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ARITHMETIC
FOR
HIGH SCHOOLS
AND
COLLEGIATE INSTLLUTES
BY
J. O.'^GLASIIAN,
OTTAWA.
ROSE PUBLISHING COMPANY
1890.
Entered according to Act of the Parliament of Canada, in the year one
thouf^and eight hundred and ninety, by Rose Publishing Company,
(Limited), at the Department of Agriculture.
printed and bound bv
Hunter, Rose & Company,
Toronto.
103
PREFACE.
The following work was prepared for the use of pupils in High
Schools and Collegiate Institutes. As all pupils in these schools are
required to possess, before admission thereto, a sufficient knowledge of
arithmetic to enable them to solve easy problems such as those in Exer-
cise I and IV pp. 45 to 55 and 75 to 85 of the present work, the
author has taken for granted the possession of such knowledge by those
who will use this book. In other words, he has sought to supplement
and to continue without any unnecessary repetition the course of arith-
metic begun in the Public School Arithmetic. Furthermore, as the
book is not intended for private study but for class-instruction under
the guidance and with the intelligent assistance of competent mathe-
matical masters, the Author has endeavored to avoid encroaching on
the province proper to the instructor and has in general given only the
main outlines of proofs and investigations leaving it to the teacher to
lill in the details and to supply preliminary illustrations.
The work consists of three distinct parts. The first part, forming
chapters i to iv, treats of Notation and Computation ; the second part,
chapter v, treats of Mensuration or Metrical Geometry ; and the third
part, chapters vi, vii and viii, deals with Commercial Arithmetic.
Chapter I treats of numbers and notation and of unita of measure-
ment. The student will already be well enough acquainted with Arabic
and Roman notation and with various compound systems to be able to
use them more or less freely, but to know a subject is one thing, to
know it in words, i. e. , to know it so clearly and distinctly as to be able
to put that knowledge into words, is quite another thing ; — this chapter
will it is hoped, help the student to put into words his knowledge of
arithmetical notation and of our ordinary units of value, mass, space
and time.
Chapter I with §§ 42, 62 and 63 of Chapter II and §§119 and 120 of
Chapter IV lay a foundation on which may be built the theory of num-
bers and the rationale of the art of calculation. True the ' ' Funda-
mental Theorems " of chapters ii and iv are, strictly speaking, Postulates
defining and determining the particular kinds of addition, multiplica-
tion and involution here considered, but this is a distinction which only
those who have advanced some way in their studies can understand, and
th;3 history of mathematics teaches that the method of presenting the
subject here adopted is the easiest and the best for beginners.
The greater part of Chapter II consists of descriptions of come of
the methods of computation employed by experts. The proper place
for these is a manual on the art of teaching, but as they are not to be
found in their proper place and as many of the pupils in our High
Schools purpose becoming teachers, it has been thought better to insert
IV PREFACE.
these descriptions in the present work than to leave it a matter of chance
whether teachers shall know and practice any other than the traditional
school-room methods. Here as elsewhere throughout the book, the
Author makes no pretension to originality ; he has selected for descrip-
tion the best methods and processes with which a somewhat extensive
acquaintance with the literature of elementary mathematics has made
him acquainted.
Approximation is a part of arithmetic which has until lately been
adequately discussed in only the higher classes of text-books, being, one
might be led to conclude from their neglect of it, an unknown subject to
the writers of the average school-book. But the great practical import-
ance of the subject is at length compelling its fuller recognition in school-
work, and it will receive more and more attention in proportion
as arithmetical instruction ceases to be impractical and as teach-
ers become better acquainted with the requirements of the count-
ing-house, the workshop and the laboratory. In Chapter III, two
methods of approximation are described ; the first, Approximation by
Continued Fractions; the second, Approximation by Abridgment of
Decimal Computations. The former of these takes precedence in
historical order and also on account of its theoretical simplicity and of
the wide range of subjects to which it is applicable, — from the purely
speculative questions of Farcy's series and the partitions of numbers to
the laboratory problem of determining the formula of an organic com-
pound from its percentage composition ; — but the latter method is
superior in facility of adaptation to all ordinary computations.
Approximation by continued fractions was well known to the ancient
Greek and Indian arithmeticians, so much so that in the oldest of their
writings now extant it is introduced abruptly and used without explan-
ation as an elementary subject with which their readers are assumed to
to be already familiar. The whole theory of the subject is contained in
the single theorem, —
f lies in value between — and — , being greater
b + k b k
than one and less than the other,
and the immediate corollary therefrom, —
— » and — are in order of Tnagnitude^
b mb + nk k
a, h, hf h, m and n denoting (absolute) numbers. But in the calculation
and use of continued fractions, no proof is needed of the theorem in the
general form in which it is here stated, its truth being tested in each
separate instance of its application. Hence no reference to the general
theorem is required in Chap. Ill and no such reference is made therein.
In the arrangement of the factors in the contracted multiplication due to
Oughtred and known by his name, the figures of the multiplier are
written in reverse order, but the arrangement adopted in Example i,
p. 68 obviates the awkwardness of this reversal and is as simple as
Oughtred's in every other respect. Teachers who prefer to discuss
abridged computation before convergents and those who prefer to omit
all discussion of the latter will find that the method of treatment which
has been adopted will permit of their following their preference. Those
I
PREFACE.
who seek for a fuller treatment of the theory of contracted calculations
will find it in the Arithmetics of Munn, Cox, Sang, Serret and Beynac,
in Ruchonnet's Elements de Calcul Approximatif , Lionnet's Approxima-
tions Numeriques and Vieille's Theorie Generale des Approximations
Numeriques.
Chapter IV contains an elementary discussion of Involution, Evolution
and Logarithms. Special attention has been given to ' Horner's Method '
of Involution and Evolution not only on account of its simplicity, it
being merely an extension of the ordinary rule of ' Reduction ', but also
because of its power and generality as a process and of its great and
varied utility. The chief value of logarithms, at least in elementary
mathematics, lies in their usefulness as an instrument of calculation, buo
the surest way to enable pupils to remember how to use tables of
logarithms is to require them to compute a portion of such a table
considered as a table of exponents to base 10. Teachers who pr' fer to
have their pupils learn at first the mere mechanical use of the tables and
defer the theory of logarithms until logarithmic series is reached will
omit §§ 114 to 122 and 124 to 136 and Exercises X to XV, but it might
be well if they should note that the development of the theory of
logarithms preceded that of the logarithmic and e"xponential series,
preceded even the invention of generalized exponents, and that no large
table of logarithms was ever computed by the immediate use of logarith-
mic series.
In connection with the subject of chapters iii and iv, the following
extract is a sign of the progress now makuig in England: —
" The Council notices M'ith pleasure, as an example of what may be
done by an examining body in the way of encouraging sound mathe-
matical teaching, the following "Remarks'* in the prospectus of the
Technical College, Finsbury, with reference to the Entrance Examina-
tion:— In Arithmetic, marks will bo deducted on those answers in
which bad and antiquated methods are used; for exrmple, if the
Italian method in division is not followed ; if decimal workings are not
properly contracted ; if remainders are given in fractions instead of in
decimals ; if logarithms are not used where their use would save time.
[Logarithm Books containing Four-figure tables, are provided at the
examination.] . ...
(Candidates) should be able to work square root and cube root by Hor-
ner's method.' " Extract from the Report of the Coiincil of the Associ-
tioji for the Improvement of Geometrical Teaching ; England, January,
1889.
Chapter TV" closes the subject of pure calculation with the exception of
the short Appendix on pp. 315 to 317, in which the notation of circulating
decimals is explained without the usual hidden reference to infinite series
and the method of limits. The curious and those who care to spend
time on a subject of no practical and of but little speculative importance
may consult the Arithmetics of Sangster, Brook-Smith, Cox, Lock, and
Sonnenschein and Nesbitt, or the Traite d' Arithmetique et d' Arith-
mologie oi' P. Gallez.
Chapter V consists of. a short treatise on elementary metrical geometry.
Much of the text, especially in the stereometry consists of proofs of
important geometrical theorems which are not to be found in the
VI PREFACE.
authorized text-book of geometry. A few propositions not properly
belonging to elementary mensuration, c. g., § 201, p. 237, are given
without proofs.
The Author begs to acknowledge his obligations in this part of his
subject to Die Elemente der Mathematik of R. Baltzer, the Planimetrie
and Stereometric of F. Reidt in Schloemilch's Handbuch der Mathe-
matik, the Traite de Geometric Elemcntaire of Rouche and Comberousse
and the Theoremes et Problemes de Geometric Elemcntaire of M. Eugene
Catalan.
