
 1. Required to find whether the decimal, equivalent to the 
 common fraction 3-f ^-qt? is finite or c 
 finite figures, if any, and the repetend 
 
 Analysis. — We first reduce the 
 fraction to its lowest terms, giving 
 TtI It- ^^° ihen search for the fac- 
 tors 2 and 5 in the denominator, 
 and find that 2 is a factor 3 times ; 
 
 hence, we know that there are three Q'^f;«")»'3 ofl + C on8''4Q7l 3*?' 
 finite decimals preceding the repe- 
 tend. We next divide the numerator 83 by the denominator 9768, and 
 
 181. How do you find the finite figures and the repetend corres]>ondi!ii» 
 to any CDUuniMi fr;icli()ii ? 
 
 jirculating : required 
 
 the 
 
 OPERATION. 
 
 
 249 
 
 83 
 
 
 29304 
 
 9768 
 
 
 83 
 
 83 
 
 
 9768 
 
 2X2X2X1221 
 
 
182 PKOPERTIES OF THE 
 
 note that the repeteiid begins at the fourth place. After the ninth 
 'livision, .we find the remainder 83 ; at this point the figures begin to 
 'epeat ; hence, the rcpetend has 6 places. 
 
 2. Find the finite decimals, if any, and the rcpetend, if any, 
 )f the fraction fy^* 
 
 3. Find the finite decimals, if any, and the repetend, if any, 
 tf the fraction ■. A„ -. 
 
 4. Find the finite decimals, if any, and the repetend, if any, 
 of the fractions ^22^ _^^ _7_2_. 
 
 PROPERTIES OF THE REPETENDS. 
 
 182. There ai-e some properties of the vepetends which it is 
 important to remark. 
 
 1. Any finite decimal may be considered as a circulating 
 decimal by making ciphers to recur ; thus, 
 
 .35 = .35^0 = .35^00' = .35^000' := .35^0000', &c. 
 
 2. If any circulating decimal have a repetend of any number 
 of figures, it 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, 
 
 2.3^57' = 2.3^5757' = 2.3^575757' = 2.3^57575757', &c.; 
 
 so, the repetend 4.16^31G' may be written 
 
 4.16^316' = 4.1G^31G316' = 4.16^310316316', Sec. &c.; 
 
 and the same may 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 : 
 
 182. How may a finite decimal be made circulating 1 When a repetend 
 has a given number of places, to what otlicr farm may it be reduced! 
 How ! Into wiiat form may any circulating dociniai be transformed ? To 
 wh.%1 fvnu may two or more icjielcnds be reduced 1 
 
KEPETENDS. 183 
 
 Find the least comvion multiple of the number of places in 
 each repetend, and reduce each repetendto such number of places. 
 
 Ex. 1. Reduce .13^8, 7.5^43^ .04^354', to repetends having 
 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 Avill con- 
 tain 6 places ; that is, 
 
 .13^8=. 13^888888'; 7.5^43'.zr 7.5^434343'; and 
 0.4^354' = 0.4^3o43o4'. 
 
 Ex. 2. Reduce 2.4^8', .5^925', .008^497133', to repetends 
 having the same number of places. 
 
 3. Any circulating decimal may be transformed into another 
 having finite decimals and a repetend of the name number of 
 figures as the first. Thus, 
 
 .^57' = .575' =: .57\57' = .57575' = .5757^57'; and 
 3.4785' = 3.47^857' = 3.478^578' == 3.4785785'; 
 and hence, any two repetends may be made similar. 
 
 These properties may be proved by changing the repetends 
 mto their equivalent common fractions. 
 
 4. Having made two or more repetends similar by the last 
 article, they may be rendered conterminous by the pi-evious 
 one ; thus, two or more repetends may alicays be made similar 
 and contenninous. 
 
 1. Reduce the circulating decimals 165.'164', .'04', .037 to 
 such as are similar and conterminous. 
 
 2. Reduce the circulating decimals .5'3, .475', and 1.'757', 
 to such as are similar and conterminous. 
 
 5. If two or more circulating decimals, having several repe- 
 tends of equal places, be added together, their sum will have a 
 repetend of the same number of places ; for, every two sets of 
 repetends toill give the same sum. 
 
 6. If any circulating decimal be multiplied by any number, 
 the product will be a circulating decimal having the same 
 number of places in the repetend ; for, each repetend will be 
 taken the same number of times, and consequently must j^i'oduce 
 the same product. 
 
184 ADDITION OF 
 
 ADDITION OF CIRCULATING DECIMALS. 
 
 183. To add circulating decimals : 
 
 I. Make the reijetends, in each number to be added, similar 
 and conterminous. 
 
 II. Write the places of the same unit value in the same column, 
 and 60 mani/ figures of the second repelend in each as shall 
 indicate with certainty, how many are to he carried from one 
 repdend to the other : then add as in whole numbers. 
 
 Note — If all the figures of a repetend are 9's, omit them and add 
 to the figure next at the left. 
 
 EXAMPLES. 
 
 1. Add .1*2^5, 4.^163', r.7143', and 2.^54' together. 
 Dissimilar. Similar. Similar and Conterminous. 
 
 .12'5 — .12'5 = .12'555oo5555555' - - - 5555 
 
 4.'163' =4.16^316' = 4.16^310316316310' - - - 3103 
 
 1.7143' = 1.71^4371'= 1.71^437143714371' - - - 4371 
 
 2.'54' = 2.54^54' = 2.54^545454545454' - - - 5454 
 
 The true sum z= 8.54^854470131097' 1 to carry. 
 
 2. Add G7.3'45', 9.^051', .^25', 17.47, :5, together. 
 
 3. Add ;475', 3.75^43', 04.75', .^57', .1788', together. 
 
 4. Add :o, 4.37, 49.4\57', .4^954', .7345', together. 
 
 5. Add .^175', 42.^57', .3753', .4'954', 37^54', together. 
 
 6. Add 105, .\1W, 147.^04', 4.^95', 94.37, 4.712345' to- 
 gether. 
 
 SUBTRACTION OF CIRCULATING DECIMALS. 
 184. To subtract one circulating decimal from another. 
 I. 3fuke the repetends similar and co7iterminous. 
 ' 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 rigid hand figure. 
 
 183. How do you add Circulating Decimals ? 
 IS'l. How do you subtract Circulating Decimals ! 
 
CIRCULATING DECIMALS. 
 
 186 
 
 EXAMPLES. 
 
 1. From 11.475' take 3.45735'. 
 
 Dissimilar. Similar. Similar and Conterminous. 
 
 11.^75' =: 11.47^57' == 11.47^575757' - - - ■ 
 3.45735' = 3.45735' = 3.45735735' - - - . 
 
 The true difference = 8.01^840021' 1 to carry. 
 
 575 
 
 735 
 
 2. From 47.5^3 take 1.757'. 
 
 3. From 17.^573' take 14.57. 
 
 4. From 17.4^3 take 12.34^3. 
 
 5. From 1.12754' take. 47384'. 
 
 G. From 4.75 take .37^5. 
 
 7. From 4794 take .1744'. 
 
 8. From 1.457 take .3654. 
 
 9. From 1.4^937' take .1475. 
 
 MULTIPLICATION OF CIRCULATING DECIMALS. 
 
 185. To multiply one circulating decimal by another. 
 
 Change the circulating decimals into their equivalent common 
 fractions., and then multiply them together ; then, reduce the 
 product to its equivalent circulating decimal. 
 
 EXAMPLES. 
 
 1. Multiply 4.25^3 by .257. 
 
 OPERATION. 
 
 4 95^3 — 4-l-_25_4. _3_ — 4 4. 225 I 
 
 3 
 
 900 
 
 2 2_R 3 S28 
 
 900 — 9T0~ 
 
 45 
 
 9.A1 
 225* 
 
 Also, .257 =: -rsoo 5 lience, 
 
 - 957 y _2_5J_ _ 2 45 9A9 _ 1 0931 OT) • 
 225 '^ 1000 — 225000 — x.v^UAVU, 
 
 and since 225000 = 5x5x5x5x5x2x2x2x9; 
 there will be five places of finite decimals, and one figure in 
 the repetend (Art. 1G7). 
 
 NoTK. — Much labor will be saved in this and the next rule by Icccpinp^ 
 every fraction in its lowest terms ; and when two fractions are to be 
 vndtipHcd together, cancel all the factors common to both terms be 
 fore niakiyig the multiplicalion. 
 
 ]Sr>. How do you mulliply Circulating Decimals 1 
 
 9 
 
186 
 
 i)ivisio>r OF 
 
 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 .375'. 
 
 6. Multiply 3.45^ 6 by .42^5. 
 
 7. Multiply 1.^456' by 4.2^3. 
 
 8. Multiply 45.1^3 by .^245'. 
 
 9. Multiply .4705^3 by;.7^35'. 
 
 DIVISION OF CIRCULATING DECIMALS. 
 
 186. To divide one circulating decimal by another. 
 
 Change the decimals into their equivalent common fractions, 
 and find the quotient of these fractions. Then change the quch 
 tieni into its equivalent decimal. 
 
 EXAMPLES. 
 
 1. Divide 5G.^6 by 137. 
 
 OPERATION. 
 
 56.^6 
 
 Then, 
 
 ] 7 
 3 
 
 56 4- 6- — SJLO _ no.. 
 137 = ip X t1_ = i7fi ^ /413G2530'. 
 
 4] 1 
 
 2. Divide 24.3^18' by 1.792. 
 
 3. Divide 8.59G8 by .2^45'. 
 
 4. Divide 2.295 by .^297'. 
 
 5. Divide 47.345 by 1.76'. 
 
 0. Divide 13.5UG9533 by 4.^297' 
 
 7. Divide .H5' by .^118881'. 
 
 8. Divide .^475' by .3753'. 
 
 9. Divide 3.753' by .^24'. 
 
 CONTINUED FRACTIONS. 
 
 1. If -we take any irreducible fraction, as J-|, and divide both 
 terms by the numerator, it will take the form 
 
 15 1 1 , ,. , ^. . . 
 
 ml = V9 = T I ]4 > ^y malcmg the division. 
 
 ZJ 1.. ^ "I 15 
 
 If now, we divide both terms of ^|^ by the numerator 14, we 
 
 have 
 
 14 1 
 
 15 ~ 1 + tL 
 
 ]8G. lluw <lo you divide Circulating Deciiu.ils ? 
 
CTKCULATING DECi:V[ALS. 187 
 
 1 
 
 If, now, we replace \i by its value, —, we s 
 
 15 1 
 
 shall have 
 
 2d 1 + 1 
 
 1 -I- -J~ • 
 
 a fraction of this form is called a coniimied fraction ; hence, 
 
 187. A Continued Fraction has 1 for Us numerutor, and 
 for its deno)ninator a ivhole manber plus a fraction which also 
 has a numerator of 1, and for a denominator, a whole number 
 plus a similar fraction, and so on. 
 
 2. Reduce \^ to the form of a continued fraction. 
 
 hence, 
 
 15 
 
 1 
 
 4 1 
 
 3 1 
 
 19 
 
 "l+T^/ 
 
 15-3 + 1 
 
 ' 4~l+i 
 
 
 15 
 19 
 
 1 
 
 
 
 1 + 1 
 
 
 
 
 3 + 1 
 
 
 
 
 1 
 
 + i- 
 
 luce 
 
 ! ^^ to the form of a continued fraction. 
 
 
 829 
 347 ~ 
 
 2 + 1 
 
 2 + 1 
 
 
 
 
 1 + 
 
 1 
 
 1 + 1 
 
 3 + tV 
 4. Reduce -^-Ar to the form of a continued fraction. 
 
 ^_ 1 
 
 149-2 + 1 
 
 3 + 1 
 
 2+ 1 
 
 2 + 1 
 
 1 + h 
 
 'B7. What is a Continued Fraction 1 
 
188 CONTINUED FKACTIONS. 
 
 Note. — In a similar manner, any irreducible common fraction may 
 be placed under the i'orm of a continued fraction. 
 
 188. Let us now consider the last example. The fractions, 
 11 1 
 
 2' 2 + 1' 2 + 1' 
 
 &c., 
 
 3 + 1, 
 are called, the Jirst, second, third, &c., approximating fractions; 
 and the object in placing a common fraction under the form of a 
 continued fraction is to find its api^roximate value. 
 
 If we stop at the first approximating fraction, 1, the denomi- 
 nator 2 will be less than the true denominator ; hence, the value 
 of the first ajjproximating fraction will be too great ; that is, it 
 will exceed the value of the given fraction. 
 
 If we stop at the second, the denominator 3 will be less than 
 the true denominator ; hence, 1 will be greater than the number 
 to be added to 2 ; therefore, 2 + l is too small, and 1 -r- 1 + ^ 
 is too large : that is, it is greater than the value of the given 
 fraction. 
 
 Thus, every odd approximating fraction gives a value too 
 large, and every even one gives a value too small. 
 
 EXAMPLES. 
 
 1. Place §1^ under the form of a eontinued fraction, and find 
 the value of each of the approximating fractions. 
 
 2. Place 1^ under the form of a continued fraction, and find 
 the value of each of the approximating fractions. 
 
 3. Place 11 under the form of a continued fraction, and find 
 the value of each approximating fraction. 
 
 4. Place -I- under the form of a continued fraction, and find 
 the value of each approximating fraction. 
 
 5 Place -^-1 under the form of a continued fraction, and find 
 .he value of each approximating fraction. 
 
 1S3. ^^'hat is an nj)[)r<i\imating fraction! Is the first aiiproximaling 
 Taction too large or too small! How is the second ! How are all the 
 xld ones 1 How are all the cvoii ones 1 
 
RATIO AXD PROPORTION'. 189 
 
 RATIO AND PROPORTION. 
 
 189. Proportion is the relation whicli one number, re- 
 garded as a standard, bears to another, with respect either to 
 their difference or quotient. Two numbers may be compared in 
 two ways : 
 
 1st. By considering hoio much one is greater or less than 
 the other, which is shown by their difference ; and, 
 
 2d. By considering Jiow many times one is contained in the 
 other, which is shown by their quotient. 
 
 In comparing two numbers, one with the other, by means of 
 their difference, the less is always taken from the greater. 
 
 In comparing two numbers, one with the other, by means of 
 their quotient, one of them must be regarded as a standard 
 which measures the other, and the quotient which arises by 
 dividing by the standard, is called the ratio. 
 
 190. Every ratio is derived from two terms : the first, or the 
 standard is called the antecedent, because it is supposed to be 
 known beforehand ; and the second, the consequent ; and the 
 two, taken together, are called a couplet. The antecedent will 
 be regarded as the standard. 
 
 If the numbers 3 and 12 be compared by their difference, the 
 result of the comparison will be 9 ; for, 12 exceeds 3 by 9. If 
 they are compared by means of their quotient, the result will 
 be 4 ; for, 3 is contained in 12, 4 times ; that is, 3 measuring 
 12, gives 4. 
 
 191. The ratio of one number to another is expressed in two 
 ■ways : 
 
 1st. By a colon; thus, 3 : 12 ; and is read, 3 is to 12; or, 
 3 measurins; 12. 
 
 189. In how many ways maj' wo numbers, having the same unit, be 
 compared with each other 1 If you compare by their difference, what 
 do you da ! If you compare by the quotient, how do you regard one of 
 the numbers '' What is the ratio 1 
 
 190. From how many terms is a ratio derived ! What is the first term 
 called 1 What is the second called I Which is the standard 1 
 
 191. How may the ratio of two number-s be expre.ssed 1 How read 1 
 
190 
 
 RATIO AND PROPORTION. 
 
 2d. In a fractional form, as Jj^-; or, 3 measuring 12. 
 
 192. If two couplets have tbe same ratio, their terms ara 
 said to be proportional : the couplets, 
 
 4 : 20 and 1 : 5, 
 
 have the same ratio 5 ; hence, the terras are proportional, 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 projiortimi : hence, 
 
 A projwrtion is an expression of equality between two ratios. 
 
 What are the ratios of the proportions ? 
 
 6 
 
 24 : 
 
 8 : 
 
 32 
 
 9 
 
 36 : 
 
 10 : 
 
 40 
 
 8 
 
 72 : 
 
 12 : 
 
 108 
 
 4 
 
 48 : 
 
 5 : 
 
 60 
 
 193. The 1st and 4th terms of a proportion are called the ex- 
 tremes ; the 2d and 3d terms, the means. Thus, in any proportion, 
 
 6 : 24 : : 8 : 32, 
 6 and 32 are the extremes, and 24 and 8 the means : 
 
 24 32 
 Smce -=-, 
 
 we shall have, by reducing to a common denominator, 
 
 24 X 8 _ 32 X 6 
 6 X 8"" 6x8' 
 But since the fractions are equal, and have the same denorai- 
 nators, their numerators must be equal, viz. : 
 
 24 X 8 = 32 X 6 ; that is, 
 
 192. If two couplets have the same Peitio, what is said of the terms ! 
 How arc thoy written T How read ^ What is a proportion ? 
 
 193. Which are the extremes of a proportion \ Which the means ' 
 What is the product of the extremes equal to 1 
 
KATlU AND PKOPORTIOJ^. 191 
 
 In any projwrtion, ike product of Ike extremes is equal to the 
 product of the means. 
 
 Thus, in the proportions, 
 
 1 : 8 : : 2 : 16; we have 1 x 16 = 2x8. 
 4 : 12 : : 8 : 24; " " 4 X 24 =: 12 X 8 ; 
 
 194. Since, in any proportion, the product of the extremes is 
 equal to the product of the means, it follows that, 
 
 1st. //' the product of lite means be divided by one of the 
 extremes, the quotient will be 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. // the product of the extremes be divided by either of the 
 means, the quotient will be the other mean. 
 
 Thus, if4x24=:16x6=:96be divided by 16, the quo- 
 tient will be 6 ; or if it be divided by 6, the quotient will be 16. 
 
 EXAMPLES. 
 
 1. The first three terms of a proportion are 5, 10, and 19 ? 
 what is the fourth terra ? 
 
 2. The first three terras of a proportion are 6, 24, and 14 : 
 what is the fourth terra ? 
 
 3. The first, second and fourth terms of a proportion are 
 9, 12 and 16 : what is the third term ? 
 
 4. The first, third and fourth terms of a proportion are 16, 
 8, and 20 : what is the second term ? 
 
 5. The second, third and fourth terms of a proportion are 
 48, 90, and 45 : what is the first term ? 
 
 194. If the product of the means be divided by one of tlie extremes, 
 whnt will the quotient be ] If the product of the means be divided by 
 either extreme, what will the quotient be ? 
 
192 KATIO AX3) PEOPORTIOIf. 
 
 SIMPLE AND COMPOUND RATIO. 
 
 195. The ratio of two single numbers is called a Simple MatiOf 
 and the proportion which arises from the equality of two such 
 ratios, a Simple Proportion. 
 
 If the terms of one ratio be multipliod by the terms of ano- 
 tlier, antecedent by antecedent, and consequent by consequent, 
 the ratio of the products is called a Compound Ratio. Thus, 
 if the two ratios 
 
 3 : 6 and 4 : 12 
 
 be multiplied together, Ave shall have the compound ratio 
 
 3x4 : G X 12, or 12 : 72; 
 in which the ratio is equal to the product of the simple ratios. 
 
 A proj)oriion formed from the equality of two compound 
 ratios, or from the equality of a compound ratio and a simple 
 ratio, is called a Compound Proportion. 
 
 196. What imrl one number is of another. 
 
 When the standard, or antecedent, is greater than the num- 
 ber which it measures, the ratio is a proper fraction, and is such 
 a pai't of 1, as the number measured is of the standard. 
 
 Note. — The standard is generally preceded by the word o/, and in 
 comparing numbers, may be named second, as in examples 7, 8, 9, 
 10. and 11, but it must always be used as a divisor, and should be 
 placed first in the statement. 
 
 1. What part of 25 is 5 ? that is, what pai't of the standard 
 25, is 5 ? 
 
 -r^^ =1; or 25 : 5 : : 1 : -^ ; 
 
 that is, the standard is to the number measured as 1 to ^ ; or, 
 the number measured is one-fifth of the standard. 
 
 195. What is a simple ratio 1 What is the proportion called which 
 cooies from the equalily of two simple ratios 1 What is a compound ratio I 
 What is a compound jiroportion 1 
 
 190. When the .^t.iiidard is grcalrr than the ronsrquont, what kind of a 
 nnuilicr is the ratio ] What part is 3 of -1 ! C of 12 ? ^^'hal part of 4 is 
 IC? 12 of 301 
 
RATIO AND PEOrOIiTION. 
 
 VJ3 
 
 2. What part of 6 is 4 ? 
 
 3. What part of 10 is 5? 
 
 4. What part of 34 is 17 ? 
 
 5 What part of 450 is 300? 
 
 6. What part of 96 is 16 ? 
 
 7. 8 is what part of 12? 
 
 8. IG is what part of 48 ? 
 
 9. 18 is what part of 90 ? 
 
 10. 15 is wliac part of 1G5 ? 
 
 11. 9 is what par' >f 11 ? 
 
 SIMPLE PROPORTION— OR, SINGLE RULE OF THREE. 
 
 197. Simple Propoktion is an expression of equality 
 between two simple ratios. Hence, a simple proportion con- 
 sists of four single term,s, in which the ratio of the first to the 
 second is equal to the ratio of the third to the fonrth. If three 
 of these terms are known, the fourth can easily be found 
 (Art. 193). 
 
 198, The Rule of Three explains the method of finding, 
 from three given numbers, a fourth, to wliich the third shall 
 bear the same ratio as exists between the first and second. 
 
 1. If 8 barrels of flour cost $5G, what will 9 barrels cost, 
 at the same rate ? 
 
 Note. — We shall denote the vequired terms of the proportion by 
 the letter x. 
 
 Analysis. — The condition, '' at the same 
 rate,"' requires that the quantity. 8 barrels 
 of flour, have the same ratio to the quan- 
 tity, 9 barrels, as $.56; the cost of 8 barrels, 
 to X dollars, the cost of 9 barrels: that is, 
 
 quantity is to quantify, as cost to cost : and, 
 
 Since the product of the two extremes is 
 equal to the product of the two means 
 (Art. 193) ; we have, 
 
 8 X 0? = 56 X 9 ; 
 
 ST.\TEMENT. 
 
 bar. bar. % 
 8 : 9 : : 56 
 
 OPERATION. 
 
 $ 
 X 
 
 9 
 
 X =^ $63, 
 
 197. What is Simple Proportion? How many terms are employed? 
 How mnny terme must be Ivuown, before the rest can be found ? 
 
 198. What is the Kule of Three? 
 
194 
 
 KATIO AND PROPORTION. 
 
 and if 8 times x is equal to 56 x 49, x must be equal to this 
 product divided by 8 : hence, 
 
 The fourth term is equal to the product of tht second and third 
 terms^ divided by the fust. 
 
 2. If 36 dollars will buy 9 yards of cloth, bow many yards, 
 at the same rate, can be bought for $44 ? 
 
 Analysis. — Thirty-six dollars, the cost 
 of 9 yards of cloth, is to $44, the cost of 
 the required cloth, as 9 yards to the re- 
 quired number of yards; that is, cost is to 
 cost J as quantity to quantity. 
 
 The product of the two extremes being 
 equal to the product of the two means, we 
 place 36 and x on the left of the vertical 
 line, and 44 and 9, on the right. 
 
 STATEMENT. 
 
 S $ : : yd. yd. 
 36 : 44 9 : X 
 
 OPERATION. 
 
 4 
 
 t^ 
 
 M 
 
 11 
 
 X — ilyd. 
 
 199. Hence, we have the following 
 
 RULE. 
 
 1. Write the nvmher which is of the same kind toith the 
 ansioer for the third term, the number named in connection with 
 it for the first term, and the remaining number for the second 
 term. 
 
 II. Multiply the second and third terms together, and divide the 
 product hy tlie first term: or, 
 
 Multiply the third term by the ratio of the first and second. 
 
 Notes. — 1. If the fir.st and second terms have diHeicnl units, ihoy 
 must be reduced to the same unit. 
 
 2. If the third term is a compound denominate number, it must be 
 reduced to its smallest unit. 
 
 109. How do you state a ([nestion l>y the Rule of Three? How do you 
 find the foin-tli term? (live the entire rule. 
 
EXAMPLES. 195 
 
 3. The preparation of the terms, and writing them in their proper 
 places, is called the statement. 
 
 4. When the vertical line is used, the unknown term is always 
 WTitten at the left. 
 
 EXAMPLES. 
 
 1. If 8 hats cost $24, what will 110 hats cost, at the same 
 rate ? 
 
 2. If 2 barrels of flour cost $15, what will 12 barrels cost ? 
 
 3. If I walk 168 miles in 6 days, how far should I walk, at 
 the same rate, in 18 da}'s ? 
 
 4. If 8^/;. of sugar cost $1,28, how much will 13/^. cost? 
 
 5. If 300 barrels of flour cost $2100, what will 125 barrels 
 cost ? 
 
 6. If 120 sheep yield 330 pounds of wool, how many pounds 
 will 3G sheep yield ? 
 
 7. 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 hogsheads 
 cost? 
 
 10. If 16 men consume 560 pounds of bread in a month, how 
 much would 40 men consume ? 
 
 11. 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 $3,25, 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 7s. 6(^. ? 
 
 15. If 5 A. IR. 16 P. of land, cost $150.5, what will 126A. 
 '2R. 20P. cost ? 
 
 16. If Vdcwt. -Iqr. of sugar cost $129,93, what will be the 
 cost of ^cwt. ? 
 
 17. The clothing of a regiment of 750 men cost £2834 os.: 
 what will it cost to cloth a body of 10500 men ? 
 
196 SIMPLE PROPORTION. 
 
 18. If dyd. 2qr. of cloth cost $15,75, how much will ^yd, 
 oqr. of the same cloth cost ? 
 
 19. If .5 of a house cost $201.5, what would .95 cost ? 
 
 20. What will 26.25 bushels of wheat cost, if o.o bushels 
 cost 48.40 ? 
 
 21. If the transportation of 2.5 tons of goods 2.8 miles costs 
 $1,80, what is that per cwt.'^ 
 
 22. If f of a yard of cloth cost $2,1 6, what will I of a yard 
 cost ? 
 
 23. If ^ of an ounce cost $11, what will l^oz. cost ? 
 
 24. What is the cost of lG|/&. of sugar, if 14p. cost $lf ? 
 
 25. If $19} will buy 14^ yards of cloth, how much will 
 39| yards cost ? 
 
 26. If |- of a barrel of cider cost -jj of a dollar, what will 
 li of a barrel cost ? 
 
 27. If T^ of a ship cost $2880, what will i§ of her cost? 
 
 28. What will 116i yards of cloth cost, if 462 yards cost 
 $150,66? 
 
 29. If 7-/j- barrels of fish cost $311 what will 32^ barrels 
 cost ? 
 
 30. How much wheat can be bought for $9G|-, if ^hu. Ipk. 
 cost $1,93 1 ? 
 
 31. If '^^ of a yard of cloth cost $1|, what will 71 yards cost? 
 
 32. What will be the cost of 37.05 square yards of pavement, 
 if 47.5 yards cost $72.25 ? 
 
 33. If 3 paces or common steps be equal to 2 yards, how 
 many yards will 160 paces make ? 
 
 34. If a person pays half a guinea a week for his board, how 
 long can he board i'or £21 ? 
 
 35. If 12 dozen copies of a certain book cost $54.72, what 
 will 297 copies cost at the same rate ? 
 
 36. If $3618 worth of provisions will subsist an armv of 
 9000 men for 90 days, if tiie army be increased by 4500 men, 
 how much would last them the same time ? 
 
 37. A grocer bought a /</<rf. of nun ibr 80 cents a gallon, 
 
EXAMPLES. 197 
 
 and after adding water pold it for GO cents a gallon, when lie 
 found that the selling and buying prices were proportional to the 
 original quantity and the mixture : how much water did he add ? 
 
 38. A man failing in business, pays GO cents for every dollar 
 wliicli he owes ; he owes A 83570, and B '"5^1875 ; how much 
 does he pay each ? 
 
 39. A bankrupt's effects amount to 82328,75, his debts 
 amount to $3726 : what will his creditors receive on a dollar ? 
 
 40. If a person drinks 80 bottles of wine in 3 months of 30 
 days each, how much does he drink in a week ? 
 
 41. If 4|- yards of cloth cost 14s. 8d., New York currency, 
 what will 401 yards cost ? 
 
 42. If a grocer use a false balance giving only lA-oz. for a 
 pound, how much will 154^lbs. of just weight give, when weighed 
 by the false balance ? 
 
 43. If a dealer in liquors use a gallon measure which is too 
 small by ^ pint, what will be the true measure of 100 of the 
 false gallons ? 
 
 44. 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 overtakes him ? 
 
 45. A person owning y of a coal mine, sold j of his share for 
 $9345 ; what was the value of the whole mine ? 
 
 4G. At wiiat time between 6 and 7 o'clock, will the hour and 
 minute hands of a clock be exactly together? 
 
 47. If a staff 5 feet long casts a shadow 7 feet, what is the 
 height of a steeple, whose shadow is 196 feet at the same time 
 of dav ? 
 
 48. Two persons are 279 miles apart, start at the same time 
 and travel toward each other. A goes 5 miles an hour, and B 
 4 miles : how many miles must each travel before they meet ? 
 
 49. A can do a piece of work in 3 days, B in 4 days, and C 
 tn G days : in what time will they all do it, working together ? 
 
 50. A can build a wall in 15 days, but with the assistance of 
 C, he can do it in 9 days : in wlmt time can C do it alone .' 
 
198 IXVKltSE PBorOUTIOX. 
 
 -* 
 51. A and B take a job for which they are to receive $1 65.75 ; 
 
 A M'orks himself and emi)loys 7 hands ; B does the same and 
 
 employs 6 hands : what should eacl^receive ? 
 
 o2. A watch, which is 10 minutes too fast at 12 o'clock, on j\Ion- 
 
 day, gains Smin. lOsec. per day : what will be the time by the watch 
 
 at a quarter past ten in the morning of the following Saturday ? 
 
 53. There are two clocks, one of which gains 10 miimtes, 
 and the other loses 7^ minutes every 24 hours. They are to- 
 gether at noon on Tuesday : what will be the difference of their 
 times at 6 o'clock on Friday morning ? 
 
 54. If 15 men can be boarded 1 week for 846,25, what will 
 it cost to board 5 men and 6 boys, the same time, the boys 
 being boarded at half price ? 
 
 55. 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 min- 
 utes : how many times must A go round the wood before he is 
 overtaken by B ? 
 
 CAUSE AND EFFECT. 
 
 200. Whatever produces Effects, as men at work, animals 
 eating, time, goods purchased or sold, money lent producing 
 interest, and the like, may be regarded as Causes. 
 
 Causes are of two kinds, simple and compound : 
 
 A Simple Cause has but a single element, as men at work, 
 
 a portion of time, goods [)urchased or sold, and the like. 
 
 A Compound Cause is made up of two or more simple 
 
 elements, such as men at work taken in connection with time, 
 
 and the like. 
 
 201. The results of causes, as work done, provisions con- 
 sumed, money paid, cost of goods, and the like, maybe regarded 
 as effects. 
 
 '21)0. ^V lint are causes? How many kinds of causes arc there? AVliat 
 is a simple cause? What is a compouiul cause ? 
 
 201. What are effects V What is a simple clVect? What is a compouud 
 effect? 
 
mvp:KsE PKoi'ouTiox. 199 
 
 A Simple Effect is one which has but a single element. 
 A Compound Effect is one which arises from the com- 
 bination of two or more elements. 
 
 202. Causes which are of the same kind, that is, which can 
 be reduced to the same unit, may be compared with each other; 
 and effects which are of the same kind, may likewise be com- 
 pared with each other. From the nature of causes and effects, 
 we know that, 
 
 1st Cause : 2d Cause : : 1st Effect : 2d Effect ; 
 and, 1st Etfect : 2d Effect : : 1st Cause : 2d Cause. 
 
 203. Simple causes and simple effects give rise to simple ratios. 
 Compound causes or compound effects give rise to compound 
 
 ratios. 
 
 204. All questions involving simple ratios are classed under 
 Simple Proportion; and all questions involving compound 
 ratios, either under Inverse or Compound Proportion. 
 
 INVERSE PROPORTION. 
 
 205. It often happens, that two numbers which are compared 
 with each other, undergo, or may undergo, certain changes of 
 value, in which case they represent variable and not Jived quan- 
 tities. Thus, when we say, that the amount of work done, in a 
 single day, will be proportional to the number of men em- 
 ployed, 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 employed, the work done will also be 
 diminished. This is called Direct Proportion. 
 
 If we say that a barrel of flour will serve 12 men a certain 
 
 202. What causes are of the same kind ? What causes mar be com- 
 pared with each other ? What do we infer from the nature of causes and 
 eJfects ? 
 
 203. Wliat ratios arise from simple causes and simple effects? From 
 what do compound ratios arise ? 
 
 204. What questions fall under simple proportion ? Questions involving 
 compound ratios give rise to what kinds of proportion ? 
 
 205. When are two numbers directly proportional ? When are two 
 numbers inversely proportional ? Does their product then vary ? 
 
200 IKVKRSE PROPOKTIOX. 
 
 time, and ask hoAv long it will serve 24 men, there is a certain 
 relation between the number of men and time ; but that relation 
 is such that the time will decrease, if the number of men is 
 mcreased, and will increase, if the number of men is decreased 
 This is called, Inverse Proportion ; hence, 
 
 1. Two numbers are directly proportional, tuhen they increase 
 or decrease together ; in which case their ratio is always the same. 
 
 2. Two numbers are inversely or reciprocally projoortional, 
 when one increases as the other decreases ; in which case their 
 product in ahoays the same. 
 
 Note. — This is sometimes called Reciprocal Proportion. 
 
 206. If we refer to the numeration table of integral and 
 decimal numbers (Art. 14G), 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. The unit of the first decimal place at the right, is 1 
 tenth, a number only 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 similai'ly for all other corresponding places ; hence. 
 
 The units of jylace, taken at equal distances from the unit 1, 
 are inversely 2}roportional. 
 
 207. 1- — The floor of a room is 20 feet long : what must be 
 its breadth, in order that it may contain 360 square feet ? 
 
 Analysis. — The length of the floor, multi- opehation. 
 
 plied by its breadth, will give the area or 3()0 
 
 . / . .. ■ J- •;! 1 1 .1 -- = 18//. breadth, 
 
 contents; hence, the area, divided by the 20 *' 
 
 length, will give the breadtli. 
 
 Note. — When the area or contents of a room are known, or given, 
 
 206. What relation exists between the units of place in the integral and 
 decimal numeration table ? Give an example ? 
 
 207. What arc tlie contents of a floor equal to? What is the breadth 
 equal to? When tlio contents of a floor are given, in wbat proportion is 
 the length to the breadth? If two numbers are inversely proportional, 
 what is either equal to ? 
 
INVERSE PROPORTION. 
 
 201 
 
 the length of the room is inversely proportional to its breadth (Art. 
 205). Il' the length of a room, which is to contain a given area, bo 
 increased any number of times, the breadth of the room mu.st be dimin- 
 ished just as many times. If the length be divided by any number, 
 the breadth must be mulliplied by the same number. Thus, two 
 rooms, one 40 feet long and 9 feet wide, the other, 10 feet oneway 
 and 3(5 feet the other, have the same area, viz., 360 square feet. 
 Hence, when two numbers ar§ inversely proportional, * 
 
 Either is equal to their product divided by the other. 
 
 208. We may regard the length of a room as one cause of 
 its contents, and its breadth as another cause of its contents; 
 foi*, the contents being equal to the product of the length and 
 breadth, is the effect of them both. 
 
 In the case of the tv»'o rooms, one 40 feet long and 9 feet 
 wide, and the other 10 feet by 36, the effects are the same. 
 The causes are compound, each being composed of two elements 
 (Art. 200) ; and since the effects ai-e equal the causes are equal 
 (Art. 202) ; hence, 
 
 Whtn the causes are equcd, the elements are inversely pro- 
 portional. 
 
 1. If 3G men, in 12 days, can do a certain work, in what time 
 will 48 men do the same work ? 
 
 Analysis. — The first cause is 
 compounded of 36 men and 12 
 days, and Ls equal to 36 x 12 = 432, 
 the number of days it. would require 
 1 man to do tlie work. 
 
 The second cause is compounded 
 of 48 men and the number of days 
 it would require them to do the 
 Bame work, and is equal lo 48 x x. 
 
 Bat since the effects are the same, 
 viz., the work done, the causes must 
 be equal ; hence, the products of the 
 elements of the causes are equal. 
 Tlierefore, in the solution of all 
 like examples, 
 
 
 STATEMENT 
 
 meji. 
 
 men. ' 
 
 
 36 
 
 : 48 
 
 
 days. 
 12 
 
 days. 
 
 X 
 
 > ; 
 
 432 
 
 : AS XX : 
 
 
 OPERATION. 
 
 
 X 
 
 
 9 
 
 Ans. X = 9 days. 
 
202 EXAMPLES. 
 
 Write the elements of the cause containing the unknown ele- 
 ment on the left of the vertical line for a divisor, and the elcmenti 
 of the oilier cause on the right for a dividend. 
 
 Notes. — 1. Since the effects are equal, they may each be denoted 
 by 1 ; hence, the causes are to each other as 1 to 1 . 
 
 2. It is evident, that in this class of questions the elements of the 
 causes are inversely proportional ; and hence, such questions have 
 generally been arranged under the head of '• Rule of Three Inverse." 
 
 EXAMPLES. 
 
 1. If 3| yards of cloth will make a coat and vest, -when the 
 cloth is 1^ yards wide, how much cloth will be needed which is 
 •I yards in width ? 
 
 2. If I have a piece of land 16f rods long and 3^ rods 
 wide, what will be the length of another piece that is 7 rods 
 wide and contains an equal area ? 
 
 3. How many yards of carpeting that is three-fourths of 
 a yard wide, will carpet a room 36 feet long and 30 feet in 
 breadth ? 
 
 4. If a man can perform a journey in 8 days, walking 9 
 hours a day, how many days will it require if he walks 10 hours 
 a day ? 
 
 5. If a family of 15 persons have provisions for 8 months, 
 by how many must the family be diminished that the provisions 
 may last 2 years ? 
 
 6. A garrison of 4 GOO men have provision for G months : to 
 what number must the garrison be diminished that the provi- 
 sions may last 2 years and G months ? 
 
 7. A certain amount of provisions will subsist an army of 
 9000 men for 90 days : if the army be increased by 6000, how 
 long will the same provisions subsist it ? 
 
 208. M'hat may we regard as causing or producing the contents of a 
 room ? When two causes arc equal, how are the elements 1 Note. — It 
 the cifects are equal, by what may they be denoted 1 
 
EXAMPLES. 203 
 
 8. If G 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 ? 
 
 9. Four thousand soldiers were supplied with bread for 24 
 weeks, each man to receive liios. per day; but, by some acci- 
 dent, 210 barrels containing 200/6. each were spoiled: what 
 must each man receive in order that the remainder may last the 
 same time ? 
 
 10. Suppose 4000 soldiers after losing 210 barrels of bread 
 each containing 2001b., were to subsist on loos, each a day for 
 24 weeks ; had none been lost they would have received 14:0Z. 
 a day : what was the weight of the whole, and how much did 
 they receive ? 
 
 11. Let us now suppose 4000 soldiers to lose one-fourteenth 
 of their bread, then to receive looz. each a day for 24 weeks : 
 what was the whole weight of their bread including the lost, and 
 how much would each have received per day had none been 
 spoiled ? 
 
 12. If 4 men can do a piece of work in 80 days, how many 
 days will 16 men require to do the same Avork? 
 
 13. If 21 pioneers make a trench in 18 days, how many days 
 will 7 men require to make a similar trench ? 
 
 14. 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? 
 
 15. If a field of grain be cut by 10 men in 12 days, in how 
 many days would it be cut by 20 men ? 
 
 16. If 90 barrels of flour will subsist 100 men for 120 days, 
 how long will it subsist 75 ? 
 
 17. 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 
 would he perform the same journey ? 
 
 18. If 50 persons consume 600 bushels of wheat in a year, 
 how long would it last 5 persons ? 
 
 19. A certain work can be done in 12 days, by working 4 
 
20'i: INVEKSE PEOPOKTION. 
 
 hours each day : how many days would it require to do the same 
 work by working 9 hours a day ? 
 
 20. If 120 men can build ^ mile of wall in 15^ days, how 
 many men would it require to build the same wall in 40| days ? 
 
 21. A garrison of oGOO men has just bread enough to allow 
 24:02. 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 against the enemy ? 
 
 22. If 3 horses or 5 colts eat a certain quantity of oats in 40 
 days, in what time will 7 horses and 3 colts consume the same 
 quantity ? 
 
 23. If a person can perfoi'm a journey in 24 days of 10^ 
 hours each, in what time can he perform the same journey, 
 when the days are 12i hours long ? 
 
 24. A piece of land 40 rods long and 4 rods wide, is equiva- 
 lent to an acre : what is the breadth of a piece 15 rods long 
 that is equivalent to an acre ? 
 
 25. If a person travelling 1 2 hours a day finish one half of a 
 journey- in 10 days, in what time will he finish the remaining 
 half, travelling 9 hours a day ? 
 
 26. How many pounds weight can be carried 20 miles for the 
 same money that 4-^ hundred weight can be carried 36 miles? 
 
 27. If 20 men can perform a piece of work in 12 days, 
 working 9 hours a day, how many men will accomplish the 
 same work in one half the time, working 10 hours a day? 
 
 28. If 72 horses eat a certain quantity of hay in 7^ weeks, 
 how many horses will consume the same in 90 weeks ? 
 
 29. Bought 5000 planks, 15 feet long and 2^ inches thick; 
 how many planks are they equivalent to, of 12^2- feet long and 
 1^- inches thick ? 
 
 30. If 12 pieces of cannon, eighteen pounders, can batter 
 down a castle in 3 hours, in what time would nine twenty-four 
 ])ounders batter down the same castle, both pieces of cannon 
 being fired the same number of times, and their balls flying 
 with the same velocity? 
 
COMPOUND PROPOKTION. 
 
 205 
 
 COMPOUND PROPORTION. 
 
 209. Compound Proportion is a comparison of compound 
 ratios wlieii the terms are unequal. 
 
 It embraces that class of questions in which the causes arc 
 compound, or in which the effects are compound. In this class 
 of questions, either a cause or a single element of a cause, may 
 be required; or an effect, or a single element of an effect may 
 be required. 
 
 1. If 8 men in 12 days can build 80 rods of wall, how much 
 will 6 men do in 18 days? 
 
 or 
 
 
 STATEMENT. 
 
 
 Cause : 
 
 2d Cause : : 1st Efl^ct 
 
 : 2d Effect. 
 
 12; • 
 
 is} - '0 
 
 : X 
 
 12x8 : 
 
 18x6 : : 80 
 
 : X 
 
 Analysis. — In this example the second 
 effeet is required, and the statement may bs 
 read thus : 
 
 If 8 men in 12 days can build 80 rods of 
 wall, 6 men in 18 dajs will build how 
 many (or x) rods of wall ? 
 
 OPERATTON. 
 
 t 
 
 If. 
 
 X 
 
 1$ 
 
 $0 10 
 
 Ans. X =z 90 rods. 
 
 2. If a family of 12 persons, in 8 months, expend $864, hov/ 
 many months will $900 serve a family of 20 persons ? 
 
 or. 
 
 
 STATEMENT. 
 
 
 1} ^ 
 
 ^.r} ' '' ^^^^ 
 
 $900. 
 
 12x8 : 
 
 20xx : : $864 : 
 
 $900. 
 
 209. What is compound proportion 1 What questions docs it eml)race' 
 Wliat i.9 ahvays required ^ 
 
20G 
 
 COMPOUND PROI'OKTIUN. 
 
 Analysis. — In this example, an element 
 of the second cause is required, viz.. the num- 
 ber of months which the money will last 20 
 men. The question is thus stated : 
 
 If 12 persons, in 8 months, expend $864, 
 20 persons in how many (or x) mouths will 
 expend $900 ? 
 
 OPER.iTION. 
 
 t% 
 
 X 
 
 n 
 
 $u 
 
 $ 5 
 
 000 
 
 Alls. X := 5 months. 
 
 o. If 24 men, in G days, working 7 hours a clay, can buiJd a 
 wall 115 feet long 3 feet thick and 4 feet high, 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 in height ? 
 
 STATEMENT. 
 
 115 
 
 or, 24x6x7 : 36x12x14 
 
 115x3x4 : .rx4x5. 
 
 Analysis. — In this example, an element 
 of the second effect is required, viz.. the 
 length of the wall ; and the question may 
 be thus stated : 
 
 If 24 men, in 6 days, working 7 hours a 
 day, can build a wall 115 feet long, 3 feet 
 thick, and 4 feet high, 36 men in 12 days, 
 working 14 hours a day, will build a wall 
 how many (or x) feet long, 4 feet thick and 
 5 feet high? 
 
 OPERATION. 
 
 u 
 
 
 
 t 
 
 X 
 
 Ans. 
 
 $6 
 
 n$ 2^ 
 
 3 
 
 .r = 450/V. 
 
 210. ITcncc, we have the following 
 
 Rule. — I. Arrange the terms in the statement so that the 
 causes shall comjiose one cotq^let, and the effects the other, jii^li^'"-!? 
 X in the 2'>l'ice of the required element. 
 
 110. Give the rule for stating the question and finding the unknown 
 
 part 
 
EXAMPLES. 207 
 
 II. If -s. fall in either extreme, make the product of the means 
 a dividend, and the product of the extremes, omitting x, a dieisor ; 
 if yi fill <n cither mean, make the product of the extremes a divi- 
 dend, and the product of the means, omittinf/ x, a divisor. 
 
 EXAMPLES. 
 
 1. If 2 men can dig 125 rods of ditch in 75 days, in Low 
 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 ? 
 
 3. If a person can travel 120 miles in 12 days of 8 hours 
 each, how far will he be able to travel in 15 days of 10 hours 
 each ? 
 
 4. If a pasture of 16 acres will feed G horses for 4 months, 
 how many acres will feed 12 horses for 9 months ? 
 
 5. If 60 bushels of oats wiU 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, and 4 
 feet thick, in 4 days, in what time will 48 men build a wall 864 
 feet Ions;, 6 feet biijh, and 3 feet wide ? 
 
 7. If the freisrht of 80 tierces of sugar, each weighinoj 3i 
 hundred weight, for 150 miles, cost $84, what must be paid for 
 the freight of 30 hogsheads of sugar, each weighing 12 hundred 
 weight, for 50 miles ? 
 
 8. A family consisting of 6 persons, usually drink 15.6 gal- 
 lons 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 regiment of 
 494 men ? 
 
 10. If a garrison of 3600 men eat a certain 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 £20 worth of wine at 2.?. 
 6(/. per bottle : how many men, at the same rate, will £7 worth 
 supply, when wine is worth l6'. 9t/. per bottle ? 
 
208 OOilPOUND PKOPOKTION. 
 
 12. If the Avages of 13 men for 74- days, be $149,76, -what 
 will be the wages of 20 men for 15 i days ? 
 
 I'd. If a footman travel 20-4 miles in Gg- days of 121- hours 
 each, in how many days of 10|- hours each will he travel 120^ 
 miles ? 
 
 14. If 120 men in 3 days, of 12 hours each, can dig 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, 
 6 feet deep, and li yards broad, in 9 days of 15 hours 
 each ? 
 
 15. If 40 men can perform a piece of work in 12 days, how 
 many men will perform another piece of work three times as 
 large, in one-fifth part of the time ? 
 
 16. 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 
 remainder of the journey? 
 
 17. 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 ? 
 
 18. If a bar of iron 5/f. long, 2li>i. wide, and l^-in. thick, 
 weigh Ablbs. how much will a bar of the same mcU\l wei/di that 
 is Ifl. long, dill, wide, and 2i/». thick ? 
 
 19. 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 7 liou-s 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 ? 
 
 20. Fifty thousand bricks are to be removed a given distance 
 in 10 days. Twelve horses can remove 18000 in 6 ^%ys: how 
 many horses can remove the remainder in 4 days? 
 
 21. If 3 men, working 10 hours a day, can plant }■ field 150 
 rods by 240 rods, in 5 days, how many men, working 12 hours 
 a day, can plant a field measuring 192 rods by 30t> vod^^, in 4 
 days ? 
 
PAI^TNEUSITTP. 200 
 
 22. If 248 men, in 5 J, days of 11 hours each, dig a trench of 
 7 degrees of hardness, 232^- yards long, of wide, and 2^ deep, 
 in how many days, of 9 hours long, will 24 men dig a trench of 
 4 degrees of hardness, 337-^ yards long, 5| wide, and 31 deep ? 
 
 PARTNERSHIP. 
 
 211. Partxership is the joining together of two or more 
 persons in trade, with an agreement to share the profits or 
 losses. 
 
 Partners are those who are joined together in carrying on 
 business. 
 
 Capital, is the amount of money employed. 
 Dividend is the gain or profit : 
 Loss is the opposite of jirofit. 
 
 212. The Capital or Stock is a cause of the entire profit : 
 Each man's capital is the cause of his profit : 
 
 Tlie entire profit or loss is the effect of the cause or capital : 
 Each man's profit or loss is the effect of his capital : hence. 
 Whole Stock : Each man's Stock 
 : : Whole profit or loss : Each man's profit or loss. 
 1. Mr. Jones and Mr. Wilson are partners in trade: Mr. 
 Jones puts in, as capital, $1250, and Mr. Wilson, $750 : at the 
 end of a year there is a profit of $720 : what is the share of 
 each? 
 
 STATEIVIENT. OPERATION. 
 
 2000 : 1250 :: 720 : Jones' share. ^^^^^ 
 
 X 
 
 
 2000 : 750 :: 720 : Wilson's share. 
 
 ^ 
 
 X — $450 Ans. 
 15 
 
 X 
 
 liv- 
 
 n^ 
 
 X = 270. 
 
 211. What is a partnership? What are partners'? What is capital oi 
 <ock ? What is dividend ? What is loss ? 
 
210 PARTNEKSniP. 
 
 Hence, the following 
 
 Rule. — As f/ie iv/wie stock is to each tnan^s share, so is the 
 whole gain or loss to each man^s share of the gain or loss. 
 
 EXAJIPLES. 
 
 1. 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 ? 
 
 2. C and D have a joint stock of $4200, of which A owns 
 $3600, and B, $600 : they gain in one j-ear, §2000 : what is 
 each one's share of the profits ? 
 
 3. A, B, C, and D, have §40,000 in trade, each an equal 
 shai'c ; at the end of six months their profits amount to §16000 : 
 what is each one's share, allowing A to recfeive $50, and D, $30, 
 out of the profits, for extra services ? 
 
 4. Five persons have to share between them an estate of 
 $20000 ; A is to have one-fourth, B one-eighth, C one-sixth, 
 D one-eighth, and E what is left : what will be the share of 
 each ? 
 
 5. Three merchants loaded a vessel with flour ; A loaded 500 
 barrels, B, 700 barrels, and C, 1000 barrels ; in a storm at 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, 
 to his third, $4500, and to his fourth, $4000. But on settling 
 the estate, it was found that after paying the debts and expenses, 
 only $12000 remained to be divided : how much should each 
 receive ? 
 
 7. A widow and her two sons receive a legacy of $1500, of 
 which the widow is to have -^, and the sons each i. But the 
 oldest son dying, the whole is to be divided in the saire propor- 
 tion between the mother and youngest son : whai will each 
 receive .'' 
 
 8. Four persons engage jointly in a land speculation : D puts 
 
COMPOUND I'AETNEKSHIP. 211 
 
 in $5499 capital. Tiiey gain $15000, of which A takes $4320,50, 
 B, $5245,75, and C, $3600,75 : how much capital did A, B, 
 and C put in, and what is D's sliare of the gain ? 
 
 9. A steam-mill, valued at $4300, was entirely destroyed by fire. 
 A owned i of it, B ^, and C the remainder ; supposing it to have 
 been insured for $2500, what Avas 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 often 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, 8362,121, to C, $250,87i, and to D, $140. He is worth 
 only §614,25 : to how much is each entUled ? 
 
 12. Brown, Smith & Co., produce dealers, were obliged to 
 make an assignment on account of a falling off in prices. Their 
 eftects were valued at $2544, with which they can pay but 
 twenty cents on the dollar : how much did they owe ? 
 
 13. Four persons, A, B, C, and D agreed to do a piece of 
 work for $270. Tiiey \vere to do the work in the proportions 
 of |, A, i, and yV : what should each receive for his work ? 
 
 14. Three persons buy a piece of land for $4569, and the 
 parts for which they pay bear tiie 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, f and -^ : how much did each pay, and what part 
 did each own ? 
 
 ■ COMPOUND PARTNERSHIP. 
 When the Cause of Projit or Lass is Comj/ound. 
 213. When the partners employ their capital for different 
 periods of time, the profit of each will depend on two circum- 
 stances ; first, on the amount of capiUd he 'puts in ; and secondly, 
 on the time it is continued in business. 
 
 213 When the partners employ their capital for different periods of 
 time, on what will the profit depend ! Will the cause, then, be simple or 
 compound 1 To what will it be equal 1 Give the rule for finding each 
 shard 
 
212 COMPOUND PARTNERSHIP. 
 
 Hence, the cause of the loss or gain will be compound, and 
 the product of the elements, in each particular case, will be the 
 cause of each man's gain or loss ; and their sum AviU be the 
 cause of the entire gain or loss. 
 
 1. A put in trade $000 for 4 months, and B, $600 for 5 
 months. They gained $240 : what was the share of each ? 
 
 OPERATION. 
 
 A, $500 X 4 = 2000 
 
 B, GOO X 5 := 3000 ^^^^ ^ ^„„ ., 
 
 -— — . < 2000 : : 240 • -f ^^^ ^^• 
 
 Hence, the following 
 
 Rule. — MuUipIy each man^s stock bij the time he continued it 
 in trade: then say, as the sum of the products is to each product, 
 so is the whole gain or loss to each man^s share of the gain or 
 loss. 
 
 EXAMPLES. 
 
 1. Three men hire a pasture for $70,20 : A put in 7 horses 
 for 3 months ; B, 9 horses for 5 months ; and C, 4 horses for 
 C months : what part of tlie rent should each pay ? 
 
 2. A commenced business, with a capital of $10000. Four 
 months afterwards B entered into partnership with him, and 
 put in 1500 barrels of tiour. At the close of the year their 
 profits were $5100, of which B was entitled to $2100: what 
 was the value of tlie flour per barrel ? 
 
 3. On the 1st of January, 1856, A commenced business with 
 a capital of $23000 ; two months afterwards ho drew out 
 $1800; on the 1st of April B, entered into partnei-sliip with 
 liim, and put in $13500; four montiis afterwards hi' drew out 
 $10000 ; at tlie end of the 3'ear their profits were $8400 : how 
 much ought each to receive ? 
 
 ■I. 'I'hrcc i)er.-ons received intei'cst to the amount of $798. 
 A put out $4000 for 12 months, B, $3000 for 15 months, and 
 C, $5000 for 8 months : to how nuich interest was each 
 ^Mititled ? 
 
EXAMPLES. 
 
 213 
 
 5. C, D, and E, form a copartneisliip ; C's sioek is in trade 
 3 montlis, and he claims yV of the gain ; D's stock is in 9 
 months ; and E put in $7oG for 4 months, and chiinis l of the 
 profits : how much did C and D put in ? 
 
 6. A ship's company took a prize worth $20700, which was 
 divided among them according to their pay and the time they 
 had been on board. There were 4 officers receiving $40 a 
 month each, and 12 midsliipmen receiving $30 a month each, 
 all of whom had been on board 6 months ; there were also 110 
 sailoi's receiving $22 a month each, and who had been on board 
 5 months : what was the share of each ? 
 
 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, but at the end of 15 months he drew out $2000. 
 At the end of 12 months, C was admitted as a partner witli 
 $7333i. The gain was $7400 : how much shouldeach receive? 
 
 8. Four persons together agreed to build a barn for $34G,50. 
 A worked 14 days, 12 hours each day; B, 18 days, 10 hours 
 each day; C, 15 days, 11 hours each day; and D, 20 days, 9 
 hours each day : how much should each man receive ? 
 
 9. In a certain school, premiums to the value of $27 are to 
 be distributed in tlie following manner : The premiums are 
 divided into three grades. The value of a premium of tlie 
 first grade is twice the value of one of the second ; and the 
 value of one of the second grade twice that of the third. There 
 are 6 to receive premiums of the first grade, 12 of the second, 
 and 6 of the third : what will be the value of a single premium 
 of each grade ? 
 
 10. Three men take an interest in a mining company. A 
 put in $480 for 6 months, B, a sum not named, for 12 months, 
 and C, $320 for a time not named : when the accounts Avere 
 settled, A received $G00 for his stock and profits, B, $1200 for 
 his, and C, $520 for his : what was B's stock, and C's time ? 
 
"214: ^ PERCENTAGE. 
 
 PERCENTAGE. 
 
 214. Pkrcentage is an allowance made by the hundred, and 
 is always a part of the number on which the allowance is made. 
 
 The Base of percentage, is the number on which the per- 
 centage is reckoned. 
 
 215. Percent means by the hundred: thus, 1 per cent 
 means 1 for every hundred ; 2 per cent, 2 for every hundred ; 
 3 per cent, 3 for every hundred, &c. The numbprs denoting 
 the allowances, 1 per cent, 2 per cent, 3 per cent, &e., are called 
 rates, and may be expressed decimally, as in the following 
 
 TABLE. 
 
 1 per cent is 
 
 .01 
 
 7 per cent is 
 
 .07 
 
 3 per cent is 
 
 .03 
 
 8 per cent is 
 
 .08 
 
 4 per cent is 
 
 .04 
 
 15 per cent is 
 
 .15 
 
 5 per cent is 
 
 .05 
 
 68 per cent is 
 
 .68 
 
 6 per cent is 
 
 .06 
 
 99 per cent is 
 
 .99 
 
 ALSO, 
 
 100 per cent is 1 ; for, \^^ is equal to 1. 
 150 per cent is 1.50 ; for, |^§ is equal to 1.50. 
 140 per cent is 1.40 ; for, }^g jg equal to 1.40. 
 200 per cent is 2 ; for, ^% is equal to 2. 
 
 i per cent is .005 ; for, yoo -t- 2 is equal to .005. 
 31 per cent is .035; for, 3i^l00 = .03 + .005 = .035. 
 5| per cent is .0575; for, 5f^l00 = .05 + .0075 = .0575. 
 &c. &c. &c. &c. 
 
 examvles. 
 
 1. Write decimally, 91 per cent, and 8^ per cent. 
 
 2. Write decimally, 12-i- per cent, and 9|- per cent. 
 
 3. Write decimally, 208 per cent, 375 per cent, and 95 per cent ' 
 
 4. Write decimall}-, 6Gj per cent. 
 
 S14. What is percentage 1 What is the base 1 
 
 2 If). What docs per cent mean 1 What do you understand by 3 pei 
 aentl Wliat is the rate, or rato per cent ? 
 
EXAilPLES. 
 
 215 
 
 216. To find the -percentage of any number. 
 1. What is the percentage of $450, the rate being 6 per 
 cent ? 
 
 OPERATION. 
 
 450 
 .06 
 
 $27.00 Ans. 
 
 Analysis. — The rate being 6 per cent, is 
 expressed decimally by .06. We are then to 
 take .06 of the base, $450; this we do by 
 multiplying $450 by .06, giving $27. 
 
 Hence, to find the percentage of a number, 
 
 Multiply the nuinber by the rate expressed decimally, and the 
 product tvill be the percentage. 
 
 EXAMPLES. 
 
 1. "What is the percentage of $564, the rate being 51 per 
 cent? 
 
 Note. — When the rate cannot be re- 
 duced to an exact decimal, it is most 
 convenient to multiply by the fraction, 
 and then by that part of the rate which 
 is expressed in exact decimals. 
 
 OPERATION. 
 
 564 
 
 188 = |- per cent. 
 2820 = 5 per cent. 
 
 $3008 = 5-J- per cent. 
 Find the percentage of the following numbers : 
 
 2. i per cent of $1256. 
 
 3. -i per cent of §956,50. 
 
 4. -I per cent of 475 yards. 
 
 5. |- per cent of o24:.6cwt. 
 
 6. A pel- cent of 125.25/6. 
 
 7. If per cent of loQhush. 
 
 8. 41 per cent of $2000. 
 
 9. 9 per cent of 186 miles. 
 
 10. 10|- per cent of 460 sheep. 
 
 11. 5y*g- per cent of 540 tons. 
 
 12. 8| per cent of $3465,75. 
 
 13. 12^ per cent of 126 cows. 
 
 14. 50 per cent of 320 bales. 
 
 15. 371 per cent of 127oyds. 
 
 16. 95 per cent of $4573. 
 
 17. 105 per cent of 25005a?-. 
 
 18. 11 2J- per cent of $4573. 
 
 19. 250 per cent of $5000. 
 
 20. 305 per cent of $1267,871. 
 
 21. 500 per cent of $3000. 
 
 22. What is the difference between 4f per cent of $1000 
 and 71 per cent of $1500 ? 
 
 216. How do 3^ou find the percentage of any nunibrr 1 
 
216 PEKCENTAGE. 
 
 23. If I buy 895 gallons of molasses, and lose 17 per cent 
 by leakage, liow much have I left ? 
 
 24. A grocer purchased 250 boxes of oranges, and found that 
 he had lost in bad ones 18 per cent : how many full boxes of 
 good ones had he left ? 
 
 25. A capitalist wishes to invest $25000 ; he invests 20 per 
 cent in bank stock, 371 per cent in railroad stock, and the 
 remainder in bonds and mortgages : what per cent, and what 
 amount did he invest in the latter ? 
 
 26. A man bought a house and lot for $3250 ; in three years 
 time it increased in value 87^ per cent : what was its value 
 then ? 
 
 27. A farmer having $1572,75, purchased cows with 25 per 
 cent of it, sheep with 12^ per cent of it, and lent 50 per cent 
 of it to a friend : how much had he left ? 
 
 217. To find the 2Jer cent which one number is of another. 
 
 1. What per cent of 64 is 16 ? 
 
 Analysis. — In tliis example 16 is the per- operation. 
 
 ccntage, G4 is the base, and we wish to find 16 1 
 the rate. Since the percentage is equal to 64 4 ' 
 the base multiplied by the rate (Art. 216), 
 
 the rate is equal to the percentage divided by the base ; hence, 
 
 1 fi 1 
 
 -— = - = .25 ; therefore, the rate is 25 per cent : hence, to find what 
 64 4 ' ' ' 
 
 per cent one number is of another, 
 
 Divide the number dinoting the percetitage by the base, and 
 the two first decimal places will express the rate per cent. 
 Notes. — 1. The base is generally preceded by llic word of. 
 
 2. There are three parts in i^ercentage : 1st. The base; 2d. The 
 rate; and 3d, their product, which is the percentage. 
 
 3. The percentage divided by the rate, gives the base ; the per- 
 centage divided by the 6a.se, gives the rate. 
 
 217 How do you find the per cent which one number is of auotbor' 
 
EXAMPLES. 
 
 EXAMPLES. 
 
 1. TVlmt 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 ? 
 G. What per cent is $84 of $96 ? 
 
 7. What per cent is 275 of 440 ? 
 
 8. What per cent is 3 miles of 400 miles ? 
 
 9. Eleven is v/hat per cent of 800 ? 
 
 10. One hundred and four sheep is Avhat per cent of a drove 
 of 312 sheep ? 
 
 11. A grocer has $325, and purchases sugars to the amount 
 of $121,87^ : what per cent of his money does he expend ? 
 
 12. Out of a bin containing 450 bushels of oats, 56^ bushels 
 were sold : what per cent is this of the whole ? 
 
 13. A merchant goes to New York with $2500 ; he first lays 
 out 20 per cent for _ groceries, and then expends $1875 for dry 
 goods : what per cent of his money has he left ? 
 
 14. Two persons invested in stocks $4500 each ; one lost 
 $562,50, and the other lost $405 : what per cent more on the 
 amount invested, did one lose than the other ? 
 
 15. A and B engage in different kinds of business with 
 $5400 capital each ; A gains $1350, and B loses $540 the first 
 year : what per cent is B's money of A's ? 
 
 218. To find tlie base u'hen Ihe 2>s^'centage is added to or sub- 
 traded from the base. 
 
 1. Mr. .Jones buys 8 hogsheads of sugar, sells them at an 
 advance of 15 per cent, and receives $470 : what did he pay 
 for the sugar ? 
 
 218. How do you find the base when the percentage is subtracted from 
 tljc base 1 
 
218 PERCENTAGE. 
 
 Analysis. — The amount received, $470 opekation. 
 
 arises from adding the percentage to the 1.15) 470 ($400 
 base; that if>j it arises from multiplying the 470 
 
 base hy 1 + pbu; the rate per cent ; hence, 
 to find the base, in such cases, 
 
 Divide (he given number by 1 j>Zi<.? the roAe per cent, expressed 
 decimally. 
 
 2. A cask of Avine, out of wliich 37 per cent had leaked, 
 was found to contain 33.39 gallons : liow many gallons did the 
 cask contain ? 
 
 Analysis. — Thirty-seven per operation. 
 
 cent denotes .37 of the capacity 1 —.37 per cent = .63 per cent, 
 of the cask ; and hence, the part 
 
 of the cask that is filled is de- .63) 33.39 (53 gallons, 
 noted by 1 - .37 = .63. But 31 5 
 .63 of the cask contains 33.39 1 89 
 gallons : therefore, the entire 1 89 
 cask will contain as many gal- 
 lons as .63 is contained times in 33.39. viz., 53 ; hence, to find the 
 base, in such cases, 
 
 Divide the given number by the difference between 1 und the 
 rate p)£r cent, expressed decimally. 
 
 EXA3IPI.es. 
 
 1. A farmer bought 40 sheep, and after keeping them for 
 one year, sold them at an advance of 55 per cent, and received 
 $248 : what did he pay for tlie sheep per head ? 
 
 2. A merchant bought a lot. of goods and marked ihcni at .in 
 advance of 20 per cent : -vvhen sold, he found that they brought 
 liini $6835,50 : what did the goods cost him ? 
 
 3. A son, who inherited a fortune, spent 371 per oent of it, 
 when lie found tliat lie had only $31250 remaining: what was 
 the amount of his fortune ? 
 
 4. A grocer purchased a lot of teas and sugar, on wiiicli he 
 lost 16 i)(!r cent, by selling them for .^4200 : wliat did he pay 
 fni- the goods ? 
 
INTEREST. 219 
 
 INTEREST. 
 
 219. Interest is a payment for the use of money. 
 Principal or Base, is the money on which interest is paid. 
 
 Amount is the sum of the Principal and Interest. 
 
 For example : If I borrow $100 for 1 year, and pay 7 dol 
 lars for the use of it ; then, 
 
 100 dollars is the Principal or Base, 
 
 7 dollars is the Interest, and 
 107 dollars is the Amount. 
 
 Rate is the ratio of the principal to the interest, when the 
 time is 1 year. Thus, .07 is tlie rate in the example, being 
 the ratio of $100 to §7. This rate is read, 7 -per cent ; that 
 is, $7 for every hundred : tlie term per cent, means by the 
 hundred, and tlie term, per annum, by the year. 
 
