IN MEMORIAM 
 FLORIAN CAJORI 
 
THE 
 
 AMERICAN 
 
 PHILOSOPHICAL ARITHMETIC 
 
 DESIGNED FOR THE USE OP ADVANCED CLASSES 
 IN 
 
 SCHOOLS AND ACADEMIES; 
 
 CONTAINING 
 
 THE ELEMENTARY AND THE MORE ADVANCED PRINCIPLES OF 
 
 THE SCIENCE OF NUMBERS, AND THEIR APPLICATIONS 
 
 TO PRACTICAL PURPOSES, 
 
 TOOETHER 
 
 WITH CONCISE AND ANALYTIC METHODS OF SOLUTION, AND 
 
 ABBREVIATED METHODS OF COMPUTATION. 
 
 BY 
 
 JOHN F. STODDA.ED, A.M., 
 
 PRINCIPAL OF THB 
 
 LANCASTER COUNTY, NORMAL SCHOOL, PA. 
 
 AIJTHOR OF 
 
 *• THE JUVENILE MENTAL," " THE AMERICAN INTELLECTUAL," AND 
 " THE practical" ARITHMETICS, " READY RECKONER," EtC. 
 
 NEW YORK: 
 SHELDON, BLAKEMAN & CO. 
 
 IBXLADKLPHU, UPPLN-COTT, GRAMBO Jk CO. BOSTON, JOHN P. JEWETT & CO. 
 
 BUFFALO, PHIXXEY k CO. CI.KAELAXD, KXIGDT, KING k CO. a.XCIN- 
 
 NATI, APPU:GATE k CO. ClKa.E\TliE, A. BEACH i CO. CHICAGO, 
 
 KEENK k BRO. OT. LOITS, E. K. WOODWARD, 
 
 AND KETfH k WOODS. 
 
Entered according kt Act of Congi-3ss, in the year IKM. or 
 
 JOHN F. STODDARD, 
 
 It. th« Clerk's Office of the District Court of the United States, for tha 
 
 Southern District of New York. 
 
JT- 
 
 PEEF ACE 
 
 *♦» 
 
 I HAVE attempted in tMs work to present 
 CLEAELY and CONCISELY, all those important 
 principles dindi prop^erties of numbers, wMcli are 
 necessary to a full comprehension of the higher 
 branches of Mathematics, and their application 
 to practical business and scientific calculations. 
 The arrangement and illustration of these prin- 
 ciples will enable the pupil to think for himself, 
 independent of arbitrary rules, and summon to 
 his aid on the instant all his arithmetical 
 knowledge. This work treats extensively of 
 mercantile transactions, interest, trade and 
 bai-ter, abbre^dated arithmetical calculations, 
 <fec., and contains several principles not here- 
 tofore published, of extensive and practical 
 application. 
 
 A Key containing additional methods of 
 analysis, and answer's, (if deemed indispensable,) 
 will be prepared for the use of Teachers. 
 
 J. F. S. 
 
 UNivzRsmr of Nokthern Pennsylvaioa, I 
 July Uh, 1853. f 
 
Digitized by the I ntferhet Archive 
 
 in 2007 with funding from 
 
 IVIicrosoft Corporation 
 
 http://www.archive.org/details/americanphilosopOOstodric 
 
STODDARD'S 
 PHILOSOPHICAL ARITHMETIC. 
 
 CHAPTER I. 
 
 INTRODUCTORY DEFINITIONS. — NOTATION. NUMERATION. 
 
 Article 1. Science is knowledge systematized ; that 
 is, knowledge so classified and arranged as to be conve- 
 niently taught, easily acquired and remembered, readily 
 referred to, and advantageously applied. 
 
 Art. 2. Art is a judicious application of science to 
 practical purposes. 
 
 Art. 3. A Unit, or Unity ^ is a single thing ; as, a tree, 
 an apple, a boy, &c. 
 
 Art. 4. A Number is either a unit or composed of an 
 assemblage of units. One, two, three, &c., are numbers. 
 
 Art. 5. .Quantity is anything that will admit of meas- 
 urement. A line, a surface, time, and other things of this 
 nature, are quantities : but imagination, reason, virtue, 
 &c., are not quantities, therefore, they are not subjects of 
 mathematical investigation. 
 
 In common language, quantity or numbers are expressed 
 by the words 07ie, two, three, four, &c.; in arithmetical 
 operations, by characters called Figures. 
 
 Art. 6. Mathematics is the science that treats of the 
 properties and relations of quantity. Its fundamental 
 branches are Arithmetic, Algebra, and Geometry. 
 
 Art. 7. Arithmetic is the science of numbers, and the 
 art of computation by them. It treats theoretically and 
 practically of the nature and properties of numbers as 
 employed in calculation. 
 
6 NOTATION. [chap. I 
 
 Art. 8. Algebra is a general method of investigating 
 the relations of quantity, by means of letters and signs, or 
 
 Art 9. Geometry is the science of magnitude ; it esti- 
 mates and compares extension and form. 
 
 Art. 10, An abstract number is one that does not 
 refer to any particular denomination ; as, one, six, ten, 
 "•Jive hundred, &c. 
 
 Art. 1 i. A concrete, or denominate number is one that 
 refers to some particular thing or denomination; as," four 
 apples, ten dollars, &c. 
 
 Similar concrete numbers express the same kind of units; 
 as, ten dollars &ud Jifty dollars. 
 
 Dissimilar concrete numbers express different kinds of 
 units ; &sjlve dollars, ten horses. 
 
 NOTATION. • 
 
 Art. 1 2. Notation is the art of expressing numbers by 
 figures, letters, or other symbols. 
 
 The Romans used the seven following . letters to ex- 
 press numbers, which we now use to number Lessons, 
 Chapters, &c. : — I, for one ; Y, for five ; X, for ten ; 
 L, for fifty; C, for one hundred ; D, for five hundred ; 
 M, for one thousand. The intermediate numbers and 
 numbers greater than a thousand are expressed by repeti- 
 tions and combinations of these letters, as exhibited in the 
 following 
 
 ROMAN table. 
 
 One is represented by I. 
 Two " "II. 
 
 Three " " III. As often as a letter is re- 
 
 Four " " IV. peated, its value is repeated, 
 
 pjyg « « Y^ Thus : X, is ten , XX, twen 
 
 Six " " yi. 
 
 Seven " " YII. 
 
 Eight " " VIII. 
 
 Niao " " IX. 
 
 ty, &,c. 
 
ART. 14.] 
 
 
 NOfATION. 
 
 Ten is represented by X. 
 
 Eleven 
 
 (< 
 
 XI. 
 
 Twelve 
 
 ti 
 
 XII. 
 
 Thirteen 
 
 it 
 
 XIII. 
 
 Fourteen * 
 
 It 
 
 XIY. 
 
 Fifteen 
 
 11 
 
 XY. 
 
 Sixteen 
 
 (( 
 
 XYI. 
 
 Seventeen 
 
 (( 
 
 XYIL 
 
 Eighteen 
 
 ti 
 
 XYIII. 
 
 Nineteen 
 
 11 
 
 XIX. 
 
 Twenty 
 
 n 
 
 XX. 
 
 Thirty 
 
 tl 
 
 XXX. 
 
 Forty 
 
 n 
 
 XT,. 
 
 Fifty 
 
 tl 
 
 L. 
 
 Sixty 
 
 tl 
 
 LX. 
 
 Seventy 
 
 tt 
 
 LXX. 
 
 Eighty 
 
 tl 
 
 LXXX 
 
 Ninety 
 
 It 
 
 xc. 
 
 One hundred 
 
 tt 
 
 c. 
 
 Five hundred 
 
 tt 
 
 D. 
 
 One thousand 
 
 tt 
 
 M. 
 
 Five thousand 
 
 tt 
 
 Y. 
 
 A letter of less value 
 placed before one of greater, 
 diminishes its value as much 
 as the value of the letter 
 placed there ; if placed after 
 the same letter it increase? 
 its value by the same num- 
 ber. Thus, if before X, ten, 
 we place I, one, it becomes 
 IX, nine ; if after, it be- 
 comes XI, eleven. If before 
 L, fifty, we place X, ten, it 
 becomes XL, forty ; if after, 
 it becomee LX, sixty, &c. 
 
 A bar ( — ) placed over 
 any letter increases its value 
 a thousand fold. Thus, IV 
 is four thousand. 
 
 ARABIC NOTATION. 
 
 Art. 13. In all arithmetical calculations numbers are 
 expressed by the Arabic system of Notation. This system 
 of notation employs the following ten characters, called 
 Figures : 
 
 1, 2, 3, 4, 5, 6, 1,- 8, 9, 0. 
 one, two, three, four, five, six, seven, eight, nine, cipher, 
 
 zero, or 
 naught. 
 Hence, figures are representatives of numbers, or expres- 
 sions of quantity. 
 
 Art. 14. The first nine of the above characters, are 
 called significant figures, as each one represents a definite 
 number when standing alone, while the last has no value, 
 and is, therefore, of itself insignificant. 
 
8 NUMERATION. [?HAP. I. 
 
 Art. 15. The significant figures are also called Digits, 
 from the Latin digitus, a finger, because, many centuriea 
 ago, people used to do their reckoning by counting their 
 fingers. The use of the ten fingers is supposed to have 
 originally suggested the idea of employing ten characters 
 to express numbers. 
 
 Art. 16. Notwithstanding the 0, cipher, has no value 
 of itself, yet it is of as much importance as either of the 
 digits, as it serves in a peculiar manner- to change their 
 value by changing their locality ; hence, by some it has 
 been called a Locater. 
 
 NUMERATION. 
 
 SIMPLE AND LOCAL VALUES OF FIGURES. 
 
 Art. 17. A figure standing alone, or occupying the 
 first place on the right of a row of figures, expressing a 
 whole number, is denominated units ; that occupying the 
 second place, tens ; that occupying the third place, huii- 
 dreds, &c. 
 
 Thus, Hds. Tens. Units. 
 
 4 2 8 
 is read four hundreds, two tens, and eight units, or four 
 hundred and twenty-eight. Nine is the largest number 
 that can be expressed by a single figure ; hence, numbers 
 greater than nine must be expressed by some combination 
 of these numerals. 
 
 Art. 18. The simple value of a figure is its value when 
 occupying unit's place. 
 
 The name of a figure expresses the -simple value of that 
 figure. Each of the nine digits as referred to in Art. 13, 
 has its simple value. In the number 35, the five has its 
 simple value, fii^e, and the three a local value. 
 
 Art. 19. The local value of a figure is that which 
 arises from its location. In the number 35, (above re- 
 ferred to,) the local value of the three is three tens, or 
 thirty In the number 465, the local value of the 4 is 
 
ART. 20.] NUMERATION. 9 
 
 four hundreds, aud that of the 6 is six tens, or sixty. The 
 5 has its simple value, Jive. 
 
 Art. 20. When the nine digits occupy the second, or 
 ten's place, each will then express the same number of tens 
 that it did units, when occupying the place of units. When 
 they occupy the third, or hundreds' place, each will then 
 express as many hundreds as it did units when in the 
 place of units. 
 
 Note. — This will be rendered plain by inspecting the following 
 TABLE. 
 
 H ■ 
 
 No units and one ten, or ten, ^10 
 
 One unit " " " " eltve7L, ^11 
 
 Two units " " " " tioelve, 12 
 
 Three " " " " " thirteen, 13 
 
 Four " " " " " fourtcm, 14 
 
 Five " " " " " fifteen, 15. 
 
 Six " " " " " sixteen, 16 
 
 Seven " " " " " seventeen. It 
 
 Eight " " " " " eighteen, 18 
 
 Nine " " •' " " nineteen, 19 
 
 No " " two " " twenty, 20 ^ 
 
 Note. — The terms thirteen, fourteen, fifteen, sixteen, &c., are obviously 
 contractions of three and ten, four and ten, five and ten, six and ten, &c. 
 
 In a similar way, by contracting the expressions two tens, three tens, four 
 tens, &o., the expressions twenty, thirty, forty, &c., are derived. 
 
 Twenty-one, Twenty-two, Twenty-three, Twenty-four, 
 Twenty-five, Twenty-six, Twenty-seven, Twenty-eight, and 
 Twenty-nine, are respectively expressed by .placing in 
 regular order the digits in the place of the cipher 
 in the number 20. In a similar manner, numbers from 
 Thirty to Forty, from Forty to Fifty, from Fifty to Sixty, 
 &c , are expressed by placing the digits in the place of 
 the cipher in the numoers Thirty, Forty, Fifty, Six* 
 ty, &c. 
 
10 
 
 NUMERATION. 
 
 fcHAP. I. 
 
 No units and 
 
 three 
 
 four 
 
 five 
 
 six 
 
 seven 
 
 eight 
 
 nine 
 
 tens, or 
 
 Thirty, 
 
 Forty, 
 
 Fifty, 
 
 Sixty, 
 
 Seventy, 
 
 Eignty, 
 
 Ninety, 
 
 HP 
 30 
 
 40 
 50 
 60 
 70 
 80 
 90 
 
 The terms twenty-one. twenty -two, &c., are comi)ounded of twenty and one, 
 twenty and two, &c. Other numbers expressed by two figures are similarly 
 formed. 
 
 One hundred, 
 
 Two " 
 
 Three " 
 
 Four " 
 Five 
 Six 
 
 Seven " 
 
 Eight " 
 
 Nine, " 
 
 100 
 200 
 300 
 400 
 500 
 600 
 
 too 
 
 800 
 900 
 
 Art. 21. By inspecting the above table, it will be 
 observed, that a figure standing in the second place, or 
 place of tens, is ten times as great as though it were in the 
 first or units' place. A figure that stands in the third 
 place, or place of hundreds, is ten times as great as though 
 it were in tens' place, and one hundred times as great as 
 though it were in the place of units. Hence, ten units 
 make one ten, and ten tens make one hundred. We 
 therefore infer universally, that 
 
 Art. 22. Figures increase in value from right to left in 
 a tenfold ratio ; that is, each removal of a figure, ove place 
 towards the left increases its value ten tim£s. 
 
 Art. 23. As figures in the Arabic system of notation 
 increase in ten-fold ratio from right to left, and decrease 
 in the same ratio in an opposite 'Erection, it is called the 
 Decimal system of notation. The ^ord decimal is derived 
 from the Latin, lecem, ten. 
 
ART, 24.] KUMERATION. 11 
 
 Art. 24, Numeration is the art of reading numbers, 
 expressed by fig-ures. 
 
 Note. — By carefully studying the following Table, the pupil will soon he 
 able to read any number which requires not more than nine figures to ex- 
 press it. 
 
 TABLE. 
 
 •1 C ^. 1 
 
 <U M 3 ♦- J3 
 
 5 s 
 
 _ Figures occupying the place of units, are 
 
 i ^ o '^ S - sometimes called units of the first order, 
 
 ^ iS ° 2 2 — those occupying the place of tens, vnits 
 
 3 2 2 '^ of the second order, — those accupying the 
 
 t ^^ ° ^ place of hundreds, units of the third order , 
 
 ~ ^ &c., as shown by the Table. 
 
 S -^ w- -S ^ "I -3 r' S '^^ read these numbers, first denominate 
 
 £o g S^o |ti®° each figure by the names units, tens, &c., 
 
 c c 3 c c c c S .^ as shown by the table, and then read from 
 
 " i K ^ H = H J§ left to right as follows :~ 
 
 .... 4 Four. 
 ... 4 5 Forty-five. 
 ..453 Four hundred and fifty-three." 
 .4532 Four thousand fi.ve hundred and thirty- two 
 4 5 3 2 6^ Forty-five thousand three hundred and twen- 
 l ty-six. 
 4 5 3^67^ Four hundred fifty-three thousand two hun- 
 ( dred and sixty-seven. 
 4 5 32678^ Four millions, five hundred thirty-two thou- 
 ( sand, six hundred and seventy-eight. 
 45326781^ Forty-five millions, three hundred twenty- 
 \ six thousand, seven hundred and eighty-one 
 C Four hundred fifty-three millions, two hun- 
 453267819< dred sixty-seven thousand, eight hundred 
 C and nineteen. 
 
 Rkmark.— It would be well to write the figures of this Table on the black 
 board, and have the pupils read them individually as well as collectively. 
 
 This Table shows plainly the simple and local values of figures. Each 
 figure, except those in the place of units, has a local value, which may b« 
 named by th« pupil an th« taach«r points to tham separataly. 
 
12 
 
 NUMERATION. 
 
 [CHAI. I. 
 
 Art. 25. In the United States and continental Eu- 
 rope the French method of numeration is in general use. 
 In this method of numeration a different name is given to 
 every three figuers, counting from the right. 
 
 The first period contains units, tens of units, hundreds 
 of units, and is therefore called the period of Units. For 
 a similar reason the next left hand period is called the 
 eriod of Thousands, &c. 
 
 Art. 26. In the following Talle the words above the 
 row of figures express the particular denomination of the 
 figures over which they are placed. 
 
 To read a number expressed by figures : — 
 Denominate each figure from right to left, remembering the 
 name of each period, then read the figures of each period, he- 
 ginning at the left hand, in the same manner as those of the 
 period of units are read, and at the end of each period give 
 its name. 
 
 FRENCH METHOD OF NUMERATION. 
 
 S.2 
 
 m O; 
 
 3 0) 4i 
 
 7 i 3. 
 
 =! ^5 .E-^ 2- .^o |=§ =. 
 
 -::: ><■ 
 
 QfB 
 
 o •<— .S .^ .° .^7^ ■'^^zi -t^c 
 
 ^ wo—; "J'-Z:2 wO-rr tnOu »CO 
 
 S3 . s; 
 
 o c « 
 
 4, 6 
 
 o -'<u;r'3a>m 30)3 3(u3 3ais-' 3a)r;; 3 
 3, 8 5 7, 8 6 2, 7 3, 4 1 0, 2 6, 3 7 4, 3 
 
 "2 " 
 
 £■« 
 
 ilil 
 
 ; S E-i r-' 
 8 4 7. 
 
 6 2. 
 
 Period Period Period Period Period Period Period Period Period Period Period Pericd 
 of De- ofNo- ofOc- of Sep- of Sex- of Quin-ofQuad- of Tril of Bil- of Mil- of Thou- of 
 cUliona. nillions. tillions. tillions. tillions. tillions. rillions. lions, lions, lions. t»nda. UniU. 
 
 EXERCISES IN NUMERATION. 
 
 Kead the following numbers : — 
 
 Ex. 
 
 
 Ex. 
 
 1. 
 
 125 
 
 6. 
 
 2. 
 
 286 
 
 7. 
 
 3. 
 
 397 
 
 8. 
 
 4. 
 
 8462 
 
 9. 
 
 5. 
 
 98623 
 
 10. 
 
 
 Ex. 
 
 
 63245 
 
 11. 
 
 836478492 
 
 732123 
 
 12. 
 
 326489472 
 
 8324671 
 
 13. 
 
 8246721478 
 
 1247632 
 
 14. 
 
 448624783146 
 
 463246273 j 15. 12345678901234 
 
ART. 28.] FUNDAMENTAL RULES. 13 
 
 EXERCISES IN NOTATION. 
 
 Art. 27. To express numbers by figures . — 
 
 Begin at the left, and write the figures of the highest 
 order mentioned, observing to place in each order, the figures 
 belonging to it, and ichen no digit is mentioned, to fill the 
 flace with a cipher. 
 
 Express the following numbers by figures : — 
 
 1. Forty-three. 
 
 2. Eighty-nine. 
 
 3. Three hundred and eight. 
 
 4. Four thousand, one hundred and four. 
 
 5. Seventy-five thousand and seventy-five. 
 
 6. Six hundred and five thousand, one hundred and 
 twenty-three. 
 
 7. Eight hundred and seventy-two thousand, five hun- 
 dred and twelve. 
 
 8. Nine millions, seven hundred and sixty-five thousand, 
 four hundred and thirty-two. 
 
 9. Three hundred and forty millions, forty-three thou 
 sand, five hundred and sixty-seven. 
 
 10. Three hundred and seventy-four billions, four hundred 
 and thirty-eight millions, eight hundred and sixty-two 
 thousand, eight hundred and forty-seven. 
 
 FUNDAMENTAL RULES OF ARITHMETIC. . 
 
 Art. 28. JVotation aud IVumeration are the Frimarj -pnn- 
 ciples of the four Fundamental llules of arithmetic; namely, 
 Addition, Subtraction, Multiplication,' and Division. 
 
 These are called Fundamental Rides, because all other 
 arithmetical operations are dependent on them. 
 
 A Rule, in Arithmetic, is a prescribed method of per- 
 forming an Arithmetical operation. 
 
14 ADDITION. [chap. )l. 
 
 CHAPTER II. 
 
 ADDITION, SUBTRACTION. MULTIPLICATION. DIVISION. 
 
 AUDITION. 
 
 Art. 29. Addition is the process of finding the sum 
 of two or more numbers. 
 
 The sign of addition is a shori horizontal line bisected 
 by a perpendicular line of the same length; as, +• This 
 symbol is called plus, and when placed between two quan- 
 tities, it denotes that they are to be added. Thus, 4 + 2 
 show that four and two are to be added; and is read,/oi^r 
 plus two. 
 
 Two parallel horizontal lines, as =, means equal to or 
 eqiials, and when placed between two quantites, denotes 
 that they are equal to each other. Thus, 4 -f 2 = 6, is read, 
 4 plus 2 equals 6. 
 
 CASE I. 
 
 Art. 30. Addition of abstract numbers when the sum 
 of each column does not exceed nine. 
 
 Rkimark. — Similar concrete Mumbers are added the same as though they were 
 abstract numbers, the amount being a concrete number of the same kind. 
 Disiimilar concrete numbers cannot be added. 
 
 1. What is the sum of 223, 451 and 114 ? 
 
 Explanation. — Write the numbers 
 OPERATION. to be added, so that ;dl the figures of 
 
 ^ the same denomination shall stand in 
 
 ^ ^^ m ' the same column; and draw a line 
 
 § g 3 underneath. Then commencing at 
 
 ^ ^ ^ the column of units, add each column 
 
 ^ , ^ separately, and phice the result direct- 
 
 ^ ^ ^ ly under it. Thus, 4 and 1 are 5, 
 
 ■*■ and 3 are S. units, which place under 
 
 _ „~ the column of units. Add the column 
 
 7 8 8 Sum or Amount. ^^^^^^^^ ^^^ of hundreds in a similar way, 
 and we obtain for the amount 788. 
 Proof. — Begin at the top and add eacli column down- 
 ward, the same as you added them upward, if the sunaa 
 ajjreo the work is right. 
 
T. 30.J 
 
 ADDITION. 
 
 
 
 2. 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 134 131 
 
 171 
 
 315 
 
 413 
 
 112 
 
 512 413 
 
 403 
 
 481 
 
 142 
 
 221 
 
 342 245 
 
 315 
 
 202 
 
 234 
 
 ' 343 
 
 8. 
 
 
 9. 
 
 
 10. 
 
 1142 
 
 
 4131 
 
 
 10465 
 
 3314 
 
 
 1512 
 
 
 43110 
 
 2001 
 
 
 3024 
 
 
 22322 
 
 1330 
 
 
 201 
 
 
 13101 
 
 15 
 
 11. What is the sum of 3141,1202 and 2382 ? 
 
 12. What is the sum of 1674,3102,4011 and 112 ? 
 
 13. What is the sum of 12132,41311,23323 and 1101? 
 
 14. What is the sum of 3421,30124,313221 and 1232 ? 
 
 15. What is the sum of 1210 + 32124 + 613253 + 
 2110301 + 2101? 
 
 16. What is the sum of 1012 + 32421 + 613352 + 
 2110103 + 2011 ? 
 
 17. What is the sum of 3121 + 21 + 1603 + 1032 ?, 
 
 18. What is the sum of 413 + 32132 + 32 + 4220 ? 
 
 19. What is the sum of 12 + 3430 + 40 + 64213 + 104 ? 
 
 20. What is the sum of 12 + 321 + 4231 + 821423 + 12? 
 
 PRACTICAL QUESTIONS. 
 
 1. If a yoke of oxen is worth 125 dollars, and a cow 32 
 dollars, what is the value of both ? 
 
 2. A man bought a load of hay for 5 dollars, a load of 
 wheat for 41 dollars, and some rye for 323 dollars; what 
 was the w^hole cost ? 
 
 3. A farmer bought a span of horses for 212 dollars, a 
 yoke of oxen for 132 dollars, and farming implements to 
 the amount of 545 dollars; what was the whole cost ? 
 
 4. A merchant sold 2131 barrels of flour one month, 
 11023 barrels the next month, and 6022 barrels the follow- 
 ing month ; how many barrels did he sell during the three 
 months ? 
 
16 ADDITION. [chap. II. 
 
 5. A man bought some butter for 123 dollars, some 
 molasses for 310 dollars, some sugar for 1101 dollars, and 
 some flour for 1121 dollars; what was the^ whole cost ? 
 
 6. James has 2312 acres of land, John has 21321, and 
 Joseph has 32154; how many acres have they all ? 
 
 t. A farmer, being asked how many sheep he had, re- 
 plied, in one field I have 225, in another 2112, in another 
 1220, and in another 10120; how many had he in all ? 
 
 8. A farmer raises the following qunatities of grain oa 
 four fields, namely : on the first 2115 bushels of wheat, on 
 t^e second 1110 bushels of rye, on the third 625 bushels 
 of oats, and on the fourth 123 bushels of buckwheat; 
 how many bushels of grain did he raise ? 
 
 9. A lends to B 1313 dollars ; to C 23121 dollars, and 
 has 55125 dollars remaining ; — how much money had A 
 at first ? 
 
 10. A man bought a farm for 4120 dollars, paid 1400 
 dollars for having it improved, and sold it so as to gain 
 1150 dollars ; for how much did he sell it ? 
 
 - 11. A man traveled 214 miles one day, 232 miles the 
 next day, and 320 the third day ; how far did he travel in 
 the three days ? 
 
 12. Mr. Smith owned five farms ; the first was wof-th 
 23520 dollars, the second 11120 d-ollars, the third 3200 
 dollars, and the fourth 32100 dollars ; what is the value of 
 the four farms ? 
 
 13. A drover bought cattle to the amount of 3100 
 dollars, sheep to the amount of 642 dollars, and a fine 
 horse for 255 dollars ; how much did they all cost ? 
 
 14. A merchant bought groceries to the amount of 3210 
 dollars, dry goods to the amount of 12210 dollars, and had 
 32216 dollars remaining ; how much had he at first ? 
 
 15. A had 2310 dollars, B 13250 dollars, C 32118 dol- 
 lars, and D 321 dollars ; how much did they together 
 have ? 
 
 16. A merchant, on setthng up his business, found he 
 owed one man 12326 dollars, another 412 dollars, another 
 3141 dollars, another 821010 dollars ; what was the 
 amount of his debts ? 
 
ART. 31.] ADDITION. It 
 
 11. A has 2113413 dollars ; B, 534231 dollars ; C, 343 
 dollars ; and D, 85241002 dollars ; — how many dollars 
 have they together ? 
 
 18. In one book there are 1210 pages, in another 235, 
 and in another 1140; how many pages did the three books 
 contain ? 
 
 19. A merchant bought books to the amount of 1111 
 dollars, paper to the amount of 2231 dollars, and dry 
 
 •goods to the amount of 23225 dollars; how much did the 
 whole cost ? 
 
 20. A man has a farm worth 1522 dollars, a mortgage 
 worth 23134 dollars, and 5222 dollars of bank sock; how 
 much rs he worth ? 
 
 CASE II. 
 
 Art. 3 1 . Addition of abstract numbers in general. 
 
 1. What is the sum of 431t + 346 + 59 + 6831 + 2194 
 + 3285? 
 
 Explanation. — Write the numbers to be 
 added as directed in Art. 30. 
 
 Begin at units' column and add thus : 5 
 and 4 are 9, and 1 is 10, and 9 are 19, and 
 6 are 25, and 7 are 32 units, — equal to 3 
 TENS and 2 units ; — place the 2 units under 
 the units' column, and car-y, or add, the 3 
 tens to the tens' column, thus : 3 and 8 are 
 11, and 9 are 20, and 3 are 23, and 5 are 
 28, and 4 are 32, and 2 are 34 tens, — equal 
 to 3 HUNDREDS and 4 tens ; — place the 4 
 tens under the tens' column and add the 3 
 hundreds to the hundreds' column, thus : 3 
 and 2 are 5, and 1 is 6, and 8 are 14, and 3 
 are 17, and 3 are 20, hundreds, — equal to 2 
 THOUSANDS and hundreds ; — place the hundreds under the 
 hundreds' column and add the 2 thousands to the column of 
 thousands, thus : 2 and 3 are 5, and 2 are 7, and 6 are 13, and 
 4 are 17 thousands, — equal to 1 ten thousand and 7 thousands, 
 *- place the 7 thousands under the column of thousands, and 
 
 operation. 
 
 1 
 1 
 
 Thousands. 
 Hundreds. 
 Tens. 
 Units. 
 
 H 
 
 4327 
 
 "3 
 
 346 
 
 ?, 
 
 59 
 
 
 6831 
 
 
 2194 
 
 
 3285 
 
 
 17042 Amount. 
 
18 ADDITION. [chap. H. 
 
 the 1 ten thousand on the left of it ; and we have for the 
 amount 17042. 
 
 Proof hy the excess of 9'5. — Find the excess of 9's in the 
 sum of the digits of each of the numbers added, and if 
 the ejxcess of 9's in these excesses, equals the excess of 9's 
 in the product, the work may be considered right. 
 
 Take for illustration the preceding example. 
 
 OPERATION. 
 
 4327 = 7 excess. 
 346 =4 " 
 59 = 5 " 
 6831 =0 " 
 2194 = 7 '^ 
 3285 = " 
 
 Amount, 17042 = 5, is the excess of 9's in the above 
 
 hence the work is 
 
 right. 
 
 Rkmark. — To comprehend this method of proof, as well as that given for the 
 proof of Subtraction, Multi{>lication and Division, it is necessary to under- 
 stand the {)roperties of the number 9 explained on page 80th, Art. 74. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 3412 
 
 7310 
 
 782 
 
 3241 
 
 18243 
 
 3410 
 
 416 
 
 4164 
 
 476 
 
 32341 
 
 218 
 
 32 
 
 3123 
 
 84324 
 
 7147 
 
 436 
 
 4 
 
 7182 
 
 18472 
 
 165 
 
 1412 
 
 74- 
 
 119 
 
 31421 
 
 2342 
 
 7. What is the sum of 4862+834+46734 + 82796 f 
 9832 8763? 
 
 8. What is the sum of 144 + 7864 + 891234 6327 + 
 9879? 
 
 9. What is the sum of 78639 + 847796+864321 + 1487 
 + 987? 
 
 10. What is the sum of 186 + ^72+4638 + 64732 + 
 8634 + 9763+478? 
 
ART. 31.] ADDITION-. 19 
 
 11. Find the sura of 986+834 + t325t + t63244 8t63 
 4-9876 + 4683 + 9824. 
 
 12. Find the sum of 8632 + 84129 + 91 + 1864 + 9981 + 
 1632 + 876324. 
 
 13. Find the sura of 11468+3121 + 863+4902+816 + 
 8196+81641 + 163. 
 
 14. What is the sum of 8463 + 121 + 84632+8468 + 
 1416+8916+868411 + 9816141 ? 
 
 • 15. What is the sura of 846832 + 981649+168321 + 
 684 + 9163+84162 + 9824 ? 
 
 16. What is tlie sura of 18461 + 982+6849+131241 + 
 6824121 + 168411 + 9163+4214 ? 
 
 11. What is the sum of 16824+4168+4134+8686 + 
 9432 + 981 + 98624? 
 
 18. What is the sum of 23416+1862541+91632+8163 
 +9168+92+8416+1231 ? 
 
 19. What is the sum of 86432+68324 + 981324+83241 
 + 964? 
 
 20. What is the sura of 16841+9683+8324+8632+ , 
 149118+16832+1984683+86432141 ? 
 
 PRACTICAL QUESTIONS. 
 
 1. A father gave to his eldest son 1413 dollars, to his 
 youngest son 3249 dollars, to his oldest daughter 1298 
 dollars, to his youngest daughter- 3998 dollars, and had 
 remaining 1968 dollars ; — how much money had he at 
 first ? 
 
 2. Several persons contributed towards building a 
 church. A gave 184 dollars, B gave 213 dollars, C gave 
 843 dollars, D gave 195 dollars, and E gave 395 dol- 
 lars ;— how much did they together contribute ? 
 
 3. Five brothers had the following sums of money ; A 
 9189 dollars, B 15450 dollars, C 899 dollars, D 3499 dol- 
 lars, and E 9999 dollars ; — how much did they together 
 have ? 
 
 4. A drover bought 491 sheep one week, 841 the next 
 week, 943 the third week, 1496 the fourth week, and 18550 
 the fifth week ; — how many sheep did he buy in all ? 
 
20 ADDITION. [chap. II. 
 
 5. A gentleman owns a farm wortli 3450 dollars, a build- 
 ing lot worth 3759 dollars, a store and lot worth 5868 
 dollars, a fine horse and carriage worth 715 dollars ; 
 what is the amount of his property ? 
 
 6. From New York to Kingston is 90 miles, from King- 
 ston to Albany is 60 miles, from Albany to Rochester is 
 251 miles, from Rochester to Buffalo is 75 miles, and from 
 Buffalo to Niagara Falls is 21 miles ; how far is it from 
 New York to Niagara Falls ? 
 
 7. An individual owns a farm worth 2463 dollars, a 
 wood-lot worth 1342 dollars, a store and lot worth 2465 
 dollars ; what is the amount of his property ? 
 
 8. A gentleman willed his estate to his wife, three sons, 
 and four daughters ; to his daughters he willed 3496 dol- 
 lars apiece; to his sons, each 5785 a piece; and to his wife 
 4698 dollars ; — how much was his estate ? 
 
 9. The distance on the New York and Erie railroad 
 from New York to Goshen is 59 miles ; from Goshen to 
 Narrowsburgh is 63 miles ; from Narrowsburgh to Owego 
 is 114 miles ; from Owego to Friendship is 137 miles ; and 
 from Friendship to Dunkirk is 87 miles. How many miles 
 from New York to Dunkirk ? 
 
 10. A boy gave for a slate 22 cents ; for an arithmetic 
 50 cents ; for an algebra 75 cents ; for a grammar 56 
 cents; and for a geography 125 cents. How much did he 
 give for them all ? 
 
 11. A butcher sold to one man 436 pounds of meat ; 
 to another 3695 pounds ; to another 9899 pounds ; to 
 another 12485 pounds ; and to another 879 pounds. How 
 many pounds did he sell in all ? 
 
 12. A, B, C, D and E enter into partnership ; A puts 
 in 475 dollars ; B 846 dollars ; C 1495 dollars ; D 985 
 dollars ; and E 7864 dollars. How much stock have they 
 in trade ? 
 
 13. Four persons deposit money in a bank ; the first 
 deposits 4490 dollars ; the second 5685 dollars ; the third 
 9947 dollars ; and the fourth 12470 dollars. How many 
 dollars did they all deposit ? 
 
 14. Bought of A 346 cords of wood ; of B 846 cords ; 
 
ART. 31.] ADDITION. 21 
 
 of C 395 cords ; of D 836 cords ; of E as much as of A 
 and C both ; and of F as much as of B and E both. 
 How many cords of wood did I buy in all ? 
 
 15. A produce-dealer has in store at one place 146 
 bushels of corn, 876 bushels of oats, 395 of rye, and 1247 
 bushels of potatoes ; at another place 1846 laushels of 
 corn, 3246 bushels of oats, 846 bushels of rye, and 437 
 bushels of potatoes ; and at another place 199 bushels of 
 corn, 847 bushels of oats, and 849 bushels of potatoes ;— 
 how much produce has he in store .'' 
 
 16. Macedon was founded 794 years B. C. by Caranus ; 
 Sparta was founded 606 years before Macedon, by Selex ; 
 Corinth, 4 years before Sparta, by Lysippus ; Thebes, 89 
 years before Corinth, by Cadmus. In what year was 
 Sparta, Corinth, and Thebes founded respectively ? 
 
 17. The population of the United States in 1790 was 
 3729326 ; in 1800 it was 1580427 more ; 1810 it had in- 
 creased 1930150 more ; in 1820, 2398377 more ; in 1830, 
 3218241 more ; and in 1840, 4244165 more. What was 
 the population in each of the above mentioned years ? 
 
 18. Mr. Harvey, the discoverer of the circulation of 
 the blood, was born in 1578, at Folkstone, in Kent ; George 
 Edwards, the ornithologist, was born 116 years later ; 
 William Herschel, the astronomer, was born 44 years after 
 Edwards ; Henry Clay, the American statesman, was 
 born 39 years after Herschel; — in what year was each 
 of the above named individuals born ? 
 
 19. At the battle of Moskowa there were 13000 Rus- 
 sians killed, 5000 taken prisoners, about 27000 wounded, 
 and 40 generals either killed, wounded or taken prisoners; 
 2500 of Napoleon's army were killed, 7500 wounded, 
 and 15 generals either killed or wounded. What was 
 the total loss ? 
 
 20. At the battle of Waterloo the French lost 40000 
 men; the Prussians 38000; the Belgians and Dutch 8000; 
 the Hanoverians 3500; and the English about 12000; — ■ 
 how many men were killed in all ? 
 
23 SUBTRACTION. [CHAP. II 
 
 SUBTRACTION. 
 
 Art. 32. Subtraction is the method of finding the dif- 
 ference between two numbers. 
 
 In subtraction there are three terms, the Mimiend, Sub- 
 trahend, and Remninde?- . Any two of these being given, 
 the remaining one can be found. 
 
 The number from which the other is to be taken is 
 called the Minuend; the number to be subtracted from it, 
 the Suhtraheiid; and the result obtained by the operation, 
 the Remainder. 
 
 A short horizontal line, thus, — , is called minus, and is 
 the sign of suMraction, When it is placed between two 
 numbers, it shows ihat the number on the right of it is to 
 be taken from the one on the left. Thus, t, (the minuend) 
 — 5, (the subtrahend) = 2, the remainder,. 
 
 CASE I. 
 
 Art. 33. Subtraction of abstract numbers, when each 
 figure of the subtrahend is less than its corresponding 
 figure in the minuend. 
 
 Remark. — The diflerence of two similar concrete numbers is a concrete 
 number of the same kind, and is found in the same way as though they were 
 abstract numbers. But two dissimilar concrete numbers can not be taken, 
 the one from the other. 
 
 1. From 946 subtract 524. 
 
 Explanation. — Write the less num- 
 operation. ber under the greater, with units unv 
 
 ^ der units &c., and draw a linfe under- 
 
 "i neath. then proceed thus : 4 units from 
 
 6 units, leave 2 units : vrrite the 2 units 
 in units' place, 2 tens from 4 tens leave 
 ,.. J c\)ia 2 tens, vehich write in tens' place, 5 
 
 ^'u^^l' A lol hundreds from 9 hundreds leavQ 4 
 
 .Subtrahend, bZi hundreds : write the 4 hundreds in 
 
 ,, .J Ann the place of hundreds, and we have 
 
 Kemamder, 4 2 2 ^^^^ ^{J^ ,emainder 422. 
 
 Proof. — Add the remainder and subtrahend together; if 
 their sum is equal to the minuend, the work is right. 
 
 p: 0- « 
 
ART. 33.] 
 
 SUBTRCTIOS. 
 
 38 
 
 2. 
 
 From 465 
 Subtract 243 
 
 From 
 Subtract 
 
 3. 
 
 842 
 511 
 
 4. 
 From 762 
 Subtract 451 
 
 5. 
 From 549 
 Subtract 334 
 
 From 
 Subtract 
 
 6. 
 465 
 143 
 
 7. 
 From 947 
 
 Subtract 837 
 
 8. From 4631 take 2310. 
 
 9. From 16820 take 3410. 
 
 10. From 9642 take 8431. 
 
 11. From 32478 take 12374. 
 
 12. From 96472 take 32361. 
 
 13. Subtract 4247 from 7449. 
 
 14. Subtract 147302 from 688925. 
 
 15. Subtract 234610 from 479824. 
 
 16. From 9867412 subtract 4243101. 
 
 17. From 1649324 subtract 443121. 
 
 18. From 256342 subtract 143242. 
 
 19. From 9864324 subtract 8432124. 
 
 20. From 9680434 subtract 45304U. 
 
 PRACTICAL QUESTIONS. 
 
 1. A boy had 36 marbles and gave 24 of them to his 
 playmate; how many had he remaining ? 
 
 2. Joseph caught 295 quails, and John caught 84; how 
 many more did Joseph catch than John ? 
 
 3. Jackson had 95 cents and Jane had 73; how many 
 more had Jackson than Jane ? 
 
 4. Elisha having 447 bushels of potatoes, sold 2S4 
 bushels of them to Perry; how many bushels had he 
 remaining ? 
 
 5. A farmer bought a span of horses for 346 dollars, a 
 yoke of oxen for 135 dollars; how much more did he 
 give for the horses than for the oxen ? 
 
 6. A drover, having 1465 sheep, sold 1235 of them ; 
 how many had he remaining ? 
 
24 SUBTRACTION. [cHAP. II. 
 
 I. A gentleman owns a store worth 4695 dollars, and a 
 grist-mill worth 2135 dollars : how much more is the store 
 worth than the grist-mill ? 
 
 8. A gentleman gave for a house and lot- 9899 dollars, 
 for a cotton factory 8495 dollars ; how much more did he 
 give for the one than the other ? 
 
 9. A speculator bought some land for 1289t dollars, a 
 tannery for 10444 dollars ; how much more did the land 
 cost than the tannery ? 
 
 10. A merchant, having 9841 yards of cloth, sold 5844 
 yards of it ; how many yards had he remaining ? 
 
 II. A drover bought cattle to the amount of 9647 dol- 
 lars, and sheep to the amount of 5434 dollars ; how much 
 more did he give for the cattle than for the sheep ? 
 
 12. A merchant sold a quantity of goods for 869*7 dol- 
 lars, and by so doing gained 1495 dollars ; how much did 
 the goods cost him ? 
 
 13. A gentleman sold an estate for 1499 dollars, and by 
 so doing gained 1084 dollars; how much did the estate 
 cost him ? 
 
 ' 14. A farm was sold for 3495 dollars, which was 1032 
 dollars more than it was worth; how 'much was it worth ? 
 
 15. A farmer had 4295 sheep, and 2145 lambs ; how 
 many more sheep had he than lambs ?. 
 
 16. A farmer, having 1346 bushels of wheat, sold 1042 
 bushels of it ; how many bushels had he remaining ? 
 
 11. A merchant, during one year, sold 1241 barrels of 
 molasses, and 2489 barrels of sugar ; how many more 
 barrels of sugar did he sell than molasses ? 
 
 18. A gentleman willed to his son 49865 dollars, and to 
 his daughter 34534 dollars ; how much more did he will to 
 his son than to his daughter } 
 
 19. A man, driving 1565 sheep to market, on his way 
 sold 435 of them ; how many had he remaining ? 
 
 20. A ship is valued at 69841 dollars, and its cargo at 
 45831 dollars ; how much more is the ship valued at than 
 the cargo ? 
 
ART. 34.1 SUBTRACTION. 25 
 
 I CASE II. 
 
 Art. 34. Subtraction of Abstract nambers in geteral. 
 1. From 728 subtract 364. 
 
 OPERATION. Explanation. — The numbers 
 
 being properly written down, we 
 
 •S -S proceed thus : 4 units taken from 
 
 •« • 2 -a « i ° units leave 4 units, which write 
 
 o § a 3 g '3 in unit's place. I cannot take 6 
 
 K/f- ^ nn^ ? /i^N o tens from 2 tens ; therefore, from 
 
 ^IT?^' ^ n 5=^ ^^^^ ^ the 7 hundreds I take 1 hundred 
 Subtrahend, 36^ ^ ^q ^^^^^ ^^^ ^^^ i^ ^^ ^^^^ 2 
 
 T, . J oo A tens, making 12 ^ews,- — 6 tens 
 
 Remamder, 3 6 4 ^^^^ 12 tens leave 6 tens, which 
 
 I write in tens' place. I have 
 taken 1 hundred from the seven hundreds, which leaves 6 hun- 
 dreds; 3 hundreds from 6 hundreds leave 3 hundreds. But for 
 convenience, it is customary to add the 1 hundred to the 3 
 hundreds, (the next figure in the subtrahend.) and take the 
 sum from the figure in the minuend under which it is placed, 
 which is the same in effect as the above. 
 
 Note. — The minuend 7 hundreds, 2 tens and 8 units, is = 6 hundreds, 12 
 t£ns and 8 units, which form it, absolutely assumes in the mind while giving 
 the above explanation; still it is not necessary to be written except to render 
 the explanation more plain. 
 
 Proof by the excess of 9'5. — Find the excess of 9's in the 
 sum of the digits of the remainder, — also of the subtrahend. 
 Then find the excess of 9's in the excesses just found, — 
 if this excess equals the excess of 9's in the vtinucnd, the 
 work is right. 
 
 Take for illustration the above example : 
 
 OPERATION. 
 
 728 = 8 excess. 
 364 = 4 " 
 
 364 = 4 « 
 
 8 excess in the subtrahend and 
 remainder, which is the same as the excess in the minuend, 
 therefore titie work is right. 
 
 2 
 
26 SUBTRACTION. [CHAP. II 
 
 2. 3 4. 5. 
 
 From 4642 From 647 From 4621 From 468 
 
 Take 2370 Take 352 Take 2432 Take 379 
 
 fi 7 R 
 
 From 68492 From * 7246 From 68243 
 
 Subtract 37508 Subtract 5839 Subtract 27359 
 
 9. From 8697 subtract 5988. 
 
 10. From 1682402 subtract 740482. 
 
 11. From 187642 subtract 94837. 
 
 12. From 9046 subtract 8074. 
 
 13. From 86432 subtract 67821. 
 
 14. Subtract 4962 from 7832. 
 
 15. Subtract 14829 from 84643. 
 
 16. From 4001 subtract 1344. 
 
 OPERATION. 2nd. Zrd. 4tli, 
 
 m Xi 
 
 ■T3 m 
 
 Minuend, 4001 = 3(10)01 = 39 (10)1=399(11) 
 Subtrahend, 1344 1344 
 
 Remainder, 2657 2657 
 
 Explanation. — The numbers being properly arranged, com 
 mence at the right and proceed thus : we cannot take 4 units 
 from 1 unit ; therefore I seek 1 from the tens' place, but find- 
 ing no tens there, I proceed to the hundreds' place, and finding 
 no hundreds there, I take 1 thousand from the 4 thousands and 
 r set it in the next place towards the right, which causes the 
 minuend to take the 2nd form. Then take 1 hundred from the 
 10 hundreds, and set it in the next place towards the right, 
 causes the minuend to take the 3rd form. Now taking 1 ten 
 from the 10 tens, and adding it to the 1 unit causes the minuend 
 to assume the 4th form, — from which we are now prepared to 
 
A.RT. 34.] SUBTRACTION. 2*1 
 
 take the subtrahend. 4 units from 11 units leave 7 units ; 4 
 tens from 9 tens leave 5 tens ; 3 hundreds from 9 hundreds 
 leave 6 hundreds ; and 1 thousand from 3 thousands leave 2 
 thousands. Hence the difference of these two numbers is 2657. 
 
 Remark.— The 2nd, 3rd and 4th forms of the minuend serve merely to ex- 
 plain the method of subtracting more clearly, and should, therefore, in practice, 
 be performed in the mind and not be written down. The sanffe result will be 
 obtained by simply adding 10 to the upper figure when it is smaller than the 
 one below it, and carrying, or adding, 1 to the next figure of the subtrahend. 
 
 ir From 41007 subtract 34138 ? 
 
 18. From 90006 subtract 9994 ? 
 
 19. How many are 10000—9 ? 
 
 20. How many are 100000—1 ? 
 
 21. How many are 89467—84732 ? 
 
 22. How many are 760743—249078 ? 
 
 23. How many are 4078603—1437908 ? 
 
 24. How many are 90807060—60708091 ? 
 
 25. How many are 97876757—79787675 ? 
 
 26. How many are 20304050—1020304 ? 
 
 27. How many are 90857565—20382468 ? 
 
 28. How many are 900000—1 ? 
 
 29. How many are 909090—1 ? 
 
 30. How many are 9080706050—16070809? 
 
 PRACTICAL QUESTIONS. 
 
 1. A gentleman willed to his son 3862 dollars, and to his 
 daughter 5324 dollars; how much more did he will to his 
 daughter than to his son ? 
 
 2. In a certain orchard there are 425 apple-trees and 
 297 plum-trees; how many more apple-trees than plum- 
 trees ? 
 
 3. A man traveled 14637 miles during one year, and 
 9843 miles the next year ; how much farther did he travel 
 the first year than the second ? 
 
 4. A merchant had 25694 pounds of pork, and sold 
 19832 pounds of it; how many pounds remained unsold ? 
 
 5. A speculator bought a quantity of cotton for 294682 
 di)llars, and sold it for 516390 dollars; how much did he 
 gain ? 
 
28 SUBTRACTION. [CHAP. II. 
 
 6. Gunpowder was invented by Schwartz, in the year 
 1330; how long was it before the birth of Bonaparte, 
 1769? 
 
 I. George Washington died in the year 1*199, at the 
 age of 6t; in what year was he born ? 
 
 8. The mariner's compass was invented at Naples in the 
 year 1302; how long before the discovery of America 1492? 
 
 9. Joseph Addison, the poet, was born 16t2, and died, 
 111 9; how old was he when he died ? 
 
 10. Sir William Blackstone, the lawyer, was born 1123, 
 and died, 1180; at what age did he die ? 
 
 II. Francis Bacon, a universal genius, died in the year 
 1626, at the age of 65; in what year was he born ? 
 
 12. Robert Burns, the poet, was born 1*159, and died 
 1196; Lord Byron, the poet, was born lt88, and died 
 1824. What was the age of each, and how long after the 
 birth of Burns was Byron born ? 
 
 13. George Edwards, the ornithologist, was born 1694; 
 how long was this before the birth of Harvey, the dis- 
 coverer of the circulation of the blood, who was born 
 1.578 ? 
 
 14. Massachusetts was settled in 1620, at Plymouth; 
 how many years before the declaration of our National In- 
 dependence 1776 ? 
 
 15. The Independence of the United States was ac- 
 knowledged in Europe in 1783; how long was that after 
 the battle of Bunker's Hill, 1775 ? 
 
 16. The first newspaper published in America, at Bos- 
 ton, was in 1704, which was 183 years after Mexico was 
 conquered by the Spaniards; in what year was Mexico 
 conquered ? 
 
 17. Michael Angelo, an Italian painter, died 1568, at 
 the age of 89; in what year was he born ? 
 
 18. Benjamin Franklin, the philosopher and statesman, 
 died 1790, at the age of 84; in what year was he born ? 
 
 ■ 19. Galileo, an Italian astronomer, died 1642, at the 
 age of 78; in what year was he born ? 
 
 20. Luther, the reformer, died 1546, at the age of 63, 
 in what year was he born ? 
 
ART. 34.] SUBTRACTION. 29 
 
 21. Raphael, the prince of painters, an Italian, was 
 born 1483, and died in 1520, which was 6 years after the 
 birth of Titian, another renowned Italian painter; to what 
 age did Raphael live, and in what year was Titian born ? 
 
 22. Cotopaxi, the highest volcano in the world, is 19408 
 feet high; how much higher is Sorato, the highest land in 
 America, which is 25380 feet high, than Cotopaxi ? 
 
 23. Benjamin West, the American painter, was bora 
 1*138; how long was this before the death of Robert Ful- 
 ton, who died in the year 1815 ? 
 
 24. Mount Ararat, (on which Noah's ark rested,) is 
 12100 feet high; now how much higher is that than mount 
 Washington in New Hampshire, which is 6234 feet in 
 height ? 
 
 25. St. Peter^s Church at Rome, is 450 feet high; how 
 much higher is that than Trinity Church, New York, 
 which is 283 feet in height ? 
 
 26. Joseph Bonaparte died 1844, at the age of 16 ; in 
 what year was he born ? 
 
 2T. Dr. Franklin was born in the year 1106, and died 
 in 1190; how old was he when he died ? 
 
 28. A man, owning 45161 acres of land, sold 23921 
 acres of it; how many acres had he remaining ? 
 
 29. A merchant, having 98012 barrels of flour, sold 
 49261 of them; how many had he remaining ? 
 
 30. In a certain town there were 24961 inhabitants, 
 which was 5084 more than there were the preceding year; 
 how many were there the preceding year ? 
 
 31. A merchant sold a quantity of goods for 38961 
 dollars, which was 813 dollars more than they cost him; 
 how much did they cost him ? 
 
 32. A man, having 21695 feet of lumber, sold 1962 feet 
 of it; how many feet had he remaining ? 
 
 33. If I borrow of my neighbor 9613 dollars, and pay 
 him 999 dollars of it; how much remains unpaid ? 
 
 34. A gentleman sold a farm for 54623 dollars, which 
 was 9240 dollars more than he gave for it; how much did 
 he pay for the farm ? 
 
 35. A farmer raised 2141 bushels of rye, and 2146 
 
30 SUBTRACTION. [CHAP. II. 
 
 bushels of corn; he sold 943 bushels of the rye, and 189 
 bushels of the corn ; — how much of it remains unsold ? 
 
 36. A and B bought a farm for 7840 dollars; A paid 
 2999 dollars, and B the remainder; — how many dollars 
 did B pay ? 
 
 37. A and B traded farms; A's farm is valued at 9863 
 dollars, and B's at 7807 dollars; — how much in equity 
 ought B to pay A ? 
 
 38. Said A to B, I have 4605 sheep; B replied, that 
 he had as many, lacking 298; — how many had B ? 
 
 39. How many years from 1496, the year in which 
 Algebra was first known in Europe; to 1808, the year in 
 which the first steamboat was put in successful operation 
 by Robert Fulton ? 
 
 40. A grocer having 346823 dollars' worth of goods, 
 shipped 196832 dollars' worth of them; how many dollars' 
 worth had he remaining ? 
 
 41. A speculator sold a factory for 35896 dollars, 
 which was 1491 dollars more than it cost him; how much 
 did it cost him ? 
 
 PRACTICAL QUESTIONS COMBINING ADDITION AND SUBTRACTION. 
 
 1. A farmer; having 4632 sheep, sold to A 785, and to 
 B 896; how many had he remaining ? 
 
 2. A farmer's yearly income was 1679 dollars; he paid 
 for repairing his house 487 dollars; for farming utensils 
 98 dollars; and for hired help 299 dollars; — how much 
 has he remaining ? 
 
 3. A man bought a span of horses and a wagon for 
 987 dollars; he then sold the wagon for 185 dollars, and 
 the horses for 736 dollars; — how much did he lose by the 
 operation ? 
 
 4. A gentleman, having 697 dollars, deposited 372 
 dollars in the bank, and spent 197 dollars of it; how much 
 had he remaining ? 
 
 5. A speculator, having 346821 acres of land, sold to 
 A 637 acres; to B 495; to C 1865; toD 26942; and to E 
 879 acres; — how many acres had he left ? 
 
ART. 34.] SUBTRACTION. 31 
 
 6. There is a farm consisting of 946 acres; 35 acres of 
 which is planted with corn and potatoes; 140 acres sown 
 with rye; 180 acres with oats; 98 with wheat; 212 is 
 pastured, and the remainder is meadow. How many acres 
 of meadow ? 
 
 7. A lady, having 467 dollars, paid for a bonnet 24 
 dollars; for a shawl 85 dollars; for a silk dress 90 dollars; 
 and for some delaines 112 dollars; — how much had she 
 remaining ? 
 
 8. A market-woman, having 234 oranges, sold to one 
 person 12 of them; to another 46; to another 54; to 
 another 32; and to another 15; — how many had she re- 
 maining ? 
 
 9. A farmer, having 89*1 sheep, sold to A 150 of them; 
 to B 160; to C 284; and to D 294;— how many had he 
 remaining ? 
 
 10. A drover, having 191 cattle, sold 112 of them, and 
 bought 81 more; how many had he then ? 
 
 11. In a certain army there are 4560 men: in a battle 
 646 of them were killed, 49t of them wounded, and 148 
 of them deserted; how many were left ? 
 
 12. A farmer, having 847 bushels of grain, sold to A 
 132 bushels; to B 112; to C 184; and gave to the poor 
 212 bushels; — how many bushels had he remaining ? 
 
 13. An individual traveled by railroad 497 miles, and 
 designed to return on foot; the first day he traveled 69 
 miles; the second 84; the third 59; the fourth 47 miles; 
 the fifth day he took the cars and arrived home. How 
 far did he go the last day ? 
 
 14. A man willed an estate of 560048 dollars to his 
 two children and wife, as follows : to his son 230645 
 dollars; to his daughter 88999 dollars; and to his wife the 
 remainder. How much did he will to his wife ? 
 
 15. A man laid out 98000 dollars in speculation; the 
 first year he gained 1847 dollars; the second year 1987 
 dollars; the third year he lost 8044 dollars. How much 
 did he lose by the operation ? 
 
 16. A merchant, having 89776 barrels of flour, sold to 
 A 967 barrels; to B 1743 barrels; to C 6842 barrels; 
 
32 MULTIPLICATION. [CHAP. II. 
 
 to D 14625 barrels; and to E the remainder. How many 
 barrels did E receive ? 
 
 IT. Four persons A, B, C and D propose to purchase 
 a manufactory, valued at 97802 dollars. A is to pay 
 4990 dollars, B 1264Wollars, C 19682 dollars, and D 
 the remainder; what sum will D have to pay ? 
 
 18. Having in my possession 8960 dollars, I wish to 
 know how much I must add to this sum, to be able to 
 purchase a farm worth 18910 dollars, and save 497 dol- 
 lars for other purposes ? 
 
 19. A had 448 oxen; B had 212 more than A; and 
 had as many as A and B together, lacking 184; — how 
 many had B and respectively ? 
 
 20. A has 470 dollars more than B, and 245 dollars 
 less than C, who has 2490 dollars; and D has as much as 
 A and B together. How many dollars have A, B and D 
 respectively ? 
 
 21. John has 240 sheep more than Joseph, and 125 
 less than James, who has 485 ; and Jackson has as many 
 as John and Joseph together, lacking 320 sheep. How 
 many sheep have John, Joseph and Jackson respect- 
 ively ? 
 
 MULTIPLICATION. 
 
 Art. 35. Multiplication is a concise method of com- 
 puting the amount of any number taken as many times as 
 there are units in another number. 
 
 There are three terms employed in multiplication; the 
 Multiplicand, the Multiplier, and the Product; any two of 
 which being given the remaining one can be found. 
 
 Art. 36. The MultplicaTid is the number taken. 
 The Multiplier is the number that shows how many times 
 the multiplicand is taken. The Product is the answer or 
 result obtained. The multiplicand and multiplier are also 
 called Factors of the product. 
 
 Art. 37. The multij^lier can never be a concrete num- 
 
XRT. 38.] 
 
 MULTIPLICATION. 
 
 88 
 
 ber, as it merely expresses the number of times the multi- 
 plicand is taken. The product will be of the same denomi- 
 nation as the multiplicand. 
 
 Art. 38. The sign of multiplication is two short lines of 
 equal length bisecting each other at an angle of 45 de- 
 grees with the horizon; thus, X, and is sometimes called 
 INTO. This sign being placed between two numbers shows 
 that they are to be multiplied, the one by the other. Thus, 
 6x8=48, indicates that 6 is to be multplied by 8, or 8 
 to be multiplied by 6, (as the case may require,) and that 
 the product equals 48. 
 
 MULTIPLICATION TABLE. 
 
 2X 
 2X 
 2X 
 2X 
 2X 
 
 0= 
 1= 2 
 2= 4 
 3= 6 
 4= 8 
 2X 5=10 
 2x 6=12 
 2X 7=14 
 2x 8=16 
 2x 9=18 
 2X10=20 
 2X11=22 
 2X12=24 
 
 3X 0= 
 
 3X 
 
 3X 
 
 1= 3 
 
 2= 6 
 
 3X 3= 9 
 
 3X 
 3X 
 
 4=12 
 5=15 
 
 3X 6=18 
 
 3X 
 3X 
 3X 
 
 7=21 
 8=24 
 
 4X 
 
 4X 1= 
 
 4X 2= 
 
 4X 3= 
 
 4X 4= 
 
 4X 5= 
 
 4X 6= 
 
 4X 7= 
 
 4X 8= 
 
 4X 9= 
 4X10= 
 
 0|5X 
 4 5X 
 
 3X10=30 
 
 3X11=33:4X11 
 
 3X12=3614X12 
 
 5X 
 5X 
 5X 
 5X 
 5X 
 5X 
 5X 
 5X 
 5X 
 5X 
 5X 
 
 0= 
 
 1= 5 
 
 2=10 
 
 3=15 
 
 4=20 
 
 5=25 
 
 6=30 
 
 7=35 
 
 8=40 
 
 9=45 
 
 10=50 
 
 11=55 
 
 12=60 
 
 6X 0= 
 
 1= 6 
 2=12 
 3=18 
 6X 4=24 
 6X 5=30 
 6=36 
 7=42 
 8=48 
 6x 9=54 
 6X10=60 
 6X11=66 
 6X12=72 
 
 6X 
 6X 
 6X 
 
 6X 
 6X 
 6X 
 
 0= 
 
 1= 7 
 2=14 
 3=21 
 4=28 
 5=35 
 6=42 
 7=49 
 8=56 
 9=63 
 fO 
 
 /X 
 
 7X 
 
 7X 
 
 7X 
 
 7X 
 
 7X 
 
 7X 
 
 7X 
 
 7X 
 
 7X 
 
 7X10 
 
 7X11=77 
 
 7X12=84 
 
 8X 0= 
 8X 1= 8 
 
 8X 2=16 
 8X 3=24 
 8X 4=32 
 8X 5=40 
 8X 6=48 
 X 7=56 
 8X 8=64 
 8X 9=72 
 8X10=80 
 8X11=88 
 8X12=96 
 
 9X 0= 
 
 9X 1= 
 
 9X 2= 
 
 9X 3= 
 
 9X 4= 
 
 9X 5= 
 
 9X 6= 
 
 9X 
 9X 8^ 
 9X 9: 
 9X10: 
 
 9X11: 
 
 9X12: 
 
 7= 
 
 10 X 
 lOX 
 lOX 
 lOX 
 lOX 
 lOX 
 lOx 
 10 X 
 lOX 
 lOx 
 lOX 
 10 X 
 lOX 
 
 0= 
 1= 10 
 
 2= 20 
 
 3= 30 
 
 4= 40 
 
 5= 50 
 
 6= 60 
 
 7= 70 
 
 8= 80 
 
 9= 90 
 
 10=100 
 
 11=110 
 
 12=120 
 
 llX 
 llX 1 
 IIX 2 
 llX 
 llX 
 llX 
 llX 
 llX 
 
 11 X 8: 
 11 X 9: 
 11X10: 
 
 11X11: 
 
 111X12: 
 
 0= 
 
 12 X 0= 
 12X 1= 
 12 X 2= 
 12 X 3= 
 12X 4= 
 12X 5: 
 12X 6: 
 12X 7= 
 12 X 8= 
 12 X 9. 
 110 I 12X10= 
 121 12x11. 
 132 ! 12X12= 
 
 99 
 
 = 
 = 12 
 : 24 
 
 : 36 
 
 = 48 
 
 : 60 
 : 72 
 : 84 
 : 96 
 :108 
 :120 
 :132 
 :144 
 
34 MULTIPLICATION. [cHAP. II. 
 
 CASE I. 
 
 Art. 39. Multiplicatimi of abstract numbers, when the 
 multiplier does not exceed 9. 
 
 I. Multiply 846 by 8. 
 
 OPERATION. Explanation — Write the numbers down, 
 
 ^ placing units under units ; then proceed 
 
 c-5 from right to left: thus, 8 times 6 units 
 
 S-S ^ =j are 48 units, or 4 tens and 8 units ;. — place 
 
 J J g 3 the 8 units in units' place, and reserve the 
 
 Tv/r 1 • V J o^'i ^ ^^^ *® add to the next product, 8 times 
 
 Mu tiplicand,8 464 tens are 32 tens, and 4 tens added are 36 
 
 Multiplier, 8 ^g^g^ ^j. 3 hundreds and 6 tens ; — place the 
 
 T. J 2 TT ^ *^°^ ^^ *^°^' place and reserve the 3 
 
 Product, 6 7 6 8 hundreds to add to the next product 8 
 
 times 8 hundreds are 64 hundreds, and 3 hundreds added are 
 
 67 hundreds, or 6 thousands and 7 hundreds, which write down ; 
 
 and we have for the product 6768. 
 
 Proof. — Multiply the multiplier by the multiplicand; if 
 the product thus obtained, equals the first product the 
 work is presumed to be right. 
 
 2. Multiply 348 by 2. 
 
 3. Multiply 483 by 3. 
 
 4. Multiply 684 by 4. 
 
 5. Multiply 6482 by 4. 
 
 6. Multiply 14682 by 5. 
 1. Multiply 18623 by 6. 
 
 ,, 8. Multiply 38943 by T. 
 9. Multiply 28462 by 8. 
 10. Multiply 8946 by T. 
 
 II. Multiply 7683 by 6. 
 
 12. Multiply 9898 by 9. 
 
 13. Multiply 6847 by 3. 
 
 14. Multiply 94762 by 6. 
 
 15. Multiply 88992 by 7. 
 
 16. Multiply 33449 by 8. 
 
 17. Multiply 884682 by 9. 
 
 18. Multiply 99999 by 5. 
 
 19. Multiply 897654 by 7. 
 
 20. Multiply 123456789 by 8. 
 
ART. 39. J MTLTIPLICATION. 85 
 
 TRACTICAL QUESTIONS. 
 
 1. A man solvl 105 sheep, at 3 dollars a piece; how much 
 did he receive for iliera ? 
 
 2. What cost 184 barrels of flour, at 6 dollars a barrel ? 
 
 3. What cost 198T ncres of land, at 9 dollars an acre ? 
 
 4. What cost 4t8C barrels of sugar, at 9 dollars a 
 barrel ? 
 
 5. In 1 mile there are 5l^S0 feet; how many feet in 5 
 miles ? '^ 
 
 6. In 1 mile there are 1160 yards; how many yards in 
 5 miles ? 
 
 I. If 9 men can mow a certain moadow in 18 days ; in 
 how many days can one man do the same ? 
 
 8. If 6 masons cq^i build a certain \v:ill in 149 days ; 
 in how many days can one mason build the same wall ? 
 
 9. If 460 bushels of oats will feed 1 horse 11 months; 
 how many bushels will be required to feed 8 horses the 
 same time ? 
 
 10. Bought 245 cords of wood, at t dollars a cord. 
 What did the whole cost ? 
 
 II. A farmer sold 8 horses, at 253 dollars a piece ; how 
 many dollars did he receive for them ? 
 
 12. A lady bought 189 yards of ribbon, at 6 cents a 
 yard; how much did it all cost her ? 
 
 18. What cost 1786 boxes of raisins, at 3 dollars a 
 box? 
 
 14. If a steamship can go 395 miles in 1 day, how far 
 can she go in 9 days ? 
 
 15. A merchant bought 2864 hats, at 4 dollars a piece; 
 how much did he pay for them all ? 
 
 16. A farmer sold 9 fat oxen, at 185 dollars a piece; 
 how much did he receive for them all ? 
 
 17. In 1 day there are 1440 minutes ; how many min- 
 utes in 8 days ? 
 
 18. In 1 day there are 86400 seconds; how many 
 seconds in 5 days ? 
 
 19. If one man receive 4 dollars a week, how much 
 will an army of 35680 men receive in 6 weeks ? 
 
36 MULTIPLICATION. [CHAP. II. 
 
 20. At 2 dollars a day, each; how much will it cost, to 
 board 685 men 1 days ? 
 
 CASE II. 
 
 Art. 40. Multiplication of abstract numbers in general ? 
 1. Multiply 437 by 56. 
 
 OPERATION. Explanation. — Write the numbers down so that 
 ^ units stand under units, tens under tens, &c. 
 
 ^ J „• w Begin at the right and proceed, thus, — 6 times 7 
 § g '3-^ units are 42 units, or 4 tens and 2 units ; write the 
 WHt" 2 units in units' place, and reserve the 4 tens to 
 4 3 7 add to the next product. 6 times 3 tens are 18 
 5 6 tens, and 4 tens added are 22 tens, or 2 hundreds 
 
 and 2 tens^ &c. We next multiply by the 5 tens. 
 
 2 6 2 2 For convenience we say 5 times 7 are 35, and place 
 218 5 the 5 under the multiplier, 5, that is in tens' place, 
 
 and reserve the three hundreds to add to the next 
 
 2 4 4 7 2 product, &c. But instead of 5 times 7, &c., it is, 
 50 times 7 units = 350 units, or 3 hundreds^ 5 tens and units. 
 I therefore placed the 5 tens in tens' place, where you perceive 
 it belongs. We proceed iifthe same way to explain why we 
 place the right hand figure of the product in the third, or 
 hundreds' place when multiplying by that figure, &c. 
 
 Proof hy the excess of 9'^. — Find the excess of 9's in 
 each FACTOR. Then if the excess of 9's in the product of 
 these excesses, equals the excess of 9's in the product of the 
 two factors, the work is right. 
 
 Take for illustration the preceding example ; 
 
 Factors, 
 
 OPERATION. 
 
 437 = 5 excess. 
 56 = 2 
 
 2622 1 , excess in the product of the excesses. 
 2185 
 
 Product, 24472 = 1 ' excesses in the product of the factors. 
 
 Explanation Commence at the left of the first factor, (the 
 
 multiplicand,) and add, thus, 4-j-3-f-7 are 14, or 1 nine and 5 
 
ART. 40.] MULTIPLICATION. 31 
 
 units, the excess of 9's in that factor. Add the second factor, 
 (the multiplier,) 5-f-6 are 11, or 1 nine and 2 units, the ex- 
 cess of 9's in that factor. The product of these excesses is 10, 
 or 1 nine and 1 unit, the excess of 9's in the product of the 
 excesses. Add the total product, 2-|-4-f-4+7-|-2 are 19, or 2 
 nines and 1, the excess of 9's in the product of the factors. 
 This excess equals the required excess; hence the work is 
 right. 
 
 2. Multiply 4624 by 35. >^ 
 
 3. Multiply 3846- by 39. 
 
 4. Multiply 8462 by 47. 
 6. Multiply 7846 by 147. 
 
 6. Multiply 3976 by 183. 
 
 7. Multiply 2243 by 144. 
 
 8. Multiply 2882 by 414. 
 
 9. Multiply 1414 by 323. 
 
 10. Multiply 2463 by 382. 
 
 11. Multiply 8632 by 132. 
 
 12. Multiply 4862 by 897. 
 
 13. Multiply 9876 by 678. 
 
 14. Multiply 4567 by 7654. 
 
 15. Multiply 1234 by 4321. 
 
 16. Multiply 8362 by 8496. 
 
 17. Multiply 146832 by 8376. 
 
 18. Multiply 36847 by 8324. 
 
 19. Multiply 1384697 by 476324. 
 
 20. Multiply 897654321 by 123456789. 
 
 PRACTICAL QUESTIONS. 
 
 1. If 15 men can build a certain wall in 235 days, how 
 long will it take 1 man to do it ? 
 
 2. If 45 men can accomplish a certain piece of work 
 in 360 days by working 8 hours a day, how many days 
 will it take one man to do the same by working 4 hours a 
 day ? 
 
 3. If 360 bushels of oats will last 185 horses 3 days, 
 how long will it last 1 horse ? 
 
88 MULTIPLICATION. [CHAP. II. 
 
 4. A drover bonglitr 685 oxea at 104 dollars a piece; 
 what was the cost of all of them ? 
 
 5. A merchant bought 25 pieces of broadcloth, each 
 piece containing 48 yards, at 9 dollars a yard. How 
 much did he pay for the whole ? 
 
 6. If a steamship can sail 18 miles in 1 hour, how far 
 can she sail in 34 days of 24 hours each ? 
 
 I. A speculator bought 8968 acres of land, at 195 dol- 
 lars an acre. How much did the whole cost him } 
 
 8. In 1 furlong there are 660 feet; how many feet in 
 8 furlongs, (1 mile)? 
 
 9 How many pounds of flour are there in 395 barrels; 
 there being 196 pounds in each barrel ? 
 
 10. What is the value of 346^shares of railroad stock, 
 at 125 dollars a share ? 
 
 II. How many pages are there in 5896 books, there 
 being 394 pages in each book ? 
 
 12. A speculator bought 302 cattle, and 293 times as 
 many sheep; how many sheep did he buy ? 
 
 13. If a garrison of men consume 98*T pounds of beef 
 in 1 day; how many pounds will a garrison containing 
 twice as many men consume in 365 days ? 
 
 14. Farmer A has 245 acres, sowed with wheat, which 
 produces 32 bushels to the acre. Farmer B has 360 
 acres, sowed with wheat, which produces 25 bushels to 
 the acre. What quantity of wheat was raised by A and 
 B respectively ? 
 
 15. A speculator bought 146 head of oxen; 230 head 
 of cows; and 69 head of calves. He made a profit of 16 
 dollars a head on the oxen; 12 on the cows; and 5 on the 
 calves. How much did he gain on the oxen, cows, and 
 calves respectively ? 
 
 16. A merchant bought 12 boxes of linen, each con- 
 'taining 25 pieces, and each piece containing 36 yards, at 
 65 cents a yard. How many pieces, and how many yards 
 did he buy, and how much did it all cost him ? 
 
 17. A has 395 acres of land, worth 2t dollars an acre; 
 and B has 493 acres, worth 19 dollars an acre. What is 
 the value of each of their farms ? 
 
4.RT. 41. J MULTIPLICATION. 89 
 
 18. In a certain orchard there are 26 rows of apple- 
 trees and 36 trees in each row. How many apples would 
 there be in the orchard, allowing 2595 apples to each tree ? 
 
 19. A farmer purchased five tracts of land, each con- 
 taining 395 acres, at 95 dollars an acre. What was the 
 whole cost ? 
 
 20. The circumference of the earth is nearly 25000 
 miles, the distance to the sun is 3800 times as much. What 
 is the distance to the sun ? 
 
