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 IN MEMORIAM 
 I FLORfAN CAJORI 
 
IV 
 
 UMREB 
 
 
 ^ mathematical ^ 
 \ Series M 
 
 Vi TWO 
 
 ^ 
 
 ^M Tt ^ 
 
 lementary Arithmetic 
 
 or THE 
 
 Octimal Notation 
 
 By GEO. H. COOPER 
 
 PublisHed by 
 TMi: \\'HITAIii:R Ca ray CO. 
 
 (iNCOnPOnATEo) 
 
 £L<ivicational PviblisHer? 
 SAN FRANCISCO 
 
X^ 4-. J^.A^ 
 
 c^ 
 
 
ELEMENTARY ARITHMETIC 
 
 OP THE 
 
 OGTIMAL NOTATION 
 
 BY 
 GEORGE H.l COOPER 
 
 SAN FRANCISCO 
 
 THE WHITAKER AND RAY COMPANY 
 (INCORPORATED) 
 
 1902 
 
CAJORI 
 
 Copyright, 1902 
 By George H. Cooper 
 
PUBLISHER'S NOTE. 
 
 This book is paged according to the Octimal System of 
 
 Numbers. 
 
 ERRATA. 
 
 Page 22. Lines 3 and 4 should read : "4 from 13 leaves 7. Write 
 
 down 7," etc. 
 Page 51. Line 3 should read : " Int. 3d yr." 
 Page 66. Last line Miscellaneous. Answer should read: " 17.726 
 
 Centimeters." 
 

 PREFACE. 
 
 History shows that many unsuccessful attempts have 
 been made to discover a system of notation which would 
 be simple and effective, and we wonder that a satisfac- 
 tory basis has not been found. It is the more surpris- 
 ing, when we consider that every part of the universe 
 suggests multiplication or division by two as a basis from 
 which to start the building of a system of notation. This 
 principle is exposed in the attempt to provide a Decimal 
 currency, for the result is an Octimal coinage, viz., the 
 Dollar, half, and quarter. And so, also, the attempt to 
 decimalize the circle, for the quadrant must be found by 
 the same process. In fact, there is nothing in the uni- 
 verse to suggest the use of ten as a radix. So many ob- 
 stacles stand in the way of the successful application of 
 the Decimal Notation, that it is astonishing it has not 
 not been abandoned long ago. 
 
 The Octimal System will be the means of consider- 
 ably reducing the work in schools, by rendering much of 
 the work unnecessary, which, owing to the defects of 
 the Decimal System, is now required, and, in addition, 
 will make arithmetical work a pleasure instead of a 
 drudgery. 
 
 Its introduction need cause no confusion, for the pres- 
 ent units in use will be retained; viz., the Inch, the 
 Pound, and the Dollar, while the methods of manipula- 
 tion are the same as those already practiced. 
 
 Any attempt to perpetuate the use of the Decimal Sys- 
 
 3 
 
 ;n;oo4nno 
 
4 Preface. 
 
 tern is nothing short of a crime against humanity, since 
 it fails in every department. 
 
 The principle of "metric" arrangement has been 
 adopted in this work, and followed out to its logical con- 
 clusion, and the desired result has been attained, — a per- 
 fect system of notation, and arrangement of weights and 
 measures. 
 
 The Arithmetic is therefore offered in the confidence 
 that it will prove an educational factor of great value, and 
 forever place the science of numbers on a sound basis. 
 
 The Author. 
 
CONTENTS. 
 
 SECTION 1. 
 
 Defixitioxs, Xotatiox, axd Xumeratiox 7-13 
 
 NoTATiox Table 7 
 
 NuMERATiox' Table 11 
 
 SECTION 2. 
 
 Additiox ''. 15 
 
 Additiox Table 15 
 
 Subtraction 20 
 
 Multiplication 23 
 
 Multiplication Table 23 
 
 Division 27 
 
 SECTION 3. 
 
 Definitions 34 
 
 Reduction 35 
 
 Weights, Measures, Money, Time. Tables 35 
 
 Compound Numbers 40 
 
 Compound Additiox 40 
 
 CoMPouxD Subtraction' 41 
 
 Compound Multiplication 41 
 
 Compound Division 42 
 
 Repetends 43 
 
 SECTION 4. 
 
 Ratio and Proportion 44 
 
 Rule of Three 45 
 
 Practice. Simple 46 
 
 Practice. Compound 47 
 
 Interest. Simple 47 
 
 Interest. Compound 50 
 
 5 
 
6 Contents. 
 
 SECTION 5. 
 
 Square Root 52 
 
 Rectangle 54 
 
 Cube Root 55 
 
 Rectangle Parallelopiped 60 
 
 Derivation of Radix 62 
 
 Elevation of Denominations 62 
 
 Elevation of Cubes and their Roots 63 
 
 Calculation by Construction 63 
 
 SECTION 6. 
 
 Longitude and Time 64 
 
 To Determine an Angle 64 
 
 Area of Circle in Terms of Square Units 65 
 
 SECTION 7. 
 
 Miscellaneous 66 
 
 Transformation of Numbers 66 
 
 Decimal Fractions — To Transform into Inferior Units ... 67 
 Decimal Numbers and Octimal Inferior Units. 70 
 
ELEiAlENTARY ARITHMETIC 
 
 OF THE 
 
 OCTIMAL NOTATION. 
 
 SECTION 1. 
 
 ARITHMETIC. 
 
 Arithmetic teaches us the use of Numbers. 
 
 A Unit, or One, is any single object or thing; as, an 
 orange, a tree. 
 
 A Whole Number, or an Integer, is a Unit, or One, or 
 a collection of ones. If a boy, for instance, have one 
 orange, and then another is given to him, he will have two 
 oranges; if another be given to him, he will have three 
 oranges; if another, four oranges, and so on. One, two, 
 three, four, etc., are called Whole Numbers, or Integers. 
 
 Notation is the art of writing any number in figures. 
 
 The Octimal Notation is the method of expressing num- 
 bers by means of the following radix of figures: — 
 
 12 3 4 5 6 7 
 
 called one, two, three, four, five, six, seven, eights, or cipher. 
 
 representing (if we express a unit by a dot; thus, .), 
 
 or one or two or three or four or five or six or seven or eights 
 unit units units units units units units units 
 
 The first seven figures have a fixed value. The value 
 of the last depends upon whether it is prefixed by another 
 figure or not, and also the value of the prefixed figure. 
 When standing alone, without an index, it has no value; 
 but when prefixed by the Index, one, its value is eight; if 
 
 7 
 
10 Elementary Arithvietic of the Octimal Notation. 
 
 by two, two eights or two-et, etc., etc. Any of the figures 
 from one to seven, when prefixed to the cipher (thus, 10), 
 indicates that the radix has been named over as many 
 times as the prefixed figure has value in units. Thus if the 
 radix has been counted through once, the figure 1 is to be 
 prefixed; and if twice, the figure 2; and if three times, 
 the figure 3, and so on until all the figures have been pre- 
 fixed. 
 
 The order of progression in the Octimal System of No- 
 tation is in powers of eight; that is to say, that when 
 eight has been counted once, that fact is to be recorded by 
 prefixing the figure 1 to the cipher (thus, 10), so that 
 while the figures of fixed value stand alone, they only 
 stand for the number of units they represent; but the 
 addition of the cipher denotes that the number to which 
 it is affixed has been counted as many eight times. 
 Thus 70 denotes that 7 has been counted eight times, or 
 8 seven times, and the figure prefixed to the cipher to 
 record that fact. 
 
 When the radix has been counted through eight times, 
 the figures which stand for one eight are to be prefixed to 
 a new cipher; thus, 100, one eight eight, the name of this 
 combination being abbreviated to etred. All numbers 
 greater than eight, but less than etred, will be repre- 
 sented by two figures; thus 11 signifies that eight has 
 been counted once, because the figure 1 appears as the 
 second figure to the left, the place to be occupied by the 
 figures which represent the number of eights which have 
 been counted, and the figure 1 to the right denotes that 
 one more than eight has been counted; and as the figure 
 1 must occupy the units' place in the number, the cipher 
 must be replaced by that figure. 
 
Arithmetic. 
 
 11 
 
 We therefore understand that the first place to the left 
 of a point belongs to the ones of units counted up to 
 seven and the cipher when the number is a multiple of 
 eight, and the second place to the left belongs to the 
 eights of units up to the seventh eight, and a cipher 
 when the number is a multiple of eight, and so on. 
 
 Example. 
 