 Hence, there are four simple parts : 1st. Principal ; 2nd. 
 Rate ; ord. Time ; 4th. Interest. 
 
 CASE I. 
 
 220. To find the interest of any principal for one or more 
 years. 
 
 1. What is the interest of $3920 for 2 years, at 7 per cent ? 
 
 Analysis. — The rate of interest being operation. 
 7 per cent, is expressed decimally by .07 : S3920 
 hence, each dollar, in 1 year, ayIII pro- .07 rate. 
 duce .07 of itself, and S3920 will pro- $274,40 int. for 1 year, 
 duce .07 of S3920, or $274,40. There- ' 3 No. of years, 
 fore, $274,40 is the interest for 1 year, $548,80 interest, 
 and this interest multiplied by 2, gives 
 the interest for 2 years : hence, the following 
 — -^ _ « ■ 
 
 219. What is interest 1 What is principalT What is amount 1 What 
 is Tate of interest 1 What does per annum mean 1 
 
 220. How do you find the interest of any principal for any number of 
 years 1 Give the analysis. 
 
220 INTEREST. 
 
 RtjLE. — Multiphj the principal by the rate, expressed decimaUy 
 and the product by the number of years. 
 
 EXAMPLES. 
 
 1. Wliat is the interest of $675 for 1 year, at 6i per cent? 
 
 OPERATION. 
 
 $675 
 Analysis. — ^We first find the interest Qgi 
 
 at \ per cent, and then the interest at ^Z" o ^ per cent 
 
 6 per cent; the sum is the interest at 4050 6 per cent. 
 
 6^ per cent. $43,875 fil per cent. 
 
 2. AYhat is the interest of $871,25, for 1 year, at 7 per cent ? 
 
 3. What is the interest of $535,50, for 7 years, at 6 per 
 cent ? 
 
 4. What is the interest of $1125,885, for 4 years, at 8 per 
 
 cent ? 
 
 5. What is the interest of -§789, 74, for 12 years, at 5 per cent ? 
 
 6. What is the intex-est of 6^2500, for 7 years, at 7i per cent ? 
 
 7. What is the interest of 83153,82, for 2 years, at A\ per 
 
 cent? 
 
 8. What is the amount of $199,48, for 16 years, at 7 per 
 
 cent ? 
 
 9. What is the amount of $897,50. for 3 years, at 8 per cent ? 
 
 10. What is the interest of $982,35, for 4 years, at 63- per 
 cent? 
 
 11. What is the amount of ^1500, for 5 years, at 5\ per cent ? 
 
 12. What is the interest of $1914,10, for 6 years, at 3|- per 
 cent ? 
 
 13. What is the interest of $350, for 21 years, at 10 per 
 cent ? 
 
 14. What is the amount of 628,50, for 5 years, at 121 per 
 cent ? 
 
 15. What is the amount of §75,50, for 10 years, at 6 per 
 cent? 
 
 16. What is the amount of 85040, for 2 years, at 1\ per 
 cent ? 
 
INTEREST. 221 
 
 Note. — When taerc are years and months, and the months are 
 an aliquot part of a year, multiply the interest for 1 year by the years 
 and the months reduced to the fraction of a year. 
 
 EXAMPLES. 
 
 1. What is the interest of $119,48 for 2 years 6 months, at 7 
 per cent ? 
 
 2. What is the interest of $250,60 for 1 year 9 months, at 6 
 per cent ? 
 
 3. Yfhat is the interest of $956 for 5 years 4 months, at 9 
 per cent ? 
 
 4. Wliat is the amount of $1575,20 for 3 years 8 months, at 
 7 ])CY cent ? 
 
 5. What is the amount of $5000 for 2 years 3 months, at 5^ 
 per cent ? 
 
 6. What is the interest of $1508,20 for 4 years 2 months, at 
 10 per cent ? 
 
 7. What is the interest of $75 for 6 years 10 months at 12^- 
 per cent ? 
 
 8. Wliat is the amount of $125 for 5 years 6 months, at 4|- 
 per cent ? 
 
 CASE II. 
 
 221. To find the interest on a given principal for any rate 
 and time. 
 
 1. What is the interest of $1752,96 at 6 per cent, for 2 years 
 4 months and 29 days ? 
 
 Analysis. — The interest for 1 year is the product of the principal 
 multiplied by the 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 2 times the interest for 1 year : the 
 interest for 4 monlht-, 4 times the interest for 1 month ; and the 
 nitere.st for 29 days, 29 times the interest for 1 day. 
 
 221. How do you find the interest for any time and rate "i How do you 
 fiud the interest for years, months, and days by the second method T 
 
222 PEKOENTAGK. 
 
 $1752;96 OPERATION. 
 
 .06 
 
 12)105,1776 int. for 1 2/r. Sl05,l776 X2 =$210,3552 2yr. 
 30)8.7 (i48 int. for Imo. 8,7648 X4 = 35,0592 4?no. 
 
 ,29216 int. for Ida. 0,29216X29= 8.47264 29c/a 
 
 Total interest, ©253,88704 
 Hence, we have the following 
 EuLE. — I. Find the interest for 1 year : 
 
 II. Divide t/iis interest by 12, and the quotient will be the 
 interest for 1 month : 
 
 III. Divide the interest for 1 month by 30, and the quotient 
 will be the interest for 1 day. 
 
 IV. Multiply the interest for 1 year by the number of years, 
 the interest for 1 month by the number of months, and the inte- 
 rest far 1 day by the number of days, and the sum of the products 
 will be the required interest. 
 
 Note. — This method of computing interest for day.s, is the one in 
 general use. It supposes tlic month to contain 30 days, or the year 
 360 days; whereas, it actually contains 365 days. 
 
 To find the exact interest for 1 day, we must regard the month as 
 containing 3j;_5 days = 30-5- days; and this is the number by which 
 the interest for one month should be divided, in order to find the exact 
 interest for one day. As the divisor, commonly used, is too small, the 
 interest found for 1 day, is a trifle too large. If entire accurac-- s 
 required, the interest for the days must be diminished by its -g-^-j 
 part = Tf 3- part. 
 
 2d method. 
 
 222. There is another rule resulting from the last analysis which 
 is regarded as the best general method of computing intei'cst. 
 
 Rule. — I. Find the interest for 1 year and divide it by 12: 
 the quotient will be the interest for 1 month. 
 
 II. Multijily 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 30 days, any number of days 
 is rulutcd to decimals of a month by dividing the number uf days by 3. 
 
INTERESl 223 
 
 
 EXAMPLES. 
 
 1. What is the interest of $655, for 3 years 7 months and 
 
 13 days, at 7 per cent ? 
 
 
 
 OPERATION. 
 
 Syrs. = SGmos. 
 
 $655 
 
 7vios. 
 
 .07 
 
 13da. = A^mos. 
 
 12)45.85 int. for 1 year. 
 
 Time — 43 Aminos. 
 
 3.82083 + int. for 1 month. 
 
 
 43.4^ time in months. 
 
 
 127361 
 
 
 1528332 
 
 
 1146249 
 
 
 1528332 
 
 165,951383 Ans. 
 
 2. What is the interest of $358,50, for 1 year 8 months and 
 6 days, at 7 per cent ? 
 
 3. What is tlie interest of $1461,75, for 4 years 9 months 
 and 15 days, at 6 per cent ? 
 
 4. What is the interest of $1200, for 2 years 4 months and 
 12 days, at 7i per cent ? 
 
 5. What is the interest of $4500, for 9 months and 20 days, 
 at 5 per cent ? 
 
 6. What is the interest of $156,25, for 10 months and 18 
 days, at 8 per cent ? 
 
 7. What is the interest of 6640, for 3 years 2 months and 
 9 days, at 6^ per cent ? 
 
 8. What is the interest of $276,50, for 11 months and 21 
 days, at 10 per cent ? 
 
 9. What is the amount of $378,42, for 1 year 5 months and 
 3 days, at 7 per cent ? 
 
 10. What is the amount of $1250, for 7 months and 21 days, 
 a.t 101^ per cent ? 
 
 11. What is the interest of $6500, for 2 months and 10 days, 
 at 9^ per cent ? 
 
 12. What is the interest of $70,50, for 10 years and 10 
 months, at 5^ per cent ? 
 
22-i PEEOENTAGE. 
 
 13. What is the amount of $45, for 12 years and 27 days, 
 at 6j per cent ? 
 
 14. What will $100 amount to in 15 years and 6 months, if 
 put at interest at 4 jier cent ? 
 
 15. How much will $475,50 gain in 5 years 9 months and 
 24 days, at 8 per cent ? 
 
 IG. What will be the interest of $45G0, for 14 months and 
 19 days, at 7 per cent ? 
 
 17. Wha^ will $128,371 amount to in 10 months and 27 
 days, at G per cent ? 
 
 18. What is the interest of $264,52, for 2 years 8 months 
 and 14 days, at 6 per cent ? 
 
 19. What is the amount of $76,50, for 1 year 9 months and 
 12 days, at 6 per cent? 
 
 20. What Avill be the interest for 3 years 3 months and 15 
 days, of $241,60, at 7 per cent? 
 
 21. What is the interest of $5600, for 30 days, at 7 per cent? 
 
 22. What will $8450 amount to in GO days, at 10 per cent ? 
 
 23. What is the interest of $4000, for 1 month and 6 days, 
 at 9 per cent ? 
 
 24. What will be the amount of $87,60, from Sept. 9th, 
 1852, to Oct. 10th, 1853, at 6i per cent? 
 
 25. What will be due on a note of $126,75, given July 8t;h, 
 1854, and payable April 25th, 1858, at 7 per cent ? 
 
 26. What is the interest of ^350, from Jan. 1st, 1856, to 15tli 
 of Sept. next following, at 5-]- per cent ? 
 
 27. Gave a note of $560,40, March 14th, 1855, on interest, 
 after 90 days : what interest was due Dec. 1st, 1856, at 10 per 
 cent ? 
 
 28. Find the interest of $1256, for 11 months and 9 days, 
 at 6 per cent. 
 
 29. What is the amount of $745,40 at 5 per cent interest, 
 being reckoned from the 5lh day of the 10th month of 1850, 
 to the 10th day of tlie 5th month of 1854 ? 
 
 30. Sept. 10th, James Trusty borrowed of Peter Credit 
 $250, and IMarch 4th, 1853, he borrowed $500 more, agreeing 
 
rNTEKEST. 225 
 
 to pay 7 per cent interest on the whole ; what was the amount 
 of his indebtedness Jan. 1st, 1854 ? 
 
 31. Ordered dry goods of A. T. Stewart & Co., at different 
 times, to the following amounts, viz., Jan. 1st, 1854, $254 ; 
 March 15th, 1854, $154,G0 ; April 20th, 1854, $424,25 ; and 
 June 3d, 1854, $75,50. I bought on time at 6 per cent in- 
 terest : what was the whole amount of my indebtedness on the 
 tirst day of Sept. following ? 
 
 32. If I borrow $475,75 of a friend at 7 per cent, what will 
 I owe him at the end of 8 months and a half? 
 
 33. In settling with a merchant, I gave my note for $127,28, 
 due in 1 year 9 months, at 6 per cent : what must be paid when 
 the note falls due ? 
 
 34. A person buying a piece of property for $4500, agreed 
 to pay for it in thi'ee equal annual instalments, with interest at 
 6-^ per. cent : what was the entire amount of money he paid ? 
 
 35. A mechanic hired a journeyman for 9 months at $40 a 
 mouth, to be paid monthly ; at the end of the time he had 
 paid nothing; he then settled, allowed interest at 7 per cent, 
 and gave his note, on interest, due in 1. year 4 months and 15 
 days : what will he pay when his note falls due ? 
 
 36. A person owning a part of a woollen factory, sold his 
 share for $9000. The terms were, one-third cash, on delivery 
 of the property, one-half of the remainder in 6 months, and the 
 rest in 12 months, M'ith 7^ per cent interest: what was the 
 whole amount paid ? 
 
 NOTES. 
 
 $382,50 Chicago, January, 1st, 1856. 
 
 1. For value received I promise to pay on the 10th day of 
 June next, to C. Hanford or order, the sum of three hundred 
 and eighty-two dollars and fifty cents, with interest fi'om date, 
 at 7 per cent. 
 
 $612 Baltimore, January 1st, 1856. 
 
 2. For value received I promise to pay on the 4th of July, 
 1858, to Wm. Johnson or order, six hundred and twelve dollars 
 with interest at 6 per cent from the 1st of March, 1856. 
 
 John Liberal. 
 
226 PERCENTAGE. 
 
 $3120 Charleston, July 3d, 1855. 
 
 3. Six months aftei* date, I promise to pay to C. Jones or 
 order, three thousand one hundred and twenty dollars with 
 interest from the 1st of January last, at 7 per cent. 
 
 Joseph Sprinps. 
 
 $786,50 New York, July 7th, 1851. 
 
 4. Twelve months after date, I promise to pay to Smith & 
 Baker or order, seven hundi-ed and eighty-six -j^^ dollars for 
 value received with interest from December 3d, 1851, at 8 per 
 cent. "" Silas Day. 
 
 $4560,72 Cincinnati, March 10th, 1856. 
 
 5. Nine months after date, for value received, I promise to 
 pay to Redfield, Wright & Co. or order, four thousand five; 
 hundred and sixty -J-^^ dollars with interest after 6 monthi?, at 
 7 per cent. Frederick Stillman. 
 
 $1854,83 Boston, July 17th, 1856. 
 
 6. Eleven months after dale, for value received, we promise 
 to pay to the order of Fondy, Burnap & Co., one thousand eight 
 hundred and fifty-four y^q dollars with interest from May 13th, 
 1856, at 6 per cent. Palmer <C- Blake. 
 
 POUNDS, SUILLINGS AND PENCE. 
 
 223. To find the interest, when the principal is pounds, shil- 
 lings and pence. 
 
 I. Reduce the shillinr/s and 2:>ence to the decimal of a pound 
 (Art. 164). 
 
 II. Then Jind the interest as though the sioii were dollars and 
 cents ; after which reduce the decimal part of the answer to shil 
 linys and pence (Art. 165). 
 
 223. How do you find tho interest when the principal is pounds, shil- 
 lings and pence ' 
 
ESTTKUEST. . 227 
 
 EXAMPLES. 
 
 1. What is the interest, at 6 per cent, of £27 15s. 9d. for 2 
 
 years ? 
 
 £27 15s. 9d. z= £27.7875. 
 
 £27.7875 x .06 x 2 = £3.3345 interest. 
 
 £3.3345 r= £3 6s: S\d. Ans. 
 
 2. "What is the interest on £203 18s. Gd., at 6 per cent, for 
 3 years 8 months 16 days ? 
 
 S.^What is the interest of £215 13s. 8:/., at 6 per cent, for 3 
 years 6 months and 9 days ? 
 
 4. Wliat is the interest of £1543 10s. 6c/., for 2 years and a 
 half, at 4 per cent ? 
 
 5. What is the amount of £1047 3s., for \p\ 4mo. 15da., at 
 6 per cent. ? 
 
 6. What is the interest on £511 Is. 4c/., at 6 per cent per 
 annum, for 6yr. G))io. ? 
 
 7. What is the interest on £161 15s. Sd., at 6 per cent, for 
 Smo. loda. ? 
 
 PROBLEMS ]N INTEREST. 
 
 224, In every question of interest there are four parts : 1st. 
 Principal ; 2d. Rate ; 3d. Time ; ^nd 4th. Interest. If any 
 three of these parts are known, the 4th can be found. 
 
 1. At what rate per cent must $325 be put at interest for 
 1 year and 6 months, to produce an interest of $34,125 ? 
 
 Analysis. — The principal, multiplied by opekation. 
 
 the rate expressed decimally, multiplied by Principal, 
 the time in years, is equal to the interest Rate, 12 
 
 (Art. 221) ; and when the time is expressed Time, Interest, 
 
 in montlis and decimals of a month, the same in months, 
 product is equal to ] 2 times the interest (Art. 221). Hence, 
 
 224. How many parts are there m every question of interest] How 
 many of these must be known before the remainder can be found ! How do 
 you find the interest when you know the Principal, Rate, and Time 1 How 
 do you find the Principal when you know the interest, rate, and time 1 How 
 do vo'i apply the formula to any ciuie ' 
 
223 PEEOKNTAQE. 
 
 I. When the time ts in months, the product of the principal 
 rate and time will be equal (o 12 times the interest. 
 
 II. When tioo of these parts and the interest are given, 12 times 
 the interest divided hy the product of the given parts will he equal 
 to the other part. 
 
 Note — Let this formula be ^Yritten on the black board, or slate, 
 and all the examples worked by it. 
 
 To apply the rule to the above example, operation. 
 
 place $325 for the principal, x for the rate, $325 
 
 18 (months) for the time, and $34,125 for ^. 
 
 the interest. Cancelling and dividing, we ^^ 
 
 2 
 
 n 
 
 $34,125 
 
 find a: = .07 : or, the rate is 7 per cent. o - i .-,- c 
 
 325x3 
 Ans. 7 per cent. 
 
 EXAMPLES. 
 
 1. What principal, at G per cent, will in 9 months give an 
 interest of $178,9552 ? 
 
 2. The interest for 2 years and 6 months, at 7 per cent, is 
 $76,9G5: what is the principal? 
 
 3. What sum must be invested, at 6 per cent, for 10 months 
 and 15 days, to produce an interest of §327,3249 ? 
 
 4. If my salary is §1500 a year, what sum invested at 5 per 
 cent, will pay it ? 
 
 ■ 5.. What sum put at interest for 4 years and 3 months, at 
 7 per cent, will gain $283,3914? 
 
 6. The interest of $2100 for 3 years 1 month and 18 days is 
 $4G0,G0 : what is the rate per cent? 
 
 7. A man invests $5420 in Eailroad stock, and receives a 
 semi-annual dividend of $244,17 : 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 ? 
 
 10. If I build a hotel at a cost of $5GO0O, and rent it for 
 67000 a year, what per cent do I receive for the investment ? 
 
INTKKEST. 229 
 
 11. The interest on $1119,48, at 7 per cent, is $195,900 : 
 what is the time ? 
 
 12. A man received $47,25 for the use of 11750 ; the rate 
 of interest being 9 per cent : what was the time ? 
 
 13. How long will it take ^7500 to amount to $7850, at 
 3J per cent per annum ? 
 
 14. How long will it take $500 to double itself, at 6 per cent, 
 simple interest ? 
 
 15. Wishing to commence business, a friend loaned me $3720, 
 at 6^ per cent, which I kept until it amounted to $5009,60 : 
 how long did I retain it ? 
 
 16. I borrowed $700 of my neighbor for 1 year and 8 months, 
 at 6 per cent ; at the end of the time he borrowed of me ^750 
 how long must lie keep it to cancel the amount of inte- 
 rest I owed him? 
 
 PARTIAL PAYMENTS. 
 
 225. We shall now give the rule established in New York 
 (See Johnson's Chancery Reports, Vol. I., page 17,) for com- 
 puting the interest on a bond or note, when partial payments 
 have been made. The same rule is also adoiJted in Massachu- 
 setts, and in most of the other states. 
 
 Rule. — I, Compiite the interest on the principal to the time 
 of tlic Jirst payment, and if the payment exceed this interest, add 
 the interest to the principal, and from, the sum subtract the pay- 
 ment : the remainder forms a new principal. 
 
 11. But if the payment is less than the interest, take no notice 
 of it until other payments are 7nade, tvhich in all, shall exceed 
 the interest computed to the time of the last payment : then add 
 the interest, so computed, to the principal^ and from the sum 
 subtract the sum of the payments : the remainder will form a 
 new princip>al on which interest is to be computed as before. 
 
 225. What is the rule for partial payments 1 
 
230 PERCENTAGE. 
 
 EXAMPLES. 
 
 $349,998 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 
 and eight mills with interest, at 6 per cent. 
 
 James Paywell 
 
 On this note were endorsed the following payments : 
 
 Dec. 25th, 1846, received $49,998 
 July 10th, 1847, " 4,998 
 Sept. Ist., 1848, " 15,008 
 June 14th, 1849, " 99,999 
 What was due April 15th, 1850 ? 
 Principal on int. from May 1st, 1846, - - - - $349,998 
 Interest to Dec. 25th, 1846, time of first pay- 
 ment, 7 months 24 days, 13,649 -f- 
 
 Amount, - - - - $363,647 •-{- 
 
 Payment Dec. 25th, exceeding interest then due § 49,998 
 
 Remainder for a new principal $313,649 
 
 Interest^of ^313,649 from Dec. 25th, 1846, to 
 
 June 14th, 1849, 2 years 5 months 19 days §46,472 + 
 
 Amount, - - - §360,121 
 
 Payment, July 10th, 1847, less~» 
 
 than interest then due, - ) ' 
 
 Payment, Sept. 1st, 1848, - - - 15,008_ 
 
 T^^^^^""^' ". I "i^20,006" 
 
 less than interest then due > 
 
 Payment, June 14lh, 1849, - - 99,999 
 
 Their sum exceeds the interest then due - - $120,005 
 Remainder for a new principal, June 14th, 
 
 1849, 240,116 
 
 Interest of $210,116 from June 14lh, 1849, 
 
 to April 15th, 1850, 10 months 1 day, $ 12.015 
 
 ToliU due, April 15th, 1850, - - - ¥252,T6r-f 
 
PARTIAL PAYMENTS. 281 
 
 $6478,84: New Haven, Feb. 6th, 1850. 
 
 2. Foi' value received I promise to pay William Jenks or 
 order, six thousand four hundred and seventy-eight dollars and 
 eighty-four cents with interest from date, at 6 per cent. 
 
 John Stewart. 
 
 On this note were endorsed 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 which 
 were endorsed the following payments, viz. : — Nov. loth, 1853, 
 $416,98; May 10th, 1854, $152: what was due March 1st, 
 1855, the interest being 6 per cent ? 
 
 $8'J74,56 New 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 are endorsed the following payments : 
 
 Feb. 16th, 1855, received $1875,40 
 Sept. 15th, 1856, " 3841,26 
 
 Nov. 11th, 1857, " 1809,10 
 
 June 9th, 1858, " 2421,04. 
 
 What will be due July 1st, 1858 ? 
 
 S345,50 Buffalo, Nov. 1st, 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. Ja^in Dor. 
 
 On this note are the following endorsements : 
 
232 
 
 
 PEKCENTAGB. 
 
 
 Juno 20tli, 
 
 1853, 
 
 received 
 
 $75 
 
 Jan. 12th, 
 
 1854, 
 
 ii 
 
 10 
 
 March 3d, 
 
 1855, 
 
 a 
 
 15,50 
 
 Dec. 13th, 
 
 1856, 
 
 u 
 
 52,75 
 
 Oct. 14th, 
 
 1857, 
 
 a 
 
 106,75 
 
 "What will there be due Feb. 4th, 1858 ? 
 
 $450 Mobile, Oct. 19th, 1850. 
 
 6. For value received we jointly and severally promise to 
 pay Jones, Mead & Co. or order, four hundred and fifty dollars 
 on demand with interest, at 8 per cent. Manning c5 Bros. 
 The following endorsements were made on this note : 
 Sept. 25, 1851, received $85,60 ; July 10, 1852, received 
 S20 ; June 6, 1853, received S150,45 ; Dec. 28, 1854, received 
 §25,12J ; May 5, 1855, received $169 : what was due Oct. 
 18, 1857 ? 
 
 LEGAL INTEREST. 
 
 226. Legal Interest is the interest which the law permits a 
 person to receive for money which he loans, and the laws do 
 not favor the taking of a higher rate. In most of the states 
 the rate is fixed at 6 per cent ; in New York, South Carolina 
 and Georgia, it is 7 ; and in some of the states the rate is fixed 
 as high as 10 per cent. 
 
 COMPOUND INTEREST. 
 
 227. Compound Interest is when the interest on a principal, 
 computed to a given time, is added to the principal, and the 
 interest then computed on this amount, as on a new principal. 
 
 NoTK. — The laws do not favor the payment of compound interest. 
 Ill most, of the states, an agreement to pay compound inlorc.-st, could 
 uot be enforced. 
 
 226. What i.s legal interest 1 
 
 227. What is compound interest T How do you computo it ' 
 
EXAMPLES. 233 
 
 Rule. — Compute the interest to the time at which it becomes 
 due ; then add it to the j^rincijxd and compute the interest on the 
 amount as on a new j^^'incijKil : add the interest again to the 
 jyrincipal and compute the interest as hefore ; do the same for all 
 the times at which payments of interest become due ; from the 
 last result subtract the 2^^'irici2}al, and the remainder will be the 
 comp>ound interest. 
 
 EXAMPLES. 
 
 1. What will be tlie compound interest of $3750 for 4 years, 
 at 7 i^er cent ? 
 
 $3750,000 principal for 1st year. 
 
 $3750 X .07 = 2G2,500 interest for 1st year. 
 
 4012,500 principal for 2tl " 
 
 64012,50 X .07 = 280,875 interest for 2d « 
 
 42y3,'375 principal for 3d " 
 
 $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 + amount at 4 years. 
 1st principal 3750,000 
 Amount of intei-est $1105,484 + 
 
 2. What will be the compound interest of 1175 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 8300, for three 
 years, at 6 per cent ? 
 
 5. What will be the compound interest of $590,74, at 6 per 
 cent, for 2 years ? 
 
 6. What will be the compound interest of $500, for 2 year, 
 nt 8 per cent? 
 
 7. What will be the compound interest of $3758,50, for 3 
 years, at 7 per cent ? 
 
 8. What will be the compound interest of $95037,50, for 7 
 
 years, at per cent ? 
 
 11 
 
234 
 
 COMPOUND INTEREST. 
 
 Note. — The operation is rendered much shorter and easier, by 
 taking the amount of 1 dollar for any time and rate given in the 
 following table, and multiplying it by the given principal ; the pro- 
 duct will be the required amount, from which subtract the given 
 principal, and the result will be the compound interest. 
 
 TABLE — Showing the amount of Si or £1, compound interest, from 1 
 year to 20 years, and at the rate of 3, 4, 5, 6, and 7 per cent. 
 
 Years 
 
 3 per cent. 
 
 4 per cent. 
 
 5 per cent. 
 
 per cent. 
 
 7 per cent 
 
 Years. 
 
 1 
 
 1.03000 
 
 1.04000 
 
 1.05000 
 
 1.06000 
 
 1.07000 
 
 1 
 
 O 
 
 1.06090 
 
 1.18160 
 
 1.102.50 
 
 1.12360 
 
 1.14490 
 
 2 
 
 3 
 
 1.09272 
 
 1.12486 
 
 1.15762 
 
 1.19101 
 
 1.22.504 
 
 3 
 
 4 
 
 1.12.550 
 
 1.16985 
 
 1.215.50 
 
 1.26247 
 
 1.31079 
 
 4 
 
 5 
 
 1.15927 
 
 1.21665 
 
 1.27628 
 
 1.33822 
 
 1.40255 
 
 5 
 
 6 
 
 1.19405 
 
 1.26531 
 
 1.34009 
 
 1.41851 
 
 1.50073 
 
 6 
 
 7 
 
 1 22987 
 
 1.31593 
 
 1.40710 
 
 1.. 50363 
 
 1.60578 
 
 7 
 
 8 
 
 1.26677 
 
 1.36856 
 
 1.47745 
 
 1.59384 
 
 1.71818 
 
 8 
 
 9 
 
 1.30477 
 
 1.42331 
 
 1.55132 
 
 1.68947 
 
 1.83S45 
 
 9 
 
 10 
 
 1.34391 
 
 1.48028 
 
 1.62SS9 
 
 1.79081 
 
 1.96715 
 
 10 
 
 II 
 
 1.38423 
 
 1.53945 
 
 1.71033 
 
 1.89829 
 
 2.10485 
 
 11 
 
 12 
 
 1.42.576 
 
 1.60103 
 
 1.79585 
 
 2.01219 
 
 2.25219 
 
 12 
 
 13 
 
 146853 
 
 I 66507 
 
 1.88564 
 
 2 13292 
 
 2.40984 
 
 13 
 
 14 
 
 1.51258 
 
 1.73167 
 
 1.97993 
 
 2.20090 
 
 2.57853 
 
 14 
 
 15 
 
 1.5.5796 
 
 1.80094 
 
 2.07892 
 
 2.39055 
 
 2 75903 
 
 15 
 
 16 
 
 1.60470 
 
 1.87293 
 
 2.18287 
 
 2. ,54035 
 
 2.95216 
 
 16 
 
 17 
 
 1.65284 
 
 1.94790 
 
 2 29201 
 
 2.69277 
 
 3.1,5881 
 
 17 
 
 18 
 
 1.70243 
 
 2.02581 
 
 2.40061 
 
 2 85433 
 
 337993 
 
 18 
 
 19 
 
 1.75350 
 
 2.10684 
 
 2.52695 
 
 3.02.559 
 
 3.61652 
 
 19 
 
 20 
 
 1.80611 
 
 2.19112 
 
 2.65329 
 
 3 20713 3 86968 
 
 20 
 
 Note. — When there are months and days in the time, find tho 
 amount for the ycarSj and on this amount cast the interest for tho 
 months and days : this, added to the last amount, v.-ill be the re- 
 quired amomit for the whole time. 
 
 9. What will be the compound interest of §75439,75, for 4 
 years? at 4^ per cent ? 
 
 10. What will SG50 amount to in 12 year.?, at 3 per cent, 
 compound intercut ? 
 
 1 1. The slave population in the United States and territories 
 in 1850, was 3204318; if the increase is 5 percent a year, 
 what will be the entire slave population in 1870 ? 
 
 12. Wliat would be the compound interest of §540,50, at 6 
 per cent, for 3 years G months and 15 days? (Soe Note). 
 
DISCOUNT. 235 
 
 13. What will $75 amount to in 10 years 4 months and 21 
 days, at 7 j^er cent, compound intei'est ? 
 
 14. What will be the compound interest of $200, for 1 year 
 7 months and 9 days, at 5 per cent ? 
 
 15. A gives B a note of $375,40, April 20, 1854, payable 
 Oct. 20, 185G : the interest is to be added at the end of every 6 
 months, and compounded at 7 per cent : what will be the 
 amount of the payment when due ? 
 
 DISCOUNT. 
 
 228. Discount is an allowance made for the payment of 
 money before it is due. 
 
 The Face of a note is the amount named in the note. 
 
 229. The PRESENT value of a note is sucli a sum as being 
 put at interest until the note becomes due, would increase to an 
 amount equal to the face of the note. 
 
 230. The DISCOUNT on a note is the difference between the 
 face of the note and its present value. 
 
 1. I give Mr. Wilson my note for SlOG, payable in 1 year: 
 what is the present value of the note, if the interest is 6 per cent ? 
 what is the discount ? 
 
 PROPORTION. 
 
 1 + its interest : $1 : : given sum : its present value. 
 
 OPERATION. 
 
 Analysis. — Since 1 dolJar in 1 year, SlOG -f- 1.06 = $100 
 at G per cent, will amount to 81,06. the 
 
 pre.'^ent value will be as many dollars as proof. 
 
 $LOG is contained times in the face of Int. SlOO lyr. = $ 6 
 
 the note; viz.. ?plOO; and the discount Principal = 100 
 
 will be SlOG — 100 = $6 : hence, Amount SlOG 
 
 Discount 6 
 
 228. What is discount 1 What is the face of a note 1 
 
 229. What is present value 1 
 
 230. What is the discount ? 
 
236 PERCENTAGE. 
 
 Rule. — Divide the face of the note by 1 dollar plus the in. 
 terest of 1 dollar for the given time, and the quotient will be the 
 present value. 
 
 Note. — When payments are to be made at different i\me,Sy find the 
 present value of the sums separately, and their sum will be the present 
 value of the note. 
 
 EXAMPLES. 
 
 Ul. What is the present value of a note of S615, due 1 year 
 4 months hence, at 7 per cent ? 
 
 2. "Wliat is the present value of $202,58, clue in 1 year 7 
 months and 18 days, at G per cent ? 
 
 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 $5160 is payable Feb. 4th, 1857, what is its 
 value Sept. lOth, 185G, 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, paya- 
 ble in 1 year 2 months and 20 days, at 7 per cent ? 
 
 7. Bouglit property to the amount of $5000, and agreed to 
 pay for it in four equal installments ; the first in 3 months ; the 
 second, in 6 months ; the third, in 9 montlis ; and the fourth, in 
 1 year : how much cash will discharge the debt ? Interest 6 
 per cent. ? 
 
 ~ 8. If I loan a sum of money on interest at G]- per cent , for 
 7 months and 15 days, and at the end of the time receive 
 $4987,50 for principal and interest, how nnicli did I loan ? 
 ;>9. A held a note of 81400 against B, payable Aug. 1st, 
 185G ; B paid it May 15th, 1856 : what sum did he pay, the 
 inlerest being 7 per cent. ? 
 
 10. A Hour merchant bought for cash 300 barrels of flour, for 
 ^10,50 per barrel ; he sold it the same day for Si 2 a barrel, 
 and took a note at 3 moijths : what was the cash value of tlie 
 Bale, and wliat his gain il'the interest is reckoned at 7 per cent. ? 
 
 11. A man purcliased a house and lot for ^lOOOO, on tlio 
 following terms : $5000 in cash, $2500 in 3 mouths, and (lie 
 
BANKING. 237 
 
 balance in G months : what was the cash value of the property, 
 interest being reckoned at 6 per cent. ? 
 
 12. A provision dealer bought for cash 78 firkins of butter, 
 of 8G pounds each, at 25 cents per pound. He sold it imme- 
 diately at 25-^ cents per pound, and took a note for tlie amount 
 at 4 months : did he gain or lose, and how much ; interest be- 
 ing reckoned at 8 per cent. ? 
 
 13. Which is the more advantageous, to buy sugar at 7-i 
 cents a pound, on 4 months, or at 8 cents a pound on G months, 
 at 6 per cent, interest ? 
 
 /114. Bought land at $10 an acre : what must I ask per acre if I 
 abate 10 per ct., and still make 20 per ct on the purchase monc}- ? 
 15. A merchant owes three notes, viz., $1000 pa3able Aug. 
 1st, 1855 ; -S-oOO. payable Oct. lOtb. 1855. and SOOO payal)lc' 
 Nov. 1st, 1855: what is-thecash value of the three notes, July 
 1st, 1855, reckoning interest at G per cent. ; and what is the 
 difference between that value and their amounts at the times 
 Avhen they fall due, if interest be reckoned from July 1st. 
 
 BANKING. 
 
 231. Banks are corporations created by law for the purpose 
 of receiving deposits, loaning money, and furnishing a paper 
 ciiTulation I'epresented by specie. 
 
 The notes made by a bank circulate as money, because they 
 are payable in specie on presentation at the bank. They are 
 called bank notes, or bank bills. 
 
 The note of an individual, or as it is generally called, a 
 promissoiy note, or note of hand, is a positive engagement, in 
 writing, to pay a given sum, either on demand or at a specified 
 time. 
 
 FORMS OF NOTES. 
 
 Negotiable Note. 
 
 $25,50. Pi-ovidence, May 1, 185G. 
 
 For value received I promise to pay on demand, to 
 Abel Bond, or order, twenty-five dollars and fifty cents. 
 
 Reuben Holmes. 
 
233 PERCENTAGE. 
 
 Note Payable to Bearer. 
 
 No. 2. 
 
 $875,39. 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 
 dollars and thirty-nine cents. Pierce Penny. 
 
 Note by two Person fi. 
 No. 3. 
 
 $659,27 Bufialo, June 2, 1856. 
 
 For value received we, jointly and severallj^, promise 
 to pay to Richard Ricks, or order, on demand, six hundred and 
 fifty-nine dollars and twenty-seven cents. Enos Allan. 
 
 John Allan. 
 
 Note Payable at a Bank. 
 No. 4. 
 
 $20,25. 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, 
 twenty dollars and twenty-five cents, for value received. 
 
 Jesse Stokes. 
 
 remarks relating to notes. 
 
 1. The person who si^nis a note, is called the drawer or malrr of 
 the note; thus, Reuben Holmes is the drawer of note No. 1. 
 
 2. The person who has the rightful pos.scssion of a note, is called 
 ihe 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 mndo payable, writes liis name on the back of 
 it, he is said to endorse the note, and he is called the endorser; and 
 when the note becomes due, the holder nnist first demand payment 
 of till' maker, Reuben Holmes, and if lie declines paying it, tlio 
 holder may then require payment of Abel Bond, the endorser. 
 
 231. Wliat arc Banks'! M'liy do (ho notes of l)nnks circulate at 
 
 nioni'V ! W'li.il are tlicy called ' W h^t is a promissory note! 
 
BANKING. 239 
 
 4. If the note is made payable to A B, or bearer, then the drawer 
 
 alone is responsible, and lie must pay to any person who holds the 
 note. 
 
 5. 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 re- 
 quired to do so, and the note will draw interest after the payment is 
 demanded. 
 
 6. AYhen a note, payable at a future day, becomes due, it will 
 draw interest, though no mention is made of interest. 
 
 7. In each of the States there is a rate of interest established by 
 law, which is called the legal interest, and when no rate is specified, 
 the note will always draw legal interest. If a rate hi"-Ler than 
 legal interest is named in the note, or agreed upon, the Cr<? Aer, in 
 most of the States, is not bound to pay the note. 
 
 8. If two persons jointly and severally give their not% (see No. 3). 
 it may be collected of either of them. 
 
 9. The words ''For value received," should be ex^rer,sed in every 
 note. 
 
 10. When a note is given, payable on a fixed dr./; and in a specific 
 article, as in wheat or rye, payment must be vjfiered at the spe- 
 cified time, and if it is not, the holder can d-imand the value in 
 money. 
 
 11. D.iYS OF GRACE are days allowed for (he payment of a note 
 after the expiration of the time named on its face. By mercantile 
 usage, a note does not legally fall due until 3 days after the expira- 
 tion of the time named on its face, unless the note specifies " ivithout 
 graceP For example, No. 2 would 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 Ycar'.s day or the 4th of July, the note must be paid the 
 day before ; that is, on the second day of grace. 
 
 12. 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 btismess paper. 2d. Notes made for the 
 purpose of borrowing money, which are called accommodation nutes, 
 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." 
 
240 PERCENTAGE. 
 
 BANK DISCOUNT. 
 
 232. Bank Discount is the charge made by a bank for the 
 payment of money on a note before it becomes due. 
 
 By the custom of banks, this discount is the interest on the 
 amount named in a note, calculated from the time the note is 
 discounted to the time when it falls due ; in which time the 
 three days of grace-are always included (see remark 11). The 
 interest on notes discounted at bank is always paiCi? in advance. 
 
 X?uLE. — Add 3 days to (he time which (he no(e has to ruHf 
 and (hen calculate (he in(ercst for (hat time, a( (he given ra(e. 
 
 EXAMPLES. 
 
 1. What is the bank discount on a note of $300, for 4 months, 
 at 6 per cent per annum ? 
 
 2. What is the bank discount on a note of $200, payable in 
 5 months, at 9 per cent ? 
 
 3. \Yhat is the bank discount and proceeds of a note of $500, 
 at G^ per cent., payable in Q\ montlis ? 
 
 4. What is the cash value of a note, payable at bank, of 
 $1255,38, payable in 4 montlis, at 7 per cent. ? 
 
 5. Wiiat would be the bank discount on a note of 8500, due 
 Aug. 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 
 Eold it the same day for $1,30 a bushel on a note of 4 months. 
 If I get this note discounted, at bank, at 7 per cent, wluit do I 
 gain or lose ? 
 
 7. What is the difference between tlie true and bank discount, 
 on $7000, payable in 7 montlis, at G per cent ? 
 
 8. What is the dilference between the true and bank discount, 
 of $10000, payable in 4i months, at 8 per cent? 
 
 9. January 1st, 1855, a note was given for -i^lOOO, at 5^ per 
 cent., to be paid May 1st, next following: what was its cash 
 value at bank ? 
 
 232. What is bank discount ? How is interest calculated by the custom 
 of banks 1 When Ik tho intrrc^^ paid ' Mow do you find tlic interest. ? 
 
NOTES OF kt:qciki!:d values. 241 
 
 NOTES OF KEQUIKED VALUES. 
 
 233. — !• For what sum must a note be drawn at 4 months 
 and 12 days, so that, when discounted at bank, at G per cent, 
 the amount received shall be $400 ? 
 
 Analysis. — If we find the interest on 1 dollcir for the given time, 
 and then subtract that interest from 1 dollar, the remainder will be 
 Ihe present value of 1 dollar, due at the expiration of that time. 
 Then, the number of times Avhich the present value of the note eon- 
 tains the present value of 1 dollar, will be the number of dollars for 
 which tlie note must be drawn ; hence, 
 
 Divide ihe 2^rescnt value of the note hy the 'present value of 
 1 dollar, reckoned for ihe same time and at the same rate of 
 interest, and the quotient xoill he the face of the note. 
 
 OPERATION. 
 
 Interest of $1 for the time 4mo. 12da. + 3da. of grace = Amo. loda, 
 = $0.0225, which taken from Si, gives present value of ^1 =5^0.9775 ; 
 then, $400 -f- .09775 = $409,207 + = face of note. 
 
 PROOF. 
 
 Interest on $409,207 for 4 months and 15 days, at 6 per cent, is 
 S9,207 +, which being taken from the face of the note, leaves $400, 
 its present bank value. 
 
 2. For what sum must a note be drawn at 7 per cent, paya- 
 ble in G months, so that when discounted at a bank it shall pro- 
 duce $285,95 ? 
 
 3. How large a note must I make at a bank, at G per cent, 
 payable in G months and 9 days, to produce $674,89 ? 
 
 4. A holds a note against B for $1500, to run G 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 discount of 5 per 
 cent in payment : how much must he receive back from the 
 hank ? 
 
 'Z'.n. liow do you find ihe face of a ncjte of a rctjiiinnl pronout value 1 
 
242 COMMISSION, 
 
 5. Marsh, Dean & Co., purchase of John Jones 380 barrels 
 uf flour, at -$9,12L a barrel, for -which they give him a bank 
 note for 90 days, at 6 per cent, for such a sum that if discounted 
 he shall receive the above price for his flour : what was the 
 face of the note ? 
 
 COMMISSION. 
 
 234. Commission is an allowance made to an a^ent for 
 buying or selling, and is reckoned at a certain rate per cent on 
 the money used in the purchase or sale. 
 
 The commission for the purchase or sale of goods, in the city 
 of New York, varies from 2i to 121 per cent, and, under some 
 circumstances, even higher rates are paid. For the sale of i*eal 
 estate, the rates are lower, varying from one-quarter to 2 per 
 cent. 
 
 EXAMPLES. 
 
 1. A commission merchant sold a lot of goods for which he 
 received $7540 ; he charged 2^ per cent commission : what 
 was the amount of his commission, and Iiow nmch must he pay 
 over ? 
 
 OPERATION. 
 
 Analysis. — We find the commission S7540 
 
 as in simple percentai^e, by multiply- .02i^ 
 
 ing by the decimal which expresses the $188,50 commission, 
 rate. The principal, diminished by the $7540 
 
 commission, gives the amount to be paid 1 88.50 
 
 over. $7351,50 amt. pd. over. 
 
 Rule. — Multiply the amount invested by the rate expressed 
 decimnlly, and the product will be the commission. 
 
 2. A commission merchant receives $1399,77, to be invested 
 in groceries ; he is to receive 3 per cent commission on the 
 amount of the purcliase : what amount is laid out in groceries ? 
 
 234. What is con)mis.sion ? Whrit arc the |;eneral rales of cominission 
 in the city of New York ! How do you find tlic amount of commission 
 on a given sum ! How do you find the amount to tic invcstrd c\clu.sivo 
 of the rijnimisbion ? 
 
EXAMPLES. 243 
 
 Analysis. — Since the broker receives operation. 
 
 3 per cent, it will require $1.03 to pur- 1.03) 1399,77 ($1359 
 chase 1 dollar's worth of stock ; hence, 
 
 there will be as many dollars Avorth purchased as $1.03 is contained 
 times in $1399,77; that is, $1359. 
 
 Rule. — Divide the amount to be expended by $1 plus the 
 commission, and the quotient ivill be the amount exclusive of 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 750 barrels of 
 flour at $9,75 a barrel : what was his commission at 2^ per 
 cent ? 
 
 5. Sold at auction 96 ho2;sliGads of su2:ar, each weif^hino; 
 Sicirt. and 50//^, at $6,50 per hundred : what was the auctioneer's 
 commission at 1|- per cent, and to how much Avas the owner 
 entitled ? 
 
 6. An agent purchased 2340 bushels of wheat at $1,75 a 
 bushel, and charged 2f per cent for buying, If per cent for 
 shipping, and the freight cost 2 per cent : what was his com- 
 mission, and what did tlie wheat cost the owner ? 
 
 7. A town collector receives 4i per cent for collecting a tax 
 of 82564,250 : what was the amount of his percentage ? 
 
 8. A bank fails, and has in circulation bills to the amount of 
 $267581. It can pay 9i per cent : how much money is there 
 on hand ? 
 
 9. I paid an attorney 6|- per cent for collecting a debt of 
 §7320,25 : how much did I receive ? 
 
 10. My commission merchant sells goods to the amount of 
 Si 000, on which I allow him 5 per cent, but as he pays over 
 the money before it comes due, I allow him li- per cent, mo e : 
 how much am I to receive ? 
 
 11. I forward $2608,625 to a commission merchant in 
 Chicago, requesting him to purchase a quantity of corn ; h^ is 
 to receive 2^ per cent, on the purchase : what does his com 
 
24:4 COMMISSION. 
 
 mission amount to, and how much corn can he buy -with the 
 remainder, at 56 cents a bushel? 
 
 12. I am obliged to sell S2640 in bills on the bank of Dela- 
 ware, upon -which there is a discount of 2| per cent : how much 
 bankable money will I receive after deducting the brokerage, 
 which is ^ per cent ? 
 
 13. My agent at Havana purchased for me a quantity of 
 sugar at 61 cents a pound, for which I allow him a commission 
 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 ? 
 
 14. A dairyman sent an agent 3476 pounds of cheese, and 
 allows him 3^ per cent for selling it : how much would he receive 
 after deducting the commission, if it were sold for 122- cents 
 per pound ? 
 
 15. A factor receives $708,75, 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 ? 
 
 IP A person having $1500 in bills of the State Bank of 
 Indiana, upon which there is a discount of 2^ per cent, and 
 $1000 of the Bank of Maryland, upon which thei^e is a dis- 
 count of 31 per cent : what will be the loss in changing the 
 amount into current money ? 
 
 *17. I sent $2204 to a ft-iend, to ])urcliase for me u number 
 of shares in the Illinois Central Railroad ; after deducting his 
 commission of '{ per cent., how many shares can he buy at 
 $109,3735 a share ? 
 
 18. A gentleman sent his son §56448,90, to invest in govern- 
 ment C\\ud< ; after deducting the brokerage of f of 2 per cent. 
 
 ■f hat sum did he invest ? 
 
 * Sec Art. 240. Paye 246. 
 
STOCKS AND BKOKEKAGE. 245 
 
 STOCKS AND BROKERAGE. 
 
 235. A Corporation is a collection of i^ersons authorized 
 i)j law to do business together. The law Avhich defines their 
 rights and powers is called a Charter. 
 
 236. Capital, or Stock, is the money paid in to carry on 
 the business of the corporation, and the individuals so con- 
 tributing are called Stockholders. This capital is divided into 
 equal parts called Shares, and the written evidences of owner- 
 ship are called Certificates. 
 
 237. When the United States Government, or any of the 
 States, borrows money, an acknowledgment is given to the 
 lender, in the form of a bond, bearing a fixed interest. Such 
 bonds ai-e called United States Stock, or State Stock. 
 
 238. The PAR VALUE of stock is the number of dollars 
 named in each share, generally 100. The market value is 
 what the stock brings per share when sold for cash. 
 
 If the market value is above the par value, the stock is said 
 to be at a premium ; but if the market value is below the par 
 value, it is said to be at a discount. 
 
 Let 1 = par value of 1 dollar ; then, 
 
 1 + premium = market value of 1 dollar, Avhen above 
 
 par : 
 1 — discount = market value of 1 dollar, when below 
 
 par. 
 
 239. A DIVIDEND is an interest or profit on a stock, and is 
 estimated at so much per cent on the par value of a share. 
 
 235. What is a corporation'! What is a charter! 
 
 236. What is capital or stock ! AMiat are shares ? 
 
 237. What are TInited States Stocks ' What are State Stocks 1 
 
 238. What is the par value of a stock 1 What is the market value 1 
 If the market is above the par value, what is said of the stock ? If it is 
 below, what is said of the stock ? What is the market value when above 
 pari What when below 1 
 
 239. What is a dividend '' On what i.-i it estimated ! 
 
24(3 pi<:kci':ntagj':. 
 
 240. Brokeuage is an allowance made to an agent wlio 
 buys or sells stocks, uncurrent money, or bills of exchange^ 
 and is generally reckoned at so much per cent on tlie pur value 
 of the stock. The brokerage, in the city of New York, is 
 generally one-fourth per cent on the par value of the stock. 
 
 241. To find the value of stock which is above or below par. 
 
 1. What is the value of $5000 of stock, reckoned at par, 
 when the stock is at a premium of 9 per cent ? 
 
 Analysis. — The question is, how much operation. 
 
 money ^Yill it take to purchase $1 of stock, at $5600 
 
 par. If the stock is above par. it will take 1 1.09 
 
 dollar plus the premium : if it is below par, it 50400 
 
 ■will take 1 dollar minus the premium ; hence, 5600 
 
 $6104.00 
 
 Multijoly the cost of the stock, at 'par, ly the price ofl dollar 
 of stock, expressed decimally, and the product will be the value. 
 
 Note. — When there is a charge for brokerage, it must be added to 
 the cost of 1 dollar of the stock. 
 
 EXAMPLES. 
 
 1. What is the cost of 56 shares of New York Central Rail- 
 road stock, at 51 per cent below par, the shares being §100 
 each, and the brokerage ^ per cent ? 
 
 2. I bought 36 shai'es of $100 each, in the Pennsylvania 
 Coal Company, at a discount of 12^^ per cent, and sold them at 
 a premium of 7 per cent, paying i per cent brokerage in each 
 case : how much did I make by the operation ? 
 
 3. Tlie par value of 257 shares of bank stock was $200 a 
 share : what is the present value of all the shares, tlie stock 
 being at a premium of $15 per cent ? 
 
 4. AYhat is the value of 120 shares of Exchan2;e Bank Stock 
 it being at a premium of 18|- per cent, and the par value be' 
 ing $150 a share ? 
 
 240. What is brokorafje ^ 
 
 241. How do you find the value of a stuck which is above par ! 
 
BTOCKS AND HKOKEEAGE. 2i7 
 
 f). What is the cost of 69 shares of Panama Railroad Stock, 
 it being at a discount of 8 per cent, and tlie par value being 
 
 $125, and the charge for brokerage f per cent ? 
 
 6. Gilbert & Co. buy for Mr. A. 200 shares of United States 
 Stock, at a premium of 6^ per cent, and charge i- per cent 
 brokerage : if the shares are $1000 each, how much money 
 does A pay for the stock ? 
 
 7. 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, reckoning the mterest of 
 money at 7 per cent ? 
 
 242. To Jind how much stock, at 2)a)- value, a given sum of 
 money tnll purchase, when the stock is at a 2'>remium or discount. 
 
 1. What value of stock, at par, can be purchased by $3045,38 
 if the stock is at a premium of 10 per cent, and -^ per cent is 
 charged for brokerage ? 
 
 Analysis. — Since the stock is at a premium of 10 per cent, and 
 the charge for brokerage is \ per 
 
 cent, it will require $1,105 to pur- operation. 
 
 chase $1 of stock, at par value; 1.105)3045.3S($2756 Ans. 
 hence, $3045.38 will purchase as 
 
 many dollars, at par, as $1,105 is contained times in $3045,38 ; viz., 
 $2756. 
 
 Rule. — Divide the given sum hy the cost of $1 of the stock, 
 expressed decimally, and the quotient u'ill denote the par value 
 of the stock p)urchased. 
 
 EXAMPLES. 
 
 1. A person wishes to invest $3000 in bank stock, which ia 
 at a discount of 15 per cent : what amount, at par value, can 
 he purchase ? 
 
 242. How do you find the sum which will purchase a given amount of 
 stock -at par value l 
 
2-18 PERCENTAGE. 
 
 2. How many shares of Galena and Chicago Raih'oad Stock 
 can be bought for $6384, at an advance of 14 per cent on tne 
 par value of $100 a share ? 
 
 S. When bank stock sells at a discount of 7\ per cent, Avhat 
 amount of stock, at par value will §3700 buy ? 
 
 4. A person has $7000, which he wishes to invest : what v.ill 
 it purchase in 5 per cent stocks, at a discount of 31 per cent, if 
 he pays i per cent brokerage ? 
 
 5. How much 6 per cent stock, at par, can be purchased for 
 $8700, at 81 per cent premium, i per cent being paid to the 
 broker ? 
 
 6. A person owning $12000 in government funds, desires to 
 purchase 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 lOi per cent : what amount of stock, at par value, 
 can he purchase, allowing the broker's charges for the pur- 
 chase to be ^ per cent ? 
 
 243. 1st. To find the rate of interest on an investment, when 
 the stock is above or below par. 2d. To find how much the 
 stock must be above or belotv par to irroduce a given rate. 
 
 1. What is the rate of interest on an investment in G per 
 cent stocks, when they are at a discount of 25 per cent ? 
 
 Analysis. — The interest on the slock is 
 computed on its par value, while the interest operation. 
 
 on the investment is computed on the amount •' 5 
 
 paid : hence, 1 dollar multiplied by the iu- 
 
 .06 
 
 terest. ou the stock will be equal to the cost x = .08 
 
 of ] dollar multiplied by the interest on the Ans. 8 per cent. 
 
 investment. 
 
 Rule. — I. Divide 1 dollar multiplied hj the interest on $1, 
 by (lie price of $1 of the slock, and the quotient tvill be the 
 rale. 
 
 243. How do you find the rate of interest on an investment when the 
 stock is above or below par 1 How do you find the value of the stock 
 whet the rite is given 1 
 
STOCKS AND BROKERAGE. 2 1:9 
 
 II. Divide 1 dollar multiplied by the interest on the stock by 
 the given rate, and the quotient will he the ijrice of $1 of the 
 stock. 
 
 EXAMPLES. 
 
 1. If I buy 7 pel :ent stock at 12-|- per cent discount, what 
 is the i-ate per cent jn the investment ? 
 
 2. At what rate of discount must I invest in 8 per cent stock 
 in order to yield me 10 per cent ? 
 
 3. The stock of the Erie Railroad is at G2^ per cent : if it 
 pays semi-annual dividends of 2i 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 7 per cent., are worth 87 per cent., and 
 the charge for brokerage is -| per cent. : what would be the 
 interest on an investment in these funds ? 
 
 5. If the par value of a stock is $100, and the interest 7 per 
 cent., what is the discount when an investment yields 12 per ct. ? 
 
 6. 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 Avill be the interest on an investment in this 
 stock ? 
 
 244. Which is the best investment? 
 
 1. I invest $1250 in State Stocks bearing an interest of 6 
 per cent, and a premium of 15 per cent. I invest the same 
 amount in State fives at 12 per cent discount : which will yield 
 the larger interest ? 
 
 OPERATION. 
 
 Analysis. — Find the rate of interest on 
 
 1.15 
 
 X 
 
 n 
 
 .06 
 
 the investment (Art. 243). ^_ q^^i + 
 
 244. How do you find which is the best investment ? 
 
250 FKKCENTAGE. 
 
 .88 
 Find the rate of interest on the second x 
 
 $1 
 .05 
 
 investment (Art. 243). The comparison 2 =.0568 4- 
 
 of these two rates will show which is the x = 5 ^-^ pr. ct. 
 
 more profitable stock. 
 
 2. "Which is the best investment, to buy sixes at par, or 
 sevens at 107 ? 
 
 3. Wliich 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 
 the same amount at G per cent, at 90, what will be the diifer- 
 ence of the proceeds of the investments at the end of 5 
 years ? 
 
 PROFIT AND LOSS. 
 
 245. Profit or Loss is a process by which merchants dis- 
 cover the amount gained or lost in the purchase and sale of 
 goods. It also instructs them how much to increase or diminish 
 the price of their goods, so as to make or lose so much per cent. 
 
 1. Bought 325 bushels of wheat at $1,37 a bushel, and sold 
 it at $1,45 a bushel : what was the profit ? 
 
 OPERATION. 
 
 Analysis. — We first find the $1,45 sold per bushel, 
 
 profit on a single bnslicl, and 1,37 cost per bushel, 
 
 then multiply by the number of ,08 profit on 1 bushel. 
 
 bu.shcls, which is 325 : the pro- then, 325 x ,08 ^$26,00 profit, 
 duct is the profit. 
 
 2. Bought a piece of broadcloth containing 36 yards, at $2,75 
 per yard ; it proving damaged, was sold at a loss of $18 : for 
 what was it sold per yard ? 
 
 215. What is Profit or Loss ! How do you find the entire profit ol 
 loss ? How do you find the profit or loss on unity \ 
 
^ItOFIT AND LOSS. 251 
 
 Analysis. — Find the cost, which is operation. 
 
 $99 ; then subtract the loss $18, and $2,75 x 3G = $99 
 then divide by the number of yards : Loss 1 8 
 
 the quotient \A-ill be the answer: hence, 36)81 ($2, 25 
 
 I. When the entire profit or loss is required : 
 
 Find the iirojit or loss on unity and multiply it by the num- 
 ber tvhich shares the profit or loss. 
 
 II. When the profit or loss on unity is required : 
 
 Find the entire profit or loss and divide it by the number 
 ivhich shares the profit or loss. 
 
 EXAMPLES. 
 
 1. Bought barrels of sugar, each weighing 250 pounds, at 
 7 cents a pound : how much profit would be made if it were 
 sold at 8^ cents per pound ? 
 
 2. If in 3 hogsheads of molasses which cost $68,04, one-third 
 leaked out, what must the remainder be sold for per gallon to 
 realize a profit of $2,52 on the whole ? 
 
 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 laiTibs, 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 ? 
 
 246. Given the per cent of the yain or loss, and the amount 
 of tJie sale, to find the cost. 
 
 1. A ffrocer sold a lot of susrars for $477,12, which was an 
 advance of 12 per cent on the cost : what was the cost ? 
 
 Analysis. — 1 dollar of the cost plus 12 per opkration. 
 cent; will be what that wliich cost $1 sold for, 1.12)477.12 
 
 viz., $1,12 : hence, there will be as many dol- $42(3 Ans. 
 
 lars of cost, as $1,12 is contained times in what 
 the goods brought. 
 
 246. How do you find the cost when you know the per cent and the 
 amount of sale 1 
 
252 PERCENTAGE. 
 
 2. Mr. A. bought a lot of sugars, but finding them of an in- 
 ferior quality, sold them at a loss of 15 per cent, and found 
 that they brought $3-10 : what did they cost him ? 
 
 Analysis. — 1 dollar of the co.^t less 15 per operation. 
 cent, Avill be what that which cost 1 dollar sold .85)340 
 
 for, viz.. $0.85 : hence, there will be as many $400 
 
 dollars of cost, as .85 is contained times in what 
 the goods brought. 
 
 Rule. — Divide the amount received by 1 plus the per cent 
 when, there is a gain, and by 1 minus the per cent when there is 
 a loss, and the quotient loill be the cost. 
 
 EXAMPLES. 
 
 1. I sold a parcel of goods for $195,50, on which I made 15 
 per cent : what did they cost me ? 
 
 2. Sold 78cwt. dqr. 14//;. of sugar, at 8 cents a pound, and 
 gained 15 per cent : how much did the whole cost ? 
 
 3. A merchant having a lot of flour, asked 33^ per cent more 
 than it cost him, but was obliged to sell it 12^ per cent less than 
 his asking price ; he received for the flour, §7.015 a barrel : 
 what did it cost him ? 
 
 4. A dealer sold two horses for $472,50 each, and gained on 
 one 35 per cent, but lost 10 per cent on the other : what was 
 the cost of each, and what was his gain ? 
 
 2-17. To find the selling price of an article so os to gain or 
 lose a certain per cent. 
 
 1. A grocer bought 12 barrels of sugar, for which he paid 
 §18,50 a barrel : what must he sell it for to yield him a profit 
 of 15 per cent ? 
 
 Analysis. — Since the amomit f »r which operation. 
 
 the sugar is to be sold exceeds its cost by $18,50 X 12 = $222 
 
 15 per cent., that amount added to the 15 per cent 33. .'JO 
 
 cost will give what it mu&t be sold for. Ans. $255.30 
 When tliere i.s a loss, subtract. 
 
PKOFIT AND LOSS. 255 
 
 Rule. — Add the iwrcentage to the cost, token there is a gain, 
 and subtract it ivhen there is a loss. 
 
 EXAMPLES. 
 
 1. A farmer sells 375 bushels of corn for 75 cents a bushel ; 
 the purchaser sells it at an advance of 20 per cent : how much 
 a busliel did he receive for the corn ? 
 
 2. A merchant buj's a pipe of wine, for which he pays 
 $322,5G, and he wishes to sell it at an advance of 25 per cent : 
 what must he sell it for per gallon ? 
 
 3. A man bought 3275 bushels of wheat, for which he paid 
 S3493,33l, but finding it damaged, is willing to lose 10 per 
 cent. : what must he sell it for per bushel ? 
 
 4. If a merchant in selling cloth at $4,70 a yard, loses G per 
 cent, on its cost, for how much must he sell it to gain 14 per 
 cent. ? 
 
 5. If I purchase two lots of land for $150,25 each, and sell 
 one for 40 per cent, more than it cost, and the other for 28 per 
 cent, less, Avhat is my gain on the two lots ? 
 
 6. 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 ? 
 
 7. 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 percent for freight, and 2 per cent commission 
 for selling, what must it sell for per bushel in New York that 
 he may realize 40 per cent net profit on the purchase ? 
 
 248. Given the gain or loss to find the iier cent. 
 
 1. Bought a quantity of goods for $200, and sold them for 
 $170 : what per cent did I lose on the purchase ? 
 
 247. How do you find the selling price of an article so as to gain or loso 
 a certain per cent 1 
 
 248. How do you find the percentage when you know the gain or loss ' 
 
 12 
 
254 PEKCENTAGK, 
 
 Analysis. — The gain or loss will opekation. 
 
 be equal to the dilierence between S200-S170 === S30 
 the cost and the amount received $30 -^ $200 = .15. 
 
 from the sale. If this difierence be Ans. 15 percent, 
 
 divided by the cost, the quotient will 
 
 denote the per cent on the cost • if it be divided by the amount of 
 the sale, the quotient will denote the per cent on the sale. 
 
 Rule. — Divide the gain or loss by the number on ivhich the 
 2)er cent is reckoned. 
 
 EXAMPLES. 
 
 1. Bought ,1 quantity of goods for $348,50, and sold tho 
 same for $425 : what per cent did I make on the amount 
 I'eceived ? 
 
 2. Bought a piece of cotton goods for G cents a yard, and 
 sold it for 71 cents a yard : what was my gain per cent ? 
 
 3. If I buy rye for 90 cents a bushel, and sell it for $1,20, 
 and wheat for $1,12-^ a bushel, and sell it for $1,50 a bushel, 
 upon which do I make the most per cent ? 
 
 4. If paper that cost $2 a ream, be sold for 18 cents a quire, 
 what is gained per cent ? 
 
 5. How much per cent would be made upon a hogshead of 
 sugar weighing Vdciut. 3qr. 14/6., that cost $8 per cwl, if sold 
 at 10 cents per pound ? 
 
 6. A hardware merchant bought 45 T'. IGcwt. 25lb. of iron, 
 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 ? 
 
 7. If 25 per cent be gained on flour when sold at $10 a 
 barrel, Avhat per cent would be gained when sold at $11, GO a 
 barrel ? 
 
 NoTK. — In this class of examples, first find the cost, as in Art. 240 : 
 then find the gain, or loss, and then divide by the number on which 
 the per cent is reckoned. 
 
 8. A lumber dealer sold 25G50 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 .'' 
 
INSURANCE. 255 
 
 9 A man sold his form for $3881,25, by which he gained 
 12^ 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 $3277,50 ? 
 
 10. If a merchant sell tea at 66 cents a pound, and gain 20 
 per cent, how much would he gain per cent if he sold it at 77 
 cents a pound ? 
 
 11. 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 ? 
 
 12. A grocer bought 3 hogsheads of sugar, each weighing 
 14121 pounds ; he sold it at 11 cents a pound, and gained 37^ 
 per cent : what was its cost, and for how much must he sell it 
 to gain 50 per cent on the cost ? 
 
 INSURANCE. 
 
 249. Insurance is an agreement, generally in writing, by 
 which individuals or companies bind themselves to exempt the 
 owners of certain property, such as ships, goods, houses, &;c., 
 from loss or hazard. 
 
 The Policy is the written agreement made by the parties. 
 
 250. The Base of insurance is the value of the property in- 
 sured. 
 
 251. Premiuji is the amount paid by him who owns the 
 property to those who insure it, as a compensation for their 
 risk. The premium is generally so much jjer cent on the pro- 
 perty insured. 
 
 252. There are four cases which may arise in questions of 
 Insurance. The principles on which these cases depend have 
 already been considered, and reference is made to the articles. 
 
 249. What is insurance 1 What is a policy 1 
 
 250. What is the base of insurance 1 
 
 251. What is a premium 1 
 
 352. How many cases are there which arise in insurance 1 What are 
 tliey 1 
 
256 I'EKCKNTAGE. 
 
 1. To find the Premium (Art. 216). 
 
 2. To find the Rate (Art. 217). 
 
 3. To find the ba.se, or sum insured (Art. 218). 
 
 4. To insure on both the base and premium. 
 
 253. To find the 2^'remiurii : 
 
 1. What would be the premium on a cargo of goods, valued 
 at $39854, the insurance being made at 41- per cent? 
 
 OPERATION. 
 
 Analysis. — This is simply a case of finding 39854 
 
 the percentage when the base and rate are given .045 
 
 (Art. 216). $1793,430 
 
 EXAMPLES. 
 
 1. What would be the premium for insuring a ship and cargo, 
 valued at $147674, at 31 per cent ? 
 
 2. What would be the insurance on a ship, valued at S47520 
 at ^ per cent ? At i per cent ? ' 
 
 3. What would be the insurance on a house, valued at Si 6800, 
 at 1^ per cent ? At |- per cent ? 
 
 4. A merchant owns |- of f of a ship, valued at $24000, and in- 
 sures Iiis interest at 2^ per cent : what does he pay for his policy ? 
 
 5. WJiat will it cost to insure a store worth $5640, at ^ per 
 cent, and the stock Avorth $7560, at -| 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 liJOIihd. of molasses, at 35 cents a 
 gallon : he gets it insured for 31 per cent on the selling 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 4^ per cent, what would be my actual loss if they were de- 
 stroyed ? 
 
 253. How (3o you find the premium 1 
 
LIFE INSUliAK(!K. 257 
 
 9. The ship Astoria is vahied at $20450, and her cargo at 
 $25 GOO ; being bound on a voyage from New York to Canton, 
 insured $12000 on the vessel, at the St. Nicholas Office, at 2|- 
 per cent, and $18500 on the cargo, at the Howard Office, at 3^ 
 per cent : if the vessel founder at sea, what will be the loss to 
 the owner ? 
 