 Art. 41. A Composite number is one that can be pro- 
 duced by multiplying two or more numbers together, each 
 of which is greater than a unit. 
 
 Thus, 15 is a composite number, as it can be produced 
 by multiplying together the numbers 3 and 5. The 3 and 
 5 are called the factors of 15. 
 
 Art. 42. When the multiplier is a composite number^, 
 resolve it into two or more factors, then multiply the mul- 
 tiplicand by one of these factors, and the product thus 
 obtained by another factor, and so on, until all the fac- 
 tors have been used as a multiplier. The last product 
 will be the answer sought 
 
 1. Multiply U3 by 35. 
 
 OPERATION. 
 743 
 
 7 Explanation. — The factors of 35 are 5 and 7; 
 
 hence, multiplying by 5 and 7, or 7 and 5, will 
 
 5201 produce the same result as multiplying by 35; 
 
 5 since 5 X 7 = 35. 
 
 26005 
 
 2. What cost 325 bushels of potatoes, at 63 cents a 
 bushel ? ' "^ 
 
 3. What cost 437 melons, at 21 cents a piece ? 
 
 4. What cost 395 yards of muslin, at 27 cents a yard ? 
 
 5. What cost 49 sheep, at 425 cents a head ? 
 
 6. What cost 77 horses, at 245 dollars a piece ? 
 
 7 What cost 15 acres of land, at 595 dollars an acre ? 
 
iO MULTIPLICATION. [CHAP. II 
 
 8. What cost to bush, of wheat, at 145 cents a bushel ? 
 
 9. What cost 18 pounds of opium, at 845 cents a pound ? 
 
 10. What cost 21 books at 95 cents a piece ? 
 
 Art. 43. MuUiplication of abstract numbers, when 
 there are ciphers on the right of the multiplier or multi- 
 plicand, or both. 
 
 1. Multiply 3464 by 2430000. 
 
 OPERATION. Explanation. — Write the numbers down, so 
 
 3464 that the right hand significant figures of the 
 
 3430000 *"^^ factors shall come one under the other ; 
 
 . ^ then multiply as in Case 2, Article 40, and 
 
 10392 bring the ciphers down on the right of the 
 
 13856 product. 
 10392 
 
 11881520000 
 
 Remark. — This method of operation is a particular case 
 under Art. 41. For in fact the number 3430000, is resolved 
 into the two factors 343 and 10000. We first multiplied 
 
 the minuend by 343, and then the product thus obtained by lOOflO, which il 
 
 done by merely adding four ciphers. 
 
 2. Multiply 2460000 by 432000 
 
 OPERATION. 
 
 Multiplicand, 2460000 
 
 Multiplier, 432000 
 
 492 
 738 
 984 
 
 Product, 1062720000000 
 
 3. Multiply 232 by 10. 
 
 4. Multiply 682 by 100. 
 
 5. Multiply 543 by 1000. 
 
 6. Multiply 4321 by 10000. 
 
 7. Multiply 3261 by 100000. 
 
 8. Multiply 246800 by 100000. 
 
 9. Multiply 2326000 by 43000. 
 
 10. Multiply 3680200 by 4863000. 
 
 11. Multiply 12360000 by 43298000. 
 
 12. Multiply 326000200 by 20046000 
 
ART. 48.] MULTIPLICATION 41 
 
 PRACTICAL QUESTIONS COMBINING ADDITION, SUBTRACTION, 
 AND MULTIPLICATION. 
 
 1. If a wagon cost 48 dollars, a yoke of oxen 3 times 
 as much, lacking 54 dollars, and a span of horses as much 
 as the wagon and oxen together; what was the cost of the 
 oxen and horses respectively, and of all ? 
 
 2. A man paid for building his house 2460 dollars; 
 for his farm 4 times as much, lacking 986 dollars; and for 
 his furniture, 122 dollars less than he paid for building 
 his house. How much did he pay for all, and for each 
 respectively ? 
 
 3. Two persons start together from the same place, and 
 travel in the same direction. One proceeds at the rate of 
 35 miles a day; the other at the rate of 42 miles a day. 
 What distance will they be apart at the end of 45 days ? 
 
 4. Bought 19t acres of land, at 47 dollars an acre; at 
 another time double the number of acres, at double the 
 price per acre, lacking 12 dollars; and at another time as 
 many acres as I had already bought, at 145 dollars an 
 acre. How many acres did I buy, and how much did it 
 cost me ? 
 
 5. A farmer purchased three tracts of land: the first 
 contained 195 acres; the second 6 times as much, lacking 
 203 acres; the third as much as the first and second 
 together, and 45 acres more. How many acres did the 
 farmer purchase, and what did the whole amount to, at 45 
 dollars an acre ? 
 
 6. A planter sold 465 bales of cotton, at 35 dollars a 
 bale; and out of the proceeds bought 18 mules, at 65 
 dollars each; 6 span of horses, at 141 dollars a span; 
 and 4 yoke of oxen, at 95 dollars a pair. How much 
 money had he left from the sale of his cotton ? 
 
 7. Mr. B.'s yearly income is 2890 dollars: he pays for 
 house-rent 265 dollars; his family expenses amount to 7 
 times as much, lacking 199 dollars. How much does he 
 save annually ? 
 
 8. A man, having 6894 dollars, paid out of it 1684 
 dollars for a farm; twice as much, lacking 1999 dollars. 
 
42 MULTIPLICATION. [CHAP. II. 
 
 for building a house; and the remainder, lacking 989 dol- 
 lars, for farmiug utensils and furnishing his house. What 
 was the cost of the farm, the house, and of the farming 
 utensils and furniture of the house, respectively ? 
 
 9. A has 789 sheep; B has 4 times as many, lacking 
 999; and D has 45 sheep more than A and B together. 
 How many sheep has B and D^respectively, and how 
 many have they all ? 
 
 10. A is worth 8967 dollars; B is worth 285 dollars 
 more th^n A; and C is worth as much as A and B to- 
 gether, lacking 3794 dollars. How much are B and 
 worth respectively ? 
 
 11. In an army of 8645 men, 1864 men were killed in 
 an action; and 4 times as many wounded, lacking the 
 number that deserted, which was 984. How many men 
 were wounded, and how many remained in the army ? 
 
 12. A certain house is worth 1460 dollars; the farm 
 on which it stands is worth 5 times as much, + 896 dollars; 
 and the stock on the farm is worth 4 times as much as the 
 house, lacking 1980 dollars. What is the value of all, 
 and of the farm and stock respectively ? 
 
 13. A lends B 12804 dollars. B let A have bank 
 stock to the amount of 2042 dollars; a farm for 5 times 
 as mucli as the bank stock, lacking 989 dollars; and is to 
 pay the remainder in cash. How much cash ought B to 
 pay A ? 
 
 14. John has 240 sheep; James 15 times as many + 
 146; and Joseph 8 times as many as both John and James, 
 lacking S999. Hbw many sheep has James and John, 
 and how many have they all t 
 
 15. If a cow cost 43 dollars; a horse 5 times as much; 
 and a farm 9 times as much as the cow and horse together, 
 lacking 36 dollars; how much more will the farm cost 
 than 5 horses and 9 cows, at the same rate ? 
 
 16. If a quantity of sugar cost 1465 dollars; a store 
 15 times as much, lacking 9999 dollars; and the lot on 
 which the store stands 2 times as much as the sugar and 
 gtore together, + 146 dollars; what will be the cost of 
 all, and of the store and lot respectively ? 
 
i 
 
 ART. 44.] DIVISION. 48 
 
 It. Said Martha to Baldwin, I am worth 245 dollars; 
 Baldwin replies, that is exactly 1 fifth as much as Ann is 
 worth, and 1 twelfth as much as I am worth. How much 
 are Ann and Baldwio together worth ? 
 
 18. If a quantity of floar cost 2864 dollars; the store 
 in which it is deposited, 14 times as much, lacking 984 
 dollars; and the lot on which the store stands 3 times as 
 much as the flour and store together, + 183 dollars; — 
 what will be the cost of all, and of the store and lot re- 
 spectively ? 
 
 19. A merchant bought 12 pieces of broadcloth, each 
 piece containing 32 yards, at 5 dollars a yard for th^two 
 pieces; 6 dollars a yard for six pieces; and 8 dollars a 
 yard for the remaining four pieces. He sold it all, at 7 
 dollars a yard, did he gain or lose, and how much ? 
 
 DIVISION. 
 
 Art. 44. Division teaches how to find the number 
 of times, or part of a time that one number is contained 
 in another. 
 
 There are three terms employed in division, the Divisor, 
 Dividend and Quotient. That which is left, (if any,) 
 after the division, is called the Remainder; — we have not 
 called it a distinct term of division, as it is a part of the 
 dividend. 
 
 The Divisor is the dividing number. The Dividend is 
 the number to be divided. The Quotient is the number 
 of times the dividend contains the divisor. 
 
 Division is indicated by the symbol, —. This sign 
 when placed between two quantities, shows that the num- 
 ber an the left is to be divided by the one on the right. 
 Thus, 8 -f- 4 = 2; shows that 8 is to be divided by 4, 
 and that the quotient is 2. In the division of concrete 
 numbers, the divisor is always considered abstractly. The 
 quotient is a concrete number of the same kind as the 
 dividend. 
 
 Division is also indicated by writing the divisor under 
 the dividend; thus, ^^2 = 3. 
 
44 
 
 SHORT DIVISION. 
 
 [chap. II 
 
 DIVISION TABLE. 
 
 1 
 2 
 
 3 
 4 
 6 
 1= 6 
 
 96-r-8= 
 
 S-r- 
 
 6-^ 
 9-r- 
 
 12 
 
 15 
 
 18-^ 
 21 -j- 
 
 24-^- 
 
 9 27-i- 
 
 :10 30 
 :11 
 
 =12 36-^ 
 
 3= 1 
 3= 2 
 3= 3 
 3= 4 
 3= 6 
 3= 6 
 3= 7 
 3= 8 
 = 9 
 3=10 
 3=11 
 3=12 
 
 4-^4 
 
 8-f-4 
 12^4 
 
 16-^4: 
 
 20^-4 
 
 24-f-4: 
 
 28-^4: 
 32-J-4: 
 
 364-4: 
 
 40-^4: 
 44-f-4: 
 
 48-j-4=12 60-j- 
 
 5-T 
 
 10-^ 
 
 15-^ 
 
 20^ 
 
 25 -f- 
 
 30 
 
 35 
 
 40-4- 
 
 45 
 
 60 
 
 55 
 
 5= 1 
 5= 2 
 5= 3 
 5= 4 
 5= 5 
 5= 6 
 5= 7 
 5= 8 
 5= 9 
 5=10 
 5=11 
 5=12 
 
 6- 
 12- 
 18- 
 24-H 
 30-f- 
 36-h 
 42-1- 
 48-^ 
 
 54-^ 
 
 60-v- 
 72-- 
 
 ■6= 1 
 ■6= 2 
 ■6= 3 
 
 14-h7 
 21-^-7 
 
 6= 4 28-i-7= 
 6= 635-^7= 
 6= 642-h7= 
 6= 749-^7= 
 6= 856-h7= 
 6= 9I63-^7= 
 6=10:70-^-7= 
 6=ll|77-T-7= 
 6=12 84-^7= 
 
 = 11 
 = 21 
 = 3 
 4 
 5 
 6 
 7 
 
 ■8= 
 
 9 
 
 18 
 27 
 36 
 45 
 54 
 63 
 72 
 81 
 90 
 99 
 108 
 
 ^9= 1 
 -j-9= 2 
 -^9= 3 
 -f-9= 4 
 -^-9= 5 
 H-9= 6 
 -j-9= 7 
 -j-9= 8 
 -h9= 9 
 -7-9=10 
 -^9=ll 
 -J-9=12 
 
 10- 
 20- 
 30- 
 40- 
 50- 
 60- 
 70- 
 80- 
 90- 
 
 100. 
 
 110. 
 
 120- 
 
 10= 1 
 10= 2 
 10= 3 
 10= 4 
 10= 5 
 10= 6 
 10= 7 
 10= 8 
 10= 9 
 10=10 
 10=11 
 10=12 
 
 11-r-l 
 
 22-^l 
 33-f.l 
 44-f-l 
 55-^1 
 66-^1 
 77-^l 
 
 88-r-l 
 
 99-j-l 
 110-i-l 
 121 -j-1 
 
 132-r-l 
 
 12-hl2= 
 
 24-^12= 
 
 36-^12= 
 
 48-T-12= 
 
 60-^12= 
 
 72-hl2= 
 
 84-^12= 
 
 96-j-12= 
 
 108-r-12= 
 
 120-hl2= 
 
 132-^12= 
 
 144-T-12= 
 
 Short Division. 
 
 Art. 45. Division of abstract numbers, when the 
 divisor does not exceed 12. 
 1. Divide 8245 by 5. 
 
 OPERATION. Explanation. — Write the divisor at the 
 
 » . left of the dividend, with a curved Hne be- 
 
 g -I tween them ; and under the dividend draw a 
 
 horizontal line. Begin at the left and pro- 
 ceed ; thus, 5 is contained in 8 thousands, 
 1 thousand times and 3 thousands remain- 
 ing. Write the 1 thousand down by pla- 
 cing the 1 under the figure divided. The 
 remainder, 3 thousands, added to 2 hun- 
 dreds, (which is the same as prefixing the 
 3 to next figure,) are 32 hundreds. 5 is contained in 32 hun- 
 dreds, 6 hundreds times and 2 hundreds remaining. Write 
 the 6 hundreds down by placing the 6 under the last figure 
 
 
 Divisor. Dividend. 
 5) 8235 
 
 Quotient, 16 4 7 
 
ART. 45.] SHORT DIVISION. 45 
 
 divided. The remainder, 2 hundreds, added to 4 tens, (which 
 is the same as prefixing the 2 to the next figure,) are 23 tens. 
 5 is contained in 23 tens, 4 tens times and 4 tens remaining 
 Write the 4 tens under the last figure divided. The remain- 
 der, 3 tens added to 5 units, are 35 units. 5 is contained in 
 35 units, 7 units times, which place under the last figure 
 divided ; and we obtain for the quotient, 1647. 
 
 Rr.MARK. — After the pupil thoroughly understands the abore explantion, the 
 following may be adopted. 
 
 OPERATION. Explanation. — 5 is contained in 8, 1 
 
 Div'sor D' *dpnd ^°^ ^ remaining. Write down the 1 and 
 
 cN 0035 * prefix the remainder to the next figure. 
 
 ^ 5 is contained in 32, 6 times and 2 remain- 
 
 Quotient, 1647 '^%. "^f^ ^^^^ f ^ ^- . ? ^^ contained 
 ' m 23 4 times and S remaining. 5 is con- 
 
 tained in 35, 7 times and no remainder. 
 
 2. Divide 1467 by 7. 
 
 OPERATION. 
 
 Divisor. Dividend 
 7) 1 4 6 7 7 
 
 Quotient, 2 9 6 — 5 remaider. 
 
 Rf.mark. — The remainder 6 may be divided by 7, and written with the quo- 
 tient; thus, 2096f or mentioned simply as a remainder, as occasion requires. 
 
 Proof. — Multiply the divisor by the quotient and add in 
 the remainder, if there be any. If this sum is equalto the 
 dividend, the work is right. 
 
 Remark. — From what we have already learned, we discover that division 
 is the reverse of multiplication, and that either may be used to verify, or 
 prove the correctness of the work of the other. 
 
 3. Divide 4682 by 2. 
 
 4. Divide 3468 by 2. 
 
 5. Divide 7639 by 3. 
 
 6. Divide 8472 by 4. 
 
 7. Divide 89631 by 4. 
 
 8. Divide 142632 by '^ 
 
 9. Divide 34682 by 6 
 
 10. Divide 24673 by 5. 
 
 11. Divide 147268 by 5. 
 
46 SHORT DIVISION. , [CHAP. II. 
 
 12. Divide 4*76846 by 9. 
 
 13. Divide 4t6342 by 8. 
 
 14. Divide 8462324 by 8. 
 
 15. Divide 8496T23 by 9. 
 
 16. Divide 846t232 by 8. 
 11. Divide 246832 by 10. 
 
 18. Divide 46t232 by 11. 
 
 19. Divide 2468324 by 12. 
 
 PRACTICAL QUESTIONS. 
 
 1. If 9 acres of land cost 2250 dollars, what will 1 
 acre cost ? 
 
 2. If 8 horses cost 1696 dollars, what will 1 horse 
 cost ? 
 
 3. If a man travel 693 miles in 9 days, how far does he 
 travel in 1 day ? 
 
 4. Divide 1648 acres of land equally among 8 indivi- 
 duals. 
 
 5. If 6 horses sell for 1332 dollars, what will be the 
 average sum received for each ? 
 
 6. A man bought 12 tons of hay for 192 dollars; how 
 much did he pay a ton ? 
 
 1. A boy sold 11 rabbits for 286 cents; how much did 
 lie receive a piece ? 
 
 8. A girl spent 342 cents for oranges, at 3 cents a 
 piece; how many oranges did she buy ? 
 
 9. Divide 68425 dollars equally among t sons. 
 
 10. How many barrels of flour, at 6 dollars a barrel, 
 can be bought for 25218 dollars ? 
 
 11. At 8 dollars a cord, how many cords of wood can 
 be bought for 1928 dollars ? 
 
 12. At 5 dollars a barrel, how many barrels of cider 
 can be bought for 1465 dollars ? 
 
 13. If in 1 week there are Y day, how many weeks are 
 there in 365 days, (one year) ? 
 
 14. A man bought a store for 3*192 dollars, which was 
 3 times as much as his house cost him; how much did his 
 house cost him ? 
 
ART. 46."] LONG DIVISION. 41 
 
 15. A drover bought 12 oxen for It 64 dollars; how 
 much was the average cost of each ? 
 
 16. A laborer worked 12 months for 288 dollars; how 
 much did he receive a month ? 
 
 IT. A is worth 15T95 dollars, which is 5 times as much 
 as B is worth, and B is worth 3 times as much as C; how 
 much are B and C worth respectively ? 
 
 18. A's house cost 2358 dollars, which is 3 times as 
 much as the furniture of the house cost; what was the 
 cost of the furniture ? 
 
 19. Says A to B, I have T4 sheep;' B replies, that is 
 just 1 tenth of my number, which is 4 times C's number; 
 how many sheep has C ? 
 
 20. Edward is worth 2000 dollars, which is 3 times 
 Luther's fortune, lacking TOO dollars: and Caleb is worth 
 4 times as much as Edward and Luther together +400 
 dollars. What is the fortune of each ? 
 
 Long Division. 
 
 Art. 46. Division of abstract numbers in general. 
 1. Divide 4379 by 24. 
 
 OPERATION. Explanation. — Write the divi- 
 
 sor on the left of the dividend; 
 and the quotient on the right, 
 separating them with a curved 
 line, and proceed thus: 24 is 
 Divisor. Dividend. Quotient, contained in 43 hundreds and 79, 
 24) 4 3 7 9(100 } hundred times ; write the 100 
 
 2400 80 ^" quotient, 100 times 24 is 
 
 2 2400, which l3eing subtracted 
 
 I Q Y Q from the dividend, leaves 1979, 
 
 10 90 18911 24 is contained in 1979, 80 times; 
 
 ^^^^' -^ ° - 2T ^rite the 80 in the quotient, 80 
 
 times 24 are 19*20, which being 
 
 subtracted from the 1979, leaves 
 
 59. 24 is contained in 59, 2 
 
 S3 "^ M " 
 
 C C C-- 
 
 59 
 
 48 
 
 Remainder, 1 1 *!«^f ' ™*^ ^^^Z '""a^^ 2"t 
 
 ' tient. 2 tunes 24 are 48, which 
 
48 LONG DIVISION. [cHAP. H 
 
 being subtracted from the 59, leaves 11. Dividing the 11 by 
 24 we have |^, which annex to the quotient. 
 
 Remark. — After the pupil comprehends the above explanation, the follow- 
 ing may be adopted. 
 
 OPERATION. Explanation. — 24 is contain- 
 
 Divisor. Dividend. Quotient, ©d in 43, 1 time. Write the 1 in 
 
 24) 4379 (182^^ *h® quotient. 1 times 24 is 24, 
 
 24 ^ which being subtracted from 43, 
 
 leaves 19. Bring down the next 
 
 197 figure of the dividend. 24 is 
 
 192 contained in 197, 8 times ; write 
 
 the 8 in the quotient. 8 times 
 
 59 24 are 192, which being sub- 
 
 48 tracted from 197, leaves 5. Bring 
 
 down the next figure of the divi- 
 
 11 dend. 24 is contained in 59, 2 
 
 times. Write the 2 in the quo- 
 tient, 2 times 24 are 48, which being subtracted from 59, leaves 
 11. Divide the remainder by 24 ; thus, ^^ ; and place it in the 
 quotient. 
 
 Proof hy the excess of 9'^. — Find the excess of 9's in the 
 divisor and quotient respectively, and also, the excess of 
 9's in the product of these two excesses; and if this last 
 excess is equal to the excess of 9's in the difference be- 
 tween the dividend and remainder, the work is right. 
 
 Take for illustration the above example : 
 
 Divisor, 24 =6 excess. 
 
 Quotient, 182 = 2 excess. 
 
 Dividend, 4379 
 Remainder, 11 
 
 Difference, 4368 = 3 J excess, 
 
 2. Divide 4368 by 13. 
 
 3. Divide 369a by 15. 
 
 4. Divide 8041 by 11. 
 
 5. Divide 5490 by 15. 
 
 6. Divide 1242 by 2t. 
 1. Divide 66384 by 24. 
 
LET. 46.] LONG DIVISION. 49 
 
 8. Divide 108220 by 28. 
 
 9. Divide 18336 by 24. 
 
 10. Divide 2841 by 29. 
 
 11. Divide 3570 by 15. 
 
 12. Divide 6048 by 72. 
 
 13. Divide 3607344 by 24. 
 
 14. Divide 949073 by 73. 
 
 15. Divide 9334949 by 307. 
 
 16. Divide 789591 by 213. 
 
 17. Divide 86431 by 342. 
 
 18. Divide 986321 by 412. 
 
 19. Divide 2364 by 82. 
 
 20. Divide 146832 by 147. 
 
 21. Divide 246832 by 432. 
 
 22. Divide 846324 by 1432. 
 
 23. Divide 98476324 by 1463. 
 
 24. Divide 1476324 by 1482. 
 
 25. Divide 47632463 by 24801. 
 
 26. Divide 476784631 by 1472 
 
 27. Divide 48468234 by 423. 
 
 28. Divide 123456789 by 846. 
 
 29. Divide 987654321 by 146. 
 
 30. Divide 987644698321 by 3223. 
 
 PRACTICAL QUESTIONS. 
 
 1. If one man can accomplish a certain piece of work 
 in 494 days, bow many days will it take 38 men to do the 
 same ? 
 
 2. If 99 sheep cost 396 dollars, what will 1 sheep cost ? 
 
 3. If 97 acres of land cost 22989 dollars, how much is 
 that an acre ? 
 
 4. What cost 1 barrel of flour, if 36 barrels cost 288 
 dollars ? 
 
 5. If in 89 books there are 28035 pages, how many 
 pages on an average in a book ? 
 
 6. If an iceberg move at the rate of 25 miles a day, 
 how many days would it be in moving from the north pole 
 to the equator, it being about 6250 miles ^ 
 
 8 
 
60 LONG DIVISION. 
 
 CHAP. II. 
 
 t. If a horse can travel 54 miles in a day, how many 
 days will it take it to travel 10854 miles ? 
 
 8. If 15 months' wages amount to 525 dollars, how 
 much is that a month ? 
 
 9. If Ml quails are sold for 128*7 cents, how much is 
 that a piece ? 
 
 10. If 38 baskets of peaches are sold for 2850 cents, 
 how much is that a basket ? 
 
 11. A drover bought cattle, at 3t dollars a-head, and 
 paid for them 8732 dollars, how many did he buy ? 
 
 12. How many barrels of molasses, at It dollars a 
 barrel, can be bought for 3604 dollars ? 
 
 13. How many pieces of cloth, at 95 dollars a piece, can 
 be bought for 3385 dollars ? 
 
 14. If 63 gallons make 1 hogshead, how many hogsheads 
 will 1449 gallons make ? 
 
 15. For 1016 dollars, how many yards of broadcloth 
 can be bought, at 8 dollars a yard ? 
 
 16. If a steamship can cross the Atlantic Ocean, a dis- 
 tance of 3000 miles, in 9 days; how many miles does the 
 ship go daily ? 
 
 17. Baldwin's income is 2555 dollars a year, how much 
 is that a day, allowing the year to consist of 365 days ? 
 
 18. Walter purchased a farm containing 235 acres, for 
 4230 dollars; how many dollars did he pay an acre ? 
 
 19. In how many days could 27 men accomplish the 
 same amount of work, that 1 man could in 594 days ? 
 
 20. If a railroad car move at the rat*^ of 625 miles 
 a day, in how many days would it go around the earth, 
 the distance being about 25000 miles ? 
 
 Art. 47. To divide one number by another, when the 
 divisor is a composite number. 
 
 Resolve the number into two or more fado's, then divide by 
 one of these factors, and the quotient thus obtained by another 
 factor, — proceed in the same way till all the factors have be 
 come, divisors, and the last quotient obtained will be the 
 answer required. 
 
 Art. 48. To find the true remainder. 
 
ART. 48.] LONG DITISION. 51 
 
 To the sum of the products of each remoXTider into all the 
 divisors preceding the one that produced it, add the first re- 
 mainder, and this sum will be the true remainder. 
 
 1. Divide 2486 by 105. 
 
 The factors of 105 are 3, 5 and t. 
 
 OPERATION. 
 
 1 3)2486 
 
 2. 5)828—2 Ist remainder. 
 
 „ 3. 7)165—3 2nd 
 
 Quotient, 23—4 3rd " 
 
 Explanation — The small figures 1, 2, 3, on the left of the 
 divisors are used to designate the numbers that have become 
 dividends. The ^d remainder is the same as the 3d dividend, 
 but a unit of the Zd dividend is equal to 5 units of the 2nd 
 dividend ; and a unit of the 2nd dividend is equal to 3 units of 
 the \st dividend, since the 1st and 2nd dividends have been 
 divided respectively by 3 and 5. Therefore, a unit of the 3d 
 remainder is equal to 5x3 units of the \st remainder, and 4 
 units of the 3d remainder is 5x3x4=60 units of the first re- 
 mainder. For a similar reason 3 units of the 2nd remainder ia 
 equal to 3x3=9 units of the Ist remainder. To the sum of 
 these products add the first remainder, and we have 60-f-9-j-2=71 
 the true remainder. 
 
 2. Divide 4898 by 21. 
 
 3. Divide 9042 by 15. 
 
 4. Divide 11128 by 1155. 
 
 5. A man bought 15 horses for 2910 dollars; how 
 much was that a piece ? 
 
 6. If 2T barrels of flour cost 250 dollars, how much is 
 that a barrel ? 
 
 7. A wealthy merchant distributed 588 yards of cloth 
 equally among 49 poor individuals; how many yards did 
 they receive a piece ? 
 
 8. A drover paid 1456 dollars for cattle, giving 56 
 doi]ars a head; how many cattle did he buy ? 
 
 9. A farmer bought 98 acres of land for 2178 dollars; 
 bow much did he pay an acre ? 
 
52 
 
 LONG DIVISION. 
 
 . [chap. II. 
 
 10. In a certain corn-field there are 5229 hills of corn, 
 and 63 rows; how many hills in a row ? 
 
 Art. 49. Division of abstract numbers, when the divi- 
 sor, or dividend, or both have ciphers on the right. 
 1. Divide 82468524 by 24500. 
 
 OPERATION. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 •7. 
 
 8. 
 
 9. 
 10. 
 11. 
 12. 
 13. 
 14. 
 15. 
 
 245!00)824685|24(33662V^ 
 735 
 
 896 
 - 735 
 
 1618 
 1470 
 
 1485 
 1470 
 
 Rkmark.— I cu| off the 
 ciphers on the rigfct of the 
 divisor, and a,s many places 
 on the right'^jf the divi- 
 dend. Alter the division, 
 affixing the remainder to 
 the quotient with the divi- 
 sor under it, and a hori- 
 zontal line between them, 
 and we have no remainder. 
 
 Remainder, 1524 
 
 Div 
 Div 
 Div 
 Div 
 Div 
 Div 
 Div 
 Div 
 Div 
 Div 
 Div 
 Div 
 Div 
 Div 
 
 de 2468 by 10. 
 
 de 374232 by 100. 
 
 de 468324 by 1000. 
 
 de 36842 by 1100. 
 
 de 468234 by 450. 
 
 de 476324 by 4810. 
 
 de 846324 by 7800. 
 
 de 14786324 by 48300. 
 
 de 246832 by 470. 
 
 de 2476800 by 470. 
 
 de 8468300200 by 47600. 
 
 de 12468300200 by 3680. 
 
 de 4780024680000 by 8496000. 
 
 de 8468476008470000 by 84000. 
 
 ABSTRACT EXAMPLES IN THE FUNDAMENTAL RULES. 
 
 Art. 50. Quantities enclosed in a parenthesis, some- 
 times called the sign of aggregation, ( ), are to be subject- 
 ed to the same operatien Thug, (3+6 — 2)x5, denotes 
 
ART. 50.] LONG DIVISION. 58 
 
 tHat the sum of 3 and 6, lacking 2 is to be multiplied by 
 5, the product of which is 35. 
 
 1. What is the value of the expression, (465 — 2*1+14:0) 
 X8.? 
 
 2. What is the value of the expression, .(846+41 + 96) 
 X25? 
 
 3. What is the value of the expression, (891— 4t+86) 
 XI— 184.? 
 
 4. What is the value of the expression, 464+ (843— 
 81 + 9) X416— 461.? 
 
 5. What is the value of the expression, 461 — 189 + 
 (88— 14 + 215)X91? 
 
 6. What is the value of the expression, (462+1—146) 
 X (84— 14 + 115).? 
 
 1. What is the value of the expression, 96+ (144 — 91) 
 X(86— 41— 189)-7-93? 
 
 8. What is the value of the expression, (41 — 23+12) 
 -7-9 + (98+4)X(144— 91)? 
 
 9. What is the value of the expression, (14 + 1)X2 + 
 (256—25)^21? 
 
 • 10. What is the value of the expression, (8 9 6 — 116) -r 
 144 + (214 + 82)X(86— 41 + 8)? 
 
 PRACTICAL QUESTIONS COMPRISING THE FOUR FUNDAMENTAL 
 RULES. 
 
 1. Henry has 4684 dollars, which lacks 248 dollars of 
 being 4 times James' fortune; and Jackson is worth 3 
 times as much as Henry and James together, lacking 3421 
 dollars. How much money have James and Jackson 
 respectively ? 
 
 2. A man bought an equal number of cows and horses 
 for 9120 dollars; for the cows he gave 234ollars a piece ; 
 and for the horses 91 dollars a piece; how many of each 
 did he buy .? 
 
 3. A merchant expended 336 dollars for an equal num- 
 ber of yards of broadcloth, consisting of three different 
 kinds; the first, at 5 dollars a yard; the second, at 1 dol- 
 
54 LONG DIVISION. [CHAP. II. 
 
 lars ; and the third, at 9 dollars a yard. How many yards 
 of each kind did he buy ? 
 
 4. A farmer sold an equal number of chickens, ducks, 
 and geese for 3540 cents; the chickens, at 12 cents each; 
 the ducks, at 37 cents each; and the geese, at 69 cents 
 each. How many of each kind did he sell ? 
 
 5. A gave 1 eighth of 89648 dollars for a farm, which 
 was 237 dollars more than it was worth; how much was 
 the farm worth ? 
 
 6. Light moves about 11550000 miles a minute ; at this 
 rate^ how long would light be in passing from the sun to 
 the earth, a distance of 95000000 of miles ? 
 
 7. The product of two numbers is 91096; and one of 
 the numbers is 472. What is the other number ? 
 
 8. The quotient arising from dividing one number by 
 another is 345: the dividend is 273585. What is the divi- 
 sor ? 
 
 9. The quotient arising from a certain division is 437; 
 the divisor is 413; and the remainder 247. What is the 
 dividend ? 
 
 10. A farmer's yearly income was 19437 dollars. He 
 paid for repairing his house 313 dollars; for hired help on 
 his farm, 5 times as much, lacking 65 dollars; and for 
 traveling expenses 2463 dollars. How much does he save 
 yearly ? 
 
 11. Bought 45 barrels of flour for 225 dollars; for what 
 must it be sold a barrel to gain 135 dollars, and what 
 will be the gain on each barrel ? 
 
 12. Bought 130 acres of land for 5850 dollars; and sold 
 112 acres of it, at 75 dollars an acre, and the remainder 
 for what it cost; how much did I gain by the bargain ? 
 
 13. Bought 150 acres of land for 9750 dollars; and 
 sold apart of it for 7140 dollars, at 85 dollars an acre; — 
 how many acres had I remaining, and how much did I 
 gain on every acre sold ? 
 
 14. A farmer sold corn for 864 dollars; wheat for 895 
 dollars; rye and oats for 3 times as much as he received 
 for the corn and wheat together, lacking 148 dollars. Out 
 of these proceeds he bought 6 span of horses, at 275 dol- 
 
ART. 60.] LONG DIVISION. 65 
 
 lars ^ span; 5 yoke of oxen, at 125 a pair; and the re- 
 mainder, lackiug 738 dollars, he paid for land, at 65 dol- 
 lars an acre. How many acres did he buy ? 
 
 15. Bought 195 acres of land, at 84 dollars an acre, 
 which cost 12 times as much as I paid for a span of fine 
 horses. I have noiw 1468 dollars remaining. JIow much 
 money had I at first ? 
 
 16. If an army of 6000 men have provisions for 5 
 months, and 4400 men be disengaged; how long will the 
 same provisions serve the remainder ? 
 
 n. A certain tradesman can earn 54 dollars a month, but 
 his necessary expenditures are 29 dollars a month. He de- 
 sires to purchase a farm containing 75 acres, worth 35 dol- 
 lars an acre. In what tune can he save money enough to 
 make the purchase ? 
 
 18. Sold to my neighbor 12 cords of wood, at 5 dollars 
 a cord; 65 barrels of corn, at 2 dollars a barrel; 45 head 
 of cattle, at 65 dollars a head. In payment, I take 5 sacks 
 of coffee, at 15 dollars a sack; 25 barrels of sugar, at 15 
 dollars a barrel; 2405 dollars in cash; and the remainder, 
 in molasses, at 26 dollars a barrel. How many barrels of 
 molasses ought I to receive ? 
 
 19. A drover bought a certain number of cattle for 
 8050 dollars, and sold a certain number of them for 6231 
 dollars, at 63 dollars each, and gained on those he sold 
 1683 dollars; how many did he buy at first, and how 
 much did he gain a piece on those he sold ? 
 
 20. A speculator gave 18810 dollars for a certain num- 
 ber of acres of land, and sold a part of it for 1990 dollars, 
 at 85 dollars an acre, and by so doing, lost 10 dollars on 
 each acre; for how much must he sell the remainder an 
 acre to gain 2180 dollars by the operation ? 
 
 21. A farmer gave 37620 dollars for a farm, and sold a 
 certain number of acres of it for 15980 dollars, at 85 dol- 
 lars an acre, and by sodding lost 20 dollars an acre; for 
 how much must he sell the remainder an acre to gain 4360 
 dollars by the operation ? 
 
66 DENOMINATE NUMBERS. [CHAP. III. 
 
 CHAPTER III. 
 
 Tables of Money, Weights and Measures. — Addition, 
 Subtraction, Multiplication, and Division of Polyno- 
 mials, OR Denominate Numbers. 
 
 Art. 5 1 • a Simple Number is either a unit, or a col- 
 lection of units considered abstractly, that is, without 
 reference to any particular thing; as 8, 16, 24, &c. 
 
 Art. 52, A Concrete, or Denominate Number, is 
 either a unit, or a collection of units having reference to 
 some particular thing; as 4 feet, 5 dollars, 8 hours, 25 men, 
 &c. The measuring unit of any quantity is a similar con- 
 crete unit, by means of which the quantity is expressed 
 numerically. 
 
 Art. 53. A Monomial in Algebra, is a quantity of one 
 term only; it may also, with propriety, be applied to an 
 Arithmetical number, when it is expressed by a single name 
 of a measuring unit; as, 5 dollars, 1 bushels, 10 men, &c. 
 
 Art. 54. A Polynomial in Algebra is a quantity con- 
 sisting of many terms; it may also be applied to denominate 
 numbers, signifying a quantity of many names; as, 2 cwt. 
 3 qrs. 15 lbs., &c. It is, however, more generally applied 
 to an abstract number consisting of many terms ; as, 
 (4 + 6 + 8 -I- a,) &c. 
 
 Table of United States Currency. 
 
 Mills 
 
 make 1 Cent, 
 
 marked c. 
 
 Cents 
 
 " 1 Dime, 
 
 " d. 
 
 Dimes 
 
 " 1 Dollar, 
 
 " $. 
 
 Dollars 
 
 " 1 Eagle, 
 
 " E. 
 
 Art. 55. It will be observed that the measftl-ing units 
 in the Dnit*ed States currency increase in a tenfold ratio, 
 as in abstract numbers. Hence this currency will be 
 treated of under Decimal Fractions. The measuring units 
 of other kinds of quantity, increase from lower te higher 
 
ART. 68.] ATOIRDUPOIS WEIGHT. 5t 
 
 orders according to the scales of increase given in the fol- 
 lowing tables. 
 
 English or Sterling Money. 
 Art. 56. English Money is the currency of England. 
 Its denominations are Pounds, Shillings, Pence, and Far- 
 things. 
 
 TABLE. 
 
 4 Farthings (far. or qr.) make 1 Penny, marked d. 
 12 Pence " 1 Shilling, " s. 
 
 20 ShilUngs " 1 Pound, " £. 
 
 Troy Weight. 
 
 Art. 57. By this weight are weighed gold, silver, and 
 jewels. 
 
 Remark. — The original of all weights used in England, was a grain of 
 wheat, taken from the middle of the ear ; 32 of these, dried, were to make 
 1 jiennyweight. 
 
 Since then it was agreed to divide the same pennyweight into 24 equal 
 parts, still called grains, being the least weight in common use. 
 
 TABLE. 
 
 24 Grains (gr.) make 1 Pennyweight, marked, pwt. 
 
 20 Pennyweights " 1 Ounce, " oz. 
 
 12 Ounces . " 1 Pound, " lb. 
 
 Avoirdupois Weight. 
 
 Art. 58. Avoirdupois Weight is used to weigh all 
 things of a course nature, as groceries, some liquids, and 
 all metals, except gold and silver. 
 
 table. 
 
 16 Drams (dr.) make 1 Ounce, marked oz. 
 
 16 Ounces - "1 Pound, " lb. 
 
 25 Pounds* " 1 Quarter, " qr. 
 
 4 Quarters " 1 Hundred Weight, " cwt. 
 
 20 Hundredweight " 1 Ton, " T. 
 
 • Note. — In buying and selling articles, it is customary to call 25 pounds, I 
 qr., instead of 28} and 100 pounds, 1 cwt., instead of 112 pounds, as was for 
 xneily done. 
 
 3* 
 
68 
 
 DENOMINATE NUMBERS. 
 
 [chap. III. 
 
 Apothecaries' Weight. 
 
 Art. 59. Apothecaries' Weight is used in compound- 
 ing, or weighing small quantities of medicines, as for pre- 
 scriptions. But medicines and drugs by the quantity, are 
 generally bought and sold by avoirdupois weight. The 
 pound and ounce Apothecaries' Weight equals the pound 
 and ounce Troy Weight. 
 
 20 Grains (gr.) 
 3 Scruples 
 8 Drams 
 
 12 Ounces 
 
 make 
 
 1 Scruple, 
 1 Dram, 
 1 Ounce, 
 1 Pound, 
 
 marked 
 
 Cloth Measure. 
 Art. 60. Cloth Measure is used in measuring cloth, 
 lace, ribbons, and all other articles sold by the yard. 
 
 21 Inches (in.) 
 4 Nails, or 9 in. 
 
 4 Quarters 
 3 Quarters 
 
 5 Quarters 
 
 6 Quarters 
 
 TABLE. 
 
 make 1 Nail, marked na. 
 
 " 1 Quarter of a yard, " qr. 
 
 " 1 Yard, " yd. 
 
 " 1 Ell Flemish, " E. FL 
 
 " 1 Ell English, " E. E. 
 
 " 1 Ell French, " E. Fr. 
 
 Long Measure. 
 Art. 61. This measure is used in measuring distances. 
 
 12 Inches (in.) 
 
 3 Feet 
 6\ Yards, or 16^ feet, 
 40 Rods 
 
 8 Furlongs 
 
 3 Miles 
 60 Geographic miles, 
 
 or 69| statute or 
 
 league miles 
 360 Degrees 
 
 
 table. 
 
 lalj 
 
 u 
 u 
 
 C( 
 
 u 
 
 e 1 Foot, 
 1 Yard, 
 1 Rod, Pol 
 1 Furlong, 
 1 Mile, 
 1 League, 
 
 
 1 Degree, 
 
 
 1 Circle, 
 
 marked 
 
 ft. 
 
 rd. 
 fur 
 
 m 
 lea 
 
 deg. or 
 
ART. 62.] 
 
 4 Inches 
 6 Feet 
 
 SUPERFICIAL, OR SQUARE MEASURE. 
 
 make 1 Hand. 
 
 59 
 
 Used in measuring 
 the height of horses, 
 
 1 Fathum, sy^^i^^r'''''''"^ 
 
 'I depths at sea. 
 
 Superficial, or Square Measure. 
 
 Art. 62. This measure is used for measuring all kinds 
 of surfaces, such as land, boards, plastering, and everything 
 else, in which length and breadiU only are considered. 
 
 4 feet. A Square is a figure 
 
 havirip; four equal sides, 
 and four equal angles, or 
 four rigi\b angles. 
 
 This diagram is called 
 four feet square^ as it is 
 four feet each way. Each 
 of the small squares, 
 (within the large square,) 
 represents 1 square foot. 
 There are 4 square feet 
 in each row, and 4 rows 
 in the whole square; 
 therefore, there are 4 
 times 4 square feet, equal 
 to 16 square feet^ in 4 feet square ; hence there is a difference 
 of 12 square feet between 4 feet square, and 4 square feet. The 
 4 square feet is represented by the squares 1, 2, 3, and 4 ; and 
 the 4 feet square, by the large square which contains 16 square 
 feet. From the above we infer that the superficial contents of 
 a square, or any rectangular figure is found by multiplying its 
 length with its width. 
 
 1 
 
 Square 
 foot. 
 
 2 
 
 3 
 
 4 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 4 feet. 
 
 TABLE. 
 
 144 Square Inche8(sq.in.) make 1 Square Foot, marked sq. ft. 
 9 Square Feet " 1 Square Yard, " sq. yd. 
 
 M. 
 
 30]- Square Yardu 
 
 " 1 Sq. Rod, or Pole, " 
 
 P. 
 
 40 Square Rodb cr Poles 
 
 " 1 Rood, 
 
 R. 
 
 4 Roods 
 
 " 1 Acre, " 
 
 A. 
 
 640 Acres 
 
 " 1 Square mile, " 
 
 S. 
 
60 
 
 DENOMINATE NUMBERS. 
 
 [chap. hi. 
 
 Surveyor's Measure. 
 In measuring land, roads, &c., Gunter's chain is used; 
 the length of which is 4 rods, or 66 feet. 
 
 TABLE. 
 
 7yVo Inches (in.) make 1 Link 
 
 marked 
 
 li. 
 
 25 Links " 1 Rod, or Pole, 
 
 u 
 
 p. 
 
 4 Poles, or 100 links " 1 Chain, 
 
 (C 
 
 cha. 
 
 10 Chains " 1 Furlong. 
 
 u 
 
 fur 
 
 8 Furlongs, or 80 chains, " 1 Mile, 
 
 c< 
 
 M. 
 
 10 Square Chains " 1 Acre, 
 
 li 
 
 A. 
 
 Solid, or Cubic Measure. 
 
 Art. 63. This measure is used in measuripg all things 
 that have length, breadth, and thickness; as timber, 
 boxes of goods, capacity of ships, &c., &c. 
 
 A cube is a solid, bounded by six equal and square 
 sides. ^ 
 
 If each of the sides of a cube is 1 foot it is called a 
 cubic foot. If each of the sides of a cube be 3 feet = 1 
 yard, it is called a cubic yard. 
 
 The annexed diagram represents 
 a cubic yard. Since each of the 
 sides of a cubic yard is 3 feet each 
 way ; each of these sides will con- 
 tain 9 square feet. If from one j^I 
 side of this cube we cut off a piece l] 
 1 foot in thickness, we evidently % 
 have 9 solid feet ; and as the whole ^ 
 block is 3 feet thick, it must con- 
 tain 3 times 9 = 27 solid feet. 
 Hence, to find the solid contents 
 of a cube, we multiply its length, breadth^ and thickness together. 
 
 3 feet=l yd. 
 
 TABLE. 
 
 1728 Cubic inches (cu. in.) make 1 Cubic foot, marked cu. ft- 
 27 " feet " 1 " yard, " cu. y<J 
 
 40 «' feet " 1 Ton, " T. 
 
 16 " feet " 1 Cord foot, " c. f^ 
 
 8 Cord feet, o- 
 128 Cubic feet 
 
 1 Cord of wood. 
 
 C. 
 
ART. 66.] 
 
 DRY MEASURE. 
 
 \ 
 
 61 
 
 Wine Measure. 
 By this measure all liquids, except beer ar^ 
 
 Art. 64 
 measured. 
 
 The wine gallon contains 231 cubic inches. 
 
 TABLE. 
 
 4 Gills (gi.) make 1 Pint, 
 
 marked pt. 
 
 2 Pints 
 
 1 Qnart, 
 
 " qt. 
 
 4 Quarts ' 
 
 1 Gallon, 
 
 gal. 
 
 31J Gallons 
 
 1 Barrel, 
 
 " bar. 
 
 42" Gallons ♦ 
 
 ' 1 Tierce, 
 
 " tier. 
 
 63 Gallons * 
 
 ' 1 Hogshead, 
 
 hhd. 
 
 2 Hogsheads ' 
 
 ' 1 Pipe, 
 
 « pi. 
 
 2 Pipes 
 
 ' 1 Tun, 
 
 " tun. 
 
 Ale, or Beer Measure. 
 
 Art. 65. By this measure ale, beer, and milk, are 
 measured. 
 
 The beer gallon contains 282 cubic inches. 
 
 table. 
 
 2 
 
 Pints (pt.) make 1 Quart, marked qt. 
 
 4 
 
 Quarts " 1 Gallon, " gal. 
 
 6 
 
 Gallons " 1 Barrel, " bar. 
 
 1| 
 
 Barrels, or 54 Gallons " 1 Hogshead, " hhd. 
 
 Dry Measure. 
 
 Art. 66. Grain, salt, coal, &c., are measured by Dry 
 Measure. 
 
 The Dry Gallon contains 268f cubic inches. The Win- 
 chester Bushel contains 2150f i cubic inches. A Cylindri- 
 cal Measure, 8 inches deep and 18^ inches in diameter, 
 contains 1 bushel. 
 
 TABLE. 
 
 2 Pints (pt.) 
 
 make 1 Quart, marked 
 
 qt. 
 
 8 Quarts 
 
 « 1 Peck, « 
 
 t. 
 
 4 Pecks 
 
 " 1 Bushel, " 
 
 36 Bushels 
 
 " 1 Chaldron " 
 
 ch. 
 
 32 Bushels 
 
 " 1 Chaldron in the United t 
 
 5tate& 
 
62 
 
 DENOMINATE NUMBERS. 
 
 [chap. IIL 
 
 Circular Measure. 
 
 Art. 67. This measure is used in noting any part of 
 the circumference of a circle; it is also used in reckoning 
 latitude and longitude^ and the revolutions of the heavenly 
 bodies. 
 
 60 Seconds (") 
 
 60 Minutes 
 
 30 Degrees 
 
 12 Signs, or 360 Degrees 
 
 TABLE. 
 
 make 
 
 1 Minute, 
 1 Degree, 
 1 Sign, 
 1 Circle, 
 
 marked 
 
 Measure of Time. 
 
 Art. 68. This measure is applied to the divisions and 
 sibdivisions of time. 
 
 60 Seconds (sec.) 
 
 
 make 
 
 1 Minute, marked min. 
 
 60 Minutes 
 
 
 a 
 
 1 Hour. 
 
 hr. 
 
 24 Hours » 
 
 
 (I 
 
 1 Day, 
 
 da. 
 
 7 Days 
 
 
 a 
 
 1 Week, 
 
 '' wk. 
 
 4 Weeks 
 
 
 ii 
 
 1 Month, 
 
 " mo. 
 
 12 Calendar months, or 
 
 52 Weeks, 1 day, and 6 hours 
 
 
 1 Year, 
 
 " yr. 
 
 )65 Days, 6 hours, (nearly,) 
 
 u 
 
 1 Year, 
 
 " yr. 
 
 The Solar year consists of 365 days, 5 hours, 48 min- 
 utes, and 5 If seconds, and is the exact time in which the 
 Earth performs one revolution around the Sun. 
 
 The Civil year consists of 365 days. Hence, the dif- 
 ference between the Solar and the Civil year is nearly 6 
 hours, making about 1 day in 4 years. 
 
 As the difference between the Solar and the Civil year 
 confused dates, Julius Caesar made the first correction of 
 the calender by introducing an intercalerary day in every 
 fourth year. This day was added to the month of 
 February making it to consist of 29 instead of 28 days. 
 This fourth year was denominated Bissextile^ and is now 
 usually called Leajp Year. As the correction should have 
 
ART. 68.J MEASURE OF TIME. 63 
 
 been 5 hrs. 48 min. and 51f sec, instead of 6 hours, by 
 considering every fourth year as consisting of 866 days, 
 there was involved an error amounting to about 18 hours 
 in every 100 years ; to remedy which every one-hundredth 
 year was considered as having only 365 days. But the 
 allowance of a whole day in every 100 years was too much, 
 by nearly one-fourth of a day, which excess in every 400 
 years amounted to an entire day. 
 
 Hence, every year, (except the centennial years,) that 
 is divisible by 4 is a Leap Year, and every centennial year 
 that is divisible by 400 is also, a Leap Year. The next 
 centennial year that will be a Leap Year is 2000. 
 
 The following are the names of the twelve calendar 
 months, which compose the civil year, and the number of 
 days In each : — 
 
 Names. Days. 
 
 « 1 
 
 1 1st month January, 
 
 31 
 
 1 1 
 
 I 2d 
 
 a 
 
 February, 
 
 28— in leap year 29. 
 
 bD 1 
 
 r 3d 
 
 
 March, 
 
 31 
 
 •g H 
 
 4th 
 
 
 April, 
 
 30 
 
 tc* 1 
 
 [ 5th 
 
 
 May, 
 
 -^ 31 
 
 *-< 
 
 r 6th 
 
 
 June, 
 
 30 
 
 s 
 
 7th 
 
 
 July, 
 
 31 
 
 m 1 
 
 [ 8th 
 
 
 August, 
 
 31 
 
 a . 
 
 { 9th 
 
 
 September, 
 
 30 
 
 _s ^ 
 
 10th 
 
 
 October, 
 
 31 
 
 <j 1 
 
 [nth 
 
 
 November, 
 
 30 
 
 ^ 1 
 
 12th 
 
 a 
 
 December, 
 
 31 
 
 " Thirty days hath September, 
 April, June, and November; 
 February 28, alone, 
 All the rest have thirty-one. 
 Except in Leap Year ; then is the time 
 When February has twenty-nine." 
 
 The following Table will enable us readily to determine 
 the number of days from one date, to any other particular 
 date in the same year : — 
 
64 
 
 DENOMINATE NUMBERS. 
 
 [chap. 111. 
 
 EXHIBITINO THE NUMBER OF DAYS FROM ANY DAY OF OWE MONTH TO THE 8AMI 
 DAY OF ANY OTHER MONTH IN THE SAME YEAR. 
 
 FROM ANY 
 DAY OF 
 
 to the same day. 
 
 Jan. 
 
 Feb.; Mar. 
 
 April May. June. 
 
 July. 
 
 Aug. 
 
 Sept.! Oct. 
 
 Nov. 
 
 Dec. 
 
 January,.... 
 
 355 
 
 31 
 
 69 
 
 90 
 
 120 151 
 
 181 
 
 212 
 
 243 , 273 
 
 304 
 
 334 
 
 February,.. 
 
 334 
 
 365 
 
 28 
 
 69 
 
 89 
 
 120 
 
 150 
 
 181 
 
 212 
 
 242 
 
 273 
 
 303 
 
 March, 
 
 306 
 
 337 
 
 365 
 
 31 
 
 61 
 
 92 
 
 122 
 
 153 
 
 184 
 
 214 
 
 245 
 
 275 
 
 April, 
 
 275 
 
 306 
 
 334 
 
 365 
 
 30 
 
 61 
 
 91 
 
 122 
 
 153 
 
 183 
 
 214 
 
 244 
 
 May, 
 
 •245 
 
 276 
 
 304 
 
 335 
 
 365 
 
 31 
 
 61 
 
 92 
 
 123 
 
 153 
 
 184 
 
 214 
 
 June 
 
 214 
 
 245 
 
 273 
 
 304 
 
 344 
 
 365 
 
 30 
 
 61 
 
 92 
 
 122 
 
 153 
 
 183 
 
 July 
 
 184 
 
 215 
 
 243 
 
 274 
 
 304 
 
 335 
 
 365 
 
 31 
 
 62 
 
 92 
 
 123 
 
 163 
 
 August 
 
 153 
 
 184 
 
 212 
 
 243 
 
 273 
 
 304 
 
 334 
 
 365 
 
 31 
 
 62 
 
 92 
 
 122 
 
 September,.. 
 
 122 
 
 153 1 181 
 
 212 1 242 
 
 273 
 
 303 
 
 334 
 
 365 
 
 30 
 
 61 
 
 91 
 
 October 
 
 92 
 
 123 1151 
 
 182 212 
 
 243 
 
 273 
 
 304 
 
 335 
 
 365 
 
 31 
 
 61 
 
 November... 
 
 61 
 
 92 120 
 
 161 181 
 
 212 
 
 242 
 
 273 
 
 304 
 
 334 ] 365 
 
 30 
 
 December,... 
 
 31 
 
 62 90 
 
 121 151 
 
 182 
 
 212 
 
 243 
 
 274 
 
 304 1 335 
 
 366 
 
 How many days from the 12tli of May to the 12th of 
 October ? In the column of months oq the left of the 
 page we find April; — passing the eye along the horizontal 
 row of figures till it comes to the perpendicular column, 
 headed " Oct.," we find 153 days to be the time. If, in- 
 stead of " 12th of Oct.," in- the above question, we sub- 
 stitute the 20th of Oct., then to the 153 days, add the 
 excess of 20 above, 12 = 8 ; and we have 153 + 8 = 161 
 days, for the number of days. 
 
 When there are 29 days in February the proper allow- 
 ance must be made, as the table considers 28 days in Feb- 
 ruary. 
 
 Books. 
 
 A sheet folded in 
 
 two leaves is called a Folio. 
 four " " "a Quarto or 4to. 
 eight " " " an Octavo, or 8vo. 
 twelve " " " a Duodecimo or 12mo 
 eighteeen " " " an ISmo. 
 twenty-four " " "a 24mo. 
 
 MISCELLANEOUS TABLE. 
 
 12 Units 
 12 Dozen 
 12 Grosd 
 20 Units 
 
 make 
 
 1 Dozen. 
 1 Gross. 
 1 Great Gross. 
 1 Score. 
 
ART. 69.] ADDITION OF DENOMINATE NUMBERS. 
 
 65 
 
 24 Sheets of paper make 1 Quire. 
 20 Quires " 1 Ream. 
 
 30 Pounds 
 
 make 
 
 1 Bushel of oats. 
 
 46 Pounds 
 
 u 
 
 1 Bush, of buckwheat or barley. 
 
 56 Pounds 
 
 (( 
 
 1 Bushel of Indian corn or rye. 
 
 60 Pounds 
 
 u 
 
 1 Bushel of wheat. 
 
 70 Pounds 
 
 (( 
 
 1 Bushel of salt. 
 
 55 Pounds 
 
 make 
 
 1 Firkin of butter. 
 
 196 Pounds 
 
 (( 
 
 1 Barrel of flour. 
 
 200 Pounds 
 
 (( 
 
 1 Barrel of pork. 
 
 200 Pounds 
 
 u 
 
 1 Barrel of beef. 
 
 200 Pounds 
 
 u 
 
 1 Barrel of shad or salmon. 
 
 14 Pounds of lead, 
 
 or iron 
 
 make 1 Stone. 
 
 21i Stone 
 
 
 " 1 Pig. 
 
 8" Pigs 
 
 
 1 Fother. 
 
 Addition of DENOMmATE Numbers. 
 
 Art. 69. Addition of dcTiominate numbers is finding 
 the sum of two or more numbers of different denominations 
 in the same kind of measure. 
 
 1. What is the sum of £S 
 15s. lOd., and Mi 10s. 3d.? 
 
 16s. 5d., £S 12s. 9d., £Q 
 
 Explanation. — Write the given num- 
 ers of the same denomination one under an- 
 other, and place 12 over the d. and 20 over 
 the 8. (to cause the pupil to bear in mind 
 that 12 pence make 1 shilling, and 20 
 shillings make 1 pound.) Then, commenc- 
 ing at the column of pence, we add it as in 
 simple addition, and obtain for the sum 27 
 pence. In 27 pence, how many shillings 1 
 There are 12 pence in Is., therefore, one- 
 twelfth of the number of pence equals 
 the number of shillings ; — 12 is contained in 27, 2 times and 
 3d. remaining. Place the 3d. under the column of pence, 
 and add the 2s. to the column of shillings, and we obtain for 
 
 operation. 
 
 
 20 
 
 12 
 
 £ 8. 
 
 d. 
 
 8 16 
 
 J 
 
 3 12 
 
 9 
 
 6 15 
 
 10 
 
 4 10 
 
 3 
 
 23 15 
 
66 DENOMINATE NUMBERS. [cHAP. III. 
 
 the sum, 55 shillings. In 55s., how many pounds ? There 
 are 2()s. in XI, therefore, one-twentieth of the number of 
 shillings, equals the number of pounds; — 20 is contained 
 in 55, 2 times and 15s, remaining. Place the 15s. under 
 the column of shillings, and add the £2, to the column of 
 pounds, and we obtain for the sum £23. This we place under 
 the column of £, and we^iave for the answer, £23 15s. 3d, 
 
 
 2. 
 
 
 
 
 
 3. 
 
 
 £ 
 
 s. 
 
 d. 
 
 far. 
 
 £ 
 
 s. 
 
 d. 
 
 far. 
 
 14 
 
 10 
 
 6 
 
 2 
 
 16 
 
 11 
 
 4 
 
 1 
 
 16 
 
 15 
 
 10 
 
 3 
 
 23 
 
 13 
 
 11 
 
 2 
 
 13 
 
 12 
 
 4 
 
 1 
 
 25 
 
 15 
 
 8 
 
 
 
 23 
 
 9 
 
 3 
 
 
 
 17 
 
 18 
 
 10 
 
 3 
 
 Troy Weight. 
 
 
 
 4, 
 
 
 
 
 5. 
 
 
 lb. 
 
 oz. 
 
 pwt. 
 
 P^r. 
 
 lb. 
 
 oz. 
 
 Pwt. 
 
 Ri"- 
 
 24 
 
 7 
 
 15 
 
 20 
 
 14 
 
 7 
 
 14 
 
 20 
 
 30 
 
 9 
 
 19 
 
 13 
 
 24 
 
 9 
 
 16 
 
 16 
 
 22 
 
 8 
 
 17 
 
 14 
 
 26 
 
 10 
 
 3 
 
 18 
 
 16 
 
 11 
 
 
 
 12 
 
 36 
 
 11 
 
 17 
 
 15 
 
 Avoirdupois Weight. - 
 
 6. 7. 
 
 T. cwt. qr. lb. oz. dr. T. cwt. qr. lb. oz. dr. 
 
 17 14 3 20 8 9 25 14 3 22 11 2 
 
 2 8 1 15 10 7 35 19 1 21 10 8 
 
 4 13 13 6 10 16 20 16 9 7 
 
 6 16 2 12 11 8 17 21 1 19 1 9 
 
 Apothecaries' Weight. 
 
 lb. 
 
 :?. 
 
 3. 
 
 ^. 
 
 Rr. 
 
 5 
 
 1 
 
 2 
 
 1 
 
 15 
 
 8 
 
 10 
 
 7 
 
 2 
 
 19 
 
 L4 
 
 10 
 
 2 
 
 3 
 
 7 
 
 5 
 
 1 
 
 2 
 
 3 
 
 16 
 
 7 
 
 9 
 
 3 
 
 1 
 
 12 
 
 lb. 5- 3. 3. gr. 
 
 10 1 6 10 
 1 2^ 1 16 
 
 13 2 7 3 20 
 
 11 5 4 13 
 17 2 4 2 18 
 
art. 69.] additon of denominate numbers. §1 
 
 Cloth Measure. 
 
 
 
 10. 
 
 
 11. 
 
 12. 
 
 
 
 
 vd. qr. na. 
 
 E.Fi 
 
 •, qr. na. 
 
 E.E qr. na. 
 
 
 
 
 '5 3 3 
 
 3 
 
 4 3 
 
 17 4 2 
 
 
 
 
 8 1 
 
 7 
 
 5 
 
 18 3 1 
 
 
 
 
 7 2 2 
 
 10 
 
 2 1 
 
 24 1 3 
 
 
 
 
 8 3 
 
 19 
 
 3 2- 
 
 17 2 
 
 
 
 
 13 1 
 
 23 
 
 1 1 
 
 15 2 
 
 
 
 
 Long Measure. 
 
 
 
 
 
 13. 
 
 
 
 14. 
 
 
 m. 
 
 fur. 
 
 rd. yd. ft. in 
 
 
 deff. m. fur. rd. yd. ft. 
 
 in. 
 
 91 
 
 7 
 
 29 4 2 10 
 
 
 122 19 
 
 4 15 
 
 9 
 
 13 
 
 3 
 
 27 1 9 
 
 
 12 7 
 
 3 23 1 
 
 9 
 
 7 
 
 7 
 
 39 1 2 4 
 
 
 27' 4 
 
 7 36 1 2 
 
 11 
 
 15 
 
 6 
 
 32 7 
 
 
 32 1 
 
 3 26 
 
 10 
 
 23 
 
 7 
 
 23 1 2 8 
 
 
 34 
 
 6 39 1 2 
 
 8 
 
 30 
 
 5 
 
 28 1 6 
 
 L, 
 
 17 4 
 
 5 37 1 
 
 7 
 
 
 
 Superficia 
 
 r Square Measure. 
 
 
 
 
 
 
 15. 
 
 
 
 
 
 S. M. A. 
 
 R. 
 
 p. sq.yd. sq. ft. sq. in. 
 
 
 
 
 123 465 
 
 1 
 
 27 20 6 
 
 14 
 
 
 
 
 12 121 
 
 1 
 
 12 8 3 
 
 122 
 
 
 
 
 36 376 
 
 2 
 
 32 7 7 
 
 100 
 
 
 
 
 37 468 
 
 1 
 
 17 2 6 
 
 132 
 
 
 
 
 19 478 
 
 3 
 
 12 8 
 
 140 
 
 
 
 
 17 300 
 
 
 
 34 5 
 
 96 
 
 
 
 Surveyor's Measure. 
 
 
 
 
 
 16. 
 
 
 
 17. 
 
 
 
 . ^■ 
 
 fur. cha. P. li. 
 
 
 m. fur 
 
 . cha. P. li. 
 
 
 
 8C 
 
 ) 2 4 2 11 
 
 
 187 2 
 
 2 19 
 
 
 
 18 
 
 ! 6 3 1 21 
 
 
 11 4 
 
 9 3 17 
 
 
 
 IS 
 
 ; 2 9 3 17 
 
 
 24 3 
 
 8 1 12 
 
 
 
 2^ 
 
 5 7 8 2 18 
 
 
 36 1 
 
 7 14 
 
 
 
 13 
 
 1 5 7 1 16 
 
 
 41 
 
 1 18 
 
 
 
 11 
 
 . 3 5 14 
 
 
 73 7 
 
 6 1 23 
 
 
68 DENOMINATE NUMBERS. [CHAP. III. 
 
 Solid, or Cubic Measure. 
 
 
 18. 
 
 
 
 
 
 19. 
 
 
 
 T. cu. ft. 
 
 cu. in. 
 
 
 
 c. 
 
 cu. ft. cu. in. 
 
 
 
 84 12 
 
 1364 
 
 
 
 12 
 
 120 1463 
 
 
 
 18 32 
 
 1431 
 
 
 
 12 
 
 121 1612 
 
 
 
 16 12 
 
 931 
 
 
 
 91 
 
 112 1316 
 
 
 
 30 28 
 
 1246 
 
 
 
 36 
 
 97 362 
 
 
 
 73 17 
 
 863 
 
 
 
 97 
 
 88 1473 
 
 
 
 96 14 
 
 1241 
 
 Wm 
 
 E Measure 
 
 12 
 
 12 784 
 
 
 
 
 ^ 
 
 Jt 
 
 
 
 
 20. 
 
 
 
 
 
 21. 
 
 
 Tun 
 
 . hhd. gal. qt. 
 
 pt. g^i. 
 
 
 
 
 hhd. gal. qt. 
 
 100 19 1 
 
 pt 
 
 94 
 
 1 26 2 
 
 
 
 
 1 
 
 4 
 
 1 21 3 
 
 1 3 
 
 
 
 
 12 37 3 
 
 1 
 
 12 
 
 32 1 
 
 2 
 
 
 
 
 32 46 
 
 
 
 17 
 
 1 47 2 
 
 1 1 
 
 
 
 
 14 37 2 
 
 1 
 
 26 
 
 53 1 
 
 
 
 
 
 
 18 17 1 
 
 1 
 
 34 
 
 60 1 
 
 1 2 
 
 OR 
 
 Beer Measur] 
 
 22 6 2 
 
 
 
 
 
 Ale, 
 
 E. 
 
 
 
 22. 
 
 
 
 
 
 23. 
 
 
 
 hhd. gal. ( 
 
 qt. pt. 
 
 
 
 bar 
 
 . gal. q^t. pt. 
 
 
 
 39 31 
 
 1 
 
 
 
 49 
 
 
 
 3 23 
 
 2 1 
 
 
 
 7 
 
 32 3 1 
 
 
 
 8 40 
 
 3 
 
 
 
 6 
 
 27 1 
 
 
 
 7 37 
 
 1 1 
 
 
 
 7 
 
 13 2 
 
 
 
 6 38 
 
 2 1 
 
 
 
 12 
 
 18 3 1 
 
 
 
 12 52 
 
 3 1 
 
 Dry Measure 
 
 14 
 
 17 1 1 
 
 
 
 
 
 
 
 
 
 24. 
 
 
 
 
 
 25. 
 
 
 
 ch. bu. pk. 
 
 qt. pt. 
 
 
 
 
 bu. pk. qt. pt. 
 
 
 
 75 1 2 
 
 2 1 
 
 
 
 
 81 1 
 
 
 
 12 35 3 
 
 7 1 
 
 
 
 
 23 3 7 1 
 
 
 
 16 25 1 
 
 5 
 
 « 
 
 
 
 12 1 
 
 
 
 10 32 2 
 
 4 
 
 
 
 
 17 2 6 
 
 
 
 11 17 1 
 
 6 1 
 
 
 
 
 10 3 5 1 
 
 
 
 22 34 
 
 3 1 
 
 
 
 
 16 1 4 1 
 
 
ART. 70.] SUBTRACTICN OF DENOMINATE NUMBERS. 
 
 69 
 
 26. 
 
 0. 
 
 s. 
 
 • 
 
 / 
 
 // 
 
 21 
 
 
 
 15 
 
 38 
 
 5 
 
 1 
 
 7 
 
 12 
 
 40 
 
 32 
 
 2 
 
 9 
 
 17 
 
 35 
 
 16 
 
 3 
 
 8 
 
 23 
 
 37 
 
 46 
 
 3 
 
 10 
 
 29 
 
 57 
 
 54 
 
 8 
 
 11 
 
 21 
 
 46 
 
 3Y 
 
 
 
 28. 
 
 
 Me. 
 
 yr. 
 
 da. 
 
 hr. 
 
 min. 
 
 sec. 
 
 99 
 
 155 
 
 1 
 
 50 
 
 44 
 
 12 
 
 10 
 
 13 
 
 42 
 
 27 
 
 16 
 
 102 
 
 18 
 
 24 
 
 36 
 
 19 
 
 8 
 
 21 
 
 54 
 
 57 
 
 23 
 
 13 
 
 19 
 
 49 
 
 48 
 
 29 
 
 18 
 
 23 
 
 58 
 
 56 
 
 Circular Measure. 
 
 0. 
 
 Measure of Time. 
 
 27. 
 
 17 11 19 53 11 
 
 1 3 21 39 52 
 
 2 9 13 42 47 
 7 -6 27 55 19 
 1 11 28 18 17 
 4 3 18 16 56 
 
 29. 
 
 wk. da. hr. min. sec. 
 
 49 
 
 3 
 
 19 
 
 36 
 
 5 
 
 3 
 
 4 
 
 20 
 
 42 
 
 17 
 
 1 
 
 6 
 
 21 
 
 47 
 
 39 
 
 2 
 
 2 
 
 19 
 
 58 
 
 52 
 
 17 
 
 4 
 
 16 
 
 48 
 
 58 
 
 23 
 
 5 
 
 12 
 
 18 
 
 19 
 
 Subtraction of Denominate Numbers. 
 
 Art. 70. Subtraction of denominate numbers is finding 
 the difference between two denominate numbers. 
 
 1. From ieSB 8s. lOd. 1 far., take £12 15s. 4d. 3 far. 
 
 Explanation. — Write the subtrahend 
 under the minuend, observing to place 
 the numbers of the same denomination 
 one under another ; and begin at the 
 right to subtract. We cannot take 3 
 farthings from 1 far., therefore, from 
 the lOd. (of the minuend) we take 1 
 penny, which equals 4 far., and add it 
 to 1 far., which makes it 5 farthings ; 3 far. from 5 far. leave 
 2 far., which place under the column of farthings. We now take 
 the 4d. from the 9d., or add 1 penny to the 4d., and take this sum 
 from lOd., which in either case gives the same remainder, 5d. 
 As we cannot take 15s. from 83., we borrow £1, = 20s., from 
 the £38, and add it to the 8s. ; from this sum, we take the 15s., 
 and obtain 13s. for a rema nder. We now take £12 from £37, 
 
 
 OPERATION. 
 
 
 
 
 20 
 
 12 
 
 > 4 
 
 
 £. 
 
 s. 
 
 d. 
 
 far. 
 
 Min. 
 
 38 
 
 8 
 
 10 
 
 1 
 
 Sub. 
 
 12 
 
 15 
 
 4 
 
 3 
 
 Rem. 25 13 5 2 
 
TO DENOMINATE NUMBERS. [cHAP. III. 
 
 or add £1 to the £12, and take the sum from the £38, which 
 in either case, gives the same remainder, £25, Hence the dif- 
 ference of the two quantities is £25 13s. 5d. 2 far. 
 
 2. 3. 
 
 £ s. d. far. lb. oz. pwt. er. 
 
 24 8 6 2 25 8 17 21 *> 
 
 16 12 7 3 • 14 7 18 23 
 
 4. 5. 
 
 T. cwt. qr. lb. oz. dr. T. cwt. qr. lb. oz. dr. 
 
 50 16 1 23 IQ 12 14 10 2 12 4 8 
 
 27 7 3 24 3 14 5 14 3 20 7 12 
 
 
 
 
 6. 
 
 
 
 
 
 7. 
 
 
 
 
 lb. 
 
 14 
 
 3. 3. 3 
 11 6 1 
 
 >. gr. 
 
 12 
 
 
 yd. 
 18 
 
 qr. na 
 
 2 1 
 
 
 
 8. 
 
 12 
 
 10 3 2 
 
 1 
 
 5 
 
 
 12 
 
 3 3 
 
 
 
 
 
 9. 
 
 
 10. 
 
 
 E.Fr, 
 
 14 
 
 ■ t 
 
 na. 
 
 2 
 
 E.E. 
 
 19 
 
 ^2 
 
 na 
 1 
 
 3eg. 
 
 21 
 
 m. 
 2 
 
 fur. rd. 
 
 7 21 
 
 yd. ft. in 
 
 3 18 
 
 10 
 
 5 
 
 3 
 
 16 
 
 4 
 
 3 
 
 18 
 
 45 
 
 3 25 
 
 4 2 10 
 
 
 
 
 11. 
 
 
 
 
 
 12. 
 
 
 
 ni. 
 16 
 
 fur. 
 
 1 5 
 
 cha. P. 
 
 3 1 
 
 lin. 
 10 
 
 
 36 
 
 [. A. R. P. 
 
 276 2 12 
 
 sq.^yd. 
 
 
 13 
 
 2 
 
 8 2 
 
 12 
 
 
 24 
 
 108 3 37 
 
 251 
 
 
 
 
 13. 
 
 
 
 
 14. 
 
 
 
 
 T. 
 
 16 
 
 en. ft. cu.ir 
 
 32 1421 
 
 1. 
 
 
 
 c. 
 16 
 
 cu ft. cu.in. 
 
 110 1612 
 
 
 
 6 
 
 37 1675 
 
 
 
 
 11 
 
 116 1719 
 
 
 
 15. 
 
 
 16. 
 
 . 
 
 Tnn. 
 32 
 
 hhd. 
 
 
 gal. qt. 
 
 24 3 
 
 ^(5- 
 
 t 
 
 
 
 hhd. gal. 
 
 62 41 
 
 8^-5* 
 
 
 15 
 
 2 
 
 52 3 
 
 1 
 
 3 
 
 
 
 49 60 
 
 2 1 
 
4.RT. 70.] QUESTIONS IN DENOMINATE NUMBERS. *ll 
 
 17. " 
 
 ch. bu. pk. qt. pt. 
 
 30 12 3 3 1 
 17 30 3' 7 
 
 wk. 
 
 36 
 26 
 
 bn. 
 
 27 
 9 
 
 18. 
 