 Thus one 
 
 is written 
 
 a 
 
 eight 
 
 etred 
 
 etand 
 
 eight etand 
 
 etred etand 
 
 million 
 
 eight million " 
 
 etred million " 
 
 billion " 
 
 u 
 
 u 
 
 1. 
 
 10. 
 
 100. 
 
 1,000. 
 
 10,000. 
 
 100,000. 
 
 1,000,000. 
 
 10,000,000. 
 
 100,000,000. 
 
 1,000,000,000. 
 
 The reversal of positions of figures alters their value. 
 
 Thus one eighth 
 one etredth 
 one etandth 
 one eight etandth 
 one etred etandth 
 one millionth 
 one eight millionth " 
 one etred millionth " 
 one billionth " 
 
 is written .1, or point 1. 
 .01, " " 01. 
 .001, " '^ 001. 
 .0001, and so on. 
 .00001. 
 .000001. 
 .0000001. 
 .00000001. 
 .000000001. 
 
 ii 
 
 u 
 
 u 
 
 u 
 
 u 
 
 (( 
 
 From the above table we see that dividing any number 
 
12 Elementary Arithmetic of the Ortimal Notation. 
 
 into periods of three figures each, beginning at the right 
 hand, the names of those periods will be — 
 
 First period, Units. 
 Second " Etands. 
 Third " MilHons. 
 Fourth " Billions. 
 Fifth " Trillions. 
 
 The names of the places in each of these periods are the 
 same. 
 
 Namely, First place. Units. 
 Second " Eights. 
 Third " Etreds. 
 
 The following plan is recommended to enable the 
 pupil to write in figures any number dictated by the 
 teacher: — 
 
 Let the pupil write on his slate a number of ciphers; 
 thus, 000,000,000,000, marking them off into periods of 
 three places each from the right; 
 
 Put U over the first period for Units; 
 E '• " second " " Etands; 
 M '' " third " " Millions; 
 B " " fourth " " Billions; 
 
 B M E U 
 
 and so on; thus, 000,000,000,000. Then when a number 
 is dictated to a pupil, all he has to do is to put each fig- 
 ure in its proper place and fill up vacancies with O's. 
 
Arithmetic, 13 
 
 Exercises. 
 
 Write the following numbers in figures: — 
 
 (1) One, four, six, three, seven, eight. 
 
 (2) Et-seven, two-et, four-et one, six-et six, two-et two. 
 
 (3) Seven-et three, etred and et-one, three etred and 
 two-et two. 
 
 (4) Etand six etred and four-et three, three etand and 
 five. 
 
 (5) Seven-et six etand three etred and five-et six. 
 
 (6) Two-et three million seven etred and six-et etand 
 and four. 
 
 (7) Four etred million three etred etand five etred and 
 two-et. 
 
 Thus two etand and three will be written thus in 
 figures: 2,003; 
 
 Six-et etand, three etred and two: 60,302; 
 
 Four etred and three-et etand, six etred and two-et one: 
 430,621; 
 
 Six etred and one millions, four etred and seven: 601,- 
 000,407. 
 
 Numeration is the art of waiting in words the meaning 
 of any number which is already given in figures; thus, 
 
 35 means three eights and five units, or three-et five; 
 
 506 means five etreds no eights and six units, or five 
 etred and six; 
 
 0604 means no etands six etreds no eights, or six etred 
 and four; 
 
 3,251,630 means three millions, two etred and five-et 
 one etands, six etred and three-et. 
 
14 Elementary Arithmetic of the Octimal Notation. 
 
 Exercises. 
 
 Write in words the meaning of — 
 
 (1) 2, 4,6, 1, 7, 12, 16, 40, 23, 75, 67, 32, 44. 
 
 (2) 63, 21, 17, 35,52, 47, 77, 34, 16, 75. 
 
 (3) 102, 165, 143, 371, 263, 542, 764, 463. 
 
 (4) 1423, 3764, 4321, 5635, 7352, 6473. 
 
 (5) 53742, 64371, 42536, 57324, 65437. 
 
 (6) 140526, 330000, 4773254, 4613025. 
 
Addition. 
 
 15 
 
 SECTION 2. 
 
 ADDITION. 
 
 Addition is the method of finding a number which is 
 equal to two or more numbers of the same kind taken 
 together. 
 
 The numbers to be added are called Addends. 
 
 The Sum, or Amount, is the number so found. 
 
 
 ADDITION 
 
 TABLE. 
 
 2 and 
 
 
 3 and 
 
 
 1 make 
 
 3, Three. 
 
 1 make 4, Four. 
 
 2 " 
 
 4, Four. 
 
 2 " 
 
 5, Five. 
 
 3 " 
 
 5, Five. 
 
 3 " 
 
 6, Six. 
 
 4 " 
 
 6, Six. 
 
 4 " 
 
 7, Seven. 
 
 5 " 
 
 7, Seven. 
 
 5 " 
 
 10, Eight. 
 
 6 " 
 
 10, Eight. 
 
 6 " 
 
 11, Et-one. 
 
 7 " 
 
 11, Et-one. 
 
 7 " 
 
 12, Et-two. 
 
 10 " 
 
 12, Et-two. 
 
 10 ' 
 
 13, Et-three. 
 
 4 and 
 
 
 5 and 
 
 
 1 make 
 
 5, Five. 
 
 1 mj 
 
 ike 6, Six. 
 
 2 " 
 
 6, Six. 
 
 2 
 
 " 7, Seven. 
 
 3 " 
 
 7, Seven. 
 
 3 
 
 " 10, Eight. 
 
 4 " 
 
 10, Eight. 
 
 4 
 
 " 11, Et-one. 
 
 5 " 
 
 11, Et-one. 
 
 
 
 " 12, Et-two. 
 
 6 " 
 
 12, Et-two. 
 
 6 
 
 " 13, Et-three. 
 
 7 " 
 
 13, Et-three. 
 
 
 " 14, Et-four. 
 
 10 " 
 
 14, Et-four. 
 
 10 
 
 " 15, Et-five. 
 
16 Elementary AritJivietic of the Octimal Notation. 
 
 6 and 
 
 1 make 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 u 
 
 li 
 
 u 
 
 i 
 10 
 
 u 
 
 it 
 
 7, Seven. 
 
 10, Eight. 
 
 11, Et-one. 
 
 12, Et-two. 
 
 13, Et-three. 
 
 14, Et-four. 
 
 15, Et-five. 
 
 16, Et-six. 
 
 7 and 
 
 1 make 10, Eight. 
 
 2 " 11, Et-one. 
 
 3 " 12, Et-two. 
 
 4 " 13, Et-three. 
 
 5 '' 14, Et-four. 
 
 6 " 15, Et-five. 
 
 7 " 16, Et-six. 
 10 " 17, Et-seven. 
 
 The sign + called plus, placed between two numbers 
 means that the two numbers are to be added together; 
 thus 2 + 2 make 4, and 2 + 2 + 1 make 5. 
 
 The sign = placed between two sets of numbers means 
 that they are equal to one another; thus, 2+2 = 4, and 
 
 2 + 2 + 1=4 + 1, and 2 + 2 + 1=5, and 4 + 1=5. The sum 
 of 2 + 2 + 1 and 4 + 1 are equal. 
 
 The sign .". means therefore. 
 
 Example 1. — ^Find the sum of 3, 5, and 2. We add 
 thus: 3 and 5 make 10, 10 and 2 make 12; .'. the sum 
 of 3 + 5 + 2 = 12. 
 
 Example 2. — Add together 5, 7, 2, 6, and 3. 5 and 7 
 make 14, 14 and 2 make 16, 16 and 6 make 24, 24 and 
 
 3 make 27; .-. 5 + 7 + 2 + 6 + 3 = 27; 
 
 or thus: — 
 
Addition. 17 
 
 Add- (1) (2) (3) (4) (5) 
 
 4 3 4 7 
 
 o 
 O 
 
 5 6 7 3 
 2 2 112 
 7 5 2 5 
 
 6 3 6 5 1 
 
 27 24 21 14 22 
 
 RULE FOR ADDITION. 
 
 Rule. — Write down the given numbers under each other, so 
 that units may come under units, eights under eights, etreds 
 under etreds, and so on ; then draw a line under the lowest num- 
 ber. 
 
 Find the sum of the column of units ; if it be less than eight, 
 write it down under the column of units, below the line just 
 drawn; but if it be greater than eight, or a multiple of eight, 
 then write down the surplus of units under the units' column, 
 and carry to the column of eights the remaining figure or figures, 
 which of course will be eights. 
 
 Add the column of eights and the figure or figures you carry 
 as you have added the column of units and treat its sum in the 
 same way you have treated the column of units. 
 
 Treat each succeeding column in the same way. 
 