 10. Sliipped from New York to the Crimea 5000 barrels of 
 flour worth $10,50 a barrel. The premium paid was $2887,50 : 
 what was the rate per cent, of the insurance ? 
 
 11. Paid $120 for insurance on my dwelling, valued at 
 $7500 : what was the rate per cent. ? 
 
 12. A merchant imported 225 pieces of broadcloth, each 
 piece containing 40 yards, at $3,50 a yard : he paid $1323 for 
 insurance : what was the rate per cent. ? 
 
 13. A merchant paid $1320 insurance on his vessel and 
 cargo, which was 51 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 2i 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 tho 
 same rate : what insurance must he pay ? 
 
 16. A store and its contents are valued at $16750. The owner 
 insures them at If per cent., and then insures the premium at 
 the same rate : what amount of insurance must he pay ? 
 
 LIFE INSURANCE. 
 
 254. Insurance for a term of years, or for the entire con- 
 tinuance of life, is a contract on the part of an authorized asso- 
 ciation to pay a certain sum, specified in the policy of insurance, 
 on the happening of an e^ent named therein, and for which the 
 association receives a certain premium, generally in the form 
 of an annual payment. 
 
 '2CA. What is a life insurance '■ 
 
 VA 
 
258 PEKCENTAGB. 
 
 255t To enable the company to fix their premiums at such 
 rates as shall be both fair to the insured and safe to the asso- 
 ciation, 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 from 
 many sources the most authentic information in regard to the 
 averwje 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 38.72 
 very nearly : hence 38.72 is said to be the " Expectation of 
 Life" at infancy. The Carlisle Tables, which are used in 
 this country and England, show the " Expectation of Life" 
 from 1 to 100 years. At 10 years old it is found to be 48.82 ; 
 at 20, 41. 4G; at 30, it is 34.34; at 40, 27.G1; at 50, it is 
 21.11; at GO, 14 years; at 70, 9.19; at 80, O.ol ; at 90, 
 3.28, and at 100, it is 2.28 years. 
 
 2.56. From the above facts, and the value of money (winch 
 is shown by the rate of interest), a comjiany can calculate with 
 great exactness the amount Avhich 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 Lisurance and Trust Company, 
 the Girard Life Lisurance, Annuity and Trust Company of 
 Philadelphia, and the Massachusetts Hospital Life Insui-ance 
 and Trust Company of Boston. The rates of insurance, iu 
 these companies, differ but little. 
 
 The PKEMiUM for life insurance is generally at so much per 
 annum on $100 ; and is always paid in advance. 
 
 2.5.'). What is ncccsKary to enable a company to fix their preiniiun.s \ 
 How is tlie expectation determined ! What do you understand by tha 
 ex|)cctation of life 1 
 
 !J.OG What may be calculated from the necessary facts 1 
 
LIFE INSURANCE. 259 
 
 EXAMPLES. 
 
 1. A person, 20 years of age, finds that tlie premium, pei 
 afiiuim, is $1.36 on $100 : what must he pay to insure his lift 
 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 $1.8G for $100 : how much 
 premium must he pay per annum on -£^12500 ? 
 
 3. A person, 38 years of age, obtains an insurance on lii.- 
 life for 5 years, at the rate of ^1,75 per annum on %100 : how 
 mucli is the annual premium on $15000? 
 
 4 A. person going to Europe, expecting to return in 2 years, 
 effects an insurance on liis life at ^ of 4 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 •'i;4,91 on 
 $100 ? 
 
 6. A person, at the age of 50 years, obtained an insurance 
 at 4| per cent, per annum on each $100 ; he insured for $1500, 
 and died at the age of 70. How mucli more was the insurance 
 than tlie payments, without reckoning interest ? 
 
 7. A gentleman, 47 years of age, going to China as ambassa- 
 dor, obtains an insurance on his life for $10000, by paying a 
 premium of $2.71 })er annum on every $100, and dies at the 
 middle of the third year : reckoning simple interest on his pay- 
 ments at 7 per cent, wliat is gained by the insurance ? 
 
 ENDOWMENTS. 
 
 257. An Exdoavmext is a certain sum to be paid at the 
 ex])iration of a given time, in case the person on whose life it is 
 taken shall live till the expiration of the time named. 
 
 » 
 2.57. What is an endowment 1 What does the table of endowments 
 
 show ] What maj- be found from the table ". 
 
200 
 
 PEIICKNTAOE. 
 
 The following table shows the value of an endowment pur- 
 chased 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. 
 
 TABLE OF 
 
 ENDOWMENTS. 
 
 
 . „_ Sums to be paid 
 '^"''- afil, if alive. 
 
 Age. 
 
 Sura to be paid 
 at 21, if alive. 
 
 Age. 
 
 Sum to be paid 
 at 21, if alivo. 
 
 Birth, - - 3376,84 
 
 5 years, 
 
 - $210,53 
 
 13 years, 
 
 - S144.12 
 
 3 months, - 344,28 
 
 6 " 
 
 198,83 
 
 14' '' 
 
 137.86 
 
 6 " - 331,46 
 
 7 " 
 
 - 188,83 
 
 15 " 
 
 131,83 
 
 9 «' - 318,90 
 
 8 " 
 
 179,97 
 
 16 " 
 
 - 125,97 
 
 1 year, - - 300,58 
 
 9 " 
 
 171,91 
 
 17 " 
 
 - 120,31 
 
 2. " - - 271,03 
 
 10 " 
 
 164.46 
 
 18 " 
 
 - 114,89 
 
 3 " - - 243,69 
 
 11 " 
 
 157,43 
 
 19 " 
 
 109,70 
 
 4 " - - 225,42 
 
 12 " 
 
 - 150,64 
 
 20 " 
 
 - 104,74 
 
 This table shows that if 8100 be paid at the birth of a child, 
 he M'ill be entitled to receive 8376,84, if he lives to attain the 
 age of 21 years. If 8100 be paid when he is ten years old, he 
 will be entitled to receive 8164,46, if he lives to attain the a^-e 
 of 21 years. And similarly for other ages. We can easily find 
 by proportion 
 
 IsL How much must be paid, at any age under 21, to pur- 
 chase a given endowment at 21 ; and 
 
 2c?. What endowment a sum paid at any age under 21, will 
 purchase ? 
 
 EXAMPLES. 
 
 1. What endowment, at 21, can be purchased for $250, paid 
 at the age of 10 years ? 
 
 2. What endowment, at 21, can be purchased for 8360, paid 
 at tlie age of 5 years ? 
 
 3. If my child is 7 years old. and I purchase an endowment 
 for 8650, what will he receive if he attains the age of 21 years? 
 
 4. If, at the birth of a daugliter, I purchase an endowment 
 for 8350, what will she receive if she attaiii* the ago of 2] 
 
 years 
 
 *? 
 
ANNUITIES. 
 
 261 
 
 ANNUITIES. 
 258. An Annuity is a fixed sum of money to be paid at 
 T'^gular periods, either for a limited time, or forever, in con- 
 rjderalion 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 
 l);iy the annuity. 
 
 The j)urchaser of an annuity should jiay more than tlie com- 
 pound interest ; for the seller cannot aiford to take the money' 
 of the purchaser, invest it, re-invest 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 
 
 Shoiving the peesent value of an annuity of %l, from 1 to 30 years, at 
 different rates of iriterct!. 
 
 Years. 
 
 5 per cent. 
 
 G per cent. 
 
 Vear.s. 
 
 5 per cent. 
 
 G per cent. 
 
 -1 
 
 0.95-2381 
 
 0.943396 
 
 16 
 
 10.837770 
 
 10.105895 
 
 
 1.859410 
 
 1.833393 
 
 17 
 
 11.274066 
 
 10.477260 
 
 3 
 
 2.723X148 
 
 2.673012 
 
 18 
 
 11.689587 
 
 10.827603 i 
 
 4 
 
 3.545950 
 
 3.465106 
 
 19 
 
 12085321 
 
 11.1.58116 
 
 5 
 
 4.329477 
 
 4,212364 
 
 20 
 
 12.462216 
 
 11.469921 
 
 6 
 
 5 075G92 
 
 4.917324 
 
 21 
 
 12,821153 
 
 11.764077 
 
 7 
 
 5.786373 
 
 5.582381 
 
 22 
 
 13.163003 
 
 12-041582 
 
 8 
 
 6.463213 
 
 6.209794 
 
 23 
 
 13488574 
 
 12.303379 
 
 9 
 
 7.107823 
 
 6.801692 
 
 24 
 
 13.798642 
 
 12.550.358 
 
 10 
 
 7.721735 
 
 7.360087 
 
 25 
 
 14.093945 
 
 12 783356 
 
 11 
 
 8.306414 
 
 7.886875 
 
 26 
 
 14.375185 
 
 13.003166 
 
 12 
 
 8.86.3252 
 
 8. .388844 
 
 27 
 
 14 643034 
 
 13.210.5.34 
 
 13 
 
 9393573 
 
 8.852683 
 
 28 
 
 14.898127 
 
 13 406164 
 
 14 
 
 9 898641 
 
 9.2949S4 
 
 29 
 
 15.141074 
 
 13.590721 
 
 15 
 
 10.3796,58 
 
 9.712249 
 
 30 
 
 15 372451 
 
 13.764831 
 
 To find the present value of an annuity for any rate, and for 
 any time, we simply multiply the present value of an annuity 
 
 2.58. What is an annuity! What is the present value of an annuity ! 
 How do you find the present value of an annuity for a given rate and time 1 
 How do you find what annuity a given sum will produce, at a given rate 
 and for a given time ? 
 
262 PKRCENTAGE. 
 
 of SI for the same rate and time, bj the annuity, and the pro- 
 duct will be its present value. 
 
 Thus, the present value of an annuity of $600 for 8 years, 
 at G per cent, is 
 
 $6.209794 X 600 = 83725.8764 ; that is, 
 
 pres. val. of $1 X annuity = pres. vaL, hence, 
 
 pres. val. ,, n 
 
 annuity = ^ ; ; therefore, 
 
 pres. val. of §1 
 
 I. To find what sum will produce a certain annuity at a given 
 rate, and for a given time. 
 
 Multiply the present value of an annuity of %1, at the given 
 rate and for the given time, hy the given annuity ; the product 
 will he that sum. 
 
 II. To find what annuity a given sum will produce at a given 
 rate and for a given time. 
 
 Divide the given sum, or present value hy the present value of 
 $1, for the given rate and time, and the quotient ivill 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 5 per 
 cent. ? 
 
 4. For what sum could Mr. Jones purchase an annuity for 
 twenty-eiglit years, of twelve hundred and twenty dollars, the 
 compound interest being reckoned at 6 per cent.? 
 
 5. What annuity, for twenty-four years, could be purchased 
 for tlie sum of twenty-seven thousand iivc luiiulred and sixty 
 dollars, the compound i-nt crest being reckoned at 6 per cent. ? 
 
ASSESSING TAXES. 263 
 
 6. Mr. Jones having a small fortune of S25000, and calcula- 
 ting tli;it he will live about 20 years, purchases an annuity at 
 six per cent., with nn agi-eement that he would pay 1)20 a year 
 to an invalid sister : what was his annual income from the 
 investment, after making that payment? 
 
 ASSESSING TAXES. 
 
 259. A Tax is a certain sum required to be paid by the in- 
 habitants of a town, county, or state, for the support of govern- 
 ment. It is generally collected from each individual, in propor- 
 tion 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. 
 
 2G0. 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 
 properry. Having done this, divide the %ohoh tax to he raised 
 by the amount of taxable jjroperty, and the quotient will be the 
 tax on $1. Then multiply this quotient by the inventory of 
 each individual, and the product will be the tax on his property. 
 
 EXAMPLES. 
 
 1. 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, 
 
 259. What is tax ] How is it generally collected 1 What is a poll-tax ? 
 
 2G0. What is the first thing to be done in assessing a tax 1 If there is a 
 poll-tax, how do you find the amount 1 How then do you find the per cent 
 of tax to be levied on a dollar I How do you find the tax to be raised on 
 each individual I 
 
264 
 
 PERCENTAGE. 
 
 B's at $2400, pays 4 polls, 
 C's at $2530, pays 2 
 D's at $2250, pays 6 
 
 u 
 
 E's at $7242, pays 4 polls, 
 F's at $1651, pays 6 " 
 G's at $1600,80, pays 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 = $4000, amount to be levied on property. 
 Then, $4000 -^ $1000000 = 4 mills on $1. 
 Now, to find the tax of each, as A's, for example, 
 
 A's inventory, 
 
 $2800 
 ,004 
 
 11,20 
 5,60 
 
 4 polls, at $1,40 each, - 
 A's whole tax, _ - - $16,80 
 In the same manner, the tax of each person in the township 
 may be found. 
 
 ASSESSMENT TABLE. 
 
 261. Having found the per cent, or the amount to be raised 
 on each dollar, form a table showing the amount which certain 
 sums Mould produce at the same rate per cent. Thus, after 
 having found, as in the last example, that four mills are to be 
 raised on every dollar, we can, by multiplying in succession by 
 the numbers 1, 2, 3, 4, 5, 6, 7, 8, &c., form the following 
 
 TABLE. 
 
 $ 
 
 $ 
 
 $ $ 
 
 $ 
 
 « 
 
 1 
 
 gives 0,004 
 
 20 gives 0,080 
 
 300 gives 1,200 | 
 
 2 
 
 •• O.OOS 
 
 30 " 0,120 
 
 400 
 
 ' 1 .600 
 
 3 
 
 • 0,012 
 
 40 " 0,100 
 
 500 
 
 ' 2,000 
 
 4 
 
 •' 0.016 
 
 50 " 0,200 
 
 600 
 
 ' 2,t()() 
 
 5 
 
 " O.O'^O 
 
 60 " 0.210 
 
 700 
 
 ' 2,800 
 
 6 
 
 " 0.024 
 
 70 " 0,280 
 
 800 
 
 ' 3.200 
 
 7 
 
 " 0.028 
 
 80 " 0,320 
 
 900 
 
 ' 3,600 
 
 9 
 
 " 0.0:12 
 
 90 " 0,3()0 
 
 1000 
 
 ' 4.000 
 
 <) 
 
 " 0,():!0 
 
 100 " 0,400 
 
 20110 
 
 • 8.000 
 
 10 
 
 " 0.040 
 
 200 " O.MOO 
 
 3000 
 
 ' 12,000 
 
 
 
 
 
 
 2G1. How do you form tlio Assessment table 1 
 
ASSESSING TAXES. 265 
 
 This table shows the amount to be raised on each sum in the 
 cohimns under $'s. 
 
 Note. — If you wish ilie tax on a sum not named in the Table, as 
 $25, it is equal to the sum of the taxes on $20 and So : and similarly 
 for other numbers. 
 
 1. To find the amount of B's tax from this table. 
 
 B's tax on $2000, is - - $8,000 
 B's lax on 400, is - - 1,000 
 
 B's tax on 4 polls, at $1,40, - 5, GOO 
 
 B's total tax is - - $15,200 
 
 2. To find the amount of C's tax from the table. 
 
 C's tax on $2000, is - - $8,000 
 
 C's tax on 500, is - - 2,000 
 
 C's tax on 30, is - - 120 
 
 C's tax on 2 polls, is - - 2,800 
 
 C's total tax is - - $12,920 
 
 In a similar manner, Ave might find the taxes to be paid by 
 D, E, &c. 
 
 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, $14G5,50 ; 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 : hovi^ much per 
 cent will be laid upon the property ? 
 
 2. In a county embracing a population of 98415 persons, a 
 tax is levied for town, county, and state purposes, amounting to 
 S100406. Of this sum, a part is to be raised by a tax of 25 
 cents on eacli poll, and the remainder by a tax of two mills on 
 the dollar : what was the amount of property on the tax list ? 
 
 3. In a county embracing a popuhuion of 5G450 persons, a 
 tax is levied for town, county, and state purposes, amounting to 
 887467 : the pergonal and real estate is valued at S4890300. 
 Each poll is taxed 25 cents : what per cent is the tax, and how 
 
266 PEKOENTAGE. 
 
 much will a man's tax be, who pays foi- 5 polls, and whose px-o- 
 perty is valued at. $5400 ? 
 
 Yy''hat is B's lux, who was assessed for 2 polls, and whose 
 j)rpperty was valued at ^3760,50 ? 
 
 4. A banking corporation, consisting of 40 persons, was taxed 
 $957.p0 ; their property was valued at §125000, and each poll 
 was assessed 50 cents each : 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 2^ per cent commission on the money col- 
 lected (Art. 218) ? 
 
 6. Allowing 4 per cent, for collection, what sum must be 
 assessed to raise $21346,75 net ? 
 
 7. In a certain township, it becomes necessary to levy a tax of 
 14423,2475, to build a public hall. The taxable property is 
 valued at $916210. and the town contains 150 polls, which are 
 each assessed 50 cents. "What amount of tax must be raised 
 to build the hall, and pay 5 per cent, for collection, and what is 
 the tax on a dollar ? 
 
 What is a person's tax who pays for 3 polls, and whose per- 
 sonal 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 li's tax, who is assessed for 1 i)oll, and $2456 ? 
 
 8. The people of a school district wish to build a new schoo. 
 house, which shall co-^^t ^2850. The taxable ])roperty of tin- 
 district is valued at $190000 : Avhat will be the tax on a dollar, 
 and what will be a man's tax, whose property is valued at $7500 ? 
 
 How much is IMr. 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 contingoucies amount to $66. They drew $41,60 
 public money, and the whole number of days attendance was 
 7688: what was D's tax, who sent 148 days? 
 
 AVhat was F's tax, wlio sent 184.V days? 
 
CUSTOM HOUSE BUSINESS. 2G7 
 
 CUSTOM HOUSE BUSINESS. 
 
 259. Persons who bring goods, or merchandise, into the 
 United States, from foreign countries, are required to land 
 them at particular places or ports, called Ports of Entry, and 
 to pay a certain amount on their value, called a Duty. This 
 duty is imposed by the General Government, and must be the 
 game on the same articles of merchandise, in every part of the 
 United States. 
 
 Besides the duties on merchandise, vessels employed in com- 
 merce are required, by law, to pay certain sums for the privilege 
 of entering the ports. These sums are large or small, in pro- 
 portion to the size or tonnage of vessels. The moneys arising 
 from duties and tonnage, are called revenues. 
 
 260. The revenues of the country are under the general 
 direction of the Secretary of the Treasury, and to secure tlieir 
 faithful collection, the government has appointed various othcera 
 at each port of entry or place where goods may be landed. 
 
 261. The office established by the government at any port 
 of entry, is called a Custom House, and the officers attached to 
 it are called Custom House Officers. 
 
 262. All duties levied by law on goods imported into the 
 United States, are collected at the various custom houses, and 
 are of two kin,ds — Specific and Ad valorem. 
 
 Specific Duty is a certain sum on a particular kind of 
 goods named ; as so much per square yard on cotton or woollen 
 cloths, so much per ton weight on iron, or so much per gallon 
 on molasses. 
 
 259. What is a port of entry ? What is a duty ? By wliom are duties 
 imposed ? What charges are vessels required to pay ! What arc the 
 moneys arising from duties and tonnage called 1 
 
 260. Under whose direction are the revenues of the country 1 
 
 261. What is a custom house \ What are the officers attached to it called 1 
 
 262. Where are the duties collected ! How many kinds are there, and 
 what are they called 1 What is a specific dutyl An ad valorem dutyl 
 
268 PERCENTAGE. 
 
 Ad valorem Duty is such a per cent on the actual cost of 
 the goods in the country from which they are imported. Thus, 
 an ad valoi-em duty of 15 per cent on Enghsh cloths, is a duty 
 of 15 per cent on the cost of cloths imported from England. 
 
 263. The laws of Congress provide, that the cargoes of all 
 TC-ssels freighted with foreign goods or merchandise, shall be 
 weighed or gauged by the custom house officers at the port to 
 ■vs'hich they are consigned. As duties are only to be paid on the 
 articles, and not on the boxes, casks, and bags which contain 
 them, certain deductions are made from the weights and mea- 
 sures, called Allowances. 
 
 Gross Weight is the whole weight of the goods, together with 
 that of the hogshead, barrel, box, bag, &c., wliich contains them. 
 
 Net Weight is what remains after all deductions are made. 
 
 Draft is an allowance from the gross weight on account of 
 waste, where thei'e is not actual tare. 
 
 lb. lb. 
 
 On 112 it is 1, 
 
 From 112 to 224 " 2, 
 
 " 224 to 336 " 3, 
 
 « 33G to 1120 " 4, 
 
 « 1120 to 2016 '' 7, 
 
 Above 2016 any weight " 9, 
 
 consequently, 9//;. is the greatest draft generally allowed. 
 
 Tare is an allowance made for the weight of the boxes, bar- 
 rels, or bags containing the commodity, and is of three kinds, 
 1st. Legal tare, or such as is established by law; 2d. Cus- 
 tomary tare, or such as is established by the custom among 
 merchants ; and 3d. Actual tare, or such as is found by re- 
 
 203. Wliat do tlie laws of Coniiress direct in relation to forciirn rroods 1 
 Why arc deductions made from their weight ? Wiiat are these dcductioiia 
 called? Wiiat is gross \vci;i;ht ? What is net weight ! "\\'hat is draft ' 
 What is the greatest draft allowed ! What is tare? What arc the dill'er 
 eiit kinds of tare 1 AVUat allowances are made on lic^uors 1 
 
CUSTOM HOUSE BUSINESS. 269 
 
 moving the goods and actually wcigliing the boxes or casks in 
 which they are contained. 
 
 On liquors in casks, customary tare is sometimes allowed on 
 the supposition that the cask is not full, or what is called its 
 actual ivants ; 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 common 
 size are estimated to contain 2|- gallons the dozen. For tables 
 of Tare and Duty, see Ogden on the Tarift'of 1842. 
 
 EXAMPLES. 
 
 1. What is the net weight of 25 hogsheads of sugar, the 
 gross weight being 66c«'i. ^qr. lUb. ; tare lllb. per hogshead? 
 
 civt, qr. lb. 
 
 66 3 14 gross. 
 
 25 X 11 = 275/6. - - 2 3 tare. 
 
 Ans.M 14 net. 
 
 2. If the tare be Alb. per hundred, what will be the tare en 
 QT. 2cwt. 3qr. Ulb. ? 
 
 Tare for QT. or UOcwf. = ASOlb. 
 2cwt. z= 8 
 Sqr. = 3 
 Ulb. = 0^ 
 
 Tare - - jM^Ib. 
 
 3. What will be the cost of 3 hogsheads of tobacco at $9,47 
 per cwt. net, the'gross weight and tare being of 
 
 ctvt. qr. lb. lb. 
 
 No. 1 - - 9 3 24 - - tare 146 
 
 « 2 - - 10 2 12 - - " 150 
 
 " 3 - - 11 1 24 - - « 158 
 
 4. At 21 cents per lb., what will be the cost of 5hkd. of coffee, 
 the tare and gross weight being as follows : 
 
570 
 
 
 1'! 
 
 1 i£Ci 
 
 ■;]SiTAGK. 
 
 
 
 cwl. 
 
 qr. 
 
 /<;-. 
 
 
 lb. 
 
 No. 1 ■ 
 
 ■ - 6 
 
 2 
 
 14 
 
 - - 
 
 tare 94 
 
 « 2 - 
 
 ■ - 9 
 
 1 
 
 20 
 
 - - 
 
 " 100 
 
 " 3 ■ 
 
 ■ - G 
 
 2 
 
 22 
 
 - - 
 
 " 88 
 
 " > ■ 
 
 • - 7 
 
 2 
 
 24 
 
 - - 
 
 " 89 
 
 " *5 - 
 
 • - 8 
 
 
 
 13 
 
 - 
 
 " 100 
 
 5. What is the net weight of IShhcL of tobacco, each weigh- 
 ing gross Sctvf. dqr. 1Mb.', tare 16/6. to the cwt."? 
 
 G. In 47'. Zciol. Zqr. gross, tare 20/i. to the cwt., Avhat is the 
 net weight ? 
 
 7. What is the net weight and value of 80 kegs of figs, 
 gross weight IT. Wcict. 3q}:, tax*e 12/6. per cwt., at $2,31 
 per civi. 
 
 8. A merchant bought Idcwt. Iqr. 24/6. gross of tobacco in 
 leaf, at $24,28 per cwl.; and 12ca'L 3qr. 19/6. gross in rolls, at 
 $28,56 per cwt. ; the tare of the former Avas 149/6., and of the 
 latter 49/6. : what did the tobacco cost him net ? 
 
 9. A grocer bought 17^^h/id. of sugar, each lOart. Iqr. 14/6., 
 draft 7/6. per civ/., tare 4/6. per cwt. What is the value at 
 $7,50 per art. net? 
 
 10. In 29 parcels, each weighing Sctvt. oqr. 14/6. gross, draft 
 8/6. per cwt., tare 4:'6. per cwt. how much net weiglit, and 
 what is the value at $7,50 per cwt. net ? 
 
 11. A merchant bought 7 hogsheads of molasses, each Aveigh- 
 ing 4rwf. 'dqr. HI/), gross, draft 7/6. per cwt., tare 8/6. per 
 hogshead, and damage in the whole 99|-/6. : what is the value 
 at $8,45 per cwt. net ? 
 
 12. Tlie net value of a hogshead of Barbadoos sugar was 
 $22,50; the custom and fees $12,49, freight $5,11, factorage 
 $1,31 ; the gross weight was llcwf. Iqr. 15/6., tare ll]-/6. per 
 cirf. : what was the sugar rated at per cwt. net, in the bill of 
 parcels ? 
 
 13. In 7h/id. of .sugar, each weighing 3cwt. 2qr. 14/6. gross, 
 tare 21/6. per cwt., what is the value at $6,25 per ca-t. ? 
 
CUST<m UOUSE BUSINESS. 271 
 
 14. I have imported 87 jars of Lucca oil, each containing 
 47 gallons : Avhat did the freight come to at $1,19 per act. net, 
 reckoning \lh. in Wlb. for tare, and 9//^ of oil to the gallon ? 
 
 15. A grocer bought bhhd. of sugar, each weighing V^cwt. 
 \qr. 12//j., at 7-^ cents a pound ; the draft was l^lh. per civi.^ 
 and the tare 5^ per cent : wliat was the cost of the net weight ? 
 
 ] 6. A wholesale merchant receives 450 bags of coffee, each 
 Aveighing 76 pounds; the tare was 8 per cent, and the invoice 
 price lOi cents per pound. He sold it at an advance of 33^ 
 per cent : what was his whole gain, and what his selling price ? 
 
 17. A merchant imported 17G pieces of broadcloth, each 
 piece measuring 46iyt/., at $3,25 a yard : what will be the duty 
 at 30 per cent ? 
 
 18. What is the duty on 547^. 13c',y^. Zqr. 2<dlb. of iron, in- 
 voiced at $45 a ton, and the duty 33^ per cent ? 
 
 19. What is the ad valorem duty, at 25 per cent, on ZhJid. 
 of molasses, at 35 cents a gallon, an allowance of 2 per cent 
 beinii: made for leakage ? 
 
 20. If I import 50 chests of tea, each weighing 140 pounds, 
 invoiced at 60 cents a pound, a deduction of IQlh. per cwt. being 
 made for tare : what were the governmental duties, at 40 per 
 cent ad valorem ? 
 
 21. What will be the duty on 225 bags of coffee, each weigh- 
 ing gross 160/6., invoiced at 6 cents per lb.; 2 per cent being 
 the legal rate of tare, and 20 per cent the duty? 
 
 22. What duty must be paid on 275 dozen bottles of claret, 
 estimated to contain 2| gallons per dozen, 5 per cent being 
 allowed for breakage, and the duty being 35 cents per gallon ? 
 
 23. A merchant imports 175 cases of indigo, each case weigh- 
 ing 196/6. gross : 15 per cent is the customary rate of tare, and 
 the duty 5 cents per lb. W^hat duty must he pay on the whole ? 
 
 24. What is the tare and duty on 75 casks of Epsom salts, 
 each weigiiing gross 2cwt. ^qr. 24//;., and invoiced at 1|- cents 
 per lb., the customary tare being 11 per cent, and the rate of 
 duty 20 per cent ? 
 
272 EQTJATION OF PAYMENTS. [ 
 
 EQUATION OF PAYMENTS. i 
 
 264. Equation op Payments is a process of finding the }| 
 average time of payment of several sums due at different times, 
 so that no interest shall be gained or lost.* 
 
 OPERATION. 
 
 $15 
 
 X 
 
 6 
 
 = 90 
 
 $18 
 
 X 
 
 7 
 
 == 126 
 
 $24 
 
 X 
 
 10 
 
 = 240 
 
 57 
 
 
 5 
 
 7)456(8 
 456 
 
 1. B owes Mr. Jones $57: $15 is to be paid in 6 months ; 
 $18 in 7 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 
 months, is the same as the interest of 
 $1 for 90 months: the interest of $18 
 for 7 mouths is the same as the interest 
 of $1 for 126 months; and the interest 
 of $24 for 10 months is the same as the 
 interest of $1 for 240 months; hence, 
 
 the sum of these products, 466, 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 ; that is, in 8 months : hence, to find the average 
 time of payment : 
 
 Multiply each "payment by the time before it becomes due, and 
 divide the sum of the products by the sum of the payments : the 
 quotient ivill be the average time. 
 
 EXAMPLES. 
 
 1. A merchant owes Si 200, of which $200 is to be paid id 
 4 months, $400 in 10 months, and the remainder in 16 months; 
 if he pays the Avliole at once, at what time must he make the 
 payment ? 
 
 264. What is equation of payments 1 How do you find the average 
 fime, of payment \ 
 
 * The mean time of payment is sometimes found by first fiiulinjr tht 
 prr.scnL value of each payment ; but the rule here <rivcn has the sanction 
 of the best authorities in this country and FJngland. 
 
 •n 
 
EQUATION OF J'Ai'MENTS. 273 
 
 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 the mean time of payment ? 
 
 3. A merchant has due him $4500 ; one-sixth is to be paid 
 in 4 months, one-tliird in 6 months, and the rest in 12 mouths: 
 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 ten months : 
 what is the average time of payment ? 
 
 5. Mr. Swain bought goods to the amount of $3840, to t)e 
 paid for as follows, viz. : one-fourth in cash, one-fourth in 6 
 months, one-fourth in 7 months, and the I'emainder in one year : 
 what is the average time of payment ? 
 
 6. A flour merchant bought at one time 150 barrels of flour, 
 at $8 a barrel ; 15 days afterwards he bought 176 barrels, at 
 $8,50 a barrel ; 25 days after that he bought 200 barrels, at 
 $9 a barrel : how many days after the first purchase would be 
 the equated time of payment ? 
 
 7. 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 pays the 
 whole at once, what would be the average time of payment ? 
 
 Notes. — 1. In finding the equated time of payments for several 
 Bums, due at different times, any day may be assumed as the one from 
 which we reckon. 
 
 2. If one of the payments is due on the day from which the 
 equated time is reckoned, its corresponding product will be nothing, but 
 the payment must still be added in finding the sum of the payments. 
 
 8. A person owes three notes : the first is for $200, payable 
 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 ? 
 
 Notes — 1. May the equated time be reckoned from any dayt 
 2. Tf one of the payments is due on the day from which the equated 
 time is reckoned, what will be the value of the corresponding product 1 
 
27J: EQUATION OI PAY3IE^-TS. 
 
 9. E. BoxD, Bought of Trust & Co. 
 1856, Aug. 1, 450 yds. muslin, at 10 cents, - - 845,00 
 
 « " 16, 800 " caljco " 12i " - - 100.00 
 
 " Sept. 5, 720 « bombazine 80 - - - 576,00 
 
 " Oct. 1, 300 " cloth, at 3,50 - - - 1050,00 
 On what day does the whole amount fall due ? 
 
 10. Mr. Johnson sold, on a credit of 8 months, the following 
 bills of goods : 
 
 April 1st, a bill of $4350, 
 May 7th, a bill of 3750, 
 June 5th, a bill of 2550. 
 At what time will the whole become due ? 
 
 11. A purchased of B the following bill of goods, on differ- 
 ent times of credit : 
 
 May 1st, 1857, a bill amounting to $800 on 3 months. 
 
 June 1st, " " " " 700 " 3 " 
 
 « 25th, " " " " 900 " 4 " 
 
 July 25th, " " " " 1000 " 6 " 
 
 "What is tlie equated time for the payment of the whole, and on 
 what day, reckoned from Aug. 1st, is the bill due ? 
 
 12. A person purchased the following bills of goods, on dif- 
 ferent times of credit : 
 
 Jan. 1st, 1855, a bill amounting to $367,20 on 4 months. 
 
 " " " 901,80 " 3 " 
 
 " " " 826,38 " 5 " 
 
 « « " 854,88 " 6 " 
 
 « " « 396,50 " 4 " 
 
 What is the average time of payment from the time the 
 first bill falls due ? 
 
 264.* To find liow long a sum of money must he at interest^ 
 to balance the interest on a given sum for a given time. 
 
 1 . If A lends B $700 for 3 months, how long ought B to 
 lend A $500 to balance tJiu interest ? 
 
 " 28th, 
 
 (( 
 
 Feb. 24th, 
 
 (( 
 
 March 30th, 
 
 ii 
 
 May 1st, 
 
 ii 
 
EQUATION OF PAYMENTS. 275* 
 
 Analysis. — Since $700, in 3 months, will pro. operat-.on. 
 
 duce as much interest as S2100 in 1 month, it '-qo x 3 = 2100- 
 
 will require as many months for $500 to pro- 2100 — 500 --44 
 duce the same interest, as 500 is contained times 
 in 2100 : \yhich is 4-^. 
 
 2. A lends B his note for $900 payable iu 5 months : how 
 long should B lend A his note for $480, to balance the favor ? 
 
 Ans. 9~ months. 
 
 3. C buys of D, 100 barrels of flour, at 87^- per barrel, and 
 in payment gives his note for 3 months ; D buys of C, 500 
 bushels of Avheat at 80 cents per bushel, and gives his note in 
 payment : how long must this note run, that each may have an 
 equal use of the other's money ? A?is. o| months. 
 
 To find hoio long the balance may he kept when 'paynitnts are 
 made before they are due. 
 
 1. A owes B, 6'800 payable in G months ; at the expiration 
 of 4 months, he pays $500 : how long beyond the 6 months 
 should A retain the balance, so that neither shall make or lose 
 interest ? 
 
 Analysis. — A has the right to retain the $800 
 for 6 months, or $4800 for 1 month. He retains operation. 
 $500 for 4 months, or $2000 for 1 month. Hence, 800x6 = 4800 
 he may still retain S2800 for 1 month, or the bal- 500X4 = 2000 
 ance, $300, as many months as 300 is contained 300 2800 
 
 times in 2800 ; or, 9|- months from the date of the 2800-f-300 = 9^ 
 debt ; or 9^—6 = 3^ months beyond the time of six 
 months. 
 
 2. C owes D, $2500 payable in 4 months, but at the end of 
 3 months pays him $1600 : how long after the payment of 
 $1600 should the remainder be retained to balance the ac- 
 count? Ans. 2^ months. 
 
 3. One merchant owes another $1600 payable in 6 months, 
 but at the end of 3 months pays $400 ; at the end of 4 months, 
 $400, and at the end of 5 months $300 ; how long, from the 
 last payment, may the balance be retained to square the ac- 
 count ? A71S. 51 months. 
 
27G* EQUATION OF PAYMENTS. 
 
 4. A note for $500 dated November 6th, 1856, payable in 
 3 months, was given by E to F. On December 3d, E paid 
 $350 : when ought the remainder to be paid to balance the 
 account? .4ns. July 3rd, 1857. 
 
 264.** An account is said to be balanced when the sum of 
 the items on the debtor side is equal to the sum of the items on 
 the credit side. When these two sums are unequal, such an 
 amount is added to the less as will make the sum equal to the 
 <Treater. This is called the balance. There are three kinds 
 of balances : 
 
 1st. The merchandize balance, in which interest on the items 
 is not considered. 
 
 2d. The interest balance, which adjusts the interest on the 
 two sides of an account ; and 
 
 3d. The cash balance, which arises from combining the 
 merchandize balance with the interest balance. 
 
 Accounts are settled either by cash or by note. In ascer- 
 taining the cash balance of an account, interest is allowed on 
 all the items of both sides ; the balance of interest makes a 
 new item, and may belong to either side of the account. 
 
 1. Ascertain the cash balance of the following account on 
 the 25th of April, 1850. 
 
 Dr. S. Snodgrass, Cr. 
 
 1850. April Is^t, To goods, $375.00 
 
 " I7th, " " 268,00 
 
 '= 25th, '' " 175,00 
 
 Cash balance, 237.93 
 
 $1055,93 
 
 April 7tli, By goods, $675,00 
 " 15th, " " 380,0t) 
 
 '• 25th, Bal. of Int. ,93 
 
 $1055^93 
 
 Analysis. — Reckoning backwards from April 25tli, we find the 
 days for which we charge interest, and these are used as multipliers. 
 The interest of $375 for 24 days is the same as the interest of $1 
 
EQUATION OP PAYMENTS. 277* 
 
 for 9000 days; and so of the other items. The difference of tho 
 sums of these products, is the number of days which $1 must be at 
 interest to produce the baUincc of interest, and the balance always 
 goes with the larger sum of the products. 
 
 OPERATION. 
 
 Debtor Items. Creditor Items 
 
 375 X 24 = 9000 
 
 675 X 18 = 12150 
 
 268 X 8 = 2144 
 
 380 X 10 = 3800 
 
 175 X = 0000 
 
 15950 
 
 11144 
 
 11144 
 
 4806 
 Then, 4806 "^tVo = ^-^3 +, balance of interest. 
 
 Rule. — I. Take the latest date of the account, or any later 
 date at which the balance is to be struck, as the point of rerJc- 
 oning, and find the days between this date and the date of 
 each item ; and consider these days as multipliers. 
 
 II. Multiply each item by its midtiplier ; then take the dljjer- 
 ence of the sums of these 2'>'>^odiicts, and multiply it by the inter- 
 est for 1 day : the result will be the interest balance, which is to 
 he added to the side having the greater sum. 
 
 III. Then find the cash balance. 
 
 Notes. — 1 . If the cash balance had been required on any day after 
 the 25th of April, the mode of proceeding would have been exactly 
 the same. 
 
 2. In the examples the rate of interest will be taken at 7 per 
 cent, and 360 days in the year. 
 
 3. After the balance of interest is found, the cash balance is ob- 
 tained, by adding the two sides of the account, taking the difference 
 of the sums and placing it on the s/naller side of the account. 
 
 4. If the cash balance is settled by a note, interest shoukl run ci 
 the note from the date of the cash balance to the time of payment. 
 
 5. Let the pupil find the interest and cash balance in each of th" 
 following examples. 
 
278' 
 
 EQUATION' OF PAYMENTS. 
 
 2. What is the balance of interest and what the cash balance 
 on the following account on Marcli 20th ? 
 
 Dr. 
 
 S. JOHXSON. 
 
 Or. 
 
 856. Jan. 1, To merch. 
 
 $500 ' 
 
 Jan. 5, By cash, 
 
 $350 
 
 ■• 16, '' cash, 
 
 450 
 
 •' 19, " merch. 
 
 780 
 
 Feb. 5, " merch. 
 
 680 
 
 " 25, '• " 
 
 250 
 
 >• 24, " " 
 
 300 
 
 Feb. 15, " cash, 
 
 600 
 
 Mar. 1, " cash, 
 
 150 
 
 cash balance, 
 
 700,65 
 
 " 16, •'•' merch. 
 
 600 
 
 
 $2680,65 
 
 Interest balance, 
 
 ,65 
 12680,65 
 
 
 
 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 ca.sh balance for this latest date : then find 
 the present value for the given date : this will be the cash balance. 
 
 3. Allowing a credit of six months on each item, what is the 
 interest and cash balance Feb. 1st, 1856 ? 
 
 R. Sherman. Cr. 
 
 Feb'y 6, By merch. $800 
 i\Iar. 7, " " 900 
 interest bal. 45,48 
 
 $2150 cash bal. 404,52 
 
 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. R. Rivers. ' Or. 
 
 Dr. 
 
 1855. July 1st, To merch., $750 
 
 " 17th, " " 600 
 
 " 25lh, " " 800 
 
 1856. May 1, To merch. $500 
 
 " 20, " 675 
 
 Jun. 6, To cash, 350 
 
 July 9; merch. 175 
 
 cash balance, 620 ,70 
 
 $232o7n) 
 
 May 6th, By cash, $400 
 
 25th. 
 
 mer. 
 
 620 
 900 
 July 20th, '■'■ mer. 400 
 interest balance, .70 
 
 June 16th, " cash, 
 
 c 
 
ALLIGATION. 275 
 
 ALLIGATION. 
 
 265. Alligation is that branch of Arithmetic which treats 
 of all questions relating to the mixing or compounding of two 
 or more ingredients of different values. It is divided into two 
 parts : Alligation Medial and Alligation Alternate. 
 
 ALLIGATION MEDIAL. 
 
 266. Alligation Medial teaches the method of findins; 
 the price or quality of a mixture of several simple ingredients 
 whose prices or qualities are known. 
 
 1. A grocer would mix 200 pounds of lump sugar, Avorth 13 
 cents a pound, 400 pounds of Havana, worth 10 cents a pound, 
 and GOO New Orleans, worth 7 cents a pound ; what should be 
 the price of the mixture ? 
 
 OPERATION. 
 
 Analysis. — The quantity, 200/6., 200 x 13 = 26.00 
 
 at 13 cents a pound, cost.s $26 ; 400 400 x 10 = 40.00 
 
 pounds, at 10 cents a pound, costs 600 x 7 = 42.00 
 
 $40 ; and 600/6. at 7 cents a pound, 1200 ) 108,00(9 cts. 
 
 costs $42 : hence, the entire mix- 
 ture, consisting of 1200/6., 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. — Multipli/ the inice or quality of a unit of each simple 
 hy the number of such units : take the sum of their products 
 and divide it by the ivhole number of units : the quotient will be 
 the price or quality of a miit of the mixture. 
 
 EXAMPLES. 
 
 1. If 1 gallon of molasses, at 75 cents, and 3 gallons, at 50 
 cents, be mixed with 2 gallons, at 37|^, what is the mixture 
 worth a gallon ? 
 
 265. What is Alligation 1 Into how many parts is it divided 1 What 
 are they 1 
 
 266 What is Alligation Medial \ IIow do you find the prict- of the 
 mixture 1 
 
276 ALLIGATION. 
 
 2. If teas at 371, 50, 62i, 80, and 100 cents per pound, be 
 mixed together, what ay ill be the value of a pound of the 
 mixture ? 
 
 3. If 5 gallons of alcohol, worth GO 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 ; GO 
 bushels of rye, at 90 cents; 36 bushels of corn, at G2i 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°, 731°, 77°, 70°, 801°, and 81° : what 
 was the average temperature for the w^eek ? 
 
 6. If gold 18, 21, 17, 19, and 20 carats fine, be melted to- 
 gether, what will be the fineness of the compound ? 
 
 7. A grocer bought 3Alb. of sugar at 5 cents a pound, 1021b. 
 at 8 cents, lS6lb. at 10 cents a pound, and 34/6. at 12 cents a 
 pound. He mixed it together, and sold the mixture so as to 
 make 50 per cent on the cost: what did he sell it for per 
 pound ? 
 
 8. A merchant sold 8//;. of tea, 11^5. of coifee, and 25lb. 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. 
 
 267. Alligation Alternate teaches the method of finding 
 wliat 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. 
 
 267. What is Alligation Alternate 1 How may Alligation Alternate b« 
 proved ! 
 
 2G8. How do you find the proportional parts 1 
 
ALTERNATE. 
 
 277 
 
 CASE I. 
 
 268. To find the 2'n'02'>ortional 2^0 fts. 
 1. A miller Avould mix wheat, worth 12 shillings a bushel; 
 corn, worth 8 shillings ; and oats, worth 5 shillings, so as to 
 make a mixture worth 7 shillings a bushel : what are the propor- 
 tional parts of each ? 
 
 OPERATION. 
 
 oats, 
 
 75. 
 
 5 s.-. 
 8,s'.J 
 
 corn, 
 wheat, 12s, 
 
 
 A. 
 
 B. 
 
 c. 
 
 D. 
 
 
 2^ 
 
 i 
 
 5 
 
 1 
 2 
 
 
 J 
 r, 
 
 
 2 
 
 
 E. 
 
 6 or 
 
 2 " 
 
 9 a 
 
 Analysis. — On every bushel put into the mixture, whose price is 
 less 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 there is to be neither gain nor loss by the mixture, the gains 
 and losses must balance each other. 
 
 A bushel of oats, when put into the mixture, will bring 7 shillings, 
 giving a gain of 2 shillings; and to gain 1 shilling, we mast take 
 half as much, or |- a bushel, which we write in column A. 
 
 On 1 bushel of wheat there will be a loss of 5 shillings ; and to 
 make a loss of 1 shilling, W"c must take -^ of a bushel, which we 
 write in column A : \ and -J- are called proportional numbers. 
 
 Again : comparing the oats and corn, there is a gain of 2 shillings 
 on every bushel of oats, and a loss of 1 shilling on every bushel of 
 corn : to gain 1 shilling on the oats, and lose 1 shilling on the corn, 
 we must take \ a bushel of the oats, and 1 bushel of the corn : these 
 numbers are written in column B. Two simples, thus compared, are 
 called a couplet : in one. the price of unity is less than the mean price, 
 and in the other it is greater. 
 
 If, every time we take i a bushel of oats we take -g- of a bushel 
 of W'heat, the gain and loss will balance ; and if every time we take 
 i a bushel of oats we take 1 bushel of corn, the gain and loss will 
 balance : hence, if the proportional numbers of a couplet be multiplied 
 bv any number^ the gain and loss denoted by the products xcill balance. 
 
 When the proportional numbers, in any column, are fractional 
 (as in columns A and B), multiply them by the least common mul- 
 tiple of their denominators, and write the products in new cohimns 
 C and D. Then, add the numbers in columns C and D, standing 
 opposite each simple, and if their sums have a common factor, reject 
 it: the last result will be the jiro; urtional luunbers. 
 
 l:{ 
 
278 ALLIGATION. 
 
 Note. — The answers to the last, and to all similar questions, will 
 be infinite in number, for t-\TO reasons : 
 
 1st. If the proportional numbers in column E be multiplied by 
 any number, integral or fraetional, the products will denote propor- 
 tional parts of the simples. 
 
 2d. If the proportional numbers of aiiy covplct be multiplied by 
 any number, the gain and loss in that couplet will still balance, and 
 the proportional numbers in the final result will be clianged. 
 
 Rule. — I. Write the prices or qualities of the simples in a 
 column, beginning ivith the loivest, and the mean p>r ice or qualitij 
 at the left. 
 
 II. Opposite the first sim2)le write the part ichich must be 
 taken to gain 1 <f the mean j^^'icc^ und opposite the other simple 
 of the couplet write the 2)ni't which must be taken to lose 1 of the 
 mean price, and do the same for each simple. 
 
 III. When the prop)ortional numbers are fractional, reduce 
 them to integral numbers, and then add those lohich stand oppo- 
 site the same simj^le : if the sums have a common factor, reject 
 it : the result will denote the proportional parts. 
 
 EXA5IPLES. 
 
 1. TVIiat proportions of coffee, at 8 cents, 10 cents, and 14 
 cent.s per pound, must be mixed together so that the compound 
 shall be worth 12 cents per pound ? 
 
 2. A merchant has teas worth 40 cents, 65 cents, and 75 
 cents a pound, from Avhich lie wishes to make a mixture worlh 
 60 cents a pound : what is the smallest quantity of each tliat 
 he can take and express the parts by whole numbers ? 
 
 3. A farmer sold a number of colts at $50 each, oxen at $10, 
 cows at $25, calves at $10, and realized an average price of 
 $30 per head : what was the smallest number he could sell of 
 each ? 
 
 4. What is the smallest quantity of water that must bo 
 mixed wiili wine wordi 14.v. and 15s. a gallon, to Ibrni a mix- 
 ture worlh i;3s. a gallon, when all the parts arc expi'csscd by 
 whole nuuibci'a? 
 
ALTEKNATK, 
 
 279 
 
 CASE II. 
 
 269. Whe7i the qu inliti/ of one of the simples is given. 
 
 1. A farmer would mix rye woi-th 80 cents a bushel, and 
 corn worth 75 cents a bushel, witli GG bushels of oats worth 45 
 cents a bushel, so that the mixture shall be worth 50 cents a 
 bushel : how much must be taken of each sort ? 
 
 OPERATION. 
 
 
 A. 
 
 B. 
 
 c. 
 
 D. 
 
 E. 
 
 F. 
 
 
 1 
 
 ] 
 
 5 
 
 G 
 
 5 
 
 11 
 
 GG 
 
 
 
 1 
 
 
 1 
 
 1 
 
 6 
 
 
 1 
 ao 
 
 25 
 
 1 
 
 
 1 
 
 6 
 
 Analysis. — Find the proportional parts, as in Case I: they are 11, 
 1 and 1. But we are to take 66 bus-bcls of oats in the mixture: 
 hence, each proportional number is to be taken 6 times ; that is, as 
 many times as there are units in the quotient of 66 -r 11. 
 
 Rule. — I. Find the 'proportional numbers as in Case I, and 
 write the given simjiles opposite its 2>i'oportional number. 
 
 II. Multiply the given simple by the ratio tvhich its propor- 
 tional number bears to each of the others, and the products will 
 denote the quantitits to be taken of each. 
 
 Note. — If we multiply the numbers in either or both of the 
 columns C or D by any number, the proportion of the numbers in 
 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 
 66, 12 and 1, numbers which will fulfil the conditions of the question. 
 
 EXAMPLES. 
 
 1. What quantity of teas at 12*-. \Qs. and Gv. must be mixed 
 with 20 pounds, at 4s. a pound, to make the mixture worth Ss. 
 a pound ? 
 
 2. How many pounds of sugar, at 7 cents and 11 cents a 
 pound, must be mixed with 75 pounds, at 12 cents a pound, so 
 that the mixture may be worth 10 cents a pound ? 
 
 269. How do you find the proportional parts when the quantity of one 
 simple is ijivcii ! 
 
280 
 
 ALLIGATION. 
 
 3. How many gcallons of oil, at 7s., 7s. Qd. and 9s. a gallon, 
 mu>t be mixed v.itli 24 gallons of oil, at 9^-. Gd. 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 §1 each, that the average price of the whole shall be 
 m? 
 
 0. A grocer mixed 50/^. of sugar worth 10 cents a pound, 
 with sugars worth 9i cents, 7-i cents, 7 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. 
 
 270. 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 17 carats fine : how 
 much must he take of each sort ? 
 
 OPERATION. 
 
 17 ^ 
 
 
 
 A. 
 
 B. 
 
 C. 
 
 D. 
 
 E. 
 
 F. 
 
 G. 
 
 H. 
 
 "I'^nO 
 
 1 
 
 2 
 
 I 
 
 2 
 
 7 
 
 5 
 
 o 
 
 15 
 
 30 
 
 20J 
 
 
 
 
 1 
 
 
 
 2 
 
 2 
 
 4 
 
 22 ) 
 
 
 1 
 
 5 
 
 3 
 
 
 2 
 
 
 2 
 
 4 
 
 24— 
 
 _y 
 
 ] 
 
 7 
 
 
 
 2 
 
 
 
 2 
 
 4 
 
 Proportional Parts : 
 15 + 2 + 2 + 2 = 21; 42 -+21 = 2. 
 
 Rule. — I. Find the proportional parts as in Case I. 
 
 II. Divide the quantity of the mixture by the sum of the pro- 
 portional jmrts, and the quotient will denote hoio many times 
 each part is to be talceii. Multiply this quotient by the parts 
 separately, and each 2>yoduct tvill denote the quantity of the cor- 
 resjiondiiig simple. 
 
 270. How do you liiul ilir proportional i)arls wlieii llic iiiiaiitity of llif 
 mixUirc is j^'ivcii 1 
 
ALLIGATION. 281 
 
 Note. — We may, as in the other cases, multiply each couplet by 
 any number we please, which will merely change the relation of the 
 proportional parts, and consequently uivc diirerent proportions of the 
 ingredients. Hence, there is an infinite number of answers^ if we 
 employ fractions, and often many answers to similar questions, in 
 whole numbers.* 
 
 EXAMPLES. 
 
 1. A grocer has teas at 5s., Os., 8?., and Os. a pound, and 
 wislies to make a compound of 88/i. worth 7^. a pound : how 
 much of each sort must be taken ? 
 
 2. A liquor dealer wishes to fill a hogshead with water, and 
 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 111, and each lamb If? 
 
 4. A merchant sold 20 stoves for 8180 ; for the largest size 
 he received $19 each, for the middle size, $7, and for the small 
 size %Q: how many did he sell of each kind ? 
 
 5. A vintner has wines at 45., 65., 8s., and IO5. per gallon > 
 he wishes to make a mixture of 120 gallons, Avorth os. per gal- 
 lon : 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, respectively : 
 how many has he of each ? 
 
 7. A jeweler melted together four sorts of gold, of 24, 22, 
 20 and 15 carats fine, so as to produce a compound of 42o2'. of 
 17 carats fine : how much did he take of each sort ? 
 
 8. A man paid $70 to 3 men for 35 days labor : to the first 
 lie paid |5 a day, to the second, $1 a day, and to the third, $^ 
 a day : how many days did each labor ? 
 
 * See an admirable article on Alhgation, published by Professor D, 
 
 Wood, in the June number of the New ifork Teacher for 1855. B\' his 
 
 permission, I have used such parts of it as seemed appropriate to a Text 
 
 Book. 
 
 1:! 
 
282 COIN'S AND CUEEEXCIES. 
 
 COINS AND CURRENCIES. 
 
 271. Coins ai-e 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 
 gold and silver only, shall be a legal tender. 
 
 The coins of a country, and those of foreign countries having 
 a fixed value established by law, together with bank notes 
 redeemable in specie, make up what is called the Currency. 
 
 272. 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 $4,83 to $4,80, seldom reaching either 
 the lowest or highest limit. The excliange value of the Eng- 
 lish pound, is $4,44|-, and was the legal value before the change 
 in our standard. This change raised the legal value of the 
 pound to $4,84, but merchants and dealers in exchange pre- 
 ferred to retain the old value, which became nominal, and to 
 add the difference in the form of a iiremium on excInnujCy 
 which is cx]ilained in Art. •287. For the values of the various 
 coins, see Table, page 3'Jl. 
 
 271. Wliat are <!i»ins ! Wliat arc tlipv callpin What is dnclarocl in 
 
EXCHANGE. 283 
 
 EXCHANGE. 
 
 273. Exchange is a term which 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. 
 
 274. 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. 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 
 draiver 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 
 buyer or remitter. 
 
 275. Bills of exchange are the proper money of commerce. 
 Suppose Mr. Isaac Wilson of the city of N'ew York, ships 1000 
 bags of cotton, worth £G000, to Samuel Johns & Co. of Liver- 
 pool ; 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. : 
 
 regard to them ? "N^'hat is provided by the Constitution of the United 
 States 1 What do you understand by Currency 1 
 
 272. How many values may a coin be said to have 1 What is the in- 
 trinsic vakie 1 ^Vhat is its Custom House value \ What is the niercantilu 
 vahie ! What is tl^e exchange value ? 
 
 273. What is Exchange ^ How is the payment generally made ? 
 
 274. What is a Bill of Exchange 1 How many parties are there to u 
 Dill of exchange 1 Name them. 
 
 275. How do bills of exchange aid commerce"! Vame all the parties to 
 the bill in this example. 
 
284 EXCHANGE. 
 
 Exchange for £G000. New York, July 30th, 1846- 
 
 Sixty days after sight of this my first Bill of Exchange 
 (second and third of the same date and tenor unpaid*) pay to 
 David C. Jones or order, six thousand pounds sterling, with 
 or without further notice. Isaac Wilson. 
 
 3Iessrs. Samuel Johns & Co., ) 
 Merchants, Liverpool. ) 
 
 Let us now suppose that Mr. James j^urchases 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 drutcer ; Messrs. Johns & Co., the drawees ; David 
 C. Jones, the payee ; and William James, the buyer or re- 
 mitter. 
 
 276. A bill of exchange is called an inland hill, when the 
 drawer and drawee both reside in the same countiy ; and when 
 they reside in different countries, it is called a foreign hill. 
 Thus, all bills in which the drawer and drawee reside in the 
 United States, are inland bills ; but if one of them resides in 
 England or France, the bill is a foreign bill. 
 
 277. The time at which a bill is made payable varies, and is 
 a matter of agreement between the drawer and buyer. Tliey 
 may either be drawn at sight, or at a certain number of days 
 after sight, or at a certain number of days after date. 
 
 278. Days of Guace are a certain number of days granted 
 to the person who pays the bill, after the time named in the bill 
 
 276. ^^'hat is an inland bill ! What is a foreign bill ? Arc bills drawn 
 between one state and another inland, or foreign 1 
 
 277. How is the time determined at which a bill is made payable 1 How 
 are bills always drawn 1 
 
 * Three bills are generally drawn for the same amount, called the first, 
 second, and third, and together they form a set. One only is paid, and 
 then the oUier two are of no value. This arranjrement avoids the acci- 
 dents and delays incident to transmiuinr; the bills. 
 
KXCIIANGE. 285 
 
 has expired. In the United States and Great Bi-itain three 
 days are allowed. 
 
 279. In ascertaining the time when a bill payable so many 
 days after sight, or ai'ter date, actually falls due, the day of 
 presentment, or the day of the date, is not reckoned. When 
 the time is expressed in months, otlendar monlha are always 
 understood. 
 
 If the month in which a bill falls due is shorter than the one 
 hi 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, would fall due on 
 the last of February, except for the days of grace, and would 
 be actually due on the third of March. 
 
 ENDORSING BILLS. 
 
 280. In exaniining 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 
 endorse it in blank; 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 endorser. 
 
 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 endorsement, and William James is the endorsee, and he 
 may either endorse in blank or write over his signature, " Pay 
 
 27S. ^Vllat are days of grace 1 How many days of grace are allowed 
 hi this country and in Great Britain ! 
 
 279.- In ascertaining the time when a bill is payable, what days are 
 reckoned \ When the time is expressed in months, wbat kind of months 
 is under.stood ! If the month in which tlie bill falls due is shorter than 
 that ill which it is drawn, what rule i.? observed 1 
 
 280. What is an cndor.scmcnt in blank 1 What is the person making il 
 called ' What is a special endorsement ! What is tl;c effect of an en- 
 dorsement ' How may a bill drawn to bearer be transft.'rred .' 
 
28() KXCHANGK, 
 
 to tlie order of Ambrose Spooner," and the drawees, Messrs. 
 Johns & Co., will then be bound to pay ihe amount to Mr. 
 Spooner. 
 
 A bill drawn payable to bearer, may be transferred by Ti:';re 
 delivery. 
 
 ACCEPTANCE. 
 
 281. When the bill drawn on Messrs. Jolms & 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 agree- 
 ment 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 acce2)tors. 
 
 LIABILITIES OF THE PARTIES. 
 
 282. 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 en- 
 dorsers notified. Such notice is called a pro/esi, and is given 
 by a public officer called a notary, or notary public. If the 
 parties are not notified in a reasonable time, they are not respon- 
 sible for the payment of the bill. 
 
 If the drawer accepts the bill, and fails to make the payment 
 when it becomes due, the parties must be notified as before, and 
 this is called protcsti/iy the bill for non-payment. If the en- 
 dorsers are not notified in a reasonable time, they are not 
 responsible for the amount of the bill. 
 
 281. ^A'llat is an arcrptancc ! How is it inado ] 
 
 282. ^\'ll(n) (Iocs the drawee of a bill become rc.spon.sible for its pay- 
 ment! If the drawee docs not accept, what nni.st tlie holder do! What 
 is .such notice called ! 13y whom is it made! If the parties to the bill 
 are not notified, what is the consequence ! If the drawee accepts the bill 
 and fails to make Ihe payment, what must then be done ! If the bill if 
 not protested, what will be the consequence ! 
 
EXCHANGE. 287 
 
 PAR OF EXCHANGE — COURSE OF EXCHANGE. 
 
 283. The intrinsic ixtr of exchange, is a term used to com- 
 pare tlie coins of different countries with each other, witli respect 
 to their intrinsic values, that is, with reference to the amount of 
 pure metal in eacli. Tims, the English sovereign, which repre- 
 sents the pound sterling, is intrinsically worth $4,8G1 in our 
 gold, taken as a standard, as determined at the Mint in Phila- 
 delphia. This, therefore, is the value at which the sovereign 
 must be reckoned, in estimating the par of exchange. 
 
 284. The commercial jxn- of exchanr/e is a comparison of the 
 coins of different countries according to their market value 
 Tiius, as the market value of the English sovereign varies from 
 $4,8.j to $4,85 (Art. 272), the commercial par of exchange 
 will fluctuate. It is, however, always determined when we 
 know the value at which the foreign coin sells in our market. 
 
 285. 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 difiers from the intrinsic par of exchange, 
 and also from the commercial par, in the same way that the 
 market price of an article differs from its natural price. Tlie 
 commercial par of exchange would at all times determine the 
 course of exchange, if there were no fluctuations in trade. 
 
 286. 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 
 
 283. What do you understand by the intrinsic par of exchange 1 \A'hat 
 is the intrinsic value of the English sovereign 1 
 
 284. What is the commercial par of exchange \ What is the comnicr 
 cial value of the English sovereign 1 
 
 285. What do you understand by the course of exchange ? How does 
 it differ from the intrinsic par and the commercial par ! What causes it to 
 differ from the commercial par ! 
 
 286. ^^'hat is said when the price of a foreign bill is above the commer- 
 cial par 1 When is it below it ? To whom is a favorable state of exchange 
 advantageous 1 To whom is it injurious \ 
 
288 EXCHANGE. 
 
 place has sold more than it has bought, and that specie must be 
 shipped to make up the difference. "When the market price is 
 beloio this par, exchange is said to be below par, 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 favorable state of exchange is 
 advantageous to the buyer but not to the seller, whose interest, 
 as a dealer in exchange, is identified with that of the place on 
 which the bill is drawn. 
 
 287. It was stated in Art. 272, that the exchange value of 
 the pound sterling is $4,44^ — 4,4444 4- ; that is, this value is 
 the basis on \\ liich the bills of exchange are drawn. Now this 
 value being 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 tlie 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 fluc- 
 tuations of trade and other accidental circumstances. 
 
 INLAND BILLS. 
 
 288. We have seen that inland bills are those in which the 
 drawer and drawee both reside in the same country (Art. 276), 
 
 EXAMI'LES. 
 
 1. A merchant at New Orleans wishes to remit to New York 
 $8405, and exchange is 1^ per cent premium. How much 
 must he pay for such a bill ? 
 
 287. What is the exchange value of the pound slerhng 1 
 
 288. What are inland bills 1 
 
EXCHANGE. 289 
 
 2. A merchant in Boston wishes to pay in Phihulelphia 
 $8746,50 ; exchange between Boston and Philadelphia is 1} per 
 cent below par. What must he pay for a bill ? 
 
 0. A merchant in Philadelphia wishes to pay $9876,40 in 
 Baltimore, and finds exchange to be 1 per cent below par : what 
 must he pay for the bill ? 
 
 ENGLAND. 
 
 289. It has already been stated that llie exchanges between 
 this country and England are made in pounds, shillings, and 
 pence, and that the exchange value of the pound sterling is 
 $4,44|-, and tliat the premiums are all reckoned from this 
 standard, 
 
 EXAMPLES. 
 
 1. A merchant in New York wishes to remit to Liverpool 
 £11 G7 10s. 6c?., exchange being at 8i per cent premium. 
 How much must he pay for the bill in Federal money ? 
 
 First, £1167 10s. 6c?. - - =£1167.525 
 
 For multiply by 8^ per cent, - .085 
 
 the product is the premium - - = 99.239625 
 
 the product added gives - - £1266.764625 
 
 which reduced to dollars and cents at the rate of $4,44f to the 
 pound, gives the amount which must be paid for the bill in 
 dollars and cents. 
 
 2. A m.erchant has to remit £36794 8s. dd. to London, how 
 much must he pay for a bill in dollars and cents, exchange being 
 7|- 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 the premium to the exchange value of 
 the pound, viz. S4,44f , which in this case gives $4.84444, and then 
 divide the amount in dollars by ihis sum, and the quotient will bo 
 tlie amount of the bill in pounds and the decimals of a pound. 
 
 289. In what currency are the excManges lietween this country and 
 England made ? A\'hat is the exchange value of the pound sterling 1 
 
290 EXOHANUE. 
 
 4. A mercbant in New York owes £1256 18s. 9c?. in London ; 
 exchange at a nominal premium of l-h per cent : how much 
 money, in Federal currency, will be necessary to purchase the 
 
 bill? 
 
 5. I have S947,8G and wish to remit to London £364 IBs. 8a'., 
 exchange being at 8i per cent : how much additional money 
 will be necessary ? 
 
 FRANCE. 
 
 290. The 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 tht 
 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 com- 
 puted on the value 18.6 cents, but is often quoted by stating 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 hundreds of a franc. 
 
 EXAMPLES. 
 
 1. A merchant in New York wishes to remit 167556 francs 
 to Paris, exchange being at a premium of 11 per cent. What 
 will be the cost of his bill in dollars and cents ? 
 
 Commercial value of the franc, - - 18.6 cents, 
 Add li per cent, - - - - .279 
 
 Gives value for remitting, - - 18.879 cents; 
 
 then, 167556 x 18.879 := $31632,89724, 
 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 = $17326,274 +, the amount. 
 Is this bill above or below par ? What per cent ? 
 
 290. In what currency are the exchanges with France conducted^ 
 What is a centime ? "What is the vahie of a franc I \\hi\l is meant Then 
 exchange on Paris is quoted at 5 francs 40 centimes 1 
 
EXCHANGE. 291 
 
 3. How much money must be paid to purchase a bill of ex- 
 change on Paris for G8097 francs, exchange being 3 per cent 
 below par ? 
 
 4. A merchant in New York wishes to remit $16785,25 to 
 Paris ; exchange gives 5 francs 4 centimes to the dollar : how 
 much can he remit in the currency of Paris ? 
 
 HAMBURG. 
 
 291. Accounts and exchanges with Hamburg are generally 
 made in the marc banco, valued, as we see in the tabic, 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 ? 
 
 ARBITRATION OF EXCHANGE. 
 
 292. 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 
 the first and last are compared. 
 
 293. When there is but one intervening currency it is called 
 Simple Arbitration ; and when there is more than one it is called 
 Conqiound Arbitration. The method of performing this is called 
 the Chain Rule. 
 
 291 In what are accounts kept at Hamburgh 1 What is the value o/ 
 the marc banco 1 
 
 292. What is arbitration of exchange ? . 
 
 293. When there is but one intervening currency, what is the exchange 
 called 1 When there is more than one, what is it called 1 
 
232 AEBITKATION OK EXCHANGE. 
 
 29 i. The pi'inciple involved iii 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. 
 