 3 7 
 
 1 
 
 19. 
 
 C. S. " ' " 
 
 20 10 15 24 32 
 9 5 24 56 52 
 
 da. 
 
 3 
 6 
 
 20. 
 
 hr. min 
 
 12 43 
 
 20 55 
 
 sec. 
 
 15 
 32 
 
 Practical Questions in Addition and Subtraction of 
 Denominate Numbers, 
 
 1. Erom a piece of cloth containing 27 yards 3 qrs. 
 
 1 na., there were taken three garments; the first contain- 
 ed 8 yds. 8 qrs. 2 nas. ; the second 4 yds. 1 qr. 8 nas. ; and 
 the third 2 yds. 8 qrs. 8 nas.; — how much remained ? 
 
 2. Bought a hogshead of sugar wieghing 9 cwt. 3 qrs. 
 21 lbs.; sold to A 1 cwt. 2 qrs. 15 lbs.; to B 2 cwt. 8 qrs. 
 24 lbs.; and to C 3 cwt. 1 qr. 15 lbs.; — how much re- 
 mained unsold ? 
 
 3. A man agrees to build 186 rods and 15 feet of stone 
 fence; — at one time he built 86 rds. 2 feet; at another 
 time 56 rds. 8 feet; and at another time 10 rds. 1 foot. 
 How much still remains to be built, 
 
 4. I agreed to let a person have 24 T. 9 cwt. 2 qrs. 
 15 lbs. of hay. He took away four loads, the weight of 
 which were as follows: the first weighed 16 cwt. 2 qrs. 
 18 lbs.; the second, 19 cwt. 3 qrs. 12 lbs.; the third, 1 T. 
 
 2 cwt. 1 qr. 21 lbs.; and the fourth, 1 T. 5 cwt. 2 qrs. 
 14 lbs. ; — to how much hay is he still entitled ? 
 
 5. How many yard of cloth in three pieces: the first 
 containing 12 yds. 3 qrs. 2 nas.; the second 6 E. English 
 2 qrs. 1 na.; the third 9 E. French, 1 qr. 8 nas.? 
 
 6. Bought three pieces of cloth: the first containing 25 
 yds. 8 qrs. 1 na.; the second 47 yds. 1 qr. 8 uasrj and the 
 third 85 yds. 8 qrs. 2 nas.; — I sold 73 yds. 8 qrs. 2 nas. of 
 it. How much remained unsold ? 
 
T2 DENOMINATE NUMBERS. [CHAP. III. 
 
 T. A merchant bought, at one time 956 bnshels and 3 
 pecks of Indian corn; at another time '759 bushels, 2 pks. 
 and t quarts; and sold 325 bush. 3 pks. and 6 qts. of it. 
 How much had he remaining ? 
 
 8. John is 23 years, 9 month, and 18 days old; James 
 is 18 years, 10 months, and 25 days old. What is the dif- 
 ference of their ages ? 
 
 9. Suppose a person was born February 29, 1*188; how- 
 many birth-days will he have seen 'on February 29, 1840, 
 not counting the day on which he was born ? 
 
 10. A merchant sold goods to the amount of iS39t 18s. 
 6d. 2 qrs.; and received in payment £199 19s. lOd. 3 qrs.; 
 how much remains due ? 
 
 11. From a pile of wood containing 423 cords, Isold at 
 one time, 56 C. 112 cu. ft.; at another time, 91 C. 113 cu. 
 ft.; at another time, 126 C. 96 cu. ft. How many cords 
 remain unsold ? 
 
 12. How long from the birth of William Shakspeare, 
 April 23, 1564, to the birth of Milton, Dec. 9, 1608 ? 
 
 13. A farmer raises 125 bush. 2 pks. 6 qts. of wheat on 
 one field; 19t bush. 1 pk. T qts. on another field: he sells 
 to one person 97 bush. 3 pks. 7 qts.; and to another per- 
 son 112 bush. 2 pks. 6 qts. How many bushels has he 
 remaining ? 
 
 14. A gentleman owned three tracts of land: the first of 
 which contained 127 acres, 3R. l5 rods; the second, 496 A. 
 1 R. 25 rods; the third, 525 A. R. 35 rods; how much 
 remained after he sold 1008 A. 2 R. 25 rods? 
 
 15. Suppose a note given Sept. 10, 1796, to be paid 
 March 5, 1808. How long was the note on interest, if we 
 count 30 days to the month ? How long if the time is 
 accurately computed ? 
 
 Multiplication of Denominate Numbers, 
 
 Art. 71. Multipligation of denominate numhers is tak- 
 ing a quantity of different denominations as many times as 
 there are units in another number. 
 
 Multiply £5 12s. 6d. by 5. , 
 
ART. 11.] MULTIPLICATION OF DENOMINATE NtJMBERS. 73 
 
 OPERATION. Explanation The numbers being properly 
 
 & s. d. written down, we begin at the right to multiply. 
 
 5 12 6 5 times 6d. are 30d., in 30d. how many shillings ? 
 
 5 There are 12d. in Is., therefore, one-twelfth of the 
 
 number ofpence equals the number of shillings. 12 is 
 
 28 2 6 contained in 30, 2 times, and 6d. remaining \ — write 
 
 the 6d. under pence, and reserve the 28. 5 times 
 
 I2s. are 60s., and 2s. added, are 62s., which equals £3 2s. ; — 
 
 write the 2s. under shillings, and reserve the £3. 5 times 
 
 £5 are £25, and £3 added are £28. Hence, &c. 
 
 2. 
 
 » 
 
 
 3. 
 
 
 
 4. 
 
 
 
 £ s. d. 
 
 far. 
 
 cwt. gr. lb. oz. 
 
 22 3 21 12 
 
 
 T. 
 
 cwt. qrs. 
 
 lb. 
 
 oz. 
 
 12 10 8 
 
 3 
 
 
 4 
 
 12 1 
 
 20 
 
 12 
 
 
 4 
 
 
 5 
 
 
 
 
 
 7 
 
 5. 
 
 
 
 6. 
 
 
 
 7. 
 
 
 m, fur. rd. 
 
 ft. 
 
 
 deg. m. fur 
 
 18 21 4 
 
 rd. 
 
 
 yds. 
 
 qrs. 
 
 nas 
 
 12 7 32 
 
 2 
 
 
 20 
 
 
 14 
 
 3 
 
 2 
 
 
 12 
 
 
 
 8 
 
 
 
 
 14 
 
 8. 
 
 9. 
 
 
 10. 
 
 
 
 yds. qrs. na. 
 
 
 cwt. 
 
 qr. lb. 
 
 3 23 
 
 T. 
 
 cwt. 
 
 qr. lb. 
 3 21 
 
 oz. 
 
 dr. 
 
 17 3 1 
 
 
 18 
 
 4 
 
 17 
 
 12 
 
 14 
 
 20 
 
 
 
 15 
 
 
 
 
 
 9 
 
 11. How much cloth will it take for 9 suits of clothes, 
 if each suit require 8 yds. 2 qrs. 2 nas. ? 
 
 12. How long will it take a man to chop 14 cords of 
 wood, if it take him 7 hours, 40 minutes, and 50 seconds 
 to chop 1 cord ? 
 
 13. What is the weight of 12 hogsheads of sugar, each 
 weighing 8 cwt. 3 qrs. 23 lb. ? 
 
 14. If a span of horses, at 1 load, can draw 1 cord 212 
 cubic feet of wood, how many cords can they draw in 14 
 loads ? 
 
 15. If a family of 6 persons, consume 10 gallons, 3 
 quarts, and 1 pint of molasses in 1 week; what quantity 
 will a family of double the number of persons consume in 
 1 year? 
 
 4 
 
14 DENOMINATE NUMBERS. [CHAP. Ill 
 
 16. What is the weight of 18 silver spoons, if each 
 weigh 5 oz. 14 pwt. 20 grs. ? 
 
 17. If 1 acre of land produce 45 bush. 3 pks. T qts. 
 1 pt. of wheat, how much will 12 acres produce ? 
 
 18. If a man walk 25 miles, 5 fur. 2t rds. in 1 day, how 
 far can he walk in 9 weeks, not counting Sunday ? 
 
 19. An estate of i£3295 15s. 6d. is divided among four 
 children: the first has £125 16s. lid.; the second twice 
 as much, lacking ^802 18s. 9d.; the third £M6 its. 9d.; 
 and the fourth the remainder. How much did the fourth 
 receive ? • 
 
 20. If a locomotive move 1 m. 25 rds., in 1 minute; how 
 far will it move in 1 day ? 
 
 Division of Denominate Numbers. 
 
 ' Art. TS. Division of denominate numhers is the process 
 
 of finding any proposed part of a given number, composed 
 
 of two or more denominations of the same kind of measure. 
 
 1. If 5 barrels of sugar weigh 9 cwt. 1 qr. 10 lbs., how 
 
 much will 1 barrel weigh ? 
 
 OPERATION. Explanation. — Write the divisor on the left of 
 cwt. qr. lb. the dividend, as in division of abstract numbers 
 5)9 1 10 5 is contained in 9, once and 4 cwt. remaining. 
 
 4 cwt. = 16 qrs., to which add the 1 qr. and it 
 
 1 3 12 equals 17 qrs., 5 is contained in 17, 3 times, and 
 
 2 qrs. remaining. 2 qrs. = 50 lbs., to which 
 
 add the 10 lbs., and it = 60 lbs. 5 is contained in 60, 12 times. 
 
 Therefore, one-fifth of 9 cwt. 1 qr. 10 lbs., is 1 cwt. 3 qrs. 12 lbs. 
 
 Note. — It is impossible to divide one concrete number by another, (See 
 Art. 44) hence in the above example we do not divide 9 cwt. 1 qr. 10 lbs. by 
 6 barrels, but we separate the 9 cwt. 1 qr. 10 lbs. into 5 equal parts j the 6 
 barrels, being considered an abstract number. 
 
 
 2. 
 
 
 
 3 
 
 
 
 
 4 
 
 .. 
 
 
 £ 
 
 8. 
 
 d. iar. 
 
 cwt. 
 
 qr. 
 
 lb. 
 
 oz. 
 
 T. 
 
 cwt. 
 
 I 
 
 lb. 
 
 )62 
 
 7 
 
 9 3 
 
 5. 
 
 6)101 
 
 1 
 
 13 
 
 8 
 
 7)32 
 5. 
 
 14 
 
 15 
 
 
 
 
 
 
 
 
 
 m. fur. 
 
 rd. ft. 
 
 
 
 deff. 
 
 m. fur 
 
 . rd. 
 
 
 
 
 12)118 1 
 
 12 
 
 
 
 9)144 
 
 26 4 
 
 25 
 
 
 

 8. 
 
 
 yds. 
 
 ft. 
 
 in. 
 
 7)196 
 
 2 
 
 11 
 
 ART. 72.] DIVISION OF DENOMINATE NUMBERS. 75 
 
 T. cwt. qr. lb. oz. dr. 
 
 9)44 1 1 1 3 14 
 
 9. 10. 
 
 A R p T. cwt. qr. lb. oz. dr. 
 
 11)346 3 37 5)19 18 3 20 12 13 
 
 11. Divide, ^£346 18s. 4d. 2 far. by 47. 
 
 OPERATION. 
 
 20 12 i,y 
 
 £ s. d. far. 
 
 47)346 18 4 2 (£7 7s. 7d. 2 far. Ans. 
 329 
 
 17 
 
 20 
 
 47) 358 (7s. 
 329 
 
 29 
 12 
 
 47)352(7d. 
 329 
 
 23 
 4 
 
 47)94(2 far. 
 94 
 
 .0 
 
 12. Divide 137 lbs. 9 oz. 18 pint. 19 grs. by 23. 
 
 13. If 451 individuals share equally 8021 T. 12 cwt. 
 1 qr. 6 lbs. 8 oz. of sugar, how much will each receive ? 
 
 14. If 13 hogsheads of sugar weigh 6 T. 8 cwt. 2 qrs. 7 
 lbs.; how much will 1 hogshead weigh ? 
 
 15. A vintner sold 33 hhds. 56 gals, of wine, to 15 dif 
 ferent men ; how much did each buy, providing they each 
 purchased an equal quantity ? 
 
 16. If a man travel 348 mi. 1 fur. 12 rds. in 28 days, 
 how far, on an average is that a day ? 
 
'^6 DENOMINATE NUMBERS. [cHAP. Ill 
 
 IT. A merchant sold 320 yds. 2 qrs. 2 nas. of broad- 
 cloth, in 19 successive days; how much did he sell daily, 
 providing he sold the same quantity each day ? 
 
 18. A produce dealer divided 132 bushels, 3 pks. t qts. 
 of wheat, equally among 23 of his poor neighbors; how 
 much did each receive ? 
 
 19. 26 men bought 645 acres 20 P. of land, and are 
 to share it equally; how much ought each to receive ? 
 
 20. A speculator bought H9 cwt. 1 qrs. 3 lb. of sugar, 
 and sold it to 36 men; how much did each receive, pro- 
 viding each bought the same quantity ? 
 
 Practical Questions combining Addition, Subtraction, 
 Multiplication and Division of Denominate Num- 
 bers. 
 
 1. A farmer having 19 cwt. 2 qrs. 19 lbs. of pork, 
 sold 5 cwt. 3 qrs. 1 lb. of it, and the remainder he put 
 into 6 barrels; how much did each barrel contain ? 
 
 2. Bought of A 91 acres, 2 R. 12 P. of land, of B 
 4 times as much, lacking *I acres, 1 R., and of C one-half 
 as much as of A and B together ; how much did I buy of 
 B and C respectively, and how much in all ? 
 
 3. A merchant bought 9 pieces of silk, each contain- 
 ing 5t yds. 3 qrs. Having sold to another merchant one- 
 third of it, and to 4 ladies, each 9 yds. 3 qrs. 3 nas,, how 
 much remains unsold ? 
 
 4. A farmer has three fields of wheat: from the first 
 he obtains 224 bus. 2 pks. 2 qts.; from the second one- 
 half as much, increased by *I6 bus. 3 pks. 1 qt.; and from 
 the third, as much as from the other two, lacking 84 bus. 
 2 pks. 1 qts. How much did he obtain from the three 
 fields? 
 
 5. From one-half of a piece of cloth containing 82 yds. 
 2 qrs., a tailor cut six suits of clothes. How much did 
 each suit contain ? 
 
 6. A, B, C, and D, having 4 cwt. 3 grs. 20 lbs. of 
 sugar, agree to divide it as follows: A is to take 15 lbs 
 
ART. 72.] PRACTICAL QUESTIONS. 11 
 
 and one-fifth of the remainder; B 1 qr. 3 lbs. and one- 
 fourth of the remainder; C 1 qr. 12 lbs. and one-third of 
 the remainder; and D is to have what now remains. How 
 much sugar should each receive ? 
 
 I. A, B, C, and D, share 840 bushels, 3 pks. of wheat 
 as follows: A takes 16 bush. 3 pks. and one-fourth of the 
 remainder; B takes 14 bush. 2 pks. and one-third of what 
 remains; C takes 13 bush. 2^ pks. and one-half of what 
 remains; and D takes what now remains. How much 
 does each receive ? 
 
 8. Bought of one man 8 bus. 2 pks. 3 qrs. of grass- 
 seed; of another man 3 times as much, and 2 bus. 2 qts. 
 more; and of another 3 times as much as of the second, 
 lacking 1 bus. 1 pk. 6 qts. How much did I buy of each 
 respectively, and how much of all ? 
 
 9. 32 men agree to construct 28 miles. 4 fur. 32 rds. of 
 road; — after completing one-half of it, one-fourth of the 
 number of men left the company. What distance did 
 each man construct before and after one-fourth of the men 
 left ? 
 
 10. A, B, C, and D, having 184 bus. 2 pks. of wheat, 
 agree to divide it as follows: A is to have one-half of the 
 whole; B is to have one-third of the remainder; C is to 
 have one-fourth of what then remains; and D is to have 
 what is left. What is the portion of each ? 
 
 II. Divide 448 acres, 3 R. 24 P. of land among A, B, 
 C, and D, so that A shall have one-eighth of the whole, 
 -\- 4 acres, 3 rds. G pis. ; B one-fifth of the remainder ; C 
 one- third of what then remains; and D the rest. How 
 much will each one have ? 
 
 12. An estate of iE2490 is to be divided among a widow, 
 two sons, and three daughters; — the widow receives one- 
 third of the whole, lacking ^£34 6; the youngest sou re- 
 ceives as much as the widow, -f £212; the oldest son re- 
 ceives as much as the widow and youngest son together, 
 lacking £335 10s.; and the three daughters share equally 
 of the remainder. How much does each receive ? 
 
IS RKDCCTION. [chap. III. 
 
 REDUCTION. 
 
 -Art. 7*3. If the quantity is to be changed from a 
 higher to a lower denomination, the process is called Re- 
 duction Descending; — if from a lower to a higher denom- 
 ination, Reduction Ascending. 
 
 Reduction Descending. 
 1. In JE34 155. Qd. how many pence ? 
 
 OPERATION. Explanation. — There are 20a. in £1; 
 
 20 12 therefore, 20 times the number of pounds 
 
 £ 8. d. equal the number of shillings. 20 times 34 
 
 34 15 6 are 680, and 15s. added = 695s. There are 
 
 20 12d. in Is.; therefore, 12 times the number 
 
 of shillings equal the number of pence. 
 
 o^^ 12 times 695 are 8340, and 6d. added =: 
 
 12 8346d. TherefoK ^34 158. 6d. =8346d. 
 
 8346 
 
 2. In £23 12s. 8d. 3 far., how many farthings ? 
 
 3. In 18 lbs. 6 oz. 15 pwt. 14 grs., how many grains ? 
 
 4. How many grains in 1 lb. 1 5 2 9 12 grains ? 
 
 5. How many drams in 1 T. 3 qrs. 21 lbs. 10 drs. ? 
 
 6. Reduce 14 cwt. 2 qrs. 20 lbs. to pounds. 
 1. Reduce 16 yards, 2 qrs. 2 nas. to nails. 
 
 8. In 12 E. Fr. 5 qrs. 1 na., how many nails ? 
 
 9. In 8 E.^. 4 qrs. 3 nas., how many nails ? 
 
 10. In 1 mile, how many feet ? 
 
 11. In 1 mi. 5 fur. 35 rds. 5 yds. 2 ft. 6 in.; how many 
 inches ? 
 
 12. How many square poles in 102 acres, 3 R. 2t P.? 
 
 13. In 5 hhd. 20 gals. 3 qts. 1 pt.; how many pints? 
 
 14. In 2 pi. 2 gals, 2 gills; how many gills ? 
 
 15. In 6 barrels, 25 gals. 3 qts. 1 pt. of beer; how 
 many pints ? 
 
 16. In 8 bushels, 2 pks. 5 qts. 1 pt. ; how many pints ? 
 
 17. In 2 weeks, 5 days, 5 hours, and 5 minutes; how 
 many minutes ? 
 
ART. 73.] REDUCTION ASCENDING. 79 
 
 18. In 1 day, how many minutes and seconds? 
 
 19. In 1 year, how many hours. 
 
 20 Tti 5 days, 4 hours, 45 seconds; how many seconds ? 
 
 21. In 1 T. 1 lb. 1 dr.; how many drams ? 
 
 22. In 1 acre; how many square feet ? 
 
 Reduction Ascending. 
 
 1. In 647d., how many pounds, shillings and pence ? ' 
 
 OPERATION. Explanation. — ^There are 12 pence in 
 
 d. Is.; therefore, one-twelfth of the num- 
 
 12)647 ber of pence equals the number of shil- 
 
 lings, which is 53s., and lid. remaining. 
 
 20)53 lid. rem. In 53s. how many pounds 7 There are 
 
 20s. in £1 ; therefore one-twentieth of 
 
 2 13s. lid. the number of shillings equals the num- 
 ber of pounds, which is £2, and 13s. 
 
 remaining. Therefore, 647d. = £2 13s. lid. 
 
 2. In 16823 far., bow many pounds, shillings, &c. ? 
 
 3. In 84672 grs. Troy Weight; how many pounds, 
 ounces &c. ? 
 
 4. How many pounds, &c., Apothecaries' Weight, in 
 569 5? 
 
 5. In 1894763 dr. Avoidupois Weight; how many 
 tons, cwt. &c. ? 
 
 6. In 89643 lbs.; how many tons, cwt. &c. ? 
 
 7. In 8467 nails; bow many yards, qrs. &c. ? 
 
 8. In 2706 nas.; how many E, E., qrs. &c. ? 
 
 9. In 4762 nas,; how many E. Fr., qrs. &c. ? 
 
 10. In 84672 feet; how many miles, &c. ? 
 
 11. In 3647 rods; how many miles, furlongs, &c. ? 
 
 12. In 1478 P.; how many acres, roods and poles ? 
 
 13. In 165 qts.; how many gallons ? 
 
 14. In 20042 gills; how many hogsheads, &c. ? 
 
 15. In 17632 gallons; how many tons, &c. ? 
 
 16. In 4007 pints of beer; how many barrels, &c. ? 
 
 17. In 147 pints; bow many bushels, &c. ? 
 
 18. In 64 pints; how many bushels .? 
 
 19. In 86400 seconds; how many days ? 
 
 20. In 3146232 secondsj how many weeks, days, &c ?' 
 
80 PROPERTY OF THE NUMBER 9. [cHAP. IV 
 
 CHAPTER TV. 
 
 Peculiar Property of the I^umber 9. 
 
 Art. 74. Any number is divisible by 9, when t/ie sum 
 of its digits is divisible by 9. Consequently, every number 
 divided by 9, will give the same remainder as the sum of its 
 digits divided by 9. 
 
 Also, if from any number, the sum of its digits be sub- 
 tracted, the remainder will be divisible by 9. 
 
 Note. The pro6r of the fundamental Rules of Arithmetic, is founded upon 
 the above properties of the number 9, which we will now consider. 
 
 Take any fiumber, as t65, which equals TOO + 60 + 5. 
 
 Now, 100 = tXl00 = Tx (99 + 1) = 1 X 99 + t 
 
 60 = 6X. 10=6X( 9 + 1) = 6X 9 + 6 
 
 5= 5 
 
 Hence, 165= 1X99 + 6x9 + 1 + 6 + 5 
 
 But, 1 X 99 + 6 X 9, which lacks the sum of the 
 digits of the number, 165, of being equal to that number, 
 is divisible by 9; since each of the expressions, 1 X 99 
 and 6x9, contains the factor 9. Hence, if the remain- 
 ing part of the number, which is the sum of its digits, is 
 divisible by 9 the number itself is divisible by 9. As 
 every number can be separated into two parts, — the sum 
 of its digits, and another number, divisible by 9, it follows 
 that the same remainder will be found by dividing the 
 number by 9, as is, by dividing its digits by 9: Also, if a 
 number be diminished by the sum of its digits, the remain- 
 der will be divisible by 9. 
 
 Multiplication of Abstract Polynomials. 
 
 1. Multiply 5 + 1 by 3 + 6. 
 
 operation. Explanation Coinmence at 
 
 517 the left, and multiply each terra 
 
 3 I g in the multiplicand successively, 
 
 by each term in the multiplier. 
 
 15 -L 21 + 30 4 42 Product. It is evident that the sum of these 
 
ART. 82.] DEFINITIONS. 81 
 
 several partial products, (15 -}- 21 -f- 30 -f- 42 := 108,) is equal 
 to the product of the sum of 5 and 7 by the sum of 3 and C. 
 
 2. Multiply 2 + 3 + 4 by 4 + 6 + 1. 
 
 3. Multiply 8 + 6+2 by 2 + 3 + 4. 
 
 4. Multiply 4 + 6+^ + 8 by 3 + 2 + 4. 
 
 5. Multiply 1 + 2 + 3 + 4 + 5 + 6 by t + 8 + 9. 
 
 6. Multiply 9 + 8 + ti- 6 + 5 + 4 by 1 + 2 + 3-1- 4. 
 
 Miscellaneous Definitions. 
 
 Art. 75. An Integer is any whole number. 
 
 Art. 76. An Even number is any integer that con- 
 tains 2 a whole number of times, without a remainder. 
 
 Art. 77. An Odd number is any integer that does not 
 contain 2 a whole number of times without a remainder. 
 Hence, an odd number differs from an even number by a 
 unit. 
 
 Art. 78. A Prime number is any integer that cannot 
 be produced by multiplying two numbers together, each 
 of which is greater than a unit; as 1, 2, 3, 5, 7, 11, 13, 17, 
 19, 23, &c. 
 
 Art. 79. A Composite number is an integer that can 
 be produced by multiplying two numbers together, each 
 of which is greater than a unit. Thus, 48 is a composite, 
 number and may be produced either by multiplying to- 
 gether the composite factors, 6 and 8, or the prime factors 
 2, 2, 2, 2 and 3. 
 
 Art. 80. The Prime factors of a number are the prime 
 numbers that are multiplied together to produce that num- 
 ber. The prime factors of 48 are 2, 2, 2, 2, and 3. 
 
 Art. 8 1 . All integers are prime numbers or composite 
 numbers ; and all composite numbers are composed of 
 prime factors ; — hence every integer is a prim£ number or 
 composed of prime factors. 
 
 Art. 82. A Square number is a composite number, 
 which is composed of two equal factors ; as 9, (=3x3) ; 25, 
 («6X5); 49, (=7X7) &c. 
 
 4* 
 
82 PRIME NUMBERS. [cHAP. IV 
 
 Art. 83. A Cube number is a composite number, that 
 is composed of three equal factors; as 8, (-=2x2x2); 2t 
 
 (=3X^X3) &c. 
 
 Art. 84. The symbol .*. , is equivalent to the word 
 therefore or consequently. 
 
 PRIME NUMBERS. 
 
 Art. 85. All prime numbers, except the digit 2, are odd 
 numbers; consequently, they terminate with an odd digit; 
 as, 1, 3, 5, *I, or 9. All numbers that end in 5 are divisible 
 by 5, since the remainder, (if any next preceding the 5,) 
 is a certain number of tinges 10, which added to 5, gives a 
 number divisible by 5, since each of the numbers composing 
 it, contains the factor 5; — therefore, all prims numbers^ 
 except 2 and 5 must terminate with 1, 3, T, or 9. 
 
 Hence, to determine whether a given number is a prime, 
 first, inspect its terminating figure, and if it differs from 
 1, 3, t, or 9, it is a composite number; if not, it may still 
 be conposite ; — now, if we can find no number between 2, 
 and another prime, the square of which is not less than the 
 given number, that will divide it, the number is a prime. 
 
 Art. 86. An odd number divided by an even number 
 gives an odd number for a remainder; hence, if any pme 
 number, except 2 and 3, be divided by 6 the remainder 
 will be 1, 3, or 5 ; but, the remainder cannot be 3, as the 
 number would then have been divisible by 3, since the 
 divisor and remaiiider are each divisible by 3. Therefore, 
 any prime number, except 2 and 3, when divided by 6 will 
 give 1 or b for a remainder. 
 
 The following table is sufficiently extended for ordinary 
 calculations. 
 
ART. 86.] 
 
 PRIME NUMBERS. 
 
 ss 
 
 Table of Prime Numbers. 
 
 1 
 
 163 
 
 383 
 
 |6iU 
 
 ' 877lll^9 
 
 1433 
 
 1697 
 
 1999 
 
 2293 
 
 2609 
 
 2 
 
 167 
 
 389 
 
 631 
 
 1 88111151 
 
 1439 
 
 1699 
 
 2003 
 
 2297 
 
 2617 
 
 3 
 
 173 
 
 397 
 
 641 
 
 883 
 
 1153 
 
 1447 
 
 1709 
 
 2011 
 
 2309 
 
 26211 
 
 5 
 
 179 
 
 491 
 
 643 
 
 887 
 
 1163 
 
 1451 
 
 1721 
 
 2017 
 
 2311 
 
 2633 
 
 1 
 
 181 
 
 409 
 
 647 
 
 907 
 
 1171 
 
 1453 
 
 1723 
 
 2027 
 
 2333 
 
 2647 
 
 . 11 
 
 191 
 
 419 
 
 653 
 
 911 
 
 1181 
 
 1459 
 
 1733 
 
 2029 
 
 2339 
 
 2657 i 
 
 ! 13 
 
 193 
 
 421 
 43f 
 
 659 
 
 919 
 
 118711471 
 
 1741 
 
 2039 
 
 2341 
 
 2659 1 
 
 IT 
 
 197 
 
 661 
 
 929 
 
 1193 
 
 UM 
 
 1747 
 
 2053 
 
 2347 
 
 2663 
 
 19 
 
 199 
 
 433 
 
 673 
 
 937 
 
 1201 
 
 14.53 
 
 1753 
 
 2063 
 
 2351 
 
 2671 
 
 28"211 
 
 439 
 
 677 
 
 941 
 
 }^IS 
 
 1487 
 
 1759 
 
 2069 
 
 2357 
 
 2677 
 
 29 
 
 223 
 
 443 
 
 683 
 
 947 
 
 1217 
 
 1489 
 
 1777 
 
 2081 
 
 2371 
 
 26S3 
 
 • 31 
 
 227 
 
 449 
 
 691 
 
 953 
 
 1223 
 
 1493 
 
 1783 
 
 2083 
 
 2377 
 
 2687' 
 
 37 
 
 229 
 
 457 
 
 701 
 
 967 
 
 1229 
 
 1499 
 
 1787 
 
 2087 
 
 2381 
 
 2689 j 
 
 41 
 
 233 
 
 461 
 
 709 
 
 971 
 
 1231 
 
 1511 
 
 1789 
 
 2089 
 
 2383 
 
 2693 i 
 
 43 
 
 239 
 
 463 
 
 719 
 
 977 
 
 1237 
 
 1523 
 
 1801 
 
 2099 
 
 2389 
 
 2699' 
 
 47 
 
 241 
 
 467 
 
 727 
 
 983 
 
 1249 
 
 1531 
 
 1811 
 
 2111 
 
 2393 
 
 2707; 
 
 53 
 
 251 
 
 479 
 
 733 
 
 991 
 
 1259 
 
 1543 
 
 1823 
 
 2113 
 
 2399 
 
 2711 
 
 59 
 
 257 
 
 487 
 
 739 
 
 997 
 
 1277 
 
 1549 
 
 1831 
 
 2129 
 
 2411 
 
 2713 
 
 61 
 
 263 
 
 491 
 
 743 
 
 1009 
 
 1279 
 
 1553 
 
 1847 
 
 2131 
 
 2417 
 
 2719 
 
 67 
 
 269 
 
 499 
 
 751 
 
 1013 
 
 1283 
 
 1559 
 
 1861 
 
 2137 
 
 2423 
 
 2729 
 
 71 
 
 271 
 
 503 
 
 757 
 
 1019 
 
 1289 
 
 1567 
 
 1867 
 
 2141 
 
 2437 
 
 2731 
 
 73 
 
 277 
 
 509 
 
 761 
 
 1021 
 
 1291 
 
 1571 
 
 1871 
 
 2143 
 
 2441 
 
 2741 
 
 79 
 
 281 
 
 521 
 
 769 
 
 1031 
 
 1297 
 
 1579 
 
 1873 
 
 2153 
 
 2447 
 
 2749 I 
 
 83 
 
 283 
 
 523 
 
 773 
 
 1033 
 
 1301 
 
 1583 
 
 1877 
 
 2161 
 
 2459 
 
 2753 
 
 89 
 
 293 
 
 541 
 
 787 
 
 1039 
 
 1303 
 
 1597 
 
 1879 
 
 2179 
 
 2467 
 
 2767 
 
 97 
 
 307 
 
 547 
 
 797 
 
 1049 
 
 1307 
 
 1601 
 
 1889 
 
 2203 
 
 2473 
 
 2777 
 
 101 
 
 311 
 
 557 
 
 809 
 
 1051 
 
 1319 
 
 1607 
 
 1901 
 
 2207 
 
 2477 
 
 2789 
 
 103 
 
 313 
 
 563 
 
 811 
 
 1061 
 
 1321 
 
 1609 
 
 1907 
 
 2213 
 
 2503 
 
 2791 
 
 107 
 
 317 
 
 569 
 
 821 
 
 1063 
 
 1327 
 
 1613 
 
 1913 
 
 2221 
 
 2521 
 
 2797 
 
 109 
 
 331 
 
 571 
 
 823 
 
 1069 
 
 1361 
 
 1619 
 
 1931 
 
 2237 
 
 2531 
 
 2801 
 
 113 
 
 337 
 
 577 
 
 827 
 
 1087 
 
 1367 
 
 1621 
 
 1933 
 
 2239 
 
 2539 
 
 2803 
 
 127 
 
 347 
 
 587 
 
 829 
 
 1091 
 
 1373 
 
 1627 
 
 1949 
 
 2243 
 
 2543 
 
 28 f 9 
 
 131 
 
 359 
 
 593 
 
 839 
 
 1093 
 
 1381 
 
 1637 
 
 1951 
 
 2251 
 
 2549 
 
 2833 i 
 
 137 
 
 353 
 
 599 
 
 853 
 
 1097 
 
 1399 
 
 1657 
 
 1973 
 
 2267 
 
 2551i2837| 
 
 139 
 
 359 
 
 601 
 
 857 
 
 1103 
 
 1409 
 
 1663 
 
 1979 
 
 2269 
 
 2557 2843 • 
 
 149 
 
 367 
 
 607 
 
 859 
 
 1109 
 
 1423 
 
 1667 
 
 1987 
 
 2273 
 
 2579 2851 i 
 
 151' 
 
 373 
 
 613 
 
 863 
 
 1117 
 
 1427 
 
 1669| 
 
 1993 
 
 2281 
 
 2591 
 
 28571 
 
 157' 
 
 379 
 
 617 
 
 871 
 
 1123 
 
 1429 
 
 16931 
 
 1997 
 
 2287 
 
 2593 
 
 2861J 
 
84 composite numbers. . [chap. 17 
 
 Resolution of Composite Nuaibers into their Primb 
 Factors. 
 
 1. What are the prime factors of 144 ? 
 
 OPERATION. Explanation. — Divide the 144 by any prime 
 
 2)144 number, greater than a unit, that is contained in it 
 
 without a remainder; and divide this quotient in 
 
 2)72 *^^ same manner, and so continue dividing until the 
 
 quotient obtained is a prime number. Then, a unit, 
 
 2)36 *he several divisors, and the last quotient will be the 
 
 prime factors required. Proceeding, thus, we find 
 
 2)18 the prime factors of 144 to be 1, 2, 2, 2, 2, 3, and 3. 
 
 3)9 
 3 •^ 
 
 2. What are the prime factors of 96 ? 
 
 3. What are the prime factors of 360 ? 
 
 4. What are the prime factors of 36 ? 
 
 5. What are the prime factors of 56 ? 
 
 6. What are the prime factors of 480 ? 
 *l. What are the prime factors of 500 ? 
 
 8. What are the prime factors of 840 ? 
 
 9. Resolve 460 into its prime factors ? 
 10. Resolve 680 into its prime factors ? 
 
 Divisors or Measures of Numbers. 
 
 Art. 87". A divisor or measure of any number is a 
 number that is contained in it an exact number of times, 
 without a remainder. 
 
 1. What are the divisors of *r2 ? 
 
 Explanation We first find the prime factors of 72, which 
 
 are 1, 2, 2, 2, 3, and 3. A number is evidently, divisible by 
 its prime factors and the products arising from every combina- 
 tion of them. A unit and the factor 2 with all the products 
 arising from 2, 2, and 2, gives 1, 2, 4, and 8. A unit and the 
 factor 3 with all the products arising from 3 an(f 3, gives 1, 3, 
 
ART. 88.] COMMON MEASURE. 85 
 
 and 9. The various products arising from the products already 
 obtained, may be found by multiphcation. Thus, 
 
 1+2+4+8 
 
 1 + 3+9 
 
 1 + 2+4 + 8 + 3 + 6+12 + 24 + 9 + 18 + 36 + 72. 
 Therefore, the divisors of 72, are 1, 2, 4, 8, 3, 6, 12, 24, 9, 18, 
 36, and 72. 
 
 2. What are the divisors of 48 ? 
 
 3. Find all the divisors of 96. 
 
 4. Find all the divisors of 144. 
 
 5. Find all the divisors of 360. 
 
 Common Measure or Divisor. 
 
 Art. 88. A Common measure or divisor of two or 
 more numbers, is any number th*t is contained in each of 
 them a whole number of times without a remainder. Thus, 
 5 is a common measure of 10 and 15. 
 
 1. Fiud all the common measures, or divisors of 144 
 and 360. 
 
 Explanation I first find the prime fac- 
 tors common to both numbers, which I 
 have marked by *. — Since, 1, 2, 2, 2, 3, 
 and 3 are the only prime factors that are 
 common to 144 and 360, it follows that 
 each of these factors, together with the 
 products arising from their various com- 
 binations will be all the divisors of the 
 two numbers, 1. 2, 4, and 8 are all the di- 
 visors arising from the common factors, 
 2, 2, and 2, 1, 3, and 9 are all the divisors 
 arising from the common factors 3 and 3. 
 The divisors arising from the combinations 
 of the above divisors are found by multipli- 
 
 OPERATION. 
 
 *2)144 360 
 
 *2) 
 
 72 
 
 180 
 
 *2) 
 
 36 
 
 90 
 
 «3) 
 
 18 
 
 45 
 
 »3) 
 
 6 
 
 15 
 
 cation, 
 
 2 5 
 
 . Thus, 
 1+2+4+8 
 1+3+9 
 
 1+2+4+8+3+6+12+24+9+18-4-36+72 
 Hence ; 1, 2, 4, 8, 3, 6, 12, 24, 9, 18, 36, and 72 are all the 
 common divisors of 144 and 360. 
 
86 GREATEST COMMON MEASURE. [cHAP. IV. 
 
 2. Find all the common divisors of 24 and 48. 
 
 3. Find all the common divisors of 48, 96, and 120. 
 
 4. Find all the common divisors of 180, 360, and 480. 
 
 5. Find all the common divisors of 60, 120, and 180. 
 
 Greatest Common Measure. 
 
 Art. 89. The greatest common measure, or the great- 
 est common divisor of two or more numbers, is the greatest 
 number that is contained in each of them a whole number 
 of times without a remainder. Thus, 7 is the greatest 
 common measure of 35 and 42. 
 
 1. What is the greatest common measure of 126, 294, 
 and 462 ? 
 
 • OPERATION. 
 
 ( 126=2* X 3* X 3 X7* 
 The prime factors of \ 294=2*x3*x7*x7 
 (462=2*x3*x7*xll 
 
 Explanation Since a number is divisible only by its prime 
 
 factors and the various products of them, it follows that the 
 product of all the factors that are common to any two or more 
 numbers, must be the greatest common measure of these num- 
 bers. 
 
 The factors marked (^) are common to all these numbers, 
 hence their product is the greatest common measure, or divisor 
 of these numbers; which is 2x3x7=42. 
 
 2. What is the greatest common measure of 462 and 
 t70? 
 
 3. What is the greatest common divisor of 140, 105, 
 And 245? 
 
 4. What is the greatest common divisor of 210, 350, 
 and 770 ? 
 
 5. Wliat is the greatest common measure of 286, 429, 
 and 715 ? 
 
 Art. 90. The greatest common measure of two or 
 more numbers may also be found by 
 
 Dividing the larger number by Ike smaller^ and the preceding 
 divisor ly the remainder, (if there be any,) and so contiwiu 
 
ART. 90.] PRACTICAL QUESTIONS. 87 
 
 to divide the preceding divisor by the last remainder until 
 nothing remains, then will the last divisor he the greatest com' 
 mon measure. 
 
 Note— If there are more than two numbers ; first find the greatest common 
 measure of two of them, and then take this divisor and the remaining num- 
 Der and proceed as before. 
 
 6. What is the greatest commoa measure of 105 and 
 490? 
 
 OPERATION. Explanation. — If the remainder, 
 
 105)490(4 (if 3.ny after division) will divide the 
 
 420 preceding divisor, it will also divide 
 
 the dividend, as that is the sum of 
 
 70)105(1 ^ certain number of times the divi- 
 
 70 sor and this remainder : it is also 
 
 the greatest divisor of the two nuro- 
 
 35)70(2 ^6rs, as that divisor is the same 
 70 as the greatest divisor of the remain- 
 
 der and the preceding divisor. 
 
 This is rendered plain by inspec- 
 
 tion. Since 35 is contained in 70, it 
 is contained in 105, (the sum of 70 and 35,) also in 490, (the 
 sum of 70 und 4 times 105 :) and is the greatest divisor of 105 
 and 490, as it is the greatest divisor of itself and 70 the largest 
 numbers, taken at pleasure, that will produce the 105 and the 
 490. The 105=35-f 70 and 490=4x35+70. 
 
 T. What is the greatest common measure of 3094 and 
 4420 } 
 
 8. What is the greatest common measure of 296 and 40t ? 
 
 9. What is the greatest common measure of 360 and 480 ? 
 
 10. What is the greatest common measure of 268 and 
 286? 
 
 PRACTICAL QUESTIONS IN COMMON MEASURE. 
 
 1. A farmer has 120 bushels of wheat and 460 bushels 
 of rye, which he is desirous of putting into boxes of equal 
 size, without mixing the two kinds of grain. How much 
 will the largest boxes that can be used hold .'' 
 
 2. A had $480; B $960; and C $360, which they were 
 desirous of separating into different parcels, each contain- 
 
88 MULTIPLES, [chap; IV. 
 
 ing the same number of dollars. What ib the greatest 
 number of dollars that each parcel can contain ? 
 
 3. A speculator has in one place 240 acres of land, in 
 another 480, and in another 640, and -wishes to divide the 
 whole into fields that shall be of equal size, and contain- 
 ing the greatest number of acres circumstances will allow. 
 What will be the number of acres in each field ? 
 
 Multiples. 
 
 Art, 91. A Multiple of any number, is a number 
 that will contain it a whole number of times, without a 
 remainder. Thus, 21 is a multiple of 3. 
 
 Art, 92. A common multiple of any two or more 
 numbers, is a number that will, when divided by each of 
 them, give an integer for a quotient. Thus, 24 is a com- 
 mon multiple of 4 and 8. 
 
 Art. 93. The least common multiple of any two or 
 more numbers, is the smallest number that will, when 
 divided by each of them give an integer for a quotient. 
 Thus 24 is the least common multiple of 6, 8, and 12. 
 
 Art. 94. To find the least common multiple. 
 
 Place the numbers in a horizontal line. Then divide by 
 any prime number greater than a unit that will divide the 
 most of the given numbers without a remainder, and 
 place the quotients thus obtained, with the undivided 
 numbers in a liue beneath; thus continue to divide until 
 no number greater than a unit will divide any two or more 
 of them without a remainder. Then the product of all 
 the divisors, the last quotients, and the undivided num- 
 bers will be the least common multiple. 
 
 1. What is the least common multiple of 6, 9, and 30 ? 
 
 OPERATION. 
 
 2)6 9 30 
 
 3)3 9 15 
 
 13 5 
 
 2 X 3 X 3 X 5 = 90 is the least common multiple. 
 
ART. 94.] ABBREVIATED OPERATIONS. 89 
 
 Explanation. — Since the numbers 6, 9, and 30 are composed 
 of the prime factors 2, 3, 3, and 5, or a certain number of 
 them, it follows that their product will be a common multiple 
 of these numbers; — and as all these factors are necessary to 
 produce the above numbers, their product must be their least 
 common multiple. 
 
 2. What is the least common multiple of 12, 16, and 20 ? 
 
 3. What is the least common multiple of 15, 30, and 9 ? 
 
 4. What is the least common multiple of 4, 8, 12, 16, 
 and 20 ? 
 
 5. What is the least common multiple of 24, 48, 12 ? 
 
 6. What is the least common multiple of 18, 54, 2T, 
 and 12 ? 
 
 7. What is the least common multiple of 12, 90, and 45 ? 
 
 8. What is the least common multiple of 25, 45, 90 
 and 5 ? 
 
 9. What is the least common multiple of 64, 8, 81, 24, 
 and 12 ? 
 
 10. What is the least common multiple of 60, 120, 48, 
 36, 24, 12, 6, and 4 ? 
 
 PRACTICAL QUESTIONS IN COMMON MULTIPLE. 
 
 1. What is the smallest sum of money for which I could 
 purchase a number of hogs, at $9 each; a number of 
 cows, at $2t each; or a number of horses, at $60 each; — ■ 
 and how many of each could I purchase for that sum ? 
 
 2. What is the smallest number of bushels of corn that 
 will fill a number of barrels, each containing 3 bushels; a 
 number of sacks, each containing 6 bushels; or a number 
 of boxes, each containing 25 bushels ? 
 
 3. If one team can haul 12 barrels of sugar, at a load; 
 another 15; and another 20; — what is the smallest num- 
 ber of barrels that will make a number of full loads for 
 any of three teams ? 
 
 Abbreviated Operations in Arithmetical Calculations. 
 There are many abbreviated methods of calculation, in 
 
90 ABBREVIATED OPERATIONS. [cHaP. IV 
 
 particular cases, which will be of interest to the student, 
 and of much importance to business men. We will men- 
 tion a few of them to awaken a habit of observation on 
 the part of the learner, that he may be enabled to discover 
 others as circumstances may require. 
 
 Art. 95. To multiply by 13, 14, &lg., to 19. 
 
 Write the product of the unit's figure and the multipli- 
 cand, under the multiplicand, one place to the right and 
 then add them. 
 
 Multiply 364^2 by 16. 
 
 OPERATION. 
 
 3G472 X 16 
 
 218832 
 
 583552 
 
 Art. 96. If the multiplier is a unit followed by one 
 or more ciphers and a significant figure, the multiplicatioQ 
 can be performed by writing the product of tiie units' 
 figure and the multiplicand as many places to the right of 
 the multiplicand as there are intervening ciphers -|- 1. 
 
 Multiply 3642T3 by 104. 
 
 OPERATION. 
 
 364273 X 104 
 
 1457092 
 
 37884392 Ans. 
 Multiply 8468327 by 10007. 
 
 OPERATION. 
 
 3468327 x 10007 
 
 24278289 
 
 34707548289 Ans. 
 
 Art. 97. .To multiply by 21, 31, 41, &c. to 91. 
 
 Place the product of the tenh figure and the multiplicand, 
 under the multiplicand, so that its unit figure shall be 
 under the tens of the multiplicand. 
 
 Multiply 3476 by 41. 
 
ART. 99.] ABBREVIATED OPERATIONS. 91 
 
 OPERATION. 
 
 3476 X 41 
 13904 
 
 142516 
 
 Should there be ciphers between the unit and the other 
 significant figure of the multiplier; write the product of 
 the significant figure one more place towards the left for 
 every cipher. 
 
 Multiply 36U32 by tOl. 
 
 OPERATION. 
 
 367432x701 
 •2572024 
 
 257569832 Ans. 
 Multiply 46^321 by 80001. 
 
 OPERATION. 
 
 467321 X 80001 
 3738568 
 
 37386147321 Ans. 
 
 Art. 98. To multiply by any number of 9's. From 
 the multiplicand with as many ciphers annexed as there 
 are 9's in the multiplier, subtract the multiplicand. 
 
 Multiply 34682 by 9999. 
 
 OPERATION. 
 
 346820000 
 34682 
 
 346785318 Ans. 
 
 Explanation — 9^99= 10000 — 1, consequently, by annexing 
 four ciphers to the multiplicand, we have taken it one time 
 more than we should have done, hence by subtracting the 
 multiplicand gives the correct result. 
 
 Art. 99. If the multiplier is an Aliquct Part of 
 any number of tens, hundreds, or thcAisands; multiply by 
 the number of tens^ hundreds, or thousands^ of which the mul- 
 tiplier is an aliquot jpart, then take the same jpart of the 
 product thus found. 
 
ABBREVIATED OPERATIONS. CHAP. IV 
 
 Aliquot Parts. 
 
 12i = I of 100 
 16f = :^ of 100 
 25 = j of 100 
 50 = I of 100 
 75 = I of 300 
 331 = t of 100 
 13^ = i of 40 
 
 3 3 
 
 &C. &C. 
 
 1. Multiply 3248 by 12i. 
 
 121 X 8 = 100, therefore 12^ is one-eighth of 100. 
 
 OPERATION. 
 
 8)324800 
 
 40600 Ans. 
 
 2. Multiply 86432 by 25. 
 
 25 X 4 = 100, therefore 25 is one-fourth of 100. 
 
 operation. 
 4)8643200 
 
 2160800 Ans. 
 
 3. Multiply 846828 by ^^. 
 
 33^ X 3 = 100, therefore 33^ is one-third of 100. 
 
 OPERATION. 
 
 3)84682800 
 28227600 Ans. 
 
 Art. 100. Any number ending in 5, that is expressed 
 by two figures, can be squared mentally. 
 
 Tht two right hand figures of the square number will 
 always he 25, the remaining figures on the left, will he the 
 product of the digit in terHs 'place and a figure a unit greater. 
 
 1. What is the square of 25. 
 
 25 X 25 = 625 
 
 By inspecting the following multiplication, the reason 
 of this method of squaring a number, expressed by two 
 figures, that ends in 5, will become evident. This meth- 
 od of squaring a quantity will apply to a number expressed 
 
IRT. 101.] ABBREVIATED OPERATIONS. 93 
 
 by three, or more, figures ; providing the figure occupying 
 the unit's place is 5. 
 
 OPERATION. 
 
 25 = 20 + 5 
 25 == 20-1-5 
 
 Remark — ^Commence at 
 
 5x20-f-25 the right and multiply by 
 
 20x20-f- 5x20 each figure separately. 
 
 ' 20 x20-f-10x 20-1-25 
 The product is composed of (10 + 20) times 20, + 25, 
 or 30 times 20 + 25 = 600 + 25 = 625. 
 45 squared = 2025 
 85 squared = 7225 
 &c. &c. 
 
 *125 squared = 15625. 
 &c. &c. 
 
 Art. 101. The square of any number and a half is 
 equal to the ^product of that number and a number a unii 
 greater, increased by one-fourth. 
 
 9-J squared = 9xl0+^=90|- 
 
 8i « = 8x 9+1=72^ 
 
 &c. &c. 
 
 EXAMPLES IN ABBREVIATED MULTIPLICATION. 
 
 The pupil should be required to give the reason of all 
 abbreviated operations. 
 
 1. Multiply 46234 by 13. 
 
 2. Multiply 8647 by 16. 
 
 3. Multiply 84672 by 19. 
 
 4. Multiply 46732 by 103. 
 
 5. Multiply 68472 by 107. 
 
 6. Multiply 723246 by 1009. 
 
 7. Multiply 67234 by 21. 
 
 8. Multiply 846232 by 41. 
 
 9. Multiply 8467231 by 81. 
 
 10. Multiply 102324 by 701 
 
 11. Multiply 347234 by 6001. 
 
 12. Multiply 4726846 by 80001. 
 
 * Consider the 12 on the left of the 5 as one number, and multiply it by « 
 number a unit greater. 
 
94 ABBREVIATED OPERATIONS. [CHAP. IV. 
 
 13. Multiply 4862321 by 12»-. 
 
 14. Multiply 846232 by 33^. 
 
 15. Multiply *r23246 by 16|. 
 
 16. Multiply 8462342 by 25. 
 lY. Square 25 mentally. 
 
 18. Square 35, 45, 55, 65, T5, 85, aud 95, mentally, 
 
 19. Square 125, 135, 145, and 155, mentally. 
 
 20. Square ^, 5^, 6^ H, 8i, 9i, 10^, 11^, and 12i, 
 mentally. 
 
 Art. 102. Any number is divisible by another when it 
 contains the same prime factors as that numbe? 
 
 Hence, to divide one number by another, resolve them 
 into their prime factors and reject equal factors from each ; — 
 the product of the remaining factors of the dividend will 
 be the quotient. Should the divisor not be a measure of 
 the dividend, there will be factors remaining in the divisor 
 also. In such cases, the product of the remaining factors 
 in the dividend, divided by the product of the remaining 
 factors in the divisor, will give the quotient. 
 
 1. Divide 1260 by 84. 
 
 OPERATION. 
 
 The prime factors of < 
 
 1260 = 2X2X3X3X5X1. 
 84 =2X2X3X1 
 
 I^i^'^d^^^' ^X^X3X3X 5x;^ ^ 15 tient. 
 Divisor, ^ x ^ X $ X ^ 
 
 2. Divide 3780 by 420. 
 
 3. Divide 6615 by 315. 
 
 4. Divide 46305 by 63. 
 
 5. Divide 2205 by 378. 
 
 OPERATION. 
 
 mi ' e A. v^ 2205 = 3X3X5X7X7. 
 The prime factors of | 3^3 ^ 3 ^ 3 >< 3 >< 3 ^ t. 
 
 Dividend, $ X^ X 5 X ^ X 7 ^ 35 ^ ^^ ^^^ 
 Divisor, 2 X $ X $ X^X^ 6 ' 
 
 Remark. — The pupil, from what has been said, will readily discover otk«r 
 useful lavthods of abbreviating the operations of the Fundamenal rules, 
 
ART. 108.] FRACTIONS. 95 
 
 Properties of NuiiBERS. 
 
 Art. 103. All numbers terminating on the right with 0, 
 2, 4, 6, or 8, are divisible by 2 ; since each of the numbera 
 2, 4, 6, 8 and 10 contain a factor 2. 
 
 Art. 104. All numbers terminating on the right with 0, 
 or 5, are divisable by 5; since each of the numbers, 5 and 10, 
 contain a factor 5. 
 
 Art. 105. If the two right-hand figures of any number 
 are dimnble by 4, the whole number will be divisible by 4. 
 For, if there is any remainder next preceding the two 
 fi(i:ures on the right, it will be a certain number of times 
 100; and 4 is a measure of 100, since the 100 contains the 
 same prime factors as 4 ; and as it is also, a measure of 
 the two right-hand figures; it is a measure of the whole 
 number. 
 
 Art. 106. If the three right-hand figures of any number 
 are divisible by 8, the whole number will be divisible by 8. 
 For the remainder, (if any,) next preceding the three figures 
 on the right, will be a certain number of times 1000; and 
 8 is the measure of 1000, since the 1000 contains the same 
 prime factors as 8; the 8 being also, a measure of the three 
 right-hand figures; — it must be a measure of the whole 
 number. 
 
 Art. lOT. When the sum of the digits of any number 
 is divisible by 3 or 9, the number itself is divisible bj! 3 or 9. 
 (For the reason, see Art. 74.) 
 
 CHAPTER Y. 
 
 FRACTIONS 
 
 Abt. 1 08. A Fraction is an expression denoting one 
 or more of the equal parts into which a unit, or any col- 
 lection of units may be divided. 
 
 There are two kinds of fractions employed in Arithme- 
 
96 FRACTIONS. [chap. V. 
 
 tical calculations; namely, Common Fractions and Decimal 
 Fractions. 
 
 The Common Fractions have generally been called Vul- 
 gar Fractions ; the word vulgar, meaning common. 
 
 Common Fractions. 
 
 Art. 109. A Common Fraction consists of two num- 
 bers, one written above the other, with a short horizontal 
 line between them. 
 
 The number above the line is called the Numerator^ and 
 shows how many of these parts are considered, or taken. 
 
 The number below the line is called the Denominator, 
 and shows into how many equal parts the unit or integer is 
 divided. 
 
 I, |, 4, f , &c., are Common Fractions and are read 
 
 Numerator. 1 OuC 
 
 Denominator. 3 Third Of One. 
 Numerator. 2 TwO > ce^x, e n 
 
 " TTi"^!- If > or one fiitn of 2. 
 
 Denominator. 5 Fifths Of OnC, J 
 
 or one seventh of 4. 
 
 Numerator. 4 FOUT 
 
 Denominator. 7 Sevenths of onc. 
 
 Numerator. 5 FivC > • ^i y. r 
 
 r o- XT, e / or one sixth of 5. 
 
 Denominator. 6 Slxths of OUC, J 
 
 By inspecting the above expressions, it will be observed 
 that they are unperformed operations in division. The 
 denominators being the divisors, and the numerators the 
 dividends. Hence, a Common Fraction may be considered 
 a method of expressing division. (See Remark 2, Art, 45.) 
 
 In the fraction f , the numerator, 5, is the dividend, and 
 the denominator, 9, is the divisor 
 
 Art. no. When the numerator of a fraction is less 
 than the denominator, the value is less than a unit; as, f. 
 
 2. When the numerator of a fraction is equal to the de- 
 nominator, the value is a unit; as, f = 1. 
 
 3. When the numerator of a fraction is greater than 
 
ART. 112.] COMMON FRACTIONS. 91 
 
 the denominator, tlie value is greater than a unit; as, 
 
 J=ii. 
 
 Art. 111. There are five kinds of Common Frac- 
 tions, namely; Proj^er, Im'projper, Simple, Compound, and 
 Complex. 
 
 A Proper Fraction is one, the numerator of which is 
 less than the denominator; therefore, its value is less than 
 a unit; as, |, f, |, &c. 
 
 An Improper Fraction is a fraction, the numerator of 
 which is equal to, or greater than the denominator; there- 
 fore, its value is equal to, or greater than a unit; as, |-, |, 
 ^, &c. 
 
 A Simple Fraction is one in which the numerator and 
 denominator each consist of an integer ; and may be either 
 a proper or an improper fraction. * 
 
 A Compound Fraction is a fraction of a fraction, or any 
 number of fractions connected by the word of; as, f of f 
 of I of |. 
 
 A Complex Fraction is a fraction that has a fraction, 
 or a mixed number, in the numerator or denominator, or 
 
 in both; as — ; — &c. 
 4J 31 
 
 A Mixed Number consists of an integer and a fraction; 
 as, H.» 24f , &c. 
 
 Art. 112. The Terms of a fraction are two in num- 
 ber, the numerator and the denominator. 
 
 To invert a fraction, cause the numerator and the 
 denominator to change places. Thus, | when inverted, 
 becomes f . 
 
 Any whole number may be expressed fractionally by 
 writing a unit below it for a denominator. Thus, 
 
 4 = f , and is read 4 ones, or four. 
 
 7= ^, " " " 7 ones, or seven. 
 
 9 = f , " " " 9 ones, or nine. 
 
 12 == V, " " " 12 ones, or twelve. 
 
98 , FRACTI0N3. [cHAP ▼. 
 
 Reduction of Common Fractions. 
 
 Art. 113. Reduction of Fractions is changing them 
 from one form to another while their value remains the 
 1 same. 
 
 Art. 114. Reduction of Mixed Numbers to Improper 
 Fractions. 
 
 1. In 25| how many thirds ? 
 
 Solution. — In 1 there are f , and in 25 there are 25 times 
 %=\^, which added to |= V, consequently, 25|=y. 
 
 2. In 43f , how many fourth ? 
 
 3. In 14:6f , how many sevenths ? 
 
 4. In 2361, how many halves ? 
 
 5. Reduce 684| to an improper fraction. 
 
 6. Reduce 783|- to an improper fraction. 
 
 1. Reduce 1862y'^ to an improper fraction. 
 
 8. Reduce 2864y\ to an improper fraction. 
 
 9. Reduce 86232^^ to an improper fraction. 
 
 10. Reduce 76432yy8 to an improper fraction. ♦ 
 
 Art. 1 15. Reduction of Improper Fractions to Mixed 
 
 Numbers. 
 
 1. Reduce ^^^ to a mixed number. 
 
 Solution. — In one there are f ; therefore, 1 third of the 
 'number of thirds, equals the number of whole ones. 1 
 third of 24t=82i. Hence, 2|i=82i. 
 
 2. Reduce -^-f^ to a mixed number. 
 
 3. Reduce ^f ^ to a mixed number. 
 
 4. Reduce ^|^ to a mixed number. 
 6. Reduce -^-f^ to a mixed number. 
 6. Reduce -V^-^ to a mixed number. 
 1. Reduce -i-V^ to a mixed number. 
 
 8. Reduce -f f-^ to a mixed number. 
 
 9. Reduce -V/-^ to a mixed number. 
 10. Reduce a.ii|AX to a mixed number. 
 
ART. 116.] PROPOSITIONS. 99 
 
 Propositions. 
 
 Art. 116, Proposition!. — Multiplying the numerator 
 of a fraction hy any number, the denominator remaining the 
 same, multiplies the value of the fraction by that number. 
 
 The denominator of a fraction shows into how many equal parts 
 the quantity is divided, and therefore, designates the size of the 
 parts compared with that quantity. The numerator shows how 
 many of these parts are taken ; hence, multiplying the numerator 
 increases the value of the fraction as many times as there are units 
 in the multiplier, if the denominator, that is, the size of the parts 
 remains the same. 
 
 Proposition 2. — Dividing the denominator of a fraction 
 by any number, the numerator remaining the same, multiplies 
 the. value of the fraction by that member . 
 
 The numerator of a fraction shows how many parts are taken, 
 and the denominator measures the size of these parts, compared 
 with the quantity referred to ; if we divide the denominator by 
 any number it diminishes the number of parts into which the thing 
 is divided, and, therefore, increases their size proportionally • 
 hence, the value of the fraction is multiplied by the same number, 
 if the numerator, that is, the number of parts taken, remains the 
 same. 
 
 Proposition 3. — Multiplying the denominator of a frac- 
 tion by any number, the numerator remaining the same, divides 
 the value of the fraction by that number. 
 
 The numerator of a fraction shows how many parts are taken, 
 and the denominator shows into many equal parts the unit or 
 thing is divided, and therefore, designates the size of these parts 
 compared with the unit or thing divided. Multiplying the denom- 
 inator by any number, increases the number of parts into whitch 
 the thing is divided, as many times as there are units in the mul- 
 tiplier, and necessarily diminishes their size proportionally ; there- 
 fore, the value of the fraction is divided, if the numerator remains 
 unchanged. 
 
 Proposition 4. — Dividing the numerator of a fraction by 
 'any number, the denomi7iator remaining the same, divides the 
 value of the fraction by that number. 
 
 Since, the denominator shows into how many equal parts the 
 unit or thing is divided, and the numerator shows how many of 
 these parts are taken ; it follows, that dividing the numerator. 
 
100 FRACTIONS. '^ [chap IV. 
 
 divides the value of the fraction, as it diminishes the number of 
 parts taken while their size remains the same. 
 
 Remark. — By inspecting proposition 1 and 3, we deduce 
 
 Proposition 5. — Multiplying the numerator and denorri' 
 inator of any fraction by the same number, does not change 
 the value of the fraction. 
 
 Remark. — By, inspecting proposition 2 and 4, we deduce 
 
 Proposition 6. — Dividing both numerator and denomina- 
 tor of any fraction by the same number, does not change the 
 value of the fraction. 
 
 Multiplication of Fractions by Integers. 
 
 Art. IIT, According to propositions 1st and 2d, to 
 
 to multiply a fraction by any integer ; Multiply the numera- 
 tor of the fraction by that number, — or divide its de7iominator 
 by the same number. 
 
 1. If 1 bushel of apples" cost $4, what will 15 bushels 
 cost? 
 
 Solution. — If 1 bushel cost $4> 1^ bushels will cost 
 15 X $4 = V = $8f 
 
 2. If 1 barrel of sugar cost $12|-, (equal to %^^,) what 
 will 3 barpels cost ? 
 
 Solution. — If 1 barrel cost %^\^, 3 barrels will cost 3 
 times $113. = 112 _ 1371, 
 
 3. What cost 25 bushels of peaches, at $f a bushel } 
 
 4. What cost 18 yards of broadcloth, at $6| a yard ? 
 6. What cost 4t barrels of flour, at $5| a barrel ? 
 
 6. What cost 15 cows, at 25f each ? 
 1. What cost 52 barrels of cider, at $11 a barrel ? 
 8. What cost IT hogsheads of molasses, at %i1^ a hogs- 
 head ? 
 
 Division of Fractions by Integers. 
 
 Art. 118. According to Propositions 3d and 4th, to 
 
 divide a fraction by any integer ; Divide the nu7nerator of 
 the fraction by that number, — or muitiply the denominator by 
 the same number. 
 
ART, 119.] DIVISION OF FRACTIONS BY INTEGERS. 101 
 
 1. If 16 yards of cloth cost $3ti, what will 1 yard cost ? 
 
 2. If 13 yards of ribbon cost 32^ cents, how much is 
 that a yard ? 
 
 3. If 8 books cost $13|, how much is that a piece ? 
 
 4. If 14 lbs. of sugar cost 93f cents, how much is that 
 a pound ? 
 
 5. If It barrels of sugar cost $282^, how much is that 
 a barrel ? 
 
 6. What cost 1 horse, if 19 horses cost $t824f ? 
 1. What cost 5 oranges, if 15 cost 35|^ cents ? 
 
 8. What cost 6 acres of land if It acres cost $403f ? 
 
 9. What cost 3 bushels of flax-seed, if 7 bushels 
 cost $2tf ? 
 
 10. What cost 4 horses if 12 cost $2104^ ? 
 
 Art. 119. According to proposition 6th, to reduce a 
 fraction to its lowest terms: 
 
 Divide both numerator and denominator by any number 
 greater than a unit, that is contairied in them both without a 
 remainder ; proceed in the same way with the successive results, 
 until the operation can be carried no farther. (See Arti- 
 cles 103 to 107, Properties of numbers.) 
 Or, 
 
 Find the greatest common measure of the numerator and 
 denominator (by Art. 89 or Art. 90) and divide them by it. 
 Or, 
 
 Resolve both numerator and denominator into their prime 
 factors and reject equal factors from each; (See Art. 102) 
 the result will be the fraction reduced to its lowest terms. 
 1. Reduce |f^ to its lowest terms. 
 
 Operation by the last method. 
 
 The prime factors of \ ^— 
 ^ I 600= 
 
 2. Reduce f f f to its lowest terms. 
 
 3. Reduce -j-f ^ to its lowest terms. 
 
 4. Reduce -f f to its lowest terms. 
 
 5. Reduce f || to its lowest terms. 
 
 6. Reduce |;^| to its lowest terms. 
 
102 FRACTIONS. [chap. V. 
 
 *l. Reduce j%W to its lowest terms, 
 
 8. Reduce ||f to its lowest terms. 
 
 9. Reduce |m to its lowest terms. 
 10. Reduce {UH ^o its lowest terms. 
 
 Art. 120. Reducton of Compound Fractions to 
 Simple ones. 
 
 Remark. — The word of in the following questions, is equivalent to the 
 sign of multiplication; therefore, in its stead the sign X, may be used. 
 
 1. Reduce | of f to a simple fraction. 
 
 OPERATION. Solution.— i of ^ is yV and ^ of | is 4 
 
 2 4 8 A *i°^6s yV, which is y\ ; and if ^ of f is y\, 
 
 oX^=^R ^^^ f of ! is twice y^^, which are yV There- 
 fore, f off =yV 
 
 Remark. — From the above solution we observe, that to reduce a compound 
 fraction to a simple one, we multiply all themimerators together for a new numera 
 tor, and all the denominators for a neto denominator. 
 
 2. Reduce | of | to a simple frastion. 
 
 3. Reduce | of y\ to a simple fraction. 
 
 4. Reduce | of | of y^^- to a simple fraction. 
 
 5. Reduce ^ of i of | to a simple fraction. 
 
 6. Reduce ^ of | of f to a simple fraction. 
 
 Cancellation. 
 
 Art. 121. To reduce a Compound Fraction to a 
 Simple one by Cancellation. 
 
 Write the fraction down with the sign of multiplication 
 between them, and cancel or reject all the factors that are 
 common to the numerators and deTwminators, (which by Pro- 
 position 6th, under Art. 116, does not change the value of the 
 fraction ;) then multiply the remaining numerators together 
 for a n£w numerator, and the remairdng denominators for 
 a new denominator. 
 
 Take for illustration the 6th example. 
 1 I ^_}_ 
 
ART. 122.] CANCELLATION. 103 
 
 Explanation.— First cancel the 3 of the numerator against 
 the 3 of the denominator, by drawing a line across them ] then 
 cancel the 4 of the numerator against the 4 of the denomina- 
 tor in the same manner. As there are no more factors com- 
 mon to both numerator and denominator, — multiply the remain- 
 ing numerators together for a new numerator; and the remain- 
 ing denominators, for a new denominator. 
 
 Art. 122. If any numerator and denominator have a 
 common divisor, divide them hoth by this divisor, and use ihi 
 quotients as a neiv fraction. 
 
 6. Reduce 4 of j\ of |f to a simple fraction. 
 
 OPERATION. 
 
 *1 2 
 
 8 3 
 
 Explanation. — 5 being a divisor of the 5 in the numerator 
 and the 15 in the denominator, we divide them both by 5 and 
 cancel the 5 and the 15, and consider the quotients, 1 and 3, 
 arising from this division, instead of the 5 and 15. Next can- 
 cel the 7 in the numerator and the 7 in the denominator. We 
 observe that 4 is a common measure of the 8 in the numerator, 
 and of the 12 in the denominator ; therefore, we divide by it, 
 and cancel the 8 and 12, and place the quotients in their pro- 
 per places. As there are no more factors common to both 
 numerator and denominator, nor any number that will divide 
 them both without a remainder, we multiply all the remaining 
 numerators together for a new numerator, and all the remain- 
 ing denominators for a new denominator, and obtain for the 
 answer f . 
 
 T. Reduce f of |- of j\ of j\ to its simplest form. 
 
 8. Reduce 4 of | of |f of | to its simplest form. 
 
 9. Reduce | of | of If of f of V of /, to its simplest 
 form. 
 
 10. Reduce | of f of | of j% of f of ff to its simplest 
 form. 
 
 11. What is the value of the compound fraction, | of ^ 
 ofiofyVofHof^V? 
 
 * In practice we do not write the quotient when it is a unit. 
 
104 FRACTIONS. [chap. V 
 
 12. What is the value of the compound fraction 4 of 
 
 2J of 12. of i^ of 25 of 15 ? 
 6 "^ 3 "^ 4 5 "^ S 3 "^ 2 5 • 
 
 Remark.— All whole and mixed numoers that oecur in compound fractions, 
 must be changed to improper fractions before the required reduction is per- 
 formed, 
 
 13. Reduce ly\ of 3 of y% of 2f to its simplest form. 
 
 14. Reduce yVj of 3^ of ly»j of l^ to its simplest form. 
 
 15. Reduce j\ of 4^ of 3^ of /g of ^ of li of 3^ to its 
 simplest form. 
 
 16. Reduce | of 9i of 3^ of |f of "^f to its simplest form. 
 It. Reduce j\ of 3y\ of y^ of j\ of 3^ to its simplest 
 
 form. 
 
 18. Reduce yV of 1} of /^ of 12f of y\ to its. simplest 
 form. 
 
 19. Reduce j\\ of 3"^ of 4 of if of ^ to its simplest 
 form. 
 
 A Common Denominator. 
 
 Art. 123. Two or more fractions have a Common 
 Denominator, when they have the same number for a 
 denominator. 
 
 1. Reduce f and | to equivalent fractions having a com- 
 mon denominator. 
 
 OPERATION. 
 
 3 4_15 16 ,, _ 15, 16 
 ^' "^ 20' 20 20 
 
 Explanation. — I first multiply the denominators together for 
 a common denominator, — 4 times 5 are 20, the common denom- 
 inator. Since, I have multiplied the denominator 4, of the 
 fraction f by 5, to preserve the value of the fraction, I multiply 
 the numerator 3, by the same number. 5 times 3 are 15 ; 
 therefore, f equals -^-f. I have multiplied the denominator 5, 
 of the fraction | by 4, and to preserve the value of the fraction, 
 1 multiply the numerator 4, by the same number. 4 times 4 are 
 16^ therefore, f equals ^f . 
 
 From the above explanation, to reduce fractions to equivalent 
 ones have a common denominator, we infer that we should 
 Multiplu all the denominators together for a common denominator .^ 
 and each numerator by all the denominators except its own, for a 
 new numerator. 
 
ART. 124.] THE LEAST COMMON DENOMINATOR. 105 
 
 Remark.— It is readily observed that, by the above process, both numerator 
 and denominator of each fraction is multiplied by the same number, which by 
 Proposition 5, under Art. 116. does not change the value of the fraction. 
 
 2. Reduce f and | to equivalent fractions having a 
 common denominator. 
 
 3. Reduce 4 and ^ to equivalent fractions having a 
 common denominator. 
 
 4. Reduce f and f to equivalent fractions having a 
 common denominator. 
 
 5. Reduce -f and ^ to equivalent fractions having a 
 common denominator. _ 
 
 6. Reduce -,% and y\ to equivalent fractions having a 
 common denominator. 
 
 1. Reduce |, |, and f to equivalent fractions having 
 a common denominator. 
 
 8. Reduce |, f , and | to equivalent fractions having 
 a common denominator. 
 
 9. Reduce |, f and f to equivalent fractions having 
 a common denominator. 
 
 10. Reduce i, i, i, and f to equivalent fractions having 
 a common denominator. 
 
 The Least Common Denominator. 
 
 Art. 124. The least common denominator of two or 
 more fractions, is the least common multiple of their denom- 
 inators. Hence, to find the least common denominators 
 of two or more fractions, reduce compound fractions to 
 simple ones, whole and mixed numbers, to improper fr actions , 
 and all to their lowest terms ; then find the least common mul- 
 tiple of the, denominators of the fractions, (hy Art. 9 4, J and 
 it will he their least common denominator. 
 
 1 . Reduce y5_, f , and y\, to equivalent fractions having 
 the least denominator. 
 
 OPERATION. 
 
 i, i, 1 = L^ so y, or write them this, 1^?^^ 
 2) 12 6 18 36' 36 36 36 
 
 3) 6 3 9 
 
 2 13 
 
 2 X 3 X 2 X 3 = 36, the least common denominator. 
 
 6* 
 
106 FRACTIONS. [chap. V. 
 
 Solution. — The remaining part of the work is to reduce each 
 of the given fractions to thirty sixths, without changing their 
 value. This can be done by multiplying the terms of each 
 fraction by a number that will cause its denominator to become, 
 3G. (See Art. 116, Proposition 5.) To find what number 
 I must multiply 12 by to produce 36, I divide the 36 by 12, and 
 find it to be 3. Multiply both numerator and denominator of 
 f3 by 3, gives if = -^\. Proceed in the same way with the 
 remaining fractions. 
 