Chapters VI, VII and VIII complete the course of elementary Com-
mercial arithmetic begun in the Public School Arithmetic. In selecting
questions for Exercises I and IV, it was assumed that pupils could solve
by the so-called Unitary Method, the simpler problems in commercial
arithmetic, including simple interest and discount ; but that method,
however excellent as a mere answer-obtaining process, has an almost
irresistible tendency to withdraw the attention of those who make ex-
clusive use of it, from the general principles upon which all methods are
founded. Numerous easy problems have therefore been proposed in the
earlier exercises of Chap. VII, which it is hoped the teacher will take
advantage of to endeavor to raise his pupils at this stage of their studies
from infantile dependence on the unitary crawling-on-hands-and-knees
method, and lead them to make direct application of general principles,
including that widest of all principles, The Substitution of Equivalents.
For further information concerning promissory notes and bills of ex-
change, teachers should consult the Bills of Exchange Act of 1890.
Data for an unlimiljcd number of problems on stocks, bonds and deben-
tures will be found in the Stock Exchange Year-Book by Th. Skinner.
Students who wish to advance to the higher questions on interest, annui-
ties and life insurance, will find an elaborate discussion of these subjects
in the Institute of Actuaries' Text-Book ; Part I, Interest, including
Annuities-Certain, by Wm. Sutton; Part II, Life Contingencies, in-
cluding Life Annuities and Assurances, by Geo. King ; to these two
volumes may be added Ackland and Hardy's Graduated Exercises and
Examples.
The Author desires to express his general indebtedness to the writings
ofDe Morgan, Duhamel, Grassmann, Schlegel and Houel. He also takes
pleasure in acknowledging his special indebtedness to Professor R. R.
Cochrane, of Wesley University, Winnipeg, and Mr. Robert Gill,
Manager of the Ottawa Branch of the Canadian Bank of Commerc-?, for
much valuable advice and assistance, and for many practical problems,
and he tenders these gentlemen his grateful thanks for their kindness.
CONTENTS
PAGE
Numbers and Notation :
Numeration - - 1
Notation --. 5
Prime Units - 9
Tables of Values, Weights and Measures - - - 10
Metric System of Weights and Measures - - - 17
The Four Elementary Operations :
Addition and Subtraction 21
Multiplication and Division ----- 26
Miscellaneous Problems 45
Approximation :
Convergent Fractions .-.-.. 56
Approximate Calculations - - * - - - - 66
Miscellaneous Problems 75
The Three Higher Operations :
Involution ------..-86
Evolution 97
Incommensurable Numbers ------ 116
Logarithmation -------- 126
Use of Tables of Logarithms - - - - . 143
Computation by Help of Logarithms - . - . 1-18
Mensuration :
Introduction 159
Similar Rectilineal Figures 161
Definitions - 164
Areas of Trapezoidal Figures 168
Volumes of Prismatic Solids 177
Triangles, — Lengths of Sides 204
Area - 210
Circle, — Length of Circumference - - - - 213
viii CONTENTS.
Mensuration : PA.GE
Circle and Ellipse,— Area - 225
Cylinder, Cone and Sphere, — Area .... 233
" <' " Volume - - - 241
Proportional and Irregular Distribution - -^ - - 251
Partnership _ - . 266
Percentage :
Introduction - - - 269
Profit and Loss 272
Insurance - - - - - - - - - 275
Commission and Brokerage . . . . . 279
Discount - - - 282
Promissory Notes and Inland Bills of Exchange - - 284
Interest - - - - - - - - ' 289
Simple Interest - - - - - - - - 290
Averaging Accounts ------- 293
Partial Payments - - 295
Compound Interest ------- 297
Stocks and Bonds - - - - - - - 302
Exchange, — Foreign - - 308
Appendix, — Circulating Decimals . - - . - 315
Tables of Logarithms -.--... 318
The following selected course may be taken by candidates preparing
for the primary examination and by all who have no special aptitude for
mathematics: — Pages 1 to 55, 66 to 73, 75 to 93, problems 1 to 7 (f Ex.
vi, pp. 97 to 105 omitting last six lines, problems "1 to 12 of Ex". ix,
pp. 251 to 298 and 302 to 314, and the portions of chapter v covering
the requirements in mensuration required for the primary examin;\tion,
but substituting verification by inspection of models for purely logical
demonstration of the geometrical theorems quoted.
ARITHMETIC.
li
CHAPTER I.
OF NUMBERS AND NOTATION.
1 . The simplest expression of a Quantity consists of two factors
or components. One of these factors is the name of a magnitude
which has been selected as a standard of reference and which
is necessarily of the same kind as the quantity to be expressed.
The other factor expresses how many magnitudes, each equal to the
standard magnitude, must be taken to make up the required
quantity. The standard magnitude is termed a Unit and its cofactor,
the other component of the expression, is termed the Numerical
Value of the quantity. Hence, —
A Unit is any standard of reference employed in counting any
collection of objects, or in measuring any magnitude.
A Number is that which is applied to a unit to express the
comparative magnitude of a quantity of the same kind as the unit.
A Number is the direct answer to the question, ' How many V
Thus, when it is said that a certain slate is ten inches long, the
number ten applied to the unit-length, inch, indicates the magnitude
of a certain length, that of the slate, compared with the unit-length,
an inch.
2. The number and the unit together indicate the absolute
magnitude of the quantity ; the number indicates the relative
m.agnitude, or, as it is termed, the Batio of the qiuintity to the
uidt.
3. Numeration is counting, or the expressing of numbers in
words.
The ordinary system of numeration is the Decimal System
(Latin, decern ten), so called because it is based on the number ten.
4. The names of the first group of numbers in regular succession
are one, two, three, four, five, six, seven, eight, nine. Other
2 ARITHMETIC.
number-names are ten, hundred, thousand, million, billion, trillion,
quadrillion, quintillion, sextillion, , tenth, hundredth,
thousandth, millionth, billionth, and so on, forming names from
the Latin numerals.
5. The number one applied to any unit denotes a quantity which
consists of a single unit of the kind named.
The number two applied to any unit denotes a quantity which
consists of one such unit and one unit more.
The number three applied to any unit denotes a quantity which
consists of two such units and one unit more.
And so on with the numbers four, five, six, seven, eight, nine ;
applied to any unit they denote quantities increasing regularly by
one such unit with each successive number.
6. The number next following nine is ten, which applied to any
unit denotes a quantity consisting of nine such units and one unit
more.
Counting now by ten units at a time, as before we counted by
single units, we get the numbers ten, twenty (twen-tig, twain-ten),
thirty {three-tig, three-ten), forty (four-ten), ninety
(nine-ten).
The names of the numbers between ten and twenty are, in
order, eleven (endlufon from en, dn, one and lif ten), twelve (twd
two and lif ten), thirteen (three and ten), fourteen (four and ten),
nineteen (nine and ten).
The names of the numbers between twenty and thirty, thirty
and forty, are formed by placing the names of the
numbers one, two, three, nine, in order after twenty,
thirty, ninety.
7. The number hundred applied to any unit denotes a quantity
which consists of ten ten-units.
Counting now by a hundred units at a time, as before we counted
by single units, we get the numbers one hundred, two hundred,
nine hundred.
The names of the numbers between one hundred and two
hundred, two hundred and three hundred, , .-. . . are formed by
placing the names of the numbers from one to ninety-nine in
regular succession after one hundred, two hundred, nine
hundred.
NUMBERS AND NOTATION. 8
8. The number thousand applied to any unit denotes a quantity
which consists of ten hundred-units.
Counting now by a thousand units at a time as before we counted
by single units, we get the numbers one thousand, two thousand,
nine thousand, ten thousand, eleven thousand, twelve
thousand, twenty thousand one hundred thousand
two hundred thousand, nine hundred and
niniBty-nine thousand.
The names of the numbers between one thousand and two
thousand, two thousand and three thousand, are formed
by placing in order the names of the numbers from one to nine
hundred and ninety-nine, — the numbers preceding a thousand, —
after one thousand, two thousand, nine hundred and
ninety-nine thousand.
9. The number million applied to any unit denotes a quantity
which consists of a thousand thousand-units.
The number billion applied to any unit denotes a quantity
which consists of a thousand million-units.
The number trillion applied to any unit denotes a quantity which
consists of a thousand billion-units.