 1. Suppose it were required to exchange 109150 pence into 
 dollars, by first changing them into shillings, then into pounds, 
 and then into dollars. 
 
 OPERATION. 
 
 109150^. = 109150 X j\s. 
 109150^. = £109150 X Jg- X 2V 
 109150(;. = $109150 X Jj X Jo X 4-444 = $201,924. 
 
 Analysis. — Since 12 pence make 1 shilling, there will be one- 
 twelfth as many shillings as pence : since 20 shillings make 1 pound, 
 there will be one-twentieth as many pounds as shillings ; and since 
 there are ^4.444 in a pound, there will be as many dollars as result 
 from taking the pounds, 4.444 times; that is, S201.924. 
 
 2. Let it be required to remit $G570 to London, by the way 
 of Paris, excliange on Paris being 5 francs 15 centimes for 
 $1, and the exchange from Paris to London, 25 francs and 80 
 centimes for £1 : what will be the value of the remittance at 
 London ? 
 
 OPERATION. 
 
 $6570 = G570 x 5.15 francs. 
 $6570 = $6570 x 5.15 x j^'-r-q = =£131 3^. lOfc/. 
 
 Analysis. — Since 5 francs and 15 centimes are equal to Si. there 
 ■will be as many francs at Paris as are equal to 6570 taken 5.15 
 times ; and since £l at London is equal to 25.80 francs, there will 
 be as many pounds as 25.80 is contained times in the last product; 
 that is, jCl31 lis. 10|ri. Hence, the following, Avhich is called the 
 Chain Rule : 
 
 Miildph/ the Stan to be remitted by the foUoioina quotienls: 
 By a certain amount at the second place divided by its equivalent 
 
 2!)4. What principle is involved in the arbitration of exchange 1 What 
 is till! chain rule ? Give the rule. 
 
AKBITRATION OF EXCHANGE. 293 
 
 at Ihe first ; by a certain amount at the third place by its equiva- 
 lent at the second ; by a certain amount at the fourth 2'>loce divi- 
 ded by its equivalent at the third, and so on to the last place. 
 
 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. Now, the 
 exchange between Paris and London is 25 francs 80 centimes 
 for £1 sterling, and between Hamburg and London 13|- marcs 
 banco for £1 sterling. How had he better remit ? 
 
 1st. To London direct. 
 
 The amount to be remitted is $4888,40, The exchange 
 value of £1 is $4,444, and since the exchange is at a premium 
 of 10 per cent, the value of £1 is $4,444 + ,4444 r= $4,8884 : 
 hence, 
 
 $4888,40 X T.sVsT = ^1000 : 
 
 hence, if he remits direct he will obtain a bill for £1000. 
 
 2d. Exchange through Paris. 
 1.03 
 
 4888,40 X ^ X j^ ^ £975,7852 = £975 15s. 8}d. 
 5.16 
 
 Analysis. — Since 5.15 francs are equal to 1 dollar, the first mul- 
 tiplier will be this amount divided by $1 ; and since £l is equal to 
 25.80 francs, the second multiplier will be £^1 divided by this amount. 
 Then, by dividing by 5 and multiplying, we find that the amount 
 remitted by the second method would be worth, at London, £^975 
 t5s. S^^. 
 
 3d. Exchange through Hamburg. 
 
 $4888,40 X .3J3 X 13^.^5 =^1015.771 = £1015 15s. 5d. 
 
 Analysis. — Since 1 marc banco is equal to 35 cents, it is 35 hun- 
 dredths of a dollar : hence, the first multiplier is 1 marc banco 
 
294 ARBITRATION OF EXCHANGE. 
 
 divided by .35, and the second, 1 divided by 13.75. By tlii.s course 
 of exchange the remittance at London would be worth £1015 1 5s. 5d. 
 
 Hence, the best way to remit is through Hamburg, then 
 direct, and the least advantageous, through Paris. 
 
 2. A merchant in London has sold goods in Amsterdam to 
 the amount of 824 pounds Flemish, which could be remitted 
 to London at the rate of 34s. 4rf. Flemish per pound sterling. 
 He orders it to be remitted circuitously at the following rates, 
 viz. : to France at the rate of 48(/. Flemi.sh per crown ; thence 
 to Vienna at 100 crowns for 60 ducats ; thence to Hamburg at 
 lOOd. Flemish per ducat ; thence to Lisbon at 50c/. Flemish 
 per crusado of 400 reas ; and lastly, from Lisbon to England 
 at 5.y. 8'J. per milrea : does he gain or lose by the circular 
 exchange ? 
 
 48(/. Flemish = 1 crown, 
 
 100 crowns = GO ducats, 
 
 1 ducat =: 100(7. Flemish, 
 
 50d. Flemish = 400 reas, 
 
 1 milrea or 1000 reas = G8d. sterlinjr. 
 
 o~- 
 
 *o- 
 
 a:0 $ 17 
 
 1 00 100 00 ^$ 824x17 14008 
 ^$ i00 1 00 1000 ~ 25 ~ 25 
 
 $ m 
 
 25 
 = £560 6s. 4frf. 
 
 The direct exchange would give, 
 
 82i X ^-i j:^. — = 824 X If"- = £480 sterhng. 
 
 34s. 4c?. Flemish '^ ** ° 
 
 Hence, the amount gained by circuitous exchange wouliJ b" 
 £80 6s. |rf. 
 
GENERAL AVEKAGE. 295 
 
 GENERAL AVERAGE. 
 
 295. Average is a term of commerce and navigation, to 
 signify a contribution by individuals, wliere tlie goods of a par- 
 ticular merchant are thrown overboard in a storm, to save the 
 ship from sinking, or wliere the masts, cables, anchors, or other 
 furniture of the ship are cut away or destroyed for the preser- 
 vation of the whole. In these and like cases, where any sacri- 
 fices are deliberately made, or any expenses voluntarily incurred, 
 to prevent 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 freiglit, and the cargo, so 
 that the loss may fall proportionably on all. The amount sacri- 
 ficed is called the jeidsoji. 
 
 296. 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 par- 
 ticular sacrifice is made for the sake of the ship only or the 
 cargo only, these losses must be borne by the parties imme- 
 diately interested, and are consequently defrayed by a -particular 
 average. There are also some small charges called pjetty or 
 accustomed averages, one-third of which is usually charged ta 
 the ship, and two-thirds to the cargo. 
 
 No general average ever takes place, except it can be shown 
 that tlte danger was imminent, and that the sacrifice was made 
 indiqoensahle, or sup)-posed to be so by the captain and officers, 
 fur the safety of the ship. 
 
 297. In different countries different modes are adopted of 
 valuing the articles which are to constitute a general average. 
 
 295. \Miat does the term average signify 1 
 
 296. How many kinds of average are there ! What are the small 
 charges called 1 Under what circumstances will a general average take 
 place 
 
296 GENERAT> AVERAGE. 
 
 In general, hoTvever, the value of" the freightage is held to be 
 the clear sura which the ship has earned after seamen's wages, 
 pilotage, and all such other charges as came under the name 
 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 Avoukl have brought in ready money at tlie j^loce of delivery, 
 on the ship's arriving there, freight, duties, and all other charges 
 being deducted : indeed, they bear their proportions, the same 
 as the goods saved. The ship is valued at the price she would 
 bring on her arrival at the port of delivery. But when the loss 
 of masts, cables, and other furniture of the ship 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, Avas 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 - $300 ) gOQ 
 Deduct one-third, . - - 100 ) 
 
 Expenses of getting the ship off the sands, 56 
 
 Pilotage and port duties going in and out | jqq 
 
 of the harbor, commissions, &c., - ) 
 
 Expenses in port, ----- 25 
 
 Adjusting the average, - - - - 4 
 
 Postage, - - - - - - - 1^ 
 
 Total loss, $1186 
 
GENERAL AVERAGE. 297 
 
 ARTICLES TO CONTRIBUTE. 
 
 Goods of A cast overboard, - - - - -$500 
 
 Value of B's goods at N. O., deducting freight, &c., 1000 
 
 " of C's " « « « 500 
 
 " of D's « « « « 2000 
 
 " of E's « « • « « 5000 
 
 Value of the ship, 2000 
 
 Freight, after deducting one-third, - - - - 800 
 
 S1L800 
 
 Then, total value : total loss : : 100 : per cent of loss. 
 $11800 : 1180 :: 100 : 10; 
 
 hence, each loses 10 per cent on the value of his interest in the 
 cargo, ship, or freight. Therefore, A loses 850 ; 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 aix3 to receive their disbursements from the contribu- 
 tion, 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 wiU 
 actually receive $200. Hence, the account will stand : 
 The owners are to receive . - - _ _ $200 
 A loses 8500, and is to contribute $50 ; hence, he 
 
 he-) 
 
 y 450 
 
 receives - - - - - "- " ) 
 
 B loses $200, and is to contribute $100 ; hence, he "> 
 
 ' y 100 
 
 receives ------- t 
 
 Total to be received, . - - $750 
 
 C $ 50 
 
 C, D, and E, have lost nothing, and are to pay -l D 200 
 
 E 500 
 
 Total actually paid, - - - - $750 ; 
 
 297. How is the freight valued 1 How much is charged on account o( 
 the seamen's wages 1 How is the cargo valued 1 Does the part lost beaj 
 its part of the loss 1 How is the ship valued 1 When parts of the ship 
 are lost, how arc they compensated for 1 How do you explain the example ' 
 
 5 
 
298 TONNAGE OF VESSELS. 
 
 60 that the total to be paid is just equal to the total loss, as it 
 should be, and A and B get their remaining and injured goods, 
 and the three others get theirs in a perfect state, after paying 
 their ratable proportion of the loss. 
 
 TONNAGE OF VESSELS. 
 
 298. There are certain custom house charges on vessels, 
 which are made according to their tonnasre. The tonage of a 
 vessel is the number oi, tons weight she will carry, and this is 
 determined by measurement. 
 
 [From the " Digest," by Andrew A. Jones, of the N. Y. Custom House]. 
 
 Custom house charges on all ships or vessels entering from any foreign 
 
 port or place. 
 
 Ships or vessels of the United States, having three-fourths of 
 
 the crew and all the officers American citizens, per ton, SO, 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 
 
 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 slate to a 
 port in another state, other than an adjoining state, unless the officers 
 and three-fourths of the crew are American citizens : to ascertain 
 •which, the crew's 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 tc 
 more than one port of the United States, she may proceed coastwise 
 with a certified manifest until her voyage is completed. 
 
 298. What is the tonnage of a vessel ] What are the custom house 
 charges on the dilTercnt classes of vessels trading with foreign countries f 
 To what charges arc coasters subject 1 
 
TONNAGE OF VESSELS. 299 
 
 299. The government estimate the tonnage according to one 
 rule, while the ship carpenter wlio builds the vessel uses another. 
 
 Government Rule. — I. Measure, in feet, above the vpper 
 deck the length of the vessel, from the fore part of the main stem 
 to the after part of the stern-post. Then measure tlte breadth 
 taken at the ividest part above the main ivale on the outside, and 
 the depth from the under side of the deck-plank to the ceiling in 
 the hold. 
 
 11. From tlte length take three-fifths of the breadth and mul- 
 tiply the remainder by the breadth a?id depth, and the product 
 divided by 95 will give the tonnage of a single decker ; and the 
 same for a double decker, by merely making the depth equal to 
 half tlte breadth. 
 
 Carpenters' Rule. — Multiply together the length of the 
 keel, the breadth of the main beam, and the depth of the hold, 
 and the product divided by do will be the carpenters^ tonnage for 
 a single decker ; and for a double decker, deduct from the depth 
 of the hold half the distance betiveen decks. 
 
 EXAMPLES. 
 
 1. What is the government tonnage of a single decker, whose 
 length is 75 feet, breadth 20 feet, and depth 17 feet ? 
 
 2. What is the carpenters' tonnage of a single decker, the 
 length of whose keel is 90 feet, breadth 22 feet 7 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, 
 after deducting half between decks, 14 feet 8 inches ? 
 
 4. What is the government tonnage of a double decker, the 
 length being 103 feet, breadth 25 feet 6 inches ? 
 
 5. What is the carpenters' 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 ? 
 
 299. What is the government rule for finding the tonnage 1 What the 
 fchipbuilders' rule 1 
 
800 INVOLUTION. 
 
 INVOLUTION. 
 
 300. A POWER is the product of two or more equal factors. 
 Either of the equal factors is called the root of the power. 
 
 The root, in Algebra, is Q.dl\eAi\\Q first poiver. 
 The second lyower is the j^roduct of the root by itself: 
 The third power is the product when the root is taken 3 times 
 as a factor : 
 
 The fourth poioer, when it is taken 4 times : 
 The fifth power, when it is taken 5 times, &;c. 
 
 301. 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 root or base. 
 
 3- = 3 X 3 = 9 the 2d power of 3. 
 
 33 = 3 x3x3i^ 27 the 3d power of 3. 
 
 3* = 3x3 x3x3= 81 the 4th power of 3. 
 
 35 = 3 X 3 x 3 X 3 X 3 = 243 the 5th power of 3. 
 
 Involution t's /^e operation of finding the powers of numhers. 
 
 Note. — 1. There are three things connected with every power : 
 1st, The root; 2d, The exponent : and 3d, The .power or result of Iho 
 multiplication. 
 
 2. In finding any power, one multipliccation gives the 2nd power: 
 hence, iha number of multiplications is 1 less than the exponent. 
 
 Rule. — Multiply the mimber i?tto itself as many times less 1 
 as there are units in the exponent, and the last 'product will be 
 the power. 
 
 300. What is a power 1 What is the root of a power 1 What is the 
 first power ! Wliat is the second power ] The third power ? 
 
 301. What is the exponent of the power ? How is it written 1 What 
 is Involution ; How many things are connected with every power \ How 
 do you find the power of a number ? 
 
EVOLUTION. 
 
 '601 
 
 EXAMPLES. 
 
 Find the power of the following numbers : 
 
 1. The square of 4? 
 
 2. The square of 15? 
 
 3. Tlie square of 26 ? 
 
 4. The square of 142 ? 
 
 5. The square of 4G3 ? 
 G. Tlie square of 1340? 
 
 7. The square of 24. G ? 
 
 8. The square of .526 ? 
 
 9. The square of 3.125 ? 
 
 10. The square of .0524 ? 
 
 11. The square of 246.25 ? 
 
 12. The square of -I ? 
 
 13. The square of |^? 
 
 14. The square of -g-? 
 
 15. Tlie square of }|-? 
 
 16. The square of -| j ? 
 
 17. The square of yjy ? 
 
 18. The square of 24 ? 
 
 19. The square of 7|- ? 
 
 20. The square of 15-j\ ? 
 
 21. The square of 225^%? 
 
 22. The cube of 6 ? 
 
 23. The cube of 24 ? 
 
 24. The cube of 72 ? 
 
 25. The cube of 125 ? 
 
 26. The cube of 136? 
 
 27. The 4th power of 12 ? 
 
 28. The 5th power of 9 ? 
 
 29. The value of (4.25)^? 
 
 30. The value of (1.8)'? 
 
 31. The value of (32.4)3? 
 
 32. The valu« of (.45)5 p 
 
 33. The value of (If)^? 
 
 34. The cube of (|) ? 
 
 35. The 4th power of | ? 
 
 36. The cube of 14f? 
 
 37. The value of (21)^? 
 
 38. The value of (|f)^? 
 
 39. The value of (24|)3 ? 
 
 40. The value of (.25)6 ? 
 
 41. The value of (142.5)3? 
 
 42. The value of (3.205)- ? 
 
 EVOLUTION. 
 
 302. Evolution is the operation of finding the root when 
 know the power. 
 
 The Square Root of a number is the factor which multi- 
 plied by itself 07ice will produce the number. 
 
 The Cube Eoot of a number is the factor which multiplied 
 by itself taice will produce the number. 
 
 302. What is Evolution! What is the square root of a number 7 \^'hat 
 IS the cube root of a number 1 How do you denote the square root of a 
 number ! How the cube rooti 
 
 14 
 
302 EXTRACTION OF THE SQUARE ROOT. 
 
 Thus, 8 is the square root of 64, because 8 X 8 = 64 ; and 
 3 is the cube root of 27, because 3 X 3 x 3 = 27. 
 
 The sign J is called the radical sign. "When placed before 
 a number it denotes that its square root is to be extracted. 
 Thus, yST = 6. 
 
 "VVe denote the cube root by the same sign with 3 written 
 over it : thus, ^/^, denotes the cube root of 27, which is equal 
 to 3. The small figure 3, placed over the radical, is called the 
 index of the root. 
 
 EXTRACTION OF THE SQUARE ROOT. 
 
 303. The square ro t of a number is one of the two equal 
 factors of that number. To extract the square root is to find 
 this factor. The first ten numbers and their squares are 
 
 1, 2, 3, 4, 5, G, 7, 8, 9, 10. 
 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. 
 The numbers in the first line are tlie square roots of those in 
 the second. The numbers 1, 4, 9, 16, 25, 36, &c., having two 
 exact equnl factors, are called perfect squares. 
 
 A perfect square is a number which has two exact equal factors. 
 A perfect square is a number which has two exact factors. 
 
 Note. — The square root of a number less than 100 will be less 
 than 10, while Ihc square root of a number greater than 100 i\ill be 
 greater than 10. 
 
 304, To find the square root of a nunxber. 
 
 1. "What is the square of 36 = 3 tens + 6 units ? 
 
 Analysis. — The square of 36 is found by- 
 taking 3fi thirty-six times; and this is done 
 by first taking it (5 units times and then 3 tens 
 times, and adding the products. 3G taken 
 6 units times, gives 3 x G + 6^ : and taken 
 3 tens times gives 3= + 3 X 6. and their sum 3" + 2(3 X fi) + G" 
 is 3* + 2 (3 X G) + 6" : that is, 
 
 303. What is the square root of a number \ What is a perfect square'? 
 How many perfect squares arc there between 1 and 100 ] How do the 
 roots of numbers less than 100 compare with 10 ! 
 
 ;J0'1. Into wliat parts may every inmibcT be decomposed ! Wlicn so 
 J-ffiMiiiiuscd what is its square equal to ! 
 
 OPERATION. 
 
 
 3 + 6 
 
 
 3 + 6 
 
 3 
 
 X 6 + 6= 
 
 3^+3 
 
 X 6 
 
EXTRACTION OF THE SQUAKE ROUT. 
 
 303 
 
 F 
 
 30 
 
 I 
 
 
 
 30 
 
 
 6 
 
 
 
 6 
 
 
 6 
 
 TT 
 
 
 180 
 
 
 3G 
 
 
 
 30 
 
 E 
 
 
 O 
 
 
 
 o 
 
 
 n 
 
 
 80 
 30 
 
 900 
 
 C8 
 
 30 
 6 
 
 180 
 
 D 
 
 K 
 
 M 
 
 A 
 
 30 
 
 B 
 
 T/ie square of a nicmher is equal to the square of the tens, vlus 
 twice the 'product of the tens hy the units, jjIus the square of ''.'C 
 units. 
 
 The same may be shown by a diagram : 
 
 Let the line AB re- 
 present the 3 tens or 30, 
 and BC the six units. 
 
 Let AD be a square 
 on AC, and AE a square 
 on the ten's line AB. 
 
 Then ED will be a 
 square on the unit line 
 6, and tlie rectangle EF 
 ■will be the product of 
 HE, which is equal to 
 the ten's line, by IE, 
 which is equal to the 
 unit line. Also, the 
 
 rectangle BK will be the product of EB. which is equal to the ten's 
 line, by the unit line, BC. But the whole square on AC is made up 
 of the .'^quare AE, the two rectangles FE and EC and the square ED . 
 hence, it is equal to 
 
 The square of the tens plus t/rice the product of the tens hy the 
 units, plus the square of the units. 
 
 1. Let it nov/ 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 hundred, it follows that the square of the tens of the 
 required root must be found in the two figures on the left of 25. 
 Hence, beginning at the right, we point off the number into periods 
 of two figures eacli. 
 
 We next find the greatest square contained in 
 20 tens, which is 4 tens or 40. We then square 
 4 tens which gives 16 hundred, and then place 
 16 under the second period, and subtract ; this 
 takes away the square of the tens, and leaves 
 425, which is twice the product of the tens by the 
 units plus the square of the units. 
 
 
 OPERATION. 
 
 
 20 
 
 2'c 
 
 (85 
 
 
 16 
 
 
 
 8 
 
 5)42 
 
 5 
 
 
 
 42 
 
 5 
 
 
o04 EXTRACTION OF THE SQUARE ROOT. 
 
 If now, we double the tens and then divide the remainder exclu- 
 sive 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 augmented divisor, 
 and then multiply the whole divisor tiius increased by it. the pro- 
 duct will be twice the tens by the units plus the square of the 
 units : and hence, we have found both figures of the root. 
 
 This process may also be illustrated by the diagram. 
 
 Suppose AC = 45. Then, subtracting the square of the tens is 
 taking away the square AE, and leaves the two rectangles FE and 
 BK, together with the square ED on the unit line. 
 
 The two rectangles FE and BK representing the product of units 
 by tens, can be expressed by no figures less than tens. 
 
 If then, "we divide the number 42, at the left of 5. by twice the 
 tens, that is, by twice AB or BE, the quotient will be BC or EK, the 
 unit of the root. 
 
 Then, placing BC or 5, in the root, and also annexing it to the 
 divisor doubled, and then multiplying the whole divisor 85 by 5. wc 
 obtain the two rectangles FE and CE, together with the square ED 
 
 305. Hence, for the extraction of the square root, we nave 
 the following 
 
 lluLE. — I. Separate tlie given number into periods of tico 
 figures each, hy setting a dot over the pfoce of units, a second 
 over the place of hundreds, and so on for each cdternate figure 
 to the left. 
 
 II. Note the greatest square contained in the j^eriod on the left, 
 and place its root on the rigid after the manner of a quotient in 
 division. Subtract the squai-e of this root from the first period, 
 and to the remainder bring down the second period for a divi- 
 dcnd. 
 
 III. Double the root thus found for a trial divisor and place 
 ii on the left of the dividend. Firid hoiv 7nang times the trial 
 divisor is contained in the dividenJ, exclusive' of the right-hand 
 
 305. What is the first step in extracting the square root of numliors \ 
 Wlr.t is tlic second 1 What is the third 1 Wliat the fourth ? \Vlial tho 
 iW'ili \ (Ilvo tiic entire rule] 
 
EXTKACTION OF Tnii SQUARE BOOT. 305 
 
 figure, and place the quotient in the root and also annex it to the 
 divisor. 
 
 IV. Midtiply the divisor thus increased, hi/ the last figure of 
 the root; subtract the product from the dividend, and to the 
 remainder bring down the next period for a new dividend. 
 
 V. Boxdjle the whole root thus found, for a new trial divisor, 
 and contimie the operation as before, until all the periods are 
 brought down. 
 
 EXAMPLES. 
 
 1. What is the square root of 425104? 
 
 Analysis. — We first place a dot over the operation. 
 
 4, making the right-hand period 04. We 42 51 04(652 
 
 then put a dot over the 1. and also over the 36 
 
 2, making three periods. 125)651 
 
 The greatest perfect square in 42 is 36, 625 
 
 the root of which is 6. Placing 6 in the 1302)2604 
 root, subtracting its square from 42, and 2604 
 
 bringing down the next pcriod"51. we have 
 
 651 for a dividend, and by doubling the root we have 12 for a trial 
 divisor. Now, 12 is contained in Go., 5 times. Place 5 both in the 
 root and in the divisor; then multiply 125 by 5 ; subtract the pro- 
 duct and bring down the next period. 
 
 We must now double the whole root 65 for a new trial divisor; or 
 we may take the first divisor after having doubled the last figure 5; 
 then dividing, we obtain 2. the third figure of the root. 
 
 Notes. — 1. Tlie left-hand period may contain but one figure ; each 
 of the others will contain two. 
 
 2 If any trial divis^or is greater than its dividend, the correspond- 
 ing quotient figure will be a cipher. 
 
 3. If the product of the divisor by any figure of the root exceeds 
 the corresponding dividend, the quotient figure is too large and must 
 be diminished. 
 
 4. There will be as many figures in the root as there are periods 
 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 casoj 
 
 periods of ciphers may be annexed, forming new periods, each of 
 
 which will give one decimal place in the root. 
 
 14 
 
306 
 
 EXTEACTION OF THE SQUARE KOOT. 
 
 2. What is the square root of 758692 ? 
 
 OPERATION. 
 
 75 86 92(871.029 +. 
 64 
 
 Note. — After using all the 
 periods of the given number, 
 we annex periods of decimal 
 ciphers, each of which gives 
 one decimal place in the quo- 
 tient. 
 
 167)11 
 11 
 
 86 
 69 
 
 1741)17 92 
 17 41 
 
 174202)510000 
 348404 
 
 1742049)16159600 
 15678441 
 
 481159 Rem, 
 
 306. To extract the square root of a fraction 
 
 1. "What is the square root of .6 ? 
 
 Note. — We first annex one cipher to make 
 even decimal places; for. one decimal multi- 
 plied by itself will give two 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 ? 
 
 Note. — The square root of a fraction 
 is equal to the square root of the nume- 
 rator divided by the square root of the 
 denominator. 
 
 3. What is the square root of |^ ? 
 
 NoTK. — When the terms are not per- 
 fect square.", reduce the common fraction 
 to a decimal, and then extract the square 
 root of the decimal. 
 
 OPERATION. 
 
 .60(.774 + 
 
 49 
 
 147)1100 
 
 1029 
 
 1544)7100 
 
 6176 
 
 924 rem. 
 
 OPERATION. 
 
 fxS. Vl 6 _ 4 
 
 2^ -V2^-** 
 
 OPERATION. 
 
 f = .75 ; 
 yf'= y/^ = .8545 •+• 
 
 SOf). How do you extract the square root of a decimal number \ How 
 of a coiiirnon fraction \ 
 
EXTKACTION OF THE SQUARE KOOT. 
 
 307 
 
 Rule. — I. If ike fraction is a decimal, point off the periods 
 from the decimal point to the rir/lU, annexing cipliers if neces- 
 sary, so that each period shall contain two places, and then ex- 
 tract the root as in integral numbers. 
 
 11. If the fraction is a common fraction, and its terms perfect 
 squares, extract the square root of the numerator and denomina- 
 tor separately ; if they are not perfect squares, reduce the frac- 
 tion to a decimal, and then extract the square root of the result. 
 
 EXAMPLES. 
 
 What are the square roots of the following numbers ? 
 
 1. Square root of 49 ? 
 
 2. Square root of 144 ? 
 
 3. Square root of 225 ? 
 
 4. Square root of 2304 ? 
 
 5. Square root of ff ? 
 
 6. Square root of o^^^ ? 
 
 7. Square root of .0196? 
 
 8. Square root of 6.25 .'' 
 
 9. Square root of 278.89? 
 
 10. Square root of 6275025 ? 
 
 11. Square root of 7994 ? 
 
 12. Square root of .205209 ? 
 
 13. Square root of | ? 
 
 14. Square root of ^|? 
 
 15. Square root of Jg- ? 
 
 16. Square root of .60794 ? 
 
 17. Value of ^-022201 ? 
 
 18. Value of -^^25.1001 ? 
 
 19. Value of -v/196.425 ? 
 
 20. Value of VTS ? 
 
 21. Value of VffH ? 
 
 22. Value of ^ ? 
 
 23. Value of 
 
 25 • 
 
 24. Value of yT35 ? 
 
 25. Value of -^19000 ? 
 
 26. Value of yT784? 
 
 27. Square root of 5647.5225 ? 
 
 28. Square root of 160048.003 6! 
 
 APPLICATIONS IN SQUARE ROOT. 
 
 307. A triangle is a plain figure Avhich has three sides and 
 three angles. 
 
 If a straifjht line meets another straio;ht line, 
 
 C5 7 
 
 making the adjacent angles equal, each is called 
 a right angle ; and the lines arc said to be per- 
 pendicular to each other. 
 
 '.507. Wliat is a triangle ? What is a iglit angle 
 
308 
 
 EXTKACTION OF THE SQUAKE KOOT. 
 
 308. A right angled triangle is one which 
 has one right angle. In the right angled tri- 
 angle ABC, the side AC opposite the right 
 angle B, is called the hypotheniise ; the side 
 AB the base ; and the side BC the perpen- 
 dicular. 
 
 309. In a right angled triangle the square described on the 
 hypothenuse is equal to the sum of the squares described on the 
 other two sides. 
 
 Thus, if ACB be a right 
 angled triangle, right an- 
 gled 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 carpen- 
 ter's theorem. By count- 
 ing 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 numbers 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. 
 
 310. When the base and perpendicular are known, to find the 
 hrjuotlienusc. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 D 
 
 
 — 
 
 
 
 
 
 
 
 
 
 
 308. What is a ri^ht angled triangle ? Which side is the hypothenuse] 
 
 309. In a right angled triangle, what is the square on the hyjjolhenuso 
 fq\ial to 1 
 
 310. How do you find the hypothenuse when you know the bise and 
 perpendicular 1 
 
EXTRACTION OF THE SQUARE KOOT. 309 
 
 1. "Wishing to know the distance from A 
 to the top of 11 tower, I measured the lieight 
 of the tower and found it to be 40 feet ; also 
 the distance from A to B, and found it 30 
 feet : Avhat was the distance from A to C ? 
 AB = 30 ; AB- = 30^ = 900 
 BC = 40 ; BC2 = 402 ^ iGOO 
 AC- = AB2 + BC2 = 900 +1600 
 AC = y'2500 =: 50 feet. 
 
 Rule. — Square the base and square the 2Jei-pendicular, add 
 the results, ami then extract the square root of their su)7i. 
 
 311. To Jiiid one side ivhen we know the hypothenuse and the 
 other side. 
 
 1. The length of a hidder which will reach from the middle 
 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 s-trect and the 
 height of the house, the square of the length of the ladder diminished 
 by the square of half the width of the street will be equal to the 
 square of the heighl of the house : hence, 
 
 EuLE. — S'juare the hypothenuse and the hnoion side, and take 
 the difference ; the square root of the difference will be the other 
 side. 
 
 EXAMPLES. 
 
 1. A general having an army of 117649 men, Avished 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 centre of a square garden, there is an artificial 
 circular pond covering an area of 810 square feet, which is y^ 
 
 311. When j'ou know the hypothenuse and one side, how do you find 
 lh(.- other i;ide ! 
 
310 EXTK ACTION OF THE SQUARE EOOT. 
 
 of the whole garden : how many rods- of fence will enclose tho 
 garden ? 
 
 4. Let it be required to lay out 67 J. 2R. 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 3200 dwarf pear 
 trees. He has a field which is twice as long as it is wide which 
 he appropriates to this purpose, setting the trees 12 feet apart- 
 each way : how many trees will there be in a row, each way, 
 and how much land will they occupy ? 
 
 G. There is a wall 45 feet- high built upon the bank of a 
 stream GO feet wide : how long must a ladder be that will reach 
 from the outside of the stream to the top of the wall ? 
 
 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 loot 
 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 Q)b 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 higli 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 tlie part standing ? 
 
 10. What will be the distance from corner to corner, tln-ough 
 the centre of a cube, whose dimensions are 5 feet on a side ? 
 
 1 1 . Two vessels start from the same point, one sails due 
 north at the rate of 10 miles an liour, 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, llian if it were in the form of a square, the cost of thu 
 fence being $2.50 a rod ? 
 
CUBE KOOT. 311 
 
 13. "^Yhat is the diameter of a cylindrical resei'voir contain- 
 ins: 9 limes as much water as one 25 feet in diameter, the 
 heiijhts beino; the same ?* 
 
 14. If a cylindrical cistern 8 feet in diameter will hold 120 
 barrels, what must be the diameter of a cistern of the same depth 
 to hold 1500 barrels? 
 
 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 
 discharo-e IGOO gallons in the same time? 
 
 IG. "What length of rope must be attached to a halter 4 feet 
 long that a horse may feed over 21 acres of ground ? 
 
 17. Three men bou2;ht a si-rindstone, Avhich was four feet in 
 diameter : how much must each grind off to use up his share 
 of the stone ? 
 
 CUBE ROOT. 
 
 312. The Cube Root of a number is one of three equal 
 factors of the number. 
 
 To extract the cube root of a number is to find a factor whicli 
 multiplied into itself twice, will produce the given number. 
 
 Thus, 2 is the cube root of 8 ; for, 2 X 2 X 2 =: 8 : and 3 b 
 the cube root of 27 ; for, 3 x 3 x 3 = 27. 
 
 1, 2, 3, 4, 5, 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 number is a iX'rfccl cube when 
 
 313. What is the cube root of a number \ "When is a number a perfect 
 cube 1 How many perfect cubes are there between 1 and 1000 ] 
 
 * 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 iheir diameters, or other like dimensions. 
 
312 CUBE ROOT, 
 
 it has three exact equal factors. By examining the numbers 
 in the two lines we see, 
 
 1st. That the cube of units cannot give a liigher order than 
 hundreds. 
 
 2d. That since the cube of one ten (10) is 1000 and the cube 
 of 9 tens (*J0), 729000, tJie cube of tens ivill not give a lower 
 denomination than thousands, nor a higher denomination 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 sis, 
 its I'oot Avill contain more than two, and so on ; every additional 
 three figures giving one additional figure in the root, and the 
 figures Avhich remain at the left hand, although less than three, 
 will also give a figure in the root. This law explains the 
 reason for pointing off into periods of three figures each. 
 
 313. Let us now see how the cube of any number, as 10, is 
 formed. Sixteen is composed of 1 ten and 6 units, and may be 
 ■written, 10 + 0. To find the cube of 10 = 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, - - - 1000 + 1200 + 300 
 
 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 chIjc of the tens ; that is, 
 
 10 X 10 X 10 = 1000. 
 
 16 = 10 + 
 10 + 
 60 + 
 100 + 60 
 
 6 
 6 
 
 3G 
 
 100 + 120 + 
 10 + 
 
 36 
 6 
 
 yi3. Of how many parts is the cube of a number composed 1 \Vhal 
 arc I hey ! 
 
CUBE ROOT. 313 
 
 2. The second part 1800 is three times the square of the tens 
 multiplied by the units ; that is, 
 
 3 X (10)2 x6 = 3xl00xG = 1800. 
 
 3. The thii-d part 1080 is three times the square of the units 
 multiplied by the tens ; that is, 
 
 3 X 62 X 10 = 3 X 36 X 10 = 1080. 
 
 4. The fourth part is the cube of the units ; that is, 
 
 63 = 6x6x6 = 216. 
 1. "What is the cube root of the number 4096 ? 
 
 Analysis. — Since the number contains more than three figures, we 
 know that the root will contain at 
 least units and tens. operation. 
 
 Separating the three right-hand ' 4 096(16 
 
 figures from the 4, we know that 1 
 
 the cube of the tens will be found 1* X 3 = 3)3~0 (9-8-7-6 
 
 in the 4 ; and 1 is the greatest cube 16' = 4 096. 
 
 in 4. 
 
 Hence, we 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., 1800, 
 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 hundreds 
 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 Mere not so, 
 the 30 hundreds, divided by three times the square of the tens, 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 6859. Then trying 
 8 we find the cube of 18 still too large; but when we take G we find 
 the exact number. Hence, the cube root of 4096 is 16. 
 
 314, Hence, to find the cube root of a number: 
 
 Rule. — T. Separate the given number into pteriods of three 
 fiyures each, by placing a dot over the place of units, a second 
 
81-i OUBK KOOT. 
 
 over the ])lace of thovsa7ids, and so on over each third figure ta 
 the left: the left hand period will often contain less than three 
 places of figures. 
 
 II. JV^ote the greatest perfect cube in the first period, and set 
 its root on the right, after the manner of a quotient in division. 
 Subtract the cube of this number fro )n the first period, and to the 
 remainder bring down the first figure of the next jieriod for a 
 dividend. 
 
 III. Take three times the square of the root just found for a 
 trial 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 found, and if their cube be greater 
 than the first tzvo j^eriods of the given nujuber, diminish the last 
 figure, but if it be less, subtract it from the first two periods, and 
 to the remainder bring dotvn the first figure of the next period for 
 a new dividend. 
 
 IV. — Tahe 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 the 
 first three jyeriods of the given numhir when 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. 
 
 1 What is the cube root of 2070 G875 ? 
 
 OPERATION. 
 
 20 796 875(275 
 2'= 8 
 2' X 3 = 12)127 
 First two periods, ... 20 796 
 (27)' = 27 X 27 X 27 = 19 6S3 
 
 3 X (27)' = 2217)1 i 138 
 Tirst three periods, - - - 20 796 875 
 (275)' = 275 X 275 X 275 = 20 796 875 
 
 314. WTiat is the rule for oxtracting the cube root of a number? 
 
CUBE KOOT. 313 
 
 Find the cube roots of the followinnr numbers : 
 
 1. Cube root of 1728? 
 
 2. Cube root of 117649? 
 
 3. Cube root of 46G56? 
 
 4. Cube root of 15069223 ? 
 
 5. Cube root of 5735339 ? 
 
 6. Cube root of 48228544 ? 
 
 7. Cube root of 84604519 ? 
 
 8. Cube root of 28991029248? 
 
 315. To extract the cube root of a decimal fraction : 
 Annex ciphers to the decimal, if necessary, so /hat it shall con- 
 sist q/ 3, 6, 9, ^c, decimal places . Then put the first point over the 
 'place of tlioiisaudths, the second over the place of milliontlts, and 
 so on over every third place to the rigid ; after which extract 
 the root as in whole numhers. 
 
 Notes. — 1. There will be as many decimal places in the root as 
 there are periods in the given number. 
 
 2. Tlie same rule applies when the given number is composed of a 
 "whole number and a decimal. 
 
 3. 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. 
 
 Find the cube roots of the following numbers : 
 
 5. Cube root of .387420489 ? 
 
 6. Cube root of .000003375 ? 
 
 1. Cube root of 8.343 ? 
 
 2. Cube root of 1728.729? 
 
 3. Cube root of .0125 ? 
 
 4. Cube root of 19683.46656? 
 
 7. Cube root of .0066592? 
 
 8. Value of y8Tr7"29? 
 
 316. To extract the cube root of a common fraction, 
 
 I. Reduce compound fractions to simple ones, mixed numhers 
 to improper fractions, and then reduce the fraction to its lowest 
 terms. 
 
 314. What is the rule for extracting the cube root of a number 1 
 
 315. How do you extract the cube root of a decimal fraction '^ How 
 many decimal places will there be in the root \ W"\\\ the same rule apply 
 when there is a whole num!)er and a decimal I If in extracting the root 
 of any number you find a decimal, how do you proceed T 
 
816 CUBE ROOT. 
 
 II. Extract the cube root of the numerator and denominator 
 separately, if they have exact roots ; but if either of them has not 
 an exact root, reduce (he fraction to a decimal, and extract the. 
 root as in the last case. 
 
 EXAMPLES. 
 
 Find the cube roots of the following fractions : 
 
 1. Cube root of y"^ ? 
 
 2. Cube root of III? 
 
 3. Cube root of SIJ^? 
 
 4. Cube root of 911? 
 
 5. Cube root of 141? 
 
 6. Cube root of yf^ ? 
 
 7. Cube root of oVVrir ? 
 
 8. Cube root of If ^f^? 
 
 9. Cube root of 7f ? 
 10. Cube root of 56f ? 
 
 APPLICATIONS. 
 
 1. What must be the dimensions of a cubical bin, that its 
 A'ohime or 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 : w'hat would 
 be the side of a cube of equal solidity ? 
 
 4. A person wished to make a cubical cistern, which should 
 hold 150 barrels of water ; what must be its depth ? 
 
 5. A farmer constructed a bin that would contain 1500 bush- 
 els of grain ; its length and breadth were equal, and each half 
 the height ; what were its dimensions ? 
 
 6. AVhaL is the difference betAveen half a cubic yard, and a 
 rube whose edge is half a yard ? 
 
 7. 7l merchant paid $911,25 for some pieces of muslin. He 
 paid as many cents a yard as there were yards in each piece, 
 and there were as many pieces as there were yards in one 
 piece : how many yards were there, and how much did he pay 
 a yard ? 
 
 Notes. — 1. Bodies are said to be similar when they have the same 
 form and have their like parts proportional. 
 
 2. It is proved in Geometry, that the volumes or weights of similar 
 bodies are to each other ns the cubes of their like dimensions. 
 
 3. Those bodies which arc named in the same example arc sup 
 posed to bo similar. 
 
ARITHMETICAL PROGRESSION. 317 
 
 8. If a sphere 3 feet in diameter contains 14.1372 cubic feet, 
 what are the contents of a sphere 6 feet in diameter ? 
 
 33 : 63 :: 14.1372 : 113.0976. Ans. 
 
 9. If a ball 2^ inches in diameter weighs 8 pounds, how much 
 will one of the same kind weigh, that is 5 inches in diameter ? 
 
 10. What must be the size of a cubical bin, that will contain 
 8 times as much as one that is 4 feet on a side ? 
 
 11. How many globes, 6 inches in diameter, will it require 
 to make one 12 inches in diameter? 
 
 12. If a ball of silver, 1 unit in diameter, be worth $8, what 
 will be the value of one 51 units in diameter ? 
 
 13. If a plate of silver, 6 inches long, 3 inches wide, and 
 i inch thick, be worth $100, what will be the dimensions of a 
 similar plate of the same metal worth $800 ? 
 
 14. If one man can dig a cellar 12 feet long, 10 feet wide, 
 and 41 feet deep, in 3 days, what will be the dimensions of a 
 similar cellar that requires him 24 days to 3ig it, working at 
 the same rate, and the ground being of the same degree of 
 hardness ? 
 
 15. If I put 2 tons of hay in a stack 10 feet high, how high 
 must a similar stack be to contain 16 tons ? 
 
 16. Four women bought a ball of yarn 6 inches in diameter, 
 and agreed that each should take her share sejiarately from the 
 surface of the ball : how much of the diameter must each wind 
 off? 
 
 ARITHMETICAL PROGRESSION. 
 
 317. If we take any number, as 2, we can, by the continued 
 addition of any other number, as 3, form a series of numbers : 
 thus, 
 
 2, 5, 8, 11, 14, 17, 20, 23, Sec, 
 
 in which each number is formed by the addition of 3 to the 
 preceding number. 
 
 ,S17. What is an arithmetical progression 1 What is the number added 
 or subtracted called ? 
 
515 AKmrMETICAL PROGRESSION. 
 
 This series of numbers may also be formed by subtracting 3 
 continually from a larger number : thus, 
 
 23, 20, 17, 14, 11, 8, 5, 2. 
 
 An Arithmetical Progression is a series of numbers in 
 which each is derived from the preceding by the addition or 
 subtraction of the same number. 
 
 The number which is added or subtracted is called the com- 
 mon difference. 
 
 318. When the series is formed by the continued addition 
 of the common difference, it is called an increasing series ; and 
 when it is formed by the subtraction of the common diflference, 
 it is called a decreasing series : thus, 
 
 2, 5, 8, 11, 14, 17, 20, 23, is an increasing series. 
 23, 20, 17, 14, 11, 8, 5, 2, is a decreasing series. 
 
 The several numbers are called terms of the progression 
 the first and last terms are called the extremes, and the interme- 
 diate terms are called the tncaas. 
 
 319. In every arithmetical progression there are five parts, 
 any three of wliich being given or known, the remaining two 
 can be determined. They are, 
 
 1st : The first term ; 
 
 2d : The last term ; 
 
 3d : The common difference ; 
 
 4tli : Tlie number of terms ; 
 
 5th : The sum of all the terms. 
 
 318. When the common difference is added, what is the series called 1 
 What is it called when the common difl'crencc is subtracted 1 What are 
 the several numbers called 1 What are the first and last called 1 What 
 arc the intermediate ones called ! 
 
 319. How many parts are there in every arithmetical progression ^ What 
 are they! How many parts must be given before the remaining ones can 
 be found 1 
 
AKITIIMETICAL PROGRESSION. 319 
 
 CASE I. 
 
 320. Hating given the first term, the common difference, and 
 the nuinher of terms, iojind the lust term. 
 
 1. The first term of an increasing progression is 4, the com- 
 mon difference 3, and the number of terms 10 : what is the 
 last term ? 
 
 Analysis. — By considering the manner in which the increasing 
 progression is formed, we see that the 
 2d term is obtained by adding the operation. 
 
 common difference to the 1st term ; 9 no. less 1 
 
 the 3d, by adding the common differ- 3 com. diff. 
 
 ence to the 2d ; the 4th, by adding the 27 
 
 common difference to the 3d, and so 4 1st term, 
 
 on; the number of additions^ in every 31 last term, 
 
 case^ being 1 less than the ymmber of 
 
 terms found. Instead of making the additions, we may multiply the 
 common difference by the number of additions, that is, by 1 less than 
 tlie number of terms, and add the first term to the product. 
 
 Rule. — Multiply the common difference by 1 less than the 
 mimber of tei-ms : if the 2^^'ogression is increasing, add the pro- 
 duct to the first term, and the sum 'will be the last term ; if it is 
 decreasing, subtract the product from the first term and the dif- 
 ference will be the last term. 
 
 EXAMPLES. 
 
 1. What is the 18th term of an arithmetical progression, of 
 which the fii'st term is 4, and the common difference 5 ? 
 
 2. A man is to receive a certain sum of money in 12 pay- 
 ments : the first payment is $300, and each succeeding pay- 
 ment is less than the previous one by $20 : what will be the 
 last payment ? 
 
 3. What will $200 amount to in 15 years, at 7 per cent 
 simple interest : the first year it increases $14, the second, $28, 
 and so on ? 
 
 320. When you know the first term, the common difference and the 
 iiuiuber of terms, how do you find the last term 1 
 
820 ARITHMETICAL PROGRESSION. 
 
 4. A man bus a piece of land 35 rods in length, which 
 tapers to a point, and is found to increase ^ rod in width,, for 
 every rod in length : what is the width of the wide end ? 
 
 5. James and John have 100 marbles. It is agreed between 
 them that John shall have them all, if he Avill place them in a 
 straight line half a foot apart, and so that he shall be obliged 
 to travel 300 feet to get and bring back the farthest marble ; 
 and also, if he will tell, without measuring, how far he must 
 travel to bring back the nearest. 
 
 CASE II. 
 
 321. Knoiving the two extremes of an arithmetical 'progreS' 
 sion and the number of terms, to find the common difference. 
 
 1. The two extremes of a progression are 4 and 68, and the 
 number of terms 17 : what is the common difference ? 
 
 Analysis. — Since the common difference multiplied by 1 less than 
 the number of terms gives a product equal 
 to the difference of the extremes, if we operation. 
 
 divide the difference of the extremes by 1 68 
 
 less than the number of terins^ the quo- 4 
 
 Xieni vf\\\ he WiQ common difference : hence, 17 — 1 = 16)64(4 
 
 Rule. — Subtract the less extreme from the greater, and divide 
 the remainder by 1 less than the number of terms : the quotient 
 will be the common difference. 
 
 EXAMPLES. 
 
 1. A man started from Chicago and travelled 15 days ; each 
 day's journey was increased by the distance which he travelled 
 the first day : what was his daily increase if he travelled 75 
 miles the last day ? 
 
 2. A merchant sold 14 yards of cloth, in pieces of 1 j-ard 
 each ; for the first yard ,he received $1, and for the last $2G^ . 
 what was the diflference in the price per yard ? 
 
 321. ^Vllrn you know the extremes and number of terms, how do you 
 find tile oonniion dLfTercujio 1 
 
2 5 8 
 .23 20 17 
 
 11 
 14 
 
 14 
 11 
 
 17 
 8 
 
 20 
 5 
 
 23 
 2 
 
 25 25 25 
 
 25 
 
 25 
 
 15 
 
 25 
 
 25 
 
 ARITHMETICAL PROGRESSION. 321 
 
 3. A board is 17 feet long: it is 2-^ inches wide at one end, 
 and 144 at the other : what is the average increase in width 
 per foot in length ? 
 
 CASE III. 
 
 322. To find the sum of the terms of an arithmetical prO' 
 gression. 
 
 1. What is the sum of the series whose first term is 2, com- 
 mon difference 3, and number of terms 15 ? 
 Given series, 
 Same, order inverted. 
 Sum of both series. 
 
 Analysis. — The two series are the same ; hence, their sum is 
 equal to twice the given series. But their sum is equal to the sum 
 of the two extremes, 2 and 23. taken as many times as there are 
 terms J and the given series is equal to half this sum, or to the sura 
 of the extremes multiplied by half the number of terms. 
 
 Rule. — Add the extremes together and multi2'>Iy their sum by 
 half the number of terms ; the product will he the sum of all the 
 t4rms. 
 
 EXAMPLES. 
 
 1. What debt could be discharged in a year, by weekly pay- 
 ments in arithmetical progression, the first payment being $5, 
 and the last $100 ? 
 
 2. A person agreed to build 56 rods of fence ; for the first 
 rod he was to receive 6 cents, for the second, 10 cents, and so 
 on : what did he receive for the last rod, and how much for the 
 whole ? 
 
 3. If a person travel 30 miles the first day, and a quarter of 
 a mile less each succeeding day, how far will he travel in 30 
 days ? 
 
 4. If 120 stones be laid in a straight line, each at a distance 
 of a yard and a quarter from the one next to it, how far must a 
 person travel who picks them up singly and places them iu a 
 
 a23. How do you find the sum of the series 1 
 
822 ARITHMETICAL PEOGEESSION. 
 
 heap, at the distance of 6 yards from the end of the line and in 
 its continuation ? 
 
 CASE IV. 
 
 323. Having given the first and last terms, and the common 
 difference, to find the number of terms. 
 
 1. The first term of an arithmetical progression is 5, the com- 
 mon difference 4, and the last term 41 : what is the number of 
 terms ? 
 
 Analysis. — Since the last term is equal operation. 
 
 to the first term added to the product of the 41 — 5 = 36 
 common dilTerence, by 1 less than the num- 4)36( = 9 
 
 ber of terms (Art. 320), it follows that, if 9 + 1 = 10 No. terms, 
 the first term be taken from the last term, 
 
 the difference will be equal to the product of the common difference 
 by 1 less than the number of terms : if this be divided by the com- 
 mon difference, the quotient will be 1 less than the number of terms.' 
 
 Rule. — Divide the difference of the tico extremes hy the com- 
 mon difference, and add 1 to the quotient : the sum will be the 
 number of terms. 
 
 EXABIPLES. 
 
 1. A farmer sold a number of bushels of wheat ; it was 
 agreed that for the first bushel he should receive 50 cents, and 
 an increase of 9 cents for each succeeding bushel, and for the 
 last he received $500 : how many bushels did he sell ? 
 
 2. A person proposes to make a journey, and to travel 15 
 miles the first day, and 33 miles the last, with a dally increase 
 of 11 miles : in how many days did he make the journey, and 
 what was the Avhole distance travelled ? 
 
 3. I owe a debt of $2325, and wish to pay it in equal install- 
 ments, the first payment to be $575, the second $500, and 
 decreasing by a common diflcrencc, until the last payment which 
 is $200 : what will be the number of installments ? 
 
 323. Having given the first and last terms and the common difference, 
 Low do \ou find the number of terms ! 
 
GEOMETRICAL PKOaKESSIOJ^. 328 
 
 GEOMETRICAL PROGRESSION. 
 
 324. A Geometrical Progression is a series of terms, 
 each of which is derived from the preceding one, by muUiply- 
 ing it by a constant number. The constant multiplier is called 
 the ratio of the progression. 
 
 32.5. If the ratio is greater than 1, each term is greater than 
 the preceding one, and the series is said to be increasing. 
 
 If the ratio is less than 1, each term is less than the preced- 
 ing one, and the series is said to be decreasing ; thus, 
 
 1, 2, 4, 8, 16, 32, &c. — ratio 2 — increasing series : 
 32, 16, 8, 4, 2 1, &c. — ratio \ — decreasing series. 
 
 The several numbers resulting from the multiplication arc 
 called terms of the progression. The first and last are called 
 the extremes, and the intermediate terms are called means. 
 
 326. In every Geometrical, as well as in every Arithmetical 
 Progression, there are five parts : 
 
 1st : The first term ; 
 
 2d : The last term ; 
 
 3d : The common ratio ; 
 
 4th : The number of terms ; 
 
 5th : The sum of all the terms. 
 
 If any three of these parts are known, or given, the remain- 
 ing ones' can be determined. 
 
 324. What is a geometrical progression 1 What is the constant multi- 
 plier called 1 
 
 325. If the ratio is greater than 1, how do the terms compare with each 
 other 1 What is the series then called ! If the ratio is less than 1, how 
 do they compare T What is the series then called ? What are the several 
 numbers called ? What are the first and last terms called 1 What are tho 
 intermediate ones called l 
 
 326. How many parts are there in every geometrical progression \ 
 What are they "! How many must be known before 'lie others can be 
 found \ 
 
324 GEOMETKICAL PROGRESSION. 
 
 CASE I. 
 
 327. Having given the first term, tJie ratio, and the number 
 of terms, to find the last term. 
 
 1. The first term is 4, and the common ratio 3 : what is the 
 5th term? 
 
 Analysis. — The second term is formed by operation. 
 
 multiplying the first term by the ratio; the 3 X 3 X 3 x 3 = 81 
 third term by multiplying the second term 4 
 
 by the ratio, and so on ; the number of mul- Ans. 324 
 
 tiplications icing 1 less than the number of 
 terms : thus, ■ 
 
 4 = 4, 1st term, 
 3x4= 12, 2d term, 
 3x3x4== 36, 3d term, 
 3x3x3x4= 108, 4th term, 
 3x3x3x3x4= 324, 5th term. 
 
 Therefore, the last term is equal to the first term multiplied 
 ly the ratio raisvd to a jioioer whose cxpomnt is 1 less than the 
 number of terins. 
 
 Rule. — Raise the ratio to a poicer tchose exponent is 1 less 
 than the number of terms, and then midtiphj this power by the 
 first term, 
 
 EXAMPLES. 
 
 1. The first term of a decreasing progression is 2187 ; the 
 ratio is l, and the number of terms 8 : what is the last term ? 
 
 Note, — The 7lh power of -J- i.s -rrx^y; this operation. 
 
 multiplied by the first term, 2187, gives 1, (-J-)' = •jxVt 
 
 the last term. (ttVt x 2187 = 1. 
 
 2. The first term of an increasing geometrical series is 8, 
 the ratio 5 : wliat is the 9th term ? 
 
 3. The first term of a decreasing geometrical series is 729, 
 the ratio i : what is the 10th term ? 
 
 M'Zl. Knowing the first term, the ratio, and the number of tenns, how Jo 
 you find the last term ? 
 
OKOMETRICAL PROGRESSION. 325 
 
 4. If a farmer should sell 15 bushels of wheat, at 1 mill for 
 the first bushel, 1 cent for the second, 1 dime for the third, and 
 60 on ; what would he receive for the last bushel ? 
 
 5. A man dying left 5 sons, and bequeathed his estate in the 
 following manner ; to his executors $100 ; his youngest son was 
 to have twice as much as the executors, and each son to have 
 double the amount of the next younger brother : what was the 
 eldest son's portion ? 
 
 6. A merchant engaging in business, trebled his capital once 
 in 4 years ; if he commenced with $2000, what will his capital 
 amount to at the end of the 12th year ? 
 
 7. A farmer wishing to buy 16 oxen of a drover, finally 
 agreed to give him for the whole the cost of the last ox only. 
 He was to pay 1 cent for the first, 2 cents for the second, and 
 doubling on each one to the last : how much would they cost 
 him ? 
 
 CASE 11. 
 
 328. Knowing the two extremes and the ratio, to Jind the sum 
 of the terms. 
 
 1. What is the sum of the terms of the progression 2, C, 18, 
 54,162? 
 
 OPERATION. 
 
 6 + 18 + 54 + 162 + 486 = 3 times. 
 
 2 + 6 + 18 + 54 + 162 =z 1 tim e. 
 
 486 - 2 = 2 times. 
 486 - 2 484 
 
 2 
 
 = 242 sum. 
 
 Analysis. — If we multiply the terms of the progression by the 
 ratio 3. we have a second progression, 6, 18, 54, 162, 486, which is 
 3 times as great as the first. If from this we subtract the first, the 
 remainder. 486 — 2, will be 2 times as great as the first; and if this 
 remainder be divided by 2. the quotient Avill be the sum of the terms 
 of the first progression. 
 
 328. Knowing the two extremes and the ratio, how do you find the cum 
 
 of the terms ! 
 
 IS 
 
g26 GEOMETlilCAL PKOGKESSION. 
 
 But 48P is the product of the last term of the given progression 
 multiplied by the ratio, 2 is the first term, and the divisor 2, 1 less 
 than the ratio : hence, 
 
 Rule. — Multiply the last term by the ratio ; take the differ- 
 ence between this 2}roduct and the first term and divide the remairir- 
 der by the difference between 1 and the ratio. 
 
 Note. — When the progression is increasing, the first term is sub- 
 tracted from the product of the last term by the ratio, and the di\'isor 
 is found by subtracting 1 from the ratio. When the progression is 
 decreasing, the product of the last term by the ratio is subtracted 
 from the first term, and the ratio is subtracted from 1 . 
 
 EXAMPLES. 
 
 1. The first term of a progression is 4, the ratio 3, and the 
 last term 78722 : what is the sum of the terms ? 
 
 2. The first term of a progression is 1024, the ratio i, and 
 the last term 4 : wliat is the sum of the series ? 
 
 3. What debt can be discliarged in one year by monthly 
 payments, the first being $2, the second S8, and so on to the 
 end of the year, and what will be the last payment ? 
 
 4. A gentleman being importuned to sell a fine horse, said 
 that he would sell him on the condition of receiving 1 cent for 
 the first nail in his shoes, 2 cents for the second, and so on, 
 doubling the price of every nail : the number of nails in each 
 shoe being 8, how much would he receive for his liorse ? 
 
 5. A laborer agreed to thresh 64 days for a farmer on the 
 condition that he should give him 1 grain of wheat for the first 
 day's labor, 2 grains for the second, and double each succeeding 
 day : what number of bushels would lie receive, supposing a 
 pint to contain 7G80 grains, and wliat number of ships, each 
 carrying 1000 tons burden, might be loaded, allowing 40 bushels 
 to a ton ? ( 
 
ANALYSIS. 
 
 327 
 
 ANALYSIS. 
 
 J. If 12 yards of cloth cost S48,36, what will 7 yards cost? 
 
 Analysis. — One yard of clolh will cost -^ as much as 12 yards : 
 •ince 12 yards cost S48,36, one yard will cost Jg- of $48,36 = $4,03 
 7 yards will cost 7 times as much as 1 yard, or 7 times -^ of $48, 3G 
 = 828.21 ; therefore, if 12 yards of cloth cost -148,36, 7 yards will 
 cost $28,21. 
 
 OPERATION. 
 
 4,03 
 4,03 
 
 i$,$$ 1 7 ^„^^, 1^ 
 
 ^S^^i 
 
 $28,21 ; or 
 
 MM 
 7 
 
 I $28,21 Ans. 
 
 3. If 27 pounds of butter will buy 45 pounds of sugar, how 
 much butter will buy 36 pounds of sugar ? 
 
 Analysis. — One pound of sugar will buy -^ as much butter as 45 
 pounds, or -^-^ of 27lbs. of batter; 3fc pounds of sugar will buy 36 
 times as much butter as 1 pound of sugar, or 36 times -^ of 27/65., 
 which is ^^Ibs. = 21|/65. 
 
 OPERATION. 
 
 ^:^ 1 36 
 
 3. "What will 6| cords of wood cost, if 2^ cords cost $7^ ? 
 
 Analysis. — Since 2| cords = ^ cords of wood costs S7|- ~. S-^ 
 one cord will cost as many dollars as -l^ is contained times in 'y 
 or $3 : C-| cords = ?j? cords, will cost ^-^ times as much as 1 Qmd 
 that is. $3 X 3J = $8ji ::= $20.25. 
 
 0PER.4TI0N. 
 
 108 
 
 5 
 
 27 
 
 . =2ims.; or 
 
 
 108 
 
 
 
 21^/55- 
 
 a ■ ^ ii) ^ 4 ~ 4 
 
 $201; or, 
 
 $ 
 
 4 
 
 $ 
 27 
 
 4 81,00 
 
 $20,25 Ans. 
 

 12 
 
 
 3 
 
 $ 
 
 ^ 
 
 n 
 
 $ 
 
 
 5 
 
 7 
 
 180,00 
 
 328 ANALYSIS. 
 
 Note. — The fractional pirt of a dollar may always be reduced to 
 cent.s by annexing two ciphers, and to mills by annexing three, and 
 then dividing by the denominator. 
 
 4. A farmer sold a number of cows, and had 12 left, which 
 was i of the number sold ; if the number sold be divided by 
 I of 91, the quotient will be 1 the number of dollars he received 
 per head : how much did he receive per head for his cows ? 
 
 Analysis. — 12 is ^ of 3 times 12 = 36, the number of cows sold ; 
 36 divided by | of 9J=7, the quotient, 8^, is | of 5 times ^=-i-fA 
 
 = S25f 
 
 OPERATION. 
 
 12 3 4x3 5 180 ^^,^ „ 
 
 T^1^3^28^i=-T==^2^^'^^^; 
 
 $25,71|- Ans: 
 
 5. What will 20 bushels of barley cost, in dollars and cents, 
 at 7 shillings a bushel, New York currency ? 
 
 Notes. — 1. Although United States money is expressed in dollars, 
 cents, and mills, still in most of the States the dollar (always valued 
 at 100 cents) is sometimes reckoned in pounds shillings and pence; 
 thus. 
 
 2. In the New England States, in Indiana, Illinois, Missouri, 
 Virginia, Kentucky, Tennesce, Mississippi, and Texas, the dollar is 
 reckoned at 6 shillings ; in New York, Ohio, and Michigan, at 8 shil- 
 lings ; in New Jersey, Pennsylvania, Delaware, and Maryland, at 
 
 • 75. Gd.; in South ^Carolina and Georgia, at 45. 8fZ. ; in Canada and 
 Nova Scotia, at shillings. 
 
 3. It often occurs that the retail price is given in shillings and 
 pence, and the result or cost is required in dollars and cents. 
 
 Analysis. — Since 1 bu.shel of barley costs 7 shillings, 20 busliela 
 will cost 20 times 7 shillings, or 140 shillings; and as 8 shillings 
 make 1 dollar, New York currency, there will be as many dollars ub 
 }< i.^ contained times 140 = $17 50. 
 
ANALYSIS. 
 
 829 
 
 OPERATION. 
 
 20 X 7 -i- 8 = $171 ; or 
 
 n 
 
 35,00 
 
 "$17,50 Ans. 
 
 6. What will be the cost of 72 bushels of potatoes, at 35. M. 
 per bushel, New York currency ? 
 
 4 
 
 13 
 
 Ans. 
 
 OPERATION. 
 
 Or, 
 
 ^0^ 
 
 3 
 
 n 
 
 39 
 
 4 
 
 117,00 
 
 4 
 
 117,00 
 
 
 $29,75 
 
 
 $29,75 Ans 
 
 Note. — When the pence is an aliquot part of a shilling, the price 
 may be reduced to an improper fraction, which will be the multiplier 
 m the denomination of shillings ; thus, 35. 3c?. = 'i\s. = ^^^5. : or, the 
 shillings and pence maybe reduced to pence; thus, 3s. 3rf. = 39rf., 
 tn which case the product will be pence, and must be divided by 96, 
 the number of pence in $1. 
 
 7. What will 121 pounds of tea cost at 65. Sc?. a pound, 
 Pennsylvania currency ? 
 
 OPERATION. 
 
 3 ^ 
 
 20 
 
 9 
 
 100 
 
 $111 Ans. 
 
 Or, 
 
 00 
 9 
 
 25 
 
 4 
 
 100,00 
 
 $11,11j \ns 
 
 Note. — In the last example the multiplier is 65. 8J. = 6>5. — \ 
 or %Qd. The divisor is 75. Gc/. = 7^.s. = ^s.^ or 90d Hence, to fiii 
 fjie cost of articles in dollars and cents, when the price is in shillings 
 and pence, 
 
 Muliiplij the commodity hy the price, and divide the product 
 by the value of a dollar, ex2')resscd in the unit of the price. 
 
330 
 
 ANALYSIS. 
 
 8. How many daj^s work at 10s. 6c/. a day, must be given 
 for 18 bushels of corn at os. lOd. a bushel? 
 
 
 
 1$ 
 
 2 
 
 OPERATION. 
 
 Or, 
 
 m 
 
 10 days Ans. 
 
 10 
 
 10 days Ans. 
 
 Note. — The same rule applies in this as in the preceding examples, 
 except that the divisor is the price of the articles received in pay- 
 ment, reduced to the same unit as the price of the article bought. 
 
 9. What will 5cwt. of coffee cost, at I5. 4c?. per pound, New 
 York currency? 
 
 OPERATION, 
 
 > 
 
 25 
 
 4 
 
 t 
 
 Or, 
 
 n 
 
 250 
 
 25 
 
 250,000 
 
 883,3331 Ans. 
 
 $83,3331 Ans. 
 
 Note. — Reduce the cwts. to Ihs. by multiplying by 4, and then by 
 25, after which proceed as in the preceding examples. 
 
 10. A merchant bought a number of bales of cloth, each 
 containg 1331 yards, at the rate of 12 yards for $11, and sold 
 it at the rate of 8 yards for $7, by which he lost $100 in the 
 trade : how many bales were there ? 
 
 Analysis. — Since he paid Su for 12 yards, for 1 yard he paid -A 
 of $11, or W of %\ \ and since he received $7 for 8 yard.«. for 1 yard 
 he received \ of $7, or \ of $1, He lost on 1 yard the difference 
 between \\ and |- — -jjtj. of a dollar. Since his whole loss was Si 00, 
 he had as many yards as ^ is contained times in 100 = 2400 yards; 
 and 1 here were as many bales as 133J (the number of yards in ] 
 bale) is contained times in the whole number of yards = 18 bales. 
 
 OPERATION. 
 
 _1_ 
 
 '2 4 
 
 (100 - ^v) - 133'^ = 18 ^«5. ^''' m_ 
 
 100 
 
 6 
 
 3 
 
 18 bales Ano. 
 
ANALYSIS. 831 
 
 11. A can mow an acre of grass in 7^ hours ; B, in 5 hours ; 
 C, in 5| hours ; how many days working 6|- hours, would 
 they requix'e to mow lo| acres ? 
 
 Analysis. — Since A can mow an acre of grass in 7^ hours, 
 B, in 5 hours, and C in 5|, A can mow ^, B, \, and C. -^-^ of 
 an acre, in 1 iiour. Together, they can mow t'5+i+4^^=-4l "' "•' 
 acre in 1 hour: and to mow 1 a^.TC, they will require as many Imurs- 
 as |-| is contained times in 1 = -ff hours: to mow 13-|- acres, they 
 will require 13-| times -1^ = 27 hours, and working 6|- hours each 
 day. will require 4 days. 
 
 OPKRATION. 
 
 15 5 45 4o fit 
 
 Or ^ 
 
 00 4 , , ^'' n 
 
 X -— X 
 
 n $ 
 
 — = 4 days 
 
 4 
 
 4 days Ans. 
 