 2. Reduce f, y\ and /^ to equivalent fractions having 
 the least common denominator. 
 
 3. Reduce |, |, and y\ to equivalent fractions having 
 the least common denominator. 
 
 4. Reduce y^g, g^, and -/p to equivalent fractions having 
 the least common denominator. 
 
 5. Reduce -f, 9y\, and |f to equivalent fractions hav- 
 ing the least common denomhlator. 
 
 6. Reduce y'V, 2|, and Sy^^ to equivalent fractions hav- 
 ing the least common denominator. 
 
 7. Reduce 2|, 3y\, and 3 -^^ to equivalent fractions 
 having the least common denominator. 
 
 8. Reduce f of |, ^ of y\, and | of | to equivalent 
 fractions having the least common denominator. 
 
 9. Reduce i of f of |, f of |, | of f , and f of f to 
 equivalent fractions having the least common denominator. 
 
 10. Reduce 2f of f , y^^ of f , 3| of ^^ of f and i of f 
 of I to equivalent fractions having the least common 
 denominator. 
 
 Addition of Common Fractions. 
 
 Art. 125. Addition of common fractions is the process 
 of finding the sum of two or more fractions. 
 
 1. What is the sum \, f , | and | ? 
 
 OPERATION. 
 
 1 + f +1 +f=V,or2|. Ans. 
 
 2. What is the sum of f , f , f , V» ^nd | ? 
 
 3. What is the sum of ^, f, f , V, and f ? 
 
ART. 126.] SUBTRACTION OF COMMON FRACTIONS. 107 
 
 4. What is the sum of j\, y\, \\, |f , |f , and |4 ? 
 
 5. What is th^^ sum of If, If, ff, H, H, H, and || ? 
 
 Remark. — Reduce con' pound fractions to simple ones, and mixed numbers to 
 improper fractions, and o/i to their lowest terms. Also, reduce fractions that 
 have diderent denominatois to equivalent ones having the least common denom- 
 inator. 
 
 6. What is the sum of f , f and j\ ? 
 
 OPERATION. 
 
 3 7 7 18 4-21+14 _53_o5 Ans 
 
 2 )2, 4, 6 
 1; 2, 3 
 2x2x2x3 = 24, the least comoion denominator. . 
 
 7. What is the sum of |-, f , and f'2 ? 
 
 8. What is the sum of |, 4f , and 2^ ? 
 
 9. What is the sum of 4i, 8|, and 8f ? 
 
 10. What is the sum of f of 4 of f , and | 6f f ? 
 
 11. What is the sum of | of W, f of \% \ of f , and f ? 
 
 12. What is the sum of ^ of 5i, 6^*3- off and ^ of /^ ? 
 18. What is the sum of i, i, i, f , 1, and f ? 
 
 14. What is the sum of i, |, f, f , I, f , f, and | ? 
 
 Remark. — When two fractions are to be added, the numerator of each being 
 a unit, it may be done mentally, by taking the sura of the denominators for a 
 new numerator and their product for a denominator. 
 
 Thus, the sum of 1 and i — I-±4 = ^ 
 ^ ' 7x5 35 
 
 15. What is the sum of ^ and ^ ? 
 
 16. What is the sum of J- and ^ ? 
 It. What' is the sum of \ and \ ? 
 
 18. What is the sum of | and \ ? 
 
 19. What is the sum of ^ and | ? 
 
 Subtraction of Common Fractions. 
 
 Art. 126. Subtraction of common fractions is the 
 method of finding the difference between two fractions. 
 1. From I subtract 3f . 
 
108 , FRACTIONS. [chap. V. 
 
 OPERATION. 
 
 I - f = f . Ans. 
 
 2. From f take f . 
 
 3. Fromf take f * 
 
 4. From |f take y\. 
 
 5. From jf take j\. 
 
 I Remark. — Reduce compound fractions to simple ones, and mixed numher to t»». 
 
 ' proper fractions, and all to their lowest terms. Also, reduce fractions that 
 have different denominations to equivalent ones having the least common de- 
 nominator. 
 
 6. From f take /j- 
 
 OPERATION. 
 
 =i 9i_in n 
 
 Ans, 
 
 OPERATION. 
 
 7 5 _21— 10_11 
 2)8 12"" 24 ""24 
 
 2)4 6 
 
 2 3 
 2x2x2x3x=24, tlie least common denominator. 
 
 1. From I take f . 
 
 8. From f take |. 
 
 9. From 1^ take |. 
 
 10. From 4i take 1|. 
 
 11. From 8| take 6f . 
 
 12. From ^f take 6f 
 
 13. From | of f take } of |. 
 
 14. From f of y^. take f of 4 of 1. 
 
 15. From | of 1^ of 4 f of U of yV- 
 
 16. From 9yV of 4} take i of | of f of 4^. 
 
 Remark. — When both the fractions have a unit for their numerator, the 
 subtraction may be performed mentally by placing the product of the denom- 
 inators under their diflference. 
 
 Thus, i — i = 
 
 8 — 53 
 
 • 4 ="f JT' 
 
 11. From 1 take i. 
 
 
 18. From i take i. 
 
 
 19. From i take yV- 
 
 
 20. From I take yV- 
 
 
 21. From 4 take -jV. 
 
 
ART. 128.] MULTIPLICATION OF COM1|ON FRACTIONS. 109 
 
 22. From j\ take Jg- 
 
 23. From j\ take j\. 
 
 Multiplication of Common Fractions. 
 
 Art. 127". Multiplication of common fractions is the 
 method of finding the product of two or more fractions, 
 or of integers and fractions. 
 
 Art. 128. To multiply one fraction by another, or an 
 integer by a fraction. First: 
 
 Reduce compound fractions to simple ones, and whole or 
 mixed numbers to improper fractions. Then proceed as in 
 the reduction of compound fractions. (See Art. 121.) 
 1. Multiply f , 4, If, If, and ||, together. 
 
 OPERATION BY CANCELLATION. 
 
 2 $ ^ 
 ^ $ X^ X^ $$ 2 
 -X-X— X— X— =- Ans. 
 
 d ^ x$ x$ t$ s 
 
 2. Multiply I by |. 
 
 3. Multiply f by ||. 
 
 4. Multiply I by ||. 
 
 5. Multiply ^ by 41. 
 
 6. Multiply together f , |, |, and -f. 
 
 t. Multiply together f |, |, |f , and |. 
 8. Multiply together 21, ||, Si and jf 
 9. ^Multiply together 4i, yV, 5^, 8^, and /j. 
 10. Multiply together |, 9i, 7|, yV, If, and H. 
 
 PRACTICAL QUESTIONS IN MULTIPLICATION OF FRACTIONS. 
 
 Remark. — In business transactions it is customary to add 1 cent when the 
 fraction is equal to or greater than a half of a cent, and to omit it when it la 
 less than the half of a cent. 
 
110 FRACTIONS. [chap. V. 
 
 1. What cost 42 bushels of apples, at 63f cents a 
 bushel ? 
 
 2. What cost Tf dozens of eggs, at 12^ cts. a dozen ? 
 
 3. What cost 13f bushels of turnips, at 3t^ cents a 
 bushel ? 
 
 4. What cost lOf yards of calico, at 15i cents a yard ? 
 
 5. What cost t5^ pounds of sugar, at If cts. apouni? 
 
 6. What cost Sf^tons of hay, at $12f a ton ? 
 
 T. What cost 6| bushels of apples, at 31^ cents a 
 bushel ? 
 
 8. What cost 13|- pounds of fish, at 9f cts. a pound ? 
 
 9. What cost lt5 pounds wool, at 39f cts. a pound ? 
 
 10. What cost 18f yards of ribbon, at 23^ cents a yard ? 
 
 11. What cost 18 pocket handkerchiefs, at f of a dollar 
 each ? 
 
 12. What cost 22| yards of selicia, at 81| cents a yard ? 
 
 13. What cost 35| pounds of raisins, at 18| cents a 
 pound ? 
 
 14. What cost t5f bushels of wheat, at $1| a bushel ? 
 
 15. What cost 23| cords of wood, at $3f a cord ? 
 
 16. What^cost 212| pounds of beef, at 7^ cents a pound ? 
 It. What'cost 14f barrels of vinegar, at $1 Of a barrel ? 
 
 18. What cost 22f barrels of sugar, at $15f a barrel ? 
 
 19. What cost 35i tons of coal, at $9f a ton ? 
 
 Division of Common Fractions. 
 
 Art. 129. Division of common fractions is the method 
 of dividing one fraction by another, or whole numbers and 
 fractions by each other. 
 
 Remark. — When the fractions have a common denominator, division can 
 be performed by dividing the numerator of the one by the numerator of the 
 other. 
 
 1. Divide f by f . 
 
 OPERATION. 
 
 2. Divide |f by /y. 
 
 3. Divide ^j by y\. 
 
ART. 130.] PRACTICAL QUESTIONS. Ill 
 
 4. Divide j^ by y\. 
 
 5. Divide ^^ by ^\. 
 
 6. Divide | by f . 
 
 Solution. — 1 is contained in |, | times ; and if 1 is con- 
 tained in I, I times, J- is contained in |, 4 X | times; and 
 f i^ contained in it i of 4 X | times = | x | = f times. 
 
 Art, 130. Hence, to divide a fraction by a fraction, 
 or fractions and whole numbers, by each other, we merely, 
 Invert the divisor and ^proceed as in multiplication, after hav- 
 ing reduced comjpound fractions to simple ones, and whole and 
 mixed numbers to improper fractions. 
 
 7. Divide f by f. 
 
 8. Divide f by f . 
 
 ' 9. Divide 21 by 4. 
 
 10. Divide 31 by li. 
 
 11. Divide H by 6i 
 
 12. Divide 8^ by 6|. 
 
 13. Divide 1 of f by 2f . 
 
 14. Divide f of | by 2^. 
 
 15. Divide 3^ of | by | of 1^. 
 
 16. Divide 4^ times 3| by 8|. 
 
 17. Divide f of -J-f by -f of if. 
 
 18. Divide 2% of H of 8f by 4i times 6f . 
 
 19. Divide 34 of 81 by 61 of 3i: 
 
 20. Divide yV of |i of 81 times 4 by f of 4^^. 
 
 PRACTICAL QUESTIONS IN DIVISION OF FRACTIONS. 
 
 1. At $1 a bushel, how many bushels of apples can be 
 bought for $20 ? 
 
 2. At f of a cent a piece, how many oranges can be 
 bought for 14| cents ? 
 
 3. If I pay 4 1 cents for riding 1 mile, how many miles 
 can I ride for 280 cents ? 
 
 4. A butcher expended $25t|^ for sheep, at $lf a head; 
 how many sheep did he buy } 
 
112 FRACTIONS. [chap V. 
 
 5. How many pounds of tea, at $lf a pound, can be 
 obtained for $19f ? 
 
 6. A lady bought 3^f yards of calico for 561 cents; 
 how much did it cost a yard ? 
 
 7. A merchant bought 96 sheep for $99|i; how much 
 did he give a head ? 
 
 8. How many tons of coal, at $8f a ton, can be bought 
 for $97 ? 
 
 9. A man paid $565^ for a farm, giving $21f an acre; 
 of how many acres did the farm consist ? 
 
 10. At $1^ a day, how many days must a man work for 
 
 Complex Fractions. 
 
 Art. 131. To reduce complex fractions to simple ones; 
 we lirst, Reduce compound fractions to simple ones, and whole 
 and mixed numbers to improper fractions. Then consider 
 the denominator of the complex fraction a divisor and 
 proceed as in division of fractions. 
 
 3 
 
 1. Reduce | to a simple fraction. 
 
 5 
 
 OPERATION. 
 
 i zrz ? X X = " = If Ans. 
 2 Reduce I to a simple fraction. 
 
 3 
 2. 
 
 3. Reduce ^ to a simple fraction. 
 
 4 
 
 - of ^ 
 
 4. Reduce ^ ? to a simple fraction. 
 
 5. Reduce 5_2_I to a simple fraction. 
 
 I off 
 
 6. Reduce — ?_ to a simple fraction. 
 
 ioff 
 
 52- 
 
 *l Reduce 3— to a simple fraction. 
 
 1 of 51 
 
A.RT. 132.] LEAST COMMON MXLTIPLIE OF FRACTIONS. 113 
 
 8 Reduce ^^ ^^ ^^ to a simple fraction. 
 H Of 3f 
 
 J. 4- 1 
 9. Reduce ^ ^ * to a simple fraction, 
 a 
 
 3 4 
 
 10. Reduce ^ "^ ^ ^ ^ to a simple fraction, 
 f ofl + f 
 
 Least Common Multiple of Fractions. 
 
 Art. 132, The Least Common Multiple of any two 
 or more fractions, is the smallest number that will, when 
 divided by each of them, give an integer for a quotient. 
 
 Since, in dividing any number by a fraction, the denomi- 
 nator of that fraction becomes a multiplier ^ and the nume- 
 rator a divisor of that number ; it is evident, — Tha:t the 
 least common multiple of any two or more fractions, after 
 reducing mixed numbers to improper fractions, compound 
 fractions to simple ones, and all to their lowest terms^ will be 
 the quotient arising from dividing the least common multiple 
 of their numerators by the greatest common measure of their 
 denomhmtors. 
 
 1. What is the least common multiple of 4f f , SyV^, and 
 
 Solution. — The above mixed numbers, changed to improper 
 fractions become ^-^^^ f^f, and 2//. These fractions when 
 reduced to their lowest terms become, '^f, Vi', and \y. The 
 least common multiple of the numerators, {65, 143, and 117) 
 of these fractions is 6435. See Art. 94. 
 
 The greatest common measure of the denominators, (14, 
 28, and 42,) of these fractions is 14. See Art 89. Hence, 
 6435, the least common multiple of the numerators of these 
 fractions, is 14 times larger than the least common multiple 
 of the fractions. Consequently ^f|^ or 459y^j is the least com- 
 mon multiple required. 
 
 2. What is the least common multiple of 4y\, 1\ and 2/^? 
 
 3. What is the least common multiple of 4f , 9yV> and 12i? 
 
 ♦ Note.— The student will readily observe that the Greatest €ommom Measukk 
 of any two or more fractions, after being reduced to their simplest form, will be 
 the QUOTTEXT arising from dividing the greatest common measure oi their numerators 
 by the least common multiple of their denominators. 
 
 What is the greatest common measure of J, ||, and IfP 
 
il4 FBiCTIONS. [chap. V, 
 
 4. What is the least common multiple of }|, |f , i|, 
 and f f ? 
 
 5. What is the least common multiple of 2|-, 16f , and 
 
 10/3? 
 
 PRACTICAL QUESTIOXS IN MULTIPLES. 
 
 1. What is the smallest sum of money for which a per- 
 son could purchase, either a number of geese, at $1^ a 
 piece ; or a number of turkeys, at $2^ a piece, and how- 
 many of each could be bought, — the entire sum to be 
 employed in either purchase ? 
 
 2. A can travel 6| miles in a day; B lly^j miles ; 
 20/0 miles ; and D SOa^j miles in a day. What is the least 
 number of miles that will afford a number of whole days^ 
 travel for any of the four, and how many days would it 
 take each to accomplish the journey ? 
 
 3. What is the least number of bushels of grain that 
 will fill a number of hogsheads, each containing 10/j 
 bushels; a number of boxes, each containing 23|f bushels; 
 or a number of bins, each containing 25 }| bushels; and 
 how many times would it fill each of them ? 
 
 4. What is the smallest sum of money for which I could 
 purchase a number of cows, at $20^^; a number of oxen, 
 at $47|^; or a number of horses, at $51y^j; and what 
 number of each could I purchase for that sum ? 
 
 5. Three vessels A, B, and C start from the same place, 
 at the same time, and sail in the same direction around 
 an island 30 miles in circumference; A at the rate of 3, 
 B 11, and C 23 miles an hour. How many hours before 
 they will all meet at the place from which they started ? 
 How many hours before they will first meet, and at what 
 point ? Suppose they continue sailing, how. often will they 
 all be together ? 
 
 Solution. — A moves at the rate of 3 miles an hour; 
 consequently, to move 1 mile it will take ^ of an hour, and 
 to move 30 miles, (once around the island,) it will take 
 y or 10 hours. In a similar way, we find B will move 
 
ART. 154.] PRACTICAL QUESTIONS IN MULTIPLES. Il5 
 
 once around the island in ff of an hour; and C, in ff of 
 an hour. Now it is evident that the least common multiple 
 of 10, f f, and |f will express the number of hours that 
 must elapse before they will all meet at the place from 
 which they started ; which is 30. hours. 
 
 How long before they will first meet ? C gains on B, 
 23 — 11= 12 miles in 1 hour; consequently, to gain 1 mile 
 it will take -^-^ of an hour, and to gain 30 miles, (the dis- 
 tance he must gain before he overtakes B,) 30 times -^^ = 
 ^f , or f of an hour. — B gains on All — 3 = 8 miles in 
 1 hour; hence to gain 1 mile it will take | of an hour, and 
 to gain 30 miles, (the distance he must gain before he 
 overtakes A,) 30 times | = y, or \^of an hour. Since 
 C will overtake B in f of an hour; and B will overtake 
 A in y of an hour; it is evident that the number of hours 
 that must elapse before C will overtake B, at the same 
 time that B overtakes A will be the least common multiple 
 of f and y , which is 7^ hours. 
 
 If they are together in t^ hours, 
 A must sail 22i miles, or times around the island + 22^ miles. 
 B " '* 82^ " or 2 " " " '* + 22i " 
 
 C " " 172i "• or 5 " ♦* " *' +22i « 
 
 Consequently 22^ miles from the place from which they 
 started is the place where they first will be together. 
 
 If they continue traveling they will be together every 
 t| hours. 
 
 6. If four men A, B, C, and D, start from the same 
 place at the same time, and walk around an island 2t 
 miles in circumference; A at the rate of 4, B 12, C 20, 
 and D 28 miles a day; how many miles will each have to 
 travel before they meet, and how many days before they 
 will all meet at the place from which they started ? 
 
 7. There are three wheels, A, B, and C, each lOf feet 
 in circumference, standing with their axes in a right line, 
 with a letter M on the circumference of each, which are 
 also in a right line. If these wheels be set in motion ; A 
 at the rate of 5|, B 7^, and C 131 feet in 1 second, how 
 long before the M's. on the ch-cumference of the wheels 
 
116 FRACTIONS. [chap. V. 
 
 will all be in a right line again, and how many revolutions 
 will each have made ? 
 
 PKACTICAL QUESTIONS IN FRACTIONS. 
 
 1. Reduce 23 ly^ to an improper fraction. 
 
 2. Reduce 478yVy to an improper fraction. 
 
 3. Reduce -f f ^ to a mixed number. 
 
 4. Reduce 2-iiS- to a mixed number. 
 
 5. Reduce ^^ to its lowest terms. 
 
 6. Reduce |||ff to its lowest terms. 
 
 *r. Reduce |- of f of yV of |f of ^f to its simplest form. 
 
 8. Reduce |f of if of f | of f f to its simplest form. 
 
 9. Reduce |, f , |, and -f to equivalent fractions, hav- 
 ing a common denomination. 
 
 10. Reduce y\, |f and \l to equivalent fractions, hav- 
 ing the least common denomination. 
 
 11. What is the sum of |, |, |, f, and ii ? 
 
 12. What is the sum of y^o of 4^, | of 3|, and ^ of 6| ? 
 
 13. From 8f subtract 6f . 
 
 14. From | of 8^ subtract f of 2f . 
 
 15. Divide 3^ by 41-. 
 
 16. Divide | of 2| by | of 4f . 
 
 IT. A has 4f times $25, and B has ^ times $8|; how 
 much more has A than B. 
 
 18. A has I of $16; Bfof$4T3; Cf of $8621; andD 
 I of f of f of $168|. How many dollars have they together. 
 
 19. A had | of f of 12^ times $8643^, and paid f of 
 ■|- of it for a farm ; how much had he remaining ? 
 
 20. A had $864T2, which was 6f tpimes as much as B 
 had; how much had B ? 
 
 21. A has 1278 sheep, which is 188 more than | of 3^ 
 times B's number; how many sheep has B ? 
 
 22. A and B own 680 acres of land; | of A^s number 
 of acres equals f of B's; — how many acres have each ? 
 
 23. A farmer has 495 bushels of wheat and rye together. 
 
f 
 
 ART. 132.] PRACTICAL QUESTIONS. lit 
 
 and f of the number of bush, of wheat equals ^ of the num- 
 ber of bush, of rye. How many bushels of each has he ? 
 
 24. A speculator bought 688 geese and turkeys; how 
 many of each did he buy, providing there were only | as 
 many geese as turkeys ? 
 
 25. A and B together own 824 sheep; how many has 
 each, providing A has 1| times as many as B .? 
 
 26. A owns j\ of a certain tract of land, containing 
 98600 acres; B owns ^f of the remainder; C owns j\ as 
 much as A and B together; and D owns the remainder. 
 How much does each own ? 
 
 27. A merchant expended $463 for dry goods; | of 
 ^j of the remainder for groceries; and what then re- 
 mained, which was $4680, he expended for a store and 
 lot. How much did the groceries cost ? 
 
 28. A gentleman invested | of his fortune in specula- 
 tion, and the remainder, which was $1630 more than the 
 half of his fortune, he put out on interest. At the end 
 of the year he gained by speculation y\ as much as he 
 laid out, and his interest was /^ of the principal; how 
 much was his fortune, and how much did he gain during 
 the year ? 
 
 29. A man being asked the value of his horse, replied, 
 that its value increased by its -f and $268| more, equaled 
 $864. What was the value of the horse ? 
 
 30. A farmer has -f of the number of his sheep in one 
 field ; and the remainder, which is 46 more than the half 
 of his flock in a second field. How many sheep has he in 
 each field, and how many in both ? 
 
 31. A certain sum of money was divided between two 
 brothers, James and Jackson ; James took | of it, lacking 
 $145 ; and Jackson the remainder. It now appears that 
 each has the same sum. How much did each receive ? 
 
 32. From a certain flock of sheep A purchased f of 
 them ; B f of them ; C f of them ; and D the remain- 
 der, which was 116. How many sheep were then in the 
 field, and how many did A, B, and C, buy respectively ? 
 
 33. A has | of 1-f times $2660 which is 3^ times as 
 many again as B has. How many dollars has B ? 
 
,118 FRACTIONS. [chap. V, 
 
 34. A and B together own a farm ? A owns y\- of it ; 
 and B /_ of it. If B should sell to A 12| acres, they 
 would then each have the same number of acres. How 
 many acres has each ? 
 
 35. An estate was divided among A, B, and C. A 
 had ^ of it ; B ^ of it ; and C the remainder. A, by 
 this division, received $180 more than B. How much 
 was the estate, and how much did each receive ? 
 
 36. Divide $9926 among A, B, C, and D, so that A 
 shall have -f of it, lacking $1812 ; B | of the remainder, 
 lacking $858 ; C f of what now remains, lacking $1880 ; 
 and D what is left ? 
 
 3t. A gentleman's house cost $4800, and If times its 
 cost, is 3i times -f of the cost of the furniture contained 
 in it ; what was the cost of the furniture ? 
 
 38. Henry had f of a certain fortune ; Perry ^^ Of it ; 
 and Elisha the remainder, which was $1600. How much 
 was the fortune, and how much did Henry and Perry re- 
 ceive respectively ? 
 
 39. Bought at one time 460 acres of land, at $25|- an 
 acre ; at another time 345 acres, at $43^ an acre. If | 
 of the whole quantity were sold, at $21 an acre, and the 
 remainder, at $34 an acre, what would be the gain or loss ? 
 
 40. A merchant purchased 120 yards of cloth for $780, 
 and sold | of it at a profit of $lf a yard ; and the re- 
 mainder, at a loss of $f a yard. How much did he gain 
 by the operation ? 
 
 41. A person bought 38 barrels of flour at $4| a bar- 
 rel. Having sold 21|- barrels of them, at $4f a barrel, at 
 what price a barrel must the remainder be sold to gain 
 $25^ on the whole. 
 
 42. James, Henry, and Joseph were employed to hoe a 
 field of corn for $32.10. James could hoe a row in 2 Of 
 minutes; Henry in 25| minutes; and Joseph in 32y^o min- 
 ntes. It so happened that when they all first came to the 
 end of a row at the same instant, that the work was com- 
 pleted. How long were they engaged in the field; how 
 many rows did the field contain; and how m'ich in equity 
 ought each to receive ? 
 
ART. 134.] DECIMAL FRACTIONS. 119 
 
 CHAPTER YI. 
 DECIMAL FRACTIONS. 
 
 Art. 133. A Decimal Fraction is one in which the 
 denominator is not expressed, but is understood to be a 
 unit followed by one or more ciphers ; or such a fraction 
 the successive orders of which increase from right to left in 
 a tenfold ratio, consequently decrease from left to right in 
 the same ratio. 
 
 Decimal Fractions originate from dividing 1 into 10 
 equal parts ; each of these parts into 10 other equal parts ; 
 and each of the parts thus obtained into 10 other equal 
 parts, and so on, indefinitely. Thus, \ — 10 = yV ; yV "^ 
 10 = y^o ; y^o- -h 10 = yoVoj &c., wMch are expressed 
 in Decimals as follows : — 
 
 tV = -1 ; T^o = 01 ; y^Vo = -001, &c. 
 
 Art. 134. In expressing Decimal Fractions, the nu- 
 merator only is written with a point before it, called a 
 Decimal 'point or Separatrix, to distinguish it from whole 
 numbers*; the denominator being understood. Thus, 
 
 y\ = 't tenths. 
 Too = '^^ hundredths. 
 To\o = "00 1 thousandth's. 
 
 _7 
 
 
 
 = 'OOOt ten-thousandths. 
 r= 0223 ten-thousandths. 
 
 By inspecting the above fractions, it is observed that 
 tenths occupy the first place at the right of the decimal 
 point ; that hundredths occupy the second place; that thovr- 
 sandths occupy the third place, &c. 
 
 We also observe that each removal of a figure one place 
 towards the right, decreases its value in a tenfold ratio. 
 Hence, 
 
120 DECIMAL FRACTIONS. [cHAP. VI. 
 
 Every cipher placed on the left of a decimal figure dimin- 
 ishes its value in a tenfold ratio. Thus, '9 = -^q, '09 ^- 
 rW, aid -009 = y/oo, &c. 
 
 If a cipher be placed on the right of a decimal figure, 
 it does not change its value, as the figure still occupies 
 the same place. Thus, '9 = '90 = "900, y\ = rVo = 
 
 JULQ_ Jirn 
 To 0> *^^' 
 
 ^ Numeration of Decimal Fractions. 
 
 Art. 135. A whole number and a decimal fraction, 
 when considered together, is called a mixed number ; the 
 relation and names of which can be learned from the fol- 
 lowing 
 
 TABLE. 
 
 
 <r 
 
 
 
 
 OQ 
 
 'S 
 
 
 a 
 
 
 
 A 
 
 
 . m . 
 
 s 
 
 ■4^ 
 
 
 i 
 
 
 -5 
 
 i 
 
 1 
 
 rd 
 
 
 1 
 
 1 
 
 
 1 
 d 
 
 1 
 
 
 isand 
 dreds, 
 
 3. 
 
 i 1 
 
 
 ^ 
 
 3 
 
 S 
 
 
 rd r3 
 
 .1 s 
 
 ■-d 
 
 m 
 
 ,d 
 
 d 
 o 
 
 .2 
 
 -3 
 
 ■73 
 
 c§ 
 
 J ^ § § 
 
 •3 1 
 
 
 1 
 
 O 
 
 d 
 
 d 
 d 
 
 
 d 
 s 
 
 3 
 
 d 
 
 d 
 
 d 
 
 i 
 
 H ffi H 
 
 ^ 1 
 4 
 
 H 
 
 ffl 
 
 H 
 
 H 
 
 ffi 
 
 ffi 
 
 PQ 
 
 H 
 
 K 
 
 
 9 3 1 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 4 7 
 
 8 
 
 6 
 
 2 
 
 7 
 
 
 Whole Numbers. 
 
 
 
 
 
 Decimals. 
 
 
 
 
 
 Art. 136. To read a Decimal number expressed in 
 figures. 
 
 Read the figures as in whole numhcrs, and add the name 
 of the decimal place. Thus, "9 is read nine-tenths ; '09 is 
 read nine hundredths ; '100024 is read one hundred thousand 
 and 24 millionths, &c 
 
 Rkmark.— To ascertain the name of the right hand figure, begin at the left 
 and name each figure till you come t© the last one, which will be the name 
 required. 
 
ART. 13t.] UNITED STATES CURRENCY. 
 
 121 
 
 Read the following numbers. 
 
 1. 13 
 
 2. -06 
 
 3. -0102 
 
 4. -02202 
 
 7. 
 
 9. 
 
 46-824 
 
 36-4231 
 
 61-46212 
 
 141-8630202 
 
 11. . 102-460203 
 
 12. 14683-04000602 
 
 13. 18602-84683002 
 
 14. 10001-861320401 
 
 5. -060108 10. 1468-10002 15. 1010101-10101010101 
 
 United States Currency,* or Federal Money. 
 
 Art. 137. United States Currency is the legal Money 
 of the United States, and is expressed according to the 
 Decimal Scale of notation. 
 
 Money may be considered the measure of the value of 
 things. 
 
 f The denominations of the United States Currency are 
 Eaf^hs, Dollars, Dimes, Cents, and Mills. 
 
 The coins of the United States are the 
 Double-Eagle, 
 
 made of gold, 
 
 ► made of silver; 
 
 Eagle 
 
 Half-Eagle, 
 
 Quarter-Eagle, 
 
 and Dollar, 
 
 The Dollar, 
 
 Half-Dollar, 
 
 Quarter-Dollar, 
 
 Dime, 
 
 Half-Dime, 
 
 and Three-cent piece. 
 
 The cent is made of copper. 
 
 The mill is not coined. 
 
 Note. — By an act of Congress, January ISth, 1837, the gold 
 and silver coin must consist of y^^ pure metal, and -^-^ alloy. 
 
 Remark. — * This currency is usually called Federal Muney, because it was 
 made the currency of the United States at the time of the adoption of the 
 Federal Constitution, August 8th, 1786. 
 
 t The word Dollar is derived from a Danish word, which was derived from 
 Dale, the name of the town in which this coin was first made. The word 
 Dime is derived from a French word signifying ten; — the word Cent, from a 
 Latin word signifying one hundred; — the word Mill, from a Latin word signi- 
 fying one thousand. The terms, Dime, Cent, and Mill, are applied to coins of 
 our currency, in consequence of the relation they respectively bear to the 
 Dollar. 
 
 6 
 
122 DECIMAL FRACTIONS. ^CEAT. VI. 
 
 The alloy for gold must consist of an equiil quantity 
 of silver and copper, . and the alloy for silver of pure 
 copper. 
 
 The three-cent piece is | silver and | copper. 
 
 Before 183t, the gold for coinage consisted of || pure 
 I gold, -^j silver, and 2V copper; or, as sometimes expressed, 
 22 carats gold, 1 of silver, and 1 of copper; the word 
 ca7-at meaning one twenty-fourth. 
 
 Silver for coinage consisted of 1489 parts of pure silver, 
 and 179 parts of pure copper; or, expressed in carats, 
 2IT3V of silver, and 2//^ of copper. 
 
 Table of United States Currency. 
 
 10 Mills 
 
 make 
 
 1 Cent, 
 
 marked 
 
 C. 
 
 10 Cents 
 
 C( 
 
 1 Dime, 
 
 u 
 
 d. 
 
 10 Dimes 
 
 (I 
 
 1 Dollar, 
 
 <( 
 
 $.* 
 
 10 Dollars 
 
 « 
 
 1 Eagle, 
 
 u 
 
 E. 
 
 Art. 138. The accounts of the United States are 
 kept in Dollars Cents, and Mills. In business transactions, 
 the eagle is expressed in dollars,^-the dime in cents. Thus, 
 4 eagles, according to the preceding table, is = 40 dol- 
 lars; and, instead of saying 4 eagles and 5 dollars, we 
 say 45 dollars. Also, *I dirn^s are = 70 cents; therefore, 
 instead of saying 7 dimes and 9 cents, we say 79 cents, &c. 
 
 Art. 139. In the United States Currency, dollars are 
 Integers, and therefore, occupy the place of units. Cents 
 express hundredths of a dollar, consequently they occupy 
 the first and second place on the right of the decimal 
 point; the third place is mills; &c. Thus, $47'356, is 
 read forty-seven dollars thirty-five cents and six mills. 
 
 Art. 140. To express any number of cents less than 
 10, there should be a cipher placed between them and the 
 decimal point, as cents are hundredths of a dollar, and 
 therefore occupy two places; if mills only are expressed, 
 
 * This symbol is probably a contraction of the letter U, placed upon an 
 8, to denote U. S. (United Stateo.) 
 
ABT. 142.] REDUCTION OF DECIMALS TO COMMON FRACTIONS. 123 
 
 two ciphers should be placed between them and the 
 
 decimal point. Tiuis: 
 
 4 cents is written $*04, and is read 4 hundredths of a dollar. 
 12^ cents is written $12^, and is read 12^ hundredths of a dollar. 
 I ota cent is written $-00f , and is read | of 1 hundredths of a dol- 
 lar. 5 mills is written |-005, and is read 5 thousandths of a 
 dollar. 4 cents and 6 mills is written $-046, and is read 46 
 thousandths of a dollar. 
 
 Art. 141. It might be well to observe, that the first 
 figure after the decimal point, expresses te^iths of a dollar, 
 or tens of cents ; the second, cents; the first two places 
 taken together express hundredths of a dollar, or cents ; 
 the third viills ; the fourth tenths of mills, &c. 
 
 Reduction of Decimals to Common Fractions. 
 
 Art. 142, From what has been said of Decimal FraC' 
 tions, we infer that a decimal can be reduced to a common 
 fraction by erasing the decimal pointy and underneath writing 
 the denominator, which is a unit followed by as many ciphers 
 as there are places in the decimal ; then reduce the fraction 
 to its lowest terms. 
 
 I. Reduce '025 to a common fraction. 
 
 operation. 
 
 •025 = yV/o = tV- Ans. 
 Reduce each of the following decimals of a dollar to 
 equivalent common fractions. 
 
 2. $0625. 5. $-25. 8. $-625. 
 
 3. $-125. 6. $-5. 9. $'875. 
 
 4. $-375. 1. $-75. 10. $-5625. 
 
 II. Express 6*0625 by an integer and a common frac- 
 tion. 
 
 12. Express 12 25 by an integer and a common fraction 
 
 13. Express 145 by an integer and a common fraction 
 
 14. Express 25'625 by an integer and a common frac 
 tion. 
 
 15. Express 45'8t6 by an integer and a common fraoi 
 tion. 
 
124 
 
 DECIMAL FRACTIONS. 
 
 [chap. VI. 
 
 16. Express 3t*15 by an integer and a common frac- 
 tion. 
 
 It. Express 16 'St 5 by an integer and a common frac« 
 tion. 
 
 18. Express 34"93t5 by an integer and a common 
 fraction. 
 
 19. Express 62-5636 by an integer and a common 
 fraction. 
 
 20. Express 141*18t5 by an integer and a common 
 fraction. 
 
 Reduction of Common Fractions to Decimals. 
 1. Reduce -f to a decimal. 
 
 
 OPERATION 
 
 • 
 
 
 
 
 
 . 
 
 .d 
 
 
 
 
 
 X 
 
 ■S 
 
 
 
 
 
 '2 
 
 g c3 
 
 
 
 n 
 
 1 
 
 c 
 5 
 
 S .a 
 
 
 
 j3 
 
 
 
 o 
 
 
 
 
 5 
 
 s 
 
 Xi 
 
 
 m 
 
 1 
 
 a 
 
 o 
 
 J3 
 
 ^ 6 
 
 i2 
 
 •5 
 
 
 3 
 
 H 
 
 Ti <a 
 
 
 C 
 
 H 
 
 C 
 
 a 
 
 a 
 
 'S 
 
 (U 
 
 3 
 
 j3 
 
 
 s 
 
 t3 
 
 H 
 
 K 
 
 H 
 
 H 
 
 W 
 
 7)4. 
 
 •0 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 7 
 
 1 
 
 4 
 
 2+* 
 
 Explanation. — 7 is not contained into 
 4 units a whole number of times, there- 
 fore, we reduce 4 to tenths; 4 = 40 
 tenths. 7 is contained in 40 tenths, 5 
 tenths times and 5 tenths remaining, — 
 which equals 50 hundredths. 7 is con- 
 tained in 50 hundredth, 7 hundredtJis 
 times, and 1 hundredth remaining, which 
 equals 10 thousantlis^ &c. 
 
 Remark. — If the decimal is carried out six places, the result will be suffl. 
 ciently exact for all practical purposes. 
 
 What decimal of a dollar is equal to the following 
 fractions of a dollar ? 
 
 4. $f. 
 
 5. $f. 
 
 6. $f. 
 
 1. %h 
 
 2. $1 
 
 3. $^. 
 
 8. Reduce f 12^ to an equivalent decimal expression 
 
 9. Reduce $16^ to an equivalent decimal expression. 
 
 * The symbol, -f-; when plficed after a decimal indicates more. 
 
ART. 144.] REPETENDS. 125 
 
 10. Reduce $42| to an equivalent decimal expression. 
 
 11. Reduce $25 y\ to an equivalent decimal expression 
 
 12. Reduce llS^'^g- to an equivalent decimal expression, 
 
 13. Reduce |3t|| to an equivalent decimal expression. 
 
 Reduction of Mixed Decimals to Simple Decimals. 
 
 Art. 143. If, in the place of the Common Fraction, 
 we place its equivalent decimal without the point prefixed, 
 the value remains the same. Thus, .4f . 
 
 The f = -75, hence '41 = -475 
 
 14. Reduce '251 to an equivalent simple decimal. 
 
 15. Reduce •2t-]- to an equivalent simple decimal. 
 
 16. Reduce "SSi to an equivalent simple decimal. 
 It. Reduce '411 to an equivalent simple decimal. 
 
 18. Reduce •64f to an equivalent simple decimal. 
 
 19. Reduce '841 to an equivalent simple decimal. 
 
 20. Reduce "1461 to an equivalent simple decimal. 
 
 Repetends. 
 
 Art. 144. If a common fraction cannot be accurately 
 expressed in decimals, it is evident that the decimal figures 
 will recur in periods; and that the number of figures in 
 the period cannot exceed the number of units in th6 denom- 
 inator, less one — for every remainder must be less than the 
 denominator; and whenever a remainder occurs like one 
 previously obtained, the decimal figures will begin to 
 repeat. 
 
 Decimals that recur in this way are called repeating 
 decimals ; the figures repeated are called a repeteiid, and are 
 distinguished by a (') placed over the first and last, as, 
 1 =z -333, &c. = -3; 1 = -142857142, &c. = •142857. 
 
 If decimal figures precede the repetend, they are called 
 i\Q finite part of the decimal; as, -Jj = 0-0833, &c. The 
 finite part is -08. 
 
126 DECIMAL FRACTIONS. [CHAP. VI. 
 
 When tbe repeating period begins with the first decimal 
 figure, it is called a simple repetend. 
 
 A simple repetend that contains as many figures in the 
 repeating part as there are units in the denominator, less 
 one, is called a Perfect Repetend. Thus, 
 
 \ = 0-142851:. 
 
 tV = 0-658823529411l64i. 
 
 tV = 0-6526315^894l36842i. 
 
 &c. &c. 
 
 Compound Repetends. 
 
 The following fractions are called Compound Repetends^ 
 because they consist of a finite and a repeating part. 
 
 i =016. 
 
 tV = 0083. 
 
 tV = 0-0h4285. 
 
 &c. &c. 
 
 Art. 145. Repetends Reduced to Common Frations, 
 
 •i = X; -01 = ir. -ooi = ^1^; -oooi = ^^v». 
 
 •2 = I; -02 ^ -/^; -002 = «f^; '6002 = ^/^^. 
 . -3 =. f ; -03 = ^V; -003 = ^|^; -0003 = ^^\^. 
 &c., &c., &c., &c. 
 
 From which we observe that Uie repetend, with the 
 decimal point and useless ciphers on the left, erased, is the 
 numerator, and the denominator is as many 9's as there are 
 places in the repetend. 
 
 Thus, -004004, &c. = ^f^. 
 
 1. Redcce '142857 to a common fraction. 
 
 2. Reduce 02*1 to a common fraction. 
 
 3. Reduce 'Oli to a common fraction. 
 
JLPT. 147. I REDUCTION OF COMPOUND REPETENDS 121 
 
 4. Reduce •'12 to a common fraction. 
 
 5. Reduce •123 to a common fraction. 
 
 6. Reduce -238095 to a common fraction. 
 
 Art. 14G, Reduction of Compound Repetends 
 
 7. Reduce 'OSSSSS, &c., to a common fraction. 
 
 OPERATION. 
 
 •083333, &c. = -083 = -081 = ^ = ^ = 1. Ana. 
 
 100 300 12 
 
 8. Reduce '16 to a common fraction. 
 
 9. Reduce '06 to a common fraction. 
 
 10. Reduce '045 to a common fraction. 
 
 11. Reduce '0416 to a common fraction. 
 
 12. Reduce •0'il4285 to a common fraction. 
 
 Art. 147. We have se^ that the value of some common 
 fractions can be accurately expressed in decimals, while 
 that of others can only be approximately expressed, as the 
 process of division will never terminate. 
 
 As we annex ciphers to the numerator and divide by 
 the denominator to change a common to a decimal fraction; 
 and, as annexing a cipher to any number is the same as 
 multiplying it by 10, it follows that whenever the prime 
 factors of the denominator of a common fraction (affer 
 reducing it to its lowest terms) do not differ from 2 and 5 
 (the prime factors of 10), the division will terminate. 
 
 Hence, to determine whether a common fraction can be 
 accurately expressed in decimals, 
 
 Reduce it to its lowest terms, then resolve the denominator 
 into its prime factors ; if these factors do not differ from 2 
 and 5 {the factors of lOJ, the fraction can he accurately 
 expressed in decimals. It is also evident that the highest ex- 
 yo'iie'nt of the 2 or 5 will denote the number of decimal places 
 required tt express it. 
 
128 DECIMAL FRACTIONS. [cHAP. V 
 
 1. Cun -gf J be accurately expressed in decimals ? If so, 
 how many places will be required to express it ? 
 
 2. Can yf ^ be accurately expressed in decimals ? If so, 
 how many places will be required to express it ? 
 
 3. Can y|j be accurately expressed in decimals ? If so, 
 how many places will be required to express it ? 
 
 4. Can -^ be accurately expressed in decimals ? If so, 
 how many places will be required to express it ? 
 
 6. Can -y Jo" be accurately expressed in decimals ? 
 
 Addition of Decimals and United States Currency. 
 
 Art. 148. Since decimals increase from right to left, 
 and decrease from left to right, in the same ratio as simple 
 numbers, we can add, subtract, multiply, or divide them, in 
 the same manner as though they were abstract numbers. 
 In addition, observe to place units under units, tens under 
 tens, &c. Hence the decimal points will always come one 
 under another. 
 
 1. What is the sum of 14-623, 231-6231, 101-36, 
 8002-68023, and 1-462312 ? . 
 
 operation. 
 
 is to 
 
 '« 2 
 
 . <»■ S 2 
 
 c -a TJ c o TS 5 
 
 Si's wiS-'S !=''^'S:§ 
 
 14-6 2 3 
 2 3 1-6231 
 10 1-3 6 
 8 2-68023 
 
 7-462312 
 
 835 7-7 48642 Ans. 
 
 2. What is the sum of 12-6, 63-04, 342-021, 8462-':321 
 62-48132, and 412-16321? 
 
 3. What is the sum of $27046, $14-4041, $7-86321, 
 $324-8631, and $412382631 ? 
 
ART. 148.J PRACTICAL QUESTIONS. 129 
 
 4. What is the sum of 8-321, 9-6231, 84-n63,116-84324, 
 and 18312.83201 ? 
 
 5. What is the sum of 82-631, l-t632, 8413-0001, 
 t32-46n, 842-n, and 9802 ? 
 
 PRACTICAL QUESTIONS. 
 
 1. Sold a box of candles for $12*25 ; a barrel of flour 
 for $t-t5 ; a sack of coffee for $12121 j and a barrel of 
 sugar for $18-3t| ; required the amount I should receive, 
 
 2. What cost a horse, a yoke of oxen, a cow, and a 
 sheep, if the horse cost $141'62i; the oxen, $184-06^; 
 the cow, $46-82; and the sheep, $7-06i? 
 
 3. A merchant bought broadcloth to the amount of 
 $8t2-45; muslin and linen to the amount of $184*75; 
 sugar to the amount of $296*85 ; and flour .to the 
 amount of $38t-80. What was the whole cost ? 
 
 4. A gentleman has some young cattle worth $6t82-8t; 
 a horse worth $285-60 ; a yoke of oxen worth $196-87; 
 four cows worth $210-20; and a farm worth $8642-84. 
 What is the value of the whole ? 
 
 5. A merchant bought a quantity of goods for 
 $89785;95 ; paid for duties $897*40; and for transporta- 
 tion $38887. For how much must he sell the goods to 
 gain $346-82 ? 
 
 6. Bought a ton of hay, for $14-87| ; a cord of wood, 
 for $6-12i; a barrel of apples, for $3-06^; a barrel of 
 flour, for $8-371; and a quarter of beef, for $9*87i. Re- 
 quired the sum to be paid ? 
 
 7. A farmer's bill at the store was as follows : 4i yards 
 of cloth, $24-121; 3 pair of boots, $16-87^; a dozen 
 skeins of silk, $*87i; and 15 yards of muslin, $l*12i. 
 Required the amount of the bill ? 
 
 8. A farmer sold produce as follows: wheat, for 
 $325-871 ; corn, for $137*621; rye, for $237*85; oats, 
 for $96061; hay, for $62-62i. Required the amount of 
 the sale ? 
 
 9. What should be paid for a barrel of sugar, worth 
 $18-471;. a quarter of beef, worth $9*871; a barrel of 
 flour, worth $6*37^; a box of raisins, worth $8*20; a fir 
 
13U I>ECIMAL FRACTIONS. [cHAP. VI. 
 
 kin of butter, worth $15'9t|; and a barrel of molasses, 
 worth $12-25? 
 
 10. Bought a quantity of sugar, for $183'92 ; a quan- 
 tity of flour, for $227-621; a quantity of hams, for 
 $384'18f; a quantity of co*rn, for $38()-8Ti. For how 
 much must it all be sold so as to gain $465-85, after pay- 
 ing $120-37^ for cartage and storage ? 
 
 Subtraction of Decimals and the United States 
 Currency. 
 
 1. From 64-5 subtract 37-8046. 
 
 OPERATION. 
 
 Min. 64-5000 
 Sub. 37-8046 
 
 Rem. 26-6954 
 
 Rkmark.— In examples of this kind we annex ciphers to the minuend, which 
 does not eftect its value. (See last para^^raph, Art. 134.) 
 
 Care must be taken to place the numbers so that the decimal points shall 
 stand one under another, in order that units may be taken from units, &c.j 
 tenths from tenths, &c. 
 
 2. From 204*614 subtract 9-131. 
 
 3. From 6 subtract 4-00006. 
 
 4. From 4-4 subtract 3-00004. 
 
 5. From 1 subtract -000001. 
 
 6. From 16802-4682 subtract 981-8364. 
 
 7^ Subtract 10014-40001 from 80084-600861. 
 
 practical questions. 
 
 1, A man bought a span of horses for $465"85, and a 
 yoke of oxen for $195-38 ; how much more did he pay 
 for the horses than for the oxen ? 
 
 2. A gentleman having $18654-84, gave $2685-69 of it 
 for a store ; how much money has he remaining ? 
 
 3 A man is owing $6785-95, and has due him $9986-125; 
 how much more is due him than what he owes ? 
 
 4. A quantity of lumber was bought for $5682-18|, and 
 sold for $7631-561; how much was the gain .- 
 
ART. 149. J MULTIPLICATION OF DECIMALS. 131 
 
 6. A quantty of flour was purchased for $3896-12^, 
 and sold for $oJ:'9-18f ; how much was the loss ? 
 
 6. A grazier uonght cattle for $384'95, and sheep 
 for $135-68. He sold the cattle for $419-12^, and the 
 sheep for $109*72^; how much did he gain by these trans- 
 actions ? 
 
 7. A manufacturer purchased a quantity of cotton for 
 $387 95, which he made into cloth, at an expense of 
 $184*06| ; how much will he make by selling the cloth 
 for $600 ? 
 
 8. A speculator purchased wheat for $587 -871, and 
 pork for $968' 12|. He sold his wheat for $73918^, and 
 his pork for $78437^. Did he gain or lose by the opera- 
 tion, and how much ? 
 
 9. A speculator bought at one time 347 '35 acres of 
 land; at another, 637*25 acres; and at another, 1435'7 
 acres. He is desirous of making his purchases amount to 
 1 225*5 acre's. How much land does he still want ? 
 
 Multiplication of Decimals and the United States 
 Currency. 
 
 Art. 149. One-tenth taken. two times, or multiplied 
 by 2, gives for a product y^^; if taken once, or multiplied 
 by 1, the product will be -,V; if taken one-tenth of a time, 
 or multiplied by j\ of 1, the product must be y^ of yV 
 
 — _X_. thn« 1 V Jl— — _J • _i_ V — '_ — 1 • 1 V _' 
 
 10 0) HJUO, 10-^10 100) 100^^10 1000)1000'^lir 
 
 = Toooo» &c-> which decimally expressed becomes 'IX'l 
 = 01; -OlX'l^-OOl; "001 X •bl = -00001,&c. From which 
 we observe that the number of decimal places in the pro- 
 duct is equal to the number of ciphers (which in practice 
 is understood) in the denominators of both factors, which 
 /s always equal to the number of decimal places in the 
 two factors. Hence to multiply one decimal by another 
 we proceed as in whole numbers, and/rom the right of t/ie 
 'product, point off as many places for dccim.als as there are 
 decimal places in both multiplier and multiplicand. Should 
 there not be places enough in the product, prefix ciphers. 
 
132 
 
 DECIMAL FRACTIONS. 
 
 [chap. VI. 
 
 1. Multiply 4-86 by t-39. 
 
 2. Multiply 14-683 by 10-83. 
 
 3. Multiply 122- by 46-7832. 
 
 4. What is the product of 202*002 and 1 0002 ? 
 
 5. What is the product of 165-3701 and 47-8201 ? 
 
 6. What is the product of 3786-478 and 831-0241 ? 
 
 7. What is the product of 8602-8312 and 48-76324 ? 
 
 Art. 150. A decimal is multijplied hy 10, 100, 1000, 
 SfC.y by merely reinoving the decimal point as many places to 
 the right as there are ciphers in the multiplier. If necessary 
 auTiex ciphers to the number. 
 C 86-723 ) 
 Multiply I 14-243 } by 10. 
 r 1001-001 S 
 
 Multiply 
 
 Multiply 
 
 Multiply 
 
 : 8-076 
 
 41-3421 
 ' 716-311' 
 
 , 1-832 ^ 
 
 30-12 ' 
 
 ' 8-63412 ' 
 
 148-63 
 7-34876 
 28-31017 
 186-4 
 
 2-7 
 
 by 100. 
 
 by 1000. 
 
 1^ by 10000. 
 
 PRACTICAL QUESTIONS. 
 
 1. What cost 95 tons of hay, at $12-75 a ton ? 
 
 2. What cost 125 yards of broadcloth, at $5 -37 J a 
 yard ? 
 
 3. What cost 275 bushels of potatoes, at $'62^ a 
 bushel ? 
 
 4. What cost 384 barrels of sugar, at- |17-87^ a bar- 
 rel ? 
 
 5. What cost 312 pounds of butter, at $18^ a pound ? 
 
 6. What cost 245 barrels of molasses, at $23-18| a 
 barrel ? 
 
ART. 151.] DIVISION, OF DECIMALS. 138 
 
 1. If 25 men earn $3*T'18f in one day, how much 
 can they earn in a year, of 365 days ? (not counting Sun- 
 days. 
 
 8. A gentleman purchased a farm containing 445-5 
 acres, at $34 "12^ an acre ; how much did the farm cost 
 him ? 
 
 9. How much should be paid for 25*5 cwt. of tobacco, 
 at $12-37i a hundred weight ? 
 
 10. Bought 275 sheep, at $1-87^ a head, and sold them, 
 at $2-12^ a head; how much did I gain by the opera- 
 tion ? 
 
 Division of Decimals and the TJnited States Currency. 
 
 Art. 151. The quotient arising from dividing any num' 
 her by another of the same denominxition, is a whole number. 
 Thus, if units be divided by units, tenths by tenths, hun- 
 dredths by hundredths, or thousandths by thousandths, 
 &c., the quotient will be a whole number. Therefore, in 
 the division of decimals, when the divisor and dividend 
 each contain the same number of decimal places, the quo- 
 tient will be a whole number; and if the dividend contain 
 more decimal places than the divisor, there must of neces- 
 sity be as many decimal places in the quotient as the 
 number of decimal places in the dividend exceed the num- 
 ber of decimal places in the divisor. 
 
 We deduce the same conclusion from the following con- 
 siderations. 
 
 In the multiplication of decimals, the number of decimal 
 places in the product equals the number of decimal places 
 in both factors. In the division of decimals, the divisor' 
 and quotient are multiplied together to produce the divi- 
 dend ; therefore, there must be as many decimal places in 
 the quotient as those in the dividend exceed those in the 
 divisor. 
 
 The pupil should bear in mind that he can affix ciphers 
 to the dividend without changing its value ; and when 
 necessary, he should prefix ciphers to the quotient. 
 
 1. Divide .0016016 by 1.12. 
 
134 
 
 DECIMAL FRACTIONS. 
 
 [chap. 
 
 TI 
 
 OPERATION. Explanation. — As the number 
 
 1-12)'0016016(00143 Ans. o^ Peaces in the quotient was not 
 
 12^2 equal to the number of decimal 
 
 places in the dividend minus the 
 
 481 number of decimal places in the 
 
 448 divisor, so the two ciphers were 
 
 — — prefixed that the required number 
 
 ^•^" of decimal places could be cut off. 
 
 336 ^ 
 
 2. Divide -00144 by 1-2. 
 
 3. Divide 'OOOOOtS by "005. 
 
 4. Divide 86-4 by '24. 
 
 5. Divide 59-74514 by 13-6. 
 
 6. Divide -001728 by 4-8. 
 
 1. Divide 2549052 by 24-6. 
 
 8. Divide 2448 by '012 
 
 Art. 152. A decimal may be divided by 10, 100, 1000, 
 &c., by removing the decimal poiat as many places to the 
 left as there are ciphers ia the divisor. If necessary, pre- 
 fix ciphers to the dividend. 
 r4-36 
 Divide \ ^^I'^ll J> by 10. 
 431-2 
 
 Divide 
 
 Divide 
 
 146-34 
 3 24 
 36 741 
 14683 
 47632-1 
 1478*3 
 231-46 
 76-041 
 31046-1 
 
 by 100. 
 
 > by 1000, 
 
 PRACTICAL QUESTIONS. 
 
 1. If 128 barrels of flour be worth $784, what is the 
 value of I barrel ? 
 
ART. 152.] PRACTICAL QUESTIONS. 135 
 
 2. If 54 acres of land cost $816'75, how much is that 
 an acre ? 
 
 3. What cost 1 yard of broadcloth, if 46,. yards cost 
 $263-12? 
 
 4. What cost 1 horse, if 34 horses cost $4662-1 ? 
 
 5. What cost 1 bushel of apples, if 70 bushels cost 
 $43-75? 
 
 6. If 137 bushels of onions cost $154-12^, what will 
 1 bushel cost ? 
 
 7. If 75 quarts of strawberries cost $4*6875, what will 
 1 quart cost ? 
 
 8. If 275 bushels of corn cost $171*87^, how much will 
 1 bushel cost ? 
 
 ->• 
 
 PRACTICAL QUESTIONS IN DECIMALS AND THE UNITED STATES 
 CURRENCY. 
 
 1. What cost 8640 brick, at $4-25 a 1000 ? 
 Solution If 1000 brick cost $4-25, 1 brick will cost one 
 
 thousandth of $425, which is $-00425, and 8640 brick will cost 
 8640 times $00425, which is $36-72. 
 
 2. What will be the cost of 4832 feet of boards, at 
 $6-50 a 1000? 
 
 3. What will be the cost of 28460 feet of lumber, at 
 $2-18f a hundred ? 
 
 4. What cost 17640 feet of timber, at $9-45 a 1000 ? 
 
 5. What cost 586 feet of pine boards, at $25-12^ a 
 1000? 
 
 6. What must be paid for planing 46324 feet of boards, 
 at $1-45 a 1000? 
 
 7. What is the value of 14672 feet of hemlock boards, 
 at$6-37ia 1000? 
 
 8. A speculator bought 500 acres of land for $98 7f; 
 and 250 acres for $647f . He sold 478^ acres for $1245^. 
 How much land has he remaining, and for what must 
 he sell it per acre, so as to neither gain nor lose by the 
 operation ? 
 
 9. Having deposited in a bank $186050; I drew out 
 at one time $84*87^; at another, $47-12^; at another, 
 
136 DECIMAL FRACTIONS. [cHAP. VI 
 
 $485-18f ; and at another, $14t-31i. How much have 
 I remaining ia the bank ? 
 
 10. A gentleman, who was on a journey of 24 1^ miles, 
 traveled 4 days, at the rate of 42| miles a day; what 
 distance still remains to be traveled ? 
 
 11. Bought a house and lot for $324050, and paid for 
 improvements on the same $685"8T^. I then sold the 
 property for $4985-621. How much did I gain by the 
 transaction ? 
 
 12. A land dealer has in one farm 195* 7 5 acres; in 
 another 465f acres; in another 483f acres. He sold t5^ 
 acres from each. How many acres has he left ? 
 
 13. A drover bought cattle, for $n5'84 ; mules for 
 $286-95 ; horses, for $384-87| ; and sold them all for 
 $1847 -12i. How much did he gain by the speculation ? 
 
 14. A merchant bought cloth for $246-84 ; silks for 
 $-387-8U; and sugar for $865-18f. He sold the cloth at 
 a profit of $98-75; the silks, at a loss of $104-121; and 
 the sugar, at a profit of $146'18f. Did he gain or lose^ 
 and how much ? 
 
 15. A merchant bought 47-5 yards of cloth, at $4*75 a 
 yd.; and sold it, at $6'12i a yard. How much did he gain ! 
 
 16. Bought 285 sheep, at $2-12i each; and sold them 
 for 25 young cattle. For what must I sell the cattle 
 a head so as to make $75 by the operation ? 
 
 17. How much money must be paid for 4-5 cwt. of ham, 
 at $14-25 a cwt.; 14 barrels of flour, at $5-37i a barrel; 
 8 barrels of fish, at $9 '621 a barrel; and 5f barrels of 
 sugar, at $19'30 a barrel ? 
 
 18. What sum of money should be paid for 75-75 pounds 
 of sugar, at $-1125 a pound; 14 lbs. of tea, at $r37i a 
 pound; 15 lbs. of chocolate, at $-125 a pound; -and 5-75 
 gallons of molasses, at $-37| a gallon ? 
 
 19. A speculator bought 147i acres of land, at $27-121 
 an acre; and 232f acres, at$35f an acre. He sold the 
 first tract, at $32181 an acre; and the second, at $28-371 
 an acre. Did he gain or lose by the operation, and how 
 much ? 
 
 20. Bought 4 pieces of cloth, each contaming 47| yards, 
 
ART. 162.] PRACTICAL QUESTIONS. 13f 
 
 for $863-25; of which 25f yards have been sold, at $6-87^ 
 a yard. What will be the gain or loss on the whole, if 
 the remainder be sold, at $5*95 a yard ? 
 
 21. A drover bought 247 cattle, at $25-87^ each. He 
 sold 84 of them, at |32t5 each; 45 of them, at $2245 
 each; and the remainder, at $28*12^ each. How much 
 did he gain by the speculation ? 
 
 22. A merchant barters to a farmer, 18*75 yards of 
 broadcloth, at $7*12^ a yard; 47^ yards of muslin, at 
 $09| a yard; 6 pair of boots, at $4-37i a pair;— for 47 
 bushels of corn, at $-57 a bushel; 65f bushels of wheat, 
 at $1-121 a bushel. The difference in the value of the 
 articles exchanged, is to be paid in money. Which of 
 them must receive money, and how much ? 
 
 23. A merchant bought 1246 bushels of wheat, at 
 $137i a bushel; of which he sold to one man 463 bush- 
 els, at $1-45 a bushel; to another 384^ bushels, at $1*87^ 
 a bushel. At what price per bushel must the remainder 
 be sold so as to gain on the whole, at the rate of $56 on 
 a 1000" bushels ? 
 
 24. A person, having $46*87^ was desirous of purchas- 
 ing an equal number of pounds of tea, coffee, and sugar; 
 the tea, at $1'12J- a pound; the coffee, $'62i; and the 
 sugar, $"12^ a pound. How many pounds of each could 
 he buy ? 
 
 25. Find the amount of a store-bill for 15f yards of 
 cloth, at $3-371 a yard; 20^ yards of silk, at $l-18f a 
 yard; and 15 skeins of thread, at $-06i a skein. 
 
 26. Bought 16 barrels of sugar for $425-25, and sold 
 the same at a profit of $1'87| a barrel. At what price 
 per barrel was it sold, and what was the entire profit ? 
 
 27. A merchant bought 35 pieces of broadcloth, each 
 containing 18f yards, at $6-18f a yard; and sold it so as to 
 clear, after deducting $4-37i for his trouble, $89'87i. 
 At what price per yard was the cloth sold ? 
 
 28. What is the value of sugar a cwt. when '75 cwt. 
 cost $6-375; and what should be paid for 16f cwt. of 
 sugar, at the same rate ? 
 
 29. A merchant bought of one farmer 225| bushels of 
 
138 DECIMAL FRACTIONS. [cHAP. VI. 
 
 wheat, and of another 106|- bushels, at $l'18f a bushel. 
 He made 195 bushels of- it into flour; and sold the flour, 
 at a profit of $125*87|. Will he gain or lose, if he sells 
 the remainder of the wheat, at $*93f a bushel. 
 
 30. A drover bought 146 cows, at $2T-18f a head; 
 and 166 sheep, at $1*8T|- each. He sold 83 of the cows, 
 at $28-12^- a head; and" all of the sheep, at ll'Sl^ each. 
 At what rate per head must he sell the remainder of his 
 cows so as to make a profit of $125"93^ on the whole ? 
 
 Art. 153. Reduction op Denominate Numbers to 
 Decimals. 
 
 1. Reduce 15^. 9d. 3 far. to the decimal of a pound. 
 
 Explanation. — We annex two ci- 
 phers to the 3 far., which reduces it 
 to hundredths. 4 far. make 1 penny ; 
 therefore, ^ of the number of far- 
 
 things will equal the number of 
 
 15'8125 s. pence, which is 'lod. This being 
 
 annexed to the 9d. = 9'15d. VVe 
 
 •790625 of a pound, next divide this by 12, to reduce it 
 to the decimal of a shilling, and 
 obtain -81255. ; which, being annexed to the 156\ gives 15 8125s. 
 We now divide this by 20, to reduce it to the decimal of a 
 pound, and obtain •790625 of a pound for the answer. 
 
 2. Reduce ISs. 9d. 2 far. to the decimal of a pound, 
 sterling. 
 
 3. Reduce 1 ft. 8 inches, to the decimal of a yard. 
 
 4. Reduce 1002-15 pwt. 9 grs., to the decimal of a 
 poiind Troy. 
 
 5. Reduce 15 cwt. 3 grs. 15*45 lbs., to the decimal of 
 a ton. 
 
 6. Reduce 5 fur. 25 rds., to the decimal of a mile. 
 T. Reduce 2 R. 25*5 P., to the decimal of an acre. 
 
 8. Reduce 6 fur. 15 rds. 3 yds. 2 ft. 10 in., to the deci- 
 mal of a mile. 
 
 9. Reduce iE4 155. lOd. 1 farthings, to the decimal of a 
 pound. 
 
ART. 154.] REDUCTION OF DENOMINATE DECIMALS. 139 
 
 10. Reduce 6 T. 12 cwt. 2 qrs. 14 lbs. 10 oz. 8 dr. to 
 the decimal of a ton. 
 
 Art. 154. Reduction of Denominate Decimals, to 
 
 Whole Numbers of a lower denominations. 
 
 1. Reduce '735 of a pound, to shillings, pence and far- 
 things. 
 
 OPERATION. Explanation. — I wish to reduce '735 of a 
 
 £ '735 pound to shillings. There are 205. in £1 ; 
 
 20 therefore, 20 times the number of pounds = 
 
 the number of shillings, 20 X -735 = 14- 75. 
 
 14-700 s. In -75., how many pence ? There are 12c/. in 
 
 12 Is. ; therefore, 12 times the number of shil- 
 lings equal the number of pence. 12 X "7 
 
 8-400 d. _ 8-4^ In .4^?. how many farthings 1 There 
 
 ^ are 4 farthings in 1 penny ; therefore, 4 time^ 
 
 T~af\r\ f -. thenumber of pence equal the number of far- 
 
 i-DUU lar. ^^.^^^ ^ ^ .^^ ^ -^.g ^^^ Therefore, XO-735 
 
 -= 145. Sd. 1-6 far. 
 
 2. What is the value of "389 of a pound sterling ? 
 
 3. What is the value of '635 of a yard ? 
 
 4. What is the value of -451 of an ell French ? 
 
 5. What is the value of '832 of an ell English ? 
 
 6. What is the value of '^Sf of a mile ? 
 1. What is the value of '895 of an acre ? 
 
 8. What is the value of 975625 of a pound Troy ? 
 
 9. What is the value of '875 of a score ? 
 
 10. What is the value of -95625 of a ream of paper ? 
 
 11. What is the value of '854 of a firkin of butter ? 
 
 12. What is the value of -7575 of a great gross ? 
 
 13. What is the value of -123 of a pound sterling ? 
 
 14. What is the value of 142857 of a bushel of salt ? 
 15 What is the value of "783 of a bushel of wheat? 
 
 16. What is the value of -857142857 of a bushel of 
 corn or rye ? 
 
 17. What is the value of -083 of a pound sterling? 
 
 18. What is the value of "16 of a cwt. ? 
 
140 DECIMAL FRACTIONS. [cHAP. n. 
 
 19. What is the value of -123 of a mile ? 
 
 20. What is the value of '463 of a ton ? 
 
 PRACTICAL QUESTIONS. 
 
 1. What is the value of 3 cwt. 2 qrs. 15 lbs. of sugar 
 at $5-^5 a cwt. ? 
 
 2. What is the value of 15 gallons, 3 qt. 1 pt. of molas 
 ses, at $'87| a gallon ? 
 
 3. What is the value of 16 bushels, 2 pks. t qts. of rye 
 at $1-3H a bushel? 
 
 4. What is the value of 84 yds. 3 qrs, 3 nas. of broad- 
 cloth, at $5-8ti a yard ? 
 
 5. What is the value of 16 cwt. 2 qrs. 14'5 lbs. of 
 pork, at $14-93f a cwt. ? 
 
 6. What is the value of 84 T. 14 cwt. 2 qrs. 15 lbs. of 
 hay, at $14-18f a ton ? 
 
 t. What is the value of 34 lbs. 8^ oz. of butter, at 
 $'18f a pound ? 
 
 8. What will it cost to construct 14 miles, 5 fur. 25 rds. 
 of plank road, at $1437-621 per mile ? 
 
 9. A farmer sold 34 bush. 3 pks. *I qts. of clover-seed, 
 at $6'84-i- a bushel, and in payment received 40 bushels 
 2 pks. 1 pt. of grass-seed, at$3'8t^a bushel. How much 
 remains due ? 
 
 10. A tailor paid $1468-75 for 385 yds. 3 qrs. 3 nas. of 
 cloth; I of which he sold, at $4*37^ a yard; and the 
 remainder, at $5-93f a yard. How much did he gain by 
 the bargain ? 
 
 11. If f of a ton of hay cost $8-87i, what will 4 T. 
 15 cwt. 3 qrs. cost ? 
 
 12. A merchant bought 125 hhds. 30-5 gals. 3 qts, of 
 molasses for $1585-12^; and sold | of it for $21-75 a 
 hogshead; and the remainder, at $28-93| a hogshead. 
 How much did he gain by the operation ? 
 
 Reduction of Denominate Fractions. 
 Art. 155. A Denominate fraction is a fraction of any 
 denominate number; as | of a yard, | of a mile, &c. 
 
ART. 156.] REDUCTION OF FRACTIONS. 141 
 
 Reduction of denominate fractions is changing them from 
 one denomination to another without altering their value. 
 
 1. Reduce jf g of a gallon to the fraction of a gill. 
 
 OPERATION BY CANCELLATION, 
 gal. 
 
 :r:r-x V - V - V - = — of a gill. 
 $00 ^ 1 ^ 1 '^ 1 28 ^ 
 
 xx% 
 
 28 
 
 Explanation There are 4 quarts in 1 gallon ; therefore, 
 
 4 times the number of gallons equal the number of quarts. 
 ^I? X 4 = 2 If of a quart ; (which, for convenience, may be read 
 in the form of a compound fraction. There are 2 pints in 1 
 quart ; therefore, twice the number of quarts equal the number 
 of pints. ^lB^XtXf = TT2ofa pint. There are 4 gills in 1 
 pint ; therefore, 4 times the number of pints equal the number 
 of gills. ^%^ X t X f X f , equals the number of gills, which, 
 when cancelled, becomes ^^ of a gill., 
 
 2. Reduce -^\-^ of a pound to the fraction of a farthing. 
 
 3. What part of a grain is gsio o o^ ^ pound Troy ? 
 
 4. What part of a pint is -^^-^-^ of a bushel ? 
 6. What part of a pound is y/o^ of a ton ? 
 
 6. What part of a second is y 03F80 of a day ? 
 
 t. What part of a foot is y/j o^ ^ furlong ? 
 
 8. What part of a dram is 2 04 jo of a hundredweight ? 
 
 Art. 156. Reduction of Fractions of a Lower, to 
 THOSE of a Higher Denomination. 
 
 1. Reduce -f of a farthing to the fraction of a pound. 
 operation by cancellation. 
 
 far. 
 111 1 r -, 
 
 iy_xy — := of a Donnd. 
 
 ^ X 4 A ^^ X 20 1120 ^ 
 
 2 
 
 Explanation. — f of a farthing is what part of a penny ? 
 4 farthings make 1 penny \ therefore, | of the number of far- 
 things equals the number of pense. By a similar method of 
 
142 DECIMAL FRACTIONS. [CHAP. VI 
 
 reasoning we find yV of the number of pence equal the num- 
 ber of shillings; and ^'o «f the number of shillings equal the 
 number of pounds. 
 
 2. What part of a pound Troy, is f of a grain ? 
 
 3. What part of an acre is 1| feet ? 
 
 4. What part of 10 days is | of a minute *? 
 
 5. What part of 20 bushels is | of f of a gill? 
 
 6. What part of a rod is ^^ of 2^ inches ? 
 
 7. What part of 8 miles is f of a rod ] 
 
 8. What part of a yard is J of | of f of an ell French 1 
 
 Art. l9>7« Reduction of Simple or Denominate Num- 
 bers, TO the Fractional Part of another Simple ob 
 Denominate Number. 
 
 1. What part of £1 is 10s. Qd. 1 far. 1 
 
 operation. * 
 
 105. 6d. 1 far. = 505 far. 101 ^ 
 
 = — part. 
 
 =9(50 far. 192 
 
 Solution. — 4 farthings make 1 penny; therefore 1 far- 
 thing is I of a penny. 6|c^. = Y^- 12t/- make 1 shilling, 
 therefore ^^ of the number of pence equals the number of 
 shillings. j\X^i=Us. 10^s. = ^^^s. 20^. make £1 ; 
 therefore -^^^ of the number of shillings equals the number 
 
 of £. 2*5 X 4^ =i§2'^" 
 
 2. What part of 3 yds. is 4 E. Fr. 2 qrs. ? 
 
 3. What part of 3 cwt. 3 qrs. is 2cwt. 3 qrs. 15 lbs. 1 
 
 4. W^hat part of 3 A. 3 R. 32^1- P. is 2 A. 2 R. 30 P. ? 
 
 5. What part of 3 feet square is 3 square feet ? 
 
 Art. 158. To find tee value of a Denominate Frao 
 WON, in Whole Numbers, of a Lower Denomination. 
 
 1. What is the value of 4 of a pound sterling ? 
 
 OPEKATION 
 
 Ans. 
 
 £ 
 
 7)5 
 
 20 12 4 
 s. d. qr. 
 
 
 
 
 
 14 3 14 
 
ART. 158.] ADDITION OF DENOMINATE FRACTIONS. 148 
 
 Explanation. — T wish to find f of £1, but ^ of £1 is the 
 same as I- of £5 ; heuce, I find \ of £5. 
 
 2. What is the value of | of a shilling ? 
 
 3. What is the value of f of a cwt. ? 
 
 4. What is the value of f of a yard ? 
 6. What is the value of ff of a day ? 
 
 6. What is the value of | of a mile ? 
 
 7. What is the value of |i of a hogshead of wine ? 
 
 8. What is the value of | of a year ? 
 
 9. What is the value of -5 of an ell French ? 
 10. What is the value of j\ of a ton ? 
 
 Addition of Denominate Fractions. 
 
 Art. 159. We have learned that whole numbers of 
 different denominations connot be added; the same is true 
 of fractions of different denominations. Hence, we first 
 find the value of the given fractions by Art. 158; then add 
 them together. 
 
 1. Add |i of a pound to -f of a shilling. 
 
 OPERATION. 
 
 1^ of a pound =145. Sd. 
 f of a shilling. = lOd. l\ far. 
 
 Ans. 15s. 6fZ. 1\ far. 
 
 2. Add I of a pound to j\ of a shilling. 
 
 3. Add /o of a cwt. to | of a quarter. 
 
 4. Add 4 of a ton to \-^ of a cwt. 
 
 5. Add f of a mile to | of a furlong. 
 
 6. Add I of an acre to f of a, rood. 
 
 *7. Add f of a hogshead to | of a gallon. 
 
 8. Add together -f of a bush., | of a peck, and | of a 
 quarter. 
 
 9. Add together | of a ton, f of a cwt., and 4 of a qr, 
 10. Add together | of a month, f of a week, and -f of 
 
 a day. 
 
144 DECIMAL FRACTIONS. [cHAP. VI. 
 
 Art. 160. Subtraction of Denominate Fractions. 
 