And so on with the numbers quadrillion, quintillion, sextillion,
septillion, ; applied to any unit they denote quantities
increasing regularly one-thousand fold with each successive number.
Counting by a million units at a time, as before we counted by
single units from one to nine hundred and ninety-nine, we get the
numbers
one million, two million, ten million, one
hundred million, nine hundred and ninety-nine million.
Counting by a billion units at a time, we get the numbers
one billion, two billion, ten billion, one
hundred billion, nine hundred and ninety-nine billion.
This system is continued with the numbers trillion, quadrillion,
quintillion, &c. , counting from one of each to nine hundred and
ninety-nine of the same.
The names of the numbers between one million and two million,
two million and three million nine hundred and ninety-
nine million and one billion are formed by placing in order after
one million, two million, nine hundred and ninety-ninQ
million, the names of the numbers preceding a million.
4 ARITHMETIC.
The names of the numbers between one billion and two billion,
two billion and three billion, nine hundred and ninety-
nine billion and one trillion are formed by placing in order after
one billion, two billion, nine hundred and ninety-nine
billion, the names of the numbers preceding one billion.
This system is continued with the numbers trillion, quadrillion*
quintillion, , by placing in order after one of each, two
of each, three of each, &c., the names of all the numbers that
I^recede one of the same.
10. English arithmeticians, following the example of Locke, the inventor of
these names (An Essay concerning Human Understanding, Bk. II, Chap, xvi,
§ 6), employ billion as the name not of a thousand millions but of a million of
millions ; trillion then signifies million of billions, quadrillion means million
of trillions, and so on. This gives what is known as the English System of
Numeration. These names are however of little practical importance, being
seldom or never required in the ordinary affairs of life, while for scientific
purposes another system, to be explained hereafter, is generally employed.
1 1 . The number tenth applied to any unit denotes that quantity
of which ten make up the unit.
The number hundredth applied to any unit denotes that quantity
of which ten make up one tenth of the unit.
Consequently, one hundred of the hundredths of any unit make
up that unit.
The number thousandth applied to any unit denotes that
quantity of which ten make up one hundredth of the unit.
Consequently, one thousand of the thousandths of any unit
make up that unit.
The number millionth applied to any unit denotes that quantity
of which a thousand make up one thousandth of the unit.
The number billionth applied to any unit denotes that quantity
of which a thousand make up one millionth of the unit.
The numbers trillionth, quadrillionth, &c. ; applied to any unit
denote quantities decreasing regularly one-thousandfold with each
successive number.
12. We count by the tenth of a unit at a time, as before we
counted from one to nine by a single unit each time, thus, —
one tenth, two tenths, ..... nine tenths.
We count by a hundredth of a unit at a time, as before we
counted from one to ninety-nine by a single unit each time, thus, —
one hundredth, two hundredths, ninety-nine hundredths.
NUMBERS AND NOTATION. 5
We count the thousandths of a unit, the niillionths of a unit,
the billionths of a unit, &c. , from one of each to nine hundred and
ninety-nine of the same in like manner as we count thousands of
the unit, millions of the unit, billions of the unit, &c.
13. Notation is the art of expressing numbers by means of
certain marks or characters called numerals.
The system of notation in general use is the Arabic Notation,
so named because it was introduced into Europe by the Arabs.
Another system now employed for only a few special purposes such as
numbering the chapters in books and marking the hours on clock-
faces, is the Roman Notation, so called because it was the
system in use among the ancient Romans.
14. The Arabic Numerals, styled also Figures, are
0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
denoting nought, one, two, three, four, five, six, seven, eight, nine
respectively. The first of these is named nought, cipher or zero ;
the remaining nine are called digits. By means of these numerals
and a dot called a decimal point we can write down any number
expressed decimally. The method of doing so may be described as
follows : —
A figure immediately to the left of the decimal point denotes so
many single units.
A figure immediately to the left of the single-units figure denotes
so many tens of the units, while a figure immediately to the right
of the single-units figure denotes so many tenths of the unit.
Figures to the left of the tens-figure, taking them in order from
right to left, denote so many hundreds of the unit, so many
thousands of the unit, so many ten-thousands of the unit, so many
hundred-thousands of the unit, so many millions of the unit, &c.
Figures to the right of the tenths-figure, taking them in order
from left to right, denote so many hundredths of the unit, so
many thousandths of the unit, so many ten-thousandths of the
unit, so many hundred-thousandths of the unit, so many
millionths of the unit, &c.
15. Since the number expressed by a digit depends not only on
the particular digit made use of but also on the jilace the digit
occupies relative to the place of the single-units figure, the several
places within any number must be distinctly marked off. This is
b ARITHMETIC.
done by leaving no such place vacant ; places occupied by digits are
sufficiently marked off by these digits, — one digit, one place ;
places unoccupied by digits are filled up with 7ioughts, — one place^ one
oiought. In determining the place occupied by any of the figures of
a number expressed in Arabic notation, count right or left, as the
case may require, from the ones or single-units figure, not from the
decimal point.
16. The sole use of the decimal point is to distinguish or point
out the figure expressing single units, which figure is always the
first to the left of the point. If in any number there are no
figures to the right of the decimal point, that point is omitted and
the right-hand figure then expresses the number of single units.
17. According to our system of numeration, the figures to the left
of the decimal point are necessarily read in groups of three figures
each — ones, thousands, millions, &c. The same system of reading
may be conveniently applied to the figures to the right of the decimal
point by counting and reading each tenth of the unit as a hundred of
the thousandths of the unit, each hundredth of the unit as ten of
the thousandths of the unit ; each ten-thousandth of the unit as
. a hundred of the millionths of the unit, each hundred-thousandth
of the unit as ten of the millionths of the unit ; and so on.
18. The Arabic system of Notation may be exhibited in tabular
form, thus : —
GO
oQ cc 'O -tJ >^ rt
rt fl fl . -^ "S "2
tM O «4H O CM CS O' CM rt «*-• 5 M-l S
o;r3 o;:^ o2 t^ooj o2 ^o
I— I >-H P .ri °Q J -^
- -- - o =5 rd ^
^ ^ S ^ o
^ s m
;h ;-i }-i u C^ f-t s-i ;-i
TJ . •'Tj . •■Ti . ' n . • tTj^tJ • ''TIS . • "TJ • •
WhoWhoWhoWho WhoWhoKho
732468908519 • 431130072
The number here placed as an example beneath the table is
Seven hundred and thirty-two billion, four hundred and sixty-eight
million, nine hundred and eight thousand, five hundred and
nineteen, and four hundred and thirty-one thousandths, one
hundred and thirty millonths and seventy-two billionths.
NUMBERS AND NOTATION.
^m the Roman system was in general use in Western Europe ; but
^f it was employed only for recording numbers, not in performing
calculations. These were made by means of lines drawn on a
sand-strewn tablet, or by the movement of counters arranged on a
reckoning-board called an abacus.
20. The Roman Numerals and their equivalents in Arabic
notation are
Early Forms.
Later Forms.
Arabic Equivalents
1
I
1
A or V
V
5
X
X
10
Uy or ±
L
50
0 or ©
C
100
n
D
500
(Ti or 00
M
1,000
RN
5,000
^
/JK\
10,000
RJ^
50,000
(i^ 100,000
After the invention of printing p) and ^ and the forms derived
from them, were, for convenience in type-setting, modified to 10
and CIO and forms correspondingly derived from these.
When letter-forms were used as numerals, it was a very common practice to
distinguish a numeral from an ordinary letter by drawing a short horizontal stroke
through the numeral, thus ©•, or over it. thus r^ . The former method was
employed in early times, but the latter method superseded it in later times.
Quite recently it has been proposed to employ a short stroke over a Roman
numeral to denote a thousandfold increase in the value of the numeral ; thus
V denoting 5, V ^o^^d then denote 5,000, ^ would denote 5,000,000, &c.
As no numeral higher than M is ever now made use of, this innovation is not
needed ; and as the bar over a letter has, for more than two thousand years
past, been used solely to mark that the letter is a numeral, the innovation
is worse than useless.
21. In Roman notation a numeral standing alone has the value
assigned it in the preceding section. If the numeral be followed by
another or by others of equal or of less value, the sum of their values
IV
exjiresses four.
IX
" nine.
XL
40.
XC
90.
CMXCIX
999.
8 ARITHMETIC.
is indicated. If a numeral be preceded by another of less value, the
difference of their values is to be taken. Thus
II expresses two.