 12. A person employed three men, A, B, and C, to do a 
 piece of work for $132,66. A can do the work alone in 23^ 
 days, working 12 hours a day; B can do it in 25 days, working 
 8 hours a day; and C can do it in 16 days, working IIJ 
 hours a day. In what time can the three do it, working to- 
 gether, 10 hours a day, and what share of the money should 
 each receive ? 
 
 Analysis. — Since A can do the work in 23^ days, working 12 
 hours each day: B, in 25 days, working 8 hours each day; and C, 
 in 16 days, working lli hours each day, A can do the same v/ork in 
 280 days. B, in 200 days, and C. in 180 days, working 1 hour each 
 
 day : then A, B, and C, can do -^h + nh + li o = Trifo ^^ ^^^^ 
 work in 1 day, working 1 hour : by working 10 hours, they will do 
 10 times as much ; or, the work done by each in 1 day of 10 hours, 
 will be denoted by -^q. -^^q, and ,^^ : and the whole work done in 
 1 day by tt^tt^ ; hence, the number of days will be denoted bv the 
 number of times which 1 contains ■jWiT^'We'^ "'''"g^ days. 
 
 If the part which each does in 1 day be multiplied by the number 
 of days, viz., 7^, the product will be the part done by each ; viz., Aj 
 
 5^0 x'^W^iV^; B> ^x ' sV^tVs ■■ '^"d C, J^o_.x7^=^% ; there- 
 fore, A must have ^-^, B, ^-^ and C J^ of S132,GG. 
 
332 ANALYSIS. 
 
 OPERATION. 
 
 First: l-^Y^=.W7f=V^days. Ans. 
 
 Second: 8132,66 XTV8 = ^33,53f|=A's share. 
 $132,66 xf-^=S46,95fi=B's share. 
 8132,76 X ^^ =$52,16 |^=:C's share. 
 Total paid 3132 66" 
 
 13. If 336 men, in 5 days, working 10 hours each day, can dig 
 a trench of 5 degrees of hardness, 70 yards long, 3 yards wide, and 
 2 yards deep ; how many days of 12 hours each, will 240 men 
 require to dig a trench 36 yards long, 5 yards wide, and 3 yards 
 deep, of 6 degrees of hardness ? 
 
 Analysis. — Since 336 men require 5 days of 10 hours each, to dig 
 a trench, it will take 1 man 336 times 5 days of 10 hours each, and 
 10 times (336 X 5) days of 1 hour each^ to dig the same trench. To 
 dig a trench 1 yard loug. will require -^ as much time as to dig one 
 70 yards long ; to dig one 1 yard wide, ^ as much as 3 yards wide; 
 to dig one 1 yard deep, -^ as much as 2 yards deep ; and to dig one 
 of 1 degree of hardness \ as much as to dig one of 5 degrees of 
 hardness. 240 men can dig a trench 1 yard long, 1 yard wide. 1 yard 
 deep, and of 1 degree of hardness in ^^ of the time that 1 man can 
 dig the same, and in -jlj as many days of 12 hours each, as of 1 hour 
 each; but to dig one 36 yards long, will require 36 times as much 
 time as to dig one 1 yard long ; to dig one 5 yards wide, 5 times as 
 much as 1 yard wide ; to dig one 3 yards deep, 3 times as much as 
 1 yard deep; and to dig one of 6 degrees of hardness will require 
 6 times as much time as to dig one of 1 degree of hardness. 
 
 OPERATION. 
 
 «Ji^5xi0^1xixlxlx-Lxlx?^x*x?x?=9*,. 
 1 Tt0 1i fi $ t0 1^2 1 1 1 1 
 
 Or, 
 
 U0 
 
 m 
 
 X 
 
 $ 
 
 n 
 
 10 q 
 
 n 
 
 % 
 
 $ 
 
 % 
 
 H 
 
 $ 
 
 0^ 
 
 
 da; 
 
ANALYSIS. 
 
 083 
 
 Note. — The principle of the above analysis is this : 1st. Find how 
 many hours it will take 1 man to dig 1 cubic yard of trench ; this ia 
 done in the first part of the analysis. 2d. Find how long it will take 
 240 men. working 12 hours a day, to dig the required trench, working 
 at the same rate ; this is done in the second part of the analysis. 
 
 14. If 20 cords of wood are equal in value to 6 tons of hay, 
 and 5 tons of luiy to 36 bushels of wheat, and 12 bushels of 
 wheat to 25 bushels of com, and 14 bushels of corn to 5Q 
 pounds of butter, and 72 pounds of butter to 8 days of labor ; 
 how many cords of wood will be equal to IG days of labor? 
 
 Analysis. — Since 20 cords of wood arc equal to 6 tons of hay, 
 1 Ion of hay is equal to ^ of 20 cords of wood, or ^- cords ; 5 tons 
 are equal to 5 times ^^ or ^^ cords ; since 5 tons of hay. or ^^ of a 
 cord of AA"ood are equal to 36 bushel.s of wheat, 1 bushel of wheat is 
 equal to ^-^ of ^^ = -|f cords, and 12 bushels of wheat are equal to 
 12 limes |^ = ^J cords ; and since 12 bushels of wheat, or ^^ cords 
 of wood are equal to 25 bushels of corn, 1 bushel of corn is equal to 
 ^V ^^ y = t of a cord of wood, and 14 bushels of corn are equal to 
 14 times f ~ 2_8 cords ; and since 14 bushels of corn, or 2_8 cords of 
 wood are equal to 56 pounds of butter, 1 pound of butter is equal 
 Jg. of 2J = ^ig^ of a cord, and 72 pounds of butter are equal to 72 
 times ^Ig = 4 cords of wood ; and since 72 pounds of butter or 4 cords 
 of wood are equal to 8 days' labor, 1 day's labor is equal to -I- of 
 A =:z ^ cord of wood, and 16 daj's labor are equal to 16 times -^ of a 
 
 cord, or 8 cords of wood. 
 
 20 
 1 
 
 1 
 6 
 
 5 
 1 
 
 1 
 36 
 
 .| Xf>XiXor?Xi X4)--Xt X 
 
 12 
 
 T 
 
 1^ 
 
 25 
 
 OPERATION. 
 
 14 2 
 1 ^"56"^ 1 
 
 X^X8XY=8cord3;or, 
 
 
 
 i0 ^ 
 
 $0 
 
 $ 
 
 t$ 
 
 n 
 
 $$ 
 
 u 
 
 
 
 n^ 
 
 
 10 
 
 8 cords. Ans. 
 
 Note. — This and similar examples fall under what is called the 
 Chain Rule. In analyzing, then, always commence with the ternj 
 which is of the same name ur kind as the required answer. 
 
334 ANALYSIS. 
 
 15. A, B, and C, put in trade $5626 : A's stock was in 5 
 months, B's, 7 months, and C's, 9 months. They gained $1260, 
 wliich was so divided that A i-eceived $4 as often as B had $5; 
 and as often as C had $3. After receiving 82164,50, B ab- 
 sconded. What was each one's stock in trade, and how much 
 did A and C gain or lose by B's withdrawal ? 
 
 Analysis. — Since A received S4 as often as B had So, and as often 
 aS C had $3, if the whole gain were divided into 12 equal parls, A 
 would have ^. B, ^j. and C. y%, of $1260, or A would have S420, 
 B, $525. and C. $315. Now. if their respective gains be divided by 
 the luimber of months each one's stock continued in trade, the quo- 
 tients will represent their monthly gains, viz., A's will be $420 -r 5 
 = S84 ; B's, $525 -r 7 = $75 ; and C's. $315 -r 9 = $35, which 
 gives $194 as their whole gain for 1 month. 
 
 But since each one's share of the gain for a given time will be to 
 the whole gain for the same time, as each one's stock to the whole 
 stock: it follows that, A will have ^^r-. B, J^. nnd C, y^^. of the 
 whole stock, or A will have $2436, B, $2175, and C, $1015. When 
 B ran away he was entitled to his original stock $21 75. and his share 
 of the gain for 7 months, that is, to $2175 + $525 = $2700; but as 
 he took away only $2164,50, A and C gained $535.50 by his with- 
 drawal, which must be divided between Hicm in tlic ratio of their 
 ' If 
 
 investments, or as 4 to 3 ; therefore. A will have 4-. and C -f of B's 
 unclaimed portion, or A will have $306, and C $229.50. 
 
 OPERATION. 
 4 + 5 + 3 = 12. 
 
 A's whole gain = j\ of SI 2 60 = $420 
 B's " " = y\ « « _ $525 
 C's " " = j% " " = $315 
 
 A's monthly gain = $420 -j- 5 =: $84 
 B's " " = $525 -f 7 = $75 
 C's " " = $315 — 9 = $35 
 
 $194 
 
 A's stock = /^ of $5626 = $2436 
 B's " = yVi " " =^2175 
 C's ** = T-Vr " " = $1015 
 
ANALYSIS. So5 
 
 $2175 + $525 - $2164,50 = $535,50, what B left, 
 i- of $535,50 = $306 A's share of it. 
 3. " « = $229,50 B's share of it. 
 16. Mr. Johnson bought goods to the amount of $2400, 1 to 
 be paid in 3 months, i in 4 months, | in 6 montlis, and the 
 remainder in 8 months : what is the equated time for the pay- 
 ment of the whole ? 
 
 Analysis.— SSOO to be paid in 3 months, is the same as $1^ to bo 
 paid in 2400 months ; SGOO, in 4 months, the same as Si in 2400 
 months; $600, in 6 mouths; the same as $1, in 3600 months; and 
 $400 in 8 mouths, the same as $1, in 3200 months. Then $1, payable 
 in 2400 + 2400 + 3600 + 3200 = 11600 months, is the same as 
 $2400 in 2^0^ of 11600 months, wliich is 4| months — 4 months 26 
 days, the equated time of payment. 
 
 OPERATION. 
 
 800 X 3 = 2400 
 GOO X 4 = 2400 
 600.x 6 = 3600 
 400 X 8 = 3200 
 
 2400 11600 
 
 11600 -^ 2400 = 4|wo. = ioio. 25da. Ans. 
 
 17. What will be the interest on $60,48 for 1 year 3 months, 
 at 7 per cent ? 
 
 Analysis. — Since the interest on $1 for 1 year is 7 cents, or seven 
 hundredths of SI. the interest on $60,48 for 1 year, will be $60,48 
 X .07 = $4.2336. The interest for 1 month wnll be ■^\ as much as 
 for 1 year or ^ of $4.2236 = ^0.3528, and for \yr. 3mo. = 15 
 months, it will be 15 times as much as for 1 month, or $0,3528 x 15 
 = $5,292. 
 
 OPERATION. 
 
 ($60,48 X. 07 -f 12) X 15 = $5,292 yl?is. Or, i^ 
 
 5,04 
 
 ()0,4^ 
 • 7 
 15 
 
 I $5,292 Ans. 
 18. What will be the interest on $88,92, for 8mo. 20Ja., at 
 7 per cent? 
 
836 
 
 ANALYSia. 
 
 Analysis. — Since the interest on ^1 is 7 cents for 1 year, the in- 
 terest on $^88.92 for 1 year will be $88.92 x .07 = S6.224 ; the in- 
 terest for 1 month will be ^-^ of $6,224 = $0,5187 ; and since the 
 number of days divided by 30 will give the value of those days ia 
 decimals of a month (Art. 222) 20c/a. = .6f months. The interesi 
 for Pmo. 20cZa. = 8.6| months, will be 8.6|- times as much as for 1 
 mouth = 0.5187 X 8.6| = $4.4954. 
 
 OPERATION. 
 
 ($88,92 X .07-M2)x8.6f=$4,4954 A?is. 
 
 n 
 
 2 47 
 
 .07 Or, .It 
 
 26 ^^ $0 
 
 '^"^ 26 
 
 I $4.4954 I S4.4954 An>s. 
 
 19. A liquor mercliant mixed together 25 gallons of brandy 
 at $],60 a gallon, 25 gallons at $1,80, 10 gallons at $2,50, and 
 20 gallons of water ; wliat was the value of 1 gallon of the 
 mixture, and what was the gain on a gallon if he sold it at the 
 average price of the liquor ? 
 
 Analysis. — The value of 20 gallons of water would be ; of 25 
 gallons of brandy at $1,60 a gallon would be $1,60 x 25 = $40; of 
 25 gallons at $1.80, would be $1.80 X 2a = $45 ; of 10 gallons at 
 $2,50, would be $2.50 X 10 = $25. 25 + 25 + 10 gallons of 
 brandy + 18 gallons of water = 80 gallons, the amount of the 
 mixture; and $40 + $45 + 25 = $110, the value of the mixture; 
 hence, if 80 gallons are wortli $110, one gallon is worth -^ of 
 $110 = $1,371. But 25 + 25+10 = 60 gallons of brandy, arc 
 worth $110, and $110 -- 60 = $1.83^, the average price per gallon 
 of the brandy; therefore $1,83-J- — $1,37^ = 45|- cents, the gain oa 
 1 gallon. 
 
 OPERATION. 
 
 20 X r^ 00 
 25 X l.GO = 40 
 25 X 1.80 -^ 45 
 10 X 2.50 = 25 
 
 80 no 
 
 8110-^80 = $l,37.V value: ^1 10 -f- G0=r$1.83i averagre price, 
 81,83^ - $1,371 = e0,45;;. 
 
ANALYSIS. 
 
 337 
 
 10. A merchant has three kinds of cloth, worth $lf, $2 J, 
 fi^^a. yard : what is the least number of whole yards he can 
 sell, to receive an average price of $2}, a yard ? 
 
 Analysis. — If he sells 1 yard worth Sl|-, for $2 J, he will gain 
 j of a dollar; to gain 1 dollar he must sell as many yards as | is 
 contained times in 1, or -| yards. But since he is neither to gain or 
 lose by the operation, if he gains on one kind, he must lose an equal 
 'mm on some other ; hence, he must sell some that is worth more 
 ihan the average price. If he sell 1 yard worth $3|- for $2^^, he will 
 lose -I of a dollar; and to lose $1, he must sell -|- of a yard. There- 
 fore, to make the loss equal to the gain, he must sell ^ of a yard at 
 $3|- a yard, as often as he sells I- of a yard at Slf a yard. 
 
 If he sells 1 yard Avorth $2^, for $2i, ho gains -^ of a dollar, and 
 to gain Si he must sell 4 yards ; hence, to keep the average price, he 
 must lose as much on some other, and as he can only lose on that at 
 $3-| a yard, he must sell enough of that to lose $1, which would be 
 |- of a yard ; therefore, as often as he sells -| yards at Slf a yard, he 
 must sell I yards at $3-| a yard; and as often as he sells 4 yards at 
 $2^ a yard, he must sell -| yards at $3|- a yard. 
 
 But since it is desirable to have the proportional parts expressed in 
 the least whole numbers, we may multiply the numbers by the least 
 common multiple of their denominators, and divide the products by 
 their greatest common factor; this being done, we obtain in the 
 above example, 3 yards at Sl| a yard. 10 yards at $2^ a yard, and 
 4 yards at $3| a yard. 
 
 OPERATION. 
 
 3 
 
 10 
 
 4 
 
 21. The hour and minute hands of a clock are together at 
 12 o'clock : when are they next together ? 
 
 Analysis. — Since the minute hand passes over 60 minute spaces 
 while the hour-hand passes over 5. the minute-hand passes over 
 12 minute spaces while the hour-hand passes over 1, gaining 11 
 minute spaces on the hour-hand in 12 minutes of time ; the minme- 
 hand requiring one minute of time to pass over 1 minute of space. 
 Hence, in 1 minute of time, the minute-hand gains on the hour hand. 
 \-^ of a minute space. 
 
 
 n ' 
 
 .5 
 
 
 6 
 
 
 6 
 
 ^ 
 
 
 
 4 
 
 
 20 
 
 20 
 
 
 4 
 5 
 
 4 
 5 
 
 4 
 
 4 
 
 8 
 
OoS ANALYSIS. 
 
 When the minntc-hand returns to 12, the hour-hand will be at 1, 
 and M"ill require the niinute-hand to gain 5 minute spaces. As the 
 minute-hand passes over -^^ the space gained, to gain 5 minute spaces 
 it must pass over -i^ of 5 = ^^ = 5^j minute spaces, requirin"! 5^ 
 minutes of time = 5ini. 27-^jSCC.^ which added to 1 o'clock, gives 
 ihr. 5ini. 27-^^.sec. 
 
 Second Analysis. — In 12 hours the minute-hand passes the hour- 
 hand 11 times, consequently, if both are at 12, the minute-hand will 
 pass the hour-hand the first time in ^ of 12 hours, or ihr. 5mi. 
 27^jscc. It will pass it the second time in -^ of 12 hours, and so on. 
 
 OPERATION. 
 
 5 X -^-f = yy = h-^m'i. —. omi. 27-^scc., which added to 
 Ihr. =r l//r. omi. 27^sec. Ans. 
 
 21. An apple boy bouglit a certain number of apples at the 
 rate of 3 for 1 cent, and as many more at 4 for 1 cent, and 
 selling them again at 2 for 1 cent, he found that he had gained 
 15 cents : how many apples had he ? 
 
 Analysis. — Since he bought a number of apples at 3 for a cent, 
 and as many more at 4 for a cent, he paid ^ of a cent apiece for the 
 first, and ^ of a cent apiece for the second Jot : then, ^ + \ = ^ o( 
 a cent, what he paid for one of each, and -^-t- 2 = -^ of a ceni, the 
 average price for all he bought. Since he sold at 2 for a cent, or -J- a 
 cent a piece, he must have gained on each apple the difference be- 
 tween -^ and -^j = ^ of a cent ; hence, to gain 1 cent he must sell 
 as many apples as -^ is contained times in 1 = 4-| apples, and to 
 gain 15 cents he must sell 15 times as many, or 4-| x 15 ^ 72 apples. 
 
 operation. 
 
 Ill 7 7_L.9 7 1 7 5 
 
 3^4— T2' T^ ■ ^ ■ — TT' 2 2T — 2T' 
 
 1 -f ^\- = 4|, 4|- X 15 = 72 apples. .In*'. 
 
 22. A gentleman left to his three sons, whose ages were 
 13, 15 and 17 years, $15000, to be divided in such a maimer, 
 that each share being put at interest, at 7 per cent, should 
 give to each son the same amount when he attained the age of 
 21 years : wliat Avas the share of each ? 
 
ANALYSIS. 839 
 
 ANJi LYSIS. — By the question their respective shares would be at 
 interest 8, 6 and 4 years. 
 
 Find the present worth of $1 for 8, 6 and 4 years, respectively: 
 They are S0.G41025G +, $0,7042253 + , and $0,78125. These s'nns 
 teing put at interest at 7 per cent, will each amount to $1 at tho 
 expiration of their respective times ; and the sum of these numbers, 
 $0,6410256 + $0,7042253+^0,78125 = $2,1265009+ is the amount, 
 which being so distributed among them, will produce $1 to each. If 
 each number be divided by the sum, $2,1265009, the quotients will 
 denote the parts of $1, which according to the conditions of the 
 question, each person should receive, and which put at interest will 
 produce equal amounts at the end of their respective times ; there- 
 fore, each person will receive for his entire share 15000 like parts of 
 one dollar. 
 
 OPERATION. 
 
 $1 -^ 1.56 = $0,6410256 + present worth of $1 for 8 years. 
 
 $1 -^ 1.42 = $0,7042253 + " " " 6 " 
 
 $1 -f- 1.28 = $0.78125 " « " 4 « 
 $2,1265009 
 
 S0,6410256 -f 2.1265 x 15000 - $4521,694 
 $0,7042253 + 2.1265 x 15000 = $4967,494 
 $0,78125 -^ 2.1265 x 15000 = $5510,815 
 
 23. A, B, C, and D, agree to do a piece of work for $312. 
 A, B, and C, can do it in 10 days ; B, C, and D, in 7-i- days ; 
 
 C, D, and A, in 8 days ; and D, A, and B, in 8y days. In how 
 many days can all do it, working together ; in how many days 
 can each do it working alone ; and what part of the pay ought 
 each to receive ? 
 
 Analysis. — Since A, B, C, can do the work in 10 days, they can 
 do -^-g = -^^ of it in 1 day : since B, C, D. can do it in 7-^ days, they 
 can do ^^ = -^^ of it in 1 day; since C, D, A, can do it in 8 day.s, 
 they can do ^ = ^^^ of it hi 1 day; and since D, A, B. can do it in 
 8^ days, they can do -^q = -^^ of it in 1 day ; hence. A, B, C, and 
 
 D, by working 3 days each, will do ^2_ + ^^ + ^i^^ + ^i^ = -^^^ 
 of the work, and in 1 day they will do } of ^^^ = J^. It will then 
 tiike them as many days to do the whole as -^-^^ is contained times in 
 1 - 6^ days. 
 
310 ANALYSIS AND 
 
 By subtracting, in succession, what the three can do in 1 day, when 
 tliey work together, from what the four can do in 1 day, we shall 
 have what each one will do in 1 day ; viz., J^ — -^^ = -jl^, what 
 D will do in 1 day; ^^ — I'^o — tIo'j what A can do in 1 day; 
 ^ia-tio^ ihy ^^"li^t B can do in 1 day ; J=^ - t^o = Th: ^'^^^^ 
 C can do in 1 day. It will take each as many days to do the whole 
 work as the part which he can do in 1 day is contained times in 1 ; 
 viz., 1 -I- y4o = 40 days. A's time to do it; 1 -|- j^ = 30 days, B's; 
 1 -- ^ = 24 days, C's ; 1 -;- j^ = 17-^ days. D"s. 
 
 Now. each should receive such a part of the whole amount paid, 
 viz., $312, as he did of the whole work. This part will be denoted 
 by what he did in 1 day multiplied by the number of days he 
 worked: viz., A. ^X6^%=^-^: B, j^^x6^^=^-^; 0,^1^x0^ = 
 
 19- '''•1^0'^"l9 19- 
 
 OPERATION. 
 
 ■j^ = i\fo5 ^'l^at A, B, C, does in 1 day. 
 _2_ _ .UL "BCD " " 
 
 8 — 120' ^' -^J -^J 
 
 _7_ _ 1 4_ " 71 A R " « 
 
 fiO — 120' -^> -^5 ^> 
 
 t¥o + -i^'o + T^ + i'^ = t¥u' ^v^at A, B, C, and D, can 
 do in 3 days. 
 
 yYo -7- 3 r= jL^y, what A, B, C, and D, can do in 1 day. 
 y'^^o — y"^ = jfo^ what A can do in 1 day ; 1 -i- yf ^ — AOda. 
 
 T2 120 — 120 -^J ■*■ • TTo — ^^""* 
 
 JL9- _ JIJ_ _ _5_ « n u li 1 _i_ ,5_ _ 94 7 
 
 T20 l-'O — 120 ^» ■*■ • 120 — •<i'±((«. 
 
 t2 120 — 120 -^J '120 — ^' 7 • 
 
 Hence, the share of each will be, 
 
 S312xfiy=§ 49,2Gt?^, A's share. 
 $312Xy't=^$ 65,68-1^, B's share. 
 $3l2xtV = ^ S2,l0f||-, C's share. 
 $!312xTV = ^114'94iA D's share. 
 
 $312,00 amount paid to A, B, C, and D. 
 
PJIOMTSCUOUS EXAMPLES. 84:1 
 
 24. A person owning ^ of a vessel, sells |- of his share for 
 $1736 : Avhat was the value of the whole vessel ? 
 
 25. If a man performs a journey in 7i days, travelling 14^ hours 
 a day, in how many days will he perform the same journey, 
 by travelling. 10 5^ hours a day ? 
 
 26. If ^j- of a pole stands in the mud, 2 feet in the water, and 
 |- above the water, what is the length of the pole ? 
 
 27. After spending i of my money, and i of the remainder, 
 I had $1062 left : how much had I at first ? 
 
 28. Suppose a cistern has two pipes, and that one can fill it in 
 7J- hours, and the other in 4i hours : iu what time can both fill 
 it runninjT together ? 
 
 29. If 54 yards of ribbon cost $9, what will 26 yards cost? 
 SO. If 2 acres of land cost i of f of | of $300, what will 
 
 i of -I of 10|- acres cost,'* 
 
 31. There is a regiment of soldiers to be clothed: each suit is 
 to contain 3^ yards of cloth 1-| yards wide : how much shal- 
 loon that is |- yards wide is necessary for lining ? 
 
 32. How much tea at 7s. 61:/. a pound must be given for 234 
 bushels of oats, at 3s. 9d. a bushel. New York currency ? 
 
 S3. What will 3 pipes of wine cost at 2s. dd. per quart, New 
 England currency ? 
 
 34. A gives B 165 yards of cotton cloth, at 2s. Qd. per yard, 
 Missouri currency, for 625 pounds of lump sugar : Irow much 
 was the sugar worth a pound ? 
 
 35. If the expense of keeping 1 horse 1 day is 3s. AJ. Canada 
 currency, what will be the expense of keeping 4 horses 3 weeka 
 at the same rate ? 
 
 36. Bought 10 bales of cloth, each bale containing 14 pieces, 
 and each piece 22^ yards, at 10s. 8(/. per yard, lUinois cur- 
 rency : what was the cost of the cloth ? 
 
 37. A has l\cwt. of sugar, worth 12 cents a pound, for which 
 B gave him 12^cut. of flour : what was the flour worth a pound ? 
 
 38. What is the value of 2hhd. of molasses, at Is. 2d. per 
 quart, Georgia currency? 
 
 39. What will be the value of 3 pieces of cloth, each piece 
 
342 ANALYSIS AND 
 
 containing 24^ yards, at 4s. Gd. per yard, Pennsylvania cur- 
 rency ? 
 
 40. Bought 120 yards of cloth, at 65. 8d. a yard, New York 
 currency, and gave in payment 76 bushels of rye, at is. Qd. a 
 bushel, New England currency, and the balance in money : how 
 many dollars will pay the balance ? 
 
 41. A merchant bought 21 pieces of cloth, each piece con- 
 taining 41 yards, for which he paid $1260 ; he sold the cloth at 
 $1,75 per yard : did he gain or lose, and how much ? 
 
 42. The hour and minute hands of a watch are together at 
 12 : at what moment will they be together between 5 and 6 ? 
 
 43. How many yards of carpeting f of a yard wide will cover 
 the floor of a room 18 feet long and 15 feet wide ? 
 
 44. If 9 men can build a house in 5 months, by working 12 
 hours a day, how many hours a day must the same men work 
 to do it in 6 months ? 
 
 45. 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 ? 
 
 46. 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, working together ? 
 
 47. Divide ihe number 480 into 4 such parts that they shall 
 be to each other as the numbers 3, 5, 7 and 9 ? 
 
 48. What length of a board that is 8^- inches broad, will make 
 a square foot ? 
 
 49. The provisions in a garrison were sufficient for 1800 
 men, for 12 montlis ; but at the end of 3 months, it was rein- 
 forced by 600 men, and 4 months afterwards, a second rein- 
 forcement of 400 was sent in : how long would the provisions 
 last after the last reinforcement arrived ? 
 
 50. A merchant bought a quantity of broadclotli and baize 
 for $488,80; there was 117^ yards of broadcloth, at $3,\ i)er 
 yard; for every 5 yards of broadcloth he had 1^ yards of 
 baize : how many yards of baize did lie buy, and what did it 
 cost him per yard ? 
 
I'ROMTSCUOUS EXAMPLES. 343 
 
 ^1. If the freight of 40 tierces of sugar, aach weighing 3i 
 cui., for 150 miles, costs $42, Avhat must be paid for the freiglit 
 of lOhhd. each weighing 12cwL, for 50 miles ? 
 
 52. 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 
 of tea, when a lemon is worth 2 cents ? 
 
 53. What amount must be discounted, at 7 per cent, to make a 
 present payment of a note of $500, due 2 years 8 months hence ? 
 
 54. If the interest on $225 for 4i years is$91,12i, what 
 would be the interest on $640 at the same rate for 2i years ? 
 
 55. A farmer having 1000 bushels of wheat to sell, can have 
 $1,75 a bushel cash, or $1,80 a bushel in 90 days : which would 
 be most advantageous to him, money being worth 7 per cent ? 
 
 56. A merchant bought goods to the amount of $1575 on 9 
 months credit ; he sells the same for $1800 in cash : money 
 being worth 6 per cent, what did he gain ? 
 
 57. Three persons in partnership gain $482,62 ; A put in f 
 as much capital as B, and B put in | as much as C : what was 
 each one's share of the gain ? 
 
 58. A father divided his estate, worth $9268,60, among his 
 4 children, giving A, i of it, B, J, and C, $5 as often as he gave 
 D $6 : how much did each receive ? 
 
 59. A tax of $475,50 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 87, and D, $8 : what part of the whole tax did each 
 villa2;e nav? 
 
 60. 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 lill the 
 end of 8 weeks, what must be the daily allowance to each man ? 
 
 61. 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 '^0 days: 
 in what time will the cistern be filled if they are all allowed to 
 run at the same time ? 
 
 62. Two persons, A and B, are on opposite sides of a ■wood, 
 which is 536 yards in cii'cumference ; they begin to tra.vel in 
 
'64:4: ANALYSIS AIS'D 
 
 the same direction at the same time ; A goes at the rate ol" 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 luj over- 
 takes A ? 
 
 63. 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 16 days, and 
 and the boy could do it in 20 days : how long would it take 
 them to do it together ? 
 
 64. A owes B $500, of which $150 is to be paid in 3 months. 
 $175 in 6 months, and the remainder in 8 months : Avhat would 
 be the equated time for the payment of the whole ? 
 
 65. If 42 men, in 270 days, working 8^ hours a day, can 
 build a wall 98-| feet long, 7^ feet high, and 2i feet thick ; in 
 how many days can 63 men build a wall 451 feet long, 6^ feet 
 high, and 31 feet thick, working 111 hours a day ? 
 
 GO. 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 ? 
 
 67. A man had a bond and mort£raG;e for $2500, dated Julv 
 1st, 1854. He is not satisfied with 7 per cent annual interest, 
 and on the first day of September, 1854, he purchased 10 shares, 
 of $100 each, of railroad stock, at 115. 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 regu- 
 lar serai-annual dividend of 4 per cent, and at the end of the 
 year (January 1st, 1856,) sold his whole stock at 99 : which 
 was the more profitable investment, and how much ? 
 
 68. A landlord being asked how much he received for (he 
 rent of his property, answered, tliut after deducting 9 cents from 
 each dollar, for taxes and repairs, there remained $3014,30 : 
 wliat was the amount of his rents? 
 
 69. If 165 pounds of soap cost $16,50, for how mucli will it 
 be nfcessary to sell 390 pounds, in order to gain the cost of 3G 
 pounds ? 
 
PEOMISCUOUS EXAMPLES. 345 
 
 70. What is tlie height of a wall which is 14^ yards in length, 
 and -j\ of a yard in thickness, and -which has cost $406, it hav- 
 ing been paid for at the rate of $10 per cubic yard ? 
 
 71. A thief escaping from an officer, has 40 miles the start, 
 and travels at the rate of 5 miles an hour ; the officer in pursuit 
 travels at the rate of 7 miles an hour : how far must he travel 
 before he overtakes the thief ? 
 
 72. 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 persons and 
 10 children ; the first family used from the flour 2 wrecks, and 
 the second 3 weeks : how much ought each to pay ? 
 
 73. At iii-'42 a thousand, how much lumber should be given 
 for a farm containing 33A., 2i?., 16P., valued at $125 an acre ? 
 
 74. IIow^ many building lots, each 50 feet by 100 feet, can be 
 made out of 2^ acres of ground ? 
 
 75. A person pays $150 for an insurance on goods, at 3f 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 be- 
 sides : what was the value of the goods ? 
 
 76. A distiller purchased 5000 bushels of rye, which he can 
 have at 96 cents a bushel, ready money, or $1, with 2 months' 
 credit ; Avhich would be the more advantageous to him, to buy 
 it on credit, or to borrow the money at 7 per cent, and pay the 
 cash ? 
 
 77. A stockholder bought | of the capital of a company at 
 par ; he sold |- of his purchase at par, and the remainder for 
 125000, and by the latter sale made $5000 : what was the value 
 of the whole capital ? 
 
 78. How many bushels of grain will a bin contain, that is 
 Sf{. bin. wide, 2ft. Gin. long, and Gft. deep? 
 
 79. If the two sides of a triangle arc 75 feet and 90 feet, 
 and the perpendicular to the third side 45 feet, Avhat is the 
 length of the third side ? 
 
 80. Three travellers have 2160 miles to go before they reach 
 
''46 ANALYSIS AND 
 
 the end. of their journey ; the first goes 30 miles a day, the 
 
 second 27, and the third 24: how many days should one set out 
 alter another that they may arrive together ? 
 
 81. A house which was sold a second time for §7180, would 
 have given a profit of §420 if the second projirietor had pur- 
 chased it $130 cheaper than he did : at what price did he pur- 
 chase it ? 
 
 82. 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 ? 
 
 83. I have three notes payable as follows : one for $100, due 
 Feb. 12th, 1856, the second for $400, due March 12th, and the 
 third for $300, due April 1st: what is the average time of pay- 
 ment ? 
 
 84. How many marble slabs, 152??. square, will it take to 
 pave a floor 32 feet long, and 25 feet wide ? What will be th^ 
 cost at $3 a square yard for the marble, and 40 cents a square 
 yard for labor ? 
 
 85. A man, in bis will, bequeathed $500 to A, $425 to B, 
 $300 to C, $250 to D, and $175 to E ; but after settling up the 
 estate and paying expenses, there was but $1155 left: what is 
 each one's share ? 
 
 86. If 'dibs, of tea are worth 7lbs. of coffee, and lilbs. of 
 coffee are worth ASlbs. of sugar, and ISlbs. of sugar are worth 
 27 lbs. of soap ; how many pounds of soap are Gibs, of tea worth ? 
 
 87. "What is the hour, when the time past noon is ^ the time 
 to midnight ? 
 
 88. If f of a yard of cloth cost $|, being |- of a yard wide, 
 what is the value of f of a yard If yards wide, of the same 
 quality? 
 
 89. A i'armer sold GO fowls, a part turkeys, and a part chick- 
 ens ; for the turkeys he received $1,10 apiece, and for tlio 
 chickens 50 cents apiece, and for the whole he received $51,60 
 how many were there of each ? 
 
PllOMISCUOUS EXAMPLES. 847 
 
 90. A person hired a man and two boys ; to the man he gave 
 6 shiUings a day, to one boy 4 shillings luid to the other 3 shil- 
 lings a day, and at the end of the time he paid them 104 shil- 
 lings : how long did they work ? 
 
 91. Divide $6471 among 3 persons, so tbat as often as the 
 first gets $5, the second will get $6, and the third $7. 
 
 92. Two partners have invested in trade $1G00, by which 
 they have gained $300 ; the gain and stock of the second amount 
 to $1140 : what is the stock and the gain of each "? 
 
 93. 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 
 hiijh, chives a shade of 4.55 feet in length ? 
 
 94. 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 lirst piece 
 of work ? 
 
 95. Tm'o persons pass a certain point at an inter\ al of 4 
 hours ; the first travelling at the I'ate of 1 1^, and the second 
 174t miles an hour : how far and how long must the first travel 
 before he is overtaken by the second ? 
 
 96. 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 receives $120 
 of the gain, B $400, and C $100 : what was each person's 
 stock ? 
 
 97. 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, BIO miles, and 
 C 16 miles per day: in what time will they all be together 
 again ? 
 
 98. What length of wire, | 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 ? 
 
 99. A person having $10000 invested m G per cent, stocks, 
 sells out at 65, and invests the proceeds in 5 per cents at 82|^ • 
 what will be the difference in his income ? 
 
 100. In order to take a boat through a lock from a certain 
 
uiy ANALYSIS AND 
 
 river into a canal, as well as to descend from the canal into the 
 river, a volume of water is necessary 461 yards long, 8 yards 
 wide and 2| yards deep. How many cubic yards of water 
 will this canal throw into the river in a year, if 40 boats ascend 
 and 40 descend each day except Sundays and eight holidays ? 
 
 101. A company numbering sixty-six shareholders have con- 
 structed a bridge which cost $200000 : what will be the gain 
 of each partner at the end of 22 years, supposing that 6400 
 persons pass each day, and that each pays one cent toll, the 
 expense for repairs, &c., being $o per year for each share- 
 holder ? 
 
 102. 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 ; tlie third put in $400; 
 and every six months after he added $500 ; tlie fourth did not 
 contribute till 8 months after the commencement of the part- 
 nership ; he then put in $900, and repeated tliis sum every 6 
 months ; the fifth put in no capital, but kept the accounts, for 
 which the others agreed to allow him $800 a yeai-, to be paid 
 in advance and put in as capital. What is each one's share of 
 the gain, which was $20,000 ? 
 
 103. A general arranging his army in the form of a square, 
 finds that he has 44 men remaining, but by increasing each side by 
 another man, he wants 49 to fill up the square : how many men 
 had he ? 
 
 104. A, B and C, are to share $100 in the proportion of 
 1, i and -}, respectively ; but by the death of C, it is required 
 to divide the Avhole sum proportionally between the other two : 
 what will each have ? 
 
 105. A lady going out shopping spent at the first place she 
 stopped, one-half her money, and half a dollar more ; at tho 
 next place, half tlie remainder and half a dollar more ; and at 
 the next place lialf llio remainder and half a dollar more, when 
 she found lliat she liad l)ut three dollars left : how much had 
 she when she started ? 
 
 100. If a i)ipc of 6 inches discharge a certain quantity of 
 
PKOMISUUrOUS EXAMPLES. 349 
 
 fluid in 4 hours, in "wliat time will 4 pipes, each of 3 inches 
 bore, discharge twice that quantity ? 
 
 107. A man bought 12 horses, agreeing to pay |40 for the 
 first, and in an increasing arithmetical progression for the rest, 
 paying io70 foi' the last : what was the difference in the cost, 
 and what did he pay for them all ? 
 
 108. A, B, C and D, engaging in speculation, lost a sum of 
 money, of which A, B and C, paid $297,60 ; B, C and D, 
 $321,92 ; C, D and A, i5^375,83 ; and D, A and B, $402,50 • 
 what did each one pay ? 
 
 109. If for £3000 exchange we pay 7-|- per cent premium, 
 giving in payment notes at 4 months, 12 per cent discount, what 
 rate ought we to make the premium, giving notes at 6 months, 
 10 per cent discount ? 
 
 110. A purchase of $15000 worth of goods 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 7 per cent, how much ready money ought to pay 
 the debt ? 
 
 111. If an iron bar 5 feet long, 2^- inches broad, and 1-^- 
 inches thick, weigh 45 pounds, how much will a bar of the 
 same metal weigh, that is 7 feet long, 3 inches broad, and 2i 
 inches thick ? 
 
 112. 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 ? 
 
 113. A person passed i of his life in childhood, yV of it in 
 youth, 5 years more than -|- of it in matrimony : he then had a 
 son, whom he survived 4 years, and who readied only J the 
 a2;e of his father. At what age did he die ? 
 
 114. 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 G inches, leaving 
 the diameter of the well witliin the wall 3 feet 6 inches. If 
 
 10 
 
o50 PKOMISCUOUb EXAMPLES. 
 
 the well is 40 feet deep, how many cubic feet of stone will be 
 
 required ? 
 
 115, A surveyor measured a piece of ground in the form of 
 a rectangle, and found one side to be 37 chains, and the other 
 42 chains 1 6 links : how many acres did it contain ? 
 
 116, A, B and C, can buikl a barn in 10 days; after 4 days, 
 A leaves, and B and C go on with the work for 5 days longer, 
 when B leaves, --^q of the work being yet unfinished : C pro- 
 ceeds with the work and finishes it in 11|- days after B left : 
 how long would it take each to build the barn ? 
 
 117, A tanner bought a piece of land for ^1500, and agreed 
 to pay principal and interest in 5 equal annual instalments : 
 if the interest was 7 per cent., how much was the annual pay- 
 ment ? 
 
 118, A fountain has 4 receiving pipes, A, B, C and D ; A, 
 B and C will fill it in G^- hours ; B, C and Din 7\ hours ; C, D 
 and A in 8 hours ; and D, A and B in 15 Lours : it has also 
 4 discharging pipes, E, F, G and H ; E, F and G will empty 
 it in 7^ hours ; F, G and H in 4 hours ; G, II and E ino] hours ; 
 II, E and F in 31 hours. Suppose the fountain full of water, 
 and all the pipes open, in what time would it be emptied ? 
 
 119, IIow many planks 15 feet long, and 15 inches wide, 
 will floor a barn GOJy feet long, and 331 feet wide ? 
 
 120, If a ball 2 inches in diameter weigh 5 pounds, wjiat 
 will be the diameter of another ball of the same material that 
 weighs 78,125 pounds ? 
 
 121, A gives B his bond for $5000, dated April 1st, 1851, 
 payable in 10 equal aniuuvl instalments of $500 each, on and 
 afier the; first day of Ajjril. 1852. A afterwards agreed to 
 take up his bond on the first day of April, 1853, deduding 
 semi-annual discount, at the rate of 7 per oent, per annum, on 
 till-! several payments, which \'v\i due after the first day of 
 Api-il, i.sr>2: what sum, on the first day of April, 1853, will 
 o*UH',':l the bond ? 
 
APPLICATIONS OF AllITHMETIC. 
 
 MENSURATION. 
 
 329. Mensukation embraces all the methods of determining 
 the contents of geometrical figuies. It is divided into two parts, 
 the raeniiiration of surfaces and the mensuration of Volumes. 
 
 1 foot. 
 
 o 
 
 MENSURATION OP SURFACES. 
 
 330. 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 
 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. 
 
 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. Thei'efore, the square whose side is four feet, 
 contains sixteen square feet. 
 
 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 lo 12 x 12 = 144. 
 
 329. What is mensuration 1 
 
 330. What is a surface i What is a square 1 ^Miat is tlic luiiuber ol 
 small fciiuart's coutained ia a large square equal to 1 
 

 MElvrSUKATION 
 
 331. 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 
 angles. The side AB is the base. 
 
 When a line like CD is drawn, making 
 the angle CDA equal to the angle CDB, 
 then CD is said to be perpendicular to AB, and CD is called 
 the altitude of the triangle. Each triangle 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 altitude (Bk. IV., Prop. VI).* 
 
 OPERATION. 
 
 50 
 30 
 
 2)1500 
 Ans. 1 bO square yards. 
 
 EXAMPLES. 
 
 1. The base, AB, of a triangle is 
 50 yards, and the perpendicular, CD, 
 80 yards : what is the area ? 
 
 Analysis. — We first multiply the base 
 by the altitude, and the product is square 
 yards, which "vve divide by 2 for the area. 
 
 2. In a triangular field the base is GO chains, and the pei*- 
 pendicular 12 chains : how much does it contain ? 
 
 3. There is a triangular field, of wliich the base is 45 rods, 
 and tlic perpendicular 38 rods : what are its contents ? 
 
 4. Wliat are the contents of a triangle whos.e base is 75 
 chains, and perpendicular 36 chains ? 
 
 332. A rectangle is a four-sided figure like 
 
 a squai'e, in which the sides are perpendicular 
 
 to each other, but the adjacent sides are not 
 
 ?qual. 
 
 * All the references are to Davies' Letrendrc. 
 
 331. What is a triangle ? What is the base of a triangle'! What the 
 I'lliliulo? What is a rirjht-angled triangle 1 Which side is the hypo- 
 tlicniise 1 What is the aica of a triangle equal to \ 
 
 ■M'i. Wluit i^ a rectangle ? 
 
 \ 
 
or suKFACKs. 353 
 
 333. A parallelogram is a foux*-sided 
 figure which has its opposite sides equal 
 and parallel, but its angles not right- 
 angles. The line DE, perpendicular to - p 
 the base, is called the altitude. 
 
 334. To find the area of a square, rectangle, or parallelogram. 
 
 Multiply the hase by the 2^cfpendicular height, and the -product 
 will he the area (Bk. IV., Prop. Y.) 
 
 EXAMPLES. 
 
 1. What is the area of a square field, of which the sides are 
 each 6G.16 chains ? 
 
 2. What is the area of a square piece of land, of which the 
 sides are 54 chains ? 
 
 3. What is the area of a square piece of land, of which the 
 sides are 75 rods each ? ' 
 
 4. What are the contents of a rectangular field, the leno;th 
 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 ? 
 
 7. 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 720 feet : Avhat is the area ? 
 
 335. A trapezoid is a four-sided figure d E 
 ABCD, having two of its sides, AB, DC, 
 parallel. The perpendicular EF is call- 
 
 -^d the altitude. A F B 
 
 336. To find the area of a trapezoid. 
 
 Multiply the sicni of the two parallel sides hy the altitude, 
 
 333. What is a parallelogram T 
 
 834. How do you find the area of a square, rectangle, or parallelogram 1 
 335. ^^'hat is a trapezoid 1 
 330. How do you find the area of a trapezoid ? 
 
 If) 
 
854: 
 
 MENSURATION 
 
 OPERATION. 
 643.02 + 428.48 = 1071.50 = 
 sum of parallel sides. Then, 
 1071.50x342.32 = 366795.88; 
 and S^&l^&JSA ^ 183397.94 = 
 the area. 
 
 divide the p7-oduct hj 2, and the quotient will he the area (Bk. 
 IV. Prop. VII). 
 
 EXAMPLES. 
 
 1. Required the area or contents of the trapezoid ABCD, hav- 
 ing given AB=G43.02 feet, DC = 
 
 428.48 feet, and EF = 342.32 feet. 
 
 Analysis. — We tirst find the sum 
 of the sides, and then multiply it by 
 the perpendicular height, after which 
 we divide the product by 2. for the 
 area. 
 
 2. What is the area of a trapezoid, the parallel sides of which 
 are 24.82 and 16.44 chains, and the perpendicular distance 
 between them 10.30 chains? 
 
 3. Required the area of a trapezoid, whose parallel sides are 
 51 feet and 37 feet 6 inches, and the perpendicular distance 
 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 ? 
 
 6. What are the contents of a trapezoid, when the parallel 
 sides are 40 and 64 chains, and the perpendicular distance 
 between them 52 chains ? 
 
 337. A circle is a portion of a plane bounded by a curved 
 line, every point of which is equally dis- 
 tant from a certain point within, called 
 the centre. 
 
 The curved line AEBD is called the 
 circumference; the point C the centre ; 
 the line AB passing through the centre 
 Ti, did meter ; and C'li n radius. 
 
 TIic circiiniierence AEBD is 3.1416 
 times as great as the diiuneter AB- 
 
OF SUKFACES, 855 
 
 Hence, if the diameter is 1, the circumference Avill be 3.1416. 
 Therefore, if tlie diameter is known, the circumference is found 
 hy multiplying 3.1416 by the diameter (Bk. V. Prop. XIV). 
 
 EXA3IPLES. ' 
 
 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 
 
 ameter. » Ans. 25.1328 
 
 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 
 
 ? 
 
 338. Since the circumference of a circle is 3.1416 times as 
 great as the diameter, it follows, that if the circumference is 
 known, we may find the diameter by dividing it by 3.1416. 
 
 EXABIPLES. 
 
 1. What is the diameter of a circle whose circumference is 
 157.08? 
 
 OPERATION. 
 
 Analysis. — We divide the circumference 3.1416)157.080(50 
 by 3.1416, the quotient 50 is the diameter. 157.080 
 
 2. Wliat is the diameter of a circle whose circumference i3 
 23304.3888 ? 
 
 3. What is the diameter of a circle whose circumference is 
 13700 ? 
 
 337. What ir- a circle 1 What is the centre ? What is the circumfer- 
 ence ': What is the diameter 1 What the radius 1 How many times 
 greater is tliC circumference th.m the diameter 1 How do you find the 
 circumference when tlie diameter is known ? 
 
 33S. How do you find the diameter when the circumference is knuwu 1 
 
356 MEKSdRATION 
 
 339. To find the area or contents of a circle. 
 
 Multlphj the square of the diameter by the decimal .7854 
 (Bk. V. Prop. XII. Cor. 2). 
 
 EXA3IPLES. 
 
 1. What is the area of a circle whose diameter is 12 ? 
 
 Analysis.— We first square the diam- operation. 
 
 2 
 
 eter, giving 144, which we then multi- 12 = 144 
 
 ply by the decimal .7854 : the product 144 x .7854 = 113.0976 
 
 is the area of the circle. Ans. 113.0976 
 
 2. What is the area of a circle whose diameter is 5 ? 
 
 3. What is the area of a circle whose diameter is 14 ? 
 
 4. How many square yards in a circle whose diameter is d^ 
 feet? 
 
 340. A sphere is a portion of space 
 
 bounded by a curved surface, all the 
 
 points of which are equally distant from 
 
 a certain point within, called the centre. 
 
 The line AD, passing through its centi-e 
 
 C, is called the diameter of the sphere, 
 
 and AC its radius. 
 
 A 
 
 341. To find the surface of a sphere, 
 
 Multiply the square of the diameter by 3.1416 (Bk. VHL 
 Prop. X. Cor.) 
 
 EXAMPLES. 
 
 1. What is the surface of a sphere whose diameter is 6 ? 
 
 OPERATION. 
 
 Analysis. — We simply multiply the number 3,1416 
 
 3.1416 by the .square of the diameter : the pro- 6'= 36 
 
 duct is the surface. Ans. 1 13.076 
 
 339. How do you find the area of a circle ? 
 
 340. What is a sphrre ? \\'hat is a iliameterl What is a radius^ 
 
 341. How do you find the surface of a gpherc ? 
 
OF VOLUME. 
 
 357 
 
 3 feet = 1 yard. 
 
 2. What is the surface of a sphere whose diameter is 1 4 ? 
 
 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. 
 
 MENSURATION OF VOLUMES. 
 
 342. A SOLID or VOLUME is a portion of space h-iving tliree 
 dimensions : length, breadth, and 
 
 thickness. It is measured bj a 
 cule called the cuhic unit, or unit 
 of vohime. 
 
 A CUBE is a volume having six 
 equal faces, which are squares. If 
 the sides of the cube lie each one 
 foot long, the figure is called a 
 cubic foot. But when the sides 
 of the cube are. one yard, 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 ; and if it were 3 feet high, it would contain three 
 tiers, or 27 cubic feet. Hence, the contents of stick a f (jure arc 
 equal to the "product of its lengthy breadth, and height. 
 
 343. To find the contents of a sphere, 
 
 Multiply the surface by the diameter, and divide the product 
 by 6, the quotient will be the contents (Bk. VIII. Prop. XIV. 
 Sch. 3). 
 
 EXAMPLES. 
 
 1. What are the contents of a sphere whose diameter is 12 ? 
 
 342. What i.s a volume ] AVhat is a cube 1 W!i«t is a cubic foot ? 
 What is a cubic yard 1 How many cubic feet in a cubic yard ■! What are 
 the contents of a figure of three dimensions equal to 1 
 
 :i'iri. Hov.' di> you ihid the contetit.s of a sphered 
 
858 
 
 MENSUKATION 
 
 Analysis. — We first find the surface 
 by multiplying the square of the diam- 
 eter by 3.1416. We then multiply the 
 surface by the diameter, and divide the 
 product by 6. 
 
 OPi RATION, 
 
 12"= 144 
 mulliply by 3.1416 
 surface 452.3904 
 
 diameter ] 2 
 
 solidity 
 
 904.7808 
 
 2. What are the contents of a, sphere whose diameter is 8 ? 
 
 3. What are 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. What are the contents of a sphere whose diameter is 12 
 feet? 
 
 344. 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 pai'allelograms which bound the prism is called 
 the convex surface. 
 
 34.5. To find the convex surface of a ri"'ht 
 prism. 
 
 Multiply the perimeter of the base hi/ the perpendicular 
 height, and the product will he the convex surface (Bk. VII. 
 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. AVhat is the convex surface when there are eight equal 
 sides, each 15 feet in length, and tlie altitude is 12 feet? 
 
 344. What is a prism 1 What is the perimeter of the base 1 What ia 
 the convex surface ] 
 
 345. How do you find tlic convex surface of a prism ? 
 34G. How do you find Iho contPiit:! of a prinui 1 
 
OF VOLUM]':. 
 
 ;J59 
 
 346. To find the contents of a prism, 
 
 Multiply the area of the base by the altitude, and the product 
 will be the contents (Bk. VII. Prop. XIV). 
 
 EXAMPLES. 
 
 1. What are the contents of a square prism, each side of the 
 square -which forms the base being IG, and the ahitude of the 
 prism 30 feet ? 
 
 OPERATION. 
 
 Analysis. — We first tind the area of the square 16^— 256 
 
 which forms the base, and then multiply by the 30 
 
 altitude. Ans. 7 680 
 
 2. What are the contents of a cube, each side of Avliich is 
 48 inches ? 
 
 3. How many cubic feet in a block of marble, of which 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 solidity of a triangular prism, whose height 
 is 20 feet, and area of the base G91. 
 
 347. A Cylinder is a volume generated 
 by the revolution of a rectangle AF about 
 EF. The line EF is called the axis or alti- 
 tude — the circular surface, the convex surface 
 of the cylinder, and tlie circular ends, the bases. 
 
 348. To find the convex surface of a 
 cylinder, 
 
 Multiply the circumference of the base by the altitude, and the 
 product will be the convex surface (Bk. VIII. Prop. I). 
 
 347. What is a cylinder 1 What is the axis or altitude T What is tho 
 convex surface 1 
 
 348. How do you find the convex surface ! 
 
3 GO MENSTIRA.TION 
 
 EXASIPLES. 
 
 1. "What is the convex surface of a cylinder, the diameter 
 of whose base is 20 and the altitude 40 ? 
 
 OPKRATION. 
 
 AxALVSis. — We first multiply 3.1416 by 3.1416 
 
 the diameter, which gives the circumfer- 20 
 
 ence of the base. Then, multiplying by 62.8320 
 
 the altitude, we obtam the convex sur- 40 
 
 face. Ans. 2513.2800 
 
 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 ? 
 
 349. To find the volume of a cylinder, 
 
 3Ii(ltiph/ the area of the base by the altitude : the product will 
 he the contents or volume (Bk. VIII. Prop. II), 
 
 EXAMPLES. 
 
 1. Required the contents of a cylinder of which the altitude 
 is 11 feet, and the diameter of the base 16 feet. 
 
 Analysis. — We first find the area of the operation. 
 
 base, and then multiply by the altitude: 16^ = 256 
 
 the product is the solidity. .7854 
 
 2. What are the contents of a cylin- ^"^^^ ^^'®' 201-0624 
 
 der, the diameter of whose base is 40, 
 
 ' , . , „^^ 2211.6864 
 
 and the altitude 29 ? 
 
 3. ^Vliat are the contents of a cylinder, the diameter of whose 
 base is 24, and the altitude 30 ? 
 
 4. "V\''hat 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 of whose 
 base is 25 feet, and altitude 15 ? 
 
 U4'J IIow do you find the contents of a cyliudcr 1 
 
OF VOLTTME. 
 
 861 
 
 350. A Pyramid is a volume bounded 
 by several triangular planes united at the 
 same point S, and by a plane figure or base 
 ABODE, in which they terminate. The 
 altiude of the pyramid is the line SO, 
 drawn perpendicular to the base. 
 
 351. To find the contents of a pyramid. 
 
 3fultipli/ the area of the base by the altitude, and divide thd 
 product by 3 (Bk. VII., Prop. XVII). 
 
 EXAMPLES. 
 
 1. Required the contents of a pyramid, the 
 area of whose base is 86, and the altitude 24. 
 
 OPERATION. 
 
 86 
 
 24 
 
 3);i0tJ4 
 
 Analysis. — We simply multiply the area of the 
 base 86, by the altitude 24. and then divide the Ans. 688 
 
 product by 3. 
 
 2. What are the contents of a pyramid, the area of whose baso 
 is 3G5, and the altitude 36 ? 
 
 3. What are the contents of a pyramid, the area of whose base 
 is 207, and altitude 36 ? 
 
 4. What are the contents ef a pyramid, the area of whose 
 base i.s oG2, and altitude 30 ? 
 
 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 ahitude of 24: what are its contents ? 
 
 350. What is a pyramid 1 What is the altitude of a pyramid ] 
 
 351. How do you find the contents of a pyramid 1 
 

 GAUGING. 
 
 352. A Cone is a volume generated by 
 the revolution of a ri^'bt ani^led trianerle 
 ABC, about the side CB. The point C is 
 tli(! vertex, and the line CB is called the 
 axis or altitude. 
 
 353. To find the contents of a cone. 
 
 Multiply the area of the base hj the altitude, and divide the 
 product by 3 ; or, midtiply the area of the base by one-third of 
 the altitude (Bk. VIII., Prop. V.) 
 
 EXAMPLES. 
 
 1. Required the contents of a cone, the diameter of whose 
 
 base is 6, and the altitude 11. 
 
 OPERATION. 
 
 6^ = 36 
 
 .7854 = 28.2744 
 
 n 
 
 3)311.0184 
 Ans. 103.(j728 
 
 Analysls. — We fir.st square the diame- 
 ter, and multiply it by .7854, which gives 36 x 
 the area of the base. We next multiply 
 by the altitude, and then divide the pro- 
 duct by 3. 
 
 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. 
 354. Cask-Guaging is the method of finding the number 
 of gallons which a cask contains, by measuring the external 
 dimensions of the cask. 
 
 352. Wh&t is a cone \ What is the vertex ? What is the axis I 
 3.')3. How do you find llio contents of a cone ? 
 364. What is cask-gauging \ 
 
GAUGING. 
 
 S63 
 
 355. Casks are divided into four varieties, according to the 
 curvature of their sides. To wliicli of the varieties any cask 
 belongs, must be judged of by inspection. 
 
 1st Variety — least curvature. 
 
 2d Variety. 
 
 3d Variety. 
 
 4th Variety — greatest curvature. 
 
 356. The first thing to be done is to find the mean diameter. 
 To do this. 
 
 Divide the head diameter hy the hung diameter, and find the 
 quotient in the first column of the following table, marked Qu. 
 Then if the hung diameter he midtiplied hy the number on the 
 same line with it, and in the column answering to the proper 
 variety, the product will he the true mean diameter, or the diame 
 tcr of a cylinder having the same altitude and the same con 
 tents with the cask projMsed. 
 
 oCi5. Into how many varieties are casks divided ! 
 ^56. How do you liiid tlic mean diameter 1 
 
3(54 
 
 
 
 
 
 GAUGING. 
 
 
 
 
 Qu. 
 50 
 
 mVai. 
 
 •2dVar. 
 
 3dVar. 4thVar. 
 
 Qu. 
 
 1st Var. 
 
 2d Var. 
 
 3d Var. 
 
 4th Var. 
 
 8660 
 
 8465 
 
 7905 
 
 7637 
 
 76 
 
 9270 
 
 9227 
 
 8881 
 
 8827 
 
 51 
 
 8680 
 
 8493 
 
 7937 
 
 7681 
 
 77 
 
 9296 
 
 9258 
 
 8944 
 
 8874 
 
 52 
 
 8700 
 
 8520 
 
 7970 
 
 7725 
 
 78 
 
 9324 
 
 9290 
 
 8967 
 
 8922 
 
 53 
 
 8720 
 
 8548 
 
 8002 
 
 7769 
 
 79 
 
 9352 
 
 9320 
 
 9011 
 
 8970 
 
 54 
 
 8740 
 
 8576 
 
 8036 
 
 7813 
 
 80 
 
 9380 
 
 9352 
 
 9055 
 
 9018 
 
 55 
 
 8760 
 
 8605 
 
 8070 
 
 7858 
 
 81 
 
 9409 
 
 9383 
 
 9100 
 
 9066 
 
 56 
 
 8781 
 
 8633 
 
 8104 
 
 7902 
 
 82 
 
 9438 
 
 9415 
 
 9144 
 
 9114 
 
 57 
 
 8802 
 
 8662 
 
 8140 
 
 7947 
 
 83 
 
 9467 
 
 9446 
 
 9189 
 
 9163 
 
 58 
 
 8824 
 
 8690 
 
 8174 
 
 7992 
 
 84 
 
 9496 
 
 9478 
 
 9234 
 
 9211 
 
 59 
 
 8846 
 
 8720 
 
 8210 
 
 8037 
 
 85 
 
 9526 
 
 9510 
 
 9280 
 
 9260 
 
 60 
 
 8869 
 
 8748 
 
 8246 
 
 8082 
 
 86 
 
 9556 
 
 9542 
 
 9326 
 
 9308 
 
 61 
 
 8892 
 
 8777 
 
 8282 
 
 8128 
 
 87 
 
 9586 
 
 9574 
 
 9372 
 
 9357 
 
 62 
 
 8915 
 
 8806 
 
 8320 
 
 8173 
 
 88 
 
 9616 
 
 9606 
 
 9419 
 
 9406 
 
 63 
 
 8938 
 
 8835 
 
 8357 
 
 8220 
 
 89 
 
 9647 
 
 9638 
 
 9466 
 
 9455 
 
 64 
 
 8962 
 
 8865 
 
 8395 
 
 8265 
 
 90 
 
 9678 
 
 9671 
 
 9513 
 
 9504 
 
 65 
 
 8986 
 
 8894 
 
 8433 
 
 8311 
 
 91 
 
 9710 
 
 9703 
 
 9560 
 
 9553 
 
 66 
 
 9010 
 
 8924 
 
 8472 
 
 8357 
 
 92 
 
 9740 
 
 9736 
 
 9608 
 
 9602 
 
 67 
 
 9034 
 
 8954 
 
 8511 
 
 8404 
 
 93 
 
 9772 
 
 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 
 
 8672 
 
 8590 
 
 97 
 
 9901 
 
 9900 
 
 9851 
 
 9850 
 
 72 
 
 9162 
 
 9104 
 
 8713 
 
 8637 
 
 98 
 
 9933 
 
 9933 
 
 9900 
 
 9900 
 
 73 
 
 91 88 
 
 9135 
 
 8754 
 
 8685 
 
 99 
 
 9966 
 
 9966 
 
 9950 
 
 9950 
 
 74 
 
 9215 
 
 9166 
 
 8796 
 
 8732 
 
 100 
 
 10000 
 
 10000 
 
 10000 
 
 10000 
 
 75 
 
 9242 
 
 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 foimd in the 
 column of quotients, opposite thereto stand the numbers. 
 
 .9242 
 .919G 
 .8838 
 .8780 
 
 which being each mul- 
 >. tiplied by 82, produce 
 respectively, 
 
 20.5744 
 
 29.4272 
 28.2816 
 28.09G0 
 
 for the correspond- 
 \. ing menn diameters 
 required. 
 
 2. The head diameter of a cask is 2G inches, and the bung 
 diameter 3 feet 2 inches : what is the mean diameter, the cask 
 being of the thii'd variety ? 
 
 3. The head diameter is 22 inches, the. bung diameter 34 
 inches •• what is the mean diameter of a cask of the fourth variety? 
 
GAUGING. SG5 
 
 357. Having found the mean diameter, we multiply the square 
 of the mean diameter by the decimal .7854, and the product by 
 the length ; this -will give the contents in cubic inches. Then, 
 if we divide by 231, we have the contents in wine gallons (see 
 Art. 414), or if we divide by 282, we have the contents in beer 
 gallons (Art. 415). 
 
 Analysis. — For wine measure, we mul- operation. 
 
 tiply the length by the square of the mean I X d^ x tj^^* = 
 
 diameter, then by the decimal .7854, and I X d^ X .0034. 
 divide by 231. 
 
 If then, we divide the decimal .7854 by 231, the quotient carried 
 to four places of decimals is .003 1, 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 found operation. 
 
 in beer gallons by multiplying together the I X d- X 'l^^ = 
 
 length; the square of the mean diameter, I X d'- X .0028. 
 and the decimal .0028. 
 
 Hence, for gauging or measuring casks, 
 
 31ultiply the length by the square of the mean diameter ; then 
 midt'ply by 34 for wine, and by 28 for beer measure, and point 
 off in the product four decimal places. The product will then 
 express gallons and the decimals of a gallon. 
 
 1. How many wine gallons in a cask, whose bung diameter 
 is 36 inches, head diameter 30 inches, and length 50 inches; 
 the cask being of the first variety ? 
 
 2. What is the number of beer gallons in the last example ? 
 
 3. How many wine, and how many beer gallons in a cask 
 whose length is 36 inches, bung diameter 35 inches, und head 
 diameter 30 inches, it being of the first variety ? 
 
 4. 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 ? 
 
 t357. How do you find the contents in cubic inches 1 How do you find 
 the contents in wine gallons 1 lu beer gallons 1 
 
S6(J MECHANICAL P0WEK3. 
 
 OF THE MECHANICAL POWERS. 
 
 358. There are six simple machines, which are called 
 Mechanical 2)0wers. They are, the Lever, the Pulley, the 
 Wheel and Axle, the Inclined Plane, the Wedge, and the Screw. 
 
 359. 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. We are to consider the centre of motion, or fidcrum, 
 which means a prop. The prop or fulcrum is the point about 
 which all the parts of the machine move. 
 
 4th. We are to consider the respective velocities of the power 
 and resistance. 
 
 360. A machine is said to be in equilibrium when the resist- 
 ance exactly balances the power, in which case all the parts of 
 the machine are at rest, or in uniform motion. 
 
 We shall first examine the lever. 
 
 361. The Lever, is a bar of wood or metal, which moves 
 around a fixed point, called the fulcrum. There are three 
 kinds of levers. 
 
 1st. When the fulcrum is 
 between the weight and the 
 power. 
 
 358. How many simple machines are there ^ What are they called 1 
 
 359. What things must be considered, in order to understand the power 
 of a machine 1 
 
 300. When is a machine said to be in equilibrium? 
 
 361. What is a lever 1 How many kinds of levers are there ? Describe 
 the first kind \ Where is the weight placed in the second kind \ Where 
 ia the power placed in the third kind ] 
 
MECIIANIUAL POWERS. 367 
 
 2d. "When the weight is be- 
 tween the power and the ful- 
 crum. 
 
 3d. When the power is be- 
 tween the fulcrum and the 
 weight. 
 
 The perpendicular distance 
 from the fulcrum to the direc- 
 tions of the weight and power, 
 are called the ar/ra< of the lever. 
 
 362. An equilibrium is produced in all the levers, when the 
 weight multiplied by its distance from the fulcrum is equal to 
 tlie power multiplied bj its distance from the fulcrum. 
 That is, 
 
 The iveight is to the power, as the distance froin the poxoer 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 bOlhs. ? 
 
 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 imhs. ? 
 
 4. A lever of the first kind is 8 feet long, and a weight of 
 QOlbs. is at a distance of 2 feet from the fulcrum : what power 
 will be necessary to balance it ? 
 
 362. When is an equilibrium produced in all the levers ! What is then 
 the proportion between the weight and power 1 
 
BG3 MECHANICAL POWERS. 
 
 5. In a lever of the first kind, that is 6 feet long, a weiglit 
 of 200/is. is placed at 1 foot from the fulcrum : what power 
 will balance it ? 
 
 6. In a lever of the first kind, like the common steelyard, 
 the distance from the weight to the fulcrum is one inch : at 
 what distance from the fulcrum must the poise of lib. be placed, 
 to balance a weight of 1Z6.? A^\eightof^lbs.? 0{2lbs.?Of4:lbs.? 
 
 7. In a lever of the third kiqd, the distance from the fulcrum 
 to the power is 5 feet, and from the fulcrum to the weight 
 8 feet : what power is necessary to sustain a weight of AOlbs. ? 
 