 1. From |- of a mile subtract | of a furlong. 
 
 OPERATION. 
 
 40 6i 3 12 
 fur. rds. yds. ft. in. 
 
 ^ of a mile = 6 8 4 2 8 
 i| of a fur. = 28 3 5| 
 
 Ans. 5 20 1 2 2f 
 
 2. From f of a bushel take |f of a peck. 
 
 3. From -f of a week take | of a day. 
 
 4. From 4 of 25 yards take f of 6 E. French. 
 6. From -fi of 23 tons take 4 of 18 cwt. 
 
 6. A company agree to construct 25 miles, 8 fur. 18 rds. 
 of road, but after constructing 6 mi, 2 fur. 23 rds. and 2 ft. 
 more than | of it, they relinquish the job. How much 
 remains to be constructed ? 
 
 1. A merchant bought f of 15 hhd. 42 gals, of molasses, 
 and sold j of 2 hhd. 53 gals, of it. How much remained 
 unsold ? 
 
 8. A merchant bought 14 cwt. 3 qrs. 18 lbs. of sugar, 
 and sold | of it, lacking 4 cwt. 1 qr. 15 lbs.; how much 
 remains unsold ? 
 
 practical questions. 
 
 1. What is the value of 4 of 15 yards of cloth, at $^'62^ 
 a yard ? 
 
 2. What is the value of f of 3 bushels, 3 pks. *7 qts. of 
 gooseberries, at $06^ a quart ? 
 
 3. What cost f of 41 cords, 110 feet of wood, at $5-81^ 
 a cord ? 
 
 4. What cost 1 pound of tea, if 11^ pounds cost 
 $13-826? 
 
 5. What will 6 cwt. 3 qrs. 20 lbs. of honey cost, at 
 $18'8tiacwt. ? 
 
 6. What will 14 bushels, 2 pks. 1 qts. 1 pt. of grass- 
 Beed cost, at $6-621 a bushel ? 
 
 1. If it require 4 hours 20 minutes for a man to cut 
 
A.RT. 160.] PRACTICAL FRACTIONS. 145 
 
 I cord of wood, how many days of 8 hours and 40 minutes 
 «ach, will be required to cut 84t cords 84 feet ? ** 
 
 8. Four persons share 625 pounds of sugar as follows : 
 the first takes i of | of the whole; the second takes | of 
 3. of the remainder; the third takes f of |f of what now 
 remains; and the fourth takes what is left. How much 
 did each receive ? 
 
 9. A received | of a certain quantity of molasses; B |; 
 C I of the remainder; and D what then remained. It 
 now appears that C has 64 gals, more than A and B 
 together. How much did each receive ? 
 
 10. A farmer, owning 864 A. 3 R. 39 P. of land, divided 
 I of it equally among 4 of his sons. How much did each 
 son receive, and how many acres had the father remain- 
 ing ? 
 
 11. Bought 184 gals. 3 qts. of molasses, at $'3*r| a 
 gallon, and used 2t gals, 2 qts. of it; how must I sell the 
 remainder per gallon so as to receive $3-84^ more than 
 the whole cost ? 
 
 12. A person gave | of all his money for a horse; -i- of 
 the remainder for a colt; and | of what then remained for 
 a cow. He then had remaining $8'8t|. What was the 
 cost of each, and how much money had he at first ? 
 
 13. A merchant gave for some raisins } of all his money; 
 for some cinnamon ^ of all his money; for some sugar | of 
 what remained; for some flour | of what then remained; 
 and what still remained he gave for some butter. What 
 did each article cost him, providing the sugar cost $136*18f 
 more than the flour ? 
 
 14. A certain sum of money is to be divided among 4 
 persons; the first is to have ^ of it; the second i of it; 
 the third | of what remains; and the fourth the remainder. 
 What was the sum to be divided, and how much did each 
 receive, providing the third received $14*I'93f less than 
 the first and second together? 
 
 15. How much butter at,$-18f a pound, must be given 
 for 25 gals. 3 qts. 1 pt. of molasses, at $-3t^ a gallon ? 
 
 16. From a piece of cloth containing 147 yds. 4 E. 
 French, three suits of clothes, each requiring 6 E. English, 
 
146 DUODECIMALS. [CHAP. VI. 
 
 were taken. How much would the remainder come to, at 
 $5-18f a yard ? 
 
 It. How many inches in f of an E. E.; f of an E. Er.; 
 and f of a quarter ? 
 J 18. A merchant lost from a hogshead of molasses } of 
 it, + A of a gallon and | of a quart. How much of the 
 hogshead, expressed deciifially, leaked out, and how much 
 remained in ? 
 
 19. Bought 15 tons 14 cwt. 3 qrs. 24 lbs. of iron, and 
 sold 10 tons 5 cwt. 1 qr. 15 lbs. of it. What is the value 
 of ^ of what remains, at $'06| a pound ? 
 
 20. Bought a quantity of grain for $358"84 ; and sold 
 \^ of it to one man; f of the remainder to another man; 
 and used | of the remainder myself. What is the value 
 of the remainder ? 
 
 21. A, B, C, and D worked together on this condition: 
 A was to receive $60*06 of it, and y^ of the remainder; 
 B was to receive $70'0t and j\ of the remainder; C was 
 to receive $80'08 and j\ of the remainder; and I) took 
 what then remained. By this division each man received 
 the same sum. How much did their wages amount to ? 
 
 DUODECIMALS. 
 
 Abt. 161. Duodecimals are a kind of denominate 
 numbers, the denominations of which increase uniformly 
 in a twelve-fold ratio. Its denominations are the foot (ft.), 
 which is the unit ; the inch, or prime ('), J^ of the foot, 
 the secoTid ("), y^2 of the prime ; the third ('"), ^\ of the 
 second ; and so on, indefinitely. The accents that distin- 
 guish the denominations below feet, are called Indices. 
 
 Duodecimals are applied to the measurement of surfaces 
 and solids. 
 
 TABLE. 
 
 12 Fourths ("") make 1 X^ird, marked '" 
 
 12 Thirds '* 1 Second, 
 
 12 Seconds " 1 Prime, or Inch, " 
 
 12 Primes, or Inches '* 1 Foot, " ft 
 
art. 163.] multiplication of duodecimals. 14 1 
 
 Addition axd Subtraction of Duodecimals. 
 
 Art. 162. Duodecimals are added and subtracted the 
 same as other Denominate numbers. 
 
 1. Add together 6 ft. 4' 5" 8'", 8 ft. 4' 8" 9'", 1, ft. 
 3' 8" 9'", and 12 ft. 9' 11" 10'". 
 
 2. What is the sum of 1^ ft. 8' 9" 11"', 14 ft. 6' t", 
 8 ft. 9' 11" 4"', and 16 ft. 9' 10" 11"' ? 
 
 3. What is the sum of 20 ft. 9' 11" 6'" 1"", 14 ft. 8' 
 9" 10'", 12 ft. 9' 8" 10'" 8"", 8 ft. 11"", and 6 ft. 9' ? 
 
 4. From 84 ft. 8' 9" 11'" 3'"', subtract 66 ft. 11' 8' 
 4'" 9"". 
 
 5. What is the sum, and what is the diflference of 84 
 ft. 3' 8" 9'" 2"", and 48 ft. 9' T IV" 10"". 
 
 6. What ifi the sum, and what is the diflference of 13t 
 ft. 3' 9" 4'" 6"" and 98 ft. 9' 10" 11"' t"". 
 
 Multiplication of Duodecimals. 
 
 In Duodecimals, the foot, when used to express surfaces, 
 contains 144 sq. in., and when used to express solids, 
 1728 cu. in. Consequently, in the measurement of sur- 
 faces, 5' would equal ^^ of a square foot, instead of a linear 
 foot ; that is, j^2Xl44 sq. in. =60 sq. in. In the measure- 
 ment of solids, 5' would eq^ual ^^g of 1728 cu. in. (a cubic 
 foot) =720 cu. in. 
 
 From the .preceding remark we infer that a strip of sur- 
 face 1 inch wide and 12 inches long, makes 1' square 
 measure; and that a slab 1 inch thick, 12 inches long, ai«i 
 12 inches wide, makes 1' solid measure. 
 
 1. What is the product of 8 ft. 5' by 9 ft. r ? 
 
 OPERATION. Explanation. — 5 =: -^\, and 7' = -^-^ of a 
 
 8 ft. 5' foot- Therefore, we say^ 7' X 5' = f^\ of 
 
 9 ft.* 7' ^ foot, which is 35" = 2' 11"; we write 
 down the 11" and carry the 2' to the next 
 
 4 ft. 10' 11" product. 7' X 8 ft. = f | of a foot, which 
 75 ft* 9' is 56', and 2' added = 58', which equals 
 
 '. 4 ft. 10', this we write down. 9 ft. X 5' 
 
 80 ft. 7' 11" = tl of ^ foot, which is 45' = 3 ft. 9' ; 
 write down the 9' and carry the 3 ft. to the 
 
148 DUODECIMALS. ^CHAP. VI 
 
 next product. 9 ft. x 8 ft. = 72 ft. and 3 ft. added = 75 ft. 
 The sum of these partial products gives the required product, 
 which is 80 ft. 7' 11". 
 
 Remakk. — It has already been stated that it was impossible to multiply one 
 concrete number by another. I'he above example may appear at first 
 thought to be contrary to that statement, but«it must be remembered that the 
 multiplier is considered an abstract number. 
 
 2. What is the product of 14 ft. T 2" by 6 ft. 3' 5" ? 
 
 3. Wtiat is the area of a marble slab, the length of 
 which is 9 ft. 8' 11", and width 3 ft. 1' ? 
 
 4. How many square feet are contained in the floor of 
 a room 40 ft. 10' long, 32 ft. 8' wide ? 
 
 6. How many square feet in 10 boards, each 18 ft. 10 
 long and 1 ft. 8' wide ? 
 
 6. How many square feet of boards will it take to 
 inclose a piece of land 80 ft. 10 in. long, and 60 ft. 8 in. 
 wide, with a close fence T ft. 6 in. high ? 
 
 7. How many square yards in a floor which is 48 ft. 6' 
 long, and 36 ft. 10' wide ? 
 
 8. What will the plastering of a room cost, at 18 cents 
 a square yard, the length of which is 30 ft. 10 in., width 
 24 ft. 6 in., and height of ceiling 8 ft. 4' ? 
 
 9. In a certain building there are 32 windows; in each 
 window 16 lights; and each light is 1 ft. 10' by 11'. How 
 many square feet of glass, in the 32 windows ? 
 
 10. In a certain room 24 ft. long, 18 ft. 6' wide, and 
 10 ft. 2' high, there are 6 windows, each 6 ft. 2' long, 3 ft. 
 10' wide; and 3 doors, each 6 ft. 10' by 3 ft. What will 
 be the cost of plastering this room, at 16 cents a square 
 yard ? 
 
 11. How many solid feet in a pile of wood 24 ft. 6 in. 
 long, 6 ft. 5' high, and 4 ft. 6' wide ? 
 
 Remark.— Multiply the length, height, and width together, to find the solid 
 contents. 
 
 12. How many cubic feet in a stick of timber 32 ft. 9' 
 long, 2 ft. 2' wide, an'd 2 ft. 8' thick ? 
 
 13. How many bricks, each 8 in. long, 4 in. wide, and 
 2 in. thick, are required to build a wall 144 feet long, 6 
 ft. 6 in. high^ and three bricks wide, no allowance being 
 made for the mortar ? 
 
art. 164.] division of duodecimals. 149 
 
 Art. 164. Division of Duodecimals. 
 
 1. There are 8 ft. 5' 3" in the surface of a marble slab, 
 the length of which is 3 ft. 9' ; what is its width ? 
 
 OPERATION. Explanation. — 3 ft. is con- 
 
 3 ft. 9')8 ft. 5' 3"(2 ft. 3' Ans. tained in 8 ft. 2 times. Mul- 
 
 7 f^ Q' tiplying the whole divisor by 
 
 L 2 ft. give 7 ft. 6' for the pro- 
 
 21' 3" duct, whiclr~we subtract from 
 
 11' 3'/ the corresponding denomina- 
 
 tions of the dividend, and ob- 
 
 tain 11' for a remainder, to 
 
 which annex the next denomi- 
 nation of the dividend, and we have 11' 3". 3 ft. is contained 
 in 11', 3' times. The divisor being multiplied by this 3' give 
 11' 3", which being subtracted from the last remainder leaves 
 nothing. Therefore, the marble slab was 2 ft. 3' in width. 
 
 Remark. — If the student will bear in mind that the superficial contents oi 
 any surface is found by multiplying the length by the breadth, he will readily 
 understand that dividing the superficial contents of any surface by its length 
 will give its width, or by its width will give its length. Also, since the so- 
 lidity of a body is found by multiplying its three dimensions together, if we 
 divide its cubical contents by the product of either two of its dimensions, the 
 quotient will be the other dimension. 
 
 The number of indices to be annexed to any term of the quotient can be 
 readily determined, since the indicts of the quotient added to the indices of the 
 divisor must equal those of the dividend. 
 
 2. There are 489 sq. ft. 8' 0" 2'" 1!'", in the surface 
 of a floor. The length of the floor is 87 ft. 1' 11". What 
 is its width ? 
 
 3. There are 28 sq. ft. 3' 11" 2"', in the surface of a 
 table; the length of which is 6 ft. 9' T" ; what is its width ? 
 
 4. The area of a certain pond, the length of which is 
 43 ft. 9' 6", is 1075 sq. ft. 0' 3" 0"' 6"". What is its 
 width ? 
 
 5. A stick of timber is 3 ft. 2' wide, 2 ft. 11' thick, and 
 contains 135 cu. ft. 10' 2" 1"'. What is its length ? 
 
 6. The area of a pond is 3978 ft. 1' 6"; its length is 
 100 ft. 6'. What is its width? 
 
 7. The area of a marble slab is 27 ft. 0' 7" 9'" 6""; its 
 length is 7 ft. 6' 3". What is its width ? 
 
 8. The area of a hall is 103 ft. 4' 5" 8"' 4""; its width 
 is 6 ft. ir 8". What is its length ? 
 
1.60 REDUCTION OF CURRENCIES. fcHAP. VI. 
 
 REDUCTION OF CURRENCIES. 
 
 Art. 165. Reduction of Currencies teaches how to 
 find the value of the denominations of one currency in the 
 denominations of another. 
 
 The value of a dollar, expressed in shillings and pence, 
 is not the same in different States of the Union, and in 
 different countries. This difference may be learned from 
 the following 
 
 TABLE. 
 
 I North Carolina, i """"^y- 
 
 r New England States, "] 
 jg-. . I Virginia, ( = 6s. = £j\, called New Eng- 
 
 ^ ] Kentucky, j land currency. 
 
 [ Tennessee, j 
 
 f New Jersey, ^ 
 
 ^-. . I Pennsylvania, 1 = 75. 6d. = £|, called Penn- 
 
 * ^° I t)elaware, j sylvania currency. 
 
 i Maryland, J 
 
 ^1 . j South Carolina, ) 4s. Sd. = £/„, called Georgia 
 
 ^ ^^ i Georgia, j currency, 
 
 ^-j . ( Canada, ) 5s. = £|, called Canada cur- 
 
 ^^ ^° j Nova Scotia, [ rency. 
 
 The legal value of £1 English or Sterling money, is $4.84, 
 as fixed by an act of Congress in 1842. 
 
 The above Table gives the value of $1, expressed in the 
 fraction of a pound, in the different currencies. The value 
 of £1 in each of the above currencies is found by analysis, 
 thus,— If £| = $1, £} =z $i and £^ or £1 = 5 times i, 
 which is $f . In a similar manner from the above table, 
 we can form the following 
 
 TABLE. 
 
 £1 = $f , New York currency. 
 £1 = $y, New England currency. 
 £1 = $f , Pennsylvania currency. 
 
 £1 = $Y, Georgia currency. 
 £1 = |4; Canada currency. 
 
ART. 167.] REDUCTION OF CURRENCIES. 151 
 
 1. Redace $321*75 to its equivalent value in Penn- 
 sylvania currency. 
 
 OPERATION. 
 
 $321-75 X i = £120-65625, 
 which equals £120 135. Id. 2 far. 
 
 2. Reduce $345-25 to its equivalent value in New York 
 currency. 
 
 3. Reduce $684"12|- to its equivalent value in New 
 England currency. 
 
 4. Reduce $67*84 to its equivalent value in Georgia 
 currency. 
 
 5. Reduce $846'87| to its equivalent value, Canada 
 currency. 
 
 6. Reduce $846*625 to its equivalent value in English 
 or Sterling money. 
 
 Art. 166. Reduction of Pounds, Shillings, &c., op 
 different currencies, to federal money. 
 
 1. Reduce £75 15^. Qd. New York currency, to Federal 
 money. 
 
 OPERATION. 
 
 £75 15.S. 6d. = £75-775. 
 £75-775 X I = $189-4375. 
 
 2. Reduce £154 10^. 8^. New England currency, to 
 Federal money. 
 
 3. Reduce £346 I65. 9d. Pennsylvania currency, to 
 Federal money. 
 
 4. Reduce £843 15^. 8^. Georgia currency, to Federal 
 money. 
 
 5. Reduce £49 ISs. lid. Canada currency, to Federal 
 money. 
 
 6. Reduce £784 17^. lOd. Sterling, to Federal money. 
 
 Art. 167. The following table shows the value of 
 some of the foreign coins at their standard value : — ■ 
 
 1 Pound Sterling, or Sovereign, . . . $4- 84 
 
 1 Guinea, English, . , . . . 500 
 
 1 Crown, 1-06 
 
152 
 
 ALIQUOT PARTS, 
 
 [chap. VI. 
 
 1 Shilling piece, English, . . . , -23 
 
 1 Franc, ...... -186 
 
 1 Doubloon, Mexico, . . . . 15 60 
 
 1 Specie Dollar of Sweden and Norway, . . 1-06 
 
 1 Specie Dollar of Denmark, . . . .105 
 
 1 Thaler of Prussia and N. States of Germany, . -69 
 
 1 Florin of Austrian Empire and City of Augsburg, . -485 
 1 Ducat of Naples, .... -80 
 
 1 Ounce of Sicily, ..... 2-40 
 
 1 Pound of British Provinces, Nova Scotia, New Bruns- 
 wick, Newfoundland, and Canada, . . 4- 00 
 
 Note. — A little reflection will enable the pupil ta reduce any of these foreign 
 coins to Federal Money, or Federal Money to foreign coins. 
 
 Aliquot Parts. 
 
 Art. 168. The half, third, fourth, fifth, &c., of any 
 quantity, is an Aliquot Part of that quantity. 
 
 Art. 169. Analysis is applied to arithmetical solu- 
 tions, when the various factors of the question and their 
 relations are traced out, forming a process of reasoning. 
 
 ANALYSIS BY ALIQUOT PARTS. 
 
 1. What is the value of 4 cwt. 2 qrs. 12^ lbs. of sugar, 
 at $8.84 a cwt ? 
 
 2 qrs. = \ cwt. 
 
 121 lbs. = :^ of 2 qrs. 
 
 OPERATION. 
 
 $ 8.84 
 4 
 
 35.36 value of 4 cwt. 
 4.42 value of 2 qrs. 
 1.101 value of 121 lbs. 
 
 So'ssi Ans. 
 
 2. What is the value of 25 lbs. 5 oz 12 pwts. of silver 
 ware, at $54*18f a pound ? 
 
 3. What is the value of 6 tons 5 cwt 3 qrs. of iron, at 
 $35-371 a ton ? 
 
 4. What is the value of 16 cwt. 2 qrs 15 lbs. of sugar 
 at $9-371 a cwt. ? 
 
ART. 170.] CANCELLATION. 153 
 
 5. What is the yalue of 346 bushels 3 pks. 1 qt. of rye, 
 at $-93f a bushel ? 
 
 6. What is the value of 3 pks. 6 qts. of cherries, at 
 $1-12^ a peck? 
 
 t. A market woman bought 2 bushels 3 pks. 4 qts. of 
 strawberries, at $287 a bushel. How much did she pay 
 for them ? 
 
 8. A merchant bought 25 yds. 2 qrs. 2 nas. of silk, at 
 $1'87| a yard ; and 3t yds. 3 qrs. 3 nas. of broadcloth, at 
 $4*95 a yard. What did the whole amount to ? 
 
 9. A gentleman bought a lot of land containing 4*7 A. 
 2 R. 25 P., at $85*3T^ an acre. How much did he pay 
 for the lot ? 
 
 Cancellation. 
 Art. 170. Cancellation, in arithmetic, consists in re- 
 jeciing equal factors from a divisor and dividend, which 
 does not change the value of the fraction ; it being the 
 same as dividing both divisor and dividend by the same 
 number. (See Art. 116. Proposition 6th.) 
 
 ANALYSIS BY CANCELLATION. 
 
 1. If I of a yd. of cloth cost $f , what will | of a yd. cost ? 
 
 Analysis. — If f of a yard cost $|, ^ of a yard will cost | of 
 $f ; and § (1 yard) will cost ^ of $f. If 1 yard cost f of $|, 
 I of a yard will cost | of i of $f ] and | of a yard will cost | 
 offoflf, = $f. Ans. 
 
 OPERATION. 
 
 3 
 
 Remark. — Those who prefer can place the numerators of the fractions on 
 the right ol a perpendicular line, one under another ; and the denominators, 
 in a similar way. on the left of the same line, and thereby avoid writing the 
 sign 01 multiplication. Thus : 
 
 2 
 3 
 
 5 
 $ 
 
 $ Ans. U. 
 
154 RATIO. [chap. VII. 
 
 2. If I of a yard of cloth cost $6, what will f of a yard 
 cost? 
 
 3. How much will f of a ton of hay cost, when 4i tons 
 cost $13-39? 
 
 4. Allowing a horse to travel | of a mile in 4 minutes, 
 what distance would he travel in 48 minutes ? 
 
 5. If 6 men can perform a certain piece of work in 24'6 
 days, in what time can 24 men perform the same work ? 
 
 6. A gave towards the building of a church $140, which 
 was ^ as much as B gave, and B gave | as much as C. 
 How much did C give ? 
 
 t. If 31 bushels of corn are worth 2i bushels of rye, 
 how many bushels of corn are worth 14| bushels of rye ? 
 
 8. A has I as much money as B; and f as much as C, 
 who has ^ as much as D, who has $2400. How much 
 have A, B, and C respectively ? 
 
 CHAPTER YII. 
 
 RATIO. 
 
 Art. ITl. Two numbers or quantities of the same 
 denomination, may be compared together in two ways, — 
 
 First. By means of an Arithmetical Ratio, which is 
 expressed by their difference. 
 
 Secondly. By means of a Geometrical Ratio, which is 
 expressed by the number of ti7nes the one contains the other. 
 
 The word Ratio, when used alone, refers to a geometrical 
 ratio. 
 
 Ratio is the relation which one number, or quantity, 
 bears to another of the same denomination, and is expressed 
 by the quotient arising from dividing the first by the 
 second, or by dividing the second by the first. . 
 
 When we speak of the ratio of one number to another 
 in Arithmetic, we shall refer to the quotient arising from 
 dividing the second terra by the first, as the first term iu 
 
ART. 172.] PROPORTION, 155 
 
 a simple proportion is made the divisor. Thus, the tatio 
 of 2 feet to 8 feet is 4, or expressed in the form of a frac- 
 tion, is f. The ratio of two quantities is usually expressed 
 by, (:) being placed between them; thus, 2 : 8, which 
 equals | or 4. 
 
 A ratio cannot be a concrete or denominate number; 
 neither is there a ratio between quantities of diiferent 
 denominations. 
 
 1. What is the ratio of 5 yards to 25 yards ? 
 
 2^ What is the ratio of 4 inches to 36 inches ? 
 
 3. What is the ratio of 8 apples to 72 apples ? 
 
 4. What is the ratio of 24 sheep to 96 sheep ? 
 
 5. What is the ratio of 9 pounds to 108 lbs. ? 
 
 6. What is the ratio of 4 feet to $16 ? 
 
 7. What is the ratio of 4 sheep to 24 horses ? 
 
 PROPORTION. 
 
 Art. 172. When two quantities have the same ratio 
 as two other quanities, the four quantities are said to be 
 in Proportion. Thus, the ratio of 8 bushels to 32 bushels, 
 is the same as the ratio of $3 to $12. 
 
 Proportion is an equality of ratios of numbers compared 
 together, two and two. 
 
 Quantities are shown to be in proportion by means of 
 dots; for example, the above proportion is written, 
 
 bush. bush. $ $ 
 
 8 : 32 :: 3 : 12 
 And is read 8 busels is to 32 bushels, as $3 is to $12. 
 
 Rkmark. — The two dots placed between the first and second, also between 
 the third and fourth terms, in the above proportion, are contractions of the sign 
 of division (-;-), the horizontal line being omitted. The four dots between the 
 second and third are contracted from, and equivalent to, the sign of equality. 
 Hence, the above was formerly written, 
 
 8 bush. -T- 32 bush. := $2 -f- $K. This expression indicates that the ratio is 
 found by dividing the first term by the second. The English mathematicians 
 have adopted this method of expressing the ratio of one number to another, 
 while the French divide the second by the first, as previously directed. 
 
 The first two terms of pr9portion are called the first 
 couplet ; the second two terms, the second couplet. 
 
156 SIMPLE PROPORTION. [cHAP. VIL 
 
 The first term of each couplet is called the Anteadentj 
 and the second term is called the Consequent. 
 
 The Jlrst and fourth terms of a proportion are called 
 the Extremes, and the second and third terms are called the 
 Means. 
 
 Since, in a proportion, the quotient obtained by dividing 
 the second term by i\\Q first, is equal to the quotient obtained, 
 by dividing the fourth term by the third, we can readily 
 deduce the following 
 
 PROPOSITIONS. 
 
 1. The product of the mea7is is equal to the product of 
 the extremes. Therefore, 
 
 2. If the product of the means le divided hy one extreme, 
 the quotient will be the other extreme. Or, 
 
 3. If the product of the extremes le divided hy one mean, 
 the quotient will be the ot/ier mean. 
 
 4. The fourth term of a proportio7i is equal to the third 
 term, multiplied by the ratio of the first term to the second. 
 
 Suggestion. — These propositions being understood, the pu- 
 pil can readily determine the remaining term of a proportion, 
 if any three of them be given. 
 
 SIMPLE PROPORTION. 
 
 Art. IT'S, Simple Proportion teaches the method of 
 finding the fourth term of a proportion, by knowing the 
 other three. 
 
 Art. 174. In stating a question in Simple Proportion, 
 the FIRST and second terms must be of the same kind or 
 denomination, and the third terra like the answer sought. 
 If the answer is to be greater than the third term, the 
 larger of the two remaining terms, must occupy the second 
 place, — if smaller, the^^r^^ place. Then proceed accord- 
 ing to Proposition 2nd, or 4th. 
 
 1. If 12 bushels of wheat cost $21*60, whax will 29 
 bushels cost 1 
 
ART. 174 ] SIMPLE PROPORTION. 151 
 
 Explanation — The answer sought is to be in dollars, there- 
 fore, we have the $2100, for the titird term. The answer is to 
 be greater then the third term, because 29 bushels will cost 
 more then 12 bushels : hence, we have the larger number, 29 
 for the second term and 12 for the first. Thus : 
 
 bnsh. bush. $ 
 
 12 : 29 : : 21-60 
 29 
 
 19440 
 4320 
 
 12)626-40, the product of the means. 
 
 ^ $52-20, the other extreme, or 4th term. 
 The above question can as well be solved, by finding the ratio, 
 of the first to the second term. Thus : (See Prop. 4th). 
 
 OPERATION. 
 
 1-80 
 29 ^/•00 ^ 
 ^X—^— =$52-20. Ans. 
 
 2. What will 24T yards of cloth cost, if 25 yards cost 
 $144-60? 
 
 3. What will 347 bushels of corn cost, if 84 bushels cost 
 $66-40 ? 
 
 4. What will 384 bushels of wheat cost, if 35 bushels 
 cost $30-80? 
 
 5. What will 341 boxes of raisins cost, if 312 boxej 
 cost $436-121 ? 
 
 6. If a man travel 485 miles in 18 days, how far at this 
 rate will he travel in 125 days ? 
 
 T. A garrison of 125 men has provisions for 35 days. 
 How many of the men must be discharged, that the re- 
 mainder may be supported for 125 days ? 
 
 8. If 43 men can do a certain piece of work in 47^ days, 
 how many days will it take 15 men to do the same ? 
 
 9. A man bought cows, at $12315 for 8. How much 
 at the same rate would 35 cows cost. 
 
 10. Bought 23 pieces of delaine, each containing 41| 
 yards, at the rate of $24-45 for 45 yards. How much did 
 it all cost ? 
 
158 PROPORTION, . [chap. VII 
 
 11. If a company of 190 men consume 54 barrels of 
 flour in 6 weeks, how many barrels would it take to last 
 them 1 year ? 
 
 12. If $273 in 3 years gives $18-621 interest, how long 
 will it require to give $184 interest ? 
 
 13. If $83 in two years 8 months give $12*37^ interest, 
 what sum in the same time will give $3t5*12i interest ? 
 
 14. If 50 men. build a wall 750 rods long in 8 days, 
 how many men will be required to build 8 64 "5 rods in half 
 of the time ? 
 
 15. If a railroad car go 23 miles in 45 minutes, how far 
 will it go in 5 days of 10 hours each ? 
 
 16. If in 247^ feet there are 15 rods, how many rods 
 in 1 mile ? 
 
 17. If 47 acres of land sell for $684'48, what will be 
 the cost of a farm containing 287 '5 acres ? 
 
 18. What will be the cost of 847*56 pounds of wool, if 
 84-5 pounds cost $47-87| ? 
 
 19. If 19 sheep yield 56i pounds of wool, how many 
 pounds will 387 sheep yield ? 
 
 20. How many pounds of coffee can be bought for 
 $147-84, when 18 pounds cost $l-93f ? 
 
 21. If a tree 25 feet 4 inches in height give a shadow 
 of 50 feet 8 inches, what is the length of the shadow of 
 a tree whose height is 84 feet 9 inches ? 
 
 Remark.— After stating the question, reduce the first and second terms to the 
 id^me denominate value. ; also reduce the J!/it>rf term to its lowest denomination 
 nien'ioned : — the answer will be of the same denomination. 
 
 
 
 
 OPERATION. 
 
 
 
 ft. 
 25 
 
 in. ft. 
 
 4 : 84 
 
 in. 
 9 
 
 ft. in. 
 
 : : 50 8 : length 
 
 of shadow 
 
 required. 
 
 in. 
 304 
 
 in 
 
 : 1017:: 
 608 
 
 in. 
 608 
 
 length of shadow : 
 
 required. 
 
 
 8136 
 6102 
 
 304)618336(2034 in. = 169 ft. 6 in. Ana. 
 608 
 
ART. 174.] SIMPLE PROPORTIO?i. 159 
 
 22. If 8 horses eat 19 bushels 3 pks. of oats in a week, 
 how much would 85 horses eat iu the same time ? 
 
 23. If 12 men in 6 weeks earn £145 IO5. 9d., how 
 much can 84 men earn in half of the time ? 
 
 24. If 14 bushels 2 pks. 4 qts. of clover seed are worth 
 $6tl2i how much will 184 bush. 3 pks. 6 qts. cost? 
 
 25. If 15 horses in 4 days, consume 87 bush. 6 qts. of 
 ■oats, how many horses will 610 bush. 1 pk. 2 qts. keep the 
 same time ? 
 
 26. If the transportation of 21 cwt. 147 miles, cost 
 $23*87^, what will the transportation of 47 cwt. 3 qrs. 
 20 lbs. cost, 4 times as far ? 
 
 27. If a person accomplish a certain piece of work in 
 242 days, by working 8 hrs. a day, in how many days will 
 he accomplish the same work, by working 12f hours a day ? 
 
 28. Allowing a person to perform a certain journey in 
 26 days, when the days are 10^ hours long; in what time 
 ought he to accomplish the same journey, when the days 
 are 13 hours long ? 
 
 29. Allowing I3 A. 25 P. of land to produce 384 bush. 
 3 pks. of wheat, what number of bushels would be raised 
 from a field containing 47 A. 3 R. 30 P., at the same rate ? 
 
 30. An army of 4800 men had provisions for 8 months, 
 one-sixth of the men having been killed in battle, how 
 long ought the same provisions last the remainder ? 
 
 31. If 18 head of cattle require 25 A. 3 R. of pasture 
 ground, during the summer, how many acres ought 36 
 head to have for the same length of time ? 
 
 32. Allowing the transportation of 25 T. 18 cwt. 20 lbs., 
 a given distance, to cost $37*85; how much should be 
 charged for the transportation of 18 T. 16 cwt. 3 qrs.^ 
 10 lbs. the same distance ? 
 
 33. If a ship sail 247 leagues 1 mile 6 fur. in 15 days; 
 in how many days would she sail 3000 miles ? 
 
 34. A borrowed $250, which he kept, 3 years 6 months. 
 A subsequently, lends B $187|^. How long ought B to 
 keep this latter sum, in return for the accommodation he 
 afforded A ? 
 
 35. A merchant bought 3 pieces of cloth, each contain- 
 
160 PROPORTION. [chap. VII. 
 
 ing 23 yds. 3 qrs. for $495" 15; and sold 54 yds. 2 qrs. of 
 it for what it cost. How much did he receive for it ? 
 
 36. If 8 yards 3 qrs. of cloth cost $34-50, how much 
 will 83 E. English 3 qrs. cost ? 
 
 37. If 5 E. French 4 qrs. of cloth cost $14-60, how much 
 will 12 yds. 3 qrs. 2 nas. cost ? 
 
 38. Allowing 14 horses to consume 65 bush. 3 pks. 5 qts. 
 of oats in a week, how much would 74 horses consume in 
 the same time ? 
 
 39. If a person perform a certain journey in 14 days, 
 by traveling 9^ hours a day, how long will it take him to 
 perform the same journey by traveling 12^ hours a day ? 
 
 40. What will be the cost of 9 cwt. 3 qrs. 20 lbs. of 
 beef, if 8 cwt. cost $68 ? 
 
 41. If 36 sacks, each measuring 5 bushels, contain a 
 eiven quantity of grain ; how many sacks, each containing 
 3i bushels, will contain the same quantity ? 
 
 42. Allowing 32 head of cattle to require 23 A. 3 R 
 25 P. of pasture ground, during the summer, how many 
 acres will 145 cattle require for the same length of time ? 
 
 43. If 4 men mow 7*97^ A. of grass in a day, how many 
 men will be required to mow 63-8 A. in half the time ? 
 
 44. The capacity of a cistern is 3600 gallons, and is 
 filled with water by a pipe which pours into it 10 gals. 
 3 qts. a minute. By a leakage, 1 gal. 2 qts. 1 pt, leaks 
 out every minute during the time of filling. In what time 
 will the cistern be filled ? 
 
 45. If f of an acre of land is worth $136, how much 
 is If of an acre worth ? 
 
 Rf.mark. — Questions containing fractions, can be most conveniently salved 
 b}' finding the ratio of the first to the second term, and then multiply the 
 third term by it. (Art. 172, Proposition 4.) 
 
 OPERATION BY CANCELLATION. 
 
 I : If :: $136 : value sought. 
 3 68 
 
 |X^X— = $204 Ans. 
 
ART. 1T4.] SIMPLE PROPORTION. 161 
 
 46. If f of a farm is worth $860, how much is f of it 
 worth ? 
 
 47. If 14 of a city lot is worth $4800, how much is "| 
 of it worth ? 
 
 48. If I of a barrel of flour is worth $5-40, how much 
 is yV ^^ i^ worth ? 
 
 49. What cost 16f pounds of tea, if 6f pounds cost) 
 $8-55 ? 
 
 50. What length of board that is 16y^3 inches in width, 
 will be required to make a square foot ? 
 
 51. Bought 15^ yards of cloth for $54*90 ; what will 
 25 yards 3 qrs. cost at the same rate ? 
 
 52. If f of a ship is worth $34865, how much is the 
 whole cargo worth ? 
 
 53. If j\ enough water run into a ship by a leak, in 1 
 day 9 hrs. 15 miu., to sink her ; how long before she will 
 sink ? 
 
 54. Bought 25| barrels of flour, at $6y«y a barrel, and 
 paid for it with sheep, at $li a head ; how many sheep 
 did it take ? 
 
 55. If 6| barrels of sugar cost $112.15, how much will 
 A of a barrel cost ? 
 
 56. If 13f yards of cassimere cost $19|, what will 5| 
 yards cost ? 
 
 51. If 2f barrels of beef cost $20-*I5, how much will 1^ 
 barrels cost ? 
 
 58. If 5 pounds of butter cost 62i cents, how much 
 will If pounds cost. 
 
 59. If I of an apple cost | of a cent, what will | of an 
 apple cost ? 
 
 60. If it require 6 days for 10 men to build 360 rods of 
 wall, how many men can in i of the time build 120 rods 
 of similar wall ? 
 
 61. If 24 men in 8 days perform a certain piece of work, 
 how many men will be necessary to accomplish 3 times as 
 much work in | of a day ? 
 
 62. If it require 2 bushels of oats to feed 4 horses ^ of a 
 day, how many horses would it take to consume 144 bushels 
 in f of a day ? 
 
162 PROPORTION. [chap. VII 
 
 63. If a staff 9f feet long cast a shadow 12| feet, what 
 is the height of that steeple the shadow of which, at the 
 same time measures 285 feet ? 
 
 64. If a steamship can sail 3000 miles in 9^ days, how 
 long, at the same rate of sailing, would she require to 
 sail 24900 miles, the distance around the earth ? 
 
 65. Tiie diurnal rotation of the earth moves its equato- 
 rial portions about 24900 miles a day. (24 hours.) How 
 far is that in each minute ? 
 
 66. Admitting the earth to move in its orbit about the 
 sun 59*1000000 miles, in 365 days 6 hours ; how far on an 
 average does it move in 1 minute ? 
 
 67. If it require 35 yards of carpeting, which is | of a 
 yard wide to cover a floor, how many yards, which is 1^ 
 yards wide, will be necessary to cover the same floor ? 
 
 COMPOUND PROPORTION. 
 
 Art. ITo. Compound Proportion teaches to find a 
 required quantity in a proportion when it depends on more 
 than three terms. 
 
 1. If 6 men can earn $72 in 10 days, by working 12 
 hours a day, how many dollars can 15 men earn in 8 days, 
 by working 8 hours a day ? 
 
 Remark. — We will first solve this question by analysis. 
 
 Analysis If 6 men in a certain time earn $72, 1 man in 
 
 the same time will earn ^ of $72 = $12 ; and 15 men will earn 
 15 times $12 = $180. If in 10 days 15 men earn $180. in 1 day 
 they wiU earn yV of $180 = $18 ; and in 8 days they will earn 
 8 times $18 = $144. If in 8 days by w^orking 12 hours a day, 
 15 men earn $144, by working 1 hour a day, they will earn 
 yV of $144 = $12 ; and by working 8 hours a day they will 
 earn 8 times $12 = $96. 
 
 SOLUTION BY CANCELLATION. 
 
 IIkmakk — As the question is read the pupil will find it of assistance to 
 write it down in the following manner, as he can then more easily remember 
 the question and form the ratios. Taking the above example we proceed 
 thus ;— 
 
kRT. 115.] ' COMPOUND PROPOKTION. 163 
 
 men. $. days, hours. 
 
 6 12 10 12 
 16 8 8 
 
 X^ 3 4 
 
 Explanation If 6 men in a certain time earn $72, 1 man 
 
 will earn \ of $72^ and 15 men will earn y of $72. If 15 men 
 earn y of $72 in 10 days, in 1 day they will earn jL as much, 
 and in 8 days y^^ as much, which is ^-^ of ^ of $72. If 15 men 
 in 8 days earn y\ of */ of $72 by working 12 hours a day, by 
 working 1 hour a day they will earn y'^ as much, and by working 
 8 hours a day y^2 as much, which is y\ of y^^ of ^-^ of $72 = $96. 
 
 2. If 12 men can mow 48 acres of grass in 8 days, by 
 working 5 hours a day; how many acres can 56 men mow 
 in 5 days, by working 12 hours a day ? 
 
 3. If the wages of 36 men for 3 days be $216; how 
 many men in 4 days can earn $192 ? 
 
 4. If 15 men can cut 280" cords of wood in 16 days, by 
 working 9 hours a day, how many men will be required to 
 cut 28 cOtds in 4 days, by working 6 hours a day ? 
 
 5. If a man travel 240 miles in 14 days, by traveling 
 6 hours a day; how far can he travel in 18 days, by 
 traveling 9^ hours a day ? 
 
 6. If 15 men in 9 days, by working 6 hours a day, build 
 36 rods of stone-fence; how many men will be required to 
 build 133 3 rods in 14 days, by working 8 hours a day ? 
 
 7. If 72 men in 18 days of 12 hours each, build a wall 
 162 rods in length, 12 feet high, and 9 feet thick; how 
 many rods of wall that is 9 feet high, and 3 feet thick, can 
 40 men build in 8 days of 9 hours each ? 
 
 8. If a marble slab 20 feet long, 6 feet wide, and 4 inches 
 thick, weigh 850 pounds; what is the length of another 
 slab that is 4 feet wide and 2 inches thick, that weighs 
 212 pounds ? 
 
 9. If a family of 12 persons in 20 weeks and 4 days con- 
 sume $450 worth of provisions; how many persons will 
 $803 7 1 worth of provisions keep 45 weeks and 6 days ? 
 
164 PROPORTIOX. [chap. VII. 
 
 10. If it require 264 yds. of cloth tliat is li yds. wide, 
 to clothe 121 men; how many yards which is li yards 
 wide will be required to clothe 220 ? 
 
 11. If 210 yds. of cloth, 1 yard wide, cost $300, what 
 will 140 yds. of similar cloth cost, that is 3 quarters wide ? 
 
 12. If $250 will in 7 months gain $25, when the rate of 
 interest is 10 per cent.; at what rate per cent., will $750 
 in 9 months ^ain $67^ ? 
 
 13. If a family of 24 persons consume $120 worth of 
 bread in 8f months, when flour is worth $5 a barrel; how 
 many dollar's worth will a family of 8 persons consume in 
 6 months, when flour is worth $7 a barrel ? 
 
 14. If 240 men, by working 8 hours a day, can in 81 
 days dig 256 cellars, each 24 feet long, 27 feet wide, and 
 18 feet deep; how many men can, in 27 days of 6 hours 
 each, dig 18 cellars, each 40 feet long, 36 feet wide, and 
 12 feet deep ? 
 
 15. If 24 men, by working 8 hours a day, can in 18 days 
 dig a ditch 95 rods long, 12 feet wide, and 9 feet deep, 
 how many men, by working 12 hours a day, for 24 days, 
 will be required to dig a ditch 380 rods long, 9 feet wide 
 and 6 feet deep, in a soil that is 1| times as difficult of 
 excavation ? 
 
 CONJOINED PROPORTION. 
 
 Art. 176. Conjoined Proportion is a proportion in which 
 each antecedent is equal in value to its consequent, — each 
 consequent being of the same denomination as the preceding 
 antecedent, — and the first and last terms, of the same denom- 
 ination. 
 
 1. If 8 bushels of wheat are worth 3 cords of wood, and 
 9 cords of wood are worth 3 tons of hay, }\ow many bushels 
 of wheat are worth 6 tons of hay ? 
 
 Analysis. — If 3 tons of hay are worth 9 cords of wood, 1 ton 
 is worth § cords of wood. If 3 cords of wood are worth 8 bush. 
 
ART. lYt.] COPARTNERSHIP, 165 
 
 of wheat, 1 cord .*s worth | bushels. If 1 .^ord is worth | bush, 
 of wheat, | cords (the value of 1 ton of hay), is worth | times 
 I bushels of wheat, and 6 tons are worth 6 times f X f bushela 
 = 48 bushels of wheat. 
 
 The conditions of the above question are expressed thus : 
 
 8 bushels = 3 cords of wood. 
 
 9 cords = 3 tons. 
 
 6 tons = how many bushels of wheat ? 
 
 And may be solved by writing all the terms on the left of the 
 equality, for the numerator of a compound fraction, and those 
 on the right for the denominators. Thus : 
 
 $ 
 
 6 8 
 
 tXaXs = 48 bushels of wheat. 
 
 2. If 4 barrels of corn are worth 8 bushels of wheat, and 
 3 bushels of wheat are worth 5 bushels of rye, and 12 
 bushels of rye are worth 20 bushels of oats, how many 
 bushels of oats are worth 12 barrels of corn ? 
 
 3. A can do as much work in 3 days as B can in 6 days ; 
 and B as much in 5 days as C in 15 days. In how many 
 days could A do as much work as C in 48 days ? 
 
 4. If 48 yards of cloth in New York are worth 36 bar- 
 rels of flour in Philadelphia ; and 18 barrels of flour in 
 Philadelphia are worth 24 bales of cotton in New Orleans; 
 how many bales of cotton in New Orleans are worth 240 
 yards of cloth in New York ? 
 
 5. If 121 yards of satin cost $18*I5; and $10-25 will 
 purchase 3 yards of broadcloth; and 6^ yards of broadcloth 
 are worth 18^ yards of silk; how many yards of satin are 
 worth 120 yards of silk ? 
 
 COPARTNERSHIP.* 
 
 Art. 177. Copartnership is the association of two or 
 more individuals in the transaction of business, who agree 
 
 * CopartntrshtPi or Fellowship, is sometimes called PAKimvi; Proportiow. 
 
166 PROPORTION. [chap. VII. 
 
 to share the profits and losses in proportion to the amount 
 of capital they have in the partnership. Each individual 
 thus associated is called a Partn&r. The partners together 
 are called the Comjpany, or Firm. 
 
 The Capital Stock is the amount of money employed in 
 trade. The Dividend is the profit or loss to be shared. 
 
 1. A, B, and C entered into partnership. A put in 
 $240; B put in $400; and C put in $320. They gain $192. 
 How much is each man's gain ? 
 
 OPERATION, 
 
 A's stock, $240 
 B's " 400 
 C's " 320 
 
 Capital stock, 960 
 Therefore, A owns f f ^ = :|- of the entire stock. 
 
 BU 40 _5_ u u 
 
 Q^TT 12 
 
 C4; 320 1 U (( 
 
 9(J0 — 3 
 
 Hence, As gain is \ of $192 = $48 
 B's " j% of $192 = $80 
 C's " i of $192 = $64 
 
 2. A, B, and C enter into partnership. A puts in $360; 
 B puts in $440; and C puts in $500. They gain $t80. 
 How much is each man's gain ? 
 
 8. A, B, C, and D, hired a pasture for $12: A put in 
 12 sheep; B put in 16; C 18; and D 14. How much 
 ought each to pay ? 
 
 4. Four men traded in company and gained $1680 . 
 A's stock was $2000; B's $1600; C's $2400; and D's 
 $2000. How much is each man's gain ? 
 
 5. A farm was purchased for $7000, by A, B, and C. 
 A furnished $2500; B $3000; and C $1500. They re- 
 ceive $560 rent yearly. How much of this rent should 
 each receive ? 
 
 6. A merchant employed 4 clerks, at the annual salaries 
 of $250, $300, $400, $500, respectively. At the end of 
 the year the merchant proving bankrupt, has but $870 to 
 
ART. 17 1.] COPARTNERHIP. 16t 
 
 be divided proportionally among them. What will be the 
 portion of each ? 
 
 7. Divide $960 among three persons in such a manner 
 that their shares shall be to each other as 5, 4, and 3 re- 
 spectively ? 
 
 8. Two persons form a partnership in trade, with a cap- 
 ital of $1500, of which the first contributed $940; and 
 the second the remainder. They gain $640. How much 
 is each one's share ? 
 
 9. Divide the number 230 into three parts which shall 
 be to one another as i, |, and f . 
 
 Analysis. — the proportional terms being reduced to equiva- 
 lent fractions having a common denominator, we have ~, y^g, 
 and y^2 ; ^'^d these fractions are to one another as their nume- 
 rators 6, 8, and 9^ since they have the same denominator. 
 Hence we divide the 230 into 6 -}- 8 -|- 9 = 23 equal parts. 
 Hence 2%, aV ^^^ 2^3 ^^ ^^^ respectively, gives the required 
 numbers. 
 
 10. A, B, and C, found a purse containing $240, and 
 agreed to share it in the proportion of |, ^, and f . How 
 much should each receive ? 
 
 11. A, B, and C enter into partnership: A puts in 
 $160; B $280; and C $460. They lose $480. How 
 much is each partner's loss ? 
 
 12. A captain, mate, and 14 sailors, took a prize of 
 $24600; of which the captain takes 11 shares; the mate 
 5 shares ; and the remainder is equally divided among the 
 sailors. How much did each receive ? 
 
 13. Four partners, A, B, C, and D shipped 1280 sheep 
 for Scotland; of which A owned 240; B 160; C 400; 
 and D the remainder. In a severe storm they threw 320 
 of them overboard. How many sheep did D own, and 
 how much was each partner's loss ? 
 
 14. A, B, C, D, and E are to share $3045; A is to 
 have a certain sum; B as much again as A; C as much 
 as A and B together; D as much again as B; and E as 
 much as D and A together. How much is each to have ? 
 
 15. A, B, and C agree to contribute $620'62 towards 
 building a church, which is to be situated 2 miles from Aj 
 
168 PROPORTION. [chap. VII. 
 
 3 miles from B; and 5 miles from C. They also agree 
 that their contributions shall be proportional to the recip- 
 rocals of their distances from the church. How much 
 9ught each to contribute ? 
 
 16. A, B, and C contribute $3535- tO towards building 
 an Academy, which is to be situated 1^ miles from A; 
 If miles from B; and 21 miles from C. They also agree 
 that their contributions shall be reciprocally proportional 
 to their distances from the Academy. How much did 
 each contribute ? 
 
 It. A, B, and C found a purse containing $280' tO. 
 They agreed to divide it in such a manner that A should 
 rave I as much as B; and B |- as much as C. How much 
 should A, B, and C receive respectively ? 
 
 Rkmark. — The pupil will find the proportional terms as follows : 
 
 A's part = I of B's, 
 and B's = | of C's 
 
 Hence, | of B's = C. 
 
 Therefore, A's = y\ of B's 
 B's = if 
 
 C's = If of B's. Consequently we divide 
 the $280*70 in proportion to the numbers 8, 12, and 15. 
 
 18. A, B, and C, in partnership lose $650. A's por- 
 tion of the capital employed was | of B's, and B's was f 
 of C's. What amount of loss should each sustain ? 
 
 19. Four persons in a joint speculation gain $460, which 
 is to be divided among them so that the second shall have 
 2- as much as- the first, and the second f as much as the 
 third. How much should each receive ? 
 
 20. A farmer divided 1152 acres of land among his four 
 sons, in such a manner that | of John's number of acres 
 equals f of James'; | of Jame's equals f of Jackson's; 
 and I of Jackson's equals f Joseph's number of acres. 
 How many acres did each receive ? 
 
 21. Three men A, B, and C, agree to reap a certain 
 field of wheat, for $39-68 ; A and B calculate that they 
 can do f of the labor ; A and C, that they can do | ; and 
 B and C that they can do | of it. How much can each 
 receive according to these estimates ? 
 
ART. 118.] COMPOUND COPARTNERSHIP. 169 
 
 COMPOUND COPARTNERSHIP. 
 
 Art. 178. When the stock of the several partners is em- 
 ployed in the trade for different periods of time, it is called 
 Compound Copartnership. It is evident in such cases, that 
 the gai7i or loss must be apportioned with reference to the 
 stock and the time it has been employed in the business. 
 
 1. Three partners A, B, and C put money into trade as 
 follows : A put in $50 for 4 months ; B, $150 for 2 
 months ; and C, $250 for 3 months. They gained $250. 
 How much is each man's share of the gain ? 
 
 OPERATION. 
 
 $ m. $ 
 
 50 X 4 = 200 for 1 month. 
 150 X 2 = 300 " 
 250 X 3 = 750 " 
 
 1250 Capital Stock. 
 Explanation. — The preceding work becomes evident, by 
 considering that the interest of $50 for 4 months, is the same 
 as the interest of $200 for one month ; &c. Therefore, 
 A's part of the entire stock = -f^^^ = ^g of the whole. 
 
 ^° T2-5O 3 
 
 Hence, A's gain = /^ of $250 = $40 
 B's " = 2^ of $250 = $60 
 C's " = I of $250 = $150. 
 
 2. A, B and C hire a pasture for $240 ; A put in 16 
 cows for 10 weeks ; B, 20 cows for 1 weeks ; and C, 25 
 cows for 6 weeks. How much ought each to pay ? 
 
 3. A, B, C, and D have together performed a piece of 
 work, for which they receive $266*40. A worked 16 days 
 of 10 hours each ; B worked 20 days of 12 hours each; 
 C worked 14 days of 1 hours each; and D worked 15 days 
 of 12 hours each. How much should each man receive ? 
 
 4. A, B, C, and D engaged in partnership for 3 years. 
 A advanced $2500, B $3500, C and D, each $3800. 
 Nine months afterwards, A added $600 to his stock ; B 
 
 8 
 
ItO PROPORTION-. [chap. VII 
 
 $350; C withdrew $180 ; anclD withdrew |460. At the 
 end of the 3 years, the profits were found to be $1200 
 How much is each one's share ? 
 
 ^5. To. gather a certain field of grain, A furnished 9 
 laborers 6 days ; B 12 laborers for 4 days ; and C 14 
 laborers for 5 days. For the whole work they received 
 $54*85. How much should A, B, and C receive respec- 
 tively ? 
 
 6. An army, consisting of 3 generals, 5 colonels, 12 
 captains, and 6840 soldiers, took a prize of $89908*15, 
 which they agree to divide among themselves in propor- 
 tion to their pay and the time they have been in the 
 army. The generals and colonels have been in the army 
 9 months ; the captains 5 months ; and the soldiers, 8 
 months ; the generals have $60 a month ; the colonels, 
 $40 ; the captains, $15 ; and the soldiers, $10. How 
 much ought each to receive ? 
 
 ALMGATION MEDIAL.* 
 
 Art. 179. Alligation Medial teaches the method of 
 finding the average value of a mixture when the several 
 simples of which it is composed, and their values are 
 known. 
 
 Art. 180, Given the several ingredients and theii 
 respective values to find the average value of the compound 
 
 1. A farmer mixes together 10 bushels of oats, worth 
 40 cents a bushel; 15 bushels of corn, worth 50 cents a 
 bushel; and 25 bushels of rye, worth 70 cents a bushel. 
 What is the value of a bushel of the mixture ? 
 
 
 OPERATION. 
 
 cts. 
 
 bush. 
 
 cts. 
 
 40 
 
 X 10 = 
 
 400 
 
 50 
 
 X 15 = 
 
 750 
 
 70 
 
 X 25 = 
 
 1750 
 
 
 50 ) 
 
 2900 
 58 cts. 
 
 * Alligation Medial is sometimes called Medial Proportitn. 
 
ART. 182.] ALLIGATION MEDIAL. . xTl 
 
 cts. hush. ct.. EXPLANATION. 
 
 40 X 10 = 400 )• 10 bush, at 40 cts. a bushel is worth $4-00. 
 
 50 X 15 = 750 } 15 bush, at 50 cts. a bushel is worth $7'50. 
 
 70 X 25 = 1750 }■ 25 bush, at 70 cts. a bushel is worth $17*50. 
 
 50 ) 2900 } $29 is the entire cost of the mixture^ which 
 
 58 cts l>6i°g divided by 50, the whole number 
 
 of bushels, gives 58 cents, the average 
 
 value of 1 bushel. 
 
 2. A wine merchant mixed together 40 gallons of wine, 
 at 80 cents a gallon; 25 gallons of brandy, at ^0 cents a 
 gallon; and 15 gallons of wine, at $1*50 a gallon. What 
 is the value of a gallon of the mixture ? 
 
 3. A grocer mixed 80 gallons of rum, worth 30 cents a 
 gallon; 40 gallons of whiskey, worth 40 cents a gallon; 
 and 20 gallons of water, at the usual price. What is the 
 value of a gallon of the mixture ? 
 
 • 4. A grocer mixed 120 pounds of sugar, worth 5 cents 
 a pound; 150 pounds, worth 6 cents a pound; and 130 
 pounds, worth 10 cents a pound. What was the average 
 value of a pound of the mixture ? 
 
 5. A grocer sold 50 barrels of flour, at $7'20 a barrel ; 70 
 barrels, at $8*20 a barrel; and 80 barrels, at $5-70 a barrel. 
 How much on an average did he receive for a barrel ? 
 
 ALLIGATION ALTERNATE. 
 
 Art. 181. Alligation Alternate teaches the method 
 of finding how much of several ingredients, the values of 
 which are known, must be taken to make a compound of 
 a certain value. 
 
 CAsr I. 
 
 Art. 182. Given the values of several ingredients, to 
 make a compound of a given value. First, Place the several 
 values of the ingredients in a column, and the average value 
 on the left of this column. Join with a curved line, the value 
 of each ingredient that is less than the average value, with one 
 or more that is greater ; then place the difference between the 
 value of each ingredient and the average value, opposite the 
 price of the ingredient with which it is joined, and this dif 
 
1*72 PROPORTION. [chap. VII. 
 
 ference, or the sum of these differences, (if there is more than 
 one,) will he thz quantity required of that ingredient. 
 
 1. How much sugar worth 6, 8, and 10 cents a pound, 
 must be mixed together, so that a pound of the mixture 
 may be worth 7 cents ? 
 
 (10^ = 
 
 OPERATION. 
 
 = 1 -^ 3 = 4 of the sugar, at 6 cts. a pound. 
 1 " " " 8 cts. " 
 
 1 " " " 10 cts. " 
 
 Explanation.— By taking one pound of each kind of the 
 sugar, we shall receive on the 10 cent quality, 4 cents more 
 than the average price of the mixture, and on the 6 cent qual- 
 ity 1 cent less than the average price. The gain and the loss 
 on the different qualities of sugar are to be equal ; therefore 
 the quantities taken must be universally proportional to the 
 gain and the loss on the respective qualities. 
 
 Remark. — Questions of this kind admit of an iruhfinitt number of answers. " 
 It is obvious if we take any other quantities which are to each other, as 4, 1 
 and 1 : as 8, '2 and 2 ; 12, .3 and 3. &c., that they will each satisfy the condi- 
 tion of the question equally well. 
 
 It is evident that there may be as many answers of diflerent ratios, as there 
 are methods of connecting the several values of the ingredients. For example: 
 
 2. How many pounds of tea, at 5, 6, 9, and 12 shillings 
 4 pound, must be mixed, so that the mixture shall be 
 worth 8 shillings a pound ? 
 
ART. 183.] ALLIGATION ALTERNATE. ITS 
 
 3. How much wine, at $110 per gallon, 60 cents per 
 gallon, and 40 cents per gallon, must be mixed together, 
 so that the mixture may be worth 80 cents per gallon ? 
 
 4. How much wine, at $r60 a gallon, and water, at the 
 usual rate, must be jnixed together, so that the compound 
 may be worth $ri5 a gallon. 
 
 5. How much of each sort of grain, at 46, 54, t5, and 
 85 cents a bushel, must be mixed together so that the 
 compound may be worth 65 cents a pound ? 
 
 CASE II. 
 
 Art. 183. When one of the ingredients is limited to a 
 giyen quantity. 
 
 1. A merchant wishes to mix 60 pounds of tea, worth 
 $1"20, with three other kinds, worth $110, 70 cents, and 
 60 cents, a pound, respectively, so that the mixture may 
 be worth $0-80 a pound. How many pounds of the last 
 three kinds must be used ? 
 
 OPERATION. 
 
 go I 110x^ = 10' 3 _J 30' " " SMO 
 
 r 120-^ =20 ^ r 60 pounds, worth $120 a pound. 
 
 J llOx \ = 10 ' ^ o _1 30 " " SMO 
 
 1 70>'y = 30 f ^ '^ — 1 90 " " $0-70 
 
 [ 60-^ =40 J [120 " '' $0-60 •' 
 
 Explanation. — By Case 1, we obtain 20, 10, 30, and 40 
 pounds, respectively, which meets the requirements of the 
 question, were neither of the quantities limited ; but there is to 
 be 60 pounds of that which is worth $1*20 a pound. We there- 
 fore, multiply the 20 opposite the Sl-20 by such a number as 
 will cause the product to become 60, which I find to be 3, and 
 io preserve the value of the mixture the same per pound, we 
 multiply all the other proportional quantities by the same 
 number. 
 
 2. How much oats, at $*40 a bushel ; barley, at $'45; 
 and corn, at $'75, must be mixed with 60 bushels of rye, at 
 $'85 a bushel, so that a bushel of the mixture may be worth 
 $•60? 
 
 3. How much sugar, at 5, 8, and 10 cents a pound must 
 be mixed with 64 pounds, at 12 cents a pound, so that the 
 mixture may be worth 9 cents a pound ? 
 
114 PBOPORTION. [chap VII 
 
 4. A merchant has 40 pounds of tea, worth $150 a 
 pound, which he wishes to mix with four other kinds, worth 
 95, *I5, 60, and 40 ctjuts a pound respectively. Ilow much 
 must he take of each of these four kinds, so that the mix- 
 ture shall be worth 80 cents a pound,? 
 
 CASE III. 
 
 Art. 184. When the whole mixture is to consist of a 
 certain quantity. 
 
 1. A merchant has sugar worth 5, 6, 9, and 12 cents a 
 pound; — with a mixture of these he wishes to fill a hogs- 
 head that shall contain 220 pounds. How much of each 
 kind must he take, so, that the compound may be worth 
 8 cents a pound ? 
 
 OPERATION. 
 
 r 5-^ =4 ] ( 88 pounds at 5 cents a pound. 1 
 
 8 t^zll 22= ^1 :: t :: :: Un. 
 
 il2-^=3J 166 '^ 12 « " J 
 
 "10)220 
 
 22 J Ratio of the sum of the proportionate quantities to 
 ( the number given. 
 
 Explanation — The sum of the proportionate quantities, 
 (found by Case 1.) is 10 ; the whole number of pounds that is 
 to compose the mixture, is 220 ; therefore, I must take ^f^* 
 times as much as the sum of these proportionals, which is 22 
 times each proportionate quantity. 
 
 2. How many gallons of water, brandy, and rum, must 
 be taken, so as to make a mixture of 90 gallons, worth 80 
 cents a gallon; providing the water is of no value, the 
 brandy being worth |1'20 a gallon, and the rum, 60 cents 
 a gallon ? 
 
 Rkmark. — Archimedes employed the above in detecting the fraud respecting 
 the crown of Hiero, king of Syracuse. The king had ordered a crown of i)ure 
 gold to be made ; but s-ispecting his artist to have mixed alloy with it, he re- 
 quested Archimedes to d«^termine the fact without injuring the crown. To 
 do this, Archimedes tookr a piece of pure gold, and another of alloy, each 
 equal in weight to the crown, placing them respectively in a vessel filled 
 with water and observing the quantity of water expelled by each he readily 
 determined that the crown was composed of gold and alloy ; also the exact 
 proportion in which these ingredients were utfed. 
 
ART, 18 T.J PERCENTAGE. 175 
 
 3. Suppose the weight of the crown and of each mass 
 to be 10 pounds ; and that being placed in water, the alloy 
 expelled "92 lbs., the gold .52 lbs., and the crown "64 lbs. 
 Of how much gold, and of how much alloy, did the crown 
 consist ? Ans. 3 lbs. of alloy, and 7 lbs. of gold. 
 
 OPERATION. 
 
 •40)10-00 
 25 
 
 CHAPTER VIII. 
 PERCENTAGE 
 
 Art. 185. The term per cent, is derived from the 
 Latin words per and centum, which signify, by ike hundred. 
 Percent, therefore, is any sum or number on a hundred, 
 whatever be the denomination. Thus, 5 per cent., signifies 
 
 5 for every hundred, or 5 hundredths ; 8 per cent, signifies 
 8 for every hundred, or 8 hundredths, &c. 
 
 We have already learned that hundredths can be ex- 
 expressed either as a wnmon or as a decinal fraction ; thus, 
 
 6 hundredths =yf^ = .05; 8 hundredths^ yf- = .08, &c. 
 In all our calculations in percentage, the rate per cent, is 
 written in the decimal form. 
 
 Art. 186. Percentage is extensively used in mercan- 
 tile transactions, and more or less in the transactions of 
 all other kinds of business ; such as Assessment of Taxes, 
 Insurance, Puties, Profit and Loss, Interest, Discount, 
 &c., &c. 
 
 Art. 187. Finding the percentage on any sum or 
 quantity. 
 
 1. What is 5 per cent, of 225 barrels of sugar ? 
 
It6 PERCENTAGE. [CHAP. VIIl, 
 
 OPERATION. 
 225 
 
 .05 
 
 1125 
 
 It may also be solved thus ; 5 per cent, is yl^ = oV of the 
 given quantity. Therefore, 3^ of 225 barrels = 11-25 barrels, 
 is 5 per cent, of 225 barrels. 
 
 2. What is 6 per cent, of $140 ? 
 
 3. What is 8 per cent, of $340 ? 
 
 4. What is 35 per cent, of $380 ? 
 
 5. What is 47 per cent, of $160'35 ? 
 
 6. What is 12| per cent, of 146 yards of cloth ? 
 
 I. What is 14| per cent, of 864 gallons of molasses ? 
 
 8. What is 16| per cent, of 8472 barrels of flour ? 
 
 9. A man, having $9684, lost by an investment 12| 
 per cent, of it ; how much had he remaining ? 
 
 10. Bought 24 head of cattle at $25 a head, and sold 
 them, at 25 per cent, advance ; how much did I gain ? 
 
 II. A merchant having $8645, gave 14 per cent, of it 
 for silks; 28 per cent, of it for flour; 43 per cent.^of it 
 for broadcloth; and the remainder for sugar. How many 
 dollars did he spend for each ? 
 
 12. A farmer raising 98 T bushels of wheat, gives 9 per 
 cent, of it forgathering it; 10 per cent, of the remainder 
 for thrashing; and 10 per cent, of what now remains for 
 flouring. How much has he remaining 2 
 
 13. A merchant bought 563 barrels of cider for $2837; 
 and sold 45 per cent, of it, at $6-85 a barrel ; 35 per cent. 
 of it, at $7'12i a barrel; and the remainder for what it 
 cost. How much did he gain by the operation ? 
 
 14. A speculator invested $8640 in a speculation, and 
 lost 25 per cent.; he then invested the remainder in a 
 speculation and gained 15 per cent. ; he now invested this 
 amount in speculation and gained 24 per j3ent. How 
 much did he make by the operation ? 
 
 Insurance. 
 Art 188. Insurance is an agreement by which a 
 
AKT. 189.1 STOCKS, BROKERAGE AND COMMISSION. Ill 
 
 company, or individuals, 'obligate themselves to make good 
 any loss o: damage of property by fire, shipwreck, or other 
 casualties. 
 
 The written agreement of indemnity issued by the In- 
 surers, sometimes called the underioriters, to the persons 
 whose property is insured, is called the Policy. 
 
 The insurance is effected in consideration of a sum of 
 money, called a Premium, which is estimated at a certain 
 rate per cent, on the amount insured, and is paid before- 
 hand, to the insurers. 
 
 1. If A gets his ship and cargo insured for $86950, 
 from New York to Liverpool, at 2 per cent.; how much 
 will be the amount of the premium ? 
 
 2. An insurance of $18640 was effected on the ship 
 Baltic, at 2^ per cent. How much did the premium 
 amount to ? 
 
 3. A dwelling, yalued at $1485, was insured, at | of 1 
 per cent. How much was the premium ? 
 
 4. A steamboat, valued at $55016, has an insurance 
 effected on | of its value, at 3| per cent. How much is 
 the premium ? 
 
 5. A gentleman has his dwelling insured for $8640, at 
 29 cents on $100. What is the premium ? 
 
 6. A person at the age of 40, effects an insurance on 
 his life for 3 years for the sum of $12800, at the rate of 
 $1"95 on $160 per annum. How much is the annual pre- 
 mium ? 
 
 t. An individual, going to California with the intention 
 of returning at the expiration of 3 years, effects an insu- 
 rance of $9988 on his life, at i of | of 1^ per cent, per 
 annum. How much is the annual premium ? 
 
 Stocks, Brokerage and Commission. 
 
 Art. 189. Stocks are government funds, and the cap- 
 ital of incorporated institutions, such as banks, railroad 
 and manufacturing companies, &c. Stocks are divided 
 into shares, usually varying from $50 to $500 each, the 
 market value of which is at times variable. 
 
 8* 
 
178 PERCENTAGE. [cHAP. VIII, 
 
 The 'par value of a share is its original cost. When it 
 sells for more thaa its original cost, it is said to be abi/oe 
 par, or at an advance ; when it sells for less, it is below 
 par, or at a discount. 
 
 The rise oi fall in stocks is computed at a certain per 
 cent, on the par value of the shares. 
 
 Art. 190. Brokerage is the percentage paid to bro- 
 kers, or dealers in stocks, money, bills of credit, and for 
 the transaction of business. 
 
 Art. 191. Commission is the percentage paid to agents 
 and commission merchants, for the purchase, sale, or care 
 of property, and for the transaction of other business. 
 
 The rate per cent, of Brokerage or Commission, varies 
 in different places, and depends upon the nature of the 
 business transacted. 
 
 1. What will $9864 par value of bank stock cost, at 
 18 per cent, advance ? 
 
 Remark.— Find 18 per. cent, of $9864 and add it to the $9864 ; the sum 
 will be the amount required. 
 
 2. How much must be given for 25 shares in the Hud- 
 Boon River Railroad, at 12^ per cent, advance, the shares 
 being $340 each ? 
 
 3. What is the value of 27 share? of canal stock, at 
 18f per cent, advance, the shares being $150 each ? 
 
 4. How much will be the cost of 18 shares of bank 
 stock, at I7f per cent, below par, the shares being $240 
 each ? 
 
 5. Bought 87 shares of a certain stock, at 13^ per cent, 
 below par, and sold the same, at 17f per cent, above par; 
 how much did I gain, the original shares being $184 
 each ? 
 
 6. A gentleman paid a broker f of 1 per cent, to invest 
 $84860 in government funds. How much was the bro- 
 kerage ? 
 
 7. A lady, having $84847, paid an agent If per cent, 
 commission a year, to take care of it for her. To how 
 much did the ommission annually amount? 
 
 8. An acent sells 8484 barrels of flour, at $5-87i a 
 
ART. 192.J CUSTOM HOUSE BUSINESS. It9 
 
 barrel, and charges If per cent, commission. How much 
 money must he pay to his employer after retaining his 
 commission ? 
 
 9. A merchant, having 8646 barrels, gave an agent 2f 
 per cent, commission for selling it. How much did the 
 merchant receive, after deducting the commission, if it 
 were sold, at$15'8T^ a barrel ? 
 
 10. A bank, failing, has in circulation $984840, and is 
 able to pay only 87^ per cent. How much money has the 
 bank on hand ? 
 
 11. A broker in New York exchanged $87846 on a 
 certain bank in Ohio, for f per cent. How much was the 
 brokerage ? 
 
 12. A merchant in Cincinnati sends to a commission 
 merchant in New York $4536"42 to lay out in goods, after 
 reserving bis commission, which was 5 per cent. How 
 much was his commission ? 
 
 Solution. — It will be understood that the agent receives 5 
 per cent, or jf^^ = -^\ of the money laid out for goods only, 
 and not of his commission ; therefore, if to gV? (his commis- 
 sion,) we add |^; (the money expended for goods.) we have 
 1^ equal to the sum of the commission and amount paid for 
 the goods, which is §4536-42. Hence, ^V of the money paid 
 for the goods, (which equals the commission,) is -^j of $4536-42 
 = $20602; and ||},- the amount paid for goods, is 20 times 
 $206-02 = $4120-40. 
 
 13. A farmer sends to a broker $84&t2, to be invested 
 in government funds; after deducting the brokerage which 
 was, at 4 per cent, on the amount invested. How much 
 was invested, and how much was the brokerage ? 
 
 ^14. A commission merchant receives $14760 to pur- 
 chase silk, with what remained after deducting his com- 
 mission of 2i per cent. How many pieces of silk did he 
 buy, providing it was $32 a piece ? 
 
 Custom House Business. 
 
 - Art. 192. Duties are taxes levied by government on 
 goods imported. 
 These duties constitute the revenue of the country, and 
 
180 PERCENTAGE. 
 
 are collected bj Custom House officers, at the ports of 
 entry. 
 
 Duties are specific or ad valorem. A specific duty is a 
 certain sum imposed, on a ton, cwt., hogshead, bushel, 
 yard, &c., regardless of the value of the commodity. 
 
 An ad valorem duty is a certain percentage, on the cost 
 of the articles in the country from which they are im- 
 ported. 
 
 Gross weight is the entire weight of the commodity, 
 together with the cask, box, or bag, &c., containing it. 
 
 Tare is an allowance made for the weight of the cask, 
 box, or bag, &c., containing the meichaudise. 
 
 Draft is an allowance for waste. Leakage is an allow- 
 ance of 2 per cent, for the waste of liquors in transpor- 
 tation. 
 
 Net weight is what remains after all deductions. 
 
 The usual allowance for draft is as follows: — 
 
 
 lbs. 
 
 lb. 
 
 lbs. lbs. 
 
 On 
 
 112 
 
 1 
 
 From 336 to 1120 4 
 
 From' 
 
 112 to 224 
 
 2 
 
 '' 1120 to 2016 7 
 
 u 
 
 244 to 336 
 
 3 
 
 More than 2016 9 
 
 Note. — The draft although it is not mentioned in the question, must be do- 
 ductad. before the other stated allowances are made 
 In ad valorem duties no deduction is made. 
 
 Art. 193. To find the specific duty on goods. 
 
 From the given quantity deduct all allowance, and multi- 
 ply the remainder by the duty on a unit of the given quantity. 
 The product will he the required duty. 
 
 1. What is the duty on 12 barrels of sugar, each weigh- 
 ing 115 pounds gross, at 1^ cents a pound; tare 20 per 
 cent. ? 
 
 OPERATION. 
 
 Gross weight, 2100 lbs. 
 
 Draft subtracted, 9 lbs. 
 
 2091 lbs. 
 20 per ct. of 2091 lbs. tare, 418-2 
 
 Net weight, 1672^ lbs. x '01^ = $29274, duty. 
 