 Write down the full sum of the last column on the left hand. 
 
 The entire sum thus obtained will be the sum or amount of 
 the given numbers. 
 
 Example 1. — Add toi^ether 46,''53, and 164. 
 
 ~ 7 7 
 
 By the Rule. 46 Method of Adding. — 4 and 3 are 7, 7 
 
 r q 
 
 and 6 are 15. Write down 5 and carry 1, 
 because the sum of the units' column 
 305 is one eight units and 5 units over. 
 
20 Elementary Arithmetic of the Octimal Notation. 
 
 Then 1 and 6 are 7, 7 and 5 are 14, 14 and 4 are 20. 
 Write down under the eights' column and carry 2. 
 Then 2 and 1 are 3. Write down 3 under the etreds' 
 cohimn, which completes the calculation. 
 
 
 
 Examples. 
 
 
 
 - (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 11 
 
 15 
 
 / 
 
 21 
 
 35 
 
 56 
 
 13 
 
 6 
 
 16 
 
 32 
 
 47 
 
 44 
 
 21 
 
 23 
 
 14 
 
 16 
 
 23 
 
 73 
 
 45 
 
 46 
 
 41 
 
 71 
 
 127 
 
 215 
 
 (7) 
 
 (10) 
 
 (11) 
 
 (12) 1 
 
 (13) 
 
 (14) 
 
 52 
 
 76 
 
 134 
 
 73 
 
 321 
 
 530 
 
 73 
 
 121 
 
 62 
 
 252 
 
 763 
 
 627 
 
 104 
 
 32 
 
 251 
 
 4 
 
 222 
 
 134 
 501 ; 
 
 420 
 
 46 
 
 251 
 
 L724 
 
 1425 
 
 (15) 
 
 (16) 
 
 (17) 
 
 
 (20) 
 
 7350 
 
 1276 
 
 6325 
 
 
 
 37654 
 
 14263 
 
 43502 
 
 52403 
 
 263421 
 
 54301 
 
 73025 
 
 73526 
 
 4 
 
 37562 
 
 100134 140025 154456 1463057 
 
 SUBTRACTION. 
 
 Subtraction is the method of finding what number re- 
 mains when a smaller number is taken from a greater 
 number. 
 
Subtraction. 21 
 
 The number so found is called the remainder, or differ- 
 ence. 
 
 The sign — , called Minus, placed between two numbers, 
 means that the second number is to be subtracted from 
 the first number; thus 7 — 3 means that 3 is to be sub- 
 tracted from 7; '.*. 7—3=4. 
 
 RULE FOR SUBTRACTION. 
 
 Rule. — Write down the less number under the greater number, 
 so that units may come under units, eights under eights, etreds 
 under etreds, and so on; then draw a straight line under the 
 lower number. 
 
 Take, if you can, the number of units in each figure of the 
 lower number from the number of units in each figure of the 
 upper number which stands directly over it, and place the re- 
 mainder under the line just drawn, units under units, eights 
 under eights, etreds under etreds, and so on. 
 
 But if the units in any figure in the lower number be greater 
 than the number of units in the figure just above it, then add 
 eight to the upper figure, and then subtract the number of units 
 in the lower figure from the number of units in the upper figure 
 thus increased, and write down the remainder as before. 
 
 Add one to the next number in the lower number, and then 
 take this figure thus increased from the figure just above it, by 
 one of the methods already explained. 
 
 Go on thus with all the figures. 
 
 The whole difference, or remainder, so written down will be 
 the difference, or remainder, of the given numbers. 
 
 Example 1. — Subtract 3426 from 5345. 
 
 By the Rule. 5345 Method. — 6 from 5 I cannot take; 
 
 3426 •'• I borrow 10. Now 10 + 5 = 15. I 
 
 T^. ». .^. „ take 6 from 15, which leaves 7. Write 
 
 Dmerence=l/1 / , ,, ^ , ^ -..-r 
 
 down the / and carry 1. ^ext, 
 
22 Elementary Arithinetic of the Octimal Notation. 
 
 1 + 2 = 3. Take 3 from 4, which leaves 1. Write down 
 1 in the eights' place. Now 4 from 3 I cannot take. I 
 borrow 10. Now 10 + 3=13. 4 from 13 leaves 5. Write 
 down 5 in the etreds' place and carry 1. Next, 1+3=4. 
 Take 4 from 5, which leaves 1. Write down 1 in the 
 etands' place. 
 
 XoTE. — The truth of all sums in subtraction may be proved by 
 adding the less number and the remainder together. If the sum 
 has been worked correctly, the two numbers will be equal. 
 
 Thus proof of Example 1 : Less number + remainder =: 3426 + 
 1717 -=5345, the greater number. 
 
 
 
 ♦ 
 
 Examples. 
 
 
 
 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (<3) 
 
 (7) 
 
 From 
 
 20 
 
 24 
 
 17 
 
 73 
 
 21 
 
 75 
 
 76 
 
 Subtract 16 
 
 13 
 
 3 
 
 14 
 
 16 
 
 37 
 
 63 
 
 
 02 
 
 11 
 
 14 
 
 57 
 
 03 
 
 36 
 
 13 
 
 (10) 
 
 (11) 
 
 (12) 
 
 
 (13) 
 
 (14) 
 
 (15) 
 
 (16) 
 
 265 
 
 327 
 
 630 
 
 
 746 
 
 642 
 
 525 
 
 740 
 
 124 
 
 146 
 
 245 
 
 
 362 
 
 137 
 
 431 
 
 713 
 
 141 161 363 364 503 074 025 
 
 (17) 
 
 (20) 
 
 (21) 
 
 (22) 
 
 (23) 
 
 (24) 
 
 2653 
 
 5341 
 
 3421 
 
 4763 
 
 371246 
 
 576356 
 
 1241 
 
 1432 
 
 2675 
 
 3652 
 
 246517 
 
 537642 
 
 1412 3707 0524 1111 122527 036514 
 
Multiplication. 23 
 
 MULTIPLICATION. 
 
 Multiplication is a short method of repeated addition; 
 thus when 2 is multiplied by 3, the number obtained is 
 the sum of 2 repeated 3 times, which sum = 2 -f 2 4-2 = 6. 
 
 The number which is to be repeated or added to itself 
 is called the Multiplicand; thus in the above example, 2 
 is the multiplicand. 
 
 The number which shows how often the multiplicand 
 is to be repeated is called the Multiplier; thus in the 
 above example, 3 is the multiplier. 
 
 The number found by multiplication — for instance, 6 
 in the above example — is called the Product. 
 
 The multiplier and multiplicand are sometimes called 
 Factors, because they are factors, or makers, of the 
 product. 
 
 The sign X, called Into, or Multiplied by, placed be- 
 tween two numbers, means that the two numbers are to 
 multiplied together. 
 
 The following Multiplication Table should be com- 
 mitted to memory: — 
 
 Twice 
 
 3 times 
 
 1 are 2, Two. 
 
 1 are 3, Three. 
 
 2 '^ 4, Four. 
 
 2 " 6, Six. 
 
 8 '^ 6, Six. 
 
 3 " 11, Et-one. 
 
 4 " 10, Eight. 
 
 4 " 14, Et-four. 
 
 5 " 12, Et-two. 
 
 5 " 17, Et-seven. 
 
 6 " 14, Et-four. 
 
 6 " 22, Two-et two, 
 
 7 " 16, Et-six. 
 
 7 " 25, Two-et five. 
 
 10 " 20, Two-et. 
 
 10 " 30, Three-et. 
 
24 Elementary Arithmetic of the Octimal Notation, 
 
 4 times 
 
 5 times 
 
 1 are 4, Four. 
 
 1 are 5, Five. 
 
 2 " 10, Eight. 
 
 2 " 12, Et-two. 
 
 3 " 14, Et-four. 
 
 3 '• 17, Et-seven. 
 
 4 '' 20, Two-et. 
 
 4 " 24, Two-et four. 
 
 5 " 24, Two-et four. 
 
 5 '' 31, Three-et one. 
 
 6 " 30, Three-et, 
 
 6 " 36, Three-et six. 
 
 7 " 34, Three-et four. 
 
 7 "43, Four-et three 
 
 10 " 40, Four-et. 
 
 10 " 50, Five-et. 
 
 6 times 
 
 7 times 
 
 1 are 6, Six. 
 
 1 are 7, Seven. 
 
 2 " 14, Et-four. 
 
 2 " 16, Et-six. 
 
 3 "22, Two-et two. 
 
 3 " 25, Two-et five. 
 
 4 " 30, Three-et. 
 
 4 " 34, Three-et four, 
 
 5 " 36, Three-et six. 
 