 8. In a lever of the third kind, the distance from the fulcrum 
 to the weight is 12 feet, and to the power 8 feet : what power 
 will be necessary to sustain a weight of lOOlbs. ? 
 
 363. Rejiarks. — 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 PULLET. 
 
 364. The pulley is a wheel, having a 
 groove cut in its circumference, for the pur- 
 pose of receiving a cord which jiasses over 
 it. When motion is imparted to the cord, 
 the pulley turns around its axis, which is 
 generally supported by being attached to a 
 beam above. 
 
 363. Has the wciiiht been consiilercd in detprmining the eoiiilihrijm of 
 the levers 1 In a lever of the first kind, will the weight increase oj dimii*- 
 iih the power? How will it be in the two other kiuds 1 
 
 a(i4. What is a imllcy ! 
 
MICCUANIOAL POWEKS. 
 
 8G9 
 
 365. Pulleys are divided into two kinds, fixed pulleys and 
 movable pulleys. "When the pulley is fixed, it does -not in- 
 crease the power which is applied to raise the weight, but 
 merely changes the direction in which it acts. 
 
 3G6. A movable pulley gives a mechan- 
 ical advantage. Thus, in the movable 
 pulley, the hand which sustains the cask 
 actually supports but one-half the weight 
 of it ; the other half is supported by the 
 hook to which the other end of the cord is 
 attached. 
 
 367. 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 machines, 
 that lohat is gained in 2J0wer, is lost in time ; 
 and this is true for all machines. There is 
 also an actual loss of power, viz., the resist- 
 ance 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 diiferent machines, but 
 must always be allowed for, in calculating 
 the power necessary to do a given w^ork. It 
 would be wrong, however, to suppose that 
 
 365. How many kinds of pulleys are there ? Does a fixed pulley givo 
 any increase of power"! 
 
 ::!6';. Does a movable pulley give an}' mechanical advantage \ In a 
 eingle movable pulley, how much less is the power than the weight? 
 
 'Ml . Will an advantiige l>s gained \iy several niovalile ptilleya 1 Statu 
 
 r 
 
 \ 
 
 VI 
 
370 
 
 MECUANICAL POWEKS. 
 
 the loss was equivalent to the gain, and that no advantage is 
 derived from the mechanical powers. We are unable to aug- 
 ment our strength, but, by the aid of science we so divide the 
 resistance, that by a continued exertion of power, we accom- 
 plish 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 powei". 
 
 368. It is plain, that in the movable pulley, all the parts of 
 the cord will be equally stretched, and hence, each cord running 
 from pulley to pulley, will bear an equal part of the weight ; 
 consequently. 
 
 The power will always be equal to the weight divided hy the 
 7iumber of cords which reach from pidley to pidley. 
 
 EXAMPLES. 
 
 1. In a single immovable pulley, what power will support a 
 weight of QOlbs. ? 
 
 2. In a single movable pulley, what power will support a 
 weight of SOlbs. ? 
 
 3. In two movable pulleys with 4 cords, (see last fig.,) what 
 power will support a weight of lOOlbs.? 
 
 WINCH, OK "WHEEL AND AXLE. 
 
 369. This machine is com- 
 posed of a wheel or crank — 
 firmly attached to a cylindri- 
 cal axle. The axle is sup- 
 ported at its ends by two 
 pivots, which are of less 
 diameter than the axle around 
 wliich the rope is coileil, and 
 which turn freely about the 
 points of support. In order 
 to balance the weight, we must 
 have. 
 
MECHANICAL POWERS. 371 
 
 Tlie power to the weight, as the radius of the axle, to the length 
 of the crank, or radius of the wheel. 
 
 EXAMPLES. 
 
 1. What must be the length of a crank or radius of a -wheel, 
 in order that a power of 40/^5. may balance a weight of QOOlbs. 
 suspended from an axle of 6 inches radius ? 
 
 2, "What must be the diameter of an axle, that a power of 
 lOOlbs. applied at the circumference of a wheel of 6 feet diame- 
 ter may balance 400/fo. ? 
 
 INCLINED PLANE. 
 
 380. The inclined plane is nothing more than a slope or 
 declivity, which is used for the purpose of raising weights. It 
 is not difficult to see that a weight can be forced up an inclined 
 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 Aveight W, 
 be supported on the in- 
 clined plane ABC, by a 
 cord passing over a pul- 
 ley 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 EC : length AB. 
 
 the general principle in machines. What does the actual loss of power 
 arise from 1 What is this rubbing called 1 Does this vary in different 
 machines I 
 
 36S. In the movable pulley, what proportion exists between the cord 
 and the weight 1 
 
 369. Of what is the machine called the wheel and axle, composed? How 
 is the axle supported I Give the proportion between the power and tho 
 weight. 
 
 370. What is an inclined plane ? What proportion exists between the 
 power and weight when they are in equilibrium 1 
 
372 
 
 MECHANICAL POWEliS. 
 
 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, 
 
 The power is to the loeight as the height of the plane is to its 
 length. 
 
 exa:^iples. 
 
 1. The length of a plane is 30 feet, and its height 6 feet: 
 what power will be necessary to balance a weight of 2Q0lhs. ? 
 
 2. The height of a plane is 10 feet, and the length 20 feet : 
 what weight will a power of bOlhs. support ? 
 
 3. The height of a plane is 15 feet, and length 45 feet : what 
 power Avill sustahi a weight of l^Olbs. ? 
 
 THE WEDGE. 
 
 381. The wedge is composed of two 
 inclined planes, united together along 
 their bases, and forming a solid ACB. 
 It is used to cleave masses of wood or 
 stone. The resistance Avhich it over- 
 comes is the attraction of cohesion of 
 the body which it is employed to sepa- 
 rate. The wedge acts principally by being struck with a ham- 
 mer, or mallet, on its head, and very little effect can be produc- 
 ed with it, by mere pressure. 
 
 All cutting instruments are constructed on the principle of 
 the inclined plane or wedge. Such as have but one sloi)ing 
 edge, like the chisel, may be referred to the inclined plane, axid 
 such as ha\-e two, like the axe and the knife, to the wedge. 
 
 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 
 
 371 \^'liat is the wedge 1 What is it used for 7 "What resistance is it 
 used to overcome ! 
 
MECHANICAL POWERS. 873 
 
 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 27 inches, and the force applied measure by 250 pounds, 
 what will be the measure of the effect ? 
 
 THE SCREW. 
 
 381. 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 groove into which 
 the thread of the screw fits 
 closely. 
 
 The handle D, which projects from the nut, is a lever which 
 woj'ks 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 proportiona- 
 bly 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 attached to the 
 nut. 
 
 The screw is used for compression, and to raise heavy w^eights. 
 It is used in cider and wine-presses, in coining, and for a variety 
 of other purposes. 
 
 As the distance between the threads of a screw is to the circum- 
 ference of the circle described hy the power, so is the power em- 
 ployed to the weight raised. 
 
 381. Of how many parts is the screw composed ? Describe the screw. 
 What is the thread \ What is the nut ? What is the handle used fori To 
 what uses is the screw appUed '' What is the power of tlic screw ? 
 
Oli QUESTIONS IN PHILOSOPHT. 
 
 EXAMPLES. 
 
 1. If the distance between the thi-eads of a screw is half an 
 inch, and the circumference described by the handle 15 feet, 
 what weight can be raised bj a power denoted by 720 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 applied to 
 sustain 2 tons ? 
 
 3. "What force applied to the handle of a screw 10 feet long, 
 >vith threads 1 inch apart, working on a wedge whose head is 
 5 inches, and length of side 30 inches, will produce an effect 
 measured by lOOOOlbs. ? 
 
 4. If a^ower of 300 pounds applied at the end of a lever 
 15 feet long will sustain a weight of 282744/^5., what is the 
 distance between the threads of the screw ? 
 
 QUESTIONS IN NATURAL PHILOSOPHY. 
 UNIFORM MOTION. 
 
 382. If a moving body passes over equal spaces in equal 
 successive small portions of time, it is said to move with uni- 
 form motion, or uniformly. 
 
 383. The velocity of a moving body is measured by the 
 space passed over in a second of time. 
 
 384. 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 V, the space passed over by S, 
 and the time by T, we have 
 
 S = V X T. 
 
 382. What is a unilorm motion 1 
 
 383. Wliat is the velocity of a moving body 1 
 
 384. To wliat is tl)C space passed over in a unit of time equal 1 What 
 lb the span; passed over equal to, in uniform motion ' 
 
QUESTIONS IN PHILOSOPHY. 375 
 
 EXAMPLES. 
 
 1. A steamboat moves with a velocity of 23 feet: what space 
 does it pass over in 1\ hours ? 
 
 2. A locomotive is moving with a velocity of 32 feet : what 
 distance will it travel in 3 minutes ? 
 
 3. A horse travels uniformly a distance of 12 miles with a 
 velocity of 6 feet : what time does he require to perform the 
 journey ? 
 
 4. A carriage performs a journey of 15 miles in 2|- hours : 
 with what velocity does it move ? 
 
 5. The hammer of a pile-driver is moved upwai'd a distance 
 of *35 feet with a velocity of 1-^ feet : what time is required to 
 raise it ? 
 
 6. A ton of coal is raided from a mine 1000 feet deep in 3^ 
 minutes : with what velocity does it move ? 
 
 7. A vessel containing a criminal, after leaving a port, sailed 
 with a daily speed of 170 miles ; four days after, a clipper was 
 dispatched in pursuit, and sailed at a daily rate of 275 miles : 
 in what time did the clipper overtake the vessel ? 
 
 8. A bird flew a distance of 100 miles in 11 hours: with 
 what velocity did it travel ? 
 
 9. Sound moves with a velocity of 1127 feet. If the report 
 of a gun was heard 31.3 seconds after the flash was seen, what 
 distance was the gun from the observer ? 
 
 10. A hurricane moves with a velocity of 95 feet : what time 
 does it take to move through 3 degrees of latitude, the degree 
 being estimated at G91 miles ? 
 
 11. The velocity of light has been found, by astronomical 
 observations, and by experiments made in France, to be 191,300 
 miles : what time will it occupy to traverse the mean distance 
 of the earth from the sun, or 95000000 of miles ? 
 
 12. If a message sent by electro-magnetic telegraph 2300 
 miles requires 14 seconds for its transmission, what is the 
 velocity of the magnetic current in this telegraph line .'' 
 
376 QUESTIONS IN PHILOSOPHY. 
 
 LATVS OF FALLING BODIES. 
 
 385. A body fulling vertically downward iu a vacuum, falls 
 tlirough 16^2^;'. during the first second after leaving its place 
 of rest, 481//. during the second second, SOj^^ft. the third 
 second, and so on : the spaces forming an arithmetical pro- 
 gression of which the common difference is 32^fL, or double 
 the space fallen through during the first second. This number 
 is called the measure of the force of gravity, and is denoted 
 
 o<S6. It is seen from the above that the velocity of a body is 
 continually increasing. If H denote the height fallen through, 
 T, the time, V, the velocity acquired, and c/, the force of gravity, 
 the following formulas have been found to express the relations 
 between these quantities : 
 
 V = ^ X T ... (1). 
 
 V2 = 2r/ X H . . . (2). 
 
 H = IV X T ... (3). 
 
 B. =ig xT^ . . . (4). 
 
 From which we see, 
 
 1st. That the velocity acquired at the end of any time, is equal 
 to the force of gravity (32|^) multiplied by the time. 
 
 2d. Tliat the square of the velocity is equal to twice the force 
 of gravity multiplied by the height; or, the velocity is equal to 
 the square root of that quantity, 
 
 3d. 2'hat the space fallen through is equal to one-half the 
 velocity multiplied by the time. 
 
 4th. 2Viat the sjmce fallen through is equal to one-half the force 
 of gravity multij^lied by the square of the time. 
 
 385. If a body falls vertically, in a vacuum, how far will it fall in tlio 
 first serond of time ? How far on the second second second ! In the 
 third ] What is the common ditlercnce of the spaces 1 M'hat is the 
 measure of the force of gravity? 
 
 386. How does the velocity of a falling body change ? What is the 
 velocity acquired at the end of any time equal to ? What is the space 
 (alien through equal to ? 
 
QUESTIONS IN PHILOSOPHY. 377 
 
 387. If a body is thrown vertically upwards in a vacuum, its 
 motion will be continually retaixled by the action of gravitation. 
 It will finally reach the highest point of its ascent, and then 
 begin to descend. The height to which it will rise may be 
 found by the second formula in the preceding paragraph, when 
 the velocity with which it is projected upward is known ; for 
 the times of ascent and descent will be equal. 
 
 38S. The above laws are only approximately true for bodies 
 falling through the air, in consequence to 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 surface to the air. 
 
 EXAMPLES. 
 
 1. A body has been falling 12 seconds : what space has it 
 described in the last second, and what in the whole time ? 
 
 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 wiU it take a body to fall through a 
 space of 100 feet ? 
 
 5. Find the space through which a heavy body falls in 10 
 seconds, and the velocity acquired. 
 
 G. How far must a body fall to acquire a velocity of 1000 
 feet? 
 
 7. A stone is dropped into a well and strikes the M'ater in 
 32 seconds : what is the depth of the well ? 
 
 8. A stone is di-opped from the top of a bridge and strikes 
 the water in 2.5 seconds : what is the heio-ht of the bridge ? 
 
 9. A body is thrown vertically upward with a A'elocity of 
 1 GO feet : what height will it reach, and what wull be the time 
 of ascent ? 
 
 387. How far will a body ascend when projected upwards 1 
 
 388. Arc the above laws perfectly or only approximately true ' 
 
 17 
 
378 (iUESTIOXS IN PHILOSOPHY. 
 
 10. An arrow shot perpendicularly upwards returned again 
 In 10 seconds. Required the velocity with which it was shot, 
 and the height to which it rose. 
 
 11. If a body falls freely in vacuum, what will be its velocity 
 after 45 seconds of fall ? 
 
 12. During how many seconds must a body fall in a vacuum 
 to acquire a velocity of 1970 feet, which is that of a cannon ball? 
 
 13. What time is required for a body to fall in a vacuum, 
 from an elevation of 3280 feet? 
 
 14. From what height must a body fall to acquire a velocity 
 of 984 feet ? 
 
 15. A rocket is projected vertically upward with a velocity 
 of 386 feet : after what time will it begin to fall, and to what 
 height will it rise ? 
 
 SPECIFIC GKAVITT. 
 
 389. The SPECIFIC gravity of a body is the weight of a 
 unit of volume in terms of a unit of the standard. Distilled 
 rain water is the standard for measuring the specific gravity of 
 bodies. Thus, 1 cubic foot of distilled rain water weighs 1000 
 ounces avoirdupois. If a piece of stone, of the 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 bod}'. 
 
 If any body be Aveighed in air and then in water, it will 
 weigh less in water than in air. The difl:erence of the weights 
 will be equal to the sustaining force of the watei", which is found 
 to be equal to the weight of an equal volume of water : hence, 
 
 389. What is the specific gravity of a body'! What is the standard for 
 mpasurin<^ the specific gravity of a body ■ Wliat is the minirrical value 
 of the cubic foot of a body ! How do you find the specific gravity of a 
 body ? 
 
QUESTIONS IN PHILOSOPHr. 
 
 379 
 
 If we knoio or can find the iveight of a body in air and in 
 water, the difference of these tveights 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 water as a standard. 
 
 TABLE 
 
 OP SPECIFIC GRAVITIES. WATER 1. 
 
 NAMES OP BODIES. 
 
 SPEC. GRAV. 
 
 NAMES OF BODIES. SPEC. GRAV 
 
 METALS 
 Platinum, . 
 Gold, . . . 
 Quicksilver, 
 Lead, . 
 Silver, . 
 Copper, . 
 Bronze, 
 Brass, . 
 Steel, 
 
 L'on, . . . 
 Tin, 
 Zinc, . . . 
 
 BUILDING STONES 
 
 Hornblende, . . 
 Basalt, 
 
 Alabaster, . . 
 Syenite, . . . 
 Dolerite, . . . 
 Gniess, 
 
 Quartz, . . . 
 Limestone, . 
 Phonolite, . 
 Granite, . 
 Stone for building, 
 Trachytse, 
 
 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. GO 
 
 Porphyry, . 
 Sandstone, 
 Brick, . . . 
 
 "WOODS. 
 
 Oak, fresh felled. 
 White Willow, 
 Box, 
 Elm, 
 
 Hanbeam, 
 Larch, . 
 Pine, . 
 Maple, . 
 Ash, . . 
 Birch, 
 Fir, . . . 
 Horse Chestnut, 
 
 SOLID BODIES 
 
 Common earth, 
 Moist sand. 
 Clay, . . , 
 Flint, . . . 
 Ice, .... 
 Lime, . . . 
 Tallow, . . 
 Wax, . . . 
 
 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 
 
 By inspecting this Table, we see the Aveight of each body 
 compared with an equal volume of water. Thus, platina is 
 21 times as heavy as water; gold, 19 times as heavy; iron, 7^ 
 times as heavy, &c. 
 
3 so QUESTIONS IN PHILOSOPHY. 
 
 EXAMPI-ES ILLUSTRATING SPECIFIC GRAVITT. 
 
 1. A piece of copper weighs 93 grains in air, and 82^ 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 Aveiajhing 1000 ounces ? 
 
 3. A piece of pumice stone weighs in air 50 ounces, and 
 w'hen it is connected with a piece of copper whicli 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 : Avhat is the height of the prism above the sui'- 
 face 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 G3 pounds ? 
 
 6. A piece of glass was found to weigh in the air 33 ounces, 
 and in the water 21 ounces : wliat was its specific gravity ? 
 
 7. A piece of zinc weighed in the air 17 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 alcoliol loses 250 ounces, Avhat is the 
 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 2.837, the cubic foot of water 
 weighing 1000 ounces? 
 
 11. A jar containing air weighed 24 ounces 33 grains; tha 
 air was tlien excluded, and the jar weighed 24 ounces ; the jar 
 being tlieu filled with yxygen gas weighed 24 ounces 3G.4 
 
QUESTIONS IN PHILOSOPHY. 881 
 
 grains : wliat was the specific gravity of the oxygen, the air 
 being taken as the standard ? 
 
 12. A cylindrical vase having a base whose interior diameter 
 is 4 inches, stands upon a horizontal plane : 2G.'2 pounds of 
 mercury is poured into the vase. Kequired the height to wliich 
 the liquid will rise, the specific gravity of mercury being 13.59G. 
 
 13. A piece of alabaster weighs in the air 7.55 grains, in the 
 water 5.17 grains, and in another liquid G.35 grains : what is 
 the specific gravity of the alabaster and of the liquid ? 
 
 14. What effort will be required to prevent a cubic inch of 
 platinum, immersed in mercury, from sinking, the specific 
 gravity of the platinum being 21,5, and that of the mercury 
 13.G ? 
 
 15. What weight of mercury will a conical vase contain of 
 which the radius of the base is 9 inches and the altitude 34 
 inches, the specific gravity of the mercury being 13.59G ? 
 
 mariotte's law. 
 
 390. This laAV, which relates to air and all other gases, steam, 
 and all other vapors, was discovered by the abbe Mariotte, a 
 French philosopher, who died in 1G84. It will be easily under- 
 stood from a particular example. 
 
 Suppose an upright cylindrical vessel in a vacuum contains a 
 gas which is confined in the vessel by a piston at the upper end. 
 Suppose tlie gas or vapor fills the whole vessel, and the piston 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 : 
 
 .190. To what is the vokime uf a vapor or gas proportional 1 To what 
 is its density proportional 1 
 
382 QUESTIONS IN PHILOSOPHY. 
 
 The temjyeratwe remaining the same, the volume of a gas or 
 va2wr is inversehj proportional to the 2yrcssure which it sustains. 
 Also, the density of a gas or vapor is directly p>yop)ortional to 
 the pjresmre. 
 
 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 cer- 
 tain 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 jjressure upon the steam contained in a cylinder is 
 increased from 25 pounds upon the square inch to 47 pounds : 
 what part of the original volume will be occupied ? 
 
 5. How will the density of the steam in the last example, at 
 the second pressure, compare with that at the first ? 
 
 6. Eight quarts of hydrogen gas are contained in a vessel and 
 submitted to a pressure of 22 pounds : how many quarts of ga? 
 will there be if the pressure is changed 9i pounds .'' 
 
APPENDIX. 
 
 DIFFERENT KINDS OF UNITS. 
 
 391. There are eight kinds of units : 
 1st. Abstract Units ; 
 
 2d. Units of Currency or Coin ; 
 
 3d. Linear Units, or Units of Length ; 
 
 4th. Units of Surface, or Superficial Units ; 
 
 5th. Units of Volume, including Cubic Units and Gallons ; 
 
 Gth. Units of "Weight ; 
 
 7th. Units of Time ; and 
 
 8th. Units of Circular or Angular Measure. 
 
 ABSTRACT UNITS. 
 
 392. The abstract unit 1 is the base of all numbers, and is 
 called 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 abstract num- 
 bers formed from the unit 1, according to the scale of tens. All 
 abstract integral numbers are collections of these units. 
 
 UNITS OF CURRENCY. 
 
 393. In all civilized and commercial countries, great care is 
 taken to fix a standard value for money, which standard is 
 called the Unit of Currency. 
 
 In the United States, the unit of currency is 1 dollar ; in 
 Great Britain it is 1 pound sterling, equal to S4,84 ; in France 
 it is 1 franc, equal to 18j cents. 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. 
 
 391. How many kinds of units are there in Arithmetic 1 Name them. 
 
 393. What is said of the abstract unit 1 1 AVhat is a unit of the 2J 
 order I What of the 3d \ 4th 1 5th ! &c. How are these numbers 
 ("ormed from 1 ! 
 
884: APPEjroiX. 
 
 UNITS OF LENGTH. 
 
 394. One of the most important units of measure is that fof 
 distances, or for the measurement of length. A practical want 
 has ever been felt of some fixed and invariable standard with 
 which all distances may be compared : such fixed standard ha3 
 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. Tliese 
 two systems, being the only ones of importance, will be alone 
 considered. 
 
 395. First. — The English system of measures, to which 
 ours conforms, is based upon the law of nature, that the 
 force of gravitij is constant at the same jJoint of the carlh\s sur- 
 face, and consequently, that the length of a pendulum which 
 oscillates a certain number of times, in a given period, is also 
 constant. Had this unit been known brfore the adoption and 
 use of a system of measures, it would liave formed the natural 
 unit for division, 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 tlie pen- 
 dulum, 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 standard 
 foot, and 36 of them, an Imperial yard. The Imperial yard 
 and the standard foot are marked ujwn a brass bar, at tlie tem- 
 perature of 62^-°, and these are the linear measures from which 
 
 393. "What is a unit of ciiiTcncy 1 "What is the unit of currency in tho 
 Unitcii St;itos? Wliat in (iroat Britain ! "What in Fiance 1 
 
 394. For what is an invaiialile stanilanl of lenglli used ! 
 
 395. What is the standard unit of length in the EngUsh system 1 What 
 in ours ■> 
 
UNITS OF SURFAOK. 8 85 
 
 ours are taken. The comparison has been made by means of 
 a brass scale 82 inches long, manufactured by Troughton in 
 London, and now in the possession of the Treasury Depart- 
 ment. 
 
 396. 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 meri- 
 dian was computed, and it has been decreed by the French law 
 that the ten-millionth part of this length shall be regarded as 
 a standard French metre, and from this, by multiplication and 
 division, the entire system of linear measures has been estab 
 Ilshed. 
 
 On comparing two scales, vei'y accurately, it has been found that 
 the French metre is equal to 39.37079 English inches — differing 
 somewhat from the English yard. This relation enables us to 
 convert all measures in either system into the corresponding 
 measures of the other. 
 
 UNITS OF SURFACE. 
 
 397. 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 1 inch. 
 A square foot =144 square inches. 
 A square yard = 9 square feet. 
 A square rod = 30^ square yards. 
 &c. &c. 
 
 396. What is the standard unit of length in the French system 1 IIow 
 was it found 1 How docs the French metre compare with the Imperial 
 yard 1 
 
 397. What is the most convenient unit of surfaced What are those in 
 common use I 
 
386 Al'PENDIX. 
 
 UNITS OF VOLUME. 
 
 398. The Unit of Volume, for the measurement of solids, h 
 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 e ige is 1 foot = 1728 cubic in, 
 A cubic yard=: a cube whose edge is 1 yard = 27 cubic feet. 
 A perch of stone = 24f cubic feet ; 
 
 or a block of stone 1 rood long, 1 foot thick, and lA feet wide. 
 
 The standai-d unit of volume for the measurement of liquids 
 i3 the wine gallon, which contains 231 cubic inches. 
 
 The standard i( nit of dry measure is the Winchester bushel, 
 which contains 2150.4 cubic inches, nearly. 
 
 UNITS OF WEIGHT. 
 
 399. Having fixed an invariable unit of length, we passed 
 easily to an invariable unit of surface, and then, to an invariable 
 unit of volume. We wish now to define an invariable unit of 
 weight. 
 
 It has been found that distilled rain water is the most inva- 
 riable substance ; hence, this, at a given temperature, 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 27.701554. 
 cubic inches of distilled water. 
 
 The standard Troy -pound is the weight of 22.794422 cubic 
 
 398. What is thp unit of volume for the measurement of solids \ AVhat 
 are tiiose in common use \ ^\'h,^t is the standard unit for the iiic;i.suro« 
 ment of liquids ! What for dry measure ! 
 
 399. What is used as a standard in fixing the iinits of weight? How 
 many units of weight have we 1 How is the .standard avoirdupois iKumd 
 determined ? How tlie Troy pound ! Wliich is represented by a standard 
 at the mint 1 
 
UNITS OF TIME. 387 
 
 inches of distilled rain water. This standard is at present kept 
 in the United States Mint at Philadelphia, and is the standard 
 unit of weight. 
 
 UNITS OF TIME. 
 
 400. 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, aud is called a day. From the day, by multi- 
 plication, we form the weeks, mouths and years ; and by divi- 
 sion, the hours, minutes and seconds. 
 
 UNITS OF CIRCULAR OR ANGULAR MEASURE. 
 
 401. This measure is used for the measurement of an<Tles, 
 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 divided 
 into 60 equal parts called minutes, and these again into 60 equal 
 parts called seconds. 
 
 REJIARKS. 
 
 402. It is seen that all the units, determined by the pendu- 
 lum, depend on time as the ultimate base ; that is, the length 
 of a pendulum which A\'ill vibrate seconds determines all the 
 units of measure and weight. 
 
 Now, time is measured by motion, and the motion of the 
 earth on its axis is the only invariable m,otion. 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. 
 
 403. No class of pupils can rightly and clearly apprehend 
 the nature of numbers and the operations performed upon them, 
 
 400. How is time measured ? "What motion is uniform 1 ^^'hat is tho 
 natural unit ? 
 
 401 For what is circular or angular measure used 1 What is the uniti 
 
 402. On what do the units determined by the pendulum depend ! How 
 i time measured ] To what then arc all these units referred 1 
 
 403. How are the ideas of the absolute and relative values of the units 
 to be communicated to a class 1 What apparatus is necessary ^ 
 
388 APPENDIX. 
 
 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 b}'' means of sensible objects. 
 
 Every school room, therefore, should be provided with a 
 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 
 sui'face, the square 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 j^int, quart, gallon and bushel, 
 should be exhibited to illustrate the others. 
 
 The unit of weisrht should also be seen and handled. A 
 child even can apprehend what is meant by an ounce or a 2^oiind 
 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 Avith a com- 
 plete set of models to illusti-ace and explain the absolute and 
 relative values of the different units. 
 
 UNITED STATES MONEY. 
 
 404. United States Monet is the currency established 
 by Congress, A. D. 1786. The names or denominations of its 
 units are. Eagles, Dollars, Dimes, Cents, and Mills. 
 
 The coins of the United States are of gold, silver, and cop- 
 per, and are of the following denominations : 
 
 1. Gold : Eagle, half-eagle, three-dollars, quarter-eagle, dollar. 
 
 2. Silver : Dollar, half-dollar, quarter-dollar, dime, half-dime, 
 and three-cent piece. 
 
 3. Copper : Cent, half cent. 
 
 TABLE. 
 
 10 IMills make 1 Cent, marked ct. 
 
 10 Cents - - 1 Dime, - - d. 
 
 10 Dimes - - 1 l^Jollar, - - $. 
 
 lU Dollars - - 1 Eagle, - - B. 
 
UNITED STATES MONET. 889 
 
 Mills. 
 
 Cents. 
 
 Dimes. 
 
 Dollars. 
 
 Eagles. 
 
 10 
 
 = 1 
 
 
 
 
 100 
 
 = 10 
 
 = 1 
 
 
 
 1000 
 
 = 100 
 
 z= 10 
 
 = 1 
 
 
 (0000 
 
 = 1000 ^ 
 
 = 100 
 
 = 10 
 
 = 1 
 
 105. It is seen, fix)m the above table, that in United States 
 tuviiey, the 2^r»?2ary unit is 1 mill ; that the units of the scale, 
 in passing from mills to cents, are 10. The second unit is 1 cent, 
 and the units of the scale, in passing to dimes, are 10. The third 
 unit is 1 dime, and the units of the scale, in passing to dollars, 
 are 10. The fourth unit is 1 dollax% and the units of the scale, in 
 passing to eagles, are 10. This scale is the same as in siiiq^le 
 numbers; therefore, 
 
 27^6 nnits of United States money may he added, suhtracted, 
 rutdtipUed, and divided hy the same rides as are appUcahle to 
 simple numbers. 
 
 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 which 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. 
 
 2. The eagle contains 258 grains of standard gold, and the other 
 gold coins in the same proportion. The dollar contains 412^ grains 
 of standard silver, and the others in the same proportion The cent, 
 168 grains of puro copper. 
 
 3. If a given quantity of gold or silver be divided into 24 equal 
 parts, each part is called a ca;rat. If any number of carats be mixed 
 with so many equal caratt, nf a less valuable metal, that there be 24 
 
 404. What is United Slat ;s money 1 What are the names of its units 1 
 What are the coins of the Lnited States 1 Which gold 1 Which silver 1 
 Which copper 1 
 
 405. What is the primary? unit in United States money 1 What are tho 
 units of the scale in passing from one denomination to another ^ How 
 docs this compare with the scale in simple numbers 1 
 
S90 API'ENUIX. 
 
 carats in the mixture, then the compound is said to be as many 
 carats fine as it contains carats of the more precious metal, and to 
 contain as much alloy as it contains carats of the baser. 
 
 For example, if 20 carats of gold be mixed with 4 of silver, the 
 mixture is called gold of 20 carats fine, and 4 parts alloy. 
 
 4. Although the currency of the United States is in dollars, centa 
 and mills, yet in some of the States the old currency of pounds, 
 shillings and pence, is still nominally preserved. 
 
 In all the States the shilling is reckoned at 12 2^^nce, the pound 
 at 20 shillings, and the dollar at 100 cents. 
 
 The following table shows the number of shillings in a dollar, 
 the value of £1 in dollars, and the value of $1 in the fraction 
 of a pound : 
 
 In English currency, 4s. 6;/. - £1 = $4,84, and $1 = £4.^ 
 In N. E., Va., Ky., ) g^_ _ ^^ ^ ^3^ ^^^ ^^ ^ ^ 3 
 
 Tenn., 3 
 
 In N. Y., Ohio, N. 
 
 Carolina. 
 
 I 8s. - £1 = $21 and $1 =: £ |. 
 
 In N. J., Pa., Del, I ^^_ g^^ _ ^^ ^ ^^ ^^^ $1 ^ £ f. 
 
 Md., > •* " 
 
 In S. Carolina and Ga. 45. M. - £1 = $4f , and $1 = £ J5. 
 I Canad 
 
 Scotia, 
 
 In Canada & Nova I ^^^ . £1 = $4, and $1 = £ 1. 
 
 ENGLISH MONEY. 
 
 406. The units or denominations of English money are 
 guineas, pounds, shillings, pence, and farthings. 
 
 Notes. — 1. What is the degree of purity of the gold coins? Of the 
 silver coins ? Of the copper 1 
 
 2. How much pure gold in the eagle 1 How much pure silver in tho 
 dollar \ 
 
 3. What is a carat ? How are metals mixed by carats T 
 
 4. In what denominations is money sometimes reckoned in the different 
 ptates ? 
 
 IOC. What are the denominations of English money? 
 
ENGLISH MONEY. 
 
 391 
 
 TABLE. 
 4 farthings, marked far., make 1 penny 
 12 pence - - - - 
 
 20 shillings 
 
 - 
 
 1 pound, c 
 
 21 shillings 
 
 - 
 
 1 guinea. 
 
 far. 
 
 d. 
 
 5. 
 
 4 
 
 = 1 
 
 
 48 
 
 = 12 
 
 = 1 
 
 960 
 
 = 240 
 
 = 20 
 
 marked d, 
 1 shilling, - «, 
 
 £. 
 
 = 1 
 
 TABLE OF FOREIGN COINS WHOSE VALUES ARE FIXED 
 
 BY LAW. 
 
 Franc of Franco and Belgian, 
 
 Florin of the Netherlands, 
 
 Guilder of do. . 
 
 Livre Tournois of France, 
 
 Milrea of Portugal, . 
 
 Milrea of Madeira, 
 
 Milrea of the Azores, 
 
 Marc Banco of Hamburg, 
 
 Pound Sterling of Great Britain, 
 
 Pagoda of India, 
 
 Pteal Vellon of Spain, 
 
 Real Plate of do. 
 
 Rupee Company, 
 
 Rupee of Biiiish India, . 
 
 liix Dollar of Denmark, . 
 
 Rix Dollar of Prussia, . 
 
 Rix dollar of Bremen, 
 
 Ptoublc. silver, of Russia, 
 
 Tale of China, . 
 
 Dollar of Sweden and Norway, 
 
 Specie 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 
 laud, and Canada, ..... 
 
 4 
 1 
 
 cts. 
 
 40 
 40 
 
 12 
 
 00 
 
 83J 
 
 35 
 
 84 
 
 84 
 
 05 
 
 10 
 
 44^ 
 
 44^ 
 
 00 
 
 68^ 
 
 78f 
 
 75 
 
 48 
 
 06 
 
 05 
 
 69 
 
 40 
 
 48^ 
 
 16 
 
 16 
 
 SO 
 40 
 
 04 
 
392 
 
 APPENDIX. 
 
 TABLE OF FOREIGN COIXS "WHOSE VALUES ARE FIXED 
 
 BY USAGE. 
 
 Berlin Rix Dollar, 
 Current Marc, 
 Crown of Tuscany, 
 Elberfeldt Rix Dollar. 
 Florin of Saxony, 
 Bohemia, 
 Elberfeldt, 
 Prussia, . 
 Trieste, 
 Nuremljurg 
 Frankfort, 
 Basil, 
 St. Gaul, 
 
 (( 
 u 
 u 
 li 
 li 
 u 
 u 
 (( 
 
 Creveld, 
 
 Florence Livre, . 
 Genoa do., 
 Geneva do., . 
 Jamaica Pound, 
 Leghorn Dollar, 
 Leghorn Livre ((Ji to 
 Livre of Catalonia, 
 Neufchatel Livre, . 
 Pezza of Leghorn. 
 Rhenish Rix Dollar, 
 Swiss Livre, 
 Scud a of Malta, 
 Turkish Piastre, 
 
 the dollar), 
 
 ds. 
 
 GH 
 
 28 
 
 05 
 
 69| 
 
 48 
 
 48 
 
 40 
 
 22f 
 
 48 
 
 40 
 
 40 
 
 41 
 
 40 
 
 15 
 
 18| 
 
 21 
 
 00 
 
 90 
 
 o3| 
 
 2(1^ 
 
 90 
 
 60f 
 
 27 
 
 40 
 
 05 
 
 [The above Tables are taken from a work on the Tariff, by E. D. 
 Ogden, Esq., of the New York Custom House]. 
 
 Notes. — 1. The primary unit in EnglLsh money is 1 farthing. 
 The units of the scale, in passing from farthings to pence, are 4; ih 
 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, 
 1/ar. = \d. ; 2far. = ^l. ; Zfur. --^ \d. 
 
 3. The standard of the gold coin is 22 parts of pure gold and 2 
 parts of copper. 
 
 Note. — 1. What are the primary units of the English currency 1 Name 
 the units of the scale. 
 
UNITED STATES MONEY. 
 
 893 
 
 The standard of silver coin is 37 parts of pure silver, and 3 parts 
 of copper. 
 
 A pound of gold is worth 14.2878 times as much as a pound of 
 silver. In the copper coin 24 pence make 1 pound avoirdupois. 
 
 By reading the second table from right to left, we can see the value 
 of any unit expressed in each of the lower denominations. Thus, 
 Id. = Afar.; Is. = 12tZ. = 48/ar.; £l = 205. = 240rf. = 960/ar. 
 
 LINEAR MEASURE. 
 407. 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 IG^ feet - 
 
 - 
 
 1 rod, - - - rd. 
 
 40 rods _ - - 
 
 - 
 
 1 furlong, - - fur. 
 
 8 furlongs or 320 rods 
 
 - 
 
 1 mile, - - mi. 
 
 3 miles _ - - 
 
 - 
 
 1 league, - - L. 
 
 69i statute miles, or 
 
 60 geographical miles, - 
 
 -] 
 
 1 degree on the ) , 
 ° V deg. or . 
 
 equator, - > 
 
 360 degrees 
 
 
 a circumference of the earth. 
 
 in. ft. 
 
 yd. 
 
 rd. fur. mi. 
 
 12 =. 1 
 
 
 
 36 z= 3 = 
 
 1 
 
 
 198 = 16L = 
 
 5* 
 
 = 1 
 
 7920 = 6G0 = 
 
 220 
 
 = 40 =1 
 
 63360 = 5280 =: 
 
 1700 
 
 = 320 =8 =1 
 
 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, 
 
 3. The units of the scale, in passing from inches to feet, are 12 j 
 in passing from feet to yards, 3 ; from yards to rods, 5-J; from rods to 
 furlongs, 40 ; and from furlongs to miles, 8. 
 
 407. For what is linear measure used 1 What are its denominations? 
 Repeat the table 1 - What is a fathom \ What is a hand ? What are- the 
 I'jiits of the scale in linear measure 1 
 
894 APPESTDIX. 
 
 FOREIGN MEASURES OF LENGTH. 
 
 408. The Imperial jard of Great Britain is the one from 
 ■which ours is taken. Hence, the units of measure are identical. 
 
 " * FRENCH SYSTEM. 
 
 409. 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 metres, measured at the temperature of 32° Fahr. 
 The muhiples and divisions of it are decimals, viz. : 1 metre 
 =z 10 decimetres = 100 centimetres r= 1000 millimetres = 
 8.280899 United States feet, or 39.37079 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 = 1.03737 foot. 
 Prussia!}, 
 
 J 
 
 „, . , , I 1 foot = 12.3G1 " « = 1.0300 
 Kluneland. 
 
 Swedish, 1 foot = 11.G90 « " =0.974145" 
 ^ 1 foot — 11.034 " " = 0.9195 " 
 
 Spanish, Y league (royal) = 25000 Span. ft. = 4^ miles "i >> 
 ) " (common) = 19800 « = ^ '' \ \ 
 
 CLOTH MEASURE 
 410. Cloth measure is used for measuring all kinds of cloth, 
 ribbons, and other things sold by the yard. 
 
 TABLE. 
 2\ inches, in. make 1 nail, marked na. 
 4 nails - - 1 quarter of a yai'd, qr. 
 
 3 quarters - - 1 Ell Flemish, E. Fl. 
 
 4 quarters - - 1 yard, - - yd. 
 
 5 quarters - - 1 Ell English, - E. E. 
 
 408. How docs the Imperial yard compare with the standard in tlic 
 United States \ 
 
 409. What is the unit of tlie French system of measures ? How doci 
 the metre compare wilii our standard yard 1 
 
 410. For what is cloth measure used ! What arc its denominations 1 
 Repeat the table. What arc the units of the ucalcc ? 
 
SQUAKE MEASUKE, 
 
 395 
 
 m. 
 
 na. 
 
 qr. 
 
 ^. i^'l. 
 
 yd. 
 
 2i 
 
 = 1 
 
 
 
 
 9 
 
 = 4 
 
 = 1 
 
 
 
 27 
 
 = 12 
 
 = 3 
 
 = 1 
 
 
 36 
 
 = 16 
 
 = 4 
 
 -1^ 
 
 = 1 
 
 45 
 
 = 20 
 
 = 5 
 
 = lf 
 
 = U 
 
 KK 
 
 == 1 
 
 Note. — Tho units of the scale, iu this measure, are 2^, 4, 3, J, 
 and -J. 
 
 SQUARE MEASURE. 
 
 411. Square measure is used in measuring land, or anything 
 in which length and breadth are both considered. 
 
 A square is a figure bounded by four equal 
 lines at right angles 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. 
 
 
 1 Foot. 
 
 o 
 o 
 
 
 
 I— < 
 
 
 
 1 yard = 3 feet. 
 
 CO 
 
 1 
 
 
 
 II 
 
 
 
 
 1— t 
 
 
 
 
 If the sides of the square be each one 
 yard, the square is called a square yard. 
 In the large square there are nine small 
 squares, the sides of which are each one 
 foot. Therefore, the square yard con- 
 tains 9 square feet. 
 
 The number of small squares that is contained 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. 
 
 411. For what is square measure used ^ What is a square'? If each 
 side be one foot, what is it called 1 If each side be a yard, what is it 
 called 1 How many square feet does the square j^ard contain ? How is 
 the number of small squares contained in a large square found 1 Rc()cat 
 the table. What are the units of the scale 1 
 
39G APPENDIX. 
 
 TABLE. 
 144 square inches, sg. in. make 1 square foot, Sq.ft. 
 
 9 square feet - - 1 square yard, Sq. yd. 
 
 30^ square yards - - 1 square rod or perch, P, 
 40 square rods or perches 1 rood, - - - K. 
 
 4 roods - - - 1 acre, - - - A. 
 
 M. 
 
 A. 
 
 640 acres 
 
 - 
 
 1 square mile. 
 
 Sq. in. 
 
 Sq. ft. 
 
 Sq. yd. P. P. 
 
 144 
 
 = 1 
 
 
 1296 
 
 = 9 
 
 = 1 
 
 39204 
 
 = 272i 
 
 = 30J =1 
 
 15G8160 
 
 = 10890 
 
 =z 1210 =40 = 1 
 
 6272G40 
 
 = 43560 
 
 = 4840 =160 =4 
 
 = 1. 
 
 Note. — The units of the scale in this measure are 144, 9, 30^. 40, 
 
 and 4. 
 
 SURVEYORS' MEASURE. 
 
 412. 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^^ inches make 1 link, marked - - I. 
 
 4 rods =66//;, = 100 links 1 chain, - - - 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, arc 7^^^, 4. 80, 1, 
 
 and 10. 
 
 CUBIC MEASURE. 
 
 413. Cubic measure is used for measuring stone, tlmbei, 
 eartli, and sucli otlier things as have the three dimensions, 
 length, breadth, and thickness. 
 
 412. What chain is used in land surveying '> What is its length 1 How 
 is it divided 1 Repeat the table 1 In wliat is land generally estimated 1 
 Wiiat are the units of the scale f 
 
CUBIC MKASURK. 
 
 897 
 
 
 TABLE. 
 
 
 
 1728 cubic inches, Cu. in. 
 
 , make 
 
 1 cubic foot, 
 
 - 
 
 Cu.ft. 
 
 27 cubic feet - 
 
 - 
 
 1 cubic yard, 
 
 - 
 
 Cu.yd. 
 
 40 feet of round or 
 50 feet of hewn timber, 
 
 } 
 
 1 ton. 
 
 - 
 
 T, 
 
 42 solid feet - 
 
 - 
 
 1 ton of shipping, 
 
 T. 
 
 8 cord feet, or ) 
 128 cubic feet 1 
 
 - 
 
 1 cord. 
 
 - 
 
 C. 
 
 241 cubic feet of stone 
 
 - 
 
 1 perch, - 
 
 - 
 
 P. 
 
 C3 
 
 CO 
 
 
 Notes. — 1. A cord of -wood is a pile 4 feet wide, 4 feet hish, 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 figure 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. The base of a cube is the face on 
 which it stands. If the edge of the cube 
 is one yard, it will contain 3X3=9 
 square feet* therefore, 9 cubic feet can 
 be placed on the base, and hence, if the 
 solid were 1 foot thick, it would contain 
 
 9 cubic feet ; if it were 2 feet thick it would contain 2 tiers of cubes, 
 or 18 cubic feetj if it were 3 feet thick, it would contain 27 cubic 
 feet ; hence, 
 
 The contents of such a figure are found by multiplying the length, 
 breadth, and thickness together. 
 
 7. A ton of round timber, when square, is supposed to produce 40 
 cubic feet : hence, one-fifthjs lost by squaring. 
 
 3 feet = 1 yard. 
 
 413. For what is cubic measure used! What are its denominations? 
 
 What is a conl of wood 1 AVhat is a cord foot ? What is a cube' What 
 
 is !i cubic foot 1 What is a cubic yard ? How many cubic feet in a cubic 
 
 yard 1 What are the contents of a volume equal to ? Repeat the table. 
 
 What are tlic units of the scale 1 
 
 18 
 
898 
 
 APPEISTPIX. 
 
 
 WINE MEASURE. 
 
 
 414. "Wine measure is 
 
 used for measuring all li 
 
 [quids except 
 
 ale, beer and milk. 
 
 
 
 
 
 TABLE. 
 
 
 4 gills, gi. 
 
 make 
 
 1 pint, marked 
 
 pt. 
 
 2 pints 
 
 - 
 
 1 quart, 
 
 qt. 
 
 4 quarts - 
 
 - 
 
 1 gallon, 
 
 gal. 
 
 31 1 gallons - 
 
 - 
 
 1 barrel. 
 
 bar. or hbl 
 
 42 gallons - 
 
 - 
 
 1 tierce. 
 
 tier. 
 
 63 gallons - 
 
 - 
 
 1 hogshead, 
 
 hhd. 
 
 2 hogsheads 
 
 - 
 
 1 pipe, 
 
 2n. 
 
 2 pipes or 4 hogsheads 
 
 1 tun. 
 
 tun. 
 
 gi. pL 
 
 qt. 
 
 gal. bar. tier. hhd. 
 
 . pi. tun. 
 
 4 =1 
 
 
 
 
 8 =2 
 
 = 1 
 
 
 
 32 =8 
 
 = 4 
 
 = 1 
 
 
 1008 = 252 
 
 = 126 : 
 
 = 311 ^1 
 
 
 1344 = 336 
 
 = 168 : 
 
 = 42 =1 
 
 
 2016 ^ 504 
 
 = 252 : 
 
 = 63 = 11 = 1 
 
 
 4032 = 1008 
 
 =.504 : 
 
 = 126 =3 =2 
 
 = 1 
 
 8064 z^ 2016 
 
 zzz 1008 : 
 
 = 252 =6=4 
 
 = 2=1. 
 
 Notes.— 1. The standard unit, or gallon of wine measure, in the 
 United States, contains 2.31 cubic inches, and hence, is equal to the 
 weight, avoirdupois, of 8.339 cubic inches 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. 
 
 BEER MEASURE. 
 415. Beer measure is used for measuring ale, beer, and milk. 
 
 414. What is measured by wine measure 1 AVliat arc its dcrominations 1 
 Rcpcr.t the table. What are the units of the scale 1 What is a standard 
 wine gallon 1 
 
 415. For what is beer measure used? What are its denominations 1 
 Repeat the taiilc What arc the units of the scale 1 
 
DKY MEASUKE. 399 
 
 
 TABLE. 
 
 
 2 pints, pt. make 1 quart, marked 
 
 qt. 
 
 4 quarts - 
 
 1 gallon, - 
 
 gal. 
 
 36 gallons - 
 
 1 barrel, - 
 
 bar. 
 
 54 gallons - 
 
 1 hogshead. 
 
 hhd. 
 
 pt. qt. 
 
 gal. bar. 
 
 hhd. 
 
 2 =1 
 
 
 
 8 =4 
 
 = 1 
 
 
 288 = 144 
 
 = 36 =1 
 
 
 432 = 216 
 
 = 54 =11 
 
 = 1. 
 
 Notes. — 1. The standard gallon, heer measure, contains 282 cubic 
 inches, and hence, is equal to the weight of 10.1799 cubic inches of 
 distilled rain water. 
 
 2. Milk in many places is sold by wine measure. 
 
 DRY MEASURE. 
 416. Diy measure is used in measuring all dry articles, such 
 as grain, fruit, salt, coal, &;c. 
 
 TABLE. 
 
 2 pints, pt. make 1 quart, marked qi. 
 
 8 quarts - - 1 peck, - - pk. 
 
 4 pecks - - 1 bushel, - - biu 
 
 36 bushels - 1 chaldron, - ch. 
 
 pt. qt. pk, hu. ch. 
 
 2 =1 
 
 16 =8 =1 
 
 64 =32 =4 =1 
 
 2304 = 1152 = 144 =36 =1. 
 
 Notes. — 1. The standard bushel of the United States is the Win" 
 chester bushel of England. It is a circular measure 18-J inches in 
 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. 
 
 416. What articles are measured by dry measure 1 What are its de- 
 nominations 1 Repeat the table. What are the units of the scale 1 What 
 ir the standard bushel ! What are the contents of a ffallon '' 
 
400 APPENDIX. 
 
 3. Wine measure, Beer measure, and Dry measure, and all mea- 
 sures of volume, differ from the cubic measure only in the unit which 
 is used as a standard. 
 
 AVOIRDUPOIS WEIGHT. 
 
 417. By this weight all coarse articles are weighed, such as 
 
 bay, grain, chandlers' wares, and all metals except gold and 
 
 silver. 
 
 TABLE. 
 
 16 di'ams, dr. make 1 ounce, marked oz. 
 16 ounces - - 1 pound, - - Ih. 
 25 pounds - 1 quarter, - - qr. 
 
 4 quarters - 1 hundred weight, cwt. 
 
 20 hundred weight 1 ton, - - T. 
 
 dr. oz. lb. qr, cwt. T. 
 
 16 = 1 
 
 256 nr 16 = 1 
 
 6400 r=: 400 =25 =1 
 
 25600 = 1600 =100 =4 =1 
 
 512000 = 32000 = 2000 =80 =20 =1. 
 
 Notes. — 1. The standard avoirdupois pound is the weight of 27.701 5 
 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 hundred 
 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 j from quar- 
 ters to hundreds, 4 ; and from liundreds to tons, 20. 
 
 417. For what is avoirdupois weight used 1 How is the table to be 
 read ^ How can you determine, from the second table, the value of any 
 unit in units of the lower denominations 1 
 
 NoTF.s. — 1. What is tlie standard avoirdupois pound 1 
 
 2. What is a hundred weigiit by the English method ] What is a hun- 
 dred weight by the United States method 1 
 
 3. Name the units of the scale in passing from one denominatijii to 
 another. 
 
TROY WEIGHT. 401 
 
 TROY WEIGHT. 
 418. Gold, silver, jewels, and liquors, are weighed by Troy 
 
 wciglit. 
 
 TABLE. 
 
 24 grains, gr. 
 
 make 
 
 : 1 
 
 pennyweight. 
 
 marked pu 
 
 20 pennyweig 
 
 hts - 
 
 1 
 
 ounce, 
 
 oz. 
 
 12 ounces 
 
 - 
 
 1 
 
 pound. 
 
 lb. 
 
 gr. 
 
 pwt. 
 
 
 oz. 
 
 lb. 
 
 24 
 
 = 1 
 
 
 
 
 480 
 
 = 20 
 
 
 = 1 
 
 
 57G0 
 
 = 240 
 
 
 = 12 
 
 = 1. 
 
 Notes. — 1. The standard Troy pound is the weight of 22.794377 
 cubic inches of distilled "water. Hence, it is less tlian the pound 
 avoirdupois 
 
 2. 7000 troy grains = 1 pound avoirdupois. 
 
 175 troy pounds = 144 pounds " 
 175 troy ounces = 192 ounces " 
 
 437-^ 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. 
 
 APOTHECARIES' WEIGHT. 
 419. This weight is used by apothecaries and physicians in 
 mixing their medicines. But medicines are generally sold, in 
 the quantity, by avoirdupois weight. 
 
 TABLE. 
 
 20 grains, gr. make 1 scruple, marked 9 . 
 
 3 scruples . - 1 dram, - - 3 . 
 
 8 drams - - 1 ounce, - - 5 . 
 
 12 ounces - -^ - 1 pound, - - Ife-. 
 
 418. What articles are weighed by Troy weight"! What are its denomi- 
 nations 1 Repeat the table. What is the standard Troy pound ? What 
 are the units of the scale, in passing from one unit to another 1 
 
 419. What is the use of Apothecaries' weight 1 What are its denomi- 
 nations 1 Repeat the table. What are the values of the pound and ounce 1 
 
 What are the units of the scale, in passing from one unit to another 1 
 
 18 
 
402 
 
 
 
 APPENDIX 
 
 gr- 
 
 9 
 
 3 
 
 20 
 
 = 1 
 
 
 60 
 
 = 3 
 
 r= 1 
 
 480 
 
 = 24 
 
 = 8 
 
 ft 
 
 = 1 
 
 5760 =288 =96 =12 =1 
 
 Notes. — 1. The pound and ounce are the same as the pound and 
 ounce 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. 
 
 NEW FRENCH SYSTEM. 
 
 420. The basis of this 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. It 
 is equal to 2.204737 pounds avoirdupois. The other denomi- 
 nations are as follows : 
 
 100 kilogrammes = 1 quintal ; 10 quintals = 1 ton sea water; 
 1 gi-amme = 10 hectogrammes ; 1 hectogramme = 10 deco- 
 grammes ; 1 decogramme = 10 grammes ; 1 gramme = 10 de- 
 cigrammes ; 1 decigramme = 10 centigrammes ; 1 centigramme 
 = 10 milligrammes. 
 
 COMPARISON OP WEIGHTS. 
 
 English^ 1 pound = 1.000936 pounds avoirdupois. 
 
 French, 1 kilogramme = 2.204737 " " 
 
 Spafiish, 1 pound = 1.0152 " " 
 
 Swedish, 1 pound = 0.9376 " « 
 
 Austrian, 1 pound = 1.2351 " " 
 
 Prussian, 1 pound = 1.0333 « « 
 
 MEASURE OF TIME. 
 
 421. Time is a part of duration. The time in which the 
 earth revolves on its axis is called a daij. The time in which 
 it goes round the sun is called a solar year. Time is divided 
 into parts according to the following 
 

 DA 
 
 xr.s. 40y 
 
 
 TABLE. 
 
 60 seconds, sec. 
 
 make 1 minute, marked m. 
 
 60 minutes 
 
 1 hour, - - hr. 
 
 24 hours 
 
 1 day, - - da. 
 
 7 davs 
 
 1 week, - - wk. 
 
 4 weeks 
 
 1 month, nearly, - mo. 
 
 12 calendar m'ths = 
 
 365(/rt. 1 Julian or civil year, yr. 
 
 866 days 
 
 1 leap year. 
 
 100 years 
 
 1 century, - - cent. 
 
 see. VI. 
 
 hr. da. xok. yr. 
 
 60 =1 
 
 
 3600 =60 
 
 = 1 
 
 &6400 =1440 
 
 = 24 =1 
 
 604800 =100S0" =168 =7 =1 
 
 31536000 =525600 ^8760 =365 =521 =1. 
 
 The year is divided into 12 calendar months : 
 
 No. No. 
 
 days. 
 
 No. No. days. 
 
 1st. January, - - 
 
 31 
 
 7th. July, - - - - 31 
 
 2d. February, - - 
 
 28 
 
 8th. August, - - - 31 
 
 3d. March, - - - 
 
 31 
 
 9th. September, - - 30 
 
 4th. April, - - - 
 
 30 
 
 10th. October, - - - 31 
 
 5th. May, - - - 
 
 31 
 
 11th. November,- - - 30 
 
 6th. June, - - - 
 
 30 
 
 12th. December, - - 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. 
 
 421. What are the denominations of time 1 How long is a year 1 How 
 many days in a common year 1 How many days in a Leap year ] How 
 many calendar months in a year '\ Name each, and its number, and the 
 number of days in each. How many days has February in the leap year \ 
 How do you remember which of the months have 30 days, and which 31 1 
 
 jVoTE. — 1 How are the centuries numbered 1 How are the years nuin 
 eared 1 The days 1 The hours 1 
 
404 APPENDIX. 
 
 Notes. — 1. Days are numbered in each month fiom the first day 
 of the month. 
 
 2. Montlis are numbered from January to December. 
 
 3. The centuries are numbered from the beginning of the Christian 
 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, June 2d, 1856, was the 6th month of the 
 56th year of the 19lh century. 
 
 4. The civil day begins and ends at 12 o'clock at night. In the 
 civil day, the hours are reckoned from that time. 
 
 DATES. 
 
 1. The length of the solar year is 365da. 5hr. 48m. 4S.tcc.. very 
 nearly. It is desirable to have the periods and dates of the civil 
 year correspond to those of the solar year ; else, the summer months 
 of the one would in time become the winter months of the other, 
 thereby producing great confusion in dates and history. 
 
 2. The common civil year is reckoned at 365^/a., and the solar 
 year at 365da. 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 leap 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., ar3 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 365da. 6hr. in 
 length ; but it is not; it is only 365(la. 5hr. 48m. 48scc. long : hence, 
 tt;C 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 
 cau,scs the civil and solar years to correspond very nearly, and indi- 
 cates the following rule for finding the leap years : 
 
 Every year ivhick is divisible by 4 is a leap year, unless it is 
 a centennial year, and then it is not a leap year tmless the num' 
 her of the century is also divisible by 4. 
 
 5. Tlie registration of the days, by reckoning the civil year at 
 
jnSCKIXANKOUS TABLES. 405 
 
 365da., was established by the Roman Emperor, Julius Ca5sar, and 
 hence Ihis period is sometimes called the Julian year. 
 
 The error, arising from the fractional part, continued to increase 
 until 15S2. when it amounted to 10 days; that is, as the year had 
 been reckoned too long the number of days had been too feu\ 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 
 to^ether. The new calendar is sometimes called the Gregorian 
 Calendar. 
 
 6. The method of dating by the old count, is called Qld Style ; 
 and by the new, New Style. The difference is now 12 days. la 
 Russia, they still use the old style; hence, their dates are 12 days 
 behind ours. Their 4th of January is our 16th. 
 
 CIRCULAR MEASURE. 
 
 422. Circular measure is used in estimating latitude and 
 longitude, in measuring the motions of the heavenly bodies, and 
 also in measuring angles. 
 
 The circumference of every circle is supposed to be divided 
 into 360 equal parts, called degrees. Each degree is divided into 
 60 minutes, and each minute into 60 seconds. 
 
 TABLE. 
 
 
 
 60 seconds " make 
 
 1 minute. 
 
 marked 
 
 • 
 
 60 minutes 
 
 1 degree. 
 
 - 
 
 O 
 
 • 
 
 30 degrees 
 
 1 sign, - 
 
 - 
 
 s. 
 
 12 signs or 360° 
 
 1 circle, - 
 
 - 
 
 c. 
 
 // 1 
 
 o 
 
 s. 
 
 c. 
 
 60 = 1 
 
 
 
 
 3600 = 60 
 
 = 1 
 
 
 
 lOSOOO r= 1800 
 
 = 30 
 
 = 1 
 
 
 1296000 = 21600 
 
 = 360 
 
 = 12 
 
 = 1. 
 
 422. For what is circular measure used ! How is every circle supposed 
 lu be divided 1 Repeat the table. 
 
406 
 
 APPKNDIX, 
 
 MISCELLANEOUS TABLE 
 
 12 units, or things make 
 
 12 dozen - - _ . 
 12 gross, or 144 dozen 
 
 20 things - - . . 
 
 100 pounds - - _ 
 
 196 pounds - _ _ 
 
 200 pounds 
 
 18 inches - - - - 
 
 22 inches, nearly 
 
 14 pounds of iron or lead - 
 
 21^ stones - - - _ 
 
 8 pigs - . - . 
 
 1 dozen. 
 
 1 gross. 
 
 1 great gross. 
 
 1 score. 
 
 1 quintal of fish. 
 
 1 barrel of flour. 
 
 1 barrel of pork. 
 
 1 cubit. 
 
 1 sacred cubit. 
 
 1 stone. 
 
 1 fother. 
 
 BOOKS AND PAPER. 
 
 The terras, folio, quarto, octavo, duodecimo, Sec, indicate the 
 number of leaves into which a sheet of paper is folded. 
 
 A sheet folded 
 A sheet folded 
 A sheet folded 
 A sheet folded 
 A sheet folded 
 A sheet folded 
 A sheet folded 
 A sheet folded 
 
 n 2 leaves is called a folio. 
 
 .1 4 leaves " a quarto, or 4to. 
 
 n 8 leaves 
 n 12 leaves 
 n 16 leaves 
 in 18 leaves 
 n 24 leaves 
 n 32 leaves 
 
 a 
 ii 
 ii 
 a 
 
 a 
 
 24 sheets of paper 
 20 quires - 
 
 2 reams - 
 
 5 bundles 
 
 make 
 
 an octavo, or 8vo. 
 
 a 12mo. 
 
 a 16mo. 
 
 an 18mo. 
 
 a 24mo. 
 
 a 32mo. 
 
 1 quire. 
 1 ream. 
 1 bundle. 
 1 bale. 
 
ANSWERS. 
 
 r. 
 
 EX. 
 
 1 
 
 ANS. 
 
 EX. 
 
 "8 
 
 
 A^S. 1 
 
 EX. 
 
 15 
 
 ANS. 
 
 16. 
 
 XVI. 
 
 DCCLI. 
 
 DCCCCLVII. 
 
 16. 
 
 2 
 
 XIV. 
 
 9 
 
 MLX. 
 
 16 
 
 MCCVI. 
 
 IG. 
 
 O 
 
 XVIII. 
 
 01 
 
 MMXCI. 
 
 17 
 
 CCCCXCV. 
 
 16. 
 
 4 
 
 LXIX. 
 
 11 
 
 DLXIX. 
 
 18 
 
 DCCLV. 
 
 16. 
 
 5 
 
 LXXVIII. 
 
 12 
 
 DCCXLV 
 
 19 
 
 MDCCCXLVII. 
 
 16. 
 
 6 
 
 CXV. 
 
 13 
 
 DCCCCLXI 
 
 20 
 
 MMDXX. 
 
 16. 
 
 7 
 
 CCCCIX. 
 
 14 
 
 DCXCIX. 
 
 
 
 
 
 19. 
 
 1 7 
 
 3 9000 11 
 
 5 
 
 1 961 
 
 6 7408 
 
 19. 
 
 2 80 
 
 4 
 
 93 
 
 - 
 
 
 
 20. 
 
 7 
 
 897.021 
 
 20. 
 
 8 
 
 86.029,430 
 
 20. 
 
 9 
 
 4,328,021,063 
 
 20. 
 
 10 
 
 967,040,932 
 
 20. 
 
 11 
 
 30,430,208,123 
 
 20. 
 
 12 
 
 360,030,702,010 
 
 20. 
 
 13 
 
 5,800,006,000,812 
 
 14 
 15 
 16 
 
 17 
 18 
 19 
 20 
 
 75,605,070,905,008 
 
 904,000.800,200,720 
 
 6,000.900,704,098,020 
 
 80,510,006,040,900,040,900 
 
 6,050,900,001 
 
 987,654,321.012,345,678 
 
 208,104,111,001,111,111 
 
 23. 
 23. 
 23. 
 23. 
 23^ 
 
 24. 
 24. 
 24. 
 24. 
 24. 
 24. 
 24. 
 
 2 
 
 o 
 O 
 
 4 
 5 
 
 621 
 
 6 
 
 5.702 
 
 7 
 
 8,001 
 
 8 
 
 10,406 
 
 9 
 
 65,029 
 
 10 
 
 40,000,241 
 
 59,000,310 
 
 12,111 
 
 300,001,006 
 
 69,003,000,211 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 47.000,069,000,465,207 
 
 800. 000,'000,000,429, 006,009 
 
 95,000,000,000,000,089,089,306 
 
 6,000,000,451,065,047,104 
 
 999,065,841,411 
 
 470,040,000.000,000,000,000,000,004,006,204 
 
 65.000,800,000.750,751,975,310 
 
 3L 
 
 33. 
 33. 
 
 34. 
 34. 
 34. 
 34. 
 34. 
 
 2; 7 
 
 1 
 
 \ 42600 ; 
 (426000 
 
 36b60 
 ^8,75 
 
 10|8996 
 
 ll|i;i 125. &c^. Ifar. 
 
 12 
 13 
 14 
 15 
 16 
 
 4 5445/5. 
 IT. lAc'ct. \qr. 20/6. 
 
 262155-r5. 
 
 \22lb. 2oz. ISpwt. 9gr. 
 29:]ii2gr. 
 
 17 mib. 4 § 3 5 2 9 7gr. 
 I8\2i9i?i. 
 
 1600rJ. SSOOyds. 
 
 26400/?. 316800m 
 20 r/oi/d. 2ft. 6in. 
 
 ,9 
 
403 
 
 ANSWERS. 
 
 P. 
 
 EX 
 
 21 
 
 ANS. 
 
 EX. 
 
 30 
 
 
 ANS. 
 
 
 34. 
 
 6sq. t/d. 2v7. ft. 
 
 
 78 E.E. 
 
 \qr. 
 
 
 34. 
 
 22 
 
 2 A. OH. 35P. 
 
 
 31 
 
 1008^^. 
 
 
 
 34. 
 
 23 
 
 ^5A. Q)sq. Oh. 
 
 
 32 
 
 1 Z)hhd. 
 
 I 
 
 
 34. 
 
 21 
 
 56SP. 
 
 
 33 
 
 •i02\pt. 
 
 
 
 34. 
 
 25 
 
 967 GSOcu. in. 
 
 
 34 
 
 \29har. 
 
 
 
 34. 
 
 26 
 
 o968cu.ft.. 
 
 
 35 
 
 1984;)^. 
 
 
 
 34. 
 
 27 
 
 iiOcords. 
 
 
 36 
 
 Oc/i. o2hu. o'ph. 
 
 Iqt. 
 
 34. 
 
 28 
 
 25 \ 2 )ta. 
 
 
 37 
 
 6311385C 
 
 )sec. 
 
 
 34. 
 
 29 
 
 lUj/d.. 
 
 
 38 
 
 Smo. 2ick. 
 
 
 38. 
 
 1 
 
 182630 
 
 --J 
 
 1 
 
 395873 
 
 13 
 
 32921 
 
 38. 
 
 2 
 
 87539 
 
 8 
 
 30534 
 
 14 
 
 185876 
 
 38. 
 
 
 110526 
 
 9 
 
 74716 
 
 15 
 
 93684 
 
 38. 
 
 4 
 
 79165 
 
 10 
 
 29909 
 
 16 
 
 34289 
 
 38. 
 
 5 
 
 73285 
 
 11 
 
 74022 
 
 17 
 
 243972 
 
 38. 
 
 r 
 
 41-18907 
 
 12 
 
 833516 
 
 
 
 
 J 
 
 
 39. 
 