ART. 194.] ASSESSMENT OF TAXES. 181 
 
 2. What is the duty on 4 hogsheads of sugar, each 
 weighing 1280 lbs. gross, at 2f cents a pound j tare 14 
 per cent. ? 
 
 3. What is the duty on 420 bags of coffee, each weighing 
 240 pounds, at 3 cents a pound ; tare, 3 per cent. ? 
 
 4. What is the duty on 210 bags of coffee, the gross 
 weight of each bag being 190 lbs., invoiced* at 5 cents a 
 pound; the tare being 5 per cent., and the duty 25 per 
 cent. ? 
 
 5. When there is a duty on tea, of 10 cents a pound, 
 what must be paid on 45 chests, each weighing 120 lbs. ; 
 tare 10 per cent. ? 
 
 6. At 35 per cent, ad valorem, what will be the duty 
 on 436 yards of satin, at $r75 a yard ? 
 
 7. What is the duty on 85 bags of pepper, each weighing 
 140 lbs. gross, invoiced at 6^ cents a pound, at 3^ per 
 cent. ; tare 5 per cent. ? 
 
 8. What is the ad valorem duty, at 31i per cent., on 40 
 pieces of silk, each containing 35 yards, invoiced at $2'25 
 a yard ? 
 
 9. What is the duty, at 18 cents a gallon, on 15 casks 
 of wine, each containing 75 gallons ? 
 
 10. What is the ad valorem duty, at 62^ per cent., on 
 a case of silks, invoiced at $95800 ? 
 
 11. What is the duty on 10 barrels of Spanish tobacco, 
 each weighing 145 lbs. gross; tare 8 percent., at 6| cents 
 a pound ? 
 
 12. What is the duty, at 40 per. cent, ad valorem, on 15 
 cases of French* broadcloth, each case containing 25 pieces, 
 and each piece 35 yards, invoiced at $3 95 a yard ? 
 
 Assessment of Taxes. 
 
 Art. 194. Taxes are moneys paid by the people, to 
 defray government expenses. Taxes are assessed on the 
 citizens in proportion to their real estate-\ and personal pro* 
 
 * An invoice is a list of the articles imported, and the cost thereof, 
 t Real Estate is immovable property, as lands, houses, &c. 
 
182 PEKCENTAGE. [cHAP. VIII. 
 
 'perty,^ except the poll-tax, which is so much for each male 
 individual over 21 years of age, regardless of his property. 
 
 Before taxes are assessed, an inventory of all taxable 
 property in the state, county, or town in which they are 
 to be paid, must be made ; together with a list of the 
 number of individuals liable to pay a poll-tax. 
 
 Then, from the sum to be raised, subtract the amount 
 of the poll-taxes, and divide the remainder by the amount 
 of taxable property, which will give the sura to be paid on 
 $1, and multiply this sum,, expressed in decimals, by each 
 man's inventory, and the product will be the tax on his 
 property. 
 
 1. A tax of $840't5 is to be raised in a town containing 
 65 polls. The taxable property in the town amounts to 
 $4^00. Each poll-tax is 0.75. What will be A's tax, 
 whose property is valued at $375, and who pays one poll t 
 
 Ans. $6-94 nearly. 
 
 OPERATION. 
 
 $840-75 the tax to be raised. 
 48-75 the amount of poll-taxes. 
 
 $792-00 Remainder. 
 
 $48*75 the amount of poll-taxes. 
 
 mh = '0165 the tax on $1. 
 375 X -0165 = $6- 1875 tax on property. 
 -75 poll-tax. 
 
 $6-9375 A^mount. 
 Explanation. — We find the amount of the poll-taxes to be 
 65 X $'75 = $4875, which we deduct from $84075, and have 
 $792. If on $48000 there are $792 taxes to be paid, on $1 
 there must be paid j^^^j^ of $792 = $00165, and on $375, A's 
 inventory, 375 times $0165 = $6-1875. This being increased 
 by 1 poll-tax = $6-94, A's tax. 
 
 Remark. — After having determined the amount to be paid on $1, the work 
 of determining the tnx of each particular individual may be facilitated by 
 forming the following table. If we desire to find the tax on $600. remove the 
 decimal point in the tax on $.5 two places to the right, and we have $S :25, tha 
 
 ♦ Personal Property h that which is movable, as money, furniture, cattle, &c 
 
A.RT, 
 
 195.] 
 
 PROFIT AND LOSS. 
 
 183 
 
 tax on $500. The pupil will readily understand the application of this table, 
 and will also perceive that it is the best one that can be formed, although not 
 the one usually given by arithmeticians. 
 
 
 $ 
 
 
 S 1 $ 
 
 $ 
 
 Tax on 
 
 1 
 
 is 
 
 •0165 
 
 Tax on 11 
 
 8 -1815 
 
 (( 
 
 2 
 
 (( 
 
 •033 
 
 " 12 
 
 " -198 
 
 a 
 
 3 
 
 u 
 
 •0495 
 
 " 13 
 
 " ^2145 
 
 (( 
 
 4 
 
 u 
 
 •066 
 
 " 14 
 
 " -231 
 
 u 
 
 5 
 
 u 
 
 •0825 
 
 " 15 
 
 ' -2475 
 
 u 
 
 6 
 
 C( 
 
 •099 
 
 " 16 
 
 ' -264 
 
 11 
 
 7 
 
 (( 
 
 •1155 
 
 u 17 
 
 " -2805 
 
 cc 
 
 8 
 
 " 
 
 •132 
 
 " 18 
 
 " -297 
 
 (I 
 
 9 
 
 u 
 
 •1485 
 
 " 19 
 
 " -3135 
 
 u 
 
 10 
 
 u 
 
 •165 
 
 
 
 2. By the above table, what would be the tax on $984, 
 there being 1 poll ? 
 
 3. If I pay 4 polls, and am worth |1718-40, how much 
 is my tax ? 
 
 4. How much is that man's tax, who pays 2 polls, and 
 is worth $284-86 ? 
 
 5. . How much is that man's tax, who pays 3 polls, and 
 is worth $8972-50 ? 
 
 6. How much is that man's tax, who pays 5 polls, and 
 is worth $1784-84? 
 
 7. How much is that man's tax, who is worth $1984-35, 
 and pays 2 polls ? 
 
 Profit and Loss. 
 
 Art. 195. Profit and Loss refer to the amount which 
 the merchant or other business man, gains or loses in busi- 
 ness transactions. 
 
 1. Bought 47 barrels of sugar, at $14-87i a barrel, and 
 sold it at $16-121 a barrel. How much did I gain ? 
 ' 2. Bought 184 cords of wood, at $8T8f a cord, and 
 sold it, at $4 '371 a cord. How much did I gain by the 
 operation ? 
 
 3. Bought 387 barrels of flour, at $5-93^ a barrel, and 
 sold it, at $6*7^ a barrel. How much was the gain ? 
 
 4 Bought 16 barrels- of sugar, each containing 195 
 
184 PERCENTAGE. ( CHAP. VI^ 
 
 pounds, at $13"84 a barrel, and sold it for $.09| a pound. 
 How much was the gain ? 
 
 5. Bought flour, at $6*20 a barrd, and sold it so as to 
 gain 20 per cent. ; for how much did I sell it a barrel ? 
 
 Solution If on 100 cents I gain 20 cents., on Icent. I will 
 
 gain y|^ of 20 cents = ^^,5_ = | of a cent. Therefore, I gain ^ 
 of what it cost, } of $6-20 = $1*24, which added to the cost 
 equals $7*44, what I must sell it for. Or, 
 
 Find 20 per cent, of $6-20 ; thus, $6-20 X '20 = $1-24; to 
 which add the cost, and we have $7*44, what it must.be sold 
 for a barrel. 
 
 6. Bought broadcloth, at $5*85 a yard, and sold it so 
 as to gain 25 per cent.; for how much did I sell it a 
 yard ? 
 
 t. A horse was bought for $285-^5; for how much 
 must it be sold to gain 20 per cent. ? 
 
 8. A merchant bought 185 barrels of pork, at $18-95 a 
 barrel ; but it becoming damaged, he was obliged to lose 
 35 per cent, on the sale of it. How much did he receive 
 for it all ? 
 
 9. A merchant bought 25 pieces of silk, each contain- 
 ing 37| yards, for $675'40, and sold it so as to gain 33^ 
 per cent. For how much did he sell it a yard ? 
 
 10. A quantity of butter was bought for $150, and sold 
 for $200 ; how much was the gain per cent. 
 
 Solution.— On $150 the gain is $200— $150 = $50. If on 
 $150 there is a gain of $50. on $1, the gain will be j\^ of $50 
 = j^-g\ = ^ of a dollar, or 33^ per cent. 1 
 
 11. A gentleman invested 4280 in speculation, and at 
 the end of a year realized $5350; how much per cent, did 
 he gain ? 
 
 12. A horse was bought for $240, and sold for $400; 
 how much was the gain per cent ? 
 
 13. A gentleman sold a horse for $150, and thereby 
 gained 25 per cent.; how much did the horse cost him ? 
 
 Solution. — If he gained 25 cents on 100 cents, on 1 cent, he 
 gained y^gr = j of a cent. Therefore, he gained \ of what the 
 horse cost him, which added to |, the cost of the horse, = | of 
 
ART. 195. J PRACTICAL QUESTIONS. 185 
 
 the cost of the horse, which is equal to $150, what he sold 
 the horse for ; and ], = | of $150 = $30 ] and |, the cost of 
 the horse, = 4 times $30 = $120. 
 
 14. A. quantity of salt was sold for $864, which waa 
 331 per cent, more than it cost him; how much did it cost 
 him? 
 
 15. If in 1 year the principal and interest of a certain 
 note, at 9| per cent., amount to $12000. How much was 
 the face of the note ? 
 
 16. A quantity of rye was sold for $1896, which was 
 18| per cent more than it cost. How much did it cost ? 
 
 PRACTICAL QUESTIONS IN PROFIT AND LOSS. 
 
 1. If I buy 218 yards of broadcloth, at $4'64 a yard, 
 
 and sell it at $6"95i a yard; how much do I gain by the 
 operation ? 
 
 2. If I pay $846 for a quantity of wheat ; for what 
 must I sell it to gain 23^ per cent. ? 
 
 3. Sold 149 barrels of cider, at $4-8tj a barrel, and 
 thereby gained 37| per cent. What did it cost a barrel ? 
 
 4. Bought 480 gallons of molasses, at 28 cents a gallon, 
 and sold it for $168. How much did I gain per cent. 
 
 5. A house that cost $1500, was sold for $1250. What 
 was the loss per cent. ? 
 
 6. A farm that cost $6500, was sold for $9100. What 
 was the gain per cent. ? 
 
 7. Bought raisins, at $3 a box; how much will be the 
 loss per cent, if I sell it, at $2*50 a box ? 
 
 8. Sold 280 yards of cloth for $*I00, and thereby gained 
 25 per cent., for how much should I have sold it a yard, 
 to lose 20 per cent. ? 
 
 9. If I sell 15 yards of broadcloth for $66, and thereby 
 gain 10 per cent., how ought I to have sold it a yard to 
 have lost 25 per cent. ? 
 
 10. A quantity of wheat was sold for $3 60' 90, which 
 was 10 per cent, less than its original cost.; what would 
 have been the gain per cent, if it had been sold for 
 $450-15? 
 
186 • PERCENTAGE. [cHAP. VIII 
 
 11. Sold 45 boxes of damaged raisins for $103-50, which 
 was at a loss of 8 per cent. ; how should I have sold them 
 a box to have gained '3 per cent. ? 
 
 12. A house and lot was sold for $2t00, which was 
 8 per cent, more than its value; what would have been 
 the gain per cent, if it had been sold for $28333 ? 
 
 13. A mechanic built a house for $1980, which was 10 
 per cent, less than what it was worth ; how much should he 
 have received for it so as to have made 37^ per cent. ? 
 
 14. A gentleman sold two farms for $3680 a piece; for 
 one he received 25 per cent, more than its value; and for 
 the other, 25 per cent, less than its value. Did he gain or 
 lose by the operation, and how much ? 
 
 15. A merchant sold two boxes of goods for $540 a 
 piece; on one he gained 20 per cent, and on the other he 
 lost 20 per cent. Did he gain or lose by the operation, 
 afud how much ? 
 
 16. A speculator sold two building lots for $1200 a 
 piece, on one he received 37^ #per cent, more than it was 
 worth, and on the other 25 per cent, less than what it was 
 worth. Did he gain or lose, and how much ? 
 
 SIMPLE INTEREST. 
 
 Art. 196. Interest is money due for the use of money 
 or its equivalent ; and is estimated at a certain rate "per 
 cent, 'per a7mum, which is generally fixed by law. 
 
 The Principal is the sum on which the interest is paid. 
 
 The Amount is the sum of the principal and interest. 
 
 By t. per cent, is meant 1 cents on 100 cents, $7 on $100, 
 or 7 OTi'lOO, whatever be the denomination. 
 
 The rate per cent, is different in different States. In 
 the State of New York it is *I per cent., and in the New 
 England States it is 6 per cent., &c. 
 
 CASE I. 
 
 1. What is the interest on $68C for 6 years, at t per 
 cent. ? 
 
ART. 196.] SIMPLE INTEREST. 187 
 
 OPERATION. 
 
 $•07 int-. of $1 for 1 year. 
 
 $•42 « " " " 6 years. 
 680 
 
 3360 
 252 
 
 $285-60 int. of $680 for 6 years, at 7 per cent. 
 
 Explanation. — If the interest of $1 for 1 year is 7 cents, 
 the interest for 6 years will be 6 times 7 cents, equal to 42 
 cents. If the interest of §1 is 42 cents, the interest of $680 
 is 680 times $42, equal to $285 60. 
 
 Remark. — Much care should be taken to keep the decimal point in its 
 proper place. 
 
 2. What is the interest of $4t0 for 4 years, at T per 
 cent. ? 
 
 3. What is the interest of $683 for 2^ years, at 6 per 
 cent. ? 
 
 4. What is the interest of $846-4t for 3f years, at 1 
 per cent. ? 
 
 5. What is the interest of $86*42 for 3^ years, at 8 per 
 cent. ? 
 
 6. What is the interest of $224-45 for 6| years, at 6 per 
 cent. ? 
 
 1. What is the interest of $249'98 for 4f years, at t 
 per cent. ? 
 
 8. What is the interest of $1*84 for 1 years, at 5^ 
 per cent. ? 
 
 9. What is the interest of $163^^ for 3| years, at 6^ 
 per cent. ? 
 
 10. What is the interest of $215-12^ for 4f years, at 
 8| per cent. ? 
 
 CASE n. 
 
 To find the interest on any sum of money, for any given 
 time, at 6 per ceE(%. 
 
188 PERCENTAGE. [CHAP. VIII. 
 
 The interest of $1 for 12 months, (or 1 year.) is $00G, 
 which is equal to half the number of months. Therefore, half 
 yf the number of months equals the interest, in cents, of %\ for 
 Ihesame number of montJis. The interest of $1 for 12 months 
 being $006, the interest for 2 months. (= y^^; or ^ of a year,) 
 is I X $006 = $0-01: Again, 6 days is = /^, or ^V of 2 
 months of 30 days each ; therefore, the interest of $1 for 6 
 days is y^ X $001 = $0'001. Therefore, one-sixth of the num- 
 ber of days equals the interest^ in mills, of%lfor the same numbei 
 of days. 
 
 Hence, to find the interest of $1 for any given time, at 6 
 per cent. : 
 
 Call half the number of months cents, and one-sixth the 
 number of days, mills. The interest of $1 being found, 
 multiply it by the number of dollars in the given princi- 
 pal, and the product will be the interest requ'red. 
 
 1. What is the interest of $58t*36for 2 years 4 months 
 and 24 days at 6 per cent. ? "" 
 
 operation. 
 
 2 years 4 months = 28 months. Calling the half of the 
 28 months cents, we have ; 
 
 $.14, int. of $1 for 2 years and 4 months, at 6 per ceni. 
 Calling I of the 24 days mills, we have ; 
 
 $004, int. of $1 for 24 days, at 6 per cent. 
 Hence, $ -14, int. of $1 for 2 yrs. 4 mo. at 6 per cent. 
 •004, u a u u 24 days, at 6 per cent. 
 
 $ .144, 
 
 ( gives 
 |6per 
 
 the int. of 
 
 587-36 
 
 cent. 
 
 864 
 
 
 432 
 
 
 
 1008 
 
 
 
 1152 
 
 
 
 ■720 
 
 
 
 for 2 yrs. 4 mo. 24 days, at 
 
 $84-57984 j int. of $587-36^for the given time and the given 
 I rate per cent. 
 
 2. What is the interest of $84-25 for 1 year and 6 
 months, at 6 per cent. ? * 
 
iRT. 191.] SIMPLE INTEREST. 189 
 
 3. What is the interest of $184'50 for 3 years and 8 
 months, at 6 per cent. ? 
 
 4. What is the interest of $273-84 for 2 years and 9 
 months, at 6 per cent. ? 
 
 5. What is the interest of $84*1-80 for 4 years 1 months 
 and 12 days^ at 6 per cent. ? 
 
 6. What is the interest of $684-45 for 3 years 8 months 
 and 18 days, at 6 per cent. ? 
 
 T. What is the interest of $849-95 for 5 years 5 months 
 and 6 days, at 6 per cent. ? 
 
 Remark. — To find the Amount add the principal and interest together. 
 
 8. What is the amount of $684*45 for 2 years 3 months 
 and 18 days, at 6 per cent. ? 
 
 9. What is the amount of $483 '85 for 3 years 5 months 
 and 24 days, at 6 per cent. ? 
 
 10. What is the amount of $101-01 for 6- years 8 
 months and 14 days, at 6 per cent. ? 
 
 11. What is the amount of $849-&'7i for 2 years 9 
 months 25 days, at 6 per cent. ? 
 
 12. What is the interest of $88*88 for 4 years 11 
 months and 22 days, at 6 per cent. ? 
 
 CASE III. 
 
 Art. lOT. To find the interest on any given sum for 
 any given time, at any given rate per cent. First, find 
 the interest of $1 for the given time, at 6 per cent., (See 
 Case 2;) then take as many sixths of the interest as are equal 
 to the given per cent., which will be the interest of $1 for the 
 given time and rate per cent.; then multiply this interest ly 
 the principal. 
 
 If the interest is at t, 9, or 11 per cent., &c., it is 
 evident that, if to the interest of $1, at 6 per cent., we add 
 its J, I, or f, &c., it will give the interest of $1, at 1, 9, or 
 11, per cent., &c., respectively. If the interest is at 2^3, 
 or 5 per cent., &c.'; then, from the interest of $1, at 6]per 
 cent., we must take its |, |, or \, &c., which will give the 
 interest of $1, at 2, 3, or 5 per cent., &c., respectively. 
 
190 PERCENTAGE. [CHAP. VIII 
 
 1. What is the interest of $260 for 1 year 6 months and 
 18 days, at 8 per cent. ? 
 
 OPERATION. 
 
 $•093, int. of $1 for the given time, at 6 per cent. 
 •031, ""two-sixths of the above interest. 
 
 $•124, int. of $1 for the given time, at the given rate per cent 
 
 260 
 
 7440 
 248 
 
 $32-240 interest required. 
 
 2. What is the interest of $84*15 for 2 years and 10 
 months, at 1 per cent. ? 
 
 3. What is the interest of $65' 65 for 1 year 11 months 
 and 23 days, at 1 per cent. ? 
 
 4. What is the interest of $384'3ti for 2 years and 9 
 months and 16 days, at 8 per cent. ? 
 
 5. What is the interest of $284'95 for 3 years 8 months 
 and 20 days, at 1 per cent. ? 
 
 6. What is the interest of $84V3'?^ for 4 years 1 
 months, at T per cent. ? 
 
 I. What is the interest of $1284'62| for 2 years 10 
 months and 4 days, at 7 per cent. ? 
 
 8. What is the interest of $884*88 for 4 years 5 months 
 and 5 days, at 5 per cent. ? 
 
 9. What is the interest of $841*65 for 5 years 9 months 
 and 15 days, at 5 per cent. ? 
 
 10. What is the interest of $8484*84 for 1 years 4 
 months and 20 days, at 4 per cent. ? 
 
 II. What is the interest of $1465*811 for 8 years 8 
 months and 8 days, at 3 per cent. ? 
 
 Rkmark. — If the principal be given in English money, reduce the shillings, 
 pence and farthings, to the decimal of a pound ; then proceed as it Federal 
 money. 
 
 12. What is the interest of £S4: 10s Qd. for 3 years and 
 8 months, at 1 per cent. ? 
 
ART. 198.] SIMPLE INTEREST. 191 
 
 13. What is the interest of £U5 Us. Sd. for 2 years 
 and 6 months, at 7 per cent. ? 
 
 14. What is the interest of ^284 12^. lOd. for 1 year 
 8 months and 12 days, at 8 per cent. ? 
 
 15. What is the interest of ^£384 10.^. 6^^. for 3 years 
 8 months and 24 days, at t per cent, ? 
 
 Art. 198. The following method of computing interest 
 avoids the use of fractions, and may, therefore, be preferred 
 by some. 
 
 We shall in accordance with general usage, reckon 30 
 days to the month, and 12 months to the year. 
 
 1. What is the interest of $460 for 2 years 1 months, 
 at 9 per cent. ? 
 
 OPERATION. 
 
 $460 
 •09 
 
 $41-40, interest for 1 year. 
 31 
 
 4140 
 12420 
 
 12)1283-40 
 
 $106-95 interest required. 
 
 Explanation. — 1 find the interest of $460 for 1 year, (12 
 months.) at 9 per cent, to be $41*40. In the given time there 
 are 31 months. If the interest of $460 for 12 months is $41-40, 
 for 1 month it is jL as much : and for 31 months, it is 31 times 
 r^ = ^of: $41-40 = $106 95. Hence, to find the interest of 
 any sum, when the time is given in years and months, 
 
 Multiply the interest of the principal for 1 year hy the 
 number of months and divide the product iy l'2i. 
 
 For a similar reason, when the time is given in years, 
 months and days, 
 
 Multiply the. interest of the principal for 1 year hy the 
 number of days and divide the product by 360, the quotient 
 will be the interest required. 
 
192 PERCENTAGE. [cHAP. VIII. 
 
 It may be inferred from what has already been remarked, 
 that none of the preceding methods of computing interest is 
 strictly correct 5 however they are in general use. The follow- 
 ing correct method is adopted by many bankers and brokers. 
 
 Art. 199. Muliiply t/i£ interest of the principal for 1 
 year by the exact number of days it has been on interest, and 
 divide the product by 365, the quotient will be the interest 
 required. 
 
 2. What IS the interest of $t20 for 2*years 9 months 
 and 25 days, at 8^ per cent. ? 
 
 OPERATION. 
 
 $720 
 •08^ 
 
 5760 
 360 
 
 61 '20 interest for 1 year. 
 2 years 9 months 25 days = 1015 days. 
 
 30600 
 6120 
 6120 
 
 360)62118-00($172-55 interest required. 
 360 
 
 2611 
 2520 
 
 918 
 720 
 
 &c. 
 
 Remark. — The abov.e question is solved by the method given under Art. 
 198. The pupil should also solve the same and the following questions by 
 Art. 199, that he may discover the difl'erence between the correct and the 
 (ncorrect method of calculation. 
 
 3. What is the interest of $14-40 for 3 years 7 months, 
 at 7 per cent, ? 
 
 4. What is the interest of $25*20 for 4 years 5 months 
 and 17 days, at 9 per cent. ? 
 
ART, 200.] 
 
 SIMPLE lOTEREST. 
 
 19S 
 
 5. What is the interest of $100'80 for 5 years 9 months 
 and 20 days, at 5 per cent. ? 
 
 6. What is the interest of $201*60 for 3 years 1 months 
 and 25 days, at 6 per cent. ? 
 
 7. What is the interest of $403-20 for 3 years 8 months 
 and 8 days, at 8^ per cent. ? 
 
 8. What is the intereot of $806-40 for 4 years 5 months 
 and 21 days, at 3^ per cent. ? 
 
 9. What is the interest of $720 for 6 years 6 months 
 and 6 days, at 5| per cent. ? 
 
 10. What is the interest of $1440 for 1 year 9 months 
 and 15 days, at 8^ per cent. ? 
 
 Art. 200. Many prefer to calculate interest by mulH- 
 plying the principal by the rate per cent., and this product by 
 the number of years ; then add the interest for the months and 
 days, found by raeans of aliquot parts, to the last product. 
 
 TABLE. 
 
 ALIQUOT i>ARTS OF 
 
 A YEAR OR MONTH. 
 
 mo. yr. 
 
 days. mo. 
 
 2 = i^ 
 
 3 = y 
 
 til 
 
 10 = ^ 
 12 = f 
 
 18 = 1 
 
 9 = '( 
 
 10 = 1 
 
 20 = 1 
 
 11 = Ii 
 
 &c., &c. 
 
 Remark. — It is customary in the calculation of interest, to reckon 30 days 
 to the month, and 1*2 months to the year, although this is not true, as some of 
 the months contain more, and one of them less than 30 days ; hence, the results 
 obtained in these calculations are sometimes too large, and at other times too 
 small yet they are sufficiently correct for all practical purposes. But should 
 it he desired to compute the interest with more accuracy, it may be done by 
 finding the number of days the principal has been on interest, by the table, and 
 consider this number of days as such a part of 365, a year. {See Akt. 1'j9.) 
 
 ■ 1. What is the interest of $240*50 for 3 years 4 months 
 and 15 days, at 8|- per cent. ? 
 
 9 
 
194 
 
 PERCENTAGE. 
 
 [chap. VIII 
 
 OPERATION. 
 
 4 mo. = ^ yr. 
 
 15 days = i mo. 
 or I of 4 mo. 
 
 $ 240-50 
 
 •OSi 
 
 19-2400 
 
 1-2025 
 
 $20-4425 interest for 1 year. 
 3 
 
 $61-3275 interest for 3 years. 
 6 -81411 interest for 4 months. 
 -8517f interest for 15 days. 
 
 $68-9934-}- interest required. 
 
 1. What is the interest of $1200'12i for 6 years and 4 
 months, at 5 per cent. ? 
 
 2. What is the amount of $8t"95 for 2 years 3 montha 
 and 20 days, at 7 per cent. ? 
 
 3. What is the amount of $47*84 for 4 years 1 month 
 and 25 days, at Q^ per cent. ? 
 
 4. What is the amount of $144'44 for 3 years 6 months 
 and 18 days, at 7| per cent. ? 
 
 5. What is the amount of $650*30 for 3 years 7 months 
 and 12 days, at 7| per cent. ? 
 
 6. What is the amount of $460*40 for 4 years 8 months 
 and 15 days, at 8f per cent. ? 
 
 7. What is the interest of $640*12i from Jan. 24frh, 
 1840, to March 28th, 1841, at 6i per cent. ? 
 
 8. What is the interest of $485*9Hf from Feb. 5th, 
 1842, to Aug. 20th, 1844, at 7^ per cent. ? 
 
 9. What is the interest, at 5f per cent., of $846*84, 
 from Jan. 8th, until Nov. 20th ? 
 
 10. What is the interest, at 8| per cent., of $384*25 
 from Jan. 12th, 1853, to April 4th, 1854 ? 
 
 11. What is the amount of $144*45 from Aug. 29th 
 1852, to Nov. 28th, 1853 ? 
 
 12. What is the interest of $1200-121 from May 22Qd 
 1852, to Sept. 9th, 1854 ? 
 
 13. What is the interest, at 9| per cent, of $145*60 
 from July 14th, 1851, to Sept. 9th, 1853 ? ' 
 
ART. 201.] PROBLEMS IN INTEREST. 1&5 
 
 .14. What is the interest of $846-80 from Sept. 8th, 
 1847, to Aug. 8th, 1853 ? 
 
 1q. What is the interest of $t84-93f from Feb. 2nd, 
 1850, to April 24th, 1854 ? 
 
 PROBLEMS IN INTEREST. 
 
 Art. 201. The Principal, Time, Rate per cent., and 
 Interest, have such a relation to one another, that any 
 three of them being given, the remaining one can readily 
 be found by analysis. , 
 
 Note. — For a complete analysis of Interest, Discount and Percentage of every 
 description, see the last chapter in the "American intellectual Arithmetic." 
 
 Problem 1. — Given the rate per cent., time and interest 
 to find the principal. 
 
 1. What principal will, in 2 years and 6 months, at 6 
 per cent, give $6' 18 interest ? 
 
 Solution 2 years and 6 months equals | years. The in 
 
 terest of $1 for 1 year is 6 cents, and for J of a year, ^ of 6 
 cents = 3 cents ; and for 4 years, 5 times 3 cents = 15 cents. 
 If the interest on 100 cents is 15 cents, on 1 cent it is y|^ of 
 15 cts. = ^% = 2^ of a cent. Therefore, 2^,5^ of the principal 
 equals the interest, which is $6*18 ; and u\ of the principal = 
 i of S618 = $2-06, and |^, the principar= 20 times $2-06 = 
 $41-20. 
 
 Remark. — A similar method of analysis without further Uustration can 
 be readily applied by the pupil to all the following problems. 
 
 The interest on any sum is as many times greater than 
 the interest on $1, as that sum is greater than $1. Hence, 
 questions like the above ma,y be solved by dividing the 
 given interest by the interest of $1, at the given rate per cent, 
 for the given time. „ 
 
 2. What principal will, in 4 years and 9 months, at 8 
 per cent., give $19'38 interest ? 
 
 3. What principal will, in 3 years 8 months and 15 
 days, at 7 per cent., give $177-551 interest? 
 
 4. What principal will, in 4 years 9 months and 18 
 days, at 6 per cent., give $86-688 interest ? 
 
196 PERCENTAGE. [CHAP. VIII. 
 
 6. What principal will, in 10 years 10 months and 20 
 days, at 6|- per cent., give $1411653 interest ? 
 
 Problem 2. — Given the principal, the rate per cent., 
 and the interest, to find the time. 
 
 1. In what time will $26, at 6 per cent,, give $1*95 
 interest ? 
 
 Art. 202. The interest on a given principal is in pro- 
 portion to the time^ other things remaining the same. 
 Hence, to find the time, the other three things being given; 
 Divide the given interest by the interest of the given jprindjpal, 
 at the given rate 'per cent, for 1 year; or solve it by 
 Analysis. 
 
 2^In what time will $300, at 8 per cent., give $20 
 interest ? 
 
 3. In what time will $90-25, at 6 per cent., give $4't5 
 interest ? 
 
 4. In what time will $284"75, at 5f per cent., give 
 $18-^5 interest? 
 
 5. In what time will $114'95, at t^ per cent., give 
 $34-8ti interest? 
 
 Problem 3. — Given the principal, the tim€, and the 
 interest, to find the rate per cent ? 
 
 1. The interest of $65, for 10 months is $3 25. What 
 is the rate per cent. ? 
 
 Art. 203. The interest on a given principal is in pro- 
 portion to the rate per cent., other things remaining the 
 same. Hence, to find the rate per cent., the remaining 
 three things being given ; Divide the given interest by the 
 interest of the given principal, at 1 per cent., for the given 
 time. 
 
 2. The interest of $120 for 2 years 9 months and 12 
 days, is $13"36. What is the rate per cent. ? 
 
 3. The interest of $3t5 for 3 years and 6 months, is 
 $9t'125. What is Ihe rate per cent. ? 
 
AKT. 205.] DISCOUNT. 19T 
 
 4. The interest of $248 for 2 years 1 month and 20 
 days, is $29-194. What is the rate per cent. ? 
 
 5. The interest of $184-85 for two years 8 months and 
 18 days, is $31-84.- What is the rate per cent. ? 
 
 Problem 4. — Given the amount, time, and rate per cent., 
 to find the principal ? 
 
 1. What principal will, in 4 years 6 mouths, at 8 per 
 cent., amount to $430 ? 
 
 Art. 204. The amount of different principals, for the 
 same time, and at the same rate per cent., are to each 
 other as those principals. Hence, Dividing the given 
 amount by the amount of $1, at the given rate per ant., for 
 the given time, will give the principal. 
 
 2. What principal will, in t years and 6 months, at 8 
 per cent., amount to $2600 ? 
 
 8. What principal will, in 2 years and 4 months, at 6 
 per cent., amount to $640 ? 
 
 4. What principal will, in 5 years 8^ months, at 1 per 
 cent., amount to $2100 ? 
 
 5. What principal will, in 4 years 4. months, at 6 per 
 cent,, amount to $3800 ? 
 
 DISCOUNT. 
 
 Art. 205. Discount is an allowance, according to the 
 rate per cent., made for the payment of money before it is 
 due. 
 
 The present worth of a debt, payable at some future time, 
 without interest, is such a sum as will, in the given time, 
 and at the given rate per cent,, amount to the debt. 
 Hence, the present worth of any sum of money, payable at 
 some future time, without interest, is equal to the quotient 
 arising from dividing that sum hy the amount o/ $1, at the 
 given rate per cent., for the given time. 
 
 The Discount equals the amount, mm\xs,i\\Q present worth. 
 
 1. What is the present worth of $644, due 4 years 9 
 months and 18 days hence, at 6 per cent.? 
 
198 PERCENTAGE. [CHAP. VIII. 
 
 ExPLANA noN. — $1-288 is the amount of $1 for the given time, 
 and the given rate per cent. Now we have the proportion 
 $1288, amount: $644, amount:: $1, present worth : presgni 
 worth, required. This, solved gives $500 for the required pre- 
 sent worth. 
 
 2. What is the present worth of $840, due 3 years and 
 4 months hence, at 6 per cent. ? 
 
 3. What is the present worth of $1140, due 2i years 
 hence, at 6 per cent. ? 
 
 4. What is the discount on $450, due 2 years and 9 
 months hence, at t per cent. ? 
 
 5. What is the discount on $1200, due 3 years, 4 
 months hence, at 4f per cent ? 
 
 6. What is the discount on $84*25, due 3 years 8 months 
 and 24 days hence, at 8 per cent. ? 
 
 t What is the present worth of $9632, due 1 year 8 
 months and 12 days hence, at 6 per cent. ? 
 
 8. What is the present worth of $52 32, due 6 years 1 
 month 18 days hence, at 6 per cent. ? 
 
 9. What is the discount on $464*80, due 3 years 8 months 
 and 15 days hence, at 7 per cent. ? 
 
 10. Bought $984-45 worth of goods on a credit of 9 
 months. How much money would discharge the debt, at 
 the time of receiving the goods, interest being 9 per cent. ? 
 
 11. A merchant bought goods to the amount of $3328: 
 1 of \i) was on a credit of 6 months, and the remainder on 
 a credit of 9 months. How much money would discharge 
 the debt, interest being 8| per cent. ? 
 
 12. A merchant bought goods to the amouut of $2480: 
 $812 of which was on a credit of 3 months; $832, on a 
 credit of 8 months; and the remainder on a credit of 9 
 months. How much ready money would discharge the 
 debt, interest being 6 per cent. ? 
 
 13. A merchant bought goods to the amount of $1600: 
 i of which was on a credit of 3 months; ^ on a credit of 
 9 months ; and the remainder on a credit of 1 year. How 
 much ready money would discharge the debt, interest being 
 8 per cent. ? 
 
*rt. 206.] partial paiments. 199 
 
 Partial Payments. 
 
 Art, 206. Partial Payvients are payments, or indorse 
 ments,* made at various times, of a part of a note, hond^ 
 or obligation. 
 
 Tiie method adopted by the Supreme Court of th~e United 
 States for the calculation of interest on notes, and other 
 obligations, where partial payments have been made, is as 
 follows ; — 
 
 " Apply the payment, in the first place, to the discharge of 
 the interest then due. If the payment exceed the interest, tht 
 surplus goes towards discharging the principal, and the sub- 
 seque7Lt interest is to be computed on the balance of the principal 
 rtmaining due. If the payment be less than the interest, the 
 surplus of interest must not be taken to augment the principal ; 
 hut interest co7itinues on the former principal until the period 
 ivhen the payments taken together exceed the interest due, and 
 then the surplus is to be applied towards discharging the 
 principal ; and interest is to be computed on the balance, as 
 aforesaid." 
 
 1800. Bethany, Sept. 8th, 1850. 
 
 [1.] On demand, I promise to pay Thomas Brooking, 
 or bearer, eight hundred and sixty dollars, with interest. 
 Value received. John Jackson. 
 
 On this note are the following indorsements : — 
 
 Nov. 20th, 1851, received $382-24. 
 May 8th, 1853, " $23845. 
 
 How much is due Dec. 29th, 1853, allowing t per cent, 
 interest ? 
 
 Rkmark.— It will be of some assistance to the pupil to arrange the date of the 
 note, the payments, time of settlement, and the intervals of time between pay- 
 ments, together with the interest of $1 for the given time, at 6 per cent., as 
 follows : — 
 
 ♦ Deriv ?si from a Latin phrase signifying " upon the back ;" as the paymwits 
 are written across the back of the note. 
 
200 PERCENTAGE. [cHAP. VIIX. 
 
 Intervals 
 
 of time. Int. (f$l,at 
 
 years, mo. da. y. mo. aa raymer^ts. 6 per cent. 
 
 Date of note, . . 1850 9 8 
 
 1st payment, . 1851 11 20 12 12 $382-24 |0-072 
 
 2(i payment, . . 1853 5 8 1 5 18 238-45 0-088 
 
 Time of settlement, 1853 12 29 7 21 00385 
 
 Face of the note, or principal, . . . $86000 
 
 Int. on the same, at 7 per cent., to Nov. 20th; 1851, 7224 
 
 Amount due on the note, . " " " $932-24 
 
 First payment, ..... 382-24 
 
 Amount remaining due, — 2nd principal, . $550-00 
 
 Int. on the same, from Nov. 20, 1851. to May 8, 1853, 56-46 
 
 Amount due, May 8th, 1851, . . . $606-46 
 
 Second payment, .... 238-45 
 
 Amount remaining due,— 3rd principal, . . $36801f 
 
 Int. from May 8th, 1853, to the time of settlement, ' 16 -534- 
 
 Amount due Dec. 29th, 1853. . . $384-54| 
 
 $786//o . New York, Jan. 13th, 1848. 
 
 [2.] On demand, I promise to pay to the order of 
 JEenry Morton, seven hundred and thirty-six dollars, with 
 interest, at 7, per cent. Yalue received. 
 
 Kace B. Bonhoovan. 
 
 On this note are the following indorsements :— 
 
 Received Oct. 7th, 1849, $275-45. 
 
 " Aug. 25th, 1850, $386-38. 
 
 How much remains due Sept. 19th, 1852 ? 
 
 $684yVo- Cincinnati, July 26th, 1849. 
 
 [3.] On demand, I promise to pay James Benort, or 
 bearer, six hundred and eighty-four dollars, with interest, 
 at 6 per cent. Value received. John P. Trumbal. 
 
ART. 208.] 
 
 ANNUAL INTEREST. 
 
 201 
 
 On this note are the following indorsements. 
 
 Received Jan. 20th, 1850, $284-75. 
 March 14th, 1851, $84-75. 
 July 26th, 1853, $384-37|. 
 
 How much remains due Sept. 8th, 1854 ? 
 
 ANNUAL INTEREST. 
 
 In the computation o^ Annual Interest, the yearly increase 
 of the principal is equal to the annual interest of the first 
 principal. 
 
 If a note or obligation for |500 should be made payable 
 in 5 years with annual interest at 6 per cent., and another 
 for $500, payable in 1 year with annual interest at 6 per 
 cent., and no payment whatever should be made on either 
 until the expiration of 5 years, the amount of the obligations 
 would be equal. 
 
 1. What is the annual interest of $520 for 3 years, at 5 
 per cent. 1 
 
 OPERATION. 
 
 First Principal, - $520 
 
 -05 
 
 " Interest, 
 
 Second Principal, 
 " Interest, 
 
 Third Principal, 
 " Interest, 
 
 $26-00 
 
 $520 
 26 
 
 $546 
 •05 
 
 $27-30 
 
 $546 
 26 
 
 $572 
 •05 
 
 $28-60 
 
 First year's interest, $26*00 
 
 Second year's interest, $2Y"80 
 
 Third year's interest, $2860 
 
 Annual interest of $520 for 3 years, - - - $81-90 
 
 2. What is the annual interest of $814 for 5 years, at 6 
 per cent. ? 
 
 3. What is the annual interest of $840 for 6 years, at 7 
 per cent. ? 
 
202 COMPOUND INTEREST. ^ [CHAP. VIIL 
 
 COMPOUND INTEREST. 
 
 Compound Interest is interest on both principal and in- 
 terest together; the interest, annually or semi-annually, be- 
 ing successively taken to augment the principab 
 
 1. What is the compound interest of $520 for 3 years, 
 at 5 per cent. 1 
 
 Principal, 
 Per cent. 
 
 $520 
 •05 
 
 Interest for 1 year, 
 
 $2600 
 520 
 
 Amount for 1 year or 2nd principal, 
 
 $546 
 •05 
 
 Interest of $546, for 1 year. 
 
 - $27-30 
 546 
 
 Amount the 2nd year, or 3rd principal 
 
 , $573-30 
 -05 
 
 Interest of $573-30 for 1 year. 
 
 $28-6650 
 $573-30 
 
 Amount the 3rd year, ' - 
 Original principal, 
 
 $601-965 
 $520 
 
 Compound interest required, - $8r96|- 
 
 2. What is the compound interest of $384*50 for 3 years, 
 at 8 per cent. *? 
 
 3. What is the compound interest of |840, for 2 years, 
 mterest payable semi-annually, at 8 per cent. ? «i 
 
 4. What is the compound interest of $460, for 3 years^ 
 interest payable half yearly, at 6 per cent. ? 
 
 5. What is the difference between the annual and com 
 pound interest of $850 for 8 years, at 6 per cent. 1 
 
 Banking and Notes. 
 
 Art. 209. A Bank is. an institution created bylaw 
 for the purpose of issuing bank notes, or bank bills, which 
 circulate as money, and are redeemable in specie, on pre- 
 sentation to the bank; also for loaning money, receiving 
 deposits, and dealing in exchange. 
 
 The Capital Stpck of the bank, is divided into shares 
 
ART. 209. j FORMS OF NOTES. 203 
 
 which are owned by various individuals called stock- 
 holders. 
 
 The Stockholders, annually elect a board of Directors, 
 to manage the concerns of the Bank. This board elects a 
 Cashier^ and one of their number as President of the bank. 
 The President and Cashier sign all bills issued by the 
 Bank. 
 
 A promisory note is a positive engagement in writing, 
 to pay a certain sum at a specified time, to a person desig- 
 nated in the note, or to his order, or to the bearer. 
 
 Forms of Notes. 
 [No. 1.] 
 
 $T8yVo- Liberty, July 3d, 1849. 
 
 On demand, I promise to pay Edward Fox, seventy- 
 eight and yVo dollars, with interest. Yalue received. 
 
 Edward Everts. 
 
 [No. 2.] 
 
 Negotiable Note. 
 
 -j-^^. Kingston, Aug. 25th, 1850. 
 
 U4JL 
 
 One year after date, I promise to pay to the order of 
 Moses Morton, ninety-nitie and yVo dollars, with interest. 
 Value received. John Frontz. 
 
 [No. 3.] 
 Note Payable to Bearer. 
 
 $365tVo- Ellenville, Sept. 10th, 1851. 
 
 Six months after date, I promise to pay Isaac Ingraham, 
 or bearer, three hundred sixty-five and y^^ dollars, with 
 interest. Yalue received. Simeon Sa.wyer. 
 
 [No. 4.] 
 Note Payable at a Bank. 
 
 $47yVo. Buffalo, Oct. 15th, 1851 
 
 Forty days after date, I promise to pay to the order of 
 
204 PERCENTAGE. [cHAP. VIII. 
 
 Joseph Langhorn, at the Union Bank, in Sullivan Co., 
 N. Y., forty-seven and jW dollars, with interest. Yalue 
 received. Henry Mifflin. 
 
 The r^ rawer or Maker of a note is the person who 
 signs it. 
 
 Note No. 1, can be collected by Edward Fox only, 
 therefore, it is not negotiable. 
 
 Note No. 2, becomes collectable by any person holding 
 it, after Moses Morton writes his name on the back of it, 
 which is called indorsing the note. Moses Morton is now 
 called the Indorser. When this note becomes due pay- 
 ment must be demanded of the Drawer, and if he refuses 
 or neglects to pay it, notice must be given without delay 
 to the Indorser, demanding payment of him. 
 
 The payment of note No. 3, can be demanded by any 
 person holding it; the Drawer alone is responsible. 
 
 If Joseph Langhorn writes his name on the back of 
 note No. 4, payment may then be demanded of him, if 
 the drawer refuses or neglects to pay it at the specified 
 time. 
 
 A note that has not the words " Value received ^^ on it, 
 is invalid. ^ 
 
 Bank Discount. 
 
 Art. 210. Bank Discount is the sum paid to a bank 
 for the payment of a note before it becomes due. 
 
 The amount named in a note, is called the face of the 
 note. The discount is the interest on the face of the note 
 for 3 days more than the time specified, and is paid in ad- 
 vance.* These 3 days are called days of grace, as the bor- 
 rower is not obliged to make payment until their expira- 
 tion. Hence, to compute bank discount. 
 
 Find the interest on the face of the note for 3 days more 
 than the time specif ed ; this will he the discount. From the 
 face of the note, deduct the discount, and the remainder will 
 he the present value of the note. 
 
 * Taking interest in advance is usurious, and has been discontinued by 
 many banks ; and instead 'liereof, they deduct the true discount, found by 
 Articlo 205. 
 
ART. 211.] BANK DISCOUNT. 205 
 
 1. What is the bank discount on $240 for 6 months, at 
 T per cent. ? 
 
 2. What is the bank discount ou $460 for 4 months, at 
 8 per cent. ? 
 
 3. What is the bank discount on $150-50 for 2 months 
 and 15 days, at 6 per cent. ? 
 
 4. What is the bank discount on $4t5'85 for 3 months, 
 at 1 per cent. ? 
 
 5. What is the bank discount of a note of $8t5'50, for 
 8 months 21 days, at t percent. ? 
 
 6. What sum must a bank pay for a note of $385*^5, 
 payable in 6 months, discount, at 7 per cent. ? 
 
 7. What is the present value of a note of $875"25, dis- 
 counted at a bank for 8 months and 9 days, at 6 per cent. ? 
 
 8. What is the present value of a note of $84650, dis- 
 counted at a bank for 4 months and 15 days, at 7 per 
 cent. ? 
 
 9. What is the present value of a note of $8484*50 dis- 
 counted at a bank for 7 months and 9 days at 7 per cent. ? 
 
 Art. 211. GiYen the present value of a, hankMe note, 
 the rate per cent., and the time for which it is to be dis- 
 counted, to find the face of the note. 
 
 1. What must be the face of a bankable note so that 
 when discounted for 4 months and 15 days, at 6 per cent., 
 it shall give a present value of $1954 ? 
 
 Solution. — The discount of $1 for the given time, at the 
 given rate per cent., is $023; hence, $l_$-023=$-977, the 
 the present value of §1 for the given time, at the given rate 
 per cent. 
 
 We now have the proportion, $-977, present value : $1954, 
 present value : : $1 amount, : required amount, which is $2000, 
 the face of the note. 
 
 Hence, we infer in general, that if we Divide the given 
 PRESENT VALUE by the PRESENT VALUE of $1 foT the given 
 time and at the given rate per cent., hank discount ; the quo- 
 tient will he the amount or face of the note. 
 
 2. What must be the face of a bankable note, so that 
 when discounted for 6 months and 10 days, at 6 per cent., 
 it will gire $85, present value ? 
 
206 AVERAGE. [chap. VIII 
 
 3. What must be the amount of a banlfable note, so 
 that when discounted for 4 months and 21 days, at 1 per 
 cent , it shall give $84 '95 present value ? 
 
 4. What must be the amount of a bankable note so that 
 when discounted for 4 months and 9 days, at 8 per cent., 
 the borrower shall receive $384 ? 
 
 5. What must be the amount of a bankable note, so 
 that when discounted for 6 months and 27 days, at t oer 
 cent., the borrower shall receive $580 ? 
 
 AVERAGE. 
 
 Art. 2-12. If the sum of a series o^ 'promiscuous quan- 
 tities, be divided by the number of quantities, the quotient 
 will be one of a series of equal quantities, whose sum will 
 equal the sum of the former series. This quotient is called 
 the AVERAGE of thc given quantities. 
 
 1. What is the average of 12, 16, and 20 ? 
 
 2. During six successive months a laborer saved $12, 
 $18, $25, $30, $20, and $27 a month respectively. How 
 many dollars did he average a month ? 
 
 3. A locomotive made 4 successive trips over a track 
 20 miles in length, in the following times : 30 minutes 25 
 seconds ; 25 minutes 15 seconds ; 33 minutes 10 seconds ; 
 and 24 minutes 30 seconds. What was the average time 
 of 1 trip, also of running 1 mile ? 
 
 MERCANTILE CALCULATIONS. 
 
 Equation of Payments. 
 
 Art. 213. Equation of Payments is the process of 
 finding the average time for the payment of several sums 
 due at different times, without loss to either party. 
 
 The rules applying to mercantile calculations will be given 
 for the accommodation of book-keepers. 
 
 Art. 214. The equated time for the payment of any 
 sum, when parts of it are payable at different times, may 
 be found by the following 
 
ART. 214.] MERCANTILE CALCULATIOXS. 20"7 
 
 Multiply each jpaymejit by the time that must elapse before 
 it becomes due ; then divide the, sum of these products bf-4he 
 sum of the payments. The quotient will be the average 
 time required. 
 
 1. I purchased goods to the amount of $1200 ; $300 
 of which I am to pay in 2 months ; $400 in 3 months ; 
 and $500 in 6 months. How long a credit ought I to 
 receive, if I pay the whole at once ? 
 
 Explanation. — A credit on $300 for 2 months 
 is the same as the credit on $1 for 600 months. 
 
 A credit on $400 for 3 months is the same as 
 the credit on $1 for 1200 months. 
 
 A credit on $500 for 6 months is the same as 
 the credit on $1 for 3000 months. 
 
 Tijerefore. on the whole sum, $1900, 1 should 
 receive the same as the creditor the interest on 
 $1 for 4S00 months; the $1200 wiU give the same 
 interest in one-twelve hundredth of 4800 month."? 
 =4 months, the time in which the whole amount 
 averages due. 
 
 OPERATION. 
 
 $ mo. 
 
 mo. 
 
 300x2 = 
 
 = 600 
 
 400x3 = 
 
 = 1200 
 
 500x6 = 
 
 = 3000 
 
 1200) 
 
 4800(4 mo. 
 4800 
 
 
 
 2. If I owe $900 ; $200 of which is due in 2 months ; 
 $300 in 4 months; and the remainder in 6 months. What is 
 the average time for the payment of the whole ? 
 
 3. If you owe a man $150, payable in 2 months ; $260 
 payable in 4 months ; $490 payal)le in 8 months; at what 
 time may you in equity pay the whole ? 
 
 4. A merchant bought goods to the amount of $400, on 
 a credit of 4 months; another quantity for $500, on a credit 
 of 5 mo.; and another quantity for $800, on a credit of 8 
 mo. What is the average time for the payment of the whole ? 
 
 5. A gentleman owes a certain sum of money; ^ of which 
 is due in 3 months; | in 4 months; } in 12 months. What 
 is the average time of payment ? 
 
 Remark — We will suppose the amount owed is $1, as it can make no diffe- 
 vence what that amount is, since certain fractional parts of it become due at 
 particular times. 
 
 6. A merchant bougbi goods to the amount of $3000 ; 
 
208 AVERAGE. [chap. VIII. 
 
 I of wliicli he paid in cash at the time of receiving the 
 goods ; I is to be paid in 6 months; and the remainder in 
 1 year and 3 months. What is the average time for the 
 payment of the whole ? 
 
 7. A man purchased a farm for $3200, and agreed to 
 pay $500 of it at the expiration of 3 months ; $1200 at 
 the expiration of 9 months ; and the remainder in 12 months. 
 What is the equated time for the payment of the whole ? 
 
 Art. 215. In mercantile transactions it is customary to 
 give a credit of from 3 to 9 months, on bills of sale.' 
 
 Art. 216. To determine the average time of payment 
 of several sales, on different terms of credit, we have the 
 following 
 
 RULE. 
 
 Multiply the amount of each sale by t/ie tiTue intervening 
 betwee7i tJie date on which the first amount falls due, and the 
 date on which each sum falls due. Then divide the sum 
 of these products by the whole amoimt of debt, and the qivotient 
 will be the averaged time of payment, to be counted forward 
 from the date of the first amount falling due. 
 
 1. Purchased goods of Stiiwell, Brown & Co., at dif- 
 ferent dates and on different terms of credit ; as below 
 stated. 
 
 JFeb. 2, 1853, a bill amounting to ^460 on 3 months' credit. 
 
 Feb. 5, " " " $680 on 4 
 
 March 28, " " " $560 on 5 " 
 
 April 12, " " " i840 on 5 
 
 I wish to make one payment of the whole debt. When, 
 per average, will it become due ? 
 
 Solution. — The above bills come due respectively as follows : 
 
 days. days. " 
 
 Due May 2, $460 X 00 = 00000 
 " June 5, $680 X 34 = 23120 
 " Aug. 28, $560 X 118 = 66080 
 " Sept. 12, $840 X 133 = 111720 
 
 $2540) 200920(79 days. 
 &c. &c. 
 
ART. 216.] MERCANTILE OALCULATIONS. 209 
 
 The $460 become due May 2nd, 1853; the $680 become 
 due 34 days from May 2nd; the $560, 118 days from 
 May 2nd; and the $840, 133 days from May 2nd. By 
 equation of payments I find these bills will average due in 
 19 days from May 2nd, which is July 20th, 1853. 
 
 If it were required to know how much money would 
 balance the account any time previous to July 20thj as April 
 15th, it is evident that the present worth of $2540 from April 
 15th to July 20th, would be the sum required. * 
 
 When the different sales are made on the same terms 
 of credit, the ave]*a.ge time for the payment of the whole 
 debt, may be found as taught by the following question : 
 
 2. A merchant sold goods to one of his customers, at 
 different dates; as belov^ stated : 
 
 April 8, 1853, a bill amounting to $470 on 6 month's credit. 
 May 17, " " " $840 on 6 
 
 June 23, " ♦' «« #980 on 6 «' 
 
 July 10, «.« *' ♦' ^580 on 6 " 
 
 What is the average time for the paymei^ of the above' 
 bills ? 
 
 OPERATION. 
 
 April 8, 1853, $470 x 00 = 00000 
 May 17, " $840 X 39 = 32760 
 June 23, " $980 X 76 = 74480 
 July 10, « $580 X 93 = 53940 
 
 $2870) 161180(5^ days. 
 
 From a little reflection the pupil will discover that the above 
 bills will average due in 56 days from the time the first falls 
 due, which is Dec. 3, 1853. 
 
 3, A merchant sold to one of his customers several 
 parcels of goods, at sundry times, and on different terms 
 of credit; as follows : 
 
 Feb. 1, a bill amounting to ^300 on 4 months' credit. 
 March 7, " " $185 on 5 " " 
 
 April 15, " «« $280 on 4 " 
 
 May 20, " « $210 on 3 " 
 
 What is the equated time for the payment of all these bills ? 
 
210 AVERAGE. [chap. VIII, 
 
 4. Purchased goods of a merchant at different times, 
 and on different terms of credit as follows : 
 
 March 3, 1853, a bill amounting to $847 '10 on 3 months' credit. 
 April 5, " " " $'645.60 on 4 " 
 
 May 10, " " " :||!584-75 on 6 
 
 June 15 " «' " $475-84 on 8 '« 
 
 Aug. 18 " " " P84-.95 on 4 " 
 
 What is the equated time for the payment of the above 
 bills ? 
 
 5. Purchased goods at sundry times, and on different 
 terms of credit, as follows : 
 
 June 4, 1853, a bill amounting to ^485-90 on 8 month's credit. 
 
 July 12, " " " $675-25 on 4 " 
 
 Aug. 15, " " " i;81212^on5 «' 
 
 Sept. 22, " " " $895-25 on 6 ♦' 
 
 Nov. 20, " " '* $896-70 on 4 " 
 
 What is the equated time for the payment of all these bills ? 
 
 6. Bought goods of J. B. Smith & Co., at sundry times, 
 as shown by the statement annexed. 
 
 March 2, 1853, a bill amounting to $684"20 on 6 months' credit. 
 
 March 28, " ♦' " $875-54 " 
 
 April 10, " " " $484-40 " 
 
 May 20, " " " $795-45 '« «« " 
 
 June 30, '4 " " $840-60 *' 
 
 How much money will balance the account, July 4, 
 1854? 
 
 t. Bought goods of C. B. Hill & Co., at different times, 
 and on different terms of credit, as shown by the state- 
 ment annexed ? 
 
 June-12, 1853, a bill amounting to $340-65 on 5 months' credit. 
 
 July 8, " " " $595-75 " 
 
 Aug. 10, " " " $784-85 6 
 
 Oct. 14, " " " $987-90 8 
 
 Nov. 15, " " " $878-98 4 
 
 Dec. 19, " ♦' " $999-99 2 
 
 How much money will balance the account, Jan. 20, 
 1854 ? 
 
 8. Bought goods of R, Lancaster & Co. as shown by 
 the statement annexed; 
 
ART. 211.] MERCANTILE CALCULATIONS. 211 
 
 May 4, 1853, a bill amounting to ^432-95 on 3 months' credit. 
 
 May 18, " " . " $843-45 2 
 
 Jnne20, " " " .$732-46 6 
 
 July 8, " " " $-846-75 7 
 
 Aug. 20, " " " ^784-78 6 
 
 Sep. 24, " " " $976-34 4 
 
 What is the equated time for the payment of all these 
 bills ; and how much money would balance the account, 
 Nov. 12, 1854? 
 
 Art. 217". The rule given for the Equation of Pay- 
 ments is the one usually adopted by merchants, although 
 not strictly correct, still it is sufficiently accurate for all 
 practical purposes, when small sums and short periods of 
 time are considered. 
 
 This inaccuracy will become evident by inspecting the 
 following example : — 
 
 A owes B $4480 ; $2240 of which is due in 2 years, and 
 the remainder in 10 years. What is the equated time for 
 the payment of the whole, interest 6 per cent. ? 
 
 The average time of payment found by the rule above 
 referred to, is 6 years. From which we observe that 
 $2240 is not paid until 4 years after it is due, also that 
 $2240 is paid 4 years before it is due ; — these two condi- 
 tions are considered to mutually counter-balance each other, 
 although, they do not. It is evident, that, for deferring 
 the payment of the first $2240 for 4 years, A should pay the 
 amount of $224:0 ioY the same time, which is $2777-60 ; 
 but for the remainder, which he pay^ 4 years before it is 
 due, he should pay the present worth of $2240 for the 
 same time, which is $1806"45. Hence the Rule occasions 
 an error of $277760 + $1806-45— $4480=$10405. 
 
 Justice demands that interest should be required on all 
 sums from the time they become due until the payment is 
 made, and the present worth of all sums paid before thej 
 become due. Hence the following accurate 
 
 RULE. 
 
 Find the present worth of each of the given amounts due, 
 then, find in what time the sum of these present worths will 
 amo^mt to the sum of all the payments. 
 
212 AVERAGE. [chap. VIII. 
 
 Art. 218. When a debt due at some future period 
 has received partial payments before the time due, to find 
 how long after this time, the remainder may in equity re- 
 main unpaid. 
 
 RULE. 
 
 Divide the sum of the 'products of each payment into th( 
 time it was paid before due^ by the sum remaining unpaid 
 The quotient will be the required time. 
 
 1. A owes $1200, due in 6 months; five months before" 
 it is due $200 is paid; and 3 months before it is due, $300 
 is paid. How long after the expiration of the 6 months 
 may the remaining $500 in equity remain unpaid ? 
 
 operation. 
 
 mo. mo. ( Explanation. — A credit on $200 for 
 
 $200 X 5 = 1000 } 5 months is the same as a credit on 
 
 / $1 for 1000 months. 
 
 !A credit on $500 for 3 months is 
 the same as a credit on $1 for 1500 
 months. 
 $500) 2500 ^ , 
 
 The money paid in advance affords 
 
 5 mo. a profit equal to the interest of $1 
 
 for 2500 ; the balance remaining due, $500, will afford the 
 same interest in one-five hundredths of 2500 months, which is 
 5 months. 
 
 2. A person owes $400, due at the epd of 10 months. 
 At the end of 4 months he pays $100; 3 months after 
 that he pays $50. How long after the expiration of the 
 10 months may the balance remain unpaid ? 
 
 3. A merchant lends to a farmer $1600, payable in 12 
 months. At the end of 4 months $200 of it is paid ; 3 
 months after that $400 more is paid ; and 1 month before 
 the expiration of the 12 months $200 more is paid. How 
 long after the expiration of the 12 months may the balance 
 in equity remain unpaid ? 
 
 4. A lends B $1200 for 6 months ; at another time 
 $1800 for 8 months. For how long a time ought B to 
 lend A $2t00, to balance the favor ? 
 
IRT. 219.] MERCANTILE CALCULATIONS. 213 
 
 5 I borrowed of my neighbor $900 for 5 months ; at 
 another time $800 for 9 months. For how long a time 
 ought I to lend my neighbor $850 to balance the favor ? 
 
 Compound Equation of Payments. 
 Art. 219. Compound Equation of Payments teaches tho 
 method of ascertaining the time on which the balance of 
 an account that contains debit and credit becomes due ; 
 having first learned, by Rule under Art. 216, when the 
 debit and credit of said account falls due, respectively, 
 without regard to their relation to each other. 
 
 1st. Multiply the number of days between the dates of 
 equated time by the amount that first falls due ; and divide 
 this product by the difference between the debit and credit of 
 the account ; the quotient will be the tdae for consideration. 
 
 2d. If the larger amount comes due first, the time is counted 
 BACK from the latest date ; but if the smaller amount comes 
 due first, the time is counted forward from the latest date. 
 
 1. By equation of payments it is found that A^s account 
 with B. is as follows : 
 
 Dr. Cr. 
 
 Due June 4th, . . $400 1 Due July 24th, . . $900 
 
 When will the balance of the account become due ? 
 
 operation 
 Amount of Cr. $900 due July 24th. 
 " Dr. $400 due June 4th. 
 
 Balance $500 From June 4th to July 24th, is 50 days. 
 
 days. 
 
 50 
 400 
 
 500)20000 
 
 40 days from. July 24th, which is Sept. 2d, the balance 
 becomes due. 
 
 Explanation — It is evident that B should receive the mterest 
 on $400 from June 4th- to /uly 24th. Therefore B should retain 
 
214 AVERAGE. 1_CHAP. Vm 
 
 the balance ($500) sufficiently long after it becomes due, to re- 
 ceive the same amount of interest on it as he would have re- 
 ceived on the $400 from June 4th to July 24th. 
 
 If $400 in 50 days give a certain interest, $1 will give the 
 same interest in 400 times 50 days = 20000 days; and $500 
 (the balance) will give the same interest in -^ of 20000 days 
 =40 days; consequently, in 40 days from July 24th, the balance 
 becomes due. 
 
 2. Suppose the above account to stand as follows ? 
 
 Dr. Cr. 
 
 Due June 4, . . . $900 | Due July 24, . . . $400 
 
 At what time must a note for the balance be dated, to 
 balance the account ? 
 
 Solution. — It is evident that B should receive the interest 
 on $900 from June 4th to July 24th, Therefore, to balance 
 the account, B should receive a note of $500 (the balance), of 
 such a date that the interest on it should, on the 24th day of 
 July, equal the interest on $900 for 50 days, the time from 
 June 4th to July 24th. 
 
 If $900 in 50 days give a certain interest, $1 will give the 
 (Same interest in 900 times 50 days = 45000 days ; and $500 
 will give the same amount of interest in -^l-^ of 45000 days. 
 Hence, 90 days previous to July 24th, which is April 25th, the 
 note should be dated. 
 
 3. B has with C the following account : — 
 
 1853. Dr. I 1854. Cr. 
 
 Nov. 12. Due . . $840. | Jan. 20. Due . . 
 
 When will the balance of the account become due ? 
 
 4. C has with D the following account : — 
 
 1853. Dr. I 1853. Cr. 
 
 July 20. Due . - . $987. | Sept. 4. Due . . $507 
 
 At what time must a note of the balance be dated U 
 balance the account ? 
 
 5. At what time will the balance of the following account 
 become due ? 
 
 1853. Dr. I 1854. Cr. 
 
 Oct. 26. Due . . $1280. | Jan. 16. Due . . $840. 
 
 6. When will the balance of the following account 
 become due ? 
 
ART. 220.] MERCANTILE CALCULATIONS. 215 
 
 1853. Dr. 1 1853. Cr. 
 
 April 21. Due . . $845 | June 15. Due . . $1685 
 
 Cash Balance. 
 
 Art. 220. To find the cash balance of an account 
 consisting of various items of debit and credit, of different 
 dates, at any specified time. 
 
 RULE. 
 
 Place on the debtor or credit side, such a sum, (which may 
 he called merchandise balance, J as will balance the account. 
 
 Multiply the number of dollars in each entry hy the num^ 
 her of days from the time the entry was made to the time of 
 settlement ; and the merchandise balance by the number of days 
 for which credit was given. Then midti'ply the difference 
 between the sum of the debit, and the sum of the credit products 
 by the interest of %1 for 1 day ; this product will he the 
 \nterest balance. 
 
 When the sum of the debit products exceed the sum of the 
 redit products, the interest balance is in favor of the debit 
 side ; hut when the sum of the credit products exceed the sum 
 of tJie debit products, it is in favor of the credit side. Now 
 to the merchandise balance add the interest balance, or subtract 
 it, as the case may require, and you obtain the cash balance. 
 
 1. A has with B the following account : — 
 
 1849. Dr. 
 
 Jan. 2. To merchandise $200 
 April 20. " " 400 
 
 1849. Cr. 
 
 Feb. ^0. By merchandise $100 
 May 10. " " . 300 
 
 If interest is estimated at t per cent., and a credit of 60 
 days is allowed on the different ^ums, what is the cash 
 balance August 20, 1849 ? 
 
 Explanation. — Without interest, the cash balance would be 
 
 $200. 
 
 If no credit had been given, the debits should be increased 
 by the interest of $200 for 230 days, at 7 per cent. ; and the 
 interest of $400 for 122 days, at 7 per cent. The credits should 
 be increased by the interest of $100 for 181 day, at 7 per cent. ; 
 and the interest of $300 for 102 day^. at 7 per cent. 
 
 Since a credit of 60 days is given on all sums, it is evident 
 
2ie 
 
 AVERAGE. 
 
 [chap. VIII. 
 
 by the above calculation, that we shall increase the debits by 
 the interest of the sum "of the debits, $600, for 60 days more 
 than justice requires. Also, that we should increase the credits 
 by the interest of the sum of the credits, $400, for 60 days more 
 than we should do. 
 
 Now, instead of deducting these items of interest from the 
 amount of debit and credit interests, it is plain, that it will be 
 more convenient and equally just, to diminish the debit inter- 
 est by the interest of the merchandise balance for 60 days, which 
 can be most readily accomplished by adding the interest on 
 the merchandise balance for 60 days, to the credit items of 
 interest. 
 
 From which we discover that the interest balance is equal to 
 the difference between the sum of the debit interests, and the 
 sum of the credit interests increased by the interest of the mer- 
 chandise balance for the time for which credit was given. 
 
 DEBITS. 
 
 $ Days. 
 
 200 X 230 = 46000 
 400 X 122 = 48800 
 
 94800 
 60700 
 
 OPERATION. 
 
 CREDITS. 
 $ Days. 
 
 100 X 181 = 18100 
 
 300 X 102 = 30600 
 
 Balance, 200 x 60 = 12000 
 
 60700 
 
 0^ 
 365 
 
 X 34100 =r $6-54 Interest balance, nearly. 
 
 Therefore, the foregoing account becomes balanced as 
 follows : — 
 
 1849. Dr. 
 
 Jan. 2. To merchandise, $200-00 
 April 20. " " 400 00 
 
 Aug. 20. " balance of interest, 6-64 
 
 $606-54 
 
 1849. 
 Feb. 20. By merchandise. 
 May 10. " 
 Aug. 20. " balance, 
 
 Cr. 
 
 $10000 
 
 300 00 
 
 206 64 
 
 $60654 
 
 Aug. 20. " Cash balance, $-206 64 
 
 Note. — It is customary in practice, when the number of cents in any of the 
 ptries, are less than 60. to omit them, and to add $1 when they are 80 or more. 
 
 2. A has with B the following account : — 
 
 1852. Dr. 
 
 Jan. 8. To merchandise, $400 
 April 24. " " 800 
 
 1852. Cr. 
 
 Feb. 10. By merchandise, $300 
 May 24. " " 500 
 
TRADE AND BARTER- 21 1 
 
 If interest is estimated, at T per cent, and a credit of 
 60 days is allowed on the different sums, what is the cash 
 balance Sept. 25th, 1852 ? 
 
 3. B has with C the following account : — 
 
 1853. 
 
 Ih: 
 
 1853. 
 
 Cr. 
 
 Feb. 12. To merchandise, 
 
 $840 
 
 March 16. By merchandise, 
 
 $640 
 
 July 25. " 
 
 980 
 
 May 14. ^ - 
 
 780 
 
 Aug. 14. " 
 
 C40 
 
 Sept. 20. ♦^ •• 
 
 430 
 
 I 
 
 If interest is estimated at 8 per cent., and a credit of 
 90 days is allowed on the different sums, what is the cash 
 balance Jan. 10th, 1854 ? 
 
 TEADE AND BARTER. 
 
 Art. 221. Trade and Barter is the exchange of one 
 commodity for another without loss to either party. 
 
 1. How many yards of muslin, at $'12^ a yard, must 
 be given for 380 lbs. of butter, at $'16 a pound ? 
 
 2. How many bushels of rye, at $93^ a bushel, must 
 be given for 187| lbs. of tea, at $'62^ a pound ? 
 
 3. A merchant exchanged 630 yards of cloth, for 15 
 hogsheads of wine, at $1*10 a gallon. How much was 
 the cloth a yard ? 
 
 4. A farmer gave 20^ cwt. of hops, at $6*80 per cwt., 
 for 8 cwt. 3 qrs. 20 lbs. of sugar, and $80 in money; at 
 how much was the sugar valued per pound ? 
 
 5. A farmer received for 25 cwt. 2 qrs. 22 pounds of 
 cheese, at $081 a pound, 18 yards of cloth, at $250 a 
 yard; 16 yards of muslin, at 5^ cents a yard; 5 pair of 
 boots, at $2-75 a pair; 85 gallons of molasses, at %'l^^ a 
 gallon, and the balance in sugar, at $'09^ per pound. 
 How many pounds of sugar did he buy ? 
 
 6. A grocer barters 860 bushels of oats, which cost htm 
 $•25 a bushel, at $-37i a bushel, for cloth that cost $2-86| 
 a yard. What is the bartering price of the cloth, and 
 how many yards did the grocer receive ? 
 
 1. A farmer has 184^ bushels of rye, which is worth 
 $'84 per bushel; but in barter he is willing to put it at 
 
 10 
 
218 TRADE AND BARTER. ^CHAP. VIII. 
 
 $•56 a bushel, providing his neighbor will let him have 
 wheat worth $1-24 per bushel for $-91. Will he gain or 
 lose by the bargain, and what per cent. ? 
 
 8. Two farmers bartered: A had 240 bushels of wheat, 
 at $1-50 per bushel, for which B gave him 200 bush, of 
 corn, at $'65 per bush., and the balance in buckwheat, at 
 $•80 a bushel. How much buckwheat did A receive of B ? 
 
 9. A farmer has 380 bushels of wheat, worth $1^20 a 
 bushel; but in barter he will have $144 a bushel. A 
 merchant has broadcloth worth $.3^60 a yard ; and linen 
 worth $1'40; at what price per yard ought the merchant 
 to rate his broadcloth and linen to be equivalent to the 
 farmer's bartering price, and how many yards will the 
 farmer receive for his wheat, providing he takes an equal 
 number of yards of each ? 
 
 10. A and B barter: A has 560 bushels of wheat worth 
 $1^20, but in barter he will have $1^60 a bushel; B has 
 broadcloth worth $4^20 a yard; how must B sell his 
 broadcloth a yard in proportion to A's bartering price for 
 his wheat, and how many yards are equal in value to A's 
 wheat ? 
 
 11. A had 450 yds. of cloth, worth $1^20 a yard, which 
 he bartered with B, at $1^45 a yard; taking flour, at 
 $^•50 a barrel, which is worth but $6. How much flour 
 will pay for the cloth; and who gets the best of the bar- 
 gain ? 
 
 12. A farmer sold to a merchant one yoke of oxen for 
 $125; 184 bushels of corn, at $-3Ti a bushel; 45 bushels 
 of wheat, at $-93f a bushel, April 14th, 1853. In pay- 
 ment he received 125 lbs. of raisins, at lOf centsa pound; 
 584 pounds of sugar, at 9-]- cents a pound; and 54 gal- 
 lons of molasses, at 13| cents a gallon, Nov. 8th, 1853 
 How much remains due; interest 6 per cent. ? 
 
 13. A farmer took to market 26 tO lbs. of wheat, worth 
 .$'93f a bushel; and in payment takes $5^1 1| to pay his 
 taxes; and for the remainder he is to receive an equal 
 number of yards of muslin, at *J\ cents a yard; bleached 
 muslin, at 12^ cents; calico, at 16| cents; and linen, at 
 31^ cents. How many yards of each did he buy ? 
 
ART. 221.] TRADE AND BaRTER. 219 
 
 . 14. A farmer, Mr. Smith, lent his neighbcr, April 1st, 
 16 busliels of superior wheat, on condition that it should 
 be paid in wheat of equal quality, on the first of the 
 following November, after adding 3 per cent, to it for its 
 nse. The wheat his neighbor returned was 7 per cent, 
 inferior to that which he received. How many bushels of 
 wheat should Mr. Smith in equity receive ? 
 