 5 " 43, Four-et three, 
 
 6 " 44, Four-et four. 
 
 6 " 52, Five-et two. 
 
 7 " 52, Five-et two. 
 
 7 " 61, Six-et one. 
 
 10 " 60, Six-et. 
 
 10 " 70, Seven-et. 
 
 10 times 
 
 10 times 
 
 10 are 100, Etred. 
 
 100 are 1,000, Etand. 
 
 1,000 times 
 
 1,000 are 1,000,000, Million. 
 
 RULE FOR MULTIPLICATION WHEN THE MULTIPLIER 
 IS NOT GREATER THAN SEVEN. 
 
 Rule. — Place the multiplier in the units' place under the 
 multiplicand. Draw a line under the multiplier. Multiply each 
 figure of the multiplicand, beginning with the units, by the fig- 
 
Multiplication. 25 
 
 lire of the multiplier (by means of the ^Multiplication Table). 
 Write down and carry as in Addition. 
 
 Example 1. — Multiply 325 by 2. 
 
 By the Rule. 325 Twice 5 units makes 12 units. Write 
 
 2 2 in the units' place in the product, 
 
 g-9 and carry 1. Next, twice 2 makes 4. 
 
 Add to the 4 the 1 wliich was carried, 
 
 which makes 5. Write down 5 in the eights' place. Next, 
 
 twice 3 makes 6. Write the 6 down in the etreds' place. 
 
 
 Examples. 
 
 
 
 (1) 
 
 {'2) 
 
 (3) 
 
 (4) 
 
 Multiply 231 
 
 357 
 
 426 
 
 734 
 
 Bv 2 
 
 3 
 
 o 
 
 6 
 
 462 1315 2556 5450 
 
 (5) (6) (7) (10) 
 
 7342 6321 52264 563421 
 
 5.7 3 4 
 
 45152 54667 177034 2716104 
 
 RULE FOR MULTIPLICATION AVHEX MULTIPLIER IS 
 A NUMBER LARGER THAN SEVEN. 
 
 Rule. — Place the multiplier under the multiplicand, units 
 under units, eights under eights, etreds under etreds, and so 
 on ; then draw a line under the multiplier. Multiply each fig- 
 ure in the multiplicand, beginning with the units, by the figure 
 in the units' place of the multiplier (by means of the Multiplica- 
 tion Table) ; write down and carry as in Addition. 
 
 Then multiply each figure of the multiplicand, beginning with 
 the units, by the figure in the eights' place of the multiplier, 
 
26 Elementary Arithmetic of the OrtiriKil Notation. 
 
 placing the first figure h^o obtained under the eights of the line 
 above, the next figure under the etreds, and so on. Proceed in 
 the same way with each succeeding figure of the multiplier. 
 
 Then add up all the results thus obtained, by the rule of 
 Addition. 
 
 Example 2. — Multiply 4362 by 342. 
 
 4362 
 342 Since 342 = 300 + 40 + 2, we really multiply 
 
 . first by 2, next by 40, and last by 300, and if 
 
 ^ the sums are placed under each other and 
 
 added together, the product will appear as 
 
 shown. 
 
 1762644 
 
 To multiply by eight, all that is necessary is to affix 
 a cipher to the right of the number, and by etred, two 
 ciphers, and so on. 
 
 When any other Multiplier terminates with one or 
 more ciphers, multiply by the remaining figures, and to 
 the product add the same number of ciphers; thus: — 
 
 Example 3, — Multiply 42650 by 2300. 
 
 42650 
 2300 
 
 15037 
 10552 
 
 122557000 
 
 Multiplication by 6 may be accomplished with the 
 same result by its factors 2 and 3 in succession, and so 
 with the factors of any other number. 
 
Division 
 
 27 
 
 Numbers which cannot be broken up into factors are 
 called Prime Numbers, — 3, 5, 7, 13, etc. 
 
 
 Examples. 
 
 
 
 (1) 
 
 {'^) 
 
 (3) 
 
 (4) 
 
 Multiply 3642 
 
 5340 
 
 4263 
 
 57312 
 
 By 231 
 
 132 
 
 246 
 
 6421 
 
 3642 
 
 12700 
 
 32062 
 
 57312 
 
 13346 
 
 20240 
 
 21314 
 
 136624 
 
 7504 
 
 5340 
 
 10546 
 
 275450 
 
 1107722 
 
 751300 
 
 1322022 
 
 434274 
 
 
 465506552 
 
 
 DIVISION 
 
 • 
 
 
 Division is a short method of repeated subtraction; or, 
 it is the method of finding how often one number, called 
 the Divisor, is -contained in another number, called the 
 Dividend. The number which shows this is called the 
 Quotient. 
 
 Thus the dividend 11 divided by the divisor 3 gives 
 the quotient 3; and for this reason 3 + 3 + 3 = 11, and 
 therefore if we subtract 3 from 11, and then a second 3 
 from the remainder 6, and then a third 3 from the re- 
 mainder 3, nothing remains. 
 
 If, however, some number be left after the divisor has 
 been taken as often as possible from the dividend, that 
 number is called the Remainder; thus 10 divided by 3 
 gives a quotient 2, and a remainder 2; for, after subtract- 
 
30 Elementary Arithmetic of the Octimal Notation. 
 
 ing 3 from 10 once, there is a reroainder 5; after subtract- 
 ing 3 a second time from the remainder 5, there is a 
 remainder 2. 
 
 The sign h-, called By, or Divided by, placed between 
 two numbers, signifies that the first figure is to be divided 
 by the second. 
 
 Division is the opposite of multiplication. By the Mul- 
 tiplication Table, 3X4-14, and 14-^4=3, or 14^3=4. 
 
 RULE FOR DIVISION WHEN THE DIVISOR IS A NUM- 
 BER NOT LARGER THAN SEVEN. 
 
 Rule. — Place the divisor and dividend thus : — 
 
 divisor ) dividend 
 quotient 
 
 Take off from the left hand of the dividend the least number 
 of figures which make a number not less than the divisor. Find 
 by the Multiplication Table how often the divisor is contained in 
 this number ; write the quotient under the units' figure of this 
 number; if there is a remainder, affix to it, in the units' place, 
 the next figure in the dividend, and proceed as before. AVhen- 
 ever there is no remainder, and the next figure does not contain 
 the divisor, place a cipher in the next place to be filled in the 
 quotient. If there be a remainder at the end of the operation, 
 write it distinct from the quotient, and write the divisor under it 
 for a record. 
 
 Example 1. — Divide 1652 b}^ 4. 
 
 By the Rule. 4)16d2 Method. — 4 into 1 I cannot; 
 
 oroi therefore affix the next figure to 
 
 the 1. Then 4 into 16 will go 3 
 
 times, leaving a remainder 2; wa-ite down the 3 under 
 
Division. 31 
 
 the units' place of the number just dealt with, and to 
 the remainder affix the next number in the dividend; 
 then 4 into 25 will go 5 times, leaving a remainder 1. 
 "Write down five in the quotient; next affix the last num- 
 ber in the dividend to the remainder; then 4 into 12 w'ill 
 go twice, leaving 2 as a remainder. Write down the 2 
 under the dividend. Write the remainder and divisor 
 separately. 
 
 Examples. 
 
 (1) (2) (3) (4) 
 
 245^3 4321^.5 6732^6 73215-^4 
 
 3)245 5)4321 6)6732 4)73215 
 
 67 7034 1117 166434 
 
 o 
 
 RULE FOR DIVISION WHEN THE NUMBER IS GREATER 
 
 THAN EIGHT. 
 
 Rule. — Place the divisor and dividend thus : — 
 
 divisor ) dividend ( quotient 
 
 leaving space for the quotient to the right of the dividend. Take 
 off from the left hand of the dividend a number not less than 
 the divisor. 
 
 Find how many times the divisor is contained in this number. 
 Place the number so found in the quotient ; multiply the divisor 
 by the quotient and bring down the product under the number 
 taken off from the left of the dividend, and subtract. 
 
 On the right of the remainder bring down the next figure in 
 the dividend. Find how many times the divisor is contained in 
 this number. If it is less than the divisor, write a cipher in the 
 quotient, and bring down another figure from the dividend and 
 affix also to tlie remainder ; repeat the process until all the 
 figures in the dividend have been brought down. 
 
32 Elementary Arithmetic of the Octimal Notation. 
 
 Example 1. — Divide 1432 by 31. 
 