 18 
 
 39. 
 
 19 
 
 39. 
 
 20 
 
 39. 
 
 21 
 
 39. 
 
 22 
 
 39. 
 
 23 
 
 39. 
 
 24 
 
 39. 
 
 25 
 
 39. 
 
 j26 
 
 §991,546. 
 
 •^85,465. 
 
 4770,560, 
 
 ^525.892. 
 
 f;9638.495. 
 
 £223 2s. 5d. If. 
 
 1295/6. lOoz. 2jnct. 
 
 4531b 9 ! 3 3 
 
 2ciol.ogr.8lb.8oz. 5d>: 
 
 27j 
 
 2s: 
 
 29 
 
 3o: 
 
 31 
 
 32 
 
 .. .■> 
 oo 
 
 34 
 
 43 T. 2cict. Qqr. lib. 
 312i/d. Qqr. 2da. 
 2d\ E. E. \qr. 3na. 
 143jL. 2)7ii. 6 fur. 
 4 fur. -ird. Oi/d. \ft. 
 i22A. \R. IP. 
 2224 T. Ohlid. 5gal. 
 oqt. 
 
 Tin. 
 
 19 gal. 
 
 40. 
 40. 
 40. 
 40. 
 40. 
 
 35 
 36 
 37 
 38 
 39 
 
 2o0chal. 2bbu. opk. Aqt. 
 
 823 ?/r. I0??i0. 
 
 9Q4da. \8hr. Imi. 
 
 2 2\lAcivt.lqr.2Qlb. l5oz. 
 
 ,23592550. 
 
 40 
 41 
 42 
 43 
 44 
 
 -S137915910. 
 
 68056. 
 
 12 Imi. 4fiir. 8rd. 5ft. 
 
 519190. 
 
 1124749. 
 
 41. 
 
 i 
 
 15 
 
 $22,009. 
 
 
 1 
 
 50 ; 29026. 
 
 41. 
 
 46 
 
 $27,740. 
 
 
 51 8209.75. 
 
 41. 
 
 47 
 
 2tMn. 2hhd. 29gal. 
 
 2qt. Ojn. 
 
 52 26326421. 
 
 41. 
 
 48 
 
 §20308675. 
 
 
 53 29714. 
 
 41. 
 
 49 
 
 30569S53. 
 
 
 54 50110025. 
 
 42. 
 
 55 
 
 5980^512 
 
 60 
 
 Alb. ooz. (jjnct. 
 
 42. 
 
 56 
 
 2T. Acict. 2qr. lib. 
 
 61 
 
 1053420 
 
 42. 
 
 57 
 
 205 acre a. 
 
 62 
 
 1089507 
 
 42. 
 
 OH 
 
 S75002,295 
 
 63 
 
 32341 
 
 42. 
 
 59 
 
 $7425 
 
 — 
 
 
ANSWERS. 
 
 409 
 
 P. 
 
 EX. 
 
 ANS. 
 
 e:c. 
 
 ANS. 
 
 43. 
 43. 
 43. 
 43. 
 43. 
 
 64 
 
 65 
 CG 
 67 
 68 
 
 $27131,23 
 
 $28,105 
 39//f/. }qr. 
 
 $180,825 
 $35068,807 
 
 69 
 70 
 71 
 72 
 
 481125 
 66585383 
 
 $1019,10 
 $33800 
 
 
 44. 
 44. 
 
 73 
 
 74 
 
 380 bu. \2^k. 
 $458,342 
 
 75 
 76 
 
 £57 lis. 2d. ofar. 
 
 58G0 
 
 47. 
 47. 
 47. 
 47. 
 
 1 
 2 
 
 o 
 O 
 
 4 
 
 363296 
 
 56579 
 
 733071 
 
 17 11 0O7 
 
 5 
 
 6 
 
 7 
 8 
 
 41923288 rods. 
 
 $7838180 
 
 106026 vtills. 
 4-^01 /)/v.. 
 
 9 
 10 
 11 
 
 62786/;^. 
 198621115m. 
 3591 75765 b;>. 
 
 
 
 
 
 
 
 48. 
 48. 
 4S. 
 48. 
 48. 
 48. 
 48. 
 48. 
 48. 
 48. 
 48. 
 48. 
 48. 
 48. 
 
 12 
 
 13 
 14 
 
 15 
 16 
 17 
 
 18 
 
 19 
 
 120 
 
 21 
 
 2-2 
 
 23 
 
 ;25 
 
 4199675 cords. 
 
 S87877S<ral. 
 
 99999977//>. 
 
 $3-143.641 
 
 $806,384 
 
 $4853673,758 
 
 £14 18s. 3^/. l//r. 
 
 3 T. 8cioL 2qr. 7 lb. 
 
 i\7 yds. 2qr. \na. 
 
 odL. \mi. ofar. 28rd. 
 
 8t,un Ihhd. bogal. 2>qt. 
 
 89 A 2R. 37 P. 
 
 976bu. Ipk. 6qt. 
 
 124 cords 5Sft.. 522in. 
 
 26 
 27 
 28 
 29 
 30 
 31 
 
 36 
 37 
 3.8 
 
 25 E. E. \cjr. 3;i«. 
 79fe 10 ! 6 3 
 12 3 4 5 2 9 
 124 £. E. 3qr. 3}ia. 
 9GE. F. Iqr. Ina. 
 12 T. newt. 3qr. 
 2cwL 2/r. 22/6. 
 G9qr. 2lb. lioz. 
 l34//>. l4oz. \odr. 
 10 A 2R. 18P. 
 37 A. 2R. 34P 
 ] 47 da. 2\hr. 5(Jmi. 
 o2kr. 50)11. 54isec. 
 
 49.1 
 
 39 
 
 49.! 
 
 40 
 
 49.! 
 
 41 
 
 49.1 
 
 42 
 
 49.! 
 
 43 
 
 49. 
 
 4^ 
 
 49. 
 
 45 
 
 $8759,625 
 
 183666662 
 
 (■>?/;•. 9mo. 3ick. Wda. 
 
 8b fe ! 6 3 
 
 $8,20 
 
 $39,808 
 
 $10,626 
 
 46 
 47 
 48 
 19 
 50 
 51 
 52 
 
 £121 175, 
 
 6///. Qmo. 
 
 6353870 
 
 5747 
 
 $6020 
 
 25712808,91 
 
 $36190 
 
 Od. \far. 
 Owk. Gda. 
 
 9hr. 2tni 
 
 odJ 
 50.1 
 50.1 
 50.' 
 5,). 
 5^ 
 
 51. 
 51 
 
 Ob 
 
 54 
 ■55 
 56 
 57 
 
 GS3021 
 
 107445034 
 
 6274 
 
 4 T. 3civL 2qr. 
 
 £19 19.y. 2d. 
 
 23lb. 
 ydr. 
 
 58\2299mi. 2far. 4.rd. 
 
 59 
 60 
 61 
 62 
 63 
 64 
 
 >; 19 9,625 lost. 
 
 S175,875 
 
 $3,25 
 
 19987563 
 
 2899248 
 
 i;73675 
 
 ' 65 
 
 66 
 
 22815 
 $198,625 
 
 67 
 68 
 
 80///-. 87710. Oda. 3hr. 30m. 
 $655,125 
 
410 
 
 ANSWERS. 
 
 p-i 
 
 EX. 
 
 ANS. 
 
 EX. 
 
 AXS. 
 
 51. 
 
 G9 
 
 249?//-. Imo. Wda. 
 
 73 
 
 §7398 
 
 51. 
 
 70 
 
 17877 
 
 74 
 
 -$23G0 
 
 5!. 
 
 71 
 
 S731U756 
 
 75 
 
 526 
 
 51. 
 
 72 
 
 $62727794 
 
 76 
 
 6274 
 
 77 
 
 78 
 79 
 80 
 81 
 
 82 
 
 $356.35 gained. 
 
 3J.. 2ii. 39P. 
 
 41 cords 5 cord feet. 
 
 $3280,105 
 
 $44161,987 
 
 $14352,50 
 
 83 
 84 
 
 85 
 86 
 87 
 
 2?/A 8?reo. I9(^a. 
 
 iOgal. 2qt. Ipi. 
 
 50062 
 
 15550 
 
 12° 23' 53'' 
 
 88 
 
 $161,175 gained. 
 
 ,, 90 
 
 89 
 
 2271707 
 
 i: 91 
 
 32y(^. O^-;-. 2«a. 
 £950 2s. 8d. 
 
 1 
 2 
 
 »"> 
 o 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 6776368 
 
 68653214 
 
 3422454 
 
 1952883 
 
 4354224 
 
 1028.540646 
 
 24668698404 
 
 $70,84 
 
 $12517,764 
 
 >;96 1662,960 
 
 201638228149 
 
 4281770760 
 
 174809600 
 
 141301144560000 
 15|610071000 
 
 16 14783518400 
 
 17 £81 6s. 8d. 
 
 1 8 24 T. TcivL oqr. 
 W\UQyd. \fl. 2>in. 
 20ill4° 26' 15" 
 2\\5i6hhd. Igal. 2qt. 
 22 [598 E. F. 
 
 23 1 865 T. 11 cm;^. Syr. 20/5. 
 24]320y?-. 2mo. Oiv/c. Ida. lohr. I2f)u 
 2514896 
 
 Opt. 
 
 26 
 27 
 26 
 29 
 30 
 31 
 32 
 
 34 
 35 
 36 
 37 
 
 38^ 
 
 234048 
 
 4482566 
 
 314986464 
 
 320021195962 
 
 556321146761 
 
 1747125213301 
 
 23246S4880333 
 
 71109696192112 
 
 90012355857332 
 
 549600 
 
 670460; 6704600 
 
 570 1900; 57049000 
 
 j 498U496000 ; 
 
 1 49804960000 
 
 ,59 
 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 51 
 52 
 
 9072040000 ; 907204000000 
 
 74040900; 740409000 
 
 67493600; 67493600000 
 
 129359360000 
 
 13729103000000 
 
 664763206000000 
 
 879923S229600000 
 
 25264260 17908695000000 
 
 10936893 68 145U8437S777040 
 
 16714410677359581583737 
 
 $61975 
 
 3240WU. 
 
 §2097 
 
 \33t/d. 3qr. 2na, 
 
ANSWKKS. 
 
 411 
 
 p. 
 
 EX. ANS. 
 
 EX. 
 
 54 
 
 ANS. 
 
 62. 
 
 Do £3 1 95. 4d. 2far. 
 
 11031,68 
 
 62. 
 
 55 
 
 62. i 
 
 56 
 
 62.1 
 
 57 
 
 62. 
 
 58 
 
 62.1 
 
 59 
 
 62.' 
 
 60j 
 
 $15 
 
 $506,88 
 
 $6336 
 
 $5545 
 
 $16763832 
 
 496mi. \fur. 24:rd. 
 
 61' 
 62 
 63 
 64 
 65 
 66 
 
 63. i 
 
 67 
 
 63. 
 
 68 
 
 63. 
 
 69 
 
 63. 
 
 70 
 
 63. 
 
 71 
 
 63. 
 
 72 
 
 63. 
 
 '^O 
 
 63. 
 
 JO 
 
 146484 yards. 
 
 427816 barrels. 
 
 $84,26 
 
 $16875,60 
 
 2T. \8cw(. Iqr. 2\lb. 
 
 $971,04 
 
 461 barrels left. 
 
 $1315 cost. 
 
 74 
 75 
 76 
 77 
 
 78 
 
 <oda. 6hr. 37m. 
 
 QoldoJlars. 
 
 $24,375 
 
 SQSmiles, 
 
 71fe 2§ 3 7 9l2yrs. 
 
 411/Ow. Vpk. Oqt. 
 
 $1417 
 
 $65962788,75 
 750 days. 
 $13500 
 $243 
 
 64. 
 64. 
 64. 
 64. 
 64^ 
 65. 
 65, 
 65 
 
 66, 
 
 79 
 
 80 
 81 
 82 
 83 
 
 $4770,755 
 $61 
 $672 
 $11914 
 286yr. 9mo. 
 
 84 
 85 
 86 
 87 
 
 SArd. Uft. 
 
 50 
 
 24:Cords. 
 
 $92 
 
 88 
 89 
 90 
 
 216 men. 
 $149,25 
 37816 tons. 
 
 91 : $34.88 
 
 92 
 93 
 94 
 
 Abar. 32ya/. 3-7/. 
 669hhd. 40(/al. 2qt. 
 $13650000 
 
 95 
 90 
 97 
 
 $202,50 
 $21,475 
 $927,35 
 
 71. 
 71. 
 71. 
 
 72^ 
 72. 
 72. 
 
 72. 
 72. 
 72. 
 72. 
 72. 
 72. 
 72. 
 72, 
 72 
 
 98| $18844,01 II 99| $132,9*35 || 100 | £175 18s. 6d 
 
 G579 
 
 36842 
 
 269368 
 
 4 
 5 
 
 275155 
 794S312 
 
 6 
 
 7 
 
 1147187 
 72331642 
 
 £15 195. 9d. 
 \A OB. 33 P. 
 9i/d. 2jr. Ina. 
 $79,3445 
 
 $209,728 
 
 $66862,18 
 
 153114091-1 
 
 237132 
 
 177242 
 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17:68 
 
 18 
 19 
 
 44670 
 071 4 
 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 •^l 
 2S 
 29 
 30 
 31 
 
 $17,4512 
 
 32 
 
 $3,842^-11^ 
 
 33 
 
 $1,125 
 
 34 
 
 $0,375 
 
 35 
 
 $0,81 
 
 36 
 
 $5,01 
 
 37 
 
 $52,88 
 
 38 
 
 9 
 
 39 
 
 95 
 
 40 
 
 $8 
 
 41 
 
 763521 
 
 42 
 
 407294ifff 
 
 i43 
 
 13 195 133^1 44 
 125139201145?,-^- 
 
 269577255882yVvVi? 
 1424375774S^ff^^' 
 
 1 53959 19if fJ-^- 
 SOOOlOOO/Z-rW 
 131S09655iff|-|^ 
 300335575^3X18 
 99481 57977i8_L6 05 
 
 590S5714^\V 
 
 1258127i|^|| 
 
 119191753-/^'3i-//^ 
 
412 
 
 ANSWERS. 
 
 P. 
 
 EX. 
 
 44 
 
 72. 
 
 72. 
 
 45 
 
 72. 
 
 46 
 
 72. 
 
 1 47 
 
 AX?. 
 
 17^1. 3B. IP. 
 Ida. 12hr. olm. SOsec. 
 3omi. Ofiir. 29rd. 
 4-%a/. S^^^qt. 
 
 KX. 
 
 48 
 49 
 50 
 
 AXS. 
 
 2hi(sh. Opk. Iqt. 
 
 ?=2o,2o 
 
 2^. Ad. 
 
 73. 
 73, 
 73. 
 73. 
 73. 
 73. 
 
 51 
 52 
 53 
 54 
 55 
 56 
 
 22mi. \fitr. 8rd. 
 31 6 J. IR.SoP. 
 
 $27, 397 + 
 98765 
 $11250 
 HSUlSlfl- 
 
 o (> O 
 
 57i$4.75 
 
 58,'$ 
 
 JO 
 
 59,757 ISByLS, 
 60$1,625 
 
 61 
 
 )65r/a. 
 
 62S00008 
 
 63J47 ririffs. 
 64\lT. Ucu't. 2qr. 
 65 45c-. /■/. 9954c. in. 
 66b0liff/on.s-.' 
 67 44242f^-oZ5. 
 
 74. 
 
 2 
 
 74. 
 
 3 
 
 74. 
 
 4 
 
 74. 
 
 5 
 
 7175 
 4600 
 168525 
 76850 
 
 2725 
 
 387321 
 
 4413840 
 
 15423 
 
 2674584 
 
 280082 
 
 75. 
 75. 
 75. 
 
 1 
 
 4800 1 
 
 4| 
 
 2 
 
 5950 
 
 1 
 
 3 
 
 185000 
 
 2 
 
 83S0225 
 
 559750661 
 
 493574661 
 
 3558504001 
 148072400 
 7408000 I 
 
 2 219917G000 
 
 3 242601500 
 
 4 17573500 
 
 76. 
 76. 
 76. 
 76. 
 
 2 
 
 o 
 O 
 
 254 
 
 26251-2-0- 
 291147 
 
 5 211488i6J^ 
 
 6 978 
 
 7 852 
 
 8 954 
 
 9 
 
 140848 
 
 13 
 
 10 
 
 2025 
 
 14 
 
 11 
 
 39252 
 
 15 
 
 12 
 
 475542 
 
 16 
 
 242172 
 2250 
 48126 
 16215 
 
 77. j 
 77. 
 
 1 
 
 o 
 
 387 
 1548 
 
 532 
 
 804 
 
 5 
 6 
 
 15911 
 1935 
 
 7 
 8 
 
 1809 
 3216 
 
 78. 
 78. 
 78. 
 
 79. 
 79. 
 
 80, 
 
 80. 
 
 2 
 3 
 
 1322^ 
 1740| 
 218^ 
 
 1 j 
 2| 
 
 /I 2 9 -J fi 
 ^4200 
 146 
 
 75 
 1 5 
 
 26 
 
 9£ 
 
 ■S 4 
 00 
 
 4 
 5 
 
 83253f^V 
 2459|f| 
 
 6 
 
 7 
 
 4n7"l-6-A2_ 
 
 '^'^^ '7560 
 
 o 
 4 
 
 Ql 27 1 606 
 ^^80)400 
 
 Joo/o4:-y2o-ooi 
 
 5 
 6 
 
 ''"■-' 1 4 6000 
 
 99.175 a:; 9 H 
 
 ^ '63 000 
 
 3117^i 
 
 o 
 4 
 
 6109611- 
 
 ''i'409 
 
 5 
 6 
 
 9095f^| 
 69921|| 
 
 8i4079-A^ 
 
 81. 1 1 §142 1 2|§;17 ||3 !$14 |4|,t35!|6 $l72||7j$120i;8jC-;90 
 
 82. 1 1 $121,615 ] 2 $67,50 | 3 | $118,9145 
 
 83.| 1 1 $3,024 II 2 $12,8915 3 | $9,198 || 4| $18,22765 
 
 85. 1 
 
 3 5° 13' difT in long. 
 
 2 
 
 lA?-. 2//^ 8i-cc. P.M. 
 
 
 
 86. 3 
 86. 4 
 
 13° 23' 
 4w. 36.SYC. 
 
 5 
 6 
 
 8/^r. 1 2m. A. M. 
 10° 34' 
 
ANSWERS. 
 
 413 
 
 p. 
 
 EX. 
 
 7 
 
 ANS. I 
 
 1 
 
 ANS. 
 
 86. 
 
 35° 11' 
 
 $128 
 
 86. 
 
 8 
 
 P5° 48' West, 
 
 2 
 
 2bu. \pk. 
 
 86. 
 
 \ 10 /.r. 17m. 4Ssfc.P M. 
 
 3 , 
 
 $53.28 
 
 86. 
 
 9 
 
 120° 
 
 4 
 
 32 barrels. 
 
 86. 
 
 10 
 
 1/ir. 2m. 2G.'?«c. Fast. 
 
 5 
 
 463684 
 
 87. 
 
 6 
 
 1 41GG6|f/rt//o«s. 
 
 15 
 
 $812.25 
 
 87. 
 
 7 
 
 57979fl| 
 
 16 
 
 $147,9375 
 
 87. 
 
 8 
 
 5552-1 
 
 17 
 
 £14 145. 
 
 87. 
 
 9 
 
 Imo. \ivk. A^da. 
 
 18 
 
 £166 2s. Qd. 
 
 87. 
 
 10 
 
 12 years. 
 
 19 
 
 6d. 
 
 87. 
 
 11 
 
 ^mo. Ow. 5d. 14/i. 40m. 
 
 20 
 
 $6,95175 
 
 87. 
 
 12 
 
 765 barrels. 
 
 21 
 
 $8,64 
 
 87. 
 
 13 
 
 $72 
 
 22 
 
 $93. 
 
 87.1 
 
 14 
 
 $0- 
 
 
 
 
 88. 
 
 23 if 
 
 ?7,875 2 
 
 8 SI 9 
 
 33 
 
 7680 
 
 88. 
 
 24 1 
 
 8 cents. 2 
 
 9G780C. ft 
 
 34 
 
 1/6. 7oz. 12pwt. Ugr 
 
 88. 
 
 25 :: 
 
 6 3 
 
 $773, 39£ 
 
 ) -io 
 
 SIO 
 
 88. 
 
 2G';1 
 
 olb. 6or. lipivf. 3 
 
 1 $4.2408 
 
 36 2/>M. \pk. Iqt. 
 
 88. 
 
 27'$50 3 
 
 2 $16,702^ 
 
 ) 37 $0,75 
 
 89. 
 
 38 
 
 104 sheep. 
 
 44 
 
 $59b2Sl 
 
 89. 
 
 39 
 
 12 days 
 
 45 
 
 31680 times. 
 
 89. 
 
 40 
 
 16 caunisters. 
 
 46 
 
 130 farms. 
 
 89. 
 
 41 
 
 52gal. Iqt. 
 
 47 
 
 n9/4¥3Ta«i-^s. 
 
 89. 
 
 12 
 
 1424 
 
 48 
 
 S44397293 
 
 89. 
 
 43 
 
 96 acres. 
 
 49 
 
 11/tr. Ami. 32.SW., A.M. 
 
 90. 601 
 
 127° '30 
 
 57 
 
 5''ll-/fl?.l fur.oArd.2yd. 
 
 90. 
 
 51 
 
 1 
 
 J 67° 35' jVs Long. 
 
 \ 9hr. I9in. P. M., B's time. 
 
 58 
 
 1000000>. 
 
 90. 
 
 59 
 
 \o82Arods. 
 
 90. 
 
 52 
 
 lOcords, ^ cord ft. \^c.Jt. 
 
 GO 
 
 36100 
 
 90. 53 
 
 \cwt.. '3qr. 9lb. lOoz. 
 
 61 
 
 291111. ofur. 2rd. 16/?. 
 
 90. 54 
 
 $164,475 
 
 62 
 
 1 Oacres. 
 
 90. 55 
 
 282?/r. 6mo. 8da 
 
 63 
 
 c.'oyards. 
 
 90.'56 
 
 6i:al. 2(jt. Gpt. 2r/i. 
 
 
 
 
 91. 
 
 G4[. 
 
 3//c/. Iqr. oiia. 
 
 69 2o}ir. 6mo. IGc/a. 'Jhr. 
 
 91. 
 
 6oi 
 
 33 of each. 
 
 70 
 
 $10591021,60 
 
 91. 
 
 r,6 
 
 13209 + 
 
 71 
 
 ) $2478 widow's share 
 ) $1239 each child's " 
 
 91. 
 
 67 $11,88 
 
 91. 
 
 1 G 8 ' 1 ?/ ;•. 2 5^/a . 1 Ih r. 1 5 m. 
 
 72 
 
 $9 
 
 92. 
 
 73 
 
 130C8 shuigies. 1 . 
 
 J 107° 47' diti: iu long. 
 
 ) Ihf 1 i «j. ftsvr (lifT. in timfi. 
 
 92. 
 
 
 
 
 1 ! 
 
 ( 
 
 ■ 
 
 
 " 
 
414: 
 
 ANSWEK5. 
 
 P. 
 
 EX. 
 
 75 
 
 ANS. 
 
 KX. 
 
 78 
 
 
 AXS. 
 
 92. 
 
 1/i;-. 11?«. 8sec.,P. M. 
 
 $2 
 
 92. 
 
 7G 
 
 j Ahr. 5(J7n., P. M. 
 
 79 
 
 4333|- schoolhouses. 
 
 92. 
 
 \ 2^° from N. York. 
 
 SO 
 
 AQ^lbs. 
 
 92. 
 
 77 
 
 48 hours. 
 
 81 
 
 14 days. 
 
 93. 
 
 82 
 
 28bar. 6gaL 
 
 
 I 4S2A«. \pk. -I'jis. — l:,t. 
 
 93. 
 
 S3 
 
 24:bar. ]9gals. 
 
 90 
 
 ] U\{)hu.opk.Qqt. \^j)t. — 2A. 
 
 93. 
 
 84 
 
 $85,33i 
 
 
 I o2lbu. 2pk. \qL ^pt. =zod. 
 
 93. 
 
 85 
 
 1 1 2. rolls. 
 
 91 
 
 I 40° 50' East. 
 
 93. 
 
 86 
 
 lini. 6/ur. 20rd. 
 
 I 35^ days. 
 
 93. 
 
 87 
 
 8750 pounds. 
 
 
 f $2400 = Captain's share. 
 
 93. 
 
 88 
 
 $18,025 
 
 (-^ j $2000 = 2 Lieut:s " 
 •^"^ ] $3600 = 6 Midship. " 
 
 93. 
 
 89 
 
 2500 barrels. 
 
 93. 
 
 
 
 
 [$ 200 :nr each sailor s " 
 
 
 94. 
 
 93 
 
 87° 30' 
 
 98 
 
 514 eagles. 
 
 94. 
 
 94 
 
 9hr. 33m. 146rc., A. M 
 
 . 99 
 
 2011 bushels. 
 
 94. 
 
 95 
 
 lOhr. 54.m. IGsec. A. J 
 
 n 100 
 
 $7410 
 
 94. 
 
 96 
 
 19° 
 
 101 
 
 Ij/r. 338da. 22hr. 
 
 94. 
 
 97 
 
 4800//f^s. 
 
 — 
 
 
 
 96. 
 
 1 
 
 3x3; 2x5; 2x2x3; 2x7; 2x2x2x2; 
 
 96. 
 
 
 3x3x2; 2x2x2x3; 3x3x3; 2x2x7. 
 
 96. 
 
 2 
 
 2x3x5; 2x11; 2x2x2x2x2; 3x3x2x2; 
 
 96. 
 
 
 2x19; 2x2x2x5; 3x3x5; 7x7; 
 
 96. 
 
 3 
 
 2x5x5; 2x2x2x7; 2x29; 2x2x3x5- 
 
 96. 
 
 
 2x2x2x2x2x2; 2x3x11; 2x2x17; 
 
 96. 
 
 
 2x5x7; 2x2x2x3x3. 
 
 96. 
 
 4 
 
 2x2x19; 2x3x13; 2x2x2x2x5; 2x41; 
 
 96. 
 
 
 2x2x3x7; 2x43; 2x2x2X11; 2x3x3x5. 
 
 97. 
 
 6 
 
 2x2x23; 2x47; 2x2x2x2x2x3; 2x7x7; 
 
 97. 
 
 
 3x3x11; 2x2x5x5; 2x3x17; 2x2x2x13. 
 
 97. 
 
 6 
 
 3x5x7; 2x53; 2x2x3x3x3; 2x5x11; 
 
 97. 
 
 
 5x23; 2x2x29; 2x2x2x3x5; 5x5x5. 
 
 97. 
 
 7 
 
 2x151; 5x01 ; 2x2x151; 5x5x5x7; 
 
 97. 
 
 
 3x5x5x13 ; 5x131. 
 
 97. 
 
 8 
 
 5 X 3 X 2. 
 
 97. 
 
 9 
 
 2 X 3 X 7. 
 
 97. 
 
 10 
 
 3x5x7. 
 
 97. 
 
 11 
 
 2x3x7. 
 
 97. 
 
 12 
 
 2 
 
 97. 
 
 13 
 
 2x3x5x7. 
 
 lUO 
 
 .11 2 1 18 3 12 1 4 1 5 11 5 6 6 j 10 7 ] 28 | 8 j 14 
 
ANSWEKS. 
 
 415 
 
 P. 
 
 EX. 
 
 101. 
 
 1 
 
 101. 
 
 2 
 
 101. 
 
 o 
 O 
 
 101. 
 
 4 
 
 101. 
 
 5 
 
 ANS. 
 
 16 
 
 7 
 
 22 
 
 124 
 62 
 
 EX. 
 
 6 
 7 
 8 
 9 
 10 
 
 ANS. 
 
 81 
 
 45 bushels. 
 25 acres. 
 12 feet. 
 3 bushels. 
 
 EX. 
 
 11 
 
 ANS. 
 
 $22 jyer head. 
 13, A bought. 
 21, B " 
 
 29, a " 
 
 103. 
 
 1 
 
 1260 
 
 5 
 
 10500 
 
 9 
 
 103. 
 
 2 
 
 7200 
 
 6 
 
 lOfcOO 
 
 10 
 
 103. 
 
 3 
 
 12G0 
 
 7 
 
 540 
 
 11 
 
 103. 
 
 4 
 
 1008 
 
 8 
 
 420 
 
 12 
 
 336 
 1176 
 1 AArods, 
 
 $1680 
 
 12 
 
 112 men at $15 
 
 105 " $16 
 
 80 " $21 
 
 70 " $24 
 
 104. 
 
 
 r 210 b 
 
 ushel 
 
 1 
 
 
 
 60 days. 
 
 104. 
 
 
 bags 105 times. 
 
 
 
 A, 3 times. 
 
 104.' 
 
 13 
 
 ^ barrels 70 " 
 
 14 
 
 ■< 
 
 B, 4 " 
 
 104. 
 
 
 boxes 30 " 
 
 
 
 C, 5 " 
 
 104.! 
 
 
 hhds. 14 " 
 
 
 
 L X*, 6 " 
 
 106.' 
 
 1 
 
 32 
 
 6 
 
 4t 
 
 11 
 
 9 dozen. 
 
 lOf). 
 
 2 
 
 33 
 
 7 
 
 8 
 
 12 
 
 36 pounds. 
 
 106. 
 
 3 
 
 14 
 
 8 
 
 -/t 
 
 13 
 
 46 bushels. 
 
 106. 
 
 4 
 
 48 
 
 9 
 
 61^ 
 
 14 
 
 4 firkins. 
 
 106. 
 
 5 
 
 8f 
 
 10 
 
 27 
 
 
 1 
 
 
 107. 
 
 15 
 
 107. 
 
 16 
 
 107. 
 
 17 
 
 16|- t/a?/5. 
 8 pieces. 
 471^ bushels. 
 
 IS 
 19 
 20 
 
 15 barrels, 
 6210 bushels. 
 6|- bushels. 
 
 21 
 
 17-^ bushels. 
 1 1 i rfc/ys. 
 boxes. 
 
 41 
 
 111. 
 111. 
 
 i_5 . 3J7^ 
 
 1 9 ' 4 9 • 
 
 27 . 9 5. . 1_0_6 
 
 40 ' 40 ' 4 
 
 £1- 
 4 
 
 . 4 
 ' 4 
 
 ^■! 
 
 45 . 5.6 , ^_h_ . 9_i . 3X 
 
 68' 68' 68' 68' 68" 
 
 ^ J>_ • 8_7 . TLA • 6A • 8_5 . 9 a . \Vi_Q 
 
 '-^190' 90' 90' 90' 90' 90' 90 ' 
 
 114. 
 
 1 
 
 18. 21 
 
 8 1 8 
 
 2 
 
 
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 114. 
 
 2 
 
 "4 . 77 
 
 
 
 
 3 . 9 . 3 . 9 . 9 
 
 4 ' 8 ' 2 ' 4 ' 3 • 
 
 
 114. 
 
 o 
 
 124. 129 
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 4 
 
 7 . 7 . 7 . 7 . 
 5 ' 6" ' 3 ' 2 ' 
 
 
 114. 
 
 4 
 
 ^ 9 . •:i2 3 
 
 5 
 
 17. 17. 17. J 7 . 1 7 . 17 . 17. 
 24'16'12' 8' R'4'^' 
 
 ¥• 
 
 114. 
 
 5 
 
 9|; 12tV 
 
 6 
 
 6 . 6.6.6.6 
 20' 10' 8' 4' 2' 
 
 
 114. 
 
 6 
 
 5/9 ; m 
 
 7 
 
 7 . 7 
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 114. 
 
 7 
 
 8H; i6ff. 
 
 8 
 
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 114. 
 
 8 
 
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 114. 
 
 1 
 
 11-91 
 
 
 
 
 
 
 
 115. 
 
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 4 
 
 12.10. 6 . 4 . 3 
 26 ' 26 ' 26 ' 26 ' 2 6" 
 
 
 115. 
 
 2 
 
 7.2.1 
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 115. 
 
 3 
 
 10. 4 . 5 . 
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 8 . 4 . 3 . 2 
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416 
 
 ANSWERS. 
 
 P. 
 
 EX. 
 
 115. 
 
 7 
 
 115. 
 
 8 
 
 115. 
 
 1 
 
 115. 
 
 2 
 
 115. 
 
 3 
 
 ANS. 
 
 116. 
 116. 
 116. 
 116. 
 116. 
 116. 
 
 1 
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 119. 
 119. 
 119. 
 119. 
 119. 
 119. 
 119. 
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 120^ 
 120. 
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 1 
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 3 
 4 
 5 
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 7 
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 29 ' 
 9 
 
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 29 ' 
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 59 ' 59 ' 
 
 3 . 
 24 ' 
 
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 45 ' 
 
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 28 ' 
 
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 68 ' 
 
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 29* 
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 32- 
 
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 59' 
 
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 4 
 5 
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 7 
 8 
 
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 3:1 
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 12 . 
 99 ' 
 
 121- 
 
 2.8 . ±2 
 
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 112 
 
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 18 9' 216' 1 W2' 
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 4 
 
 7 
 
 21 
 
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 118. 
 
 12 
 
 187 
 515 
 
 22 
 
 123 
 
 9 
 
 G40y|-g- acres. 
 
 118. 
 
 13 
 
 H 
 
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 10 
 
 5273 
 
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 118. 
 
 14 
 
 69 
 349 
 
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 12 
 
 11 
 
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 118. 
 
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 309 
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 225 
 
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 191-1 
 
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 18 
 
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 91 
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 95 
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 17 
 
 15 
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 10 
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 ~"73"- 
 
ANSWKKS, 
 
 417 
 
 p. 
 
 121" 
 121, 
 121. 
 121, 
 
 EX. 
 
 9 
 
 126. 
 136. 
 126. 
 
 ANS. 
 
 T2 
 
 7 
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 _9_ 
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 11 
 
 
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 147 
 
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 122.1 
 
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 63 • 448 . 72 
 84 ' 34 ' 84 
 
 
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 630" ' 630 ' 630 
 
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 122. 
 
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 220 . 399 
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 40 ' 
 
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 40 ' 
 
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 122. 
 
 7 
 
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 72 ' 
 
 64 . 
 
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 204 . 99 
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 8 
 
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 36 . 
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 9 
 
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 30 ' 30 ' 30 ' 
 
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 30 
 
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 44 ' 
 
 33 . 
 
 44 ' 
 
 38 . 22 
 
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 122. 
 
 10 
 
 672. 264. 833 
 16 8' 168' 168 
 
 
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 150 
 60 
 
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 122. 
 
 11 
 
 266 . 9 . 129. 
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 1 29 
 24 
 
 .106. 7 
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 123. 
 
 2 
 
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 5 
 
 254 
 3 
 
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 123. 
 
 3 
 
 88 . 10. 9 
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 642 
 66 
 
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 124. 
 
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 8 
 
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 354.405. 4 
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 124. 
 
 9 
 
 96 
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 78 
 
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 416.452.14. I 
 72 ' 72 ' 72 » 77 
 
 124. 
 
 11 
 
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 226 . 37 
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 12 
 
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 5 
 
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 125. 
 
 G 
 
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 19 
 20 
 
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 _2_2_. JLL. 
 12 0' 240 
 
 16.19 
 6 3' 90 
 
 1 5 . 
 
 54 ' 
 
 ±3 
 40 
 
 21 
 22 
 23 
 
 q25 
 
 ^28" 
 1 ^337 
 ^"600 
 
 ' ^ 20 
 
 24 
 25 
 26 
 
 26^ 
 
41S 
 
 ANSWERS. 
 
 r. 
 
 EX. 
 
 27 
 28 
 29 
 30 
 
 ANS, 
 
 "l 10 
 64 3 
 
 944 8 9 
 
 iqi 1 3 
 
 EX. 
 
 31 
 
 32 
 
 ANS. 
 
 17 
 
 80 
 
 $1 
 
 89 
 
 ^168 
 
 JU_ 
 1 26 
 
 °24 
 
 EX. 
 
 AIsS. 
 
 60 
 
 36 
 37 
 
 38 
 
 2\2^^ pounds. 
 6o^^2\) pounds. 
 
 39 
 
 40 
 
 1 
 
 891-?- acres. 
 
 4 o 
 
 f 347]4 bushels. 
 H-fL- inches. 
 
 2 
 
 o 
 O 
 
 4 
 
 2i/t/. HJ/t/-. 
 
 Icdi'i. 2'7?'. 2lb. VZoz. 
 2oz. lOjjwt. l2(/r. 
 
 6 
 G 
 7 
 8 
 9 
 10 
 
 12 
 13 
 14 
 15 
 
 dcwt. Iqj: bib. 8^02. 
 
 20bu. \pk. 5^qt. 
 
 Shhd. '61(jaLoqt.Qpt.\igi. 
 
 ooda. 2hr. Aim. ohsec. 
 
 2R.20F.llsq.fi.5S^sq.{n. 
 
 1 inches. 
 
 13s. lOff/. 
 
 2da. 2hr. 30 to. 45,sec. 
 
 7/wr. 2//. ^in. 
 
 222da. Ihr. 2Am. 
 
 \2cwt. \qr. llb.ooz. O-i-^f/r. 
 
 7 02. l/Jiot. 2ogrs. 
 
 5 signs IG" 16' 40^^". 
 
 16 
 
 17 
 
 \S\\yd. Oqr. 2f «a, 
 
 19: 
 
 20' 
 
 21 
 
 99 
 
 .24 
 25 
 
 1 coro?//. llc.//.466fc.m. 
 
 2 rorrf.v, 4 cord ft. 2c. ft. 
 '3gd. 2qr, 
 ■3A. 2R. 
 
 \\cwt.2,qr.2\lb.nozAldr 
 ofwr. Orel. 2ft. Gin. 
 2wk. Ada. Ilir. 49m. 
 
 33>P. 
 
 1 
 2 
 
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 4 
 5 
 6 
 7 
 8 
 9 
 
 2. 
 
 7 
 
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 _5_ 
 48 
 2 
 195 
 
 •^"9 
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 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 43 
 
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 8 
 
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 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 
 914 
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 62J 
 "40 
 
 8i 
 
 ^8 
 
 I sold ;\i left. 
 
 §72 
 
 I8IJ- gallo7is. 
 18-}|- cords. 
 33i,''jj pounds. 
 
 28 
 29 
 
 $22JL 
 
 30j 18|yan^s. 
 1| 
 
 934 
 
 
 6^^ 
 
 "3 8 
 
 1 82-?-^ 
 
 1 
 2 
 3 
 4 
 5 
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 9oz. Ipu't. \2firs. 
 Icwt. \qr. 2Alb. 80; 
 29^a/. 3|-7^ 
 \vii. \fur. \Q>rd. 
 Is. ?>d. 
 38' 34f ". 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 563.4. OR. 35|P. 
 lOc;t'A \qr. 22lh 9\\oz. 
 Mb. 80Z. \6pu'(. IGgr. 
 2 cords, 2 cord ft. Ac. ft 
 5} in. 
 
ANSWEK3, 
 
 419 
 
 p. 
 
 EX. 
 
 ANS. 
 
 EX. 
 
 ANS. 
 
 133. 
 
 12 
 
 4 5 3 3 2 9 4^r 
 
 13 
 
 Ijjwt. I8^f/r. 
 
 l3lT 
 
 1 
 
 3f. 
 
 15 
 
 5i05. 
 
 . 11 
 
 1 
 24' 
 
 134. 
 
 2 
 
 iiiV 
 
 16 
 
 6975. 
 
 12 
 
 3 
 
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 3 
 
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 17 
 
 11725. 
 
 13 
 
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 7 
 
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 63. 
 
 17 
 
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 18 
 
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 9 
 
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 6 
 
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 19 
 
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 1584. 
 
 7 
 
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 20 
 
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 11 
 
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 8 
 
 9 
 
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 21 
 
 m- 
 
 134. 
 
 13 
 
 5987f. 
 
 9 
 
 21 
 40" 
 
 22 
 
 A- 
 
 134 
 
 14 
 
 4536. 
 
 10 
 
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 — 
 
 
 
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 — 
 
 135. 
 
 23 
 
 4 
 
 T 
 
 31 
 
 
 $5i. 
 
 39 
 
 «8tV 
 
 135. 
 
 24 
 
 I 
 1 5* 
 
 32 
 
 
 $14f 
 
 40 
 
 55 cenfs. 
 
 135. 
 
 25 
 
 02 
 
 33 
 
 
 $f. 
 
 41 
 
 $34i. 
 
 135.' 
 
 26 
 
 20. 
 
 34 
 
 
 lli/OrtS. 
 
 42 
 
 $36. 
 
 135. 
 
 27 
 
 53 
 5 6- 
 
 35 
 
 
 $221. 
 
 43 
 
 43|- shillings. 
 
 135. 
 
 28 
 
 23 
 
 84* 
 
 36 
 
 
 $3tV 
 
 44 
 
 $325. 
 
 135.i 
 
 29 
 
 9 8 i 
 
 37 
 
 
 $14f. 
 
 45 
 
 $f. 
 
 135. 
 
 30 1 
 
 540. ; 
 
 38 
 
 
 $7|1. 
 
 46 
 
 3 
 
 lo- 
 
 136.! 
 
 47 
 
 $-g- 
 
 50 
 
 22 8i cents. 
 
 136. 
 
 48 
 
 $}.}. 
 
 56 
 
 SC)32f. 
 
 136. 
 
 49 
 
 $2Uf. 
 
 57 
 
 $ioi^ 
 
 136. 
 
 50 
 
 1 2}j days. 
 
 58 
 
 $12000. 
 
 136. 
 
 51 
 
 5^ hourfi. 
 
 59 
 
 /., yl'.; ^%,B's. 
 
 136. 
 
 52 
 
 $66f. 
 
 60 
 
 h -^■■^•• 
 
 136. 
 
 53 
 
 34 miles. 
 
 61 
 
 1 
 
 ^ 120 acres, A's; 80 ac?rs, i?'s; 
 
 136. 
 
 54 
 
 456Jc/s. 
 
 i 20 «c?-e.?, (7's. 
 
 138. 
 
 1 
 
 \' 
 
 7 
 
 
 36. 
 
 13 
 
 2 
 5- 
 
 19 
 
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 138. 
 
 2 
 
 3 
 
 28- 
 
 8 
 
 
 7 
 
 14 
 
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 20 
 
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 138. 
 
 
 1 3 
 1 35' 
 
 9 
 
 
 2f. 
 
 15 
 
 16621 
 
 21 
 
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 138. 
 
 4 
 
 3 
 
 31 9- 
 
 10 
 
 
 1 23 
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 16 
 
 1363. 
 
 22 
 
 ^^^h' 
 
 138. 
 
 5 
 
 23 
 
 11 
 
 
 1 1 2 
 
 TJ5- 
 
 17 
 
 7 
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 23 
 
 40 
 133- 
 
 138. 
 
 6 
 
 7f 
 
 12 
 
 
 1 7 
 M 2S- 
 
 18 
 
 26 
 33 
 
 1 
 
 24 
 
 1 37 
 ■^2 1) • 
 
420 
 
 ANSWERS. 
 
 P. 
 
 EX. 1 
 
 138. 
 
 25 
 
 138.' 
 
 26 
 
 138. 
 
 27 
 
 138.' 
 
 28 
 
 138.! 
 
 29 
 
 138.' 
 
 30 
 
 138.* 
 
 31 
 
 138.' 
 
 32 
 
 138. 
 
 OO 
 
 ANS. 
 
 1 21" 
 3 
 
 IT* 
 
 4 
 ■JT- 
 
 1 
 To- 
 
 _9 
 52 1* 
 
 T23'* 
 40. 
 
 1120. 
 
 EX. 
 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 
 ANS. 
 
 EX. 
 
 H- 
 
 43 
 
 243 
 
 5 0' 
 
 44 
 
 1 ] 2 
 1 So- 
 
 45 
 
 il 2 
 135- 
 
 46 
 
 u. 
 
 47 
 
 825J. 
 
 48 
 
 4193-5?-. 
 
 49 
 
 16046/2. 
 
 50 
 
 7 
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 2 7- 
 
 1 
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 9-5- 
 
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 139. 
 
 51 
 
 139. 
 
 52 
 
 139. 
 
 53 
 
 139. 
 
 54 
 
 139. 
 
 65 
 
 139. 
 
 56 
 
 51/^5. 
 
 i^ybs. 
 
 4^ hours. 
 $3J-. 
 
 57 
 58 
 59 
 60 
 61 
 62 
 
 $1^. 
 
 6 ff a lions. 
 
 ■| of the whole. 
 
 21. 
 
 973. 
 ~' 8- 
 
 14-*- 
 
 ^ 4 T 
 
 63 
 
 64 
 65 
 66 
 67 
 68 
 
 _5JL 
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 19|fZ6s. 
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 $ 
 
 140.' 
 
 69 
 
 140.1 
 
 70 
 
 140.' 
 
 71 
 
 140.' 
 
 72 
 
 140.1 
 
 73 
 
 140.1 
 
 74 
 
 $13i. 
 
 10S^\5Ms/i. 
 
 4 c/ay5. — 
 
 75 
 
 2|i io/^/e.s-.! 
 
 SO 
 
 76 
 
 If rf(/y5. 
 
 81 
 
 77 
 
 SI 01 
 
 82 
 
 78 
 
 ^■ 
 
 S3 
 
 79 
 
 6. 
 
 84 
 
 
 1 
 
 
 tunes. 
 
 $^096. 
 S43i. 
 
 1'^tV 
 
 142 
 
 142 
 
 142. 
 
 142 
 
 142 
 
 142 
 
 142 
 
 142. 
 
 1 
 
 l2'4- 
 
 9 
 
 2 
 
 ]B- 
 
 10 
 
 3 
 
 962 
 '^8 1- 
 
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 4 
 
 100. 
 
 12 
 
 5 
 
 1 6 
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 6 
 
 5 
 
 7* 
 
 2 
 
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 H- 
 
 3 
 
 ' 8 
 
 
 — 
 
 350 
 
 549- 
 O 68 
 •^1 19- 
 
 53|. 
 
 
 4 
 5 
 6 
 
 9 
 
 6 4- 
 
 $15. 
 
 
 7 
 
 8 
 
 $171 
 
 
 9 
 
 n>' 
 
 lies. 
 
 10 
 
 1 — 
 
 
 jy47_9 
 ^~7 2- 
 _4J_ 
 84 0" 
 
 $26,1^. 
 1 5 yai 
 
 %\^ 
 m- 
 
 4 7 5 
 62429 
 
 ds. 
 
 3 sum i^^lJ?! (///f 
 
 *"'"i 6 242920 "i'* 
 
 93 
 20 
 
 143 
 
 143 
 
 143 
 
 143 
 
 143 
 
 143 
 
 143. 
 
 M3 
 
 111 
 
 12 
 
 13 
 
 14 
 
 15 
 
 161 
 
 \1. 
 
 18i 
 
 ^ 
 
 Lush. 
 1ml. 2fitr. \&rd. 
 
 2:Mh. 
 Ifur. 
 
 Iqr 
 
 Ann. 
 
 $20{ 
 
 272 sheep. 
 13 yards. 
 
 4oz. IS-j-j-i 
 lOrd.'dyd 
 
 dr. 
 
 0|T- 
 
 3 a 
 
 A 
 
 |19 
 20 
 21 
 
 09 
 
 24 
 
 ^5- 
 
 14 bushels. 
 
 20 {i 
 
 $27 00, yfs share. 
 
 $2800, 7i'.v 
 
 $ 800, O's " 
 
 40 
 
 ?(A/. 14///-. 30ni. 
 
ANSWKRS. 
 
 421 
 
 p. 
 
 EX. 
 
 25 
 
 1 ANJ. 
 
 1 EX. 
 
 ANS. 
 
 143. 
 
 £7 176'. od. O^far. 
 
 27 
 
 285i c/f/Y.'?. 
 
 143. 
 
 ^P \ 24 marbles to JoJoii. 
 ] 32 " to Jame-<. 
 
 28 . 
 
 J, SO;/?, 24; (7,30; 
 
 143. 
 
 D, 40 ; 66 remain. 
 
 144. 
 
 29 
 
 $467f. 3- 
 
 I ' 7g Imshtls. 
 
 144. 
 
 
 1 $2^^, luhat it sold for. „^ j $1724i A's. 
 \ $-,'\f^ = first one's gain. ^"^ \ $123 If, B's. 
 
 144. 
 
 30 
 
 144. 
 
 
 ( $2^9^ rzr s-eco/2c/ " 34 166shiep. 
 
 144. 
 
 O 1 
 
 «■>:-. 7. 1<)_ 
 
 
 
 
 145. 
 
 1 1 
 
 1ft. 2'. 
 
 6 2/. 7' 3". 
 
 145. 
 
 2 
 
 Gft 2' 6". 
 
 7 
 
 15//. 4' 10" 4"'. 
 
 14.5. 
 
 3 
 
 21//. 4' 11" 4'". 
 
 8 : 
 
 /?. 6' 5" 5'". 
 
 145. 
 
 4 
 
 5ft. 7'. 
 
 9 ^ 
 
 57/lf. Kr 7" 4"'. 
 
 145. 
 
 5 
 
 n/ or' o'/r 
 
 10 ] 
 
 183//. 5' 6" 2'". 
 
 146. 
 
 11 
 
 223/?. 8' 4" 9'". 1 
 
 1 107/?. 8' 9" 2'". 
 
 146. 
 
 12 
 
 87/7. 2' 7" 9'" 6"". . 
 
 - j 1721/?. 10' 9" 11'" sum 
 
 146. 
 
 13 '317/^. ir 0" 'V". \" 
 
 -' I 280,/?. 1' 3" 9'" c/i'/: 
 
 147. 
 
 2 
 
 ioft. 6' 6". 
 
 7 19 
 
 4/?. 4' 3"' 6"'. 
 
 147. 
 
 3 
 
 82/?. 9' 4'^ 
 
 8 39 
 
 /?. 11' 2' 3"'. 
 
 147. 
 
 4 
 
 347/i;. 10' 3'^. 
 
 9 29 
 
 6/?. 10' 6". 
 
 147. 
 
 5 
 
 oo4/?. 7' 8" 8'" 3"". 
 
 10 96 
 
 sq. yd. 2sq.ft. 8' 3" 
 
 147. 
 
 6 
 
 2917/?. 0' 0" 7'" 4""-. 
 
 
 
 - 
 
 
 148. 
 
 11 
 
 S;2Uts,Uli. 
 
 14 
 
 819/?. «' b". 
 
 16 2lD^r..yds. 
 
 148.! 
 
 12 
 
 89//. 3'. 
 
 15 
 
 $15,403 + . 
 
 17 $19.bOf. 
 
 148. 
 
 13 
 
 ^18,49J. 
 
 
 
 
 
 
 149. 
 
 1 
 
 4/. /'. 
 
 3 46ft. 6'. 
 
 5 li 
 
 /?. (i'. 7 
 
 !/■/. 7'. 
 
 149. 
 
 2 
 
 5/;'.4ai"j6 
 
 4 8/?. 7'. 
 
 6 37 
 
 ft. 3'. S 
 
 8//. 
 
 153. 
 
 1 
 
 .06 
 
 7 
 
 7.008 
 
 3 9! 
 
 )0027 
 
 153.! 
 
 2 
 
 1.7 
 
 8 
 
 9.05 
 
 4 .3 
 
 20 . 
 
 153.! 
 
 1 -3 
 
 .005 
 
 9 
 
 11.50 
 
 5 21 
 
 )0.000320 
 
 153.^ 
 
 4 
 
 .27 
 
 10 
 
 44.7 
 
 6 .3 
 
 610 
 
 153.i 
 
 5 
 
 .047 
 
 1 
 
 27.4 
 
 7 5 
 
 000003 
 
 153. 
 
 6 
 
 6.41 
 
 2 
 
 36.015 
 
 8 4( 
 
 3.0000009 
 
 154. 
 
 9 .4900 
 
 14 
 
 ] 05.00001)95 
 
 4 $275,005 
 
 154. 
 
 10 59.00G7 
 
 1 
 
 $37,205 
 
 5 $9.0U8 
 
 154. 
 
 11 1 .04 09 
 
 2 
 
 $17,005 
 
 6 $15,069 
 
 154. 
 
 12 79.000415 
 
 o 
 O 
 
 $215.03 
 
 7 $27,182 
 
 154. 
 
 13 67.0227 
 
 
 
 — 
 
 
 
 
 
 156. 
 
 1 
 
 1306.1805 
 
 4 
 
 1.5415 
 
 6 27.2Ub7 
 
 156. 
 
 2 
 
 528.697893 
 
 5 
 
 446.0924 
 
 7 88.76257 
 
 156. 
 
 3 
 
 159 37 
 
 
 
 
 
 
 
 
 19 
 
4:22 
 
 ANSWERS. 
 
 P. 
 
 EX. 
 
 ANS. 
 
 EX. 
 
 13 
 
 AXS. 
 
 EX. 
 
 ANS. 
 
 157. 
 
 8 
 
 71.01 
 
 204.0278277 18 
 
 3.8896 ions. 
 
 1.57. 
 
 9 
 
 1835.599 
 
 14 
 
 400.33269960 19 
 
 $427,835 
 
 157. 
 
 10 
 
 397.547 
 
 15 
 
 .1008879 
 
 20 
 
 $19,215 
 
 157. 
 
 11 
 
 31.02464 
 
 16 
 
 $85,463 
 
 21 
 
 $670,975 
 
 157. 
 
 12 
 
 1.110129 
 
 17 
 
 $1065.19 
 
 
 
 
 158. 
 
 22 
 
 $30,286 
 
 3 
 
 9.888899 
 
 6 
 
 1571.85 
 
 158. 
 
 1 
 
 3277.9121 
 
 4 
 
 51.722 
 
 7 
 
 .6946 
 
 1.58. 
 
 2 
 
 249.60401 
 
 5 
 
 2.7696 
 
 
 
 
 159 J 
 
 8 
 
 .89575 
 
 16 6314.9 
 
 24 103.0150 
 
 159. 
 
 9 
 
 603.925 
 
 17 365.007495 
 
 25 .4232 
 
 159.1 
 
 10 
 
 1379.25922 
 
 18 20.9942 
 
 26 171.925 acres. 
 
 159.' 
 
 11 
 
 99.706 
 
 19 260.3608953 
 
 27 $82,625 
 
 159. 
 
 12 
 
 17.94 9 
 
 20 10.030181 
 
 28 $26.60 
 
 159. 
 
 13 
 
 .099993 j 
 
 21 2.0294 
 
 29j 126.84194 tons. 
 
 159.! 
 
 14 
 
 328.9992 
 
 22 999.999 
 
 30 $761.18 
 
 159.! 
 
 15 
 
 .999 
 
 23 2499.75 
 
 31 1181.725 pounds. 
 
 160. 
 
 1 
 
 .796875 
 
 4 1.50050 
 
 5 10376 2*3913 
 
 160. 
 
 2 
 
 .263872 
 
 5 26.99178 
 
 7 275539.5065 
 
 160. 
 
 3 
 
 .0000500 - 
 
 — - 
 
 
 
 
 
 161. 
 
 8 
 
 .0206211250 
 
 16 
 
 .00715248 
 
 24 148.28125 ac.-w. 
 
 161. 
 
 9 
 
 28033.797099 
 
 17 
 
 .608785264 
 
 25 12.13035/i^. 
 
 161. 
 
 10 
 
 175.26788356 
 
 18 
 
 .02860992 
 
 26 -^24.0625 
 
 161. 
 
 11 
 
 .000432015770 
 
 19 
 
 2.435141056 
 
 27 $3192.005625 
 
 161. 
 
 12 
 
 216.91165850 
 
 20 
 
 1296 
 
 28 *210. 03125 
 
 161. 
 
 13 
 
 .000000000294 
 
 21 
 
 312.5 
 
 29x708.901875 
 
 161. 
 
 14 
 
 18616.74 
 
 22 
 
 .375 
 
 30 82.06525 (/ained. 
 
 161. 
 
 15 
 
 933.8253150761 
 
 2 23 
 
 .0036 
 
 
 
 
 16.3.1 
 
 \2 258.13007 |3 162.526 4 2757.89786 
 
 5 3566163. 
 
 165. 
 
 1 
 
 2.22 
 
 r 25.05068 
 
 r 41.622 
 
 165. 
 
 2 
 
 8.522 
 
 250.5008 
 
 416.22 
 
 165. 
 
 3 
 
 33.331 13 
 
 <^ 2505.068 . - 
 25050.68 
 
 4162.2 
 
 165. 
 
 4 
 
 1.0001 
 
 1 41622. 
 
 16.5. 
 
 5 
 
 12420.5 
 
 250506.8 
 
 416220. 
 
 165. 
 
 6 
 
 .005 
 
 
 4162200. 
 
 165, 
 
 7 
 
 4.25 
 
 r 48.65961 
 
 ( 254.7347748 
 
 16.5. 
 
 8 
 
 .007 
 
 4865.961 
 
 25473.47748 
 
 165. 
 
 9 
 
 .075 14 
 
 ■{ 48659.01 ^ 
 486596.1 
 
 254734.7748 
 
 165. 
 
 10 
 
 1.27 
 
 ] 2547347.748 
 
 165. 
 
 11 
 
 .015 
 
 4865961. 
 
 25473477.48 
 
 165. 
 
 12 
 
 17.008 
 
 1 
 
 254734774.8 
 
ANSWERS. 
 
 423 
 
 p. 
 
 EX. 
 
 165. 
 
 17 
 
 165. 
 
 18 
 
 165. 
 
 19 
 
 165. 
 
 ;^o 
 
 ANS. 
 
 .13956463 + 
 1918.515 + 
 .U0473O 
 174.412+ 
 
 KX. 
 
 21 
 
 90 
 
 24 
 
 ANS. 
 
 69.7125 
 
 1.36S32 + 
 12976.81 + 
 .004958 + 
 
 EX. 
 
 25 
 26 
 
 27 
 28 
 
 ANS. 
 
 6.165c. yd. 
 
 $9,875 
 
 $2.15 
 
 $0.62 
 
 166. 
 166. 
 i6G. 
 166. 
 
 167. 
 167. 
 167. 
 
 29 
 30 
 31 
 32 
 
 18 2)oujids. 
 8 suits. 
 14 days. 
 55.5 bush. 
 
 269 acres. $13573.204 cost. 
 $50,458 + , average jvice per acre. 
 $7631.8855, elder s share. 
 $5723.914125, each of others. 
 
 10970 
 G0200 
 1000 
 
 100 
 
 10 ; 100; 1000 ; 30; 20; 2000; 12; 
 
 1200; 500000. 
 
 168. 
 
 170. 
 170. 
 
 170. 
 170. 
 170. 
 170. 
 170. 
 
 ,375 
 
 8.311+ II 4 1.563 + 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 
 1.1604+ I 6 16.119 + 
 
 79.1188 
 35.2843 
 11.5834036 
 
 ;5i3202.8S69 
 
 .25 ; .5 ; .75 
 .8; .875; .3125 
 
 015625; .2666 + 
 125; .003 
 2571+ ; .4411 + 
 23903 + 
 07157 + 
 4375; .078125 
 00448 
 
 11 
 12 
 
 13 
 
 14 
 15 
 10 
 17 
 
 ,536; .372 
 .9 
 ,7333 + 
 
 ,48375 
 ,5128 
 
 ,5375;. 0056 + 
 ,1666-f 
 
 171. 
 
 18 
 
 171. 
 
 19 
 
 171. 
 
 20 
 
 171. 
 
 21 
 
 171. 
 
 22 
 
 1.000 + 
 
 .15909 + 
 $100.80 
 $17.R5 
 30.011 + 
 
 24 
 1 
 2 
 
 o 
 O 
 
 2.9166 + 
 2.8412 
 
 JL. 3 
 
 4 ' 4 
 
 8 ' 8 
 2 1. 1 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 16 03 
 2 0"0 
 .= 4 7 
 800 
 
 3 
 
 1 fiO 
 
 903 
 
 4000 
 
 200 ' 400 
 
 R4 
 
 jiOJl 
 5000 
 
 172. 
 172. 
 172. 
 
 172. 
 172. 
 172. 
 
 .0546875/6. 
 £.325. 
 
 .olQda. 
 71.1511 + wz, 
 
 .6625/6. 
 
 / 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 .15375 ions. 
 £.1225 
 .26175^. 
 .100511+ m?, 
 .&Acivt. 
 .91111+ lb. 
 
 1 '^ 
 
 14 
 
 15 
 
 16 
 
 17 
 
 .875yd. 
 
 .01587+ hhd. 
 .7129975c/a 
 .2325 ions. 
 £.9729 + 
 
 173.' 
 
 18 
 
 173. 
 
 19 
 
 173. 
 
 20 
 
 173.; 
 
 21 
 
 173. 
 
 22 
 
 173.: 
 
 23 
 
 173 
 
 24 
 
 .48125.4. 
 .5'oE.E. 
 
 .0016177 w^^. 
 .25625^ 
 .0041956 7: 
 .10416 cAa/. 
 .00994318m/, 
 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 
 .791666 yr. 
 
 .9111/6. 
 
 .3375 
 
 .3125 chal. 
 
 .0409»ii. 
 
 .01875 ream. 
 
 .02026rd 
 
 00 
 34 
 
 35 
 36 
 
 .19672y?-. 
 .3489 fe 
 .01537 + /iArf. 
 .005A 
 .ol25yd. 
 3.390625/i!. 
 
 174. 
 174. 
 
 1 J 2qr. nib. 4oz. 
 
 2 I Ih/id. V2,gal. oAAqt. 
 
 o 
 
 4 
 
 16.s\ Id. 2.99/ar, 
 2gal. \qt. 
 
424: 
 
 ANSWERS. 
 
 P. 
 
 EX. 
 
 5 
 
 ANS. 
 
 EX. 
 
 ANS. 
 
 174. 
 
 ( Iwk.Ada. 2ohr. 59m 
 \ 56.0SCC. 
 
 14 
 
 ( o2mi. \fiu: Urd. 
 I Uft. 9.4 0S/«. 
 
 174. 
 
 174. 
 
 6 
 
 b.F. 
 
 15 
 
 2/1. 1.5in. 
 
 174. 
 
 7 
 
 6cirL 3qr. 
 
 16 
 
 4 5 13 19 9.(Sgr. 
 
 174. 
 
 8 
 
 l/jhd. Algal. \qt. 
 
 17 
 
 '6R. \P. \3.3 \sq.1jd. 
 
 174. 
 
 9 
 
 20!/al. 1<7('.+ 
 
 18 
 
 9 sheets. 
 
 174. 
 
 10 
 
 lOos. ]8pwt.l5.99gr. 
 
 19 
 
 nibs. 
 
 174. 
 
 11 
 
 oqr. I. Qua. 
 
 20 
 
 Id. 2/ur. 
 
 174. 
 
 12 
 
 5/1. 11.9 4- ill. 
 
 21 
 
 IE. UP. 
 
 174. 
 
 13 
 
 j 24:F. 23i!q.yd. 5sq.ft. 
 
 22 
 
 286da. nhr. 18' 36''' 
 
 174. 
 
 \ 82.4832sg. in. 
 
 
 
 176. 
 
 1 
 
 .06 
 
 5 
 
 .029729 + 
 
 176. 
 
 2 
 
 .09285 + 
 
 6 
 
 .034 
 
 176. 
 
 o 
 O 
 
 .034370 
 
 7 
 
 .028 
 
 176. 
 
 4 
 
 .013281 + 
 
 8 
 
 
 
 .043055 + 
 
 179.1 
 
 2. . _fi_ 
 -■} ' 3 7 
 
 1 . 
 
 15. • 1_ 
 3 7' 11 
 
 4 
 
 143 
 
 _4_ • 1 
 1 1 ' 7 
 
 180. 
 180. 
 
 4 
 5 
 
 J}. • 
 
 3fi ' 
 34 . 
 
 269 
 
 29 
 
 _v. .^. ..« . •'^74 
 
 ' 4" 9 "5 ' "BTTe ' "^ ' 9 
 
 2 1 7 . 1_ . A 1 2A6 
 I 45' 495' 75' 16fi5 
 
 9 . 223 . 7 54 31 
 0' 330' 99 9 99* 
 
 . ] 6 3 . 4J. 
 
 ' 1 fi 5 ' 9 
 
 182.11 2 |.]875'.||3|.0'0344827 + ||4P09756' ; .'592';. 5^3 
 
 2.4M8181S' 
 .5M)25025' 
 .008^497133' 
 
 165. urn 61 U/ 
 .0P0I0 104' 
 .03"777777' 
 
 ;; ' • > Q o .1 o .-> ^ 
 .0 00 O.J 00 
 
 .4'757o75' 
 J.7'577o77' 
 
 9o.2'829647' 
 
 09.74^203112' 
 
 55.6^209780437503' 
 
 
 6 
 
 47.3'763']90' 
 216.2^542870' 
 
 185. 
 
 2 
 
 457^757' 
 
 6 
 
 185. 
 
 3 
 
 2.9^957' 
 
 7 
 
 185. 
 
 4 
 
 5.09 
 
 8 
 
 185. 
 
 5 
 
 .6o'370016280907' 
 
 9 
 
 186.1 
 
 2 
 
 186.! 
 
 3 
 
 186. 
 
 4 
 
 186. 
 
 5 
 
 186. 
 
 6 
 
 186. 
 
 7 
 
 186. 
 
 8 
 
 186. 
 
 9 
 
 5.53780^5 , 
 
 1.093^)86' 
 
 1.6411^7 
 
 1.7183\39' 
 
 1.1710^037' 
 
 G.rOSG' 
 
 ll.V)(;8735102' 
 
 .81G54U68350' 
 
 4.37^4 
 4.619^525' 
 1.0923^7 
 1.3! 62' 937' 
 
 o 
 
 4 
 5 
 
 6 
 7 
 8 
 9 
 
 13.570413'96103S' 
 
 35.024 
 
 7.719'54' 
 
 26.7837M28571' 
 
 3.r45' 
 
 3.'82a5294 117647058 
 
 1.2' 6 
 
 15.43'423' 
 
ANSWERS. 
 
 425 
 
 loi. 
 
 193. 
 193. 
 193.1 
 193.1 
 
 38 
 
 2 56 
 
 12 
 
 4 40 
 
 96 
 
 o 
 
 4 
 
 
 6 
 7 
 
 8 
 
 i 
 
 6 
 2 
 3 
 1 
 3 
 
 9 
 
 •10 
 
 11 
 
 5 
 
 ■A- 
 
 9 
 TT 
 
 195. 
 
 195 
 
 195. 
 
 ]95. 
 
 195. 
 
 195. 
 
 195. 
 
 19.5. 
 
 195. 
 
 1 
 2 
 3 
 
 4 
 5 
 
 6 
 
 7 
 8 
 9 
 
 $330 
 
 $90 
 
 504 miles. 
 
 $2,08 
 
 $875 
 
 99 pminds. 
 
 $2762,50 
 
 $20 
 
 $122,85 
 
 10 
 
 *> 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 1400 pounds. 
 16485 miles. 
 $121,871. 
 216 shillings. 
 