 15. A farmer, Mr. Jackson, owes a merchant $560, May 
 1st, 1853. On the above, Mr. J. paid, Aug. 4th, 1853, 
 4f bushels of clover-seed, at $6-75 a bushel, and a yoke 
 of oxen for $95. On Nov. 15th, 1853, Mr. J. paid 63^ 
 bushels of rye, at $-93f a bushel, and the remainder in 
 wheat, at $ri2i a bushel. How many bushels did it take, 
 interest 8 per cent. ? 
 
 16. A Mr. Judson sold to Mr. Wilson, April 10, 1852 ; 
 4 cows, at $2375 each • 15 bushels of oats, at $"37^ a 
 bushel ; 24 cwt. 3 qrs. of hay, at $1075 a ton ; and 1 
 wagon, at $84'95. Mr. Wilson, in payment, sold Mr. J., 
 March 15th, 1853, 3 plows, at $6 37^ each; 2 cultivators, 
 each $5-18f ; 12^ yards of broadcloth, at $4-85 per yard ; 
 2 barrels of sugar, at $17 75 a barrel ; 5 sacks of salt, at 
 $3*12i a sack. Allowing 7 per cent, interest, how does the 
 account stand, Oct. 12th, 1853. 
 
 17. A speculator, Mr. Manning, bought of Mr. Bron- 
 8on a house and lot for $2400, January 1st, 1853, | of 
 which was payable at the time the purchase was made, and 
 the remainder was to be paid, with interest at 7 per cent, 
 in 3 equal payments ; the first in 4 months; the second in 
 8 months; and the third in 12 months. Mr. B. sold Mr. M. 
 485| bushels of potatoes, at $'62|- a bushel, April 15th, 
 1853 ; and August 12th, 1853, 748 bushels of corn, at 
 $•47^ a bushel. They are desirous of settling, Nov. 18th, 
 1853. How much in equity should Mr. Manning pay Mr. 
 Bronson ? 
 
 18. Mr. Mathews sold to Messrs. Arnold & Co., May 12, 
 ?853, 465 lbs. of pork, at $'11^ a pound, and a span of 
 horses and pleasure wagon for $684*50, on 6 months' credit. 
 June 15, 1853, Mr. M. bought of Messrs. A. & Co. 47-75 
 acres of land, at $23'25 an acre, on 3 months' credit. Mr 
 
TRADE AVD BARTEB. lCHAP. VIII. 
 
 M. sold to Messrs. A. & Co. 184 bushels of wheat, at $-93f 
 a busiiel, for which no credit is allowed. They settled 
 accounts Nov. 15th, 1853. Which was in debt to the other, 
 and how much ? 
 
 19. A farmer sold to a merchant 41|- bushels of corn, at 
 $-65 a bushel ; 84^ bushels of rye, at $'87^ a bushel ; 36f 
 bushels of buckwheat, at $"93f a bushel ; and in payment 
 received 12^ lbs. of tea, at $ri2i a pound ; 15^ pounds 
 of coffee, at $16| a pound ; 135 pounds of sugar, at $-9| 
 a pound ; 25 gallons of molasses, at 35|- a gallon ; 36|- 
 yards of linen, at $'15|- a yard; 25^ yards of calico, at $16| 
 a yard ; 24^ yards of broadcloth, at $3'85^ a yard ; 15 
 pair of shoes, at $2*54 a pair ; and a set of spoons, knives, 
 and forks, for $19' 84. Which owed the other, and how 
 much? 
 
 20. A farmer sold a mechanic, April 1st, 1853, a span 
 of horses for $284; a yoke of oxen for $184; 148^ bushels 
 of grain, at $-93f a bushel; and 4 cows, each $25-75, on a 
 credit of 9 months. The mechanic sold the farmer, May 1, 
 1853, a lumber wagon for $184; a pleasure wagon for 
 $325; and 4 plows, each $6-75, on a credit of 3 months. 
 They settle accounts Sept. 1st, 1853; which is in debt, 
 and how much, interest 6 per cent. ? 
 
 21. A farmer sold to a merchant, Jan. 3d, 1852, 1764 
 pounds of pork, at 8| cents a pound; 1683 pounds of beef, 
 4f cents a pound; 847 pounds of ham, at 10^ cts. a pound; 
 March 12th, 1852, 485 bushels of oats, at $"43| a bushel; 
 184 bushels of rye, at $-62i a bushel; 284 bushels of wheat, 
 at $-93f a bushel; and 487 pounds of cheese, at 11^ cents 
 a pound. The farmer received of the merchant, March 
 25th, 1853, merchandise to the amount of $684-75; June 
 15th, 1853, merchandise to the amount of $84645. In 
 •the above transaction 6 months' credit was given on all 
 the articles. Balance the account July 3d, 1853, at 6 
 oer cent, interest. 
 
 22. Mr. Smith bought of a speculator a farm containing 
 o72 acres, at $40 an acre, and was to pay for it in ten years 
 as follows : | of the whole the first year; | of the remain- 
 der the second year; | of the remainder the third yearj 
 
ART. 222.] ARITHMETICAL PROGRESSION. 221 
 
 i of the remainder the fourth year; ^ of the remainder the 
 fifth year; i of the remainder the sixth year: and the 
 remainder in four equal and annual payments, without 
 interest. The first payment was made in wheat, at $1*12^ 
 a bushel; the second in wheat, at $125 a bushel; the 
 third in wheat, at $ 93f a bushel; the fourth in wheat, at 
 $'95 a bushel; the fifth in wheat, at $1-10 a bushel; the 
 sixth in wheat, at $r20 a bushel; and the remainder iu 
 wheat, at $1"15 a bushel. What was the amount of each 
 payment, and the number of bushels of wheat paid annu- 
 ally ? If no payment had been made until the end of the 
 ten years, how many bushels of wheat, at $r35 a bushel, 
 would have balanced the account, interest, at 1 per cent. ? 
 
 CH'APTER IX. 
 PROGRESSION 
 
 Arithmetical Progression. 
 
 Art. 222. A series of numbers that increase or de- 
 crease by a constant difference, is said to be in Arithmetical 
 Progression. When the terms are constantly increasing, 
 the series is called an Ascending Arithmetical Progression ; 
 as, 1, 3, 6, t, 9, 11, 13, 15, &c. 
 
 When they are constantly decreasing, the series is called 
 a Descending Arithmetical Progression ; as, 
 
 45, 43, 41, 39, 3t, Zb, 33, 31, &c. 
 
 The first and last terms of a Progression are called the 
 extremes, and the other terms are called the means. 
 
 From the nature of an Arithmetical Progression, it fol- 
 lows, that the sum of the extremes is equal to the sum of 
 any other two terms equally distant from them, or to twice 
 the middle term ; if the number of terms is unequal. This 
 
222 , PROGRESSION. [CHAP. IX. 
 
 wil] appear more plain by iuspecting the following Pro- 
 gression : — 1, 3, 5, 1, 9, 11, 13, 15, and 17. 
 
 1, 3, 5, 7, 9 ^ Here the sum of the extremes and 
 
 17, 15, 13, 11, 9 ! the terms equally distant from them, 
 — — — — — I are added, and found to be equal, aa 
 
 18, 18, 18, 18, 18 J above stated. 
 
 In Arithmetical Progression there are five distinct 
 terms to be considered : — 
 
 a, The first term; 
 
 1, The last term; 
 
 n, The number of terms; 
 
 d, The common difference; and 
 
 s, The sum of all the terms. 
 
 These terms are so related that any three of them bemg 
 known, the remaining two may be found. Since there are 
 five terms, and only three of them necessary to be known, 
 to find a fourth, it follows that there may be twenty dis- 
 tinct cases in Arithmetical Progression. We shall, how- 
 ever, notice but few of them, and refer the student to 
 Algebra for the others. 
 
 CASE I. 
 
 Art. 223. Given the first term, the common difference, 
 and the number of terms, to find the last term. 
 
 1. What is the 25th term of an arithmetical progres- 
 sion, the first term of which is 6 and the common dif- 
 ference 4 ? 
 
 Explanation. — The second term of an arithmetical progres- 
 sion ascending is equal to the first term plus the common differ- 
 ence ; the third term is equal to the first term plus twice the 
 common difierence : and so on. Therefore, when we have given 
 the first term, the common difierence and the number of terms, 
 the last term is found by Adding the first term to the product of 
 the common difierence into the number of terms less one. 
 
 2. A man bought 25 acres of land, giving $2 for the 
 first acre, $8 for the second, $14 for the third, and so on, 
 
ART. 224.] ARITHMETICAL PROGRESSION. 223 
 
 iigreasiug iif arithmetical progression. What did the 
 last acre cost at tj^is rate ? 
 
 3. x\ merchant bought 18 pieces of cloth, giving $3 for 
 the first piece; $5 for the second; $7 for third, and so on, 
 
 Jncreasiiig in arithmetical progression. What, at this rate, 
 did the last piece cost ? 
 
 4. A tapering board, 3^ inches wide at the narrow end, 
 and 14 feet long, is found to increase in width 1| inches 
 for every foot in length. What is the width of the wide 
 end? 
 
 5. In a certain orchard there are 34 rows; in the first 
 row there are 20 trees; in the second 24; in the third 
 28; and so on, the number of trees in each row continu- 
 ing to increase in an arithmetical ratio. How many trees, 
 at this rate, are there in the last row ? 
 
 CASE II. 
 
 Art. 224. Given the first terra, the last term, and 
 the number of terms, to find the sum of all the terms. 
 
 1. The first term of an arithmetical progression is 5, 
 the last term is 85, and the number of terms is 12. What 
 is the sum of all the terms ? 
 
 Explanation. — The sum of the extremes of an arithmetical 
 progression being equal to the sum of any two terms equally 
 distant from them, it follows that the terms must average halt 
 the sum of the extremes; hence, the Sum of all the terms equals 
 the product of the number of terms by half the sum of the extremes. 
 
 2. A man bought 25 acres of land: for the first acre 
 he gave $i ; for the last, $244^ ; the prices of the suc- 
 cessive acres form an arithmetical series. How much did 
 the 25 acres cost at this rate ? 
 
 3. A merchant bought 25 pieces of cloth ; for the first 
 piece he gave $3 ; for the last piece, $63 ; the prices of 
 the pieces form an arithmetical progression. How much 
 at this rate, did the cloth cost him ? 
 
 4. In a certain field there are 44 rows of corn: in the 
 first row there are 10 hills ; and in the last, 139 hills; 
 the number of hills in the successive rows form an arith- 
 
224 PROGRESSION. [CHAP. IX. 
 
 metical progression. How many hills are there in ^e 
 field ? • 
 
 • CASE III. 
 
 Art. 225. Given the extremes and the common dif- 
 ference, to find the number of terms. 
 
 1. The first term of an arithmetical progression is 8; 
 the last term 83 ; and the common difference 5. What is 
 the number of terms ? 
 
 Explanation. — Since the last term of an arithmetical pro- 
 gression equals, the product of the number of terms less on© 
 into the common difference, increased by the first term, (see 
 Case 1;) it follows that the number of terms equals the quotient^ 
 increased by 1, arising from dividing the difference of the extremes 
 by the common difference. 
 
 2. A man going a journey traveled the first day 1 miles, 
 the last day 67 miles, and each .day increased his journey 
 by 4 miles. How many days did he travel ? 
 
 3. A merchant bought a certain number of pieces of 
 cloth, the prices of which increased by $2. The first 
 piece cost $3, and the last piece $43. How many pieces 
 did he buy ? 
 
 Geometrical Progression. 
 
 Art. 226. A series of numbers that increase or de- 
 crease by a constant fmiUiplier, is said to be in Geometrical 
 Progression. 
 
 When the constant multiplier, which is called the ratio, 
 is greater than a unit, the series is called an Ascending 
 Geometrical Progression ; as, 
 
 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, &c. 
 
 When the ratio is less than a unit, the series is called a 
 Descending Geometrical Progression ; as, 
 
 512, 256, 128, 64, 32, 16, 8, &c.; or, 
 
 -*■) "a* 4> Ji T6» ii2'» "e*' ^^' 
 In Geometrical, as in Arithmetical Progression, there 
 are five terms to be considered ; — 
 
ABT. 228.] GEOMETRICAL PROGRESSION. 225 
 
 a, The first term; 
 
 1, The last term; 
 
 D, The number of terms; 
 
 r, The ratio; and 
 
 s, The sum of all the terms. 
 
 These terms are so related that any three of them being 
 known, the remaining two may be found. Since there are 
 five terms, and only three of them necessary to be known, 
 to find a fourth; it follows that there may be twenty dis- 
 tinct cases in Geometrical Progression. We shall, how- 
 ever, notice but two of them, and refer the student to 
 Algebra for the others. 
 
 CASE I. 
 
 Art. 227. Given the first term, the number of terms, 
 and the ratio, to find the last term. 
 
 1. The first term df' a geometrical progression is 2, the 
 ratio is 3, and the number of terms is 8. What is the 
 last term ? 
 
 Solution. — From the nature of geometrical progression, it 
 is evident, that the second term equals the first term, multiplied 
 by the ratio ; the third term equals the first, multiplied by the 
 ratio squared ; the fourth term equals the firsts multiplied by 
 the ratio cubed, and so on for the follo^ving terms ; Hence, the 
 first term multiplied by tliat power of the ratio denoted by the 
 number of terms, less one, will give the last term. 
 
 2. A person traveling, goes 3 miles the first day, 6 
 miles the second day, 12 miles the third day, and so on, 
 increasing in geometrical progression, for 6 days. How 
 far did he go the last day ? 
 
 3. An individual commenced business with a capital of 
 $20, and was so fortunate as to double it once in every 
 two years; what was his capital, at the end of 25 years? 
 
 CASE II. 
 
 Art. 228. Given the first term, the last term, and the 
 ratio, to find the sum of all the terms. 
 
 10* 
 
226 PROGRESSION. [cHAP. IX. 
 
 1. The first term of a geometrical progression is 4, the 
 last term is 12500, and the ratio is 5. What is the sum 
 of all the terms ? 
 
 Explanation. — If from ratio times any series, as 2, 8, and 
 32, we subtract the scries, the remainder will be ratio times 
 32 — 2. It is also evident, that- if from ratio times the series, 
 we subtract, once the series there will remain, (ratio — 1) times 
 the series, which must equal ratio times 32 — 2. Therefore, 
 the sum of the series equals 4 times 32 — 2-i-(4 — 1.) Hence, 
 Multiply the last term by the ratio ; from the product subtract the 
 first term and divide the remainder by the rath diminished by one, 
 and it will give the sum of all the terms. 
 
 2. A gentleman engaged a horse and carriage to ride 
 LO miles, and agreed to pay ^ of a cent for the first mile; 1 
 k?ent for the second; two for the third ; and so on, increas- 
 ing in geometrical progression. How much &t this rate, 
 did the 10 miles' ride cost him ? 
 
 3. A speculator sold 10 horses o^this condition : that 
 he should pay $3 for the first horse; $9 for the- second; 
 $27 for the third, and so on, increasing in geometrical 
 ratio. What did the last horse cost, and what did they 
 all cost ? 
 
 Summation of an Infinite Decreasing Series. An 
 Infinite Series is one that, being continued, would run on 
 ad infinitum. If a decreasing geometrical series, as 1, ^, 
 T' i' tV» <^^-» ^® continued to infinity the last term evi- 
 dently may be considered 0. The sum of such a series 
 may be determined as follows . 
 
 Divide the, first term by a unit diminished by the ratio. 
 
 1. What is the sum of the infinite series, 1, i, i, -i, &c.? 
 
 2. What is the sum of the infinite series, 1, i, i, &c. ? 
 
 3. What is the sura of the infinite series, 1, |, f, &c. ? 
 
 4. What is th^ sum of the infinite series, 1, |, -^\, yf;^, 
 &c.? 
 
^T. 
 
 229.] 
 
 INVOLUTION. 
 
 227 
 
 I 
 
 CHAPTER X. 
 INVOLUTION AND EVOLUTION. 
 
 Inyolution. 
 
 Art. 229. Involution, teaches the method of raising 
 a number to any proposed power. 
 
 The number to be raised to a given power is called the 
 Jirst power, or root. The product obtained by multiplying 
 that number by itself, is called the square, or second jpower 
 of that number. 
 
 4 inches. 
 
 Remark. — We generally say, the 
 square of a number, instead of the second 
 power of that number, because the 
 surface, or superficial contents of a 
 geometrical square is obtained by mul- 
 tiplying the number of linear units 
 expressing one of its sides by itself. 
 Thus, the side of the adjacent figure is 
 expressed by 4 linear units, (4 inches) 
 and its superficial contents, by 4x4=16 
 square indies. 
 
 If the square or second power of a number be multiplied 
 by the first power of that number, the product is called 
 the CUBE, or third power of that number. 
 
 Remark. — We generally say, the 
 cube of a number, instead of the third 
 power of that number, because the 
 solid contents of a geometrical cube is 
 expressed by the third power of the 
 number expressing one of its sides. 
 Thus, the solidity of the annexed cube 
 is expressed by 3 X 3 X 3 = 27 solid 
 feet. 
 
 The power to which a number is to be involved is some- 
 times expressed by a small figure, called an exponent or 
 
228 EVOLUTION, [chap. X. 
 
 iiidex, placed above and a little to the right of that num- 
 ber. Thus, 
 
 52 = 5 X 5 = 25 the square of 5. 
 
 53 = 5 X 5x 5 = 125 the cube of 5. 
 
 54 = 6X5X5X5 = 625 the fourth power of 5-. 
 
 &c,, &c., &c. 
 
 The exponent of a quantity shows how many times that 
 uantity mters as a factor. 
 
 Art, 230. A quantity is involved to any given power 
 by muliplying it by itself as many times as there are units in 
 the expone7it, less one. 
 
 1, What is the square of each of the following numbers : 
 1, 8, 9, 12, 25, 38, 274, and 487 ? 
 
 2, What is the cube of 23, 84, 96, and 273 respectively ? 
 
 3, What is the fourth power of 16, 18, 24, 147, and 263, 
 respectively ? 
 
 4, What is the fifth power of 3*2, 41-5, 82'5 and 829, 
 respectively ? 
 
 5, What is the fifth power of |, |, | and j, respectively ? 
 
 6, What is the cube of 3^, 2|, 4i, and 14|, respectively? 
 
 Evolution, 
 
 Art. 231. Evolution is the reverse of Involution. 
 It teaches the method of resolving a number into equal 
 factors, either of which is termed a root. 
 
 The square root of 49 ( = 7x7) is 7, since 7 is one of 
 the two equal factors of 49. 
 
 The cube root of 27 ( = 3x3X3) is 3, since 3 is one of 
 the three equal factors of 27, 
 
 Numbers whose roots can only be approximately ob- 
 tained, are called surd numbers. 
 
 The square mot is indicated by the symbol ^, and is 
 called the radical sign. Thus, ^9 = 3 ; ^25=5, 
 
 The cube root is indicated by placing the figure ^ above 
 the radical sign. Thus, ^27=3 j ^64=4. 
 
ART. 233.] 
 
 SQUARE ROOT. 
 
 229 
 
 SQUARE ROOT. 
 
 Art. 232. The square root of any number, which is not 
 a surd, may be determined by 
 
 Resolving the number into its prime factors — the mntinued 
 product of every other one of these different factors judUI he thz 
 root required. 
 
 1. What is the square root of 5184 ? 
 
 OPERATION. 
 
 5184 = 2x2*X2x2*x2x2*x3x3*x3x3* 
 
 Explanation. — Every other one of the different prime fac- 
 tors of 5184 is marked by *; the product of which is 2 X 2 X 
 2 X 3 X 3 = 72, the square root of 5184. 
 
 2. What is the square root of 900 ? 
 
 3. What is the square root of 18225 ? 
 
 4. What is the square root of 396900 ? 
 
 Art. 233. The square root of any quantity which is 
 not a surd, and is expressed by not more than four figures, 
 can be ascertained by inspection. 
 
 First, square the nine digits respectively, and observe 
 figure of each square number. 
 
 the terminating 
 
 The terminating figures that are alike 
 are linked together. 
 
 We observe that all square numbers end 
 in 1, 4, 9, 6, or 5; also, if the number ends 
 in 9, the figure in the root occupying the 
 unit's place must be either 3 or 7 ; if in 4, 
 the figure in the root occupying the unit's 
 place must be either 2 or 8, &c. The fig- 
 ures occupy JDg the hundreds, or the hunr 
 
230 SQUARE ROOT. [cHAP. X 
 
 dred's and thousa.id^s place, will rnable us to determine the 
 figure of the root occupying the ten's place ; and by the 
 excess of the given quantity above the square of the ten's 
 figure, we are enabled to tell which of the two figures that 
 will produce the terminating figure of the quantity, is the 
 root. 
 
 1. What is the square root of 5184 ? 
 
 Remark — In accordance with what we have already learned, we know the 
 figure in the*root occupying unit's place must be 2 or 8 j and the one occu- 
 pying the ten's place must be 7, as its square, 49, is the largest square number, 
 which is less than 61 ; and since the excess of the 51 above 49 is so small, we 
 take tne figure 2 for the unit's figure of the root. Hence, the square root of 
 the above number is 72 Should the number have been 6084. then the excess 
 of 60 above 49 would have been so great, we should have taken the 8 for the 
 unit" s figure of the root. Hence, we would have 73 for the square root of 
 6084. 
 
 2. What is the square root of 6t6 ? 
 
 3. What is the square root of 2209 ? 
 
 4. What is the square root of 1225 ? 
 
 5. What is the square root of 2916 ? 
 
 6. What is the square root of 3969 ? 
 t. What is the square root of 5041 ? 
 
 8. What is the square root of 1921 ? 
 
 9. What is the square root of 8464 ? 
 10. What is the square root of 9025? 
 
 Art. 234. The square of 1, (the smallest digit,) is 1. 
 The square of 9, (the largest digit,) is 81. Hence, the 
 square of any digit is expressed by either om or two 
 figures. 
 
 The square of 10 (the smallest number denoted by two 
 figures,) is 100. The square of 99, (the largest number 
 denoted by two figures,) is 9801. Hence, the square of 
 any number denoted by two figures, is expressed by either 
 three or four figures; in the same manner it may be shown, 
 that the square of any number denoted by three figures, 
 will be expressed by either Jive or six figures, &c. 
 
 Hence, the square of any number will contain twice as 
 many figures as that number, or twice as m^ny, less one. 
 Therefore, to extract the square root, we first separate the imrri' 
 her into jperiods of tivo figures each, commmcing at the right 
 
ART. 236.] 
 
 SQUARE ROOT. 
 
 231 
 
 Art. 235. As Evolution is the reverse of Involution, 
 
 we will involve a few quantities by considering them decom- 
 posed into UNITS, TENS, HUNDREDS, &c., froui wliich we will 
 deduce a general rule for the extraction of the square root. 
 
 The square of a binomial, that is, a quantity consisting 
 of two terms, is equal to, The, square of the first term, plus 
 twice the first term iiito the second, plus the square of the 
 second term. 
 
 1. What is the square of 35 ? 
 
 35 = 30 + 5. Consider 30 the first_term and 5 the 
 
 second; then by the above rule we have, 35'^=(30 + 5) = 
 30'+2x30x5 + 5'=:1225. 
 
 The evolution by multiplication is as follows : — 
 
 30 4-5 
 30 4-5 
 
 3024-30 X 5 
 
 30 X 5 + 5* 
 
 30'4-2x30x5-f 5' 
 
 This involution may be geometri- E 
 cally illustrated thus : — Suppose the 
 square ABCD, to be 30 inches each 
 way; then its superficial contents 
 is expressed by 30^ This square 
 may be increased by the two rec- 
 tangles ABFE andBCIH, each equal 
 in length to the side of the square, 
 and in width to BH, (or 5,) the 
 quantity by which the square has 
 been'increased ; hence, the area of 
 each of these rectangles is expressed by 30 X 5 ; also the little 
 square BHGF, whose side is BH, (or 5) ; hence, ^s area is 52. 
 
 Art. 236. The square of any polynominal is equal to, 
 The square cf the first term, plus twice the first term into the 
 second, plus the square of the second ; plus twice the sum of 
 the first two into the third, plus the, square of the third ; and 
 so on. 
 
232 
 
 SQUARE ROOT. 
 
 [chap. X. 
 
 (4004-50) X2 
 
 ?1 
 
 400x50 
 
 50^ 
 
 
 
 
 
 X 
 
 o 
 
 ^ — \ 
 
 
 \o 
 
 ^ ! 
 
 400' 
 
 X 
 
 o 
 
 + 
 
 
 o 
 
 o 
 
 
 ■^ 
 
 ^ 
 
 1. What is the square of 452 ? 
 
 452 = 400 + 50 + 2. Consider 400 the first term, 
 50 the second term, and 2 the third term; then by the 
 above theorem we have, (400 + 50 + 2'i2=:400'+2x400 
 X50+50'+2X(40Cf50)X2 + 22. 
 
 The following diagram exhibits 
 the above involution geometri- 
 cally. 
 
 By reversing the above process of involution, we obtain 
 for exTfacting the square root, the following 
 
 GENERAL RULE. 
 
 Commencing at the right, separate the number into periods 
 of two numbers each. 
 
 Find the greatest square number in the first period on the 
 left, and place its root at the right of the number, in the form 
 of a quotient ; also, on the left separating it from the num- 
 ber by a perpendicular line. Then subtract the square of 
 this root from the period on the left ; and to the remainder 
 annex the second period ; which will form the first dividend. 
 
 Double the root already found, (which is placed at the 
 left of the number) ; to this product annex a cipher, and it 
 will form the first trial divisor. The number of times the 
 trial divisor is contained m the first dividend^ will be the 
 next figure of the root, which must be added to the trial 
 divisor, to form the true divisor. Multiply the true divisor 
 by the figure of the root last obtained ; subtract the product 
 from the dividend, and to the remainder annex the n£xt period 
 for a NEW dividend. 
 
 To the last divisor, add the last figure of t/ie root found , 
 
ART. 236.] 
 
 SQUARE ROOT. 
 
 233 
 
 this sum with a djpher annexed will be the next trial divisor. 
 Then proceed as before, until all the periods have been brought 
 
 down. 
 
 Note. — When any dividend is not so large as its trial divisor, place a cipher 
 for the next figure of the root ; also, place a cipher at the right of the divisor, 
 and form a new dividend by annexing a new period. 
 
 1. What must be the length of the side of a square 
 pond that shall contain 54756 square feet ? 
 
 OPERATION. 
 Number. Root. 
 
 Linear ft. Sq. ft. Linear ft. 
 
 Ist trial divisor, 
 
 200 
 400 
 
 true divisor, 430 
 2d trial divisor, 460 
 
 true divisor, 464 
 
 54756(200 + 30 + 4 = 234. 
 40000 
 
 14756 
 12900 
 
 1856 
 1856 
 
 N 
 
 K 
 
 H 
 
 M 
 
 Explanation The re- A J I 
 
 quirement of the above ques- 
 tion was to determine the 
 side of a square that should 
 contain 54756 square feet. 
 
 It is evident that the side 
 of the square must be more 
 than 200 linear feet, since 
 the square of 200 is less than 
 54756 : also, that it must be 
 less than 300 linear feet, 
 since the square of 300 is 
 greater than 54756. There- D 
 fore, 2 is the greatest num- 
 ber whose square is contain- 
 ed in 5, (the left hand period,) and is the first, or hundreds' 
 figure of the root. 
 
 Let CDEL be a square whose side is 200 linear feet. Then 
 its area is 200'^ = 40000 square feet, which being taken from 
 the given number, leaves 14756 square feet, to be added to the 
 square DL. We first add the two rectangles CN and EM 
 
 200 
 
 E F G 
 
234 SQUARE ROOT. [cHAP. X. 
 
 each, equal in length to the side of the square DL, which has 
 already been found to be 200 feet. Therefore, the length of 
 the two rectangles is 400 feet, which forms the 1st trial divisor. 
 The 14750 being divided by the 1st trial divisor, gives a quo- 
 tient of 30, which is the width of the rectangles CN and EM, 
 also, the length of the side of the small square LMKN. Add- 
 ing to 400, (the length of the two rectangles CN and EM,) 30, 
 (the length or side of the square LK,) gives 430, the true 
 divisor. Multiply 430, the length of these three pieces by 30, 
 their width, gives 12900 square feet, the amount by which the 
 square DL, has been increased. Subtract this amount from 
 14756 square feet leaves 1856 square feet, to be added to the 
 square BDFK. 
 
 We now add to the square DK, the two rectangles BJ and 
 FH, each equal in length to 200, (the side of the square DL.) 
 plus 30, (the width of the rectangles just added to the square 
 DL). Therefore, 2 (200 + 30) == 460 linear feet, is the length 
 of the two rectangles BJ and FH, which forms the 2d trial 
 divisor. Divide 1856, (the number of square feet remaining 
 to be added to the square DK.) by the 2d trial divisor, gives 4 
 for the width of the two rectangles ; also the side of the small 
 square KHIJ. Hence, the length of the two rectangles BJ and 
 FH, increased by the length of the square KHIJ, is 464, which 
 forms the last true divisor ; this length being multiplied by 4, 
 the width of the three pieces, gives 1856 square feet, which 
 being taken from 1856 leaves no remainder. Therefore, the 
 square ADGl, the side of which is 200 -f- 30 -f- 4 = 234 linear 
 feet, contains 54756 square feet. 
 
 By omitting the ciphers the foregoing operation will 
 take the following condensed form : — 
 
 OPERATION. 
 Number. Root. 
 Linear ft. Sq. ft. Linear ft. 
 
 2 
 
 43 
 
 464 
 
 54756(234 
 4 
 
 147 
 129 
 
 1856 
 1856 
 
ART. 238.] ' SQUARE ROOT. 23jGt 
 
 2. What is the square root of 85264 ? 
 
 3. What is the square root of 55696 ? 
 
 4. What is the square root of 1971216 ? 
 
 5. What is the square root of 5499025 ? 
 
 6. What is the square root of 269222464 ? 
 1. What is the square root of 6497004816 ? 
 
 8. What is the square root of 609596100 ? 
 
 9. What is the square root of 4164081009664 ? 
 
 Remark. — The roots of the above numbers can also be determined by Art 
 232. 
 
 Art. 237. To extract the square root of a decimal. 
 
 Commencing at the decimal paint, separate the number into 
 periods of two figures each; then proceed according to the 
 General Rule^ and to the number annex ciphers^ until the 
 desired number of figures in the root are obtained. 
 
 1. What is the square root of 223024 ? 
 
 2. What is the square root of 64-1601 ? 
 
 3. What is the square root of 84-65901 ? 
 
 4. What is the square root of 187'20924 ? 
 
 5. What is the square root of 5296 ? 
 
 6. What is the square root of 2 ? 
 
 7. What is the square root of 3 ? 
 
 8. What is the square root of 5 ? 
 
 9. What is the square root of 6 ? 
 
 10. What is the square root of 7 ? 
 
 11. What is the square root of 9 ? 
 
 Atr. 238. To extract the square root of a common 
 fraction. 
 
 Reduce the fraction to its simplest form; then extract 
 the root of the numerator and denominator separately, if 
 they have an exact root; if not, reduce the fraction to a 
 decimal, and proceed as in Art, 237. 
 
 1. What is the square root of -//t ^ 
 
 2. What is the square root of ||J| ? 
 
 3. What is the square root of f | ? 
 
 4. What is the square root of | ? 
 
 5. What is the square root of | ? 
 
236 SQUARE ROOT. [CHAP. X. 
 
 QUESTIONS INVOLVING THE PRINCIPLES OP SQUARE ROOT. 
 
 Art. 239. A triangle is a 
 figure having three sides, and 
 therefore three angles. When 
 one of the angles is right, like the 
 corner of a square, (that is, con- 
 tains 90°,) the triangle is called ease. 
 a right-angled triangle. The 
 
 side opposite the right-angle, is called the hypothenuse ; one 
 of the remaining two sides is called the last, and the other 
 the perpendicular. 
 
 Art. 240. Two sides of a right-angled triangle being 
 given, the third side can be found by means of the follow- 
 ing theorem. 
 
 It is an established theorem of geometry., that the square of the 
 hypothenuse is eqimlto the sum. of the squares of the other two sides. 
 
 Tlierefore, the square of one of the sides is equal to tlie square of 
 the hypothenuse, diminished by the square of the other side. 
 
 1. How long must a ladder be to reach to the top of a 
 tree 52 feet high when the foot of it is 39 feet from the 
 tree? 
 
 OPERATION. 
 
 62' = 2704 
 39' = 1521 
 
 ^^4225 = 65 feet, the length of the ladder. 
 
 2. It is ascertained that a ladder 95 feet in length, 
 standing on the bank of a river 57 feet in width, reaches 
 to the top of a tree standing on the opposite bank. What 
 is the height of the tree ? 
 
 3. Two ships start from the same place and sail, the one 
 North and the other East. How far apart will they be 
 in 6 days, providing they sail, at the rate of 72 and 96 
 miles an hour, respectively ? 
 
 4. A man standing 39 paces, of 3 feet each, from a tree 
 which is 95 feet high and 6 feet in diameter, shoots a 
 pigeon from its topj how far did the ball move before it 
 
ART. 241.] 
 
 PRACTICAL QUESTIONS. 
 
 23t 
 
 reached the pigeon, providing the man's eye, the place 
 from which the ball started, is five feet above the ground ? 
 5. What is the distance between the opposite corners 
 of a parallelopipedon, the length of which is 8 feet, and 
 the width and depth, each 6 feet ? 
 
 Remark. — The question will be more readily comprehended by inspecting 
 the following diagram 
 
 From the right-angled triangle IHE ^ jj 
 
 we determine the hypothenuse, IE. 
 Then, from the right-angle triangle 
 lEB, we determine the hypothenuse, 
 IB, which is the distance betwee^ the 
 opposite corners of the parallelopipe- 
 don, DE. 
 
 Art. 241. The three smallest integers that can accu- 
 rately express the length of the sides of a right-angled 
 triangle, are 3, 4, and 5. 
 
 \^ / 
 
 \^ 
 
 
 / 
 
 s 
 
 
 /^\! 
 
 Thus- 
 
 A 
 
 If we multiply these three numbers by 2, it will give a 
 right-angled triangle, the sides of' which are 6, 8, and 10; 
 if by 3, another, the sides of which are 9, 12, and 15. In 
 the same way, any number of triangles may be obtained, 
 the sides of which are expressed by integers. 
 
 Hence, by knowing two sides of a right-angled triangle, 
 the sides of which are to each other as 3, 4, and 5 we can 
 readily determine the remaining side, mentally. 
 
 1. What must be the length of a ladder to reach to the 
 top of a tree, 48 feet high, when its foot is placed 36' feet 
 from its base ? 
 
 2. What is the distance, between the opposite corners 
 of a rectangular field, the length of which is 32 rods, and 
 the width 24 rods ? 
 
 3. What is the length of a rectangular field, the dis- 
 tance between the opposite corners of which is 30 rods, 
 and the width of which is 18 rods ? 
 
238 
 
 SQUARE ROOT. 
 
 [chap. X. 
 
 Mechanical Application of the Foregoing. 
 
 Art. 242. Mechanics generally make use of a right- 
 angled triangle, the sides of which are 6, 8, and 10 feet, 
 respectively, in squaring the walls for the foundation of a 
 building, &c. This is done by placing an upright stick 
 where we design the corner of the building to be, with a 
 cord about it so as to form a plain angle ; then measure 
 off 6 feet on one end of the cord, and 8 feet on the other, 
 and holding the cord horizontal, place the terminating 
 point of the 6 feet (which may be Tuarked by stickii g a 
 pin through the cord,) at one extremity of a ten-foot 
 pole, and the terminating point of the 8 feet at the other 
 extremity. The triangle thus formed will be a right- 
 angled triangle. 
 
 The following diagram will render 
 the above remark more plain. P re- 
 presents the upright stick, about 
 which the cord is placed. PA, the 
 6 feet measured off, PB, the 8 feet, 
 and AB, the ten-foot pole. 
 
 Length of Braces. 
 
 AB is the corner post of a build- 
 ing; DG a girth ; and CE a brace. 
 The triangle CDE is a right-angled 
 triangle : hence, the length of tire 
 brace CE is found by extracting the 
 square root of the sum of the squares 
 of the two lengths DE and DC. (See 
 Art. 240.) 
 
 Art. 243. The length of any brace, v.ihen DC and 
 DE are of the same length, is equal to the length DC 
 -\- as many times 5 inches as DC is feet in length. This 
 fact is of great practical utility to carpenters. 
 
 1. What is the length of a brace, when the two sides 
 DC and DE are each 3 feet long ? 
 
ART. 
 
 24.4.] 
 
 LENGTH OF BRACES. 
 
 239 
 
 The length of the brace will be 3 feet + 3 times 5 
 inches, equal to 3 feet + 15 inches = 4 feet 3 inches. 
 
 2. If the sides DC and DE are each 4 feet long, 3^ feet 
 long, 4i feet long, 5 feet long, 5^ feet long, or 6 feet long, 
 what would be the length of tlie braces for the several 
 conditions, respectively. 
 
 Length of Rafters, &c. 
 
 Art 244. To find the length 
 of a rafter. 
 
 FB is called the base line of the 
 roof, and HM the height of the 
 pitch of the roof. 
 
 If the height of the pitch is equal 
 to om-half of the base line; HM=: 
 MB; hence, the length of the raf- 
 ter, HB, is found in the same man- 
 ner we found the length of a brace, 
 under Art. 243. 
 
 A roof is said to be one-fon,rth, two-fifths, three-sevenths^ 
 &,c., pitch, when HM = ^, f , -f , &c., of the base line, FB. 
 The length of the rafters of such roofs, is found by extracting 
 the square root of the sum of the squares of JIM and MB. 
 (See Art. 240.) 
 
 1. In a three-eighths pitch roof, the base of which is 40 
 feet, what is the length of the rafters ? 
 
 -In this example the height of the pitch HM= 
 
 Solution. 
 15 feet. 
 
 _ The MB = 20 feet. 
 152 = 225 feet. 
 202 = 400 feet. 
 
 The square root of 625 = 25, the length of the rafters. 
 
 2. In a two-seventh pitch roof, the length of the base 
 of which is 28 feet, what is the length of the rafters ? 
 
 3. In a one-fourth pitch roof, the base of which is 24 
 feet, what is the length of the rafters ? 
 
240 SQUARE ROOT. [CHAP. X. 
 
 4. In a three-fifths pitch roof, the base of which is 40 
 feet, what is the lenorth of the rafters ? 
 
 Art. 245. It is an established theorem of geometry, t/iat 
 all similar surfaces, or areas, are to each other as the squares 
 of their like dimensions. 
 
 Hence, the like dimensions of similar figures are to each 
 othtt as the square roots of their areas. 
 
 1. There are two circular fish-ponds ; one of which is 20 
 rods in diameter, and the other 4 rods in diameter. How 
 much more surface in the one than in the other ? 
 
 2. A farmer has a rectangular piece of land containing 
 61 acres, the width of which is 10 rods, and the length 
 100 rods. His neighbor has a similar piece of land con- 
 taining 9 acres. Required the length and breadth of hi8 
 neighbor's piece of land. 
 
 3. Suppose a horse to be tied to a post in the centre of 
 a field, by a rope 7'13 rods in length, and is thereby ena- 
 bled to graze upon 1 acre. How long should the rope be 
 to allow it to graze upon 6^ acres ? 
 
 4. By observation I find that 11-i- gallons of water will 
 flow through an orifice of 1^ inches in diameter in 1 second. 
 How large should the orifice be so as to discharge 2J- gal- 
 lons in the same time. 
 
 5. If it require 156^ yards of carpet to cover a floor 
 that is 25 feet in length, and 20f feet in width ; what 
 must be the dimensions of a similarly shaped floor, that 
 requires 56|- yards of the same kind of carpet to cover ? 
 
 6. Five men purchased a grindstone 40 inches in dia- 
 meter. How much of the diameter must each grind off, 
 so as to have \ of the stone ? 
 
 Remark. — After the first has ground off his share, | of the 
 Btone remains, and its diameter will be 40^i = 8^20, &c. 
 
 Art. 246. When the base and the sum of the height 
 and hypothenuse of a right-angled triangle are given, to find 
 the hypothenuse : 
 
 Add the square of the height and hypothenuse to the square 
 of the base, and divide their sum by twice tJie height and 
 
4RT. 241] 
 
 CUBE ROOT. 
 
 241 
 
 1. There is a tree 80 feet in height, standing by the 
 bank of a river 50 feet in width. Where must this tree 
 break off, so that the top will reach across the river, while 
 the broken parts remain in contact ? 
 
 CUBE ROOT. 
 
 Art. 247. Whenever the cube root of a quantity is 
 expressed by a whole number, it may be found by, 
 
 Resolving the number into its prime factors. The product 
 of every third factor of these different factors, will he the 
 root required. 
 
 1. What is the cube root of 129000 ? 
 
 OPERATION. 
 
 2)729000 
 
 ( 
 
 Explanation. — Taking the continued product 
 of every third one of these different factors, 
 (which are marked by * ) we have 2x3x3 
 X 5 == 90, which is the cube root of 729000. 
 
 2)364500 
 
 ^2) 182250 
 
 3)91125' 
 
 3)30375 
 
 »3) 10125 
 
 3)3375 
 
 3)1125 
 
 *3)375 
 
 5)125 
 
 5)25 
 
 *5 
 
 2 What is the cube root of 4741632 ? 
 
 3 What is the cube root of 98611128 ? 
 4. What is the cube root of 621875 ? 
 
 11 
 
242 CUBE ROOT. [chap. X. 
 
 5. What is the cube root of 2388t8t2 ? 
 
 6. What is the cube root of 5639752 ? 
 
 1. What is the cube root of 5936493568 ? 
 
 Art. 248. The cube root of any quantity which is 
 not a surd and is expressed by not more than six figures, 
 can be ascertained by inspection. 
 
 First, cube the nine digits respectively, and observe tho 
 terminating figure of each cube number. 
 
 Digits. Their 
 Cubes. 
 
 13 =s 1 It will be observed that the terminating figure 
 
 2' = 8 of each of the cubes of the nine digits is either 
 
 33 = 2 7 1, 2;, 3, 4, 5, 6, 7, 8, or 9 ; hence, every cube num- 
 
 43 = 6 4 ber must terminate with one of the nine digits, 
 
 53 = 12 5 consequently the figure of the root occupying the 
 
 63 = 21 6 unit's place is readily determined by inspection. 
 
 73 = 34 3 The figure of the root occupying the tens place 
 
 8^ = 51 2 is determined by inspecting the number, con- 
 
 93 = 72 9 sidered as units, that preceed the first three figures. 
 
 1. What is the cube root of 614125 ? 
 
 ExPLANATio,N. — As this number ends in 5 the figure in the 
 root occupying the unit's place must be 5. 8 is the largest 
 number, the cube of which is less than the number expressed 
 by the figures on the left of the first three figures, which ia 
 614; hence, the cube root of 614125 is 85. 
 
 2. What is the cube root of 8593t ? 
 
 3. What is the cube root of 226981 ? • 
 
 4. What is the cube root of 117649 ? 
 6. What is the cube root of 50653 ? 
 6. What is the cube root of 110592 ? 
 1. What is the cube root of 405224 ? 
 
 8. What is the cube root of 438976 ? 
 
 9. What is the cube root of 778688 ? 
 
 Art. 249. Before we attempt to explain the usual 
 method of extracting the cube root, we will involve a 
 number, consisting of units and ^erw, and of units, tmsf 
 and hundreds to its third power. 
 
ART. 260.] ' CUBE llOOT. 243 
 
 Remark.— The cube of I, the smallest digit is 1. The cube of 9, the largest 
 digit is 7-29. Therefore, the cube of any digit is expressed by one. two, or tinet 
 figures. 
 
 The cube of 10. the smallest number denoted by two figures, is 1000. The 
 cube of 99, the largest number denoted by Iwo figures, is 970299. Therefore, 
 the cube of any number denoted by two figures is expres.sed by fouk, five, 
 or SIX tigtires. In the same maniier it may be shown, that the cube of a num- 
 ber denoted by three figures is expressed by sevk.n, Eir.H r, or ninf., figures, &c. 
 
 Hence, if a number be denoted by one. two, or th-ee figures, its cube root will 
 be expressed by one figure ; if hy four, Jive, or six figures, its cube root will 
 be expressed by two figures, &c. 
 
 In general the cube will contain three times as many figures as theroot, or three 
 times as many less one or two. Therefore, to extiuct the cube root, we first 
 separate tlie number into periods of three figures each, commencing at the right. 
 
 Art. 250. The cube of a Binomial, (that is, a num- 
 ber consisting of two terms,) is, the cube of the first, or 
 left hand term, plus three times the square of the first term 
 into the second, plus three times the first term into the square 
 of the second term, plus the cube of the second term. 
 
 In general, the cube of any Polynomial is equal to the 
 cube of the first, or left-hand term, plus three times the square 
 of the first term into the second, plus three times the first into 
 the square of the second, plus the cube of the second ; plus 
 three times the square of the sum of the first two into the 
 third, plus three times the sum of tJie first two into the square 
 of the third, plus the cube of the third, Sfc. 
 
 1. What is the cube of 89 ? 
 
 89 = 80 + 9; ami by Art._250, we have 
 
 (80 + 9) 3 = 80' + 3 X 80' X 9 + 3 X 80 X 92 + 9» 
 
 2. What is the cube of 3ot ? 
 357 = 300 + 5.0 + 7j_therefore; 
 
 (300 +_50 + 1)3= 3~00'+ 3 X 300' X 50 + 3 X 300X 
 60' + 50' X 3(300 + 50)2 ^ 7 +3(30(^ + 50) X 1*^ + 1' 
 
 3. What is the cube of 468 ? 
 
 468=400 + 60 + 8;.-., _ _ 
 
 (400 + 60+8)3=400'+3x400'x60 + 3X400x60'+60^ 
 + 3(400 + 60)^x8+3(400 + 60) X 82+83 
 
 The involution of example 1, by multipJ»caMoa. is as 
 follows: — (the second, &c., is similar to it). 
 
244 
 
 CUBE ROOT. 
 
 [chap. X 
 
 80 + 9 
 80 + 9 
 
 80'+80X9 
 
 80x9 + 95 
 
 (80 + 9)2=80'+2X 80x9 + 92 
 80+9 
 
 80^+2X80x9+80X92 
 
 80 X9 + 2X80X92+93 
 
 (80 + 9)3=80'+3x80'x 9+3x80x9^+93 
 
 We will now illustrate geometrically the involution of the 
 first example. 
 
 How many cubic feet in a cube, each side of which is 
 89 feet ? 
 
 89=80 + 9. 
 
 Fig. 1. D 
 
 Suppose each side of the cube AD, 
 (fig. 1,) to be 80 feet; then its solid 
 
 contents will be 80^ = 512000 cubic 
 feet. 
 
 To increase the size of the cube 
 AI), we will first add the three 
 square slabs, AC, BD, and CE, 
 each of the sides of which is 80 
 feet, (the side of the cube AD,) 
 and the thickness of each, 9 feet. 
 Hence, the solid contents of one 
 of these square slabs is 80^^ X 9, 
 and the three, 3 X 80^^ X 9 = 
 172800 cubic feet. 
 
ABT. 260.] 
 
 CUBE ROOT. 
 
 245 
 
 Fig. 2. 
 
 The cube AD, increased by the 
 three square slabs, AC, BD, and 
 CE, is represented by Fig. 2 ; — 
 the contents of which is 612000 
 4-172800 = 684800 cubic feet. 
 
 We will now increase Fig. 2, 
 by the three equal corner-pieces, 
 AB, BC, and CD, the length oi 
 each being 80 feet, (the side of 
 the cube AD,) and the width and 
 thickness of each, 9 feet. Hence, 
 the solid contents of one of these 
 pieces is 80 X 92, and of the 
 three, 3 X 80 X 9^ = 19440 cubic 
 feet 
 
 Fig. 2, increased by the three 
 pieces, AB, BC, and CD, is rep- 
 resented by Fig. 3 ; — the contents 
 of which is 512000 -f 172800 X 
 19440 = 704240 cubic feet. 
 
 We will now increase Fig. 3, 
 by the small cube XY, each side 
 of which is 9 feet ; therefore, its 
 contents is 9^ = 729 cubic feet ; 
 
 This cube is then represented by 
 Figure 4, the contents of which is 
 512000 -f- 172800 -f 19440 -f 
 729 = 704969 cubic ""eet, and 
 the side of which is 8t feet. 
 
 1 '• / 
 
246 CUBE ROOT. [chap. X. 
 
 By reversing the above process, we obtain for extracting 
 the cube root, the following 
 
 GENERAL RULE. 
 
 CommeTicing at units, separate the number into periods of 
 three figures each. 
 
 Then find the largest digit, the cube of which shall not 
 exceed the left-hand period. Place this digit, which is called 
 the first figure of the root, on the right, in the form of a 
 quotient ; also, on the left, for the first term of a first column, 
 and its square for tM first term of a second column, and from 
 the left-hand period of the given number, subtract its cube. 
 Then to the rcTnainder, annex the mxt period, for the first 
 DIVIDEND. JYow double the term in the first column, for its 
 second term, and add its product into the root already found, 
 to the first term of the second column, for the first trial 
 DIVISOR. Consider two ciphers annexed to the trial divisor, 
 and write the number of times it is contained in the divi- 
 dend, for the next figure of the root ; also, annex it to 
 Hue sum of the last term in the first column, and the first 
 figure of the root ; — this will be the next term of the first 
 column. Add the product of this term into the digit of the 
 root last found, advancing it two places to the right, to the 
 last term of the second column, for its next term ; this will be 
 the TRUE DIVISOR. From the dividend, subtract the product 
 of the true divisor into the digit of the root found ; and to 
 the remainder annex the nzxt period, for the second dividend. 
 
 Proceed in a similar way until all the periods have been 
 used. 
 
 Remark. — By carefully examining the foregoing involution, the pupil will 
 be able to deduce other rules for the extraction of the cube root, some of 
 which may perhaps, appear more plain than the one I have just given, as this 
 is more readily deduced from Algebraic involutions. I have given this rule, 
 as it will be less laborious to extract the cube root of large numbers by it, 
 than by many other rules usually given ; also, because it keeps distinct the 
 three geometrical magnitudes — lines, surf acec, and solids. 
 
 'J'he Jirst rule, however, is the most simpk;, and will ^e found of much im- 
 
 1)0 tance in reducing surd Quantities to their simplest form, (as will hereafter 
 )f fifxplained,) or in determining the roots of rational quantities. 
 
 1. What is the cube root of 104969 ? 
 
RT. 260.] 
 
 CUBE ROOT. 
 OPERATION. 
 
 2 
 
 First Col. 
 
 Sfcond Col. 
 
 NuMBKR. Root. 
 
 Unear ftet. 
 
 S<iuare feet. 
 
 Cubic feet. Linear feet. 
 
 80 
 160 
 
 6400 
 19200, trial divisor. 
 
 704969(80 -f 9 = 89 
 512000 
 
 249 
 
 21441, true divisor. 
 
 192969, 1st dividend. 
 192969 
 
 24T 
 
 
 
 Explanation. — We first find the greatest cube contained in 
 the left-hand period. We know that this number must be 
 more than 80, since 80^ = 512000, which is less than 704969; 
 also, that it must be less than 900, since 90^ = 729000, which 
 is greater than 704969. Hence, the first, or left-hand figures 
 of the root, is 8 ; whose cube is 512, which is the greatest cdbe 
 contained in 704, the first or left-hand period. 
 
 Fig. 1. D 
 
 Let each side of the cube AD, 
 represented by Fig. 1, be 80 
 linear feet ; then its cubical con- 
 tents will be 80^ = 512000 cubic 
 ft. , and 704969—512000=192969 
 cubic feet, which is still to be 
 added to the cube AD. 
 
 We first add the three square 
 slab pieces AB, BC, and CD, 
 whose length an"d breadth are 
 each respectively 80 feet, (the 
 side of the cube AD.) The area 
 of the face of the first piece, AB, 
 is 80^ = 6400 square feet. The 
 length of the other two pieces, 
 BC, and CD, is 80 -f- 80 = 160 
 feet, and their width 80 feet. 
 Hence, their superficial contents 
 is 160 X 80 = 12800 square feet, 
 which added to 6400 square feet, 
 the superficial contents of the 
 piece, AB, gives 19200 square 
 
248 
 
 CUBE ROOT. 
 
 [chap. 
 
 feet, the superficial contents of 
 the three pieces, AB, BC, and 
 CD. As these three square slabs 
 make up by far the greatest 
 amount of the whole increase, if 
 we divide 192969, (the number 
 of cubic feet remaining to be 
 added,) by 192000, the number 
 of square feet in the three pieces, 
 AB, BC, and CD, (which may be 
 called the trial divisor,) it will 
 give their thickness ; which we 
 find to be 9 feet. 
 
 Figure 2, represents the cube 
 AD, with the three pieces AB, 
 BC, and CD, added. 
 
 We now add the three corner- 
 pieces, E(j, HF, and HX, whose 
 lengths are respectively 80 feet, 
 (the side of the cube AD,) and* 
 whose width and thickness are 
 each 9 feet respectively ; also, 
 the corner-piece AW, whose 
 length, width, and thickness, are 
 each 9 feet. Therefore, the length 
 of the three pieces, EG, HF, and 
 HX, is 240 feet ; which being in- 
 creased by the length of Ihe cor- 
 ner-piece, AW, gives 249 feet for 
 the length of the four pieces. 
 Their width is 9 feet ; therefore, 
 249 X 9 = 2241 square feet, is 
 their superficial contents ; which 
 being increased by 19200 square 
 feet, the superficial contents of 
 the three pieces already added 
 on, gives 21441 square feet, (the 
 superficial contents of the seven 
 pieces added on,) which being 
 multiplied by 9, their thickness, 
 gives 192969 cubic feet, their 
 solid contents, which being sub- 
 tracted from 192969 cubic feet, 
 the quantity that remained to be 
 added to the cube AD, leaves no 
 remainder ; therefore, a cube 
 represented by Figure 3, whose 
 side is 80 -f 9 = 89 feet, will 
 cantain 704969 cubic feet. 
 
 Fig. 3. 
 
 ^=- 
 
 
 
 
 7^ 
 
 ^ 
 
 
 ^ 
 
 
 ijl|iil!l:JI!:liiii 
 
 |:iillli!lll:';llii 
 
 !l!lll! 
 
 
 lii-' 
 
 
 l.:i! 
 
ART. 251.] CUBE ROOT. 249 
 
 This work may be condensed by omitting the ciphers and 
 unimportant terms. 
 
 First Col. Second Col. Niimhei: Root. 
 
 8 64 704969(89 
 
 16 192 512 
 
 249 21441 192«69 
 
 192969 
 
 2. What is the cube root of 12895213625 ? 
 
 Remark.— To render the method of extracting the cube root familiar, when 
 there are a number of figures in the root, we will perform the above example 
 
 First Col. Second Col. Number. Root. 
 
 2 4 12895213625(2345 
 
 8 
 
 4 12 
 
 63 1389 4895 
 66 1587 4167 
 
 694 161476 728213 
 698 164268 645904 
 
 7025 16461925 82309625 
 
 82309625 
 
 Remark. — When the trial divisor is greater than its corresponding dividend, 
 place for the next figure of the root, and bring down the next period. Then 
 use the same trial divisor with two more ciphers annexed. 
 
 3. What is the cube root of 46964099891t ? 
 
 Remark. — Whenever the number has not an exact root, there will be a re- 
 mainder after the last period has been brought down. The process may b* 
 continued, and the true root more nearly obtained, by annexing ciphers to 
 new periods. The figures thus obtained will be decimals. 
 
 4. What is the cube root of 1860867 ? 
 
 5. What is the cube root of 469640998917 ? 
 
 6. What is the cube root of 58050510848 ? 
 
 7. What is the cube root of 84672374 ? 
 
 8. What is the cube root of 3 ? 
 
 9. What is the cube root of 5 ? 
 
 10. What is the cube root of 7 ? 
 
 11. What is the cube root of 493780134751068073294S « 
 
 Art. 251. To extract the cube root of a decimal. 
 First, commmcing at the decimal 'point, separate the numbers 
 
 11* 
 
250 CUBE ROOT. [chap. X. 
 
 into periods of three figures each ; if necessary, annex cipher s^ 
 so that tibe decimal may he separated into equal periods. Thmi 
 -proceed as usual. 
 
 12. What is the cube root of 561-515625 ? 
 
 13. What is the cube root of 460-099648 ? 
 
 14. What is the cube root of -1U649 ? 
 
 15. What is the cube root of t-256313856 ? 
 
 16. What is the cube root of 41-86 ? 
 
 Art. 252. To extract the cube root of a fraction, or 
 mixed number, first reduce either of them to its simplest 
 form; then find the root of the numerator and denominator 
 separately, if their roots can be accurately found ; if' not, 
 reduce the fraction to a decimal, and extract the root as 
 above directed. 
 
 It. What is the cube root of -|||| ? 
 
 18. What is the cube root of ifffffff- ? 
 
 19. What is the cube root of ^\^ ? 
 
 Art. 253. A rational quantity is one that can be ex- 
 pressed in numbers. Thus, ^"11 is rational. 
 
 An irrational or surd quantity is one that has not an 
 exact root, or which cannot be expressed in numbers. 
 Thus, ^,3/3 is a surd. 
 
 When the cube root of a large quantity is required, it 
 will be found more convenient to find the root of the 
 rational part of the number by Art. 247, and the root of 
 the surd part by General Rule. - 
 
 Thus, ^40=^2X2 x2x"5 = 2^5 = 2xl'7099764- = 
 3'419952 + . 
 
 1. What is the cube root of 19208000 ? 
 
 OPERATION. 
 
 2)19208000 
 
 2)9604000 
 *2)48020'00 
 
 [over. 
 
ART. 253.] PRACTICAL QUESTIONS. 251 
 
 2)2401000 
 2)1200500 
 ^2)600250 
 
 5)300125 
 
 7)2401 
 
 Remark. — Every third factor is marked thus, *. 
 Their product will be the cube root of the ra- 
 5)60025 tional part, which is 2x2x5x7 = 140. This 
 ^5) 12005 multiplied by ^/TTwill equal the cube root of the 
 above number. Thus, 1404/7=140 X 19 12931-f- 
 =267-81034-1-. 
 7)343 
 
 *7)49 
 
 2. What is the cubfe root of 128625 ? 
 
 3. What is the cube root of 61631955000 ? 
 
 4. What is the cube root of 257250 ? 
 
 5. What is the cube root of 84035 ? 
 
 6. What is the cube root of 2401000 ? 
 
 PRACTICAL QUESTIONS IN CUBE ROOT. 
 
 Art, 253. It is a theorem of geometry, that all similar 
 folids are to each other as the cubes of their like dimensions. 
 Therefore, 
 
 The dimensions of similar solids are to each otJier as thz 
 CUBE ROOTS of their solidity. 
 
 1. If a ball of iron 2 inches in diameter weigh 5 pounds, 
 what will a ball of the same metal weigh, the diameter of 
 which is 7 inches. 
 
 OPERATION. 
 
 2^ : 73 : : 5 : weight required. 
 8 : 343 : : 5 : 214^ pounds. 
 
 2. If the Earth is 8000 miles in diameter, and Mercury 
 3200 miles, how many times larger is the Earth than 
 Mercury ? 
 
252 GAUGING. fCHAP. X. 
 
 3. If a ball | of an inch in diameter weigh ^ of a pound, 
 what must be the diameter of another ball of the same 
 metal to weigh 24 pounds ? 
 
 4. What is the diameter of a globe of gold that is 
 worth $9T85036"80, providing a pound of gold (avoirdu- 
 pois weight,) is worth $256 ? 
 
 It will be remembered that the specific quantity of gold is 1S| ; and that a 
 cubic foot of water weighs 62^ pounds. 
 
 6. If a man 5^ feet in height, weigh 125 pounds, how 
 tall is that man who weighs 216 pounds ? 
 
 6. A cask 60 inches long and 36 inches at the bung 
 diameter, contains a certain number of gallons ; what 
 must be the dimensions of a similar cask that shall contain 
 \ as much ? 
 
 1. A carpenter wishes to make a cubical cistern tl^t 
 shall contain 14088 cubic feet of water; what must be the 
 length of one of its sides ? 
 
 8. If a cellar 12 feet long, 8 feet deep and 8 feet wide 
 contain 768 cubic feet ; what is the dimensions of a similar 
 cellar that shall contain 20736 cubic feet? 
 
 9. What will be the dimensions of a rectangular box, 
 which shall contain 1845480 cubic feet, the length, 
 breadth, and depth being to eaeh^ther as 7, 5, and 3 ? 
 
 10. A ball of fine thread 5 inches in diameter is owned 
 by five women ; what portion of the diameter must each 
 wind off so as to share equally of the thread ? 
 
 Gauging. 
 
 Art. 254. Gauging teaches the method of finding the 
 contents of any regular vessel, in gallons, bushels, &c. 
 
 Art. 255. To find the number of gallons or busliels in 
 a square vessel. 
 
 Take the dimensicms in inches, and divide the product 
 arising from mvltiplying the length, breadth, and height 
 together by 2S2 for ale gallons, 231 fyr wine gallons, and 
 21 50-42 /or hushels. 
 
ART. 257.] GAUGING. 253 
 
 1. How many wine gallons will a cubical box contain, 
 that is 5 feet long, 2^ feet wide and 2 feet high ? 
 
 2. How many ale gallons will a vess el contain, that is 
 9 feet long, 4 feet wide, and 3 feet high ? 
 
 3. How many bushel of grain will a bin contain, that 
 is 16 feet long, 12 feet wide, and 6 feet high ? 
 
 Art. 256. To find the contents of casks. 
 
 Multiply the 'product of the, square of the mean diaTmter 
 and the length in inches, hy '0034, and it will give the con- 
 tents in wine gallons ; if by "0028 instead of *0034, it will 
 give the contents in beer gallons. 
 
 Remark. — The mean diameter is equal to the head diameter incrsased hy 
 
 -i. of the difference between the head and bunec diameters when the slaves 
 
 1 TT 
 
 are m%ich curved, or by adding 1 the difference, when but little curved ; and 
 
 1. How many wine gallons does a cask contain whose 
 length is 34 inches, bung diameter 28 inches, and head dia- 
 meter 20 inches ? 
 
 2. How many wine gallons does a cask contain, whose 
 length is 45 inches, bung diameter 32 inches and its head 
 diameter 22 inches 1 
 
 Art. 257. To find the contents of a round vessel, 
 wider at one end than the other. 
 
 To i of the square of tM difference of tht diameters, add 
 their product and multiply this sum hy the height. Then 
 multiply hy "0034 for wine gallons, and hy '0028 for ale 
 or beer. 
 
 1. How many wine gallons will a vessel contain, that is 
 48 inches in diameter at the bottom and 32 inches at the 
 top ; the length of which is 60 inches. 
 
 2. How many beer gallons will a tub contain, that is 
 40 inches in diameter at the bottom and 30 inches at the 
 top ; the length of which is 48 inches. 
 
254 
 
 MENSURATION. 
 
 [chap. XL 
 
 CHAPTER XI. 
 
 MENSURATION. 
 
 Geometrical Definations. 
 
 1. A Point is that which has position, but not magnitude. 
 
 2. A Line is length without width or thickness, as AB. 
 
 a ^B 
 
 3. Parallel Lines are those which are everywhere 
 equally distant, as AB and BO. ^ » 
 
 4. A Surface is that which has r 
 length and breadth without thickness, [ 
 as ABCD. B 
 
 5. A Solid Body is that which has length, 
 breadth, and thickness : and therefore com- 
 
 bines 
 AB. 
 
 the three dimensions of extension, as 
 
 Plane Figures. 
 
 Art. 258. 1. A Plane Figure is a plane surface ter- 
 minated on all sides by lines, either straight or curved. 
 
 2. A Polygon, or rectilineal figure, is a phme terminated 
 on all sides by straight lines. The sum of these bounding 
 lines is called the contour or jperimeter of the polygon. 
 
 3. A Triangle is a b 
 figure having three sides 
 and three angles, as ABC. 
 Its altitude is a line let 
 fall from the vertex per- 
 pendicular to the base, 
 asBD. 
 
ART. 258.] GEOMETRICAL DEFINITIOXS. 
 
 4. A Square is a figure that has all of its 
 Bides equal, and its angles right-angles, as 
 CDEG. 
 
 The line CE is called its diagonal. 
 
 5. A Parallelogram is a figure that 
 has its opposite sides parallel, as ABCF. 
 
 6. A Rectangle is an equiangular 
 parallelogram, as CYDG. 
 
 7. A Trapezoid is a figure 
 that has only two of its sides 
 parallel, as XYBZ. 
 
 Remark. — It has alreadj- been remarked, that any figure, the sides of which 
 are terminated by stiaigfit lines, is called a Polygon. 
 
 A polygon of three sides is called a Triangle ; that o{ four sides a Quadri- 
 lateral ; that of ^fe sides, a Pentagon; that of six, a Hexagon ; that of seven, 
 a Heptagon ; that of eight, an Octagon ; that of nine, a Nonagon ; that of ten, 
 a Decagon ; that of twelve, a Dodecagon, &c. 
 
 A Circle is a plane, termi- 
 nated by a curved line, every 
 point of which is equally distant 
 from a point within, called the 
 centre. The curved line is called 
 the circumference. 
 
 The Diameter of a circle is a line passing through the 
 centre, and terminated by the circumference, as AC. 
 
 The radius of a circle, is a line drawn from its centre to 
 the circumference, as BD, BE, &c.; hence, it is half the 
 diameter. 
 
256 
 
 MENSURATION. 
 
 [chap. XI. 
 
 9. An Ellipse is a plane 
 bounded by a curved line, the 
 sum of the distances from eve- 
 ry point of which to two given 
 points, is equal to a given line. 
 The line AB, is called the 
 transverse, and CD, the covr 
 jugate axis. 
 
 Solid Figures. 
 
 Art. 259. A Prism is a 
 solid, the sides of which are 
 parallelograms, and the ends 
 equal and parallel polygons, 
 as figure A. 
 
 Remarks.— When the ends of a Prism are triangular, it is called a triangular 
 •prism; when they are squares, it is called a square prism, &c. 
 
 B 
 
 2. A Cube is a j)rism, all 
 the sides of which are equal 
 squares, as figure B. 
 
 3. A Parallelopipedon is a 
 
 prism 
 
 the ends of which are 
 
 parallelograms, as figure C. 
 
 4. A Cylinder is a solid, 
 having equal circles for ends, 
 and is generated by the revo- 
 lution of a rectangle about one 
 of its sides; as figure D 
 
 J 
 
 5. A Pyramid is a solid, having for its base 
 a plane rectilinear figure ; and for its sides tri- 
 angles, whose vertices- meet in a point at the 
 top, called the vertex of the pyramid. Figure 
 E represents a triangular pyramid. 
 
ART. 
 
 261.] 
 
 MENSURATION OF SURFACES. 
 
 25t 
 
 6. A Cone is a solid, having for its base a 
 circle, and tapers uniformly to a point at the top. 
 Figure F represents a cone. 
 
 *l. A Frustum of a pyramid y 
 or a cone, is the part that remains 
 after cutting off the top by a 
 plane parallel of the base. 
 
 Fig. M represents the frus- 
 tum of a pyramid; and N, that 
 of a cone. 
 
 8. A Sphere is a solid, bounded by a 
 convex surface, every point of which being 
 equally distant from a point within called 
 a center. Figure X represents a sphere. 
 
 Mensuration of Surfaces, &c. 
 
 Art. 260. The pupil is referred to Geometry for the 
 demonstration of the following rules for measuring surfaces, 
 solids, &c. 
 
 Art. 261. The area of a figure is the number of square 
 inches, feet, or yards, &c., which it contains. 
 
 Problem 1 Given the base and altitude of a triangle, to 
 
 find its area. 
 
 Multiply the base by half the altitude. 
 
 1. What is the area of a triangle whose base is 9 feet, 
 and altitude 4 feet ? 
 
 2. What is the area of a triangle whose base is 36 rods, 
 and altitude 16 rods ? 
 
25B 
 
 MENSURATION. 
 
 [chap. XI. 
 
 Problem 2. — Given the three sides of a triangle, to find its 
 
 From half of the sum of the three sides, subtract each side 
 separately ; mid the square root of the continued product of 
 these three remainders aiid the. halj%sum will be the area. 
 
 1. What is the area of a triangle whose sides are, re- 
 spectively, 12, 18, and 20 feet? 
 
 2. What is the area of a triangular field whose sides are 
 respectively 40, 30, and 50 rods ? 
 
 Problem 3 Given the sides of a rectangle, or square, to 
 
 find its area. 
 
 Multiply the lerigth by the width. 
 
 1. How many square feet of boards will be required to 
 floor a room that is 36 feet long and 18 feet wide ? 
 
 2. How many acres in a rectangular piece of land 460 
 rods long and 380 wide ? 
 
 3. How many more acres in a piece of land 160 rods 
 square, than in a rectangular piece 40 rods in length an^ 
 32 rods in width. 
 
 Problem 4. — Given the base and altitude of a parallelogram, 
 to find its area. 
 
 Multiply the base by the altitude. 
 
 1. What is the area of a parallelogram whose length is 
 28 feet, and altitude 22 feet ? 
 
 Problem 5.— Given the altitude and the parallel bases of 
 a trapezoid, to find its area. 
 
 Multiply the altitude by half the 'mm of its parallel sides. 
 
 1. What is the area of a trapezoidal field, the parallel 
 sides of which are 18 and 24 rods respectively; — the per- 
 pendicular distance between these sides being 8 rods } 
 
 Rkmark.— This rule is of practical use to lumbermen in measuring boards. 
 
 Let ABCD be a tapering 
 board, and EG its length. 
 The half sum of its parallel 
 sides, AB and CD, is HK, the 
 width of the board nt the 
 middle point. Its area, there- 
 fore.is expressed by HKXE^O. 
 
ART. 261.] 
 
 MENSURATION OF SURFACES. 
 
 259 
 
 2. How many square feet in a tapering board 36 feet 
 long, 18 inches wide at one end and 32 in. at the other ? 
 
 Problem 6. — Given the diameter of a circle to find its cir- 
 cumference. 
 
 Miiltijply the diameter by 3'1416. 
 
 1, What is the circumference of a circle whose diameter 
 is 15 feet ? 
 
 2. What is the circumference of a circle whose diameter 
 is 24 rods ? 
 
 Remark. — The true ratio of the circumference of a circle to its diameter 
 has never yet been found, hs approximate value has been extended to more 
 than 200 places. A man by the name of Van Ceukn first extended the approxi- 
 mation to 36 places by means of continually bisectinpf the arc of a circle. 
 This was considered so great an achievement that the 36 numbers expressing 
 the circumference of a circle whose diameter is 1, was engraved on his 
 tomb-stone. 
 
 The following are the numbers : — 
 3-141592653589793238462643383279502884 
 
 Problem 7, — Given the circumference of a circle to find its 
 diameter. 
 
 Divide the circumference by 8*1416. 
 
 1. What is the diameter of a circle whose circumference 
 is 62-832 feet ? 
 
 2. What is the diameter of a circle whose circumfer 
 ence is 48'5 feet ? 
 
 Problem 8. — Given the diameter of a circle, to find its area. 
 Multiply the square of the diameter by -7854. 
 
 1. What is the area of a circle whose diameter is 15 
 inches ? 
 
 2. What is the area of a circle whose diameter is 36 ft. ? 
 
 Remark. — From the above rule, we can 
 readily determine the area contained between 
 two concentric circumferences. 
 
 It is also worthy of remark that the area of 
 a .square is to the area of an inscribed circle, 
 as 1 is to 0.7854. 
 
260 MENSURATION. [CHAP. XI. 
 
 Problem 9.— Given the diameter of a circle, to find the side 
 fan equal square. 
 
 Muliijffly the diameter Jy .8862. 
 
 1. A gentleman has a circular fish-pond that is 8 rods 
 in diameter ; what must be the side of a square pond that 
 shall contain the same area ? 
 
 2. I have a circular piece of board that is 36 inches in 
 diameter ; what must be the side of a square board that 
 shall contain the same area ? 
 
 Problem 10. — Given the diameter of a circle to find the side 
 of an inscribed square. 
 
 Multiply the diarnet^ by 0.^0*11, 
 
 1. Required the side of a square 
 that can be inscribed in the circle 
 ABO, whose diameter is 24 feet. 
 
 2. How large a square stick can be sawn from a piece 
 of round timber, that is 42 inches in diameter ? 
 
 Problem 11. — Given the diameters of an ellipse, to find its 
 area. 
 
 Multiply the product of its two axes by 0'*I854. 
 
 1. How many square rods in an elliptical garden, whos(, 
 transverse axis is 280 feet, and conjugate axis 210 feet ? 
 
 2. How many square feet in an elliptical table, the 
 transverse axis of which is 6 feet 9 inches, and the con- 
 jugate axis 3 feet 6 inches ? 
 
 Problem 12. — Given the length and the circumference of a 
 cylinder, to find its surface. 
 
 Multiply its circumference hy its length, a :\d to the product 
 add the area of the two bases. 
 
ART. 261.] MENSURATION OF SURFACES. ' 261 
 
 1. What is the surface of a cylinder whose T^ength is 22 
 feet, and diameter 4^ feet ? 
 
 Problem 13. — Given the length and the perimeter of the 
 base of a prism, or parallelopipedon, to find its surface. 
 
 Multiply its length by the perimeter of its base, arid to the 
 product add the areas of the two ends. 
 
 1. What is the surface of a square prism whose side is 
 2 foot 8 inches, and length 16 feet ? 
 
 2. What is the surface of a triangular prism whose 
 length is 12 feet, and whose sides are, respectively, 3, 4, 
 and 5 feet ? 
 
 Problem 14. — Giyen the side of a cube to find the area of 
 its surface. 
 
 Multiply the area of one of its sides by 6. 
 
 1. What is the area of a cubic block, the side of which 
 is 12 feet ? 
 
 Problem 15. — Given the slant height and the sides of the 
 base of a pyramid to find its area^ 
 
 To the area of the triangles that form its sides, add the 
 area of the base. 
 
 1. What is the area of a triangular'^pyramid, the slant 
 height of which is 12 feet, and the sides of its base 3, 4 
 and 5 feet, respectively ? 
 
 ' 2. What is the area of a square pyramid, the slant 
 height of which is 35 feet, and the sides of its base 10 
 feet? 
 
 Problem 16 Given the slant height and the diameter of a 
 
 cone, to find its surface. 
 