 By the Rule. 31)1432(373 Method. — 143 is the least 
 
 113 number taken from the left 
 
 OQ2 of the dividend into which 
 
 cyrj 31 will go; for if 31 be X 3, 
 
 the product will be 113, 
 
 which is less than the num- 
 ber taken. Multiply the divisor by 3; place the product 
 under the number from the dividend and subtract; the 
 remainder is less than the divisor; .*. place the multi- 
 plier as first figure in the quotient; bring down the next 
 figure from the dividend and affix to the remainder. 
 Find how many times the divisor is contained in the 
 new dividend. Place the multiplier 7 as before in the 
 quotient. AVrite down the product under the new divi- 
 dend and subtract. There is a remainder 23. Write it 
 in the quotient, and the divisor under it, for a record. 
 
 
 Examples. 
 
 
 (1) 
 
 14160^21 
 
 
 23651 : 34 
 
 >1) 14160(560 
 125 
 
 
 34)23651(552|i 
 214 
 
 146 
 . 146 
 
 
 225 
 214 
 
 ...0 
 
 
 111 
 70 
 
 
 « 
 
 21 
 
Division. ^^ 
 
 (3) 
 52765-^46 
 
 32651-123 
 
 46)52765(1103A 
 46 
 
 123)32651(245x^3 
 246 
 
 47 
 46 
 
 605 
 514 
 
 165 
 
 162 
 
 711 
 637 
 
 52 
 
34 Elementary Arithmetic of the Octimal Notation. 
 
 SECTION 3. 
 
 DEFINITIONS. 
 
 A Unit is a predeterminate quantity of weight, length, 
 capacity, or value. 
 
 For purposes of calculation, the Unit occupies a posi- 
 tion immediately to the left of a point. 
 
 Superior Denominations are all those units which ap- 
 pear to the left of the point. 
 
 Inferior Denominations are all those units which ap- 
 pear to the right of the point. 
 
 THE POINT. 
 A Point is a sign so placed as to distinguish and to 
 separate Superior Units from Inferior Units for the pur- 
 pose of determining their value. Thus 1.1 means one 
 unit plus one eighth of one unit. 
 
 THE FRACTION. 
 
 A Fraction is an undetermined portion of a Unit re- 
 maining after units have been subdivided to the lowest 
 convenient denomination. 
 
 Example. — Three men wish to apportion 4 dollars 
 among themselves as nearly equal as possible. The cal- 
 culation is carried out by the rule for division; thus, — 
 
 $ (j; Since 400 is not a multiple of 3, and the 
 
 3)4.00 nearest number which is a multiple of 3 is 377, 
 
 1 251 there must be a remainder of 1 cent, which 
 must be dealt with in a manner to be subse- 
 quently shown. 
 
Reduction. 35 
 
 REDUCTION. 
 
 Reduction is carried out by the use of the point, termed 
 Inspection; thus 400 cents = $4.00. 
 
 tons lbs. oz. q. pt. in. 
 
 32126 oz. = 3.212.6 265 inches Liquid = 2.6.5 
 
 yd. ft. in. 
 
 265 " Linear=2.6.5 
 Thus — 
 
 E. $ c 
 
 17464.5.32 American currency. 
 
 miles yd. ft. in. 
 
 174.645.3.2 Linear measure. 
 
 yd. ft. in. 
 
 1746.45.32 Square measure. 
 
 yd. ft. in. 
 
 17.464.532 Cubic measure. 
 
 tank pk. q. p. i. 
 
 17.464.5.3.2 Capacity liquid measure. 
 
 ton lb. 02.dr.sc.gr. 
 
 1.746.4.5.3.2 Weights. 
 
 days hr. m. s. 
 
 17.46.45.32 Time. 
 
 Kevolution ° ' " 
 
 17.46.45.32 Circle. 
 
 Table of Weights. 
 
 10 Grains make 1 Scruple. 
 10 Scruples " 1 Drachm. 
 10 Drachms " 1 Ounce. 
 10 Ounces " 1 Pound. 
 1000 Pounds " 1 Ton. 
 
 Note. — The pound avoirdupois is the unit of weight. 
 
36 Elementary Arithmetic of the Octimal Notation. 
 Table of Measures of Capacity. 
 
 10 Minims make 1 Cubic Inch. 
 10 Cubic Inches " 1 Pint. 
 10 Pints " 1 Quart. 
 
 10 Quarts " 1 Peck. 
 
 1000 Pecks " 1 Tank. 
 
 Table of Linear Measure. 
 
 10 Inches make 1 Foot. 
 
 10 Feet " 1 Yard. 
 
 1000 Yards " 1 Mile. 
 
 Note. — Tlie present unit, the Inch, is to be retained. 
 
 iii|iii|iii|in 
 
 1 1 1 1 1 1 
 
 1 1 1 1 1 
 
 III III 
 
 1 1 1 1 1 1 
 
 5 
 
 6 
 
 7 
 
 10 
 
 ONE- FOOT EULE ( SHOWN IN TWO SECTIONS). 
 
Tables of Measures, Money, Etc. 37 
 
 Currency. 
 
 10 Mills make 1 Cent. 
 10 Cents " 1 Bit. 
 10 Bits " 1 Dollar. 
 10 Dollars " 1 Eagle. 
 
 Note. — The present Dollar is the unit to be retained. 
 
 Convenient American Coins. 
 
 Gold. Eagle=Eight (10) dollars To be minted. 
 
 Silver. Dollar = Etred (100) cents Present dollar. 
 
 •• Half-dollar = Four-et (40) cents . " 50 cents. 
 
 " Quarter-dollar = Two-et (20) cents " 25 cents. 
 
 " Eighth-dollar = Eight (10) cents .To be minted. 
 
 Nickel. Four (4) cents " 
 
 Copper. Two (2) cents " 
 
 '' One (1) cent 
 
 Value of Gold Decimal Coin in Octimal Figures. 
 
 Five ($5) pieces^- Five (5) dollars. 
 Ten ($10) pieces = Et-two (12) dollars. 
 Twenty ($20) pieces = Two-et four (24) dollars. 
 
 Measures of Time. 
 
 100 Seconds make 1 Minute. 
 100 Minutes '•' 1 Hour. 
 100 Hours " 1 Day. 
 
40 Elementary Arithmetic of the Octimal Notation. 
 
 Subdivisions of Circle. 
 
 100" (Seconds) make V (Minute). 
 100' (Minutes) " 1° (Degree). 
 100° (Degrees) " 1^ (Circle). 
 
 COMPOUND NUMBERS. 
 
 Compound Numbers are those numbers which contain 
 both Superior and Inferior units. 
 
 COMPOUND ADDITION. 
 
 Rule. — Place the numbers under each other, units under 
 units, eights under eights, etc., so that the points shall be directly 
 under each other. Add as in whole numbers, and place a point 
 in the result immediately under the points in the numbers 
 above. 
 
Compound Multiplication. 41 
 
 P:xample. — Add together 431.64, 56.200, .0321. 
 
 Bij the Rule. 431.64 
 
 56.200 
 .0321 
 
 510.0721 
 
 COMPOUND SUBTRACTION. 
 
 Rule. — Place the less number under the greater, units under 
 units, eights under eights, etc., so that the points shall be directly 
 under each other. If there be fewer figures in the upper number 
 to the right of the point than there are figures in the lower num- 
 ber to the right of the point, add ciphers to the upper number 
 until the numbers of figures are equal. Then subtract as in 
 whole number, placing a point immediately under the point 
 above. 
 
 Example. — Subtract 31.2652 from 53.21. 
 
 By the Rule. 53.2100 
 31.2652 
 
 21.7226 
 
 COMPOUND MULTIPLICATION. 
 
 Rule. — Multiply the numbers together as if they were whole 
 numbers, and point off in the product as many places as there 
 are in both the multiplicand and multiplier. If there are not 
 sufficient figures, prefix ciphers. 
 
 Example. — Find the product of .2X.3; also, 1.4X2.3. 
 
 By the Rule. .2 1.4 
 
 .3 2.3 
 
 .06 44 
 
 30 
 
 3.44 
 
42 Elementary Arithmetic of the Octimal Notation. 
 
 COMPOUND DIVISION. 
 
 When the number of inferior places in the dividend 
 exceeds the number of inferior places in the divisor. 
 
 Rule. — Divide as in whole numbers, and point off in the 
 quotient a number of inferior places equal to the excess of the 
 number of inferior places in the dividend over the number of 
 inferior places in the divisor ; if there are not figures sufficient, 
 prefix ciphers, as in Multiplication. 
 
 Example 1. — Divide 23.006 by 2.35. .0023006 by 2.35. 
 