III " three.
VI *' six.
XXIII " 23.
MDCCCLXXXVIII 1888.
22. The ordinary system of numeration is based on the number
ten, but any other number might be adopted as the basis of a
system, and in fact a system founded on the number twelve is
employed to a large extent in counting small articles that are bought
and sold by number. In this, the duodecimal system, the
number twelve receives the name dozen, a dozen dozen is termed
a gross, and a dozen gross is called a grrea* grross. The transactions
in which this system of counting is adopted, do not often involve
numbers of higher order than a great gross, consequently, no
names have been coined for these higher numbers, the gross gross
or dozen great gross, the gross great gross, &c.
A similar state of affairs long prevailed in the decimal system of counting
which, until the comparatively recent introduction of the word million, had no
single-word name for any number greater than a thousand ; thus a million was
called ten hundred thousand. Even now the names above a hundred million
are not fixed, billion meaning with some writers a thousand million, with other
writers a million million, and a thousand million being named indifferently a
billion and a milliard.
23. In the notation of the duodecimal system, Arabic numerals
are made use of, but to distinguish a number expressed in this
system from one expressed decimally, the names dozen, gross and
great gross are inserted in the expression under the forms doz. ,
gro. , gr. gro. This also avoids the necessity for special symbols for
ten and eleven. Thus 5 gro. 3 doz. 7 denotes five gross three
dozen and seven, 11 doz. 5 denotes eleven dozen and five.
These numbers might otherwise be written 537xii and e5xii and
their product would be 505texii in which t and e denote ten and
eleven respectively, and the subscript xii indicates that the numbers
are expressed duodecimally. Expressed decimally these numbers
are 763, 137 and 104531 respectively.
24. Sometimes more than one unit is employed in expressing a
magnitude, as when it is said that the height of a certain doorway
NUMBERS AND NOTATION.
9
is seven feet three inches, or that a certain book weighs two pounds
five ounces. In such case one of the units is taken as the principal
or Prime Unit, and the other units, termed Auxiliary Units,
are derived from it either by repeating it a given number of times,
the resulting multiple forming a unit of a higher order, or by
dividing it into a given number of equal parts, one of these parts
forming a unit of a lower order.
Thus, if a gallon be taken as the prime unit in the quantity five
bushels three pecks and one gallon, a peck, which is equal to two
gallons, will be the unit of the first order higher than a gallon ;
and a bushel, which is equal to four pecks and therefore to eight
gallons, will be the unit of the second order higher than a gallon.
If a yard be taken as the prime unit in the length, three yards two
feet and seven inches, a foot, which is the third part of a yard, will
be the unit of the first order lower than a yard ; and an inch, which
is the twelfth part of a foot and consequently the thirty-sixth part
of a yard, will be the unit of the second order lower than a yard.
25. A Simple Quantity is a quantity expressed in terms of
1^ a single unit.
Ih a Compound Quantity is a quantity expressed in terms of
two or more units. Compound quantities are often called,
though not with strict accuracy, Compound Numbers.
26. The Prime Units of the quantities commonly treated of
in the ordinary arithmetic are : —
Value, Canadian. \
United States, j
British,
"Weight and Mass,
Length, - . . .
Area, - . . -
Volume, - - - - \
Time, - - - -
Angle, ... -
27. A tabular statement of the
Dollar.
Pound Sterling.
Pound Avoirdupois.
Yard.
Square Yard.
Cubic Yard.
Gallon.
Mean Solar Day.
Complete Revolution,
numerical relations or ratios of
any set of auxiliary units to their prime unit and to each other is
called a Table of Values, of Weights or of Measures, the special
designation depending upon the nature of the units.
10 ARITHMETIC.
TABLES OP VALUES, WEIGHTS AND MEASURES.
Canadian Money.
1,000 mills = 100 cents (ct.) = l dollar. $.
The mill is defined by statute but is not recoojnized in ordinary
commercial transactions. Its use is practically confined to stating
ra,tes of local taxation which are generally described as so many
mills on the dollar of assessed value ; thus, a rate of "015 is described
as 15 mills on the dollar.
The dollar is defined by statute to be of such value that four
dollars and eighty-six cents and two-thirds of a cent shall be equal
in value to one pound sterling. ($4-86f = £1.)
United States money is practically the same as Canadian money
both in values and in names, although in Canada the United States'
silver coins are subject to a discount from their nominal values, and
in the United States Canadian silver coins are similarly subject to
a discount. The cause of this is that the market value of the
silver in the coins, — the amount of gold for which the silver' will
exchange, — is less than the nominal values of the coins, — the values
stamped on them ; and while a coin passes current for its nominal
value in the country of issue, it is worth only its market value as
silver in any other country.
The one-dollar gold piece which is the prime unit, or standard
of value in the United States, weighs 25 '8 grains ; nine-tenths of
it is pure gold and the remaining one-tenth is an alloy of copper
and silver.
British, or Sterling Money.
4 farthings = 1 penny, (d.)
12 pence =1 shilling, (s. or /-.)
20 shillings =1 pound, (£.)
1, 2 and 3 farthings are denoted by ^d., |d., and |d. respectively.
Sterling Money is the money of account employed in Great
Britain and Ireland. The prime unit is the pound which is the
value of the coin named a sovereign. The sovereign is coined of
standard gold which is composed of 11 parts of pure gold to 1 part
lir
TABLES OF VALUES, WEIGHTS AND MEASURES. 11
f alloy. 1869 standard sovereigns weigh 480 ounces Troy of 480
grains each. Hence a sovereign should contain 123 "27447 grains of
standard gold of \h fineness, but a "remedy," or allowance for error
is permitted of "2 of a grain in weight, and of 2 parts in 1,000 in
fineness. The least current weight is 122 '5 grains ; below this the
sovereign is "light," and is not legal tender, i. e., it need not be
eceived as of full value.
Avoirdupois Weight.
7,000 grains (gr.) = 16 ounces (oz.) = l pound (lb.)
2,000 pounds = 1 ton (T.)
480 grains = 1 ounce Troy (oz. Tr. )
The one-sixteenth part of an ounce avoirdupois is named by
statute a dram, but the term is not used in commerce, fractions of
an ounce being employed instead. 100 pounds is called a cental or
huiidredweight, denoted by cwt. 2240 pounds is called a lortg torn.
The Dominion Weights and Measures Act declares that ' ' All
articles sold by weight shall be sold by avoirdupois weight, except
that gold and silver, platinum and precious stones, and articles made
thereof, may be sold by the ounce troy, or by any decimal part of
such ounce."
The prime unit or standard of weight is the avoirdupois pound
hich is determined by the weight of a certain piece of platinum-
iridium, called the Dominion standard, deposited in the Department
of Inland Revenue at Ottawa. The weight of this standard is
declared to be " 6999 '98387 grains when it is weighed in air
at the temperature of 62 degrees of Fahrenheit's thermometer, the
barometer being at 80 inches," and 7,000 such grains make one
pound avoirdupois.
Avoirdupois Weight is used in Great Britain and Ireland, the
grain, the ounce and the pound being the same as the Canadian
weights bearing these names but the hundredweight is equal to
112 1b. and the ton, equal to 20 cwt., is equal to 22401b. The
Table of British or Imperial Avoirdupois Weight is : —
7000 grains (gr.) = 16 ounces (oz.) =1 pound, (lb.)
14 pounds =1 stone, (st.)
8 stone = 1 hundredweight, (cwt. )
20 hundredweight =1 ton, (T.)
12 ARITHMETIC.
Linear Measure.
12 inches (in) = l foot, (ft.)
3 feet =lyard, (yd.)
1,760 yards =1 mile, (mi.)
In measuring land, surveyors use a chain 22 yards long, divided
into 100 equal parts called links. Hence
100 links =1 chain, (ch.)
80 chains = 1 mile.
In calculations the links are written as decimals of a chain.
The following measures are used only occasionally, or for special
purposes : —
The line = j\ inch.
The size = ^ inch, used by shoemakers.
The nai? = 2j inches =j^j5- yard, formerly used in cloth measure.
TJie word is now obsolete as a term of measurement.
The hand = 4: inches, used in measuring the height of horses.
The fathom = 6 feet and ^^^ , ^^.j^^^
The cable-length = 120 fathoms, j ^
The rod, pole, or perch = 5h yards, used in measuring land, but
not by surveyors.
The furlong = 220 yards = ^ mile.
The league, not a fixed length, but in England commonly = 3 miles.