 Multiply the half sum of the slant height and the radius 
 of the base by the circumference of the base. 
 
 1. What is the surface of a cone, the slant height of 
 which is 47 feet and the diameter 18 feet ? 
 
 Problem 17. — Given the perimeter of the bases and tho 
 
262 MENSURATION. [cHAP. XI. 
 
 slant height of the frustum of a pyramid, or cone, to find the 
 area. 
 
 Multiply the half sum of the perimeter of the bases by the 
 slant height, arid to the product add the sum of the areas of 
 the two bases. 
 
 1. Suppose the slant height of a square frustum is 24 
 feet, the side of the base 8 feet, aud of the upper base or 
 top 4 feet; what is its surface ? 
 
 2. Suppose the slant height of the frustum of a cone 
 to be 40 feet, the diameter of the base 15 feet, and that 
 of the top 5 feet ; what is its whole surface ? 
 
 Problem 18. — Given the diameter of a sphere to find its 
 area or convex surface. 
 
 Multiply its circumference by its diameter. Or, which is 
 the same thing, 
 
 Multiply the square of the diameter by 3'1416. 
 
 1. What is the surface of a sphere 48 inches in diame- 
 ter ? 
 
 2. What is the area of the earth's surface, supposing it 
 to be 8000 miles in diameter ? 
 
 Art, 262. We can readily determine the distance to 
 any visible object by means of a right-angled triangle. 
 
 Suppose a man standing at A, de- 
 siring to know the distance to any 
 object as B, on the opposite side of 
 a river ; how should he proceed to 
 determine this distance, providing he 
 has nothing but a ten-foot pole ? 
 
 Form the right-angle BAG, and 
 measure any distance, as AC = 30 
 feet. Then from C, measure towards 
 A any distance, as CD =3 feet ; also 
 DE, perpendicular to CA. Suppose 
 DE=:4 feet. Then by similar triangles 
 we have 
 
ART, 262.] MENSURATION OF SOLIDS. 263 
 
 CD : CA : : DE : AB. Or, 
 
 3 : 30 : : 4 : 40, the distance AB. 
 
 The above principle is also employed 
 in measuring the heights of trees, &c. 
 In this case let AC be the height of the 
 tree ; E the height of the man's eye, 
 which we will suppose = 5 feet ; DG a 
 perpendicular pole, and DH the height 
 above the eye Then by similar tri- 
 angles we have 
 
 EH : EB :: HD : BC. Suppose we e 
 have found the first three terms of this 
 proportion to be as follows : — 
 
 3 : 24 :: 4 : CB. Then CB = 32 ft. which being increased 
 by AB = 5 ft. we have 32 + 5=37 ft. the height of the tree. 
 
 Mensuration of Solids. 
 
 Problem 19. — Given the side of a cube, to find its solHity. 
 
 Multijply its lengthy hreadth, and depth together. 
 
 1. What is the solidity of a cube, the side of which ir S ft.? 
 
 Problem 20. — Given the length, and dimensions of t^^ end 
 of a prism, or parallelopipedon, to find its solidity. 
 
 Multiply the area of its base or end by its length. 
 
 1. What is the solidity of a triangular prism 24 ^*. 
 long, and each side, 1 i feet ? 
 
 2. What is the solidity of a stick of timber 36 feet lor,'' 
 and 2| feet square ? 
 
 Problem 21. — Given the altitude and the base of a pyramid 
 to find its solidity. 
 
 Multiply the area of its base by otie third of its altitude. 
 
 1. What is the solidity of a triangular pyramid, the al- 
 titude of which is 24 feet, and each side of the base 6 ft. ? 
 
264 MENSURATION. [CHAP. XI 
 
 2. What is the solidity of a square pyramid, the altitude 
 of which is 48 feet, and a side of the base 9 feet ? 
 
 Problem 22. — Given the altitude and the diameter of a cone, 
 to find its solidity. 
 
 Multiply the area of its base hy one-third of its .altitude. 
 
 1. What is the solidity of a cone, the altitude of which 
 is 16 feet, and the diameter of the base 3 feet ? 
 
 2. What is the solidity of a cone, the altitude of which 
 is 54 feet, and the diameter of the base 12 feet ? 
 
 Problem 23. — Given the altitude, and the dimensions of the 
 two bases of a frustum of a pyramid, or of a cone, to find its 
 solidity. 
 
 To the sum of the areas of the two bases, add the mean 
 proportional betv^een these two areas, and multiply the result 
 by one-third of the altitude of the frustum. 
 
 Remark. — A mean proportional between the areas of the two bases of a 
 cone, is the square root of the product of the squares of the two diameters, 
 into 6-7854. 
 
 1. What is the solidity of the frustum of a cone, whose 
 altitude is 15 feet, and whose bases are 10 and 5 feet in 
 diameter, respectively ? 
 
 OPBRATION. 
 
 102 X 0-7854=100 X 0-7854, the area of the lower base. 
 
 52X0-7854= 25x0*7854, the area of the upper base. 
 
 -v/lOOX 25X0- 7854= 50x0-7854, ^area of the mean proportional 
 \ between the bases. 
 
 175x0-7854 t^® sum of the area of the three 
 
 Hence, 
 
 175 X 0-7854 X^ of 15=687*225, cubic feet, the solidity of the cone. 
 
 Remark. — This rule is of much importance in determining the solidity of 
 round sticks of timber, the diajneter of the ends of which dilJer. 
 
 2. There is a stick of timber, in the form of the frustum 
 of a cone, that is 48 feet long, 4 feet in diameter at the 
 larger end, and 9 inches at the smaller end. How many 
 cubic feet does it contain ? 
 
 Art. 26^. Another method of measuring round tim- 
 ber, Is tb 
 
ART. 263.] MENSURATION OF SOLIDS. 265 
 
 Multiply the square of oTie-fourtk the girth, in inches, hy 
 the length of the stick, in feet ; then divide the product by 144. 
 The quotient will be the contents in cubic feet. 
 
 Remark.— The girth should be taken two-thirds of the distance from the 
 smaller to the larger end. 
 
 3. How many cubic feet in a stick of timber, 25 feet 
 long, and whose girth is 60 inches ? 
 
 4. How many cubic feet in a stick of timber 42 feet 
 ong, and whose girth is 80 inches ? 
 
 6. How many cubic feet in a stick of timber 36 feet long, 
 and whose girth is 48 inches ? 
 
 Problejm 24. — Given the diameter of a sphere, to find its 
 solidity. 
 
 Multiply its surface by one-sixth of its diameter. Or, 
 Multiply the cube of the diameter by 5236. 
 
 1. What is the solidity of the Earth, supposing its dia- 
 meter to be 8000 miles ? 
 
 2. How many cubic inches in a cannon ball 9 inches in 
 diameter ? 
 
 It is believed that the foregoing rules will enable the 
 pupil to solve most of the examples that may arise in 
 ordinary mensuration. 
 
 For the practical convenience of those who have occasion 
 to refer to mensuration, we subjoin the following 
 
 Table of Multiples for Mechanics. 
 
 1. Diameter of a circle X 31416 = Circumference. 
 
 2. Radius of a circle X 6-2S3185 =: Circumference. 
 
 3. Square of the radius of a circle X 3-1416 = Area. 
 
 4. Square of the diameter of a circle X 7854 = Area. 
 
 5. Circumference of a circle X 0159155 =: Radius. 
 
 6. Square root of the area of a circle X 50419 = Radius. 
 
 7. Circumference of a circle X 0-31831 = Diameter. 
 
 8. Square root of the area of a circle X 1 •12338 = Diameter. 
 
 9. Radius of a circle X 1732051 = Side of inscribed eqiiilateral triangle. 
 
 10. Diameter of a circle X 0860254 = Side of inscribed equilateral triangle. 
 
 11. Side of inscribed equilateral triangle X 577350 = Radius of circle. 
 
 12. Radius of a circle X 1-414214 = Side of inscriDed square. 
 
 13. Diameter of a circle X 0-7071 = Side of an inscribed square. 
 
266 
 
 MENSUEATIOX. 
 
 CHAP. XI 
 
 14. Side of inscribed square X 707107 = Radius. 
 
 15. Square of radius of a sphere X 12 566371 = Surface. 
 
 16. Square of the diameter of a sphere X 3 1416 r= Surface. 
 
 17. Square of the circumference of a sphere X 0-3183 -^ Surface 
 
 18. Square root of surface of a sphere X 0'28i095 = Radius. 
 
 19. Square root of the surface of a sphere X 0*56419 = Diameter 
 
 20. Square root of the surface of a sphere X 1 •772454 = Circumference. 
 
 21. Cube of the diameter of a sphere X 0-5236 = Solidity. 
 
 22. Cube of the radius of a sphere X 41888 ^ Solidity. 
 
 23. Cube of the circumference of a sphere X 0016887 = Solidity. 
 
 24. Cube root of solidity of a sphere X 0-6203505 = Radius. 
 
 25. Cube root of the solidity of a sphere X 12407 = Diameter. 
 
 26. Cube root of the solidity of a sphere X 3-8978 = Circumference. 
 
 27. Radius of a sphere X 1-1547 = Side of inscribed cube. 
 
 28. Side of inscribed cube X 0-8660254 = Radius. 
 
 29. The square of the side of a tetraedron X 1. 7320508 = Surface. 
 
 30. The square of the side of a hexaedron X 6-0000000 = Surface. 
 
 31. The square of the side of an octaedron X 3-4641016 = Surface. 
 
 32. The square of the side of dodecaedron X 206457288 = Surface. 
 
 33. The square of the side of icosaedron X 8-6602540 = Surface. 
 
 34. The cube of the side of a tetraedron X 01 178511 = Solidity. 
 
 35. The cube of the side of a hexaedron X 1000000 = Solidity. 
 
 36. The cube of the side of an octaedron X 0-4714045 = Solidity. 
 
 37. The cube of the side of a dodecaedron X 7-6631189 = Solidity. 
 
 38. The cube of the side of an icosaedron X 2-181695 = Solidity. 
 
 A slight knowledge of the principles of geometry will 
 enable the pupil to deduce the above multiplies. For 
 illustration, determine the side of a cube that is inscribed 
 in a sphere. 
 
 c D 1^6t X = the side of the in- 
 
 scribed cube. In the right- 
 angled triangle BFH, BF 
 
 T 
 
 \ 
 
 M / 
 
 \ 
 
 
 
 F 
 
 \ 
 
 1^ 
 
 
 ^-s/Sk' + FH'CseeArt. 
 E 240), or BF = -v/^2- 
 
 In the right-angled trian- 
 gle, BDF, DF = X and BF 
 = v^2X^ J therefore, BD 
 = -v/SX^- But BD = the 
 diameter of the sphere, 
 which we will call unity ; 
 ^^ hence, 1 = -s/sx*^' or 1 = 
 
 X^3 = X times 1 73205+; consequently, X, the side of the 
 
AKT. 264.] 
 
 MENSURATION OjT SOLIDS. 
 
 267 
 
 cuLe, = y.TT2T!T of the diameter; or, •57736-f-timesthe diarieter 
 of the sphere ; and since Radius equals ^ of the diameter ; X, 
 the side of the cube, will equal Radius times (2x "57736 -|-) 
 = Raclnis times 1 -154724-. 
 
 In a similar manner the pupil may deduce other multi- 
 ples that he may desire, or the teacher require. 
 
 The Five Regular Bodies. 
 
 Art. 264. A regular body is a solid bounded by a 
 certain number of similar and equal plane figures. 
 
 It is proved in Solid Geometry that only three kinds of 
 equilateral and equiangular plane figures joined together 
 can make a solid angle; hence but Jive regular bodies can 
 possibly be formed. 
 
 1. A re^ra^^r^m is a solid hav- 
 ing four triangular faces. 
 
 Rkmark.— If figures similar to those annexed to the definitions, be drawn 
 on pasteboard, and cut out, by cutting through the bounding lines, and if tlie 
 other lines be cut hall' through, and then the parts be turned up and glued 
 together, the bodies defined will be formed. 
 
 2. A Hexaedron, or cube, is a 
 solid having six square faces. 
 
 . 
 
 3. An Odaedron is a regu- 
 lar solid having eight trian- 
 gular faces. 
 
268 
 
 MENSURATION. 
 
 [chap. XI 
 
 4. A Doaecae- 
 dron is a solid 
 having twelve 
 pentagonal faces. 
 
 5. An Icosae- 
 dron is a solid 
 having twenty 
 triangular faces. 
 
 Surfaces of the Five Regular Bodies. 
 Problem 1 — Given the side of a tetraedon, to find its surface. 
 
 Multiply the square of the linear side by the ^/s. 
 1. If the side of a tetraedron is 1, what is its surface ? 
 ' 2. If the side of a tetraedron is 8 feet, what is its surface ? 
 Problem 2. — Given the side of a hezaedron to find its surface. 
 D/Ftvltijply the square of the side by 6 . 
 
 1. If the side of a hexaedron is 1, what is its surface ? 
 
 2. If the side of a hexaedron is 4 feet, what is its surface ? 
 
 Problem 3. — Given the side of an odaedron, to find its surface 
 Multiply the square of the side by 2 ^^/g 
 
 1. If the side of an octaedron is 1, what is its surface ? 
 
 2. If the side of an octaedron is 8, what is it surface ? 
 
 Problem 4. — Given the side of a dodecaed^on, to find its sur- 
 face. 
 
 Multiply 15 times the square of the side by \/l + f ^/5. 
 
ART. 264.1 
 
 MENSURATION OF SOLIDS 
 
 269 
 
 1. If the lineal side of a dodecaedron is 1, what is its 
 surface ? 
 
 2, If the side of a dodecaedron is 9, what is its surface ? 
 
 Problem 5. — Given the side of an icosaedron., to find its surface, 
 Multiply 5 times the square of the side by \/S. 
 
 1. If the side of an icosaedron is 1, what is its surface ? 
 
 2. What is the surface of an icosaedron, the side of 
 which is 6 feet ? 
 
 Remark. — From the answers of the examples given under the preceding 
 problems, in which the lineal side is 1, the following table may be formed. 
 
 TABLE 
 
 Showing the surfaces of the five regulalr bodies^ when the 
 linear side is 1. 
 
 Number 
 of sides. 
 
 Names of bodies. 
 
 Surfaces of bodies. 
 
 4 
 
 6 
 
 8 
 
 12 
 
 20 
 
 Tetraedron. 
 
 Hexaedron. 
 
 Octaedron. 
 
 Dodecaedron. 
 
 Icosaedron. 
 
 1-7320508 
 6-0000000 
 3-4641016 
 20-6457288 
 8-6602540 
 
 Problem 6.— Given the side of any of the regular bodies, to 
 find its surface. 
 
 MiiUiply the square of the length of the side by the tabular 
 area opposite the figure mentioned. 
 
 1. The side of a tetraedron is 14 in.; what is its surface ? 
 
 2. The side of a hexaedron is 7 feet; what is its surface ? 
 
 3. The side of a tetraedron is 18 feet; what is its surface? 
 
 4. The side of an octaedron is 12 inches; what is its 
 surface ? 
 
 5. The side of a hexaedron is 16 feet; what is its surface ? 
 
 6. The side of a dodecaedron is 18 inches; what is its 
 surface ? 
 
 7. The side of an octaedron is 10 feet; what is its 
 surface ? 
 
2^0 MENSURATION. [cHAP. XI. 
 
 8. The side of an icosaedroa is 20 inches; what is its 
 surface ? 
 
 9. The side of a dodecaedron is 24 feet; what is its 
 surfalce ? 
 
 10. The side of an icosaedron is 16 feet; what is its 
 surface ? 
 
 Solidity of the Keg-ular Bodies. 
 
 Problem 1. — Given the side of, a tetraedron, to find its 
 solidity. 
 
 Multiply J^ of the cube of the, lineal side by the ,,^2. 
 
 1. If the lineal side of a tetraedron is 1, what is its 
 solidity ? 
 
 2. If the side of a tetraedron is 4 inches, what is its 
 solidity ? 
 
 Problem 2. — Given the lineal side of a hexaedron, to find 
 
 its solidity. 
 
 Cube the Side. 
 
 1. If the lineal side of a hexaedron is 1, what is its 
 solidity ? ^ 
 
 2. If the side of a hexaedron is 6 feet, what is its 
 solidity ? 
 
 Problem 3. — Given the side of an octaedron, to find its 
 solidity. 
 
 Multiply the cube of the side by the ^Ji, and \ of the 
 product will be the solidity. 
 
 1. What is the solidity of an octaedron, the side of 
 which is 1 ? 
 
 2. What is the solidity of an octaedron, the side of 
 which is 4 inches ? 
 
 Problem 4. — Given the side of a dodecaedron. to find its 
 solidity. 
 
 Add 4:^ to 2l,^y5, and divide this sum by iO; then muiti' 
 ply the square root of this quotient by 5 times the cube of tht 
 side. 
 
ART. 266.J 
 
 PHILOSOPHICAL PROBLEMS. 
 
 271 
 
 1. What is the solidity of a dodecaedron, the side of 
 which is 1 ? 
 
 2. What is the solidity of a dodecaedron, the side of 
 which is 8 inches ? 
 
 Problem 5 Given the side of an icosaedron, to find its 
 
 solidity. 
 
 Divide the sum of 7 aTid 3^5 by 2; then multiply the 
 square root of this quotient by f of the cube of the side. 
 
 1. What is the solidity of an .icosaedron, the side of 
 which is 1 ? 
 
 2. What is the solidity of an icosaedron, the side of 
 which is 4 inches ? ' . 
 
 Remark.— From the answers of the examples given under the preceding 
 yrobleins. in which the lineal side is 1, the following table may be formed. 
 
 TABLE. 
 
 Number 
 of Sides. 
 
 4 
 
 6 
 
 8 
 12 
 20 
 
 Names of Bodies. 
 
 Solidity of Bodies. 
 
 Tetraedron. 
 
 Hexaedron. 
 
 Octaedron. 
 
 Dodecaedron. 
 
 Icosaedron. 
 
 .1178511 
 
 1.0000000 
 
 .4714045 
 
 7^.6631189 
 
 • 2.1816950 
 
 Problem 6 Given the side of any of the regular bodies, to 
 
 find its solidity. 
 
 Multiply the cube of the side by the solidity opposite the 
 given figure in the above table. 
 
 1. What is the solidity of the five regular bodies, 
 respectively, the side of each being 8 inches ? 
 
 2. What is the solidity of the five regular bodies, 
 respectively, the side of each being 12 inches. 
 
 PHILOSOPHICAL PROBLEMS. 
 
 Art. 265. The spaces described by bodies falling from 
 a state of rest under the influence of gravity^ are yropor^ 
 
2T2 PHILOSOPHICAL PROBLEMS. [cHAP. XI. 
 
 tioned to the squares of the times during which they art 
 
 Art. 266. The spaces described by falling bodies are 
 also proportioned to the squares of the velocities which they 
 acquire in falling over those spaces* 
 
 A body in 1 second of time will fall 16^2 ^^^^j ^^^^ i* 
 Id feet. The velocity acquired in the same time is 32 
 feet. 
 
 Let T equal the time, during which a body has been 
 falling; D, the distance it has fallen; and Y, the velocity 
 acquired. 
 
 From the above laws and facts we have the following 
 proportions, from which the general formulas relating to 
 falling bodies may be deduced. 
 
 1. 2. 
 
 P : T2 :: 16 : D ; hence, D=16T2 ; .-. , T= /^g=]VL 
 
 3. 4. 
 
 32' : V2 :: 16 : D ; hence,D=l'; .-. ^\=^Mb=^'/b 
 64 
 
 Placing the right hand member of 1 and 3 equJal cc 
 
 each other, we have, 
 
 y2 
 16 T2 = — j — From which we deduce the following : 
 64 
 
 ^- '^=32 
 6. V=32T 
 
 Pupils should become familiar with the six preceding 
 formulas, which we will arrange differently for the con- 
 venience of reference. 
 
 3. D=16T2 6. V==32T 
 
 Olmsted's Natural Philosophy. — However, these laws arc nolstr'M>.iy t rreti. 
 
ART. 268.] PHILOSOPHICAL PROBLEMS. 2t3 
 
 1. How far will a leaden ball fall in 12 seconds; 14 
 seconds; 25 seconds; and 60 seconds, resiDectively ? (See 
 formula 3d.) 
 
 2. How long will a body be in falling 1024 feet; 1600 
 feet; 10000 feet; and 722500 feet, respectively ? (See 
 formula 1st.) 
 
 3. In what time would a body acquire a velocity of 128 
 feet; 160 feet; 288 feet; 1024 feet; and 3072 feet, respec- 
 tively ? (See formula 2nd.) 
 
 4. What velocity would a body acquire in 4; 7; 9; 12; 
 25; and 60 seconds, respectively? (See formula 6th.) 
 
 5. What velocity would a body acquire in falling 1024 
 feet; 7225 feet; 625 feet; 3025 feet; and 9025 feet, 
 respectively ? (See formula 5th.) 
 
 6. Through what space would a body have fallen to ac- 
 quire a velocity of 96 feet; 192 feet; 768 feet; 288 feet; 
 and 384 feet, respectively ? (See formula 4th.) 
 
 Art. 267^. The time of tlie vibrations of pendulums are 
 to each other as the square roots of their lengths ; hence, their 
 lengths are as the squares of their times of vibration. 
 
 A pendulum that vibrates seconds is 39^ inches in 
 length. 
 
 1. What is the length of a pendulum that shall vibrate 
 3 times a second ? 
 
 2. What is the length of a^ pendulum that shall vibrate 
 once in 5 seconds ? 
 
 3. What is the length of a pendulum that shall vibrate 
 once in a minute ? 
 
 4. How often will a pendulum vibrate, the length of 
 which is 225 inches ? 
 
 5. How often will a pendulum vibrate, the length of 
 which is 144 feet ? 
 
 Art. 268. The gravity of any body above the earth's 
 surface decreases, as the squares of its distance, in semi- 
 diameters of the earth, from its centre increases. Hence, 
 
 Th£. weight of a body on the earthUs surface, is to its 
 weight at any <issignable distance above the surface of thi 
 
 12* 
 
274 MISCELLANEOUS QUESTIONS. [CAAP. XI 
 
 earth; as the square of its distance from the ear til's centre, 
 to the square of the earthUs semidiameter , and vice versa. 
 
 1. If a body weigh 75'0 pounds at the earth's surface,^ 
 how much would it weigh 20000 miles above its surface ? 
 
 2. If a body at the earth's surface weigh 3600 pounds, 
 how much would it weigh 240000 miles above its centre, 
 the distance of the moon from the earth ? 
 
 If a body at the earth's surface weighed 1800 pounds 
 but being carried to a certain height weighs only 200 
 pounds, what is that height ? 
 
 MISCELLANEOUS QUESTIONS. 
 
 Rkmark. — Analvsis has been so extensively treated of in the -'American 
 Intellectual" and the " Practical Arithmetic," that it is deemed unnecessary 
 to add anything more to what has already been givenin the preceding pages. 
 
 1. A gentleman, after losing ^ of all his money, had 
 $368 remaining. How much had he at first ? 
 
 2. Thomas has 364 sheep more than James, and they 
 together have 1588. How many have they respectively ? 
 
 3. Henry, Perry, and John, found a purse containing 
 $768, which they agree to share in proportion to the 
 numbers 3, 4, and 5. How much should each receive ? 
 
 4. A farmer gave to a certain number of laborers, $14 
 apiece, if he had given them $19 apiece it would have 
 taken $125 more. How many laborers were there ? 
 
 5. A fish-pole, the length of which was 24 feet, was 
 broken into two pieces; and f of the lengtii of the longer 
 piece equalled the length of the shorter. What was the 
 length of each piece ? 
 
 6. There is a fish whose head is 18 inches long, and 
 whose tail is as long as its head + f of the length of its 
 body, and whose body is as long as its head and tail both. 
 What is the length of the fish V 
 
 7. Henry is 18 years old, and Harvey is 14; how many 
 years since was Henry twice as old as Harvey ? 
 
 8. A and B, together, have $8645, but A has $155 
 more than 3 times as much as B. How many dollars has 
 each ? 
 
ART. 208. J MISCELLANEOUS QUESTIONS. 2"75 
 
 9. A gentleman bought a hat, a vest, and a pair of 
 pants. The hat cost $6. The hat and vest cost iwice aa 
 much as the pauts. and the hat and pants cost 3 times as 
 much as the vest. What was the cost of the vest and 
 pants, respectively ? 
 
 10. A can earn a certain sum of money in 20 days ; A 
 and B together, can earn tlie same sum in 6 days. How 
 long will it take B alone to earn the same sum ? 
 
 11. A merchant bought a c<n-tain number of yards of 
 cloth for $245, and after usin^- ;) yards of it himself, sold 
 f of the remainder for $99, which was $24 more than it 
 oost. How many yards did he bny at first ? 
 
 12. A person at a game of cards lost f of all his money, 
 and then won $144; he now lost ^ of all the money he 
 had, and found he had but $95 remaining. How much 
 bad he at first ? 
 
 13. There is a cask containing brandy and water; f of 
 the whole, + 12 gallons is water; and ^ of the whole + 
 8 gallons is brandy. How many gallons of each ? 
 
 14. Two brothers, James and Henry, have the same 
 income. James contracts an annual debt, amounting to 
 $165; Henry lives on -| of his income, and saves yearly 
 $101, after lending James enough to pay his debt. How 
 much was the yearly income of each ? 
 
 15. Two masons, A and B, together can do a certain 
 piece of work in 10 days ; how long would it take each 
 separately to do it providing A does 3 times as much 
 as B ? 
 
 16. If A and B can, together, do a certain piece of work 
 in 5 days ; A and C, in 6 days ; and B and C, in H ; 
 how many days would it require for each to perform the 
 work alone ? 
 
 17. Divide $4760 among three persons, James, William 
 and Mary, so that James' part shall be to William's as 2 
 to 3, and that Mary shall have as much as James and 
 William together lacking $920. 
 
 18. A and B can, together, do a certain piece of work 
 in 10 dys. ; A and C, in 15 dys.; and B and C, in 20 dys 
 In hi">w many days could each perform the work alone ? 
 
2T6 MISCELLANEOUS QUESTIONS. [cHAP. XL 
 
 19. A gentleman distributed a certain number of dol- 
 lars among four poor women in the following manner : — ■ 
 to the first he gave ^ of the number of dollars he had + 
 $1; to the second ^ the remainder -f $i ; in the same 
 manner he gave to the third and the fourth ; and found 
 he had~ yet one dollar remaining. How many dollars 
 had he at first, and how much did he give to each 
 woman ? 
 
 20. An estate of $12850 was was left to four brothers, 
 who are 17, 15, 13, and 9 years of age, to be so divided 
 that the respective parts, being placed out at 5 per cent, 
 simple interest, should amount to equal sums when they 
 become 21 years of age, respectively. How much was 
 each one's share ? 
 
 21. A man bought a horse for $102, which was ^ of 
 twice as much as he sold it for, lacking $2. How much 
 did he gain by the bargain ? 
 
 22. A woman bought 60 oranges. For f of them she 
 paid 5 cents for 3 oranges ; and for the remainder 3 cents 
 for five ; for how much must she sell them apiece to gain 
 331 per cent. ? 
 
 23. A farmer paid to four of his hired men tt} bushels 
 of wheat. The first earned 1 bushel as often as the other 
 three earned |, f , and | of a bushel, respectively. How 
 many bushels should each receive ? 
 
 ^4. Two men A and B were playing cards; B lost $84, 
 which was -^-^ times | as much as A then had. When they 
 commenced playing, | of A's money equalled f of B's ; 
 how much had each when they began to play ? 
 
 25. What is the interest on $685-95 from April 14th to 
 Sept. 19th? 
 
 26. What is the amount of $684-99 from April 9th, 
 1853, to July 8th, 1854? 
 
 2*1. A has with B the following account : — 
 
 1853. , Dr. I 1853. Cr. 
 
 March 12th, Due . P45-45 | Sept. 16th, Due. . $784-50 
 
 At what time is the balance of the account due ? 
 
 28. I sold t/ie following bills of goods, on tlie conditions 
 below stated : 
 
ART. 268.] MISCELLANEOUS QUESTIONS 277 
 
 March 6, 1853, a bill amounting to $480 on 4 months' credit. 
 Arxril 15, " " " $-670 on 6 " " ■ 
 
 May 25, " " " iffTBo on 4 " ' 
 
 June 28, " " " $670 on 3 " " 
 
 How much money will balance the account July 20th ? 
 
 29. Three farmers, A, B, and C, together have 1920 
 acres of land; A has 40 acres more than B; and C has as 
 many as A and B together, lacking 32. How many acres 
 has each ? 
 
 30. What is the discount on $847-50 from May 12th, 
 1852, to July 25th, 1854 ? 
 
 31. What sum of money will give $18490 interest from 
 June, 16th, 1853, to Sept. 18th, 1854? 
 
 32. If A can do a certain piece of work in 80 days, and 
 with the assistance of C, in 34f days; how long will it 
 take C to do the work alone ? 
 
 33. Three carpenters. A, B, and C, earn a certain sura 
 of money in 24 days; A and B can earn the same amount 
 in 48 days; and A and C, in 36 days. How long would 
 it take each separately to earn the same amount ? 
 
 34. An individual being requested to buy a* certain 
 number of pounds of meat, found, if he bought beef, at 
 11^ cts. a pound, he would have 90 cts. remaining; but if 
 he bought pork, at 17| cts. a pound, he would lack 15 cts. 
 of having money enough to pay for it. How many pounds 
 of meat was he requested to buy, and how much money 
 did he have ? 
 
 35. A lady bought a certain number of apples, at the 
 rate of 5 for 2 cents; and paid for them with oranges, at 
 the rate of 3 for 2 cents. How many apples did she buy, 
 providing it took 144 oranges to pay for them ? 
 
 36. Three farmers, Thomas, William, and Henry, talking 
 of their sheep; says Thomas to William, I have 4 times as 
 many sheep as you; says William to Henry, I have | as 
 many as you; and says Henry to Thomas, if I had 63 sheep 
 more, I should then have as many as you. How many 
 iiad each ? 
 
 37. A man was hired for 160 days, on this condition: 
 that for, every day he worked, he should receive $144, 
 and for every day he was idle, he should pay 96 cents foi 
 
2T8 MISCELLANEOUS QUESTIONS. [cHAP. XL 
 
 his board. At the expiration of the time he received $64 
 How many days did he work ? 
 
 38 A, B, and C formed a co-partnership : A advanced 
 $10000; B $8000; and C $7000. At the end of 6 
 months, A withdrew $3000 from the business; B with- 
 drew $1500; and C increased his stock by ^ of its 
 original amount. At the end of the year, they had gained 
 $5584'60. How much should each receive ? 
 
 39. A gentleman willed $8640 to his wife, son, and 
 daughter, to be divided among them in the proportion of 
 I, I and |. The widow dying soon after, the whole sum 
 was divided in due proportion between the two children. 
 How much did each receive ? 
 
 40. A cistern receives water from 3 pipes; the first of 
 which would fill it in 12 hours; the second in 8 hours; 
 and the third in 6 hours. In what time v/ould these 
 three pipes together fill the cistern, providing i of the 
 whole capacity of the cistern leaked out in each hour ? 
 
 41. A merchant spent | of his money for silks; | of 
 the remainder for dry goods; | of the remainder for 
 groceries; and the remainder, which was $28t"65, for 
 stationery. How much money did he expend in all ? 
 
 42. Four men contracted to grade a turnpike road for 
 $12000. In accomplishing the work, one of the men 
 furnished 45 laborers for 74 days; another, 54 laborers 
 for 66 days; another, 75 laborers for 84. days; and the 
 other, 95 laborers for 85 days. How much should each 
 contractor receive ? 
 
 43. An agent receives $5685 to invest in merchandise, 
 at a commission of 1| per cent, on the amount of purchase 
 that can be made after his percentuni is deducted. What 
 is the amount of purchase; also, his commission ? 
 
 44. An upholsterer realized a profit of 25 per cent, 
 by selling carpeting, at $1.50 a yard. What would 
 have been the loss per cent, if he had sold it at $0*80 a 
 yard ? 
 
 45. How much would a person gain or lose by borrow- 
 ing $2000 from May 12th, 1852, to Nov. 12th, 1854, at 
 7 per cent, and lending the same sum, at 6^^ per cent., 
 
ART. 26'8.] MISCELLANEOUS QUESTIONS. 219 
 
 and on such conditions as will enable him to compound 
 the interest every 6 months ? 
 
 46. A drover bought 288 head of cattle, at $42f a 
 head, and pays for them with the proceeds of a note 
 which is discounted in a bank for 90 days, at 7 per cent. 
 At the end of 25 days, he sells the cattle, at $68|- a head, 
 and puts the proceeds on interest, at 8f per cent., until 
 his note is to be paid at the bank. What profit does he 
 make by these transactions, after paying $374'65 for 
 the cattle while he had them ? 
 
 47. A merchant took a farmer's note for $585-50, due, 
 without interest. May 14th, 1853. Some time afterwards, 
 the farmer got possession of a note against the merchant 
 for $894-85, due, without interest, Nov. 25th. When, in 
 equity, ought the balance to be paid ? Suppose money to 
 be worth 7' per cent., and they desire to settle Aug. 15th j 
 how stands the matter of debt between them ? 
 
 48. A is indebted to B $885; $125 of which is due 
 May 4th; $244, June 18th; $345, Aug. 12th; and the 
 remainder, Oct. 25th, — without interest. At what time 
 might the whole, in equity, be paid at once ? 
 
 49. What must be the dimensions of a granary which 
 shall contain 2400 bushels of wheat; its length to be 
 twice its breadth, and its breadth and height equal ? 
 
 50. What is the difference in arm between two fields 
 of the same perimeter; one of which is a square, and the 
 other 85 rds. long, and 251 rods wide ? 
 
 51. An individual was requested to purchase 1084 
 bushels of grain, consisting of rye, wheat, and barley; 
 I of the number of bushels of rye was to equal \ of the 
 number of bushels of wheat, and | of the number of 
 bushels of wheat was to equal f of the number of bushels 
 of barley. How many bushels of each kind must he 
 buy? 
 
 52. A man being asked the hour of the day, replied, 
 that I of the time past noon equalled ^ of the time from 
 now to midnight. What was the time 't 
 
 53. A tree, whose length was 156 feet, was broken into 
 two pieces by falling; 1^ times the length of the top piece, 
 
280 MISCELLANEOUS QUESTIONS. [cHAP. XI. 
 
 equals l} the bottom piece, + 12 feet. What is the length 
 of the two pieces respectively ? 
 
 54. A man bought a cow, a horse, and an ox for $350. 
 For the horse he gave 4 times as much as for the ox, lack- 
 ing $40; and for the ox, twice as much as for the cow, 
 lacking $12, What did he give for each ? 
 
 55. A farmer has 299 sheep in two different fields; the 
 number in the first field equals If times the number in the 
 second field, + 48. How many are there in each field ? 
 
 56. An individual, after spending | of all his money, 
 and f of what remained, lacking $12^, had only $347^ 
 remaining. How much had he at first ? 
 
 5t. There is an island 36 miles in circumference, and 
 three men. A, B, and C start from the same point, and 
 travel the same way around it; A 4 miles an hour; B, 12; 
 and C, 20. In what time will they all be together; and 
 in what time will they all meet at the place from which 
 they started ? 
 
 58. A note of $1200, given Feb. 3d., 1851, has received 
 the following indorsements : March 12th, 1852, indorsed 
 $365-45; Nov. 14th, 1852, indorsed $285-90; Jan. 12th, 
 1853, indorsed $484-12|. How much remains due March 
 20th, 1854, interest computed at 1 per cent. ? 
 
 59. Four masons. A, B, C, and D engage to build a 
 certain piece of wall for $660. While A can build 5 rods, 
 B can build 1^, C 3|, and D 6i. When the wall is | com- 
 pleted, D ceases to labor upon it, and A, B, and C finish 
 it. How much should each receive ? 
 
 60. A market-woman bought oranges, at 10 cents a 
 dozen, half of which she exchanged for lemons, at the rate 
 of 9 oranges for T lemons; she then sold all her oranges 
 and lemons, at 1| cents apiece, and thereby gained 24 
 cents. How many oranges did she buy, and how much 
 did they cost ? 
 
 61. If A can perform a certain piece of work in | of a 
 "Hay; B | of a day; and C in y^ of a day; how many times 
 
 longer will it take C to do the work alone, than it will 
 take A and B together to do it ? 
 
 62. A traveler had stolen from him {--J of all his money: 
 
ART. 268.] MISCELLANEOUS QUESTIONS. 281 
 
 the thief was caught, but not until he had spent f of it, 
 the remainder, $647-37|^, was given back. How much 
 money had the traveler at first ? 
 
 63. Three men, A, B, and C, built, a stone wall : A 
 built 15 rods; B built as much as A; f as much as C; and 
 C built as much as A and B together, lacking 5 rods. 
 How many rods did they all build, and how many did 
 B and C build, respectively ? 
 
 64. At what time between 2 and 3 o'clock will the hour 
 and minute hands of a clock be together ? 
 
 65. A person being asked his age, replied, that if his 
 age were increased by its f , its f , and 25^ years more, the 
 sum would equal 3^ times his age. What was his age ? 
 
 66. A person being asked the time of day, replied, that 
 I of the time past noon, equal f of the time from then to 
 midnight, lacking 12 minutes and 36 seconds. What was 
 the time ? 
 
 61. When James was married, he was 3 times as old as 
 his wife, but when they had been married 60 years, he 
 was only 1| times as old. How old was each when they 
 were married ? 
 
 68. Four individuals found a purse, containing $2445, 
 which they agree to share in the proportion of f, |, f, 
 and ^. How much should each receive ? 
 
 69. A deer is a 180 leaps before a hound, and takes 
 4 leaps to the hound's 9 ; and 5 of the deer's leaps are equal 
 to 9 of the hound's. Hov/ many leaps must the hound 
 take to catch the deer ? 
 
 10. A market-woman bought a certain number of pine- 
 apples, at the rate of 3 for 40 cents, and as many more at 
 the rate of 5 for 45 cents; and sold them all at the rate 
 of 7 for 81i cents, and thereby gained $1-60, Hew many 
 pine-apples did she buy ? 
 
 71. A boy bought a certain number of apples at the 
 rate of 4 for a cent, and as many more at 5 for a cent ; 
 and sold them out, at the rate of 9 for 5 cents, and by so 
 doing gained 45 cents. How many apples did he buy ? 
 
 72. A mechanic and his two sons earned $1490 in 1 
 year ; the father earned twice as much as the elder son^ 
 
282 MISCELLANEOUS QUESTIONS. [cHAP. XI. 
 
 lacking $tO, and the younger son earned i as much as the 
 elder son -f- 160 dollars. How much did each earn ? 
 
 Y3. A woman bought a certain number of oranges, at 
 the rate of 5 for 3 cents, as many more at the rate of 7 
 for 5 cents ; and sold them all, at the rate of 15 for 11 
 cents, and thereby gained 25 cents. How many oranges 
 did she buy ? 
 
 74. A merchant bought three pieces of cloth for $639: 
 f of the cost of the first piece equals -| of the cost of the 
 second; and f of the cost of the second piece equals | of 
 the cost of the third. How much did each piece cost ? 
 
 75. A merchant bought three pieces of cloth ; the first 
 piece contained | as much as the second piece + 12 yards; 
 and I of the number of yards in tlie second piece equaled 
 f of the number of yards in the third. How many yards 
 in each piece, providing there were 8 yards more in the 
 third piece than in the second ? 
 
 76. It is found that f of A's + -^ of B's fortune 
 equals $5400 ; and that"*! of A's fortune equals 1| times 
 I of B's + $24. What is the fortune of each ? 
 
 77. A hound ran 150 rods before he caught a hare; and 
 ■j?3 the distance the hare ran before it was caught equal- 
 ed the distance it was a-head v»'hen they started. How 
 far after the chase commenced, did the hare run before it 
 was caught ? 
 
 78. A and B started from the same point, and ran in 
 the same direction ; B ran 132 rods; then /g the distance 
 A had run equaled the distance A was in advance of B. 
 How much did A gain on B in running 132 rods ? 
 
 79. A gentleman left his son a fortune ; ^ of which he 
 spent in 2 years ; ^ of the remainder lasted him 3 years 
 longer ; f of the remainder lasted him 5 years longer when 
 he had only $784912^ left. How much did his father 
 leave him ? 
 
 80. I of A's number of sheep is to f of B as | to f ; 
 and f of B's number + | of A's equals 360. How many 
 sheep has each ? 
 
 81. Find the fortunes of A, B, C, D, E and F, by know- 
 ing that B is worth $220, which is ^ as much as A and 
 
ART. 268.] MISCELLANEOUS QUESTIONS. 283 
 
 are worth, and that A is worth ^ as much as B and C ; 
 and also, that, if 76 times the sum of A's, B's, and C's for- 
 tune were divided in the proportion of f , ^, and }, it 
 would, respectively, give | of D's, | of E's, and f ot F's 
 fortune. 
 
 82. There is a park 16 rods square, and it is desired to 
 make a gravel walk around it that shall contain J-f of the 
 whole area of the park. What should be the width of the 
 gravel walk ? 
 
 83. A speculator sold flour at $5 a barrel ; } of which 
 equaled his gain. How-much would he have gained per 
 cent, if he had sold it at $6-25 a barrel ? 
 
 84. A merchant sold a quantity of goods for $6*184 ; 
 and thereby cleared y^ of this money. If he had sold 
 them for $6999, what would he have gained per cent. ? 
 
 85. A speculator sold a quantity of cotton for $8484 ; 
 and by so doing gained } of what it cost him. How much 
 would he have gained per cent, if he had sold it for $9898 ? 
 
 86. A gentleman bought f of a farm for $9000 ; and 
 sold to B i of his share ; B sold to C | of what he re- 
 ceived ; C sold to D I of what he received ; and D sold 
 to E f of what he received. What part of the farm did 
 each man buy, and how much did it cost him ? 
 
 87. I bought f of a house, valued at $18000 ; and sold 
 ^ of my share to A ; A sold | of his share to B ; and B 
 sold I of his share to C. What part of the value of the 
 house does each own, and how much does C pay for his 
 part ? 
 
 88. An individual sold two horses, at $630 apiece ; for 
 one he received 25 per cent, more than its value, and for 
 the other 25 per cent, less than its value. Did he gain or 
 lose by the bargain, and how much ? 
 
 89. B's fortune added to | of A's, which is to B's as 2 
 to 3, being put on interest for 6 years, at 8 per cent,, 
 amounts to $988. What is the fortune of each ? 
 
 90. How much grain must a farmer take to mill, that 
 he may *'etch' away 14*4 bushels, after Ihe miller has taken 
 ty\ per cent, of all he took there ? 
 
 91. The interest on the sum of i of A's + | of B's 
 
284 MISCELLANEOUS QUESTIONS. [cHAP. XL 
 
 money for 4 years, at 6 per cent., is $480. What is tlie 
 fortunf^ of each, providing i of B's money equals 3 times 
 I of A's ? 
 
 92. The amount of | of A's fortune + f of B's for two 
 years, at 5 per cent., is $4950. What is the fortune of 
 each, providing ^ of A's money equals only f of f of B's ? 
 
 93. If the interest on the sum of A's and B's fortune 
 for 1 years and 6 months, at 4 per cent., is $3213 ; and 
 I of A's fortune equals | of B's ; what is the fortune of 
 each ? 
 
 94. What will be the result, if from the sum of 3, f 
 
 31 
 31, I of 3, 3i of 3-}, 4 of ^ we subtract the sum of i, | 
 
 2— 2— 3— 
 
 of i, 1 of 31 ; ^ of 5, ~ of -gT, and 3 j; multiply this 
 
 difference by the greatest common divisor of 315 and 
 405; divide this product by the least common multiple of 
 6, 9, and 24; reduce the quotient to its lowest terms; add 
 1 of I to the result; multiply | of this sum by 2^; and 
 
 divide the product by i of i of 4i of f | of ^f ? 
 
 95. Divide $3106-50 among A, B, C, and D, in the 
 following proportion : — A, B, and C are to have |^ of it; 
 B, C, and D are to have |^ of it; A, C, and D are to 
 have y'^o of it; and A, B, and D are to have f of it. 
 According to the above estimates, how much ought each 
 to receive ? 
 
 96. An individual, for two successive years, spent f 
 more than his yearly income; and found that, in 6 years, 
 by saving -^^ of his . annual income, he was able to 
 discharge the debt, and have $80 remaining. What was 
 his annual income ? 
 
 91. How many cannon balls, 8 inches in diameter, 
 can be put into a cubical vessel, 2 feet on a side; and 
 how many gallons of wine will it contain after it is filled 
 with balls, allowing the balls to be hollow, the hollow 
 being 4 inches in diameter, and the opening leading to it, 
 to contain 1^ solid inches ? 
 
ART. 268.] MISCELLANEOUS QUESTIONS. 285 
 
 98. A farmer sold hay, at $10-50 a ton, aud cleared 
 1 of his money; but hay growing scarce, he sold it, 
 at $12 a ton. What did he clear per cent, by the latter 
 price ? 
 
 99. If 24 men, in 132 days of 9 hours each, dig a 
 trench that is 4 degrees of hardness, 337| feet long, 5f 
 feet wide, and 3^ feet deep; how many men will be 
 required to dig a trench that is t degrees of hardness, 
 2321 feet long, 3| feet wide, and 2i feet deep, in 5i days 
 of 11 hours each ? 
 
 100. From a certain sum of money I took away its |, 
 ajid in its stead placed $200; I then took from this sum 
 its i, and in its stead placed $100; I now took away its 
 f, and found I had only $480 left. How much was the 
 original sum ? 
 
 101. From a sum money, $360 more than its } was 
 taken away; from the remainder, $280 more- than its i 
 was taken away; and, from what now remained, $80 
 more than its | was taken away, and then there remained 
 only $80. What was the original sum ? 
 
 102. From a certain sum of money I took its -i, and 
 put in its stead $460; from the remainder I took its |, 
 and put in its stead $600; and from what then remained 
 I took its i, and put in its stead $840, and found 1 had 
 twice as much money as I had at first. How much had 
 I at first ? 
 
 103. Make the sura, difference, product, and quotient 
 of 15 and 45 the numerators of fractions which shalLhave 
 ^i5, 40, 750, and 60 for denominators; reduce them to 
 equivalent fractions having a common denominator ; sub- 
 tract the sum of the last two fractions from the sum of 
 the first two; multiply this difference by the first fraction; 
 divide the product by the greatest common divisor of the 
 numerators; multiply the quotient by the least common 
 muttiple of the denominators; add the first fraction 
 reduced to a decimal to this quotient; subtract the second 
 fraction reduced to a decimal from this sum; multiply 
 this remainder by the third fraction reduced to a decimal; 
 divide this product by the fourth reduced to a decimal; 
 
286 MISCELLANEOUS QUESTIONS. [cHAP. XI. 
 
 then reduce the quotient to a vulgar fraction. What is 
 the result ? 
 
 104. A merchant sold 3 pieces of broadcloth, each 
 piece containing 27 yards, at $7 a yard, on 2 months' 
 credit, and made 12 per cent, on the first cost, — it had 
 been on hand 3 months; 7 pipes of wine, at $4'50 per 
 gallon, at an advance of 18 per cent, on the first cost, 
 which had been 7 months on hand, — for which he gave a 
 credit of 3 months; and 7 bales of cotton, at 11^ cents a 
 pound, each bale containing 230 pounds, which had been 
 on hand 1 month and 15 days, aj; an advance of 20 per 
 cent, on the first cost, — for which he gave 6 months 
 credit. How much did he make by the operation, and 
 how much did he make on each article ? 
 
 105. Suppose premiums, of three grades, to the amount 
 of $24 are to be distributed among the pupils of a school. 
 The value of a premium of the first grade is twice the value 
 of one of the second grade; the value of one of the second 
 grade is twice the value of one of the third grade; and 
 there are 6 of the first grade, 12 of the second, and 6 of 
 the third. What is the value of a premium of each grade ? 
 
 106. Four carpenters built a house in company. The 
 lot on which they built it cost $1000; the lumber and 
 building materials of all kinds cost $6500; they paid for 
 mason-work $500; and for painting and glazing $350, 
 Of these expenses A paid |, B |, C }, and D the residue. 
 A worked on the house 45 days, at $1'50 a day, with 3 
 apprentices, each $0*75 a day ; B worked 75 days, at 
 $1*75 a day, with 2 journeyman, each $1*25 a day; C 
 worked 60 days, at $r62i a day, with 1 journeyman, at 
 $1'37^ a day, and 2 apprentices, each $087^ a day; and 
 D, the master workman, worked 90 days, at $2'25 a day, 
 with 2 journeyman, each $1*75 a day, and 2 apprentices, 
 each $125 a day. The house being completed it was sold 
 for $2500 more than it cost. How much in equity ought 
 each partner to receive ? 
 
 107. A deer starts 40 rods before a hound, and is not 
 perceived by him until 40 seconds afterwards; the deer 
 runs, at the rate of 10 miles an hour; and the hound after 
 
ART. 268.] MISCELLANEOUS QUESTIONS. 28t 
 
 it, at the rate of 18 miles an hour. What distance will 
 the hound run before he overtakes the deer, and how long 
 will the chase continue ? 
 
 108. Two men in New York hired a carriage for $25, 
 to go to New Haven, a distance of t2 miles, and return, 
 with the privilege of taking in three more persons. Having 
 gone 20 miles, they take in A; at New Haven they take 
 in B; and when within 30 miles of New York they take 
 in C. How much in equity ought each man to pay ? 
 
 109. A boy went to a store and spent | his money, 
 and I of a cent more for pine-apples; he then went to 
 another store and spent | the money he had remaining, 
 and ^ of a cent more for oranges; he now went to a third 
 store and spent half the money he had remaining, and ^ of a 
 cent more for lemons; and then had only 9 cents remain- 
 ing. How much money had he at first, and how much did 
 he expend for pine-apples, oranges, and lemons respectively ? 
 
 110 A father left his four sons, whose ages are 15, 11, 
 8, and 6 years respectively, $57 7 T, to be so divided that 
 the respective parts being placed out, at 6 per cent, simple 
 interest, shall amount to equal sums when they become 21 
 years of age. What are these parts ? 
 
 111. An individual, at a public-house borrowed as much 
 money as he had, and spent 12^ cents; he then went to 
 another, where he borrowed as -much money as he then 
 had, and spent 12^ cents; then went to a third, and a 
 fourth and did the same; and then had no money remain- 
 ing. How much money had he at first ? 
 
 112. An estate of $17768 is to be divided among a 
 widow, two sons, and two daughters, so that each sou shall 
 receive twice as much as each daughter, lacking $240 ; 
 and the widow as much as all the children, lacking $520. 
 What was the share of each ? 
 
 113. A, B, and C can perform a certain piece of work 
 in 24 days; how long will it take each to perform the 
 work alone, if A does 1| times as much as B, and B does 
 I as much as C ? 
 
 114. A farmer, having sheep in two dififerent fields, 
 sold ] of the number from each field, and had only 280 
 
288 MISCELLANEOUS QUESTIONS. [CHAP. XI. 
 
 sheep remaining. Kow 20 sheep jumped from the first field 
 into the second; then the number remaining in the first 
 field was to the number remaining in the second field as 
 5 to 9, How many sheep were there in each field at first ? 
 
 115. A farmer paid five laborers a certain sum of 
 money every month; to the first he paid J the whole sum, 
 lacking $16; to the second ^ of the remainder, lacking $8; 
 the third ^ of the remainder, lacking $4; to the fourth ^ 
 of the remainder, lacking $2; and to the fifth the remain- 
 der, which was $11. How much did he give them all 
 a month, and how much to each ? 
 
 116. A Californian on his way home with $4000, was 
 met by a party that robbed him of | of I of all he had ; 
 a second party met and robbed him of f of f of the re- 
 mainder ; a third party met him and robbed him of y\ of 
 ^} of what he had left ; and a fourth party took from him 
 i of f of what still remained. How much money had he 
 left ? 
 
 lit. A gentleman promised his son a new arithmetic, 
 if he would go to a certain orchard, which was entered 
 through three gates, and get such a number of apples, that, 
 on his return, he could leave at the firs* gate, ^ the 
 apples he had and ^ an apple more; at the second gate, 
 i of what he had remaining and } an apple more; and at 
 the third gate, ^ the apples he still had remaining, and 
 i an apple more, without cutting any; and then have 
 1*1 apples remaining. How many apples must he get, and 
 how many will he leave at the gates, respectively ? 
 
 118. Three men, A, B, and C, agree to do a certain 
 piece of work for $52"90; A and B calculate that they 
 can do f of the work; A and C calculate that they can 
 do yV <^f the work; and B and C ^f. They are to be 
 paid proportionately, to these estimates. How much 
 should each receive ? 
 
 119. The stock of a certain bank is divided, into 32 
 shares, and is owned equally by eight persons. A, B, C, 
 D, &c. A sells 3 of his shares to a ninth person, and B 
 sells 2 of his shares to the Company. What proportiou 
 of the whole stock does A and B respectively still own ? 
 
ART. 268.] MISCELLANEOUS QUESTIONS. 289 
 
 120. A boy, being asked how many eggs he was carry- 
 ing to market, replied, I do not know; but father said, if 
 I had 1 dozen more, and should multiply this number by 
 2, and add to the product 2 dozen ; and then sell them 
 all, at 12^ cents a dozen, I would receive for them $1'50. 
 How many eggs had he ? 
 
 121. A farmer, being asked how many sheep he had, 
 replied, that he had them in four different fields; and that 
 I of the number in the second field equalled | of the num- 
 ber in the first; f of the number in the second, equalled 
 I of the number in the third ; and | of the number in the 
 third, equalled | of the number in the fourth. How many 
 sheep in each field, providing there are 64 more sheep in 
 the third field than in the fourth, and how many in all ? 
 
 122. A boy, having some oranges, sold to one person 
 1 of all he had and 10 oranges more; to another, i of 
 the remainder and 10 more; to a third, j% of what then 
 remained and t, more; to a fourth, | of what then 
 remained and 2 more; to a fifth, | of what still remained 
 and 10 more; and to the sixth, the remainder. How 
 many oranges had he at first, and how many did he sell 
 to each individual, providing the fifth bought 12 oranges 
 more than the sixth ? 
 
 123. The interest of A's, B's, and C's fortune for nine 
 years and 4 months, at 3 per cent., is $30380, What is 
 the fortune of each, providing f of A's fortune equals | 
 of B's, and f of B's equals f of C's ? 
 
 124. There is a rectangular box, 16 feet long, 4 feet 
 wide, and 3 feet deep. What must be the length and 
 width of another rectangular box of the same depth, that 
 shall contain 5625 solid feet, providing its length and 
 width are in the same proportion ? 
 
 125. A man at his death, having a daughter in France, 
 and a son in Russia, willed, if the daughter returned, and 
 not the son," that the widow should have | of the estate ; 
 and if the son returned, and not the daughter, that the 
 widow should receive | of the estate. They both returned. 
 How much, according to the will, should each receive, 
 providing the estate amounted to $7600 ? 
 
 13 
 
290 MISCELLANEOUS QUESTIONS. [cHAP, XI. 
 
 126. A, B, and C agree to do a certain piece of work 
 for $87*87 ; A and B can do the work in 6| days ; B and 
 C in 12 days ; and A and C in 10 jiays. How much 
 should each receive, according to the above estimates ? 
 
 127. What will be the dimensions of a rectangular box, 
 which shall contain 4037250 solid inches ; the length, 
 breadth, and depth being proportional to the numbers 7, 
 3, and 2 ? 
 
 128. A thief stole a horse from a farmer, B, and made 
 off with it ; 5 days after, B got intelligence of the direc- 
 tion the thief took, and followed him at the rate of 60 
 miles a day ; and by so doing gained 20 per cent, on the 
 thief. At what rate did the thief travel ; how far must 
 B ride before he overtakes him ; and how many days will 
 it require. 
 
 129. A drover being asked how many animals he had, 
 replied, that f of the number were sheep ; | of the re- 
 mainder were hogs ; and what then rena*kined were calves ; 
 and that, if he should sell the sheep at $2| a head ; his 
 hogs at $3i ; and his calves at $5 a head, he should re- 
 ceive $519, which was $119 more than they cpst. How 
 many sheep, hogs, and calves had he, respectively ? 
 
 130. If 14 oxen eat 2 acres of grass in 3 weeks, and 16 
 oxen -eat 6 acres in 9 weeks, how many oxen would eat 
 24 acres in 6 weeks ; the grass being at first equal on 
 every acre, and growing uniformly ? 
 
 131. If 8 oxen eat 2 acres of grass in 8 weeks ; and 
 15 oxen eat 5-acres in 6 weeks ; for how many weeks can 
 15 oxen graze on 6 acres, ^e grass growing uniformly ? 
 
 132. If 3 acres of grass, together with what grew on 
 the 3 acres whilq^they were grazing, keep 13 oxen 9 weeks, 
 and in like manner, 4 acres keep 20 oxen 6 weeks, how 
 many acres will be required to keep 36 oxen 4 weeks ? 
 
 133. A general drew up his regiment in the form of a 
 square and had 94 men remaining; soon after a detach- 
 ment of 485 men more joined him, whereby he was enabled 
 to increase the side of the square by 3 men. How many 
 soldiers had he at first ? 
 
 134 A market-woman carried some butter, strawberries 
 
A.RT. 268.] MISCELLANEOUS QUESTIONS. 291 
 
 and eggs, to market ; she sold her butter, at 25 cents a 
 pound ; her strawberries at 20 cents a quart ; and her 
 eggs, at 15 cents a dozen; the whole amounted to $7"65. 
 The number of pounds of butter equalled the number of 
 dozens of eggs inci*eased by the number of quarts of straw- 
 berries ; and the number of pounds of butter increased by 
 the number of quarts of strawberries, or the number of 
 dozens of eggs, would equal 3 times as much as the remain- 
 ing number. What was the quantity of each article ? 
 
 135. A, B, C, and D agree to a certain piece of work, 
 for $945; A, B, and C can perform the work in 84 days; 
 
 A, B, and D, in 72 days; A, C, and D, in 63 days; and 
 
 B, C, and D, in 56 days. How much money should each 
 receive, providing they all work until the work is com- 
 plete ? ' 
 
 136. A, B, C, and D play cards on this condition: that 
 he who loses shall give to all the others as much as they 
 already have. First A lost, then B, then C, and then D. 
 When they began to play they had $162, $82, $42, and 
 $22, respectively; how much had each at the end of the 
 fourth game ? Suppose, when they had all lost in turn, 
 that each had the same sum of money $96; how much had 
 each when they commenced to play ? 
 
 137. For three successive years ~a merchant, annually, 
 contributed $150 for charitable purposes, and added yearly 
 to that part of his capital not thus expended, a sum equal 
 its i. At the end of the third year his original capital 
 was doubled. What was his capital ? 
 
 138. There is an island 26| miles in circumference, and 
 three men A, B, and C, start from the same point, and 
 travel in the same direction around it; A goes 2^ miles 
 an hour; B goes 8^ miles an hour; and C goes 9f miles 
 an hour. In what time will they all first be together; and 
 when will they all be together at the place from which 
 they started ? 
 
 139. Three carpenters, A, B, a^^d C, receive $26 for a 
 certain amount of labor; — f of the number of days B 
 labored equaled | of the number of days A labored, and 
 I of the number of days C labored equaled | of the num* 
 
292 MISCELLANEOUS QUESTIONS. [cHAP. XI. 
 
 ber of days B labored; and A labored as many days as C, 
 lacking 5. How many days did each work, and how 
 muct did each receive a day, providing -i of A's daily 
 wages equaled. I of B's, and | of C's equaled |- of B's ? 
 
 140. A and B paid $90 for 12 acres of pasture for 8 
 weeks, with an understanding that B should have the 
 grass that was then on the field; and A, what grew during 
 the time they were grazing. How many oxen according 
 to the above understanding can each turn into the pasture, 
 and how much should each pay, providing 4 acres of pas- 
 ture, together with what grew during the time they were 
 grazing, will keep 12 oxen six weeks; and in a similar 
 manner, 5 acres will keep 35 oxen 2 weeks ? 
 
 141. A gentleman has in one bank a certain number 
 of 20, 15, and 10 dollar bills; in another a certain num- 
 ber of 5, and 2^ dollar gold coins. The number of bills 
 and coins in both banks equal 3224. How many of each 
 has he, providing | of the number of 20 dollar bills equal 
 I of the number of 15 dollar bills, | of the number of 15 
 dollar bills equal | of the number of 10 dollar bills, and 
 I of the number of 5 dollar gold coins are 48 more than 
 I of the number of 2| dollar coins; also, that -f of the 
 number of bills equal f of the number of coins; and 
 what amount of money has he in both banks ? 
 
 142. Divide a bar of lead weighing 40 pounds into four 
 pieces, with which (and a pair of scales) any number of 
 pounds from 1 to 40 may be weighed. 
 
 143. Find the least possible whole number which being 
 divided by 28, shall leave 19 for a remainder; and being 
 divided by 19, shall leave 16 for a remainder; and being 
 divided by 15,> shall leave 11 for a remainder ? 
 
 THE ENP. 
 
CONTENTS 
 
 CHAPTER I. 
 
 P.«. 
 
 IWTRODUCTORY DEFINITIONS. NO- 
 
 
 TATION. NUMERATION, 
 
 6 
 
 Notation, 
 
 6 
 
 Roman Table, .... 
 
 6 
 
 Arabic Notation, . 
 
 7 
 
 Numeration — Simple and Local 
 
 
 Values of Figures, . 
 
 8 
 
 Table 
 
 9 
 
 Do. continued, , 
 
 10 
 
 Do. ... 
 
 11 
 
 French Method of Numeration 
 
 12 
 
 Exerci-ses in Numeration, 
 
 12 
 
 Exercises in Notation, . 
 
 13 
 
 Fundamental Rules or Arith- 
 
 
 metic, 
 
 13 
 
 CHAPTER II. 
 
 
 ADDITION. SUBTRACTION. MUL- 
 
 
 TIPLICATION. DIVISION. 
 
 
 Addition, . . . • . 
 
 14 
 
 Practical Questions, 
 
 15 
 
 Practical Questions, 
 
 19 
 
 Subtraction 
 
 22 
 
 Practical Questions, 
 
 23 
 
 Practical Questions, 
 
 27 
 
 Practical Questions, combin- 
 
 
 ing Addition and Subtrac- 
 
 
 tion, 
 
 30 
 
 Multiplication, 
 
 32 
 
 iMultiplication Table, . 
 
 33 
 
 Practical Questions, 
 
 35 
 
 Practical Questions, 
 
 37 
 
 Practical Questions, combin- 
 
 
 ing Addition, Subtraction, 
 
 
 and Multiplication, 
 
 41 
 
 Division, 
 
 43 
 
 Division Table, 
 
 44 
 
 Short Division, . 
 
 44 
 
 Practical Questions, 
 
 46 
 
 Long Division, .... 
 
 47 
 
 Practical Questions, 
 
 49 
 
 Abstract Examples in the 
 
 
 Fundamental Rules, . 
 
 52 
 
 Practical Questions compris- 
 
 
 ing the Four Fundamental 
 
 
 Rules 
 
 03 
 
 CHAPTER III. 
 
 P«ff« 
 
 Tables of Monev, "Weights and 
 
 
 Measurrs. — Addition. — 
 
 
 Subtraction. — Multiplica- 
 
 
 tion. AND Division op Poly- 
 
 
 nomials, or Denominate 
 
 
 Numbers, .... 
 
 56 
 
 Table of United States Cur- 
 
 
 rency 
 
 66 
 
 English or Sterling Monet, . 
 
 67 
 
 Table 
 
 67 
 
 Troy Weight, 
 
 67 
 
 Table 
 
 67 
 
 Avoirdupois "Weight, . 
 
 67 
 
 Table, . . . . . 
 
 67 
 
 Apothecaries' "Weight, 
 
 68 
 
 Table, . ... . 
 
 68 
 
 Cloth Measure, 
 
 68 
 
 Table, 
 
 68 
 
 ' Lon^ Measure, 
 
 68 
 
 Table, ..... 
 
 68 
 
 Superficial, or Square Measure, 
 
 69 
 
 Table 
 
 69 
 
 Surveyors' Measure, 
 
 60 
 
 Table, 
 
 60 
 
 Solid, or Cubic Measure, 
 
 60 
 
 Table 
 
 60 
 
 Wine Measure, 
 
 61 
 
 Table, 
 
 61 
 
 Ale, or Beer Measure, . 
 
 61 
 
 Table, 
 
 61 
 
 Dry Measure, .... 
 
 61 
 
 Table, 
 
 61 
 
 Circular Measure, . 
 
 62 
 
 Table, . - . 
 
 63 
 
 Measure of Time, . 
 
 62 
 
 Table 
 
 65 
 
 Table, exhibiting the number 
 of days from any day of one 
 month to the same day of 
 any other month in the 
 same year, .... 
 
 Books, 
 
 Miscellaneous Table, 
 
 Addition of Denominate Num. 
 bers, ..... 
 
CfNTENTS. 
 
 Page 
 
 Troy "Weight, . 
 
 66 
 
 Avoirdupois "Weight, 
 
 66 
 
 Apothecaries' "Weight, 
 
 66 
 
 Cloth Measure, 
 
 67 
 
 Long Measure, 
 
 67 
 
 Superficial, or Square Mea 
 
 
 sure, .... 
 
 67 
 
 Surveyors' Measure, 
 
 67 
 
 Solid, or Cubic Measure, 
 
 68 
 
 Wine Measure, 
 
 68 
 
 Ale, or Beer Measure, . 
 
 68 
 
 Dry Measure, . 
 
 68 
 
 Circular Measure, . 
 
 69 
 
 Measure ol Time, . 
 
 ' 69 
 
 Subtraction of Denominate 
 
 
 Numbers, 
 
 69 
 
 Practical Questions in Addi 
 
 
 tion and Subtraction of De 
 
 
 nominate Numbers, . 
 
 71 
 
 Multiplication of Denominate 
 
 
 Numbers, . . . . 
 
 72 
 
 Division of Denominate Num- 
 
 
 bers, 
 
 74 
 
 Practical Questions combin- 
 
 
 ing Addition, Subtraction 
 
 
 Multiplication and Divi- 
 
 
 sion of Denominate Num- 
 
 
 bers, 
 
 76 
 
 Reduction, 
 
 78 
 
 Reduction Descending, . , 
 
 78 
 
 Reduction Ascending, . 
 
 79 
 
 CHArTER IV. 
 
 Peculiar Property of the 
 
 Number 9, , 
 Multiplication of Abstract 
 
 Polynomials, 
 Miscellaneous Definitions, 
 Prime Numbers, 
 Table of Prime Numbers, 
 Resolution of Composite Num 
 
 bers into their Prime Fac 
 
 tors, .... 
 Divisors or Measures of Num 
 
 bers, . . , . , 
 Common Measure or Divisor, 
 Greatest Common Measure, 
 Practical Questions in Com 
 
 mon Measure, 
 Multiples, 
 Practical Questions in Com 
 
 mon Multiple, 
 Abbreviated Operations in 
 
 Arithmetical Calculations, 
 Examples in Abbreviated Mul 
 
 ti])lication, .... 
 Properties or Numbers, . 
 
 87 
 
 CHAPTER V. 
 
 P«g« 
 
 Fractions, .... 
 Common Fractions, 
 Reduction of Common Frac 
 
 tions 
 
 Propositions, . , . 
 Multiplication of Fractions 
 
 by Integers, 
 Divisions of Fractions by In 
 
 tegers, .... 
 Cancellation, . 
 A Common Denominator, 
 The Least Common Denomi 
 
 nator, .... 
 Addition of Common Frac 
 
 tions, .... 
 Subtraction of Common Fraa 
 
 tions, .... 
 Multiplication of Common 
 
 Fractions, ... 
 Practical Questions in Mul 
 
 tiplication of Fractions, 
 Division of Common Frac 
 
 tions, .... 
 Practical Questions in Divi 
 
 sion of Fractions, 
 Complex Fractions, 
 Least Common Multiple of 
 
 Fractions, . " . 
 Practical Questions in Mul- 
 tiples, . . . 
 Practical Questions in Frac 
 
 tions, .... 
 
 CHAPTER VI 
 
 Decimal Fractions, . 
 
 Numeration of Decimal Frac 
 tions, .... 
 
 United States Currency, or 
 Federal Money, 
 
 Table of United States Cur- 
 rency, .... 
 
 Reduction of Decimals to Com- 
 mon Fractions, . 
 
 Reduction of Common Frac 
 tions'to Decimals. 
 
 Reduction of Mixed Decimals 
 to Simple Decimals, . 
 
 Repetends, 
 
 Compound Repetends, . 
 
 Additional of Decimals and 
 United States Currency, 
 
 Practical Questions, 
 
 Subtraction of Decimals and 
 the United States Currency, 
 
 Practical Questions, 
 
 Multii)lication of Decimals 
 and the United States Cur- 
 rency, 
 
 130 
 130 
 
 131 
 
rONTENTS. 
 
 m 
 
 Practical Questions, 
 Division of Decimals and the 
 
 United States Currency, 
 Practical Questions, 
 Practical Questions in Deci 
 
 mals and the United States 
 
 Currency, 
 Practical Questions, 
 Reduction of Denominate 
 
 Fractions, 
 Addition of Denominate Frac- 
 tions, .... 
 Practical Questions, 
 Duodecimals, 
 Table, .... 
 Addition and Subtraction of 
 
 Duodecimals, 
 Multiplication of Duodeci 
 
 mals 
 
 Reduction of Currencies, 
 Table, .... 
 
 Table 
 
 Aliquot Parts, 
 Analysis by Aliquot Parts 
 Cancellation, . 
 Ajaalysis by Cancellation, 
 
 CHAPTER VII. 
 
 Ratio, .... 
 
 Proportion, 
 
 Propositions, . 
 Simple Proportion, . 
 Compound Proportion, 
 Conjoined Proportion, 
 Copartnership, . 
 Compound Copartnershif, 
 Alligation Medial, . 
 Alligation Alternate, . 
 
 CHAPTER VIII. 
 
 Percentage, 
 
 Insurance, ... 
 
 Stocks, Brokerage and Com 
 mission, 
 
 Custom House Business, 
 
 Assessment of Taxes, 
 
 Profit and Loss, 
 
 Practical Questions in Profit 
 and Loss, 
 Simple Interest, 
 
 Page 
 132 
 
 133 
 134 
 
 135 
 140 
 
 140 
 
 143 
 144 
 146 
 
 146 
 
 147 
 
 147 
 160 
 160 
 150 
 152 
 152 
 153 
 153 
 
 154 
 155 
 156 
 156 
 162 
 164 
 165 
 169 
 170 
 171 
 
 175 
 176 
 
 177 
 179 
 181 
 183 
 
 185 
 18« 
 
 V»9> 
 
 Table, showinjf the Aliquot 
 
 Parts of a Year or Month, . 193 
 
 Problems in Interest, . . 19« 
 
 Discount, .... 197 
 
 Partial Payments, , . , 199 
 
 Compound Interest, . . . 201 
 
 Banking and Notes, . . 202 
 
 Bank Discount, . . . 204 
 
 Average, . . * . 206 
 
 Mercantile Calculations. 
 
 Equation of Payments, . . 206 
 
 Trade and Barter, . . . 217 
 
 CHAPTER IX. 
 
 Progression, . . . 221 
 
 Arithmetical Progression, . 221 
 
 Geometrical Progression, . 224 
 
 CHAPTER X. 
 
 Involution and Evolution, . 227 
 
 Involution, .... 227 
 
 Evolution, .... 228 
 
 Square Root, .... 229 
 Mechanical Application of 
 
 the Foregoing, . . . 233 
 
 Length of Braces, . . . 238 
 
 Length of Rafters, . . 239 
 
 Cube Root, . . . .241- 
 
 Practical Questions in Cube 
 
 Root, 261 
 
 Guaging, .... 262 
 
 CHAPTER XI. 
 
 Mensuration 254 
 
 Geometrical Definations, . 254 
 Plane Figures, . . . 264 
 Solid Figures, . . . .256 
 Mensuration of Surfaces, &c. 257 
 Mensuration of Solids, . . 263 
 Table of Multiples for Me- 
 chanics 265 
 
 The Five Regular Bodies, . 267 
 Surfaces of the Five Regular 
 
 Bodies 268 
 
 Table, ' 269 
 
 Solid ity of the Regular Bodies, 270 
 
 Table, 271 
 
 PHiLoscrHicAL Problems, . . 271 
 
 Miscellaneous Questions, . 274 
 
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 Drawings. 
 
 The author has been a practical instructor of this science for many years ; but having 
 met with no Text Book of the kind which, in his judgment, was completely adapted to the 
 useof classes, he has prepared a small volume of about 200 pages, that can be gone thorough- 
 ly through in one term of three months, a desidacUum, in which he has presented in a most 
 lucid, concise and comprehensible manner, the entire subject, as far as it is practicable to 
 .be taught in Common Schools, Seminaries or Colleges. 
 
 This treatise is already introduced in some of the best schools and sfcademies, in New York 
 and Ohio, and ifi rapidly gaining popularity. 
 
 PHELPS'S LECTURES ON PHILOSOPHY AND CHEMISTRY, 
 
 Each 300 pp. 12mo. Are highly esteemed, and used extensively. Price 75 cents 
 
 CHEMISTRY AND PHILOSOPHY FOR BEGINNERS. By Mbs. A- 
 
 Lincoln Phelps. Each 218 pn. 18mo. Price 50 cents. 
 
 These admirable books, by the distinguished authoress of '< Lincoln's Botany," are un- 
 questionably among the very best works of their kind. The great elementary truths which 
 •re the basis of these most interesting departments of study, are presented with such direct 
 ness, clearness, and force, that the learner is comi)elled to perceive and apprehend them ; 
 at the same time he is attracted, charmed, and indelibly impressed with that indescribable 
 felicity of language, which none but an accomplished lady or mother can ever address to d<- 
 lighted and instructed youth. To be approved and adopted, these books need only to b« 
 uaiTftraally known. Though but r9e«DtIy publisa«d, thoir •ireulation, already ext«nsiv«, i« 
 mpidljr ixkorsasiog. 
 
 E 
 
YB i7435 
 
 
 THE UNIVERSITY OF CAUFORNIA LIBRARY