 Bij the Rule. 2.35)23.006(7.6 2.35). 0023006 (.00076 
 
 2113 2113 
 
 1 656 1656 
 
 1 656 1656 
 
 When the number of inferior places in the dividend is 
 less than the number of inferior places in the divisor. 
 
 Rule. — Affix ciphers to the dividend until the number of 
 inferior places in the dividend equals the number of inferior places 
 in the divisor ; the quotient up to this point of the division will 
 be a whole number. If there be a remainder, and the division 
 be carried on further, the figures after this point in the quotient 
 will be inferior units. 
 
 Example 2. — Divide 2300.6 by .235. 
 
 Bij the Rule. .235)2300.600(7600. 
 
 2113 
 
 1656 
 1656 
 
 • ••00 
 
Repetends. ' 43 
 
 Rule. — Before dividing, affix ciphers to the dividend to 
 make the number of inferior places in the dividend exceed the 
 number of inferior places in tlie divisor by three. If we divide 
 up to this point, the quotient will contain three inferior places. 
 
 Example 3. — Divide 2301.2 by .235 to 3 inferior places. 
 
 .235)2301.200000(7(301.502 
 2113 
 
 1662 
 
 1656 
 
 •••400 
 
 235 
 
 1430 
 
 1421 
 
 •700 
 
 472 
 
 206 
 
 REPETENDS. 
 
 The sign * applied to a number means that the num- 
 ber is recurring, or a Repeater; thus, .2. Repetends are 
 those numbers which periodically recur in Compound 
 Division and may be extracted to the lowest required 
 denomination by merely affixing them to the quotient; 
 thus, 4-^3 gives the quotient 1.2o; and the quotient may 
 be extended thus, 1.252o2o. 
 
44 Elementary Arithmetic of the Octimal Notation. 
 
 SECTION 4. 
 
 RATIO AND PROPORTION. 
 
 The relation of one number to another in respect of 
 magnitude is called Ratio. 
 
 The Ratio of one number to another is denoted by 
 placing a colon between them. Thus the ratios of 3 to 
 14 and 14 to 3 are denoted by 3:14 and 14:3. 
 
 When two Ratios are equal, they are said to form a 
 Proportion, and the four numbers are called Proportionals. 
 Thus the ratio of 3 to 14 and 6 to 30 are equal. The 
 Ratios being equal, Proportionals exist among the num- 
 bers 3, 6, 14, 30. 
 
 The existence of Proportion between the numbers 3, 6, 
 14, 30, is denoted thus: 3:6:: 14:30. 
 
 If four numbers be proportionals when taken in a cer- 
 tain order, they will also be proportionals when taken in 
 a contrary order; for instance, 3, 6, 14, 30, are propor- 
 tionals, as already shown; and contrariwise, 30:14 :: 6: 3. 
 
 The two numbers which form a Ratio are called its 
 terms; the former term is called the Antecedent; the lat- 
 ter, the Consequent. 
 
 Because 3X2=6, and 14X2=30, .'. are those numbers 
 proportional. Again: because the Antecedent of each 
 when multiplied by the Consequent of the other produces 
 a like number, .*. are they proportional. 
 
 If any three terms of a proportion be given, the re- 
 maining term may always be found. For since in any 
 Proportion, 
 
Rule of Three. 45 
 
 1st term X 4th term = 2d term X 3d term, 
 .-. 1st term=2dX3d^4th 
 
 2d term^lstX4th^3d 
 
 3d term = lstX4th-^2d; 
 
 4th term=2dX3d^lst. 
 
 Example 1. — Find the 4th term in the proportion 3, 
 6, 14. 
 
 3:6;: 14.4th term; .'. 4th term=6Xl4-^3 = 30, Ans. 
 
 RULE OF THREE. 
 
 The Rule of Three is a method l^y which we are 
 enabled, from three numbers which are given, to find a 
 fourth, which shall bear the same ratio to the third as 
 the second to the first; in other words, it is a rule by 
 which, when the three terms of a proportion are given, 
 we can determine the fourth. 
 
 Rule. — Find out, of the three quantities which are given, that 
 which is of the same kind as the fourth or required quantity, or 
 that which is distinguished from the other terms by the nature 
 of the question. Place this quantity as the third term of the 
 proportion. 
 
 Now consider whether, from the nature of the question, the 
 fourth term will be greater or less than the third; if greater, 
 then put the larger of the other two quantities in the second 
 term and the smaller in the first term ; but if less, put the larger 
 in tiie first term and the smaller in the second term. 
 
 Multiply the second and third terms together, and divide by 
 the first. The quotient will be the answer to the question. 
 
46 Elementary Arithmetic of the Octimal Notation. 
 Example 1. — If oats are worth $24 per ton, how much 
 
 ton lb. 
 
 can be bought for $50? Since 1.000 is of the same kind 
 
 ton lb. 
 
 as the required term, — viz., tons, — we make 1.000 the 
 third term. Since $50 will buy more tons than $24, we 
 make $50 the second term and $24 the first term: 
 
 ton lb. 
 
 $24:$50:: 1.000:4th term. 
 
 tons lb. 
 
 24-^50Xl.000=4th term=2.000. 
 
 Example 2. — A has a contract to dig a ditch in 2 
 
 days. He has 4 men employed, but at present progress 
 
 it will take 4 days to complete the work. How many 
 
 men should he have employed to finish the contract on 
 
 time? 
 
 days days men 
 
 2 : 4 :: 4 : 4th term. 
 
 2-^4X4 = 10, the number of men required. 
 
 PRACTICE. 
 
 Simple Practice. 
 
 In this case the given number is expressed in the same 
 denomination as the unit whose value is given; as, for 
 instance, 21 lb. @ $1.20 per lb. 
 
 Rule. — Multiply the numbers together without reference to 
 their names, and point off the terms of the several denominations 
 in the product. 
 
 Example. — 43 lb. of sugar @ 7(J5 per lb. The product 
 will be cents; .', 43X7=365 cents, = $3.65^ 
 
Interest. 47 
 
 Compound Practice. 
 
 In this case the given number is not wholly expressed 
 in the same denomination as the unit whose value is 
 given; as, for instance, 1 ton 235 lb. @ $63 per ton. 
 
 Rule. — State the question thus: If one ton cost $G3.00, what 
 win 1,235 lb. cost? 
 
 Multiply the second and third terms together without 
 reference to their names; thus: — 
 
 1235X6300=10250;pj2l. 
 
 From the right of the product cut off a number of 
 figures which is equal to the number of inferior units in 
 the third term. The remaining figures will be the value 
 required, of the same name as the second term. Thus, 
 $102.50({', Ans. 
 
 INTEREST. 
 
 Interest (Int.) is the sum of money paid for the loan 
 or the use of some other sum of money, lent for a certain 
 time at a fixed rate; generally at so much for each $100 
 for one year. 
 
 The money lent is called the Principal. 
 
 The interest on $100 for a year is called The Rate per 
 Cent. 
 
 The principal + the interest is called the Amount. 
 
 Interest is divided into Simple and Compound. When 
 interest is reckoned only on the principal or sum lent, 
 it is Simple Interest. 
 
50 Elementary Arithmetic of the Octimal Notation. 
 
 When the interest at the end of the first period, instead 
 of being paid by the borrower, is retained by him and 
 added as principal to the former principal, interest being 
 calculated on the new principal for the next period, and 
 this interest, again, instead of being paid, is retained and 
 added on to the last principal for a new principal, and so 
 on, it is Compound Interest. 
 
 Simple Interest. 
 
 To find the interest on a given sum of money at a 
 given rate per cent for a year. 
 
 Rule. — Multiply the princiiDal by the rate per cent and divide 
 the product by 100. 
 
 Example. — Find the simple interest of $240 for one 
 year, at 7 per cent per annum. 
 
 By the Rule. $240.00x7-100=$21.40. 
 
 Compound Interest. 
 
 To find the Compound Interest of a given sum of money 
 at a given rate per cent for any number of years. 
 
 Rule. — At the end of each year, add the interest for that year 
 to the principal at the beginning of it; this will be the principal 
 for the next year ; proceed in the same way as far as may be 
 required by the question. Add together the interests so arising 
 in the several years, and the result will be the compound interest 
 for the given period. 
 
 Example. — Find the Compound Interest and Amount 
 of $600 for 3 years, at 6 per cent per annum. 
 
Practice. 51 
 
 600X6-100=^44 Int. 1st yr. 
 600 + 44 = 644x6-h100=$47..30 Int. 2d yr. 
 644 + 47.30=713.30X6^100=$53.0420 Int. 2d yr. 
 713.30 + 53.0420= Ans. $766.34.20. 
 .-. Compound interest = $53.0420 + $47.30-f .1?44=$166.- 
 3420. 
 