 $3533,936-}" 
 $86.62 
 £39679.105. 
 $39,371 
 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 
 bo 
 
 §382 
 
 S63 
 
 $0,036 
 
 $2,52 
 
 $1,925 
 
 $2,10 
 
 $52,50 
 
 198. 
 196. 
 196. 
 196. 
 
 27 
 
 28 
 29 
 30 
 
 $7200 
 137,909 + 
 $132,589 + 
 
 31 
 32 
 
 $18,66f 
 
 35 
 
 $56,355* 
 
 36 
 
 106|- yards. 
 
 37 
 
 40 weeks. 
 
 J 
 
 $112,86 
 $5427 
 2l^a^. wafer. 
 
 197. 
 
 38 
 
 197. 
 
 39 
 
 197. 
 
 40 
 
 197. 
 
 41 
 
 197. 
 
 42 
 
 A, $2142; 
 $0,62|. 
 G|- bottles. 
 1261-1 shilli 
 168 povnds. 
 
 B, $1125, 
 
 ngs. 
 
 43 
 44 
 
 45 
 46 
 47 
 
 9o 
 55 
 
 $1 
 6/i 
 14 
 
 |- gallons. 
 2 miles. 
 7444. 
 o 
 
 0// 
 
 '6Zm. A^^jsec, 
 
 19 
 
i26 
 
 ANSWERS. 
 
 P. 
 
 197. 
 197. 
 
 198.i 
 198.1 
 
 198J 
 
 202.1 
 202.i 
 202.1 
 
 EX. 
 
 48 
 49 
 
 ANS. 
 
 A, 155 miles; B, 124 miles. 
 
 14. days. 
 
 EX. 
 
 50 
 
 ANS. 
 
 22^ days. 
 
 51 
 52 
 53 
 
 As, $88,40 ; £"s, $77.35. 
 10/i?-. 40m. 36^yec. 
 
 48??i. l^sec. 
 
 54 
 55 
 
 $24,66^ 
 
 3* 
 
 16^ times. 
 
 1 
 2 
 
 o 
 O 
 
 9 yards. 
 8-| roc/^f. 
 ] CO yards. 
 
 4 
 5 
 
 7-1- (iays. 
 10 
 
 6 
 
 7 
 
 920 
 
 54 days. 
 
 203. 
 
 203. 
 203. 
 203. 
 
 8 225 f%5. 
 
 9 13 ounces. 
 • j 588000/6. 
 
 I 546000Z/^. 
 
 10 
 
 1] 
 
 12 
 13 
 
 { 
 
 588000Z6. 
 14 ounces. 
 20 f/ays. 
 54 f/ays. 
 
 14 12 days. 
 15' 6 " 
 
 le'ieo " 
 
 17:40.47. 
 
 18 120 mn. 
 
 19 51 da;/s. 
 
 204. 
 204. 
 204. 
 204. 
 
 20 
 21 
 22 
 23 
 
 45 7nejt. 
 13f ounces. 
 
 13 jY (^ia!/^- 
 
 201 days. 
 
 24 11 Of rods. 
 
 25 ho'^days. 
 
 26 8j-L cioi. 
 
 27 136 me?z. 
 
 28 
 29 
 30 
 
 6 horses. 
 
 857 1|^ plan/cs. 
 
 3 hours. 
 
 207. 
 207. 
 
 207.' 
 20J^ 
 
 208. 
 208. 
 208 
 
 210. 
 210. 
 210. 
 210. 
 210. 
 210. 
 210. 
 210. 
 210. 
 
 1 
 2 
 
 o 
 O 
 
 4 
 
 161 dai/s. 
 
 5 
 
 7200 men. 
 
 6 
 
 \%l^mihs. 
 
 7 
 
 72 acres. 
 
 8 
 
 10 daus. 
 d2\ days. 
 $36. 
 292.5 ^aZ. 
 
 9 
 
 10 
 
 u 
 
 1 156 tailors. 
 19600 men. 
 50 wie;*?.. 
 
 12 
 13 
 14 
 
 $471,04. 
 
 ■3\^da. 
 
 180. 
 
 15j600. 
 16 14f fZa. 
 17|7i-oz. 
 
 18 
 19 
 20 
 
 971/6. 
 32 days. 
 32 horses. 
 
 215 vien. 
 22|132 days. 
 
 $1000, A's. 
 
 $1200, i;'5. 
 
 $ 800, C's. 
 $1714,28A ^V 
 $285,7 If, B's. 
 £4030, ^'s. 
 £3980, //.v. 
 £3980, C's. 
 £4010, i>'s. , 
 
 ' f=5000 As. 
 §2500 i?V 
 
 <^ .$33331 CV 
 
 §2500", D's. 
 $6666f,^'*'. 
 100, As. 
 140. B's. 
 200, (7'5. 
 
 OClOO-.r,Jl .S. 
 
 $3000, B's. 
 $3000. C"5. 
 $26G6f , Z)'s. 
 $1500,soM'.'?s/t. 
 
 $3000,?//o'.9s/i. 
 $12961,50 .fs. 
 $1 5737,25 ^'s. 
 $10802,25 (7'5. 
 ^l^ooj/sgiiin 
 
 211. 
 
 
 211. 
 
 9 
 
 211. 
 
 
 211. 
 
 
 211. 
 
 10 
 
 211. 
 
 
 211. 
 211. 
 
 11 
 
 $450. 
 
 $600. 
 
 $750. 
 
 As, $4242,50 s 
 
 7i'.v, 15939,50 
 
 C".s§.6788 
 
 $237,75, As. 
 
 $181.0025, B's 
 
 tack : $;1697 gain. 
 " §2375,80 " 
 " $2715,20 " 
 
 11 
 12 
 
 13 
 
 ($125.4375 6' 
 I $70, .D's. 
 
 $12720. 
 f $87,831 +.4 
 
 $65.06 -f- B. 
 
 $18.795+C 
 
 $68,313 + Z) 
 
ANSWERS. 
 
 427 
 
 p. 1 
 
 EX. 
 
 
 ANS. 
 
 211.! 
 
 
 (Sl015,33i 
 
 the first. 
 
 211. 
 
 14 
 
 ■} $1523,00 
 
 " second. 
 
 211. 
 
 '■ (S2030,66f, 
 
 " third. 
 
 212. 
 
 
 i §16,38, A's. 
 
 r> 
 
 ($6577.23A±3^^'5. 
 
 212. 
 
 1 
 
 { $35,10, £'s. 
 
 
 |$1S22,76L46^^'5, 
 
 212. 
 
 
 1 SI 8,72, C's. 
 
 
 ( $288, A's. 
 
 212. 
 
 2 
 
 S7. 
 
 4 
 
 \ $270, B's. 
 
 212. 
 
 — 
 
 
 
 I $2-10, C's. 
 
 
 213. 
 
 5 
 
 J $280, D's. 
 } $163, C's. 
 
 
 
 r $84, A's. 
 
 213. 
 
 
 8 
 
 $90, B's. 
 
 213. 
 
 
 ($1309,43^;^^, oj/i 
 
 :ers. 
 
 ] $82,50, C's. 
 
 213. 
 
 6 
 
 ■\ $2946,22f|f, midshipmen. 
 
 
 $90, D's. 
 
 213. 
 
 
 ( $10504,33^1^, sailors. 
 
 
 ($2, \ St grade. 
 
 213. 
 
 
 1 $2648,86-i-\. A's. 
 
 
 9 
 
 ■^ $ 1, 2d " 
 
 213. 
 
 7 
 
 n2901,13-rV, ^'5. 
 
 
 
 ( $0,50, M " 
 
 213. 
 
 
 ( $1850, CVv. 
 
 
 10 
 
 \ $800, 5'a' stock. 
 
 213. 
 
 
 
 
 \ 1 5 Hio.9. C's time. 
 
 
 214. 
 
 1 
 
 .095; . 
 
 0575. 
 
 
 3 
 
 
 2.08, 
 
 3.75 
 
 ; .95. 
 
 
 214. 
 
 2 
 
 .125 ; .09875. 
 
 4 
 
 .666f. 
 
 215. 
 
 2 
 
 $3,14 
 
 9 
 
 16.74 nid.es. 
 
 16 $4344,35 
 
 215. 
 
 3 
 
 $4,7825 
 
 10 
 
 47.725 sheep. 
 
 17l2625/;ar. 
 
 215. 
 
 4 
 
 ■i. 562 5r/ds. 
 
 11 
 
 27.54 tons. 
 
 18'$5144.625 
 
 215. 
 
 5 
 
 2.S3937 ociot. 
 
 12 
 
 $300,365. 
 
 19 $12500 
 
 215. 
 
 6 
 
 \.002lbs. 
 
 13 
 
 15,75 coius. 
 
 20 $3867.01875 
 
 215. 
 
 7 
 
 126?/. 
 
 14 
 
 1 60 bales. 
 
 2l'$15000 
 
 215. 
 
 8 
 
 $90. 
 
 15 
 
 478.1 25yi?. 
 
 22 $65 
 
 216. 
 
 23 
 
 742,85 gallons. 
 
 26 
 
 
 $6093,75. 
 
 216. 
 
 24 
 
 205 boxes. 
 
 27 
 
 $196,59375. 
 
 216. 
 
 25 
 
 .421; $10625. 
 
 
 
 
 
 " i 
 
 
 217. 
 
 1 
 
 .20 
 
 5 
 
 .25 
 
 9 
 
 .01375 
 
 13 
 
 .05 
 
 217. 
 
 2 
 
 .125 
 
 6 
 
 .875 
 
 10 
 
 .33^ 
 
 14 
 
 .035 
 
 217. 
 
 3 
 
 .075 
 
 7 
 
 .625 
 
 11 
 
 .375 
 
 15 
 
 .72 
 
 217. 
 
 4 
 
 .136 
 
 8 
 
 .0075 
 
 12 
 
 .125 
 
 
 
 218. 
 
 1 1 %'i per head. \ 2 $5425 |3 $50000 4 $5000 
 
 220. 
 
 2 
 
 $60.9875 
 
 7 
 
 $283,8438 
 
 12 
 
 $373,2495 
 
 220. 
 
 3 
 
 $221,91 
 
 8 
 
 $422,8976 
 
 13 
 
 $735 
 
 220. 
 
 4 
 
 $360,2832 
 
 9 
 
 $1112,90 
 
 14 
 
 $1016,075 
 
 220. 
 
 5 
 
 $473,844 
 
 10 
 
 $265,2345 
 
 15 
 
 $120.80 
 
 220. 
 
 6 
 
 $1312,5 
 
 D 
 
 1 
 
 1 
 
 $1893,75 
 
 10 
 
 $5796 
 
423 
 
 ANSWERS. 
 
 221 
 221, 
 221 
 
 EX. 
 
 2 
 3 
 
 ANS. 
 
 $20,9U9 
 $26,313 
 
 $458,88 
 
 EX. 
 
 4 
 5 
 6 
 
 ANS. 
 
 $1979,5013 
 
 $5618,75 
 
 $628,4162- 
 
 EX. 
 
 7 
 8 
 
 ANS. 
 
 $64,0625 
 $157,65625 
 
 o 
 
 4 
 5 
 
 $42,2432 + 
 $420,2531-1- 
 $213 
 $181,25 
 
 $11,0415 
 $132,7707-1- 
 $26,95864- 
 $416,16734- 
 
 10 
 11 
 12 
 
 1334,21874- 
 
 $120,0694- 
 
 $40,0908 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 *"> 1 
 
 ol 
 
 33 
 
 2 
 
 o 
 O 
 
 11 
 
 12 
 
 2 
 3 
 
 2 
 
 o 
 O 
 
 4 
 
 $81,67784- 
 
 $162 
 
 $221,266 
 
 $389,24 66 
 
 $135,3714 
 
 $42,94044- 
 
 $933,1574- 
 $499,3394- 
 $140,6444- 
 
 19,$84,6855 
 
 20855,66854- 
 
 2l|$32,6664- 
 
 22:$8590,8324- 
 
 23l$36 
 
 24!$93,78434- 
 
 25 i $160,44084- 
 
 26 
 27 
 28 
 29 
 30 
 
 $12,9644- 
 
 $82,0364- 
 
 $70,964 
 
 $879,467 
 
 $801,769 
 
 34 
 35 
 
 36 
 
 $5085 
 
 $403,858 
 
 $9337,50 
 
 1 
 
 2 
 
 $394.325-f 
 
 $697,986 
 
 226. 
 226. 
 
 3 
 4 
 
 $3339,613 
 $823,9024- 
 
 5 
 6 
 
 $4640,5324- 
 $1976,634- 
 
 227. 
 
 227. 
 227. 
 
 2 
 
 o 
 O 
 
 4 
 
 £45 8s. l^d. 
 £45 12.S. 4 It/. 
 £154 7s. Od. 2 far. 
 
 5 
 6 
 
 1 7 
 
 £1133 106'. 9}d.+ 
 £199 6s. 3^d. 
 £6 16s. 5cl. 
 
 $3976,7824- 
 $439,80 
 $6234,76-h 
 $30000 
 
 5 
 6 
 7 
 
 $952,576-(- 
 
 .07 
 
 .09 
 
 8 
 
 9 
 
 10 
 
 .10 
 
 .05} 
 
 .121 
 
 2yr. 67710. 
 omo. 18da. 
 
 13 
 
 14 
 
 lyr. Amo. 
 i6yr. 8mo. 
 
 15 
 
 16 
 
 Syr. 4??io. 
 lyr. 6mo. 20fl?. 
 
 $5359,3664- 
 $8925.544 4- 
 
 4 
 5 
 
 ^1127,041 
 S190,758 
 
 G I $156,204- 
 
 $25,3575 
 
 5j291,7215 
 $57,3048 
 
 5 
 6 
 
 $73,015 
 $83,20 
 
 7 
 
 8 
 
 $845,837 
 $48165,936 
 
 234. 
 234. 
 
 9 
 10 
 
 $14523,555 
 $926,744 
 
 11 
 12 
 
 $8501984,90622" 
 $124,1624 
 
 235. 13 
 
 $151,5811 14 $16,3875 | 15 $445,857 
 
 236. 
 
 1 $562.50 2 1 $184,499 4- I 3 $21 
 
\ 
 
 ANSWERS. 
 
 42\^ 
 
 p. 
 
 EX. 
 
 4 
 
 ANS. 
 
 EX. 
 10 
 
 ANS. 
 
 236. 
 
 $5000 
 
 ($3538, 083 CMA'Ava/. 
 ($388,083 gain. 
 
 236.1 
 
 5 
 
 $1902,557 + 
 
 236.i 
 
 6 
 
 ($2763,703 2^r. vol; 
 \ $236,297 disc'L 
 
 11 
 
 $9890,239 
 
 238. 
 
 12 
 
 $10,890 loss. 
 
 236. 
 
 7 
 
 $4820,537 
 
 13 
 
 .00414, a^ic/s. 
 
 236. 
 
 8 
 
 $4800 
 
 14 
 
 
 236. 
 
 9 
 
 $1379,6123 + 
 
 15 
 
 ($2369,2617c«sAmi 
 ($61,9883 diff. 
 
 236. 
 
 — 
 
 
 
 240. 
 
 1 
 
 
 ^6,15 
 
 
 
 5 
 
 84,374 
 
 
 240. 
 
 2 
 
 $7,65 
 
 6 
 
 882,591+ gain. 
 
 240. 
 
 o 
 
 j 2'i,2'^\^ discount. 
 \ $476,708^ pres. val. 
 
 7 
 
 $11,785 difference. 
 
 240. 
 
 O 
 
 8 
 
 $15,4044 diferejice. 
 
 240. 
 
 4 
 
 81 225,3555 ^9/-e6\ val. 
 
 9 
 
 8981,21 cash value. 
 
 241. 
 
 2 
 
 $296,50 1 3 $697,20 4 $474,375 
 
 242. 
 
 5 
 
 $3522,092 
 
 
 243.; 
 
 3 
 
 $34,8375 
 
 H 
 
 $25420,195 
 
 243. 
 
 4 
 
 $164,53125 
 
 9 
 
 86835,283 
 
 243. 
 
 5 
 
 $96,33 $5831,67 
 
 10 
 
 8935 
 
 243.! 
 
 6 
 
 ( $163,80 commission. 
 \ $4340,70 whole cost. 
 
 11 
 
 \ 863,625 cotn'n. 
 
 243.! 
 
 I 4544.642+ 6 wi-Ae/s. 
 
 243.' 
 
 7 
 
 8115,39 + 
 
 
 
 244. 
 
 12 
 
 $255»,16 
 
 15 
 
 15 tons. 
 
 244. 
 
 13 
 
 (158 barrels. 
 \ $2412.66 
 
 16 
 
 870 
 
 244. 
 
 17 
 
 20 shares. 
 
 244. 
 
 14 
 
 $420,922 
 
 18 
 
 $55743,289 
 
 216. 
 
 1 
 
 85320 
 
 1 3 
 
 859110 
 
 216. 
 
 2 
 
 $666 
 
 4 
 
 821375 
 
 247. 
 
 5 
 
 $7999,6875 
 
 7 
 
 $300 
 
 247. 
 
 6 
 
 $213500 
 
 1 
 
 $3529,41 + 
 
 248. 
 
 2 
 
 56 shares. 
 
 5 
 
 $6000 
 
 248. 
 
 3 
 
 $4000 
 
 6 
 
 $10432,432 + 
 
 248. 
 
 4 
 
 $7235,142 + 
 
 
 
 
 
 249. 1 1 
 
 .08 
 
 o 
 
 .08 
 
 5 
 
 .4166 + 
 
 '249. 
 
 i 2 
 
 .20 
 
 4 
 
 .03 
 
 6 
 
 .05 
 
 250. 
 
 250. 
 
 2 
 
 o 
 
 7 per cent, the best. 
 
 a i 41,^ h^r.t 
 
 4 
 
 $166,66| 
 
 o 1 o pvr i;cnc. lug (yco(. }j 
 
 
 
 251.11 1 1 $33,75 
 
 2 1 $0,56 
 
 3 I $236,25 
 
430 
 
 ANSWEKS. 
 
 P. 
 
 EX. 
 
 ANS. 
 
 EX. 
 
 ANS. 
 
 EX. 
 
 4 
 
 AXS. 
 
 252. 
 252. 
 
 1 
 
 2 
 
 $170 
 548,80 
 
 
 $6,0053 
 
 $70 gain. 
 
 253. 
 253. 
 253. 
 
 1 
 
 2 
 3 
 
 $0,90 ) 
 
 $3,20 
 
 $0,96 
 
 4 
 5 
 6 
 
 $5,70 
 ■ii; 18,03 
 $0,66 
 
 7 
 
 $1,80 
 
 
 254. 
 254. 
 254. 
 254. 
 254. 
 
 2 
 
 o 
 O 
 
 4 
 5 
 
 .18 
 
 .25 
 
 the same- 
 
 .80 
 
 .25 
 
 7 
 8 
 
 $160,34375 wliole gain. 
 ] .046+ 2->e?* cent. 
 .45 
 $25,65 lost. 
 
 263. 
 
 265. 
 265. 
 265. 
 
 1 
 o 
 
 255. 
 255. 
 
 9 
 10 
 
 .U5 ^;er cent loss. 
 .40 
 
 11 
 12 
 
 $508,50 
 
 256. 
 256. 
 256. 
 
 1 
 2 
 
 o 
 O 
 
 §50 168,59 
 S158,40; 8237,60 
 $126; $252 
 
 4 
 5 
 6 
 
 $300 
 
 $89,55 
 
 $47,8125 
 
 7 
 
 8 
 
 $1252,125 
 $163,80 
 
 257. 
 257. 
 257. 
 
 9 
 
 10 
 11 
 
 $16481,25 loss. 
 
 .051 
 
 .Olf 
 
 12 
 
 13 
 14 
 
 .04i 
 
 $2i'ooo 
 
 $9020 
 
 15 
 16 
 
 $127,4625 
 $298,2546 
 
 259. 
 259. 
 259. 
 
 1 
 2 
 
 o 
 O 
 
 $121,72 
 $232,50 
 $262,50 
 
 4 
 5 
 6 
 
 $20 
 
 $98,20 
 $120 
 
 7 
 
 $9101,635 
 
 
 
 260. 
 260. 
 
 1 
 
 2 
 
 $411,15 
 
 $757,908 
 
 O 
 
 4 
 
 $1227,395 
 $1318,94 
 
 262. 
 262. 
 262. 
 
 1 
 2 
 3 
 
 $7051.63415 
 $9049,53795 
 $23058,6765 
 
 4 
 5 
 
 $16355,52 
 $2160,90 
 
 $2159,6134- 
 'g^per cent. 
 $37901125 
 
 1^ per cent. 
 
 $82,25 
 
 $56,9075 
 
 266. 
 
 4 
 
 ^perct; $15,50 
 
 
 (.015 on $1 
 \ $112,50 
 ($18 
 
 266. 
 
 5 
 
 $5820 
 
 8 
 
 266. 
 
 6 
 
 $22236,197 
 
 
 266. 
 
 
 ( $4656,05 whole tax. 
 
 9 
 
 ($7,40 
 ■[$9,225 
 
 286. 
 
 7 
 
 \ .005 on $1 ; $27 
 
 266. 
 
 
 ( $6,8775 6^'s;$12,78ir'.v. 
 
 
 
 269.11 3 I $260,9932 
 
 4 I $713,37 
 
ANSWEKS. 
 
 431 
 
 1-. 
 
 EX. 
 
 ANS. 
 
 9 
 
 ANS. 
 
 270. 
 
 5 
 
 iVr. l^rwt.lqr. 16.G8/6. 
 
 $1190,343 + 
 
 270.! 
 
 6 
 
 3 T. Inot. 
 
 10 
 
 $744,546 
 
 270.i 
 
 7 
 
 \ GT.]3cwt.2gr. Alb. 
 
 11 
 
 $250,835 
 
 270.; 
 
 \ $308,4774 
 
 12 
 
 $4,09 + 
 
 270.' 
 
 8 
 
 $792,612 
 
 13 
 
 $125,S0| 
 
 271 
 
 271, 
 
 271, 
 
 271, 
 
 271 
 
 271. 
 
 272. 
 
 
 14 
 
 -S39S.il 99 + 
 
 20 
 
 $1512 
 
 
 15 
 
 .$'166,273 
 
 21 
 
 $423,36 
 
 . 
 
 16 
 
 $1101,24; $0,14 
 
 22 
 
 1251,453 + 
 
 
 17 
 
 $7936,50 
 
 23 
 
 $1457,75 
 
 . 
 
 18 
 
 $820,4625 
 
 24 
 
 ( 22.605cdi';. tare. 
 ($68,5856 duty. 
 
 . 
 
 19 
 
 §16,206 + 
 
 12 months. 
 
 273. 
 273. 
 273. 
 273. 
 
 2 
 3 
 
 4 
 5 
 
 9 mo. 
 
 B^mo. 
 
 1 mo. oda. 
 
 64-mo. 
 
 4 
 
 6 
 7 
 8 
 
 21+ daj/s. 
 
 6mo. Gda. 
 
 2(j^da., or on July 23 
 
 274. 
 274. 
 
 9,^1 
 
 io!s 
 
 S^'^Jjda. or on Sept. 19 
 ISl^da. or Dec. 30th 
 
 11 
 12 
 
 7Sfy(/., or Oct. 19 
 59.(57da., or June 28. 
 
 275.1! li*0,50^ 
 
 276. 
 276, 
 276, 
 
 2 
 
 o 
 O 
 
 4 
 
 80,66 
 $0,49 
 $1,00 
 
 o 
 6 
 7 
 
 75" 
 
 19 carats. 
 $0,13i 
 
 80,30 
 
 278. 
 
 278. 
 278. 
 
 1 
 2 
 
 {1 
 
 lb. at Sets. 1 lb. at 
 Qcts. 3 lb, at lActs. 
 Mb. each. 
 
 1 calf", 2 cows, 1 ox, 1 colt. 
 3 gallons of water. 
 
 279.11 I I 20 pounds of each. 
 
 2 I 75 pounds of each. 
 
 280. 
 280. 
 280. 
 
 oG(/al. at 7s., 24/7a/. at 7*'. 6c/. and at 9s. 6c/., I2gal. at 9s. 
 
 10 at $2, 15 at"'Sf. 
 
 2olb. at 5 and 7, 100 at 7Jc;s., 37^ at 9^, and 50 at \Qcts. 
 
 281.! 
 
 1 
 
 22 pound of each. 
 
 281.1 
 
 2 
 
 9gal. of water, 40^ at $2,50, 13^ at $3. 
 
 281. 
 
 3 
 
 12 calves, 12 sheep, 16 lambs. 
 
 281. 
 
 4 
 
 8 at $6, 8 at $7, 4 at 819. 
 
 281. 
 
 5 
 
 ^Qfjal. at 4s. and \Qgal. each at 6?. 8s. and 10s. 
 
 281. 
 
 6 
 
 6 vests, 12 pants, 6 coats. 
 
 281. 
 
 7 
 
 30 at 15 carats, and 4 each of 20c., 22c., 24c. 
 
 281. 
 
 8 
 
 10 at 81, 15 at 81, 10 at 85. 
 
 288. 
 
 1 
 
 88591,975. 
 
432 
 
 ANSWERS. 
 
 P. 
 
 EX. 
 
 AN 3. 
 
 EX. 
 
 ANS. 
 
 289. 
 
 2 
 
 $8637,168 + 
 
 2 
 
 $176204,4729 
 
 289. 
 
 o 
 
 S9777,636 
 
 
 
 
 £14014 17s. 7J.+ 
 
 290. 
 
 4 
 
 $6005,368 
 
 2 .07 'p<:r cent al/ovepar. 
 
 290. 
 
 5 
 
 $807,874 + 
 
 
 
 
 
 291. 
 
 3^^12286,06 
 
 
 
 (§1250,52 
 
 291. 
 
 4184097 f'7tcs66 centimes. 
 
 
 ( .03 ijer cent nearly, below par. 
 
 291.':li$0657,G93 
 
 - 
 
 
 299. 
 
 1 1 
 
 22Dj\ ions. 
 
 3|729^V5^««^'- 
 
 5 1006,57 7. 
 
 OQO 
 
 i o 
 
 43841 ^^"*'- 
 
 4 300.14 r. 
 
 i 
 
 
 ^yy. 1 ^ 
 
 
 301. 
 
 116 
 
 15 iH 1 
 
 29,76,765625 
 
 301. 
 
 2225 
 
 16 
 
 1225 
 7056 
 
 30! 10,4 976 
 
 301. 
 
 3:676 
 
 17 
 
 1 " fi 2 h 
 TT 1 f ) 9 
 
 3134012,224 
 
 331. 
 
 4 
 
 20164 
 
 18 
 
 7,84 
 
 32.0184528125 
 
 301. 
 
 5 
 
 214369 
 
 19 
 
 58,140625 
 
 '^'^14 09 6 
 
 301. 
 
 6 
 
 1795600 
 
 20 
 
 250AV 
 
 34iM 
 
 301. 
 
 7 
 
 605,16 
 
 21 
 
 51030,81 
 
 3 ^ 8 1 
 
 301. 
 
 8 
 
 .276676 
 
 22 
 
 216 
 
 36 3l54|f 
 
 301. 
 
 9 
 
 9,765625 
 
 23 
 
 13R24 
 
 -^^ ^^'10 2 4" 
 
 301. 
 
 10 
 
 .00274576 
 
 24 
 
 373248 
 
 3R 390H2 5 
 "^153 1 44 I 
 
 301. 
 
 11 
 
 60639,0625 
 
 25 
 
 1953125 
 
 39 i48a6,936 
 
 301. 
 
 12 
 
 9 
 
 26 
 
 2515456 
 
 40 ,( 
 
 )002441 40625 
 
 301. 
 
 1 • > 3 6 
 
 27 
 
 20736 
 
 41 2 
 
 893640,625 
 
 301. 
 
 1449 
 
 28 
 
 59049 
 
 12 1 
 
 0,272025 
 
 307. 
 
 1 
 
 7 
 
 8 2 
 
 .5 
 
 15 
 
 .1581 + 
 
 22 
 
 3 
 
 7 
 
 307. 
 
 2 
 
 12 
 
 9 1 
 
 6,7 
 
 16 
 
 .779 + 
 
 23 
 
 .2828 + 
 
 307. 
 
 3 
 
 15 
 
 102 
 
 505 
 
 17 
 
 .149 
 
 24 
 
 11.618 + 
 
 307. 
 
 4 
 
 48 
 
 118 
 
 9,409 + 
 
 18 
 
 5,01 
 
 25 
 
 137,84 
 
 307. 
 
 5 
 
 6 
 9 
 
 12.^ 
 
 153 
 
 19 
 
 14,015 
 
 26 
 
 .885 + 
 
 307. 
 
 6 
 
 15 
 4 
 
 13.5 
 
 33 + 
 
 20 1,2247 + 
 
 27 
 
 75.15 
 
 307. 
 
 7 
 
 .1 
 
 4 
 
 14.i 
 
 )68 
 
 24 
 
 
 
 21 
 
 53 
 
 
 
 28 
 
 400.06 
 
 309. 
 
 
 1 
 
 30//. 
 
 2 
 
 221 stones. 
 
 309.1 
 
 1 
 
 343 
 
 3 
 
 21y9j- rods. 
 
 310. 
 
 4 
 
 ( GOrds. wide. 
 I \SOrds. lonq. 
 
 8 
 
 94,708/7. 
 
 310. 
 
 9 
 
 53.331/1!. 
 
 310. 
 
 5 
 
 10J.0/^.29P.168f^5'.//. 
 
 10 
 
 8,660 //!.+ 
 
 310.! 
 
 6 
 
 15 ft. 
 
 11 
 
 825,8 mikfi. 
 
 310.! 
 
 7 
 
 1 
 
 35/V. 
 
 12 
 
 $100 
 
ANSWERS. 
 
 433 
 
 r. 
 
 EX. 
 
 AiNS. 
 
 EX. 
 
 ANS, 
 
 311. 
 311. 
 311. 
 311. 
 
 13 
 14 
 15 
 16 
 
 loft. 
 
 28.28 +/<. 
 
 6i/(. 
 
 11.041rf/5. 
 
 17 
 
 ( 4.405 + i/i., 1st man's share. 
 -' 5.739 + ni., 2d " 
 ( 13.856+?;;., 3d " 
 
 
 315. 
 
 1 
 
 12 
 
 5 
 
 179 
 
 1 
 
 2,028 + 
 
 5 
 
 .729 + 
 
 315. 
 
 2 
 
 49 
 
 6 
 
 364 
 
 2 
 
 12,0016 + 
 
 6 
 
 .0]5 
 
 315. 
 
 o 
 
 36 
 
 7 
 
 439 
 
 O 
 
 .232 + 
 
 7 
 
 .188 + 
 
 315. 
 
 4 
 
 247 
 
 8 
 
 3072 
 
 4 
 
 27,0002 + 
 
 8 
 
 4,339 + 
 
 316. 
 
 1 
 
 316. 
 
 2 
 
 316. 
 
 3 
 
 316. 
 
 4 
 
 316. 
 
 
 
 316. 
 
 6 
 
 316. 
 
 7 
 
 316. 
 
 8 
 
 316. 
 
 9 
 
 316. 
 
 10 
 
 5 
 
 9 
 
 4-1- 
 
 7. 
 B 
 9 
 2.S 
 2 7 
 f>4 
 
 35 
 
 1.987 + 
 3.83 + 
 
 1 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 { 
 
 27/V. 
 
 19//;., length of each side. 
 
 2l66s5'./(f., area 
 
 36/?., length of each side* 
 
 8.57+//. 
 
 9.77 />., length and breadth. 
 
 19.54 +/;;., height. 
 
 10.125 c?i. /it. 
 
 45 cents per yard. 
 
 2025 whole number of yards. 
 
 317. 
 
 9 
 
 317. 
 
 10 
 
 317. 
 
 11 
 
 317. 
 
 12 
 
 317. 
 
 13 
 
 317. 
 
 14 
 
 317. 
 
 15 
 
 317. 
 
 1 
 
 317. 
 
 16 
 
 317- 
 
 
 319. 
 
 64/65. 
 
 8//., length of each side. 
 
 8 jrlobes. 
 
 $1331 
 
 12m. long, 6m. wide, \m. thick. 
 
 24/. long, 20/. wide, 9//. deep. 
 
 20 feet. 
 
 .54 + Mi., 1st woman's share. 
 
 .69 + ^;., 2d " 
 
 .99 in., 3d " " 4th, 3.77m. 
 
 '89 \ 2 1 $80 II 3 I §396 
 
 320. 
 320. 
 
 321. 
 321, 
 321, 
 
 4 
 5 
 
 1 
 2 
 
 174i'(/A-. 
 
 20 !/., to bring back the nearest. 
 
 1 
 2 
 
 5 miles. 
 
 $2 
 
 fftw. 
 
 ij;2730 
 
 §64,96 
 
 79l-l-m2. 
 
 I0??z«., 7/Mr., 27rds., \\yd. 
 
 322. 
 
 i 1 
 
 5551^*2/. 1 2 1 13rfff.; 'M2ini. | 3 | 6 
 
 324. 
 
 i 2 
 
 3125000 
 
 o 
 
 4^ 
 
 
 325. 
 325. 
 
 4 
 
 5 
 
 100000000000000 
 .$3200 
 
 6 
 
 7 
 
 $54000 
 
 $327,68 
 
 
434 
 
 AKSWEK3. 
 
 P. 
 
 EX. 
 
 ANS. 
 
 EX. 
 
 4 
 5 
 
 ANS. 
 
 326.1 
 326.' 
 326. 
 
 1 
 
 3 
 
 118081 
 
 2044 
 
 811184810 
 
 $4294 9672,95 
 93S249922+ ships. 
 
 34i.';2i 
 
 $4166,40 
 
 28 
 
 2i|- hours. 
 
 32 
 
 117/A. 
 
 36 $5600 
 
 341. 
 
 ;2o 
 
 9f days. 
 
 29 
 
 
 OO 
 
 S693 
 
 37 
 
 $0,071 
 
 341. 
 
 26 
 
 36 feet. 
 
 30 
 
 $100 1 
 
 '34 
 
 1 1 cents. 
 
 38 
 
 $126 
 
 341. 
 
 27 
 
 §1770 
 
 31 
 
 5i yards. 
 
 j35 
 
 9;56 
 
 39 
 
 $44,10 
 
 342. 
 
 -10 
 
 $43 
 
 
 r 60 the first. 
 
 342. 
 
 41 
 
 $240,75 gain. 
 
 47 
 
 100 " second. 
 
 342. 
 
 42 
 
 5//.r. 21 mi. l&^jscc. 
 
 1 140 " third. 
 
 342. 
 
 \o 
 
 40 yards. 
 
 
 ISO " fourth. 
 
 342. 
 
 14 
 
 10 hours. 
 
 48 
 
 16-i inches. 
 
 342. 
 
 45 
 
 36 days. 
 
 49 
 
 2y^;j months. 
 
 342. 
 
 IC 
 
 11^ days. 
 
 50 
 
 ( 35J- yards baize. 
 
 342. 
 
 
 
 I $2,20 per yard. 
 
 343. 51 
 
 $12 1 
 
 
 r-^^2317,15.4's. 
 
 443. 
 
 52 
 
 $1,20 ' 
 
 '\<-. 
 
 $1853,725'.?. 
 
 443. oo 
 
 'S7 S. 652 + discount. 
 
 OO 
 
 i 
 
 1 $2317,15 6"6\ 
 
 343. 54 
 
 8129,60 
 
 
 827S0,58Z>'s. 
 
 343. ..- 
 
 \ $19,375 most advan- 
 
 
 f $95,10.1'5. 
 
 343. 
 343. 
 
 56 
 
 ] tajTL'ous for cash. 
 1292^.823 gain. 
 
 59 
 
 $95,l0i>"6-. 
 ' $133,1 4 (?'s 
 
 34:}. 
 
 
 (SI 22,7 O^'^'. 
 \ Si 63,60 Z^'s. 
 (§1 96,32 6"6-. 
 
 
 $l52,16/;'6'. 
 
 343. 
 
 57 
 
 60 
 
 71 ounces. 
 
 343. 
 
 
 G] 
 
 8f days. 
 
 343. 
 
 — 
 
 
 1 
 1 
 
 ;62 
 
 17 times. 
 
 
 
 344. 
 
 63 
 
 41| days. II 
 
 ( 8172,78+ more advanta- 
 
 314. 
 
 64 
 
 5 months 24 day-s. 167 
 
 < geous in bond and inort- 
 
 344. 
 
 65 
 
 68 days. i 
 
 
 ( gaae. 
 
 344. 
 
 66 
 
 126":allons. 
 
 68 
 
 $3312,417 + 
 
 344. 
 
 70 
 
 1 
 
 69 
 
 §42,60 
 
 1 
 
 345. 
 
 4 yards high. • [ 
 
 77 
 
 836000 
 
 345. 
 
 71 
 
 20 hours ; 140 miles. 
 
 78 
 
 41,183+ bushels. 
 
 345. 
 
 72 
 
 f S2 the lirst. 
 
 79 
 
 137.942+ feet. 
 
 345. 
 
 ( $6 the second. 
 
 
 
 The second. 10 dava 
 
 345. 
 
 73 
 
 100 thousand leet. 
 
 
 
 ailer the od ; thv.' 
 
 345. 
 
 74 
 
 91 39 
 ~ ' 5 
 
 80 
 
 < first, 8 days after 
 
 315. 
 
 75 
 
 $3825 
 
 
 
 the second, or I"' 
 
 345. 
 
 76 
 
 i>144, better to pay casli. 
 
 
 
 days after tlic .'i 1. 
 
ANSWERS. 
 
 435 
 
 p. 
 
 346. 
 346. 
 346. 
 346. 
 346. 
 346. 
 346. 
 346. 
 346. 
 
 EX. 
 
 ANS. 
 
 EX. 
 
 ANS. 
 
 81 
 
 §6890 
 
 86 
 
 12ibs. soap. 
 
 82 
 
 ( $13 16, whole cost. 
 
 87 
 
 5 o'clock 20m. P.M. 
 
 ( -^7, cost per acre. 
 
 88 
 
 $f = $0.66f. 
 
 83 
 
 32 days, or Mar. 1 6. 
 
 89 
 
 j 24 chickens. 
 
 84 
 
 (512 slabs; 
 
 (36 turkies. 
 
 "[ $302,222., cost. 
 
 
 
 
 1 $350 A'&; $297,50 B's; 
 
 
 
 
 
 
 85 
 
 \ $210 C's; $175 D's ; 
 f $122,50 E's. 
 
 
 
 347. 
 
 90 
 
 
 347. 
 
 
 \ 
 
 347. 
 
 91 
 
 347. 
 
 
 i 
 
 347. 
 
 
 
 347. 
 
 92 
 
 
 347. 
 
 
 347. 
 
 
 
 347. 
 
 93 
 
 
 8 days. 
 
 $1707,50 first, 
 $2157 second, 
 $2516,50 third. 
 $960 stock; 
 
 $180 gain, first. 
 $640 stock; 
 
 $120 gain, 2d. 
 49.945+ leet. 
 
 94 
 95 
 
 90 
 
 97 
 98 
 99 
 
 8 
 
 llri/i;- 
 ( $5331+ A's. 
 
 oi' a week. 
 1341 
 
 miles. 
 
 \ $8&8f + B's. 
 
 ( $177,H C's. 
 36^ days 
 
 84485,006 feet. 
 
 $200,06 in favor of" 1st invest. 
 
 348. 
 
 100: 
 
 ^3599680 cubic yards. 
 
 102 
 
 $4004,338+ 5th. 
 
 348. 
 
 101=! 
 
 ^4646,363 
 
 1032160 men. 
 
 348. 
 348. 
 348. 
 
 102 
 
 '$1555,017+ 1st. 
 
 $1354,717+ 2J. 
 
 ' $4304,663+ 3d- 
 
 104 
 105 
 
 j $57,142 A's. 
 I $42,857 B's. 
 $31 
 
 348. 
 
 1 
 
 ^$5781,263+ 4th. 
 
 106 
 
 8 hours. 
 
 ( $30 com. diff 
 i $2160 whole cost. 
 '$144,03 A's 
 
 $ 90,12 B's 
 
 $ 63,45 C's 
 
 $168,35 D's 
 
 .06397 
 
 $14467,505 
 
 111 
 
 112 
 
 113 
 114 
 
 971 pounds. 
 
 |- of a cent cost ; 
 
 I of a cent sold for; 
 
 -^^ gain on each ; 
 
 80 eggs sold. 
 84 years. 
 942.'48+ cubic feet. 
 
 155.4. -.iR. 38.72P. 
 
 A, 25 days, 
 
 B, 30 days, 
 
 C, 37-^ days. 
 $365,837 nearly 
 
 118 
 119 
 120 
 121 
 
 (j-fjf hours. 
 lOSy"^^ planks. 
 5 inches. 
 ■^4006,54 + 
 
 3o2. 
 
 2 
 .■> 
 o 
 
 36 acres. ! 
 
 5 A IR. 15P. ! 
 
 4 I ] 35 acres. 
 
43G 
 
 ANSWERS. 
 
 P- 1 
 
 EX. 
 
 ANS. 
 
 EX. 
 
 5 
 G 
 7 
 8 
 
 ANS. 
 
 353.! 
 353.! 
 353.' 
 353. 
 
 1 
 
 2 
 3 
 
 4 
 
 437 A 2R. 3lP-i- 
 291^. 2/i. IGP. 
 35^. Oil. 2oP. 
 
 20 A 
 
 40^. 
 
 \bA. 
 
 2\A, ]R. 8 P. 
 
 26 A. 3Ii. 20 P. 0,/d. 
 
 354. 
 354. 
 354. 
 
 o 
 
 4 
 
 21 A OR. 39.824P. 
 921.875.V.7./5. 
 704.12o.v^. V^. 
 
 
 6 
 
 60 A SR. 12. 8 P. 
 270 A IR. 24 P. 
 
 355.1 2 
 355. i 3 
 
 356. 
 
 o»4.33/G 
 125.664 
 
 4 
 2 
 
 17 9.U712 
 7418 
 
 4360.8354- 
 
 19.635+ I 3 I 153 9384 || 4 | 1.069 +.S7. yc?. 
 
 357.1 
 357.1 
 
 358. 
 358. 
 358. 
 
 359. 
 359. 
 
 615.7536 
 4071.5136 
 
 19699657 1.7221046^. mi 
 
 2 
 
 »■> 
 o 
 
 4 
 
 268.0832 
 
 2144.6656 c. in. 
 
 2599927920826.6374908 
 
 
 
 1 
 2 
 
 904.7808 c./(5. 
 
 giOOsy./^. 
 
 ]440.s^y./i;. 
 
 2 
 3 
 
 110592 c. f/z. 
 42|c.//. 
 
 4 
 5 
 
 315f^ gallons. 
 13820 eft. 
 
 360. 
 
 2 
 
 360. 
 
 
 
 360. 
 
 4 
 
 360. 
 
 2 
 
 361. 
 361. 
 
 233.334- 
 
 ^ sq.ft. 
 2827.44 sy. zn. 
 6283.2s(7./;;. 
 36442.56 
 
 4 
 5 
 
 13571 712 
 
 9650.9952 
 7363.125 
 
 2 
 
 o 
 O 
 
 4380 
 2484 
 
 5620 
 5760 
 
 6 
 
 7 
 
 14400 
 1800 
 
 362. i 2 9160.9056 
 
 8659.035 
 
 4 1 2827.44 
 
 364.1 
 
 2 
 
 32.4938i/z. |l 3 28.2574i;z. 
 
 365. 
 365. 
 365. 
 
 1 
 2 
 
 197.459-1- gal. wine. 
 162.613+ gal. beer. 
 
 3 
 
 4 
 
 ( 136.9209+ wine gal. 
 
 (112.7583+ beer gal. 
 
 148.3772+ wine gal. 
 
 
 3 67.11 1 I 40ZA. 
 
 36S.I 5 
 368.! 6 
 
 25/6. 
 
 50/*. 
 
 II 4 I 2U/A. 
 
 AOlb. 
 
 l/ji., lirin., 2m., Ain. 
 
 7 
 8 
 
 64//;. 
 150/6. 
 
 370.11 1 I 60//?. 
 
 2 I 40/6. 
 
 II 3 I 25/6. 
 
 371.11 1 1 Ufi. li 2 1 U//. 
 
 
 
 372.11 1 1 40/6. 11 2 1 100/6. || 3 
 
 j 60/6. II 1 
 
 576/6. 
 
 373.11 2 1 2250/6. — 
 
 
 
 374.11 1 1 259200/6.11 2 |1.47 + /6.1l 3 I 
 
 1.1 + /6.II 4 1 
 
 1.2/??. 
 
ANSWEKS. 
 
 437 
 
 p. 
 
 KX. 
 
 ANS. 
 
 EX. 
 
 ANS. 
 
 375. 
 
 1 
 
 23»z/. 2760/^. 
 
 7 
 
 Gda. iWu: 
 
 375. 
 
 2 
 
 57 GO/"/-. 
 
 8 
 
 131 A 
 
 875. 
 
 3 
 
 21ir. 56 m. 
 
 9 
 
 Gini. 050 5. ^ ft. 
 
 375. 
 
 4 
 
 8fi. 
 
 10 
 
 Shr. \2>n. 12||.5ec. 
 
 375. 
 
 5 
 
 2'3^sec. 
 
 HA 
 
 11 
 
 8m. IG.Gsrc. 
 
 375. 
 
 6 
 
 12 
 
 1 G4.285?m. 
 
 377. 
 
 1 
 
 377. 
 
 377. 
 
 o 
 
 377. 
 
 ^.< 
 
 377. 
 
 o 
 
 o 
 
 377. 
 
 4 
 
 oG9]Ui. 
 23 1 G?/. 
 3Gl8f/if. 
 4821 /■/. 
 223^ 
 21.V.C. 
 
 < ft. 
 
 nearly. 
 
 6 
 
 7 
 
 { 
 
 IGOSl//. 
 
 321 g- velocity. 
 
 2mi. 49S4j^s ft. 
 
 164.G9 ft. 
 100.52/1. 
 
 375. 
 
 10 
 
 678. 
 
 11 
 
 378. 
 
 12 
 
 1447.5//. 
 C1.24SPC. 
 
 13 
 14 
 15 
 
 14.2S+ " 
 15U5011|/i;. 
 23 1 6//. 
 
 380. 
 380. 
 380. 
 380. 
 380. 
 380. 
 
 ] 
 2 
 
 3 
 4 
 5 
 G 
 
 8.857 
 
 -m^c.ft. 
 
 .9S0 
 
 2ft. 11.383m. 
 
 idoT. lom. 
 
 2.75 
 
 7 
 
 8 
 
 9 
 
 10 
 
 U 
 
 — 
 
 7.234 
 .78G 
 .875 
 
 177//;. 5oz. 
 1.103 
 
 
 381. 
 
 381. 
 381. 
 
 12 
 
 13 
 
 4.23/«. 
 (3.172 
 
 \ .504 
 
 14 
 15 
 
 4.57202. 
 
 1418/i. 3.38410S. 
 
 382. 
 382. 
 382. 
 
 1 
 
 2 
 3 
 
 3.497/5. 
 
 37.5/6. 
 
 2.4Go-r. 
 
 4 
 5 
 6 
 
 ..^319 
 
 1 wa 
 
4 S. BARNES & COMPANy's PUBLICATIONS. 
 Page's Thtory and Practice of Teaching. 
 
 THEORY AND PRACTICE OF TEACHING? 
 
 MOTIVES OF GOOD SCHOOL-KEEPING. 
 
 BY DAVID PAGE, A.M., 
 
 LATB PRINCIPAL Or THE STATE NORMAL SCHOOL, SKW YORK. 
 
 •I received a few days since your 'Tlioory and Practice, &c.,' and a capital (A«*r» 
 Bud capital /iractir.e it is. I liMve i\-:u\ it with iiiiiiuiiv'lfcl ilclii;lit. Ev<ii il' I Hhctlld 
 look Ihi-oiigli a critic's microscope, I should iKinlly tiiul a siii'^xle seiiliiiii iil to (li.»«)nj 
 from, and cerlaiiilv not one to coikIuiiiii. The cliapters on I'rizes am! on Ctn;i(iTal 
 Puiii.</iiiir-nt are tniiy adiiiiralile. Tlicy will i^xcrt a niosi salutary iiiflui-ncc. So .,f tho 
 news s/uir.-iiiii oil moral ai;;! rt'li^nous iiislruclioii, wliich you so fanicsily niiil IV-lin^ly 
 liitiisl upon, and yi-l wiilun true Protfstanl limits. It is a okand book, A.sri I ti;ank 
 Ukavkn that yi)I' uav^ written it." — Jluu. Hurace JSlann, istcrUary uf Ihe liuard uj 
 Eduiuittva tn Mamackiisctts. 
 
 "•Were it our businoss to examine teachers we would never dismi?s a candidal* 
 without namiii',' this hook. Other things bcui;,' eipial, we woulil iricatly prf "iT a leachor 
 who has reail it and speaks of it wilh enthusiasm. In one in(liireiei.l to such a work, 
 we should cerlainly have little conlideiice, however he mii,'ht appear in other respecla 
 Would that ever) "teacher employed in Vermont this winter had 'he spirit of this bvjk 
 In Uib bosom, itslessoiis impressed upon his hviuiV— h'crmuiil. Lhroiucle. 
 
 "1 nm plea.«ed with and commenil this work to the attention of scIkhiI teachers, and 
 those who intend to embrace th.it most eslunabie prolession, tor li'.;hl and instruction 
 to g'.lide and jfovern them in the discliaixe of their dehcale and imporUul dulieS."— 
 AT. S. liciUuii, HuyeruileiuLeiil uf Cumiiwn fic/tuols, State of A'cw i'ur.'c. 
 
 Bon. S. Young says, " It is aliogether the best book on this subject 1 have evei 
 seen." 
 
 Frfsident .Vurt!,, of HamiJtnn Col/esre, says, "1 have read it with al! that absorbing 
 self-deiiyins; inu-rest, which in my younger days waa reserved for ticlion mid poetry. I 
 am delighted with the book." 
 
 ifnn. Marcus S. fininnJds says, " It will do pcreat good by showinc; the Tcncher what 
 ibould be his qualificJlious, and what may justly be requiicd and expected of him." 
 
 "1 wish you would send an asent tlirough the several towns of lliis Stcte with 
 Pages 'Theijry and Practice of Teachinu',' or take some other way of Ijrin^ing this 
 falisab'.o book to the nolice of every family and u\ every l';aclier. 1 should be rejoiced 
 to see the principles which it presents as to the motivi-s ami methods of i;oud ?chc«>}- 
 keepiii? carried ul in every school-room : and as nearly as possil)liN in llie style ia 
 wh':3h Mr. Pa^e illustrates them in his own practice, as the devoted and acconiplifhud 
 Principal of your t^tate Normal School."— //c;n-y Barnard^ SupcnnUndciil vf Comnuv 
 Sthooii for the atate uf Rhode Island. 
 
 "The 'Theory and Practice of Teachins,' by D. P. Page, is one of Ihe best ()ookB oJ 
 the kind 1 have ever met with. In it the tlieory and practice of the teacher's duties 
 ere clearly explained and happilv combined. The style is easy and fainihar, and the 
 BUgKcstious it contains are plain, jiraclical, and to the point. To leachers especial!? d 
 wUl furnish very imporUmt aid in dischiu-sing the diitie«,of Jieir hc;h and respuasiiila 
 profeMion."— /eoger 6'. Howard^ Siuieriiitendent of Ck'mnion SchuoU, Uratss I'o.y t'U 
 
A. 8. BARNES AND COMPANV's PUBLICATIONS. 
 No rthend' s Teacher and P arenl, 
 
 A NEW VOLUME FOR THE TEACHEr's LTBRAET. 
 
 THE TEACHER AND THE PARENT: 
 
 A. Treatise upon Common-Scliool Education, containing Practical Sug- 
 gestions to Teachers and Parents. By Charles Northend, A. M, 
 late, and for many years, Principal of the Epes Scliool, Salens. Now 
 Su'Derintendent of Public Schools, Danvers, Mass. 
 
 ■■•"Wfi may anticipate for tbis work a -wide circulation, among teachers and friends 
 of education. The extensive and liigb reputation of its autlior, indeed, will bespeak 
 for it more than pen of ours can do. It is a work of about three hundred and 
 twenty pages, in good size tj'pe, and presents a very plea.'^ant appearance to the eyc^ 
 BS well as the work noticed on the preceding page, both of which, for their neat 
 appearance, do great credit to the enterprising publishers. 
 
 Mr. Northend's book will prove interesting to all, and of great benefit to teach- 
 ers, especially as a chart for those just commencing to engage in the profession. 
 As a r.aOe mecum, it will prove a very ple.asant companion, lor its pagi-s are tilled 
 with the results of a large experience presented in a very pleasing form. We are 
 glad to find that the author, in furnishing to teachers so useful a work, has not 
 neglected the svaviter in modo, and has here and there thrown in a pleasant anec- 
 dote, which will enliven its character, and make it all the more acceptable. Wo 
 shall have frequent occasion to refer to it hereafter. In closing this short notice, 
 we would assure our readers that a perusal of the work will more than realize to 
 them the truth of all we have attempted to say in its favor. Appended to tho 
 volume will be found a catalogue of educational works suitable for tho teachers 
 library." — Massachusetts Teacher. 
 
 "We wish that this interesting and read.ablo volume may find a place in every 
 family, and we are certain that it ought to be on the shelf of every school library in 
 the land."— 5aZem Gazette. 
 
 "It presents a multitude of practical hints, which cannot fail to do good service In 
 enlightening all laborers in the tield of education." — £ostvn Transcript. 
 
 "We unhesitatingly commend this volume of sound, practical, common sense snj- 
 pestions. Every seboid teacher should carefully examine its paces, and ho will not 
 fail— he cannot help receiving — invaluable aid therefrom." — Boxton Atlas. 
 
 "We h.ivo examined this work with care, and cheerfully commend it t? parenti 
 Md teachers. It abounds in judicious advice and sound reasoning, and cannot fail to 
 Impart ideas in the education of children which may be acted upon with the most 
 beneficial results." — Boston Mercantile Journal. 
 
 ""Kiis is an Intelligible, practical, and most excellent treatise. The book is 
 •nlivened with numerous anecdotes which serve to clinch the sood advice given, m 
 well as to keeji awake iho attention of the advised." — Boston Traveller. 
 
 "Tub la a sterling work of groat value. It should be In every fimlljr. A!^ t««oh 
 ITS need just «uch a work." — Boston Olive Branch. 
 
A. 8. BARNES A COMPANY'S PUBI.ICATT0N8. 
 
 Mansfield on American E du ea tion. 
 
 /MERICAN education; 
 
 ITS PRINCIPLES AND ELEIIENTS. 
 
 DEDICATED TO THE TEACHERS OF THE UNITED ST/'l'I.i 
 
 BY EDWARD D. MANSFIELD, 
 Author of ^'•I'olitical G-rammar,'''' etc. 
 
 This work is suggestive of principles, and not intended to point o;-* r, 
 eourse of studies. Its aim is to excite attention to what sliould be tii« 
 elements of an American education ; or, in other words, what are thfl 
 ideas connected with a republican and Christian education in this period 
 of rapid development. 
 
 "The author could not have applied his pen to the production of a book upon a 
 subject of more importance th:m the one he has chosen. We have had occasion to 
 notice one or two new works on education recently, whi^h indicate tliat the attention 
 of authors is beini; directed toward that suljject. We trust that those who occupy tho 
 proud position of teachers of American yoiUli will find much in these worlis, wliich are 
 a sort of inlerchanc;e of opinion, to assist them in tlie discharge of their responsibie d'.ities, 
 
 "The author of the work lietbre us does not point out any particular course of sliulies 
 to be pursued, but conliues hirasell to the consideration of the principles which should 
 govern teachers. His views upon the elements of an American education, and ila 
 bearings upon our institutions, are sound, and worthy the attention of those to whom 
 they are paiticularly addressed. We commend the work to teachers." — Jiuohester 
 Daily Advertiser, 
 
 "We have examined it with some care, and are delighted with it. It discusses the 
 whole subject of American education, and presents views at once enlarged and compr& 
 hensive ; it, in fact, covers the whole ground. It is high-toned in its moral ana 
 religious bearing, and points out to the student the way in which to be a man. It 
 ehoidd be in every public and private library in the country." — Jackson Patrwt. 
 
 " It is an elevated, dignified work of a philosopher, who has written a book on tho 
 Bubject of education, which is an acquisition of great value to all classes of our 
 Countrymen. It can be read with, interest and profit, by the old and young, the 
 educaled and unlearned. We hail it in this era of superficial and ephemeral litera- 
 ture, iLM the precursor of a better future. It discusses a momentous subject; bringing 
 to bear, in its examination, the deep and labored thought of a comprehensive mind. 
 We hope its sentiments may be diffused as freely and as widely throughout our land 
 88 the air we breathe." — Kalamazoo Oazctte, 
 
 " Important and comprehensive as is the title of this work, we assure our readers 11 
 [fi no misnomer. A wide gap in the bulwark of lliis age and this country is greatly 
 lossenetl by this excellent book. In the first place, the views of the author on educa- 
 tion, irrespective of time and place, are of the highest order, contrasting strongly with 
 too groveling, time-seeking views so plausible and so popular at the present day. 
 A leading purpose of the author is, as he says in the preface, ' to turn the thoughts ol 
 those engaged in the direction of youth to the fact, thai it is the entire soul, in all ita 
 fgiculties, which needs education.' 
 
 '•The views of the author are eminently philosophical, and he does not pretend to 
 enter into the details of teaching: but his is a practical philosophy. Iiaviug to do with 
 Ufine, abiding truths, and does not sneer at utility, though it demands a utility thai 
 takes hold of the spiritual part of man, and reaches into bis immortality." — Uolden'i 
 
A. S, BARNES k COMPANY'S PUBLICATICN8. 
 J) e Tocquevill c' s American Ins t itutione. 
 
 AMERICAN INSTITUTIONS AND THEIR INFLUENCE. 
 
 BY ALEXIS DE TOCQUEVILLE. 
 
 WITH NOTES, BY HON. JOHN C. SPENCER. 1 vol. 8vo. 
 
 This book is the first part of De Tocqueville's larger work, on the Repiblio ot 
 America, and is one of llie most valuable treatises on American poiiiics that has eTM 
 t>i»n issued, and should be in every library in the land. The views of a Hbarai- 
 miaded and enlightened European statesman upon the working of our coiuitry's socia) 
 said political establishments, are worthy of attentive perusal at all times; those of a raa» 
 lite Ue Tucqueville have a higher iuti'insic value, from the fact of his residence among 
 the people he describes, and his after position as a part of the republican government 
 of France. Tlie work is enriched likewise with a preface, and carefully prepared notes, 
 by a well-known American statesman and late Secretary of the Navy. The book is on« 
 of great weight and inleresi, and is admirably adapted for the district and school library 
 as well as that of the private student. It traces the origin of the -Anglo-American* 
 treats of their social condition, its essential democracy and [lolitical consequences, tb# 
 •overeignty of the people, etc. It also embraces the author's views on the .Ameriaii 
 system of townships, counties, &c. ; federal and state powers; the judiciary ; llie cod 
 Blitution ; parties; the press ; American sociely ; jjower of the mijority, its tyrann\ 
 and the causes which mitigate it ; trial by jury; religion; the three races; the arista 
 cratic party ; causes of American commercial prosperity, etc., etc. The work is ai 
 epitome uf the entire political and social condition of the United Slates. 
 
 "M. De Tocqueville was the first foreign author who comprehended the genius c» 
 our institutions, and who made intelligible to Europeans the complicated ra.ichliierj- 
 wheel within wheel, of the stale and federal governments. His ' Democracv ic 
 America' is ackiiowled'-'ed to be the most profound and philosophical work iipor 
 rnodern reiiublieanlsm that has yet appeared. It is characterized by a rare uniiiii o 
 discenimenl, rellecliiin, and candor; and though occasionally tinged with the aiilhor'f 
 pecuiiarilies of education and faith, it may he accepted as In the main a just and iu> 
 partial criticism upon the social and political feiitures ol the United Slates. The pui> 
 lishers have now sought to adapt it as a te.\t-book for higher seminaries of learninji 
 Fi'r this purpose they have published the lirsl volume as an independent work, ihut 
 avoiding the auihur's speinilalions ujiDn our social habits and reli'-rioiis cnmlitinii. Thi» 
 volume, however, is unmiililaled — I he author ia left throu^'hout to speak for himself; bill 
 where at any pnjnl he had misappiehended our sjstem. the detect is sujiplied by iiule! 
 or para:,'raphs in brackets from the pen of one inosi thoroughly versed in the hislnri 
 the legislalioii, the adininislration, and the jurisprudence of our counlry. This woi^k 
 will supply a felt deliciency in the educational apparatus of our higher schools. Kveri 
 man wlio pretends lo a good, and much more to a liberal ediic;ition, should ma.-;i i th'f 
 5)rinciples ami philosophy of the inslitutionsorbis coiintiy. In the hands of ajuu.ciou* 
 teacher, Hits volume will be an adniir.ible te\t-bi)ok." — The huUjicndcnl. 
 
 '■' Having had the honor of a personal acqiiaiiitance with .M. De Tocqueville while h# 
 was in this country ; having discussed with him many of the topics treated of in Ibb 
 bo<jk ; having entered deeply into the feelings and sentiments which guided and iu> 
 pelled him In his Uisk, and having formed a hi;;h admiration of his chaia-uer and rf 
 this production, the editor felt under some obligation lo aid in procuring' for one whoix 
 he ventures to call his friend, a hearini; from Ihose who were Ihe olijecl-s of his ob' 
 »ervalions.' The notes of Mr. Spencer will be found to elucidate occasional niiacc* 
 ceplions of the transl.iior. It is a most judicious text-book, and ought lo be reivC 
 wrefully by all who wish to know this country, and to trace its power, position, an^ 
 nllimate deslit/y from the true source of philosophic governmi'iii, Ivejuiblicanism — th« 
 people. De Toc<iueville, believing the destinies olcivilizalion lo depend on the p<.wer 
 of the jx'ople and on the principle which so ^'randly founded iin exponent on this roB 
 linenl, analyzes with jealous care and peculiar critical acumen the tendencies of lh« 
 DOW Democracy, and candidly gives his approval of the new-born giaiil, or pointt 
 out ami warns him of daimers which his faillifiil and independent philosophy foreseen 
 We birlieve the perusal ot his observations will have Iho ell'ecl of enhancing still nior* 
 lo •■ is Ainerii:an readers the stiiiclure ol their governnifml, by the clejir and proloiin/ 
 W^-'e in which !.«* prfajtiitsi U.^^./'/mcncun Jircteu. 
 
Davies' System of Mathematics. 
 
 DAVIES' LOGIC OF MATHEMATICS. 
 Tlie Logic and Utility of Mathematics, with the best imthods of Inatruo 
 tion, explained and illustrated. By Chaules Davies, L. L. D. 
 
 "One of the most remarkable books of the month, is ' The Lo;,'ic and Utility of 
 Malhemiitics, by Cb;irles Davies, 1,. L. 1).,' pnlili'^lied by Barnes &. Co. ll is not iu- 
 tendi'd as a treatise on any special br.incli of niailieinatical science, ami demani's fm 
 its full a|i|>recialiun a general actjiiainlance with the leading iiictbods ami routine of 
 niatliemalical invesiiijation. 'I'o tho e who have a natural fondness lor this (iiirsuit 
 and enjoy the leisure for a retrospect of their fivorite studies, the pre.sent \olunie wil 
 possess a charm, not surpassed by tiie (asciiiations of a romance, it is an efiliorale 
 and Iticid exposition of the principles which lie at the foundation of pure malliematicu 
 with a hit;hly inuenions application of their results to the development of the essen- 
 tial idea id' .Arilhnietic. (Jeinoelry, Al'.'i'bra, .-Xnalytic (Jeometry, and the DuiiTi-ntial 
 and Inteural Calculus. The work 'is preceded by a ;,'eneral view of the subject of Lo;;ic, 
 mainly drawn from the writinj:s of Arclibisbop Whately and Mr. Mill, and closes «ith 
 an essiiy on the utility of mathematics. Some occasional e.xa^'geralions, in presenting 
 the clai'ms of the science to which his life has been devoted, UiUst here be pardoned 
 to the professional enthusiasm of the author. In general, the work is wriitea with 
 s'.ngnlar circuniS(jection ; the views of the best thinkers on the subject have been 
 thorouuhly digcaled, and are presented in an orij.'inal form; every thin<; bears the im- 
 press of llie intellect of the writer ; his style is for the most part chaste, simple, trans- 
 parent, and in admirable harmony with the dif;nity of the subject, and his condensed 
 generalizations are often profound ai.d always suj^gestive."— i/ar//er's A'cic jUu«£Aij 
 Maj;aiinc. 
 
 "This work is not merely a mathematical treatise to be used as a te.\t book, but a 
 complete and philosophical unfolding of the principles and truths of luathematical 
 science. 
 
 " ll is not only designed for professional teachers, professional men, and students ol 
 mathematics and philosophy, but for the general reader who desires mental improve- 
 meni. and would learn to search out the miport of language, and ac(iuire a habit of 
 notiiit; of conni'xion between ideas and their signs; also, of the relation of ideas to 
 each other. — The Student. 
 
 " Students of the Science will find this volume full of useful and deeply interesting 
 matter." — .Itbany Evening Juuruixl. 
 
 " Seldom have we opened a book so attractive .is this in its typography and style ol 
 eiccution ; and there is besides, on the margui opposite each section, an index of the 
 subject (d" whicli it treats— a great convenience to the student. I!ut the mutter is no 
 less to be commended than the manner. And we are very much mi-taken if ibis wurk 
 sliall not prove mnre popular and more useful than any which the distinguished author 
 has given to tlie public."' — Lutheran Observer. 
 
 " We have been much interested both in the plan and in the execution of the work. 
 Iinil would recommend the study of it to ilie tlieologian as a discipline in close ami 
 Kcurate tliinking. aMi in logical nielhod ar.d reasoning. It will be useful, al-o, to thf 
 general scholar and to tlie practical muchanic. We w(Uild s|iecially rcconiiiiend it M 
 those who would have nothing tauglit in our Free .Acailemy and other liiglier inslitu 
 Uons but what is directly 'practical'; nowhere have wa seen a finer iiluslraliun o< 
 the eoiuieetion between the ab.straetly scieiitific and tlie praclic.-il. 
 
 "The work is divided into three books; llie first of wliich treats of Logic, mainly 
 ipon tlie liasis of Whately; the second, of Matlieuiatical Science; aud Iho third, r f the 
 Dtility of },liilhemixtic3."—Incl^2^endent. 
 
 "Tlie authoi's style is perspicuous and concise, and he exliibits a in.istery of the 
 ibstni.se topics which he attempts to .simplify. For the inatbeiiiatical student, ^^■!l^ 
 desires an analytical kiunvled^'e of the science, and who would begin at the beginning. 
 we .should suppose the work would have a special utility. I'rof. Davies" matbemati- 
 cal works, we liolieve, have become quite popular with educators, and this di.selo.soj 
 qidte .IS much reasearch aud practical scliolarship a.s any we havp seen from Ids p^u ' 
 
 •A>i(i- Yi }'k EvangelUst 
 
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