The geographical or nautical mile, called also a minute of mean
latitude, is j^loo ^^ *^® earth's semicircumference from pole to pole.
Its length is 6,077 feet, but for rough approximations it is taken as
= 6, 000 f eet = 1, 000 fathoms.
The Paris /ooi:= 12 -79 inches.
' The French perch = 18 Paris feet = 6 '395 yards.
The arpent, or " acre " = 180 Paris feet =64 yards nearly.
The three measures last-named are used under authority of the
Dominion Weights and Measures Act for measuring lands in certain
parts of the Province of Quebec. Distances less than a mile are
often stated in that Province in *' acres."
The prime unit, or standard of length, is the distance in a
straight line between the centres of two gold plugs or pins in a
certain bronze bar deposited in the Department of Inland Revenue
at Ottawa, measured when the bar is at a temperature of 61 '91
degrees of Fahrenheit's thermometer.
II
TABLES OF VALUES, WEIGHTS AND MEASURES. 13
Surface Measure.
144 square inches (sq. in. ) = 1 square foot, (sq. ft. )
9 square feet = 1 square yard, (sq. yd. )
4,840 square yards =1 acre, (A.)
640 acres =1 square mile, (sq. mi.)
10,000 square links = 1 square chain.
10 square chains = 1 acre.
Sqiiare links and sqiuire chains are used by land-surveyors in
describing land-areas ; in calculations they are written as decimals
of an acre.
In old deeds and descriptions of property the square rod pole
or perch = 30| sq. yd., and the rood = J acre are sometimes used,
but these terms are now practically obsolete.
The prime unit or standard of surface measurement is the square
yard, that is, a square surface whose sides are each one yard in
length. Hence the prime unit of surface measurement is derived
ifrom, and is determined hy, the prime unit of linear measurement.
J Cubic, or Volume Measure.
f 1,728 cubic inches (c. in.)=l cubic foot, (c. ft.)
27 cubic feet =1 cubic yard, (c. yd.)
Firewood and rough stone are measured by the cord of 128 cubic
feet, which is equal to a pile of the material 8 feet long, 4 feet
wide, and 4 feet high. The cord is not a statutory measure, that
is, it is not defined by statute.
The prime unit of volume measure is the cubic yard, that is, a
cube whose edges are each one yard in length. Hence the prime
unit of volume 'nfieastiremetit is derived from, a/tid is determined hy,
the prime unit of linear measurement.
Measure of Capacity.
2 pints (pt.) = l quart, (qt.)
4 quarts =1 gallon, (gal.)
2 gallons =1 peck, ^ (pk-)
4 pecks =1 bushel, (bu.)
The (jill is one-quarter of a pint ; it is not defined by statute, but
the term is used in the second schedule to the Dominion Weights
and Measures Act of 1879.
14 ARITHMETIC.
The capacity of cisterns, reservoirs and the like is often expressed
in barrels (bbl.) of 31^ gallons each, or in hogsheads (hhd.) of 63
gallons each.
The legal bushel of certain substances is determined not by
measure, but by weight. These weights are given in the following
table : —
Blue Grass Seed - 14 ft). Indian Corn - - - 56 ft).
Oats 34 ft). Rye 56 ft).
Malt ------ 36 ft). Wheat, Beans, Peas,
Castor Beans - - - 40 ft). and Red Clover
Hemp Seed - - - 44 ft). Seed ----- 60 ft).
Barley - - . - - 48 ft. Potatoes, Turnips,
Buckwheat - - - 48 ft. Carrots, Parsnips,
Timothy Seed - - 48 ft. Beets and Onions 60 ft.
Flaxseed- - -^<6.5Q,ft. Bituminous Coal - 70ft).
A barrel of jlour contains 196 ft).
A barrel of pork or of beef contains 200 ft.
A quarter of wheat, or of other grain = 8 bushels = 480 pounds.
This measure is very commonly used in England but not in Canada.
A chaldron = 36 bushels, used in measuring coal, coke, and a few
other articles.
Apothecaries subdivide the pint as follows : —
60 fluid minims (n|^)=l fluid drachm - - - (fl. 5)
8 fluid drachms =1 fluid ounce - - - (fl. 3)
20 fluid ounces =1 pint ------ (O.)
The prime unit or standard measure of capacity is the gallon
containing ten Dominion standard pounds of distilled water weighed
in air against brass weights with the water and the air at the
temperature of sixty-two degrees of Fahrenheit's thermometer, and
with the barometer at thirty inches. The weight of a cubic foot of
water under these conditions is 62 "356 lb, , consequently a gallon
contains, or is equal to, 277 '118 cubic inches.
The Imperial gallon was formerly declared by statute to be of 277*274 cubic
inches capacity, which is the volume of 10 lb. of pure water at 67*5 °F., but
this part of the statute of weights and measures has been repealed.
A cubic foot of pure water at 52 ° F. weighs 62 '4 lb. =998*4 oz.,
and this is the weight usually adopted in calculations aiming at a,
II
TABLES OF VALUES, WEIGHTS AND MEASURES. 15
high degree of accuracy, but where great accuracy is not required,
62*5 lb. =1,000 oz, is taken as the weight of water per cubic foot.
This approximation is close enough for ordinary purposes, the
more so as natural waters contain mineral matter in solution and
consequently are somewhat denser than pure water.
Measures of Time.
I
^p 60 seconds (sec.) = l minute ------ (min.)
^K 60 minutes =1 hour - (hr.)
^K 24 hours =1 day - - (da.)
^B 7 days =1 week - - - - (wk.)
^B 365 days =1 common year - - - (yr.)
■p 366 days =1 leap year
The calendar year is divided into twelve parts called months ;
seven of these consist of 31 days each, four consist of 30 days each,
and one (February) consists of 28 days — in leap years of 29 days.
The lengths of the several months may be remembered from the
following rhymes : — -
^L Thirty days have September,
^^^^^^^ April, June, and November ;
^^^^^^H| February has twenty-eight alone,
^^^^^^^V All the rest have thirty-one ;
^^^^HHp But leap year coming once in four,
^^^^^^^ Gives February one day more.
^f The civil day begins and ends at 12 o'clock midnight. A.M.
denotes time before noon ; M. , at noon ; P. M. , after noon.
The prime unit of time is the day, or, strictly speaking, the
I^Kmean solar day. A solar day is the time-interval between two
successive transits of the sun's centre over the meridian ; but as
these intervals are of unequal length, we take the mean or average
of all the solar days in the year, and to this mean solar day we give,
in ordinary speech, the name day.
A year is the period of the earth's revolution about the sun,
from some determinate position back again to the same position.
If the starting point be the vernal equinox, the interval is called a
tropical year and has been found to consist of 365 '242216 mean
solar days = 365 da. 5 hr. 48 min. 47^ sec. The tropical year
16 ARITHMETIC.
determines the recurrence of the seasons, and of all the important
phenomena of vegetation and life depending thereon, but to adopt
it as the civil or calendar year, the year of ordinary business affairs,
would involve having one part of a day belonging to one year and
the remainder of the day belonging to the following year. This
partition of a day is avoided by having civil years of two different
lengths, the one of 365 days which is less than a tropical year, the
other, called bissextile or leap year, of 366 days, which is greater
than a tropical year. Now 400 tropical years would be greater
than 400 years of 365 days each by -242216 da. x 400 = 96-8864 da.,
or nearly 97 days, hence if every 400 years consist of 303 years of
365 days each and 97 years of 366 days each, the average civil year will
be practically of the length of a tropical year, and the seasons will
recur at the same times by the calendar. This is accomplished by
making every year whose date-number is exactly divisible by 4, a
leap-year, except in the case of the years whose dates are even
hundreds, the date-numbers of these must be exactly divisible by
400. Thus the years 1880, 1884, 1888 were leap-years ; 1881, 1882,
1886, 1887 were not leap-years ; 1600 was and 2000 will be a leap-
year ; 1800 was not and 1900 will not be a leap-year.
Neither the period of the earth's revolution about the sun nor
the period of its rotation on its axis is absolutely constant. The
latter is lengthening by the 39V7 part of itself per hundred years.
Angular Measure.
60 seconds (") =1 minute ------(')
60 minutes =1 degree (°)
90 degrees =1 quadrant or right angle.
4 quadrants, or 1 1 • i i i • •■
360 degrees J =^ ^'^^^^ ^^ ^^'^^^ ^^^^"^*-
The prime Unit of angular measure is one complete revolution.