 Amount, $600 + $166.3420=1766.342. 
 
52 Elementary Arithmetic of the Octimal Notation. 
 
 SECTION 5. 
 
 SQUARE ROOT. 
 
 The Square of a given number is the product of that 
 number multiplied by itself. Thus 6X6, or 44, is the 
 square of 6, or 44=6^ 
 
 The Square Root of a given number is a number which, 
 when multiplied by itself, will produce the given num- 
 ber. 
 
 The Square Root of a number is denoted by placing 
 the sign ;/ before the number. Thus -|/44 denotes the 
 square root of 44, so that y^44=6. The sign ^ placed 
 above the number a little to the right denotes that the 
 number is to be squared; thus, 6^=44. 
 
 Note. — The squares of 
 all those numbers which 
 are in geometrical progres- 
 sion appear with the order 
 of their indices reversed, 
 and their roots may be 
 extracted by mere inspec- 
 tion. 
 
 A square figure Avill al- 
 ways contain the square 
 of a number of square 
 units. 
 
 The square root of a number is one side of a square with area 
 equal to the number. 
 
 Table 1. 
 
 Table 2. 
 
 1 = 1/1 
 
 1 = 
 
 1/1 
 
 2= 4 
 
 • 
 
 2- 
 
 4 
 
 3= 11 
 
 • 
 
 4 — 
 
 26 
 
 4= 20 . 
 
 16— 
 
 106 
 
 5- 31 
 
 26 = 
 
 406 
 
 6- 44 
 
 • • 
 
 40 
 
 2606 
 
 7- 61 
 
 166— 
 
 10606 
 
 10= 100 
 
 266— 
 
 40606 
 
Square Root. 53 
 
 .*. A square containing 44 square units has a side containing 
 6 linear units of the same vahie as one side of each of the 44 
 S(juare units contained in the given square. 
 
 RULE TO EXTRACT THE SQUARE ROOT OF A GIVEN 
 
 NUMBER. 
 
 Rule. — Place a dot over the units' place of the given number ; 
 and thence over every second figure to the left of that place ; 
 and thence over every second figure to the right, when the num- 
 ber contains inferior units, annexing a cipher when the number 
 of inferior places is odd ; thus dividing the number into periods. 
 The number of dots over the superior and inferior units respec- 
 tively will show the superior and inferior places in the root. 
 
 From Table 1 take the index which is nearest (but less) to 
 the first period, and write its root in the first place in the root. 
 Now consider whether the first period is nearer the index already 
 taken, or nearer the index which comes next in the table. If 
 nearer the index taken, try a number less than four for the 
 second place in the root ; now square the number for trial. If 
 the product is greater than the given number, try a number 
 which is less ; repeat the operation until all the places in the 
 root are filled. 
 
 FlxAMPLE 1. — Find the Square Root of 1261.04. 
 
 32.2 = i/l26i.04 
 
 Sy/ll 4|/ 20 Method. — From Table 1 take the 
 
 Index 11; the next one, 20; apply 
 the root of the 11 to first place; first 
 period is near 11; .*. try 32^, also 
 33'; the product 1244 is nearest the 
 given number; .*. write 2 in the 
 
 1244 1331 root; 32^ is much nearer the given 
 
 number than 33"; .•. try 322. 
 
 32 
 
 33 
 
 32 
 
 33 
 
 64 
 
 121 
 
 116 
 
 121 
 
54 Elementary Arithmetic of the Octimal Notation. 
 
 322 
 
 322 
 
 644 
 644 
 1166 
 
 322X322 = 126104. 
 
 Square root of 1261.04=32.2. 
 
 A RECTANGLE. 
 
 A mixed number of square units will always 
 126104 be contained in a Parallelogram, one of whose 
 dimensions shall be a root number, as in Table 2. 
 
 Rule. — To conform the given number to a parallelogram, 
 divide the given number by the selected dimension. The quotient 
 will be a perpendicular. The divisor and the quotient will be the 
 required dimensions. 
 
 Example 1. — Given one side and the area of a Paral- 
 lelogram, to find the other side, one side to be 20, the 
 area to be 541. 
 
 By the Rule. 541^20=26.04; 
 .*. one dimension =20; 
 
 u 
 
 u 
 
 = 26.04. 
 
 o 
 
 o 
 
 <M 
 
 Area, 541 Units. 
 
 26.04 
 
 THIS DIAGRAM IS ONE EIGHTH ACTUAL DIMENSIONS. 
 
oo 
 
 b 
 
 Cube Boot. 55 
 
 Example 2. — Conform 541 to the required Parallelo- 
 gram, one side to be 40. 
 
 Bij the Rule. 541-^40=13.02; 
 .'. one dimension=40; 
 
 = 13.02. 
 
 Area, 541 Units. 
 
 40.00 
 
 THIS 1)[A(4RA>[ IS ONE EIGHTH ACTUAL DIMENSIONS. 
 
 CUBE ROOT. 
 
 The Cube of a given number is the product which arises 
 from multiplying that number by itself, and then mul- 
 tiplying the result again by the same number. Thus 
 6X6X6, or 330, is the cube of 6; or 330=61 
 
 The Cube Root of a given number is a number which, 
 when multiplied into itself, and the result is again mul- 
 tiplied by it, will produce the given number. Thus 6 is 
 the Cube Root of 330; for 6X6=44, and 44X6=330. 
 
 The Cube Root of a number is denoted by placing the 
 sign f before the number. Thus ]^330 denotes the 
 Cube Root of 330; so that ]f 330=6. 
 
Table 1. 
 
 Table 2. 
 
 1-fa 
 
 1-1 
 
 3/1 
 
 2- 10 
 
 2- 
 
 16 
 
 3- 33 
 
 4- 
 
 100 
 
 4- 100 
 
 10- 
 
 iooo 
 
 5- 175 
 
 20- 
 
 16000 
 
 6= 330 
 
 40= 
 
 106006 
 
 7- 527 
 
 100- 
 
 1006006 
 
 10- 1000 
 
 200- 
 
 16006006 
 
 56 Elementary Arithmetic of the Octimal Notation. 
 
 Note. — The Cubes 
 of all those numbers 
 which are in geomet- 
 rical progression oc- 
 cur in ciphers with 
 the index 1. Their 
 roots may be extracted 
 by mere inspection. 
 
 All the indices will 
 reappear when their 
 roots are bisected continuously. 
 
 Cube Root is the number of units of measure contained in a 
 line which is one edge of a Cube. 
 
 The Cube Root of a number can only be found when the given 
 number is the cube of the number of units in the base. 
 
 RULE FOR EXTRACTING THE CUBE ROOT OF A GIVEN 
 
 NUMBER. 
 
 Rule. — Place a dot over the units' place of the given number, 
 and thence over every third figure to the left, and thence over 
 every third figure to the right, when the number contains inferior 
 units, aflixing one or two ciphers when necessary to make the 
 number of inferior places a multiple of three, thus dividing the 
 given numbers into periods. The number of dots over the Avhole 
 number and inferior units, respectively, will show the whole 
 number and inferior places in the root. 
 
 Place the sign |^' over the given number. Point over the same 
 number of places to the left of the sign the same number of dots 
 that there are periods in the given number, in the form of a divisor. 
 
 Find among the indices the number which is nearest (but 
 below) the number in the first period, and place the correspond- 
 ing root number in the first place in the divisor. 
 
 Now consider whether the index just applied is nearer the 
 given number than is the index which is nearest above or greater 
 than the given number. 
 
Cube Root. 
 
 57 
 
 If the index applied is the nearest, add a number which is less 
 than four, now cube the new number, and if the product be less 
 than the given number, writedown the figure in the second place 
 in the root, but if it be greater, trj' a lower number. This opera- 
 tion is to be repeated until all the places in the root are filled. 
 
 THIS CUT SHOWS ONE EIGHTH OF THE ACTUAL DIMENSIONS. 
 
60 Elementary Arithmetic of the Octimal Notation, 
 Example. — Extract the Cube Root of 16606.305, 
 
 23.5|/16606.305 
 
 By the Rule. 2|/10 Zy'66 
 
 23 
 
 24 
 
 235 
 
 23 
 
 24 
 
 -235 
 
 71 
 
 120 
 
 1421 
 
 46 
 
 50 
 
 727 
 
 551 
 
 620 
 24 
 
 472 
 
 23 
 
 60111 
 
 2073 
 
 3100 
 
 235 
 
 1322 
 
 1440 
 
 360555 
 
 15313 
 
 17500 
 
 220333 
 140222 
 
 Method. — From Table 
 1 take the index 10; ap- 
 ply its root to the first 
 place; the index applied 
 is nearer the first period 
 than 33; .'. try 23^, also 
 24^, then write 3 in the 
 second place. The given 
 number is nearer 24^; .*. 
 try 2351 
 
 16606305 
 
 RECTANGLE PARALLELOPIPED. 
 
