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IN MEMORIAM
I FLORfAN CAJORI
IV
UMREB
^ mathematical ^
\ Series M
Vi TWO
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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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