Angles less than seconds are expressed as decimals of a second.
Angles are always measured in practice by Angular Measure, but in
many theoretical investigations another system of measurement,
called Circular Measure, is adopted.
METRIC SYSTEM OF WEIGHTS AND MEASURES. 17
THE METRIC SYSTEM OF WEIGHTS AND MEASURES.
28. The French or Metric System of Weights and Measures
which is a decimal system, or system based on ten as the common
scale of relation among each set of units of the same kind, is used
in scientific treatises. Its use is permissive in Canada, the British
Islands, f\,nd the United States, and it has been adopted absolutely as
the sole system throughout great part of Europe and South America.
29. The prime units in this system and their ratios to the prime
units of the Dominion or Imperial system are
I Length Metre =1-09362311 yards.
Area Are =119*601150 square yards.
Volume or \ -r - - _ / = '00130798582 cubic yards.
Capacity J ^^^^^ \ = '22021444 gallons.
Weight and Mass. Gramnie = 15 "43234874 grains.
30. The fundamental unit of this system is the metre which was
intended to be the ten-millionth, ('000,000,1), part of a quadrant
of latitude, i. e. , of the distance of the pole of the earth from the
equator, measured at the level of the sea. In this respect the
legal metre is not quite exact, but this is of no consequence as
practically the length of the metre is fixed in each country adopting
the metric system, by means of Standard Metres marked on metal
rods, just as the Standard Yard is determined. The original of
these rods is the French Standard Metre,* a platinum rod
deposited in the state archives at Paris.
From the metre are derived the are, the litre and the gramme.
The are is equal to 100 square metres ; the litre to the '001 of a
cubic metre ; and the gramme to the '000,001 of the weight in vacuo
of a cubic metre of distilled water at its temperature of greatest
density. In measuring wood, a Stere = 1 cubic metre = 1,000 litres
is used, and in weighing heavy articles a Millier or Metric Ton =
1,000,000 grammes is employed.
31. The names of the auxiliary units in this system are formed
by attaching certain prefixes to the names metre, are, litre and
gramme respectively ; thus : —
* The Canadian Standard Metre is defined by statute as equal to 1*0939*
standard yards. It therefore differs appreciably from the French Standard
Metre which is equal to i '09362311 standard yards, the difference amounting
to rather more than a yard in two miles,
18
ARITHMETIC.
micro —
milli —
centi —
deci —
rOO(),001
001
01
deka-
hecto —
kilo-
myria -
mega— J
metre
are
litre
^%<
>of
A -
J
T
2 E
<S) z
00 =
7
eo
«r -
i£) -
^
ei
LO z
^ E
r« =
z
-
Si z
■:
gramme
The
micron
metre
are
litre
gramme
10
100
1000
10000
Vi,ooo,oooJ
name micrometre is shortened tc
and kilogramme frequently to kilog.
100 kilogrammes is named a quintal, and the
megagramme is the Millier. The "001 part
of a micrometre is termed a micromillimetre.
32. The metric system of Linear
Measure may be tabulated as an example,
thus : —
1,000 micromillimetres, (/"/^.)
= 1 micron, (//)
1,000 microns = 1 millimetre, - (mm.)
10 millimetres = 1 centimetre, (cm.)
10 centimetres = 1 decimetre, (dm.
10 decimetres =1 metre, - - - (m.)
10 metres = 1 dekametre, - - (Dm. )
10 dekametres = l hectometre, (Hm.)
10 hectometres = 1 kilometre, (Km.)
1,000 kilometres = 1 megametre. (Mgm.)
33. The accompanying scales and diagram
may perhaps assist those accustomed to
Imperial or Dominion measures alone, in
the realization of the actual magnitudes of
the metric units. The upper of the two
scales is 4 inches in length and is divided
into inches and subdivided into sixteenths
of an inch. The lower scale is 1 decimetre
in length and is divided into 10 centimetres
and subdivided into 100 millimetres.
METRIC SYSTEM OF WEIGHTS AND MEASURES. 19
1 i r
1 Square Decimetre,
Each side of this square measures
1 decimetre, or
3^f inches, very nearly.
A litre is a cube each face of which has the dimensions of this
square.
A gramme is the weight of a cubic centimetre (see small square
above) of distilled water, weighed in vacuo at temperature of
maximum density, 39*1 F. A litre or cubic decimetre of such
water weighs 1 kilogramme or 24 lb. nearly.
34. The following approximations may be noticed
5 inches is very nearly 127 millimetres.
8 kilometres is somewhat less than 5 miles.
10 metres is nearly 11 yards.
64 metres is very nearly 70 yards.
64 miles is very nearly 103 kilometres.
43 square feet is nearly 4 centiares.
61 centiares is nearly 55 square yards.
2 hectares is nearly 5 acres.
22 gallons is nearly 100 litres or 1 hectolitre.
22 pounds is nearly 10 kilogrammes.
20 ARITHMETIC.
35. It is evident that whenever a quantity is expressed in the
decimal notation in terms of a single unit, a decimal system of
values, weights or measures is employed. Thus 23*75 lb. expresses
23 lb. 12 oz. decimally, and 3 '375 yd. is the decimal equivalent of
3 yd. 1 ft. 1^ in.
The metric system being a decimal system, it is not necessary
to employ more than a single unit in expressing any quantity
metrically. Thus, 3 dekametres 7 metres 4 decimetres 5 centimetres
and 7 millimetres is written 37*457m. which is read 37 metres
457 millimetres. If it becomes necessary to change the unit of the
expression, such change is accomplished by shifting the decimal
point, at the same time changing the unit-denomination. Thus,
37 •457m = 374 •57dm. = 3745 •7cm. = 37457mm. = 3-7457Dm. =
•037457Km.
WORD SYMBOLS.
36. Certain words and phrases recur so often in Arithmetic that
it is found convenient to represent them by easily made symbols.
These are
= , read is equal to, ivill be equal to, &c., thus y\ = § ;
= , read is, is the same as, represerits, denotes, thus V = 5,
D=500.
> , read is greater tlian, thus f > ^ ;
< , read is less than, thus | <i I
.'., read therefore, consequently, hence;
'. ', read because, since, thus '. ' f > f and § < f , . % f > f.
i€
CHAPTER II.
THE FOUR ELEMENTARY OPERATIONS.
ADDITION AND SUBTRACTION.
37. Addition is the operation of finding that quantity which
is made up as a whole of two or more given quantities as its parts.
The quantities to be added together are called addends, — or
addenda.
The result of the addition is termed the sum of the addends.
Since the sum is the whole of which the addends are the parts, —
Addends and sum must all he quantities of the same kind^ i. e.,
they must all have the same unit.
38. The sign of addition is + , read 2)lus, meaning increased hy.
The sign + placed before any quantity indicates that the quantity
is an addend. Thus 8 + 3 is read * eight plus three * and denotes
t 3 is to be added to 8. In like manner 24 + 9 + 5 is read
wenty-four plus nine plus five,' and denotes that 9 is to be added
o 24 and then 5 added to the sum.
The sum of any nuiuber of given quantities is expressed by
writing the quantities in a row in the order in which they are to be
added, with the sign + between every adjacent pair.
39. Subtraction is the operation of finding the part of a
given quantity which remains after a given part of the quantity
has been taken away.
The quantity from which a part is to be taken away is called tlie
minuend.
The jmrt of the minuend whidh is to be taken away is called the
subtrahend.
The result of the subtraction is called the remainder and also
the difference between the minuend and the subtrahend.
Since the minuend is the whole of which the subtrahend and the
remainder are the parts, —
Minuend^ sidjtrahend and remainder must all have the same unit ;
and, — If the suhtrahettd he added to the remainder the sum tmll be
the minuend.
22 ARITHMETIC.
40. The sign of subtraction is — , read minus, meaning diminished
by. The sign - placed before any quantity indicates that the quantity
is a subtrahend. Thus 8 - 3 is read * eight minus three ' and denotes
that 3 is to be subtracted from 8. In like manner 24 — 9 - 5 is read
' twenty-four minus nine minus five ' and denotes that 9 is to be
subtracted from 24 and then 5 subtracted from the remainder. So
24 + 9-5 denotes that 5 is to be subtracted from 24 + 9 while
24-9 + 5 denotes that 5 is to be added to 24-9.
41. An expression consisting of a succession of addends and
subtrahends, such as 8 + 5-3 + 6-3-4, is called an aggregate.
The several parts, the addends and the subtrahends, as 8, +5,
-3, +6, -3, -4, are called the terms of the aggregate ; and
The quantity which results from collecting the terms by performing
the indicated additions and subtractions is called the total or sum
of the aggregate.
42. The Fundamental Theorems of Addition and Subtraction
are ; —
I. If equals he added to equals, the wholes are equal.
II. If equals be suhtroAited from equals, the remainders are equal.
III. The sum of two addends will be the same whether tlie second
be added to the first or the first be added to the second.
IV. Adding to an addend adds an eqvxil quantity to the sum.
V. Sid)tracting from an addend sid)tracts an equal quantity froim
the sum.
VI. Adding to the minuend adds an equal quamtity to the remainder.
VII. Subtracting from the minuend subtracts an equal quantity
from the remainder.
VIII. Adding to the subtrahend subtracts an equal quantity from
the remainder.
IX. Subtracting from the subtrahend adds an equal quantity to
the remainder.
X. Adding zero to any quantity leaves the qua/ntity unchanged.
Theorms III to IX may be stated in a single theorem, thus, —
Changing the order of collecting the terms of any aggregatCy does
not change the total or sum of the aggregate.
43. To prove any calculation is to perform another calculation
that will test or put to proof the correctness of the results of
the first calculation.
ADDITION AND SUBTRACTION.
23
44. The simplest and best way to prove a result in addition
is to repeat the addition, adding downwards the columns that were
added upwards on the first addition and upwards the columns that
were then added downwards.
45. In the additions of tabulated numbers which are to be added
both vertically and horizontally the agreement of the grand total
of the row of partial sums with the grand total of the columns
of partial sums is, in general, a suflicient test of mere correctness,
but if a mistake has been made, it is not enough to detect
its existence, the mistake must be located in the partial sums
and there corrected. This location and correction is often greatly
facilitated by what is known as Computers' Addition. In this
method the sum of each column is set down separately, the right
hand figure of each partial sum being placed under the column from
which it is derived, and the other figures in their order diagonally
downwards to the left. These partial sums are then added together
to obtain the sum. By this arrangement the addition of any colunm
can be tested independently of that of the preceding column, no
knowledge of the ' carried ' number being required. Thus if it be
known that an error has been committed in the addition of the
himdreds, it can be discovered and corrected without adding the
tens to ascertain the ' carriage. '
In this example, the sum of the first
column is 38. The 8 is placed under the
first column and the 3 under the second
column but in the line next below that of the
8. The sum of the second column is 69.
The 9 is placed under the second column
immediately on the left of the 8 and above
the 3 of 38, and the 6 is placed on the left
of the 3. The sum of the third column
is 42. The 2 is placed under the third
column immediately on the left of the 9 of 69
and the 4 diagonally below to the left.
The sum of the fourth column is 57, of which
the 7 is written beneath the fourth column from which it was
obtained, and the 5 is placed diagonally below it to the left. These
partial sums are now added to obtain the sum, 61928.
Exampl
e.
8784
27
3295
19
2133
9
8594
26
6272
17
9585
27
7986
30
9286
25
5993
26
7298
6
5463
20
61928
24 ARITHMETIC.
46. Since in this method the columns are added independently,
the result may be tested by adding together the digits in each
horizontal row as shown in the example. The total of these sums,
— in the example, 206, — should be the same as the total of the
column-sums, — in the example, 38, 69, 42 and 57, — treated as a
row of mutually independent numbers.
47. Some computers prefer to arrange the figures of the column-
sums from right diagonally upwards to left and to add in the carried
numbers as is done in the ordinary method. Taking the preceding
example, the lowest addend and the result would
by this arrangement appear as in the margin, the ; . • •
upper addends being here omitted merely to save 5993
space.' The column-sums would be 38, 72, 49 and 61. '^^^
The 6 of 61, the last column-sum, is not written in 61928
the carriage-line but is placed at once in the sum-line.
48. Computers' Subtraction. The best way to perform
subtraction is the method based on the fundamental theorem that
the sum of the subtrahend and the remainder is equal to the
minuend.
Example. From 435,846 take 259,784.
- It is required to find what number added to 259,784 will make
435,846.
Write the subtrahend under the minuend so 4.S'i84fi
that the figures of the same decimal order in
each shall be in the same vertical column as in -. i-r^rj^o
the margin.
To 4, the right-handed figure of the subtrahend, 2 must be
added to make up 6 the right-hand figure of the minuend ; put
down this 2 as the right-hand figure of the remainder. The 8 (ten)
of the subtrahend cannot be m,ade up to the 4 (ten) of the minuend,
so make it up to 14 (ten), this requires that 6 (ten) be added ; put
down this 6 (ten) in the remainder. To the 7 (hundred) of the
subtrahend add 1 (hundred) carried from the 14 (ten), thus making
it 8 (hundred), and 0 (hundred) is required to make this 8 (hundred)
up to the 8 (hundred) of the minuend ; put 0 (hundred) in the
remainder. To the 9 (thousand) of the subtrahend add 6 (thousand)
to make up 15 (thousand) which will give the 5 (thousand) of
the minuend ; and put down this 6 (thousand) in the remainder.
ADDITION AND SUBTRACTION.
25
To the 5 (ten thousand) of the subtrahend add 1 (ten thousand) from
the 15 (thousand) already made up and then add 7 (ten thousand)
more to make up 13 (ten thousand) in the minuend, putting
down this 7 (ten thousand) in the remainder. To the 2 (hundred
thousand) of the subtrahend carry 1 (hundred thousand) from the
13 (ten thousand) and add 1 (hundred thousand) more to make up
the 4 (hundred thousand) of the minuend, putting this 1 (hundred
thousand) in the remainder.
Fancy you are doing addition with the sum at the top of the
columns of addends and work thus setting down, as you j:»ronounco
them, the figures here printed in thick-faced type : —
4 and 2, six ; 8 and 6, fourteen ; 8 and O, eight ; 9 and 6,
fifteen ; 6 and 7 thirteen ; 3 and 1 four.
After a little practice the minuend-sums need not be pronounced.
The actual character of the process will perhaps be better
comprehended by working a few examples with the subtrahend
written as the lower of two addends, and the minuend written as
their sum, the problem being to find the other
addend. Arrange the preceding example in this 259784
way, (see margin), and repeat the working given A^F^QAa
above.
49. This method is nearly always adopted in " making change"
and so lends itself to calculations involving both additions and
subtractions that it is almost universally employed by professional
comjiuters, and is generally known as Computers' Subtraction.
Example. From 9564 take 1357 + 498 + 1976 -}- 83 -h 3758.
Arrange the subtrahends in column under the minuend as
addends are arranged in addition ; — see margin.
Add the subtrahends together and ' make up '
to the minuend, setting down the ' making up '
number. Thus
1st. Column ; 11, 17, 25, 32 & 2 ; 34 ; carry 3
2nd. '' 8, 16, 23, 32, 37 & 9 ; 46 ; " 4
3rd. " 11, 20, 24, 27 & 8; 35; "3
4th. *' 6, 7, 8 & 1 ; 9.
9^64
1357
498
1976
83
3758
1892
50. To prove any subtraction add the subtrahend to the
remainder, the sum shijuld be the same as the minuend.
26 ARITHMETIC.
MULTIPLICATION AND DIVISION.
51. The simplest expression of a quantity consists of two
components, one naming the unit, the other stating the number of
such units in the quantity. But since the unit is a magnitude it
may itself be considered as a quantity and expressed in terms of
another unit which relative to it is called a primary unit.
Thus the expression of a given quantity may consist of three
components, one naming a primary unit, a second stating the
NUMBER of these primary units composing a standard quantity or
derived unit, and a third stating the number of these derived
units in the given quantity.
52. The number of primary units in such a quantity is called
the product of the number of primary units in the derived unit
multiplied by the number of derived units in the quantity.
Thus 35 marbles is the same quantity as 5 counts of 7 marbles
each, therefore 35 is the product of 7 multiplied by 5. In tliis case
the primary unit is a marble and the derived unit is a count of 7
marbles.
Again | yd. is the same quantity as f of | yd,, hence h is the
product of I multiplied by f . In this case the primary unit is a
yard and the derived unit is | of a yard.
53. Multiplication is the operation of finding the product of
two numbers ; in other words,
Multiplication is the process of finding the number of units of a
given kind in a quantity which contains a given number of standard
quantities each consisting of a given number of units of the
given kind.
The numbers to be multiplied together are called the factors of
the product.
The factor which is to be multiplied by the other is called the
multiplicand.
The factor by which the other is to be multiplied is called the
multiplier.
64. A boy who has to read 18 pages of 38 lines each wishes to
know how many lines he has to read. Here it is required to find
the number of lines in the quantity 18 pages-of-38-lines, a quantity
whose unit, a page-of-38-lines, is expressed in terms of the primaiy
MULTIPLICATION AND DIVISION. 27
unit, a line. The required number may be found by counting, or
by addition, or by multiplication. In this case the product, that of
38 and 18, may be obtained by addition.
IH^ A man is required to weigh out ^j of an article the whole weight
^K of which is ^^ of a pound. What part of a pound must he weigh
out ? Here it is required to find the number of pounds in the quantity
^ of ^j lb. , a quantity whose unit, ^j lb. , is expressed in terms of
the primary unit, a pound. The required number may be found
by counting, or by a series of additions and subtractions, or by
multiplication. In this case the product, that of y*- and ^ may
be obtained by a series of additions and subtractions.
I There are, however, cases — to be treated of hereafter, (see § 120),
^P in which the product of two factors cannot be obtained by mere
counting and in which, in consequence, multiplication cannot be
replaced by or be resolved into any number, however great, of
additions and subtractions.
Thus certain calculations may be performed by addition or by
multiplication indiflFerently ; other calculations, as is known, belong
to addition exclusive of multiplication, and still other calculations
belong to multiplication exclusive of addition.
53. In arithmetical multiplication, the multiplier must be simply
a number, for it states the number of multiplicands in the product ;
but for the purely numerical multiplicand there may be substituted
the derived unit, the quantity whose absolute magnitude is
expressed by taking as components the multiplicand proper and the
primary unit. In such case the product is the quantity whose
absolute magnitude is expressed by the purely numerical product
as one component, and the primary unit as the other. But
although the primary unit may thus appear in the multiplicand, it
is not itself operated on in any way, the m^uUiplier operating on the
numerical component of the multiplicand and on it alone.
66. The sign of multiplication is x, read ^'■multiplied by.'^
The sign x placed before any number indicates that the number is
a nmltiplier. Thus 5 lb. x 4 is read ''5 1b. multiplied by 4 " or
"4 times 5 lb. " and denotes a weight equal to 4 weights of 5 lb. each.
In like manner | yd. x | is read " | yd. multiplied by |" or
" f of I yd." and denotes a length which is f of the length, f yd.
The product of two or more factors may be expressed by writing
28 ARITHMETIC.
the factors in a row with the sign x between every adjacent pair.
If there are more than two factors and if in none of the factors
there appears a decimal point, the sign x may be replaced by a
simple dot or period ; thus 3x5x7x11x13 may be written
3.5.7.11.13, but 3-6x7 X 11-13 must not be written 3 '5. 7. 11 '13, as
the difference in position between the decimal point and the period
i^j not marked enough to prevent confusion.
57. Division is the operation of finding either of two factors,
there being given the other factor of the two and also their product.
The factor found is called the Quotient.
The factor given is called the Divisor.
The given product of the Divisor and the Quotient is called the
Dividend.
58. Division is the inverse of multiplication, for in multiplication
two or more factors are given and it is required to find their
product ; in division, on the other hand, the product of two factors
is given and also one of the two factors and it is required to
find the other factor. This being the case, division may appear
under either of two guises according as the factor to be found is the
multiplicand or the multiplier, when the dividend is recalculated as
the product of the divisor and the quotient.
In the first case, that in which the quotient is to the divisor as
multiplicand to multiplier, the divisor is simply a number and the
quotient is a quantity of the same kind as the dividend.
Examples. If 75 ct. be divided into 15 equal parts, what will be the
value of one of these parts ? Answer, 5 ct. ; for 5 ct. x 15 = 75 ct.
What is the weight of an iron rod if ^| of it weigh il^lb.?
Answer, f i lb. ; for |f lb. x U=H 1^-
In the second case, that in which the quotient is to the divisor as
multiplier to multiplicand, the divisor is a quantity of the same
kind as the dividend and the quotient is simply a number.
Examples. How many five-cent pieces will make up a sum of 75 ct. ?
Answer, 15 ; for5ct. x 15 = 75ct. What part of an iron rod will weigh
A I lb. if the whole rod weigh f f lb. ? Answer, ff ; for |f lb. x ff = iflb.
59. There are three signs of division, viz., :, -^, and /, all read
" divided by.'^ Any of these signs placed before a number indicates
that the number is a divisor. Thus 75 ct. -^ 15 is read ' ' 75 ct. divided
by 15," and denotes that 75 ct. is to be divided by 15. In like
MULTIPLICATION AND DIVISION. 29
lanner 36 : 3 : 4 is read *'36 divided by 3, divided by 4," and
'denotes that 36 is to be divided by 3 and the quotient then divided
by 4. So 36 X 3 : 4 denotes that 36 is to be multiplied by 3 and the
product divided by 4, while 36 : 3 x 4 denotes that 36 is to be
divided by 3 and the quotient multiplied by 4.
60. In an expression containing a succession of multipliers and
divisors, the operations are to be performed in order from left to
right. Thus,
9 x5-^3x 6^10-^4 = 45^3 x6-M0^4=15x6^10-^4
:90-^10-^4=9-^4=2i^
Compare this with
9 + 5—3 + 6—10—4 = 14—3 + 6—10—4 = 11 + 6—10-4
= 17_10_4=7— 4=3.
In an aggregate whose terms contain multipliers and divisors,
nthe multiplications and the divisions are to be performed before the
Additions and the subtractions are made. Thus,
[ 6 X 5 + 15 X 4+3—16+2 x 3 = 30 + 20— 24 - 26.
61, The signs + and / are employed exclusively by English-speaking nations,
all other nations denote division by the sign : alone. Furthermore, while the
laws governing the use of the sign : are definite and invariable, the signs -^ and
/ are employed in one way by one writer and in another way by another. Thus
30 -f 5 X 3 would be interpreted by one author ' ' 30 divided by 5 and the quotient
multiplied by 3," while another would interpret it "30 divided by 5x3."
The first author would write 30 + 5x3=18; the second author would write
30+5x3 = 2.
In like manner English mathematicians are not united in their views regarding
the employment of the sign x . Many authors place the multipher before the
sign X which they then read "multiplied into," or simply "into"; their
order of arrangement is thus multiplier, sign, multiplicand. Other authors,
following the uniform practice of 'continental' mathematicians, adopt the
arrangement 'multiplicand, sign, multiplier,' thus preserving the analogy in
use between the signs x and -=- and the signs + and - .
62. The Fundamental Theorems of Multiplication and
Division are : —
XI. If equals be multiplied by equals the products are equal.
XII. If equals be divided by equals the quotients are equal.
XIII. The product of two purely numerical factors will be the
same whether the first factor be multiplied by the secoiid or the second
factm- be multiplied by the first.
30 ARITHMETIC.
XIV. Multiplying a factor by any number multiplies the product
by the same number.
XV. Dividing a factor by any number divides the j^roduct by the
same number.
XVI. Multiplying the dividend by any number multiplies the
quotient by the same number.
XVII. Dividing the dividend by any number divides the quotient
by the same number.
XVIII. Multiplying the divisor by any number divides the quotient
by the same number.
XIX. Dividing the divisor by any number multiplies the quotient
by the same number.
XX. Multiplying any number by one leaves the number unchanged.
XXI. If one of the factors be zero, the product will be zero.
Theorems XIII to XIX may be stated in a single theorem,
thus : —
If an expression co^itain a succession of midtipliers and divisors,
changing the order of the multipliers and the divisors does not change
the value of the expression.
Example. 10-r5x 12-^3 = 10-^5-=-3x 12 = 10-^3x 12-^5
= 10xl2-^5-^3 = 8.
63. The Fundamental Theorems connecting the operations
of addition and subtraction with the operations of multiplication
and division are, —
XXII. Multiplying the several terms of an aggregate by any
number Wyultiplies the aggregate by that number.
XXIII. Dividing the several terms of an aggregate by any number
divides the aggregate by that number.
64. Scholars* Multiplication. Multiplications in which
both multiplier and multiplicand require many digits to express
them are generally best made by means of a table of multiples of
the multiplicand. This table may be formed by successive additions
of the multiplicand written on a slip of paper to be moved down
the column of multiples as the successive additions are made. The
multiples should extend from the first to the tenth, the last testing
the accuracy of the work ; and, for convenience