 A mixed number of cubical units will always be con- 
 tained in a Parallelopiped, two of whose dimensions shall 
 be a root number in Table 2. 
 
 Rule. — To conform a given number to a Parallelopiped, mul- 
 tiply the selected dimensions together and divide the given num- 
 ber by the product. The quotient will be the third dimension of 
 a Parallelopiped. 
 
 Example 1. — Conform 6340 to a Parallelopiped. 
 
Eectan(jle Parallelopiped, 
 
 By the Rule. Selected dimensions = 20 each; 
 •. 20X20 = 400 = area of one side; 
 6340-^400= third dimension = 14.7; 
 .-. one dimension = 20; 
 Second dimension = 20; 
 Third dimension = 14.7. 
 
 61 
 
 THIS CUT SHOWS ONE EIGHTH OF THE ACTUAL DIMENSIONS. 
 
 Example 2. — Selected dimensions are: one dimension, 
 10; one dimension, 40; 
 
 .-. 10X40=400=area of one side; 
 6340 -^400= third dimension = 14.7; 
 .'. one dimension = 10; 
 Second dimension =40; 
 Third dimension=14.7. 
 
62 Elementary Arithmetic of the Octimal Notation. 
 
 DERIVATION OF RADIX. 
 
 The Radix is suggested by the number of parts into 
 which the Cube may be subdivided, Tvhen all its dimen- 
 sions are bisected. Hence the number of symbols in the 
 Octimal Notation permit the same mechanical treatment 
 as does the Cube or any other regular solid; .'. since the 
 third mechanical subdivision of the Cube results in 10 
 Cubes, each a counterpart of the original in the ratio of 
 10 to 1, therefore will the third bisection of the radix re- 
 sult in symbols the counterpart of the original in the 
 ratio of 10 to 1. 
 
 ELEVATION OF DENOMINATIONS. 
 
 Whole numbers are raised to the next higher denomina- 
 tion by affixing a cipher; thus, 35X10=350. 
 
 Whole numbers are reduced to the next lower denomina- 
 tion by introducing a point; thus, 35-^10 = 3.5. 
 
 .•. let A represent a cube, and a, a cube, then if the 
 
 weight of A^ 
 
 :35 lb., a^ 
 
 :3.5 lb 
 
Calculation by Construction. 63 
 
 ELEVATION OF CUBES AND THEIR ROOTS. 
 
 For each time a Root is multiplied by two, raise cube 
 number one denomination by affixing a cipher. 
 
 For each time a Root is bisected, reduce its cube num- 
 ber one denomination by pointing off one figure from the 
 right of the whole number. 
 
 CALCULATION BY CONSTRUCTION. 
 
 A Cube, Sphere, or Parallelopiped may be reduced to 
 convenient limits for calculation by simply bisecting its 
 diameter to the lowest convenient denomination. 
 
 Rule. — For each time the three dimensions are bisected, 
 write down one cipher, and for the last place apply the index 1, 
 or .r. When the value of the index is found, replace the index 
 by that value. 
 
 Example. — A cube of bricks is 140.0 feet long at base. 
 How many bricks does it contain? and what is its value 
 at $7.00 per etand bricks? 
 
 ^-^ 
 
 By the Rule. 
 2)140.0 
 
 2)60.0 i Number, 12,530,000. 
 
 Ans. ' 
 
 2)30.0 ( Value, 12,530,000X$7.00=$112,550.00. 
 
 2)14.0 
 
 6.0=6.0 ft. = Index. 
 
 The Index is found to contain 1253 bricks; therefore to 
 the sum so found affix four ciphers. 
 
64 Elementary Arithmetic of the Octimal Notation. 
 
 SECTION 6. 
 
 LONGITUDE AND TIME. 
 
 The Earth's Circle is subdivided, commencing opposite 
 the meridian of Greenwich. 
 
 The Dial is subdivided to correspond from midnight. 
 Longitude is therefore always west. 
 Difference in time gives Longitude. 
 
 Example. — Meridian of Departure =0. The sun passes 
 overhead 40 ^^'- 2V later. What is your Longitude? 
 Ans. 40° (degrees) 21' (minutes). 
 
 TO DETERMINE AN ANGLE. 
 
 >\ 
 
 Example. — Required the Angle A B. 
 Ans. Angle = 12 degrees. 
 
Area of Circle in Terms of Square Units. 65 
 
 Rule. — At point of intersection raise a perpendicular. Bisect 
 the right angle. The value of the new angle -^10. Should the 
 angle exceed 10, bisect again and continue to the ratio of 1 to 10, 
 repeating the operation if necessary. 
 
 When the Angle is less than 10, the operation is similar. 
 
 AREA OF CIRCLE IN TERMS OF SQUARE UNITS. 
 
 The diameter of a circle is to its circumference as 1 
 is to 3.1. 
 
 The square of the diameter of a circle is to its area as 
 1 is to .62. 
 
 .'. Multiply the square of the diameter by .6222. 
 
 Example. — Required the area of a circle whose di- 
 amter is 16.2 inches. 
 
 By the Rule. 16.2^ = 313.04 X.6222 = 237.421710 inches. 
 
66 Elementary Arithmetic of the Octimal Notation. 
 
 SECTION 7. 
 
 MISCELLANEOUS. 
 
 1 Meter = 47. 2753412 Inches, nearly. 
 
 .•. A Centimeter is to an Inch as 1:2.43; /. InchesX 
 2.43 = number of Centimeters. 
 
 An Inch is to a Centimeter as 1:.311; .'. Cm.X.311 = 
 number of Inches. 
 
 Note. — This comparison is sufficiently accurate for all prac- 
 tical pm'poses in measurements up to one foot. 
 
 Example. — Required the value of 6.2 Inches in Centi- 
 meters. 
 
 m. 
 
 By the Rule. 6.2X2.43 = 17.723 Centimeters. 
 
 TRANSFORMATION OF NUMBERS. 
 
 Decimal numbers may be transformed into Octimal 
 numbers. 
 
 Rule. — Divide by eight continuously ; the remainders will be 
 the Octimal figures. 
 
 Example. — 1631 Decimal = 3137 Octimal. 
 
 By the Rule. 8)1631 
 
 8)203(7 . . , 
 Note. — The operation is carried 
 
 8)25(3 on in the decimal system. 
 
Decimal Fractions. 67 
 
 Octimal numbers may be transformed into Decimal 
 numbers. 
 
 Rule. — Divide by et-two continuously ; the remainders will be 
 the Decimal numbers. 
 
 Example. — 15356 Octimal=6894 Decimal. 
 By the Rule. 12)15356 
 
 12)1261(4 
 12)104(11=9 
 6)10=8 
 
 DECIMAL FRACTIONS. — TO TRANSFORM INTO IN- 
 FERIOR UNITS. 
 
 Rule. — Multiply by eight continuously. The figures to the 
 left of the point in each product will be the Octimal Inferior 
 Units required. 
 
 Example. — .125 = .l. 
 
 By the Rule. .125 
 
 8 
 
 1.000 
 
 Note. — When all the figures in the i)roduct are below the 
 point, supply a cipher in the units' place. 
 
 Example. — .0625 = .04. 
 
 By the Rule. .0625 
 
 8 
 
 0.5000 
 8 
 
 4.0000 
 
70 Elementary Arithmetic of the Octimal Notation. 
 
 DECIMAL NUMBERS AND OCTIMAL INFERIOR 
 
 UNITS. 
 
 Addition. 
 
 Rule. — Add the octimal inferior units together, carrying 
 eights. Carry the superior units, and add to the decimal num- 
 bers above the point. Continue in the decimal system. 
 
 Dec. Oct. Dec. Vulgar Fractions. 
 
 Example. — 29.3 inches=29 + | inches. 
 
 29.6 
 
 u 
 
 -29 + 1 
 
 a 
 
 
 29.23 
 
 u 
 
 -29 + i 
 
 a 
 
 4- 3 
 + "6 4' 
 
 29.76 
 
 u 
 
 -29 + 1 
 
 a 
 
 4- ^ 
 
 + 61^ 
 
 118.31 118 + 1 inches + ^x. 
 
 Multiplication. 
 
 Dec Oct. Dec. Vul. Frac. 
 
 118.31x4 = 118+1 + ^^X4 
 4 4 
 
 473.44 473 + i + e\ 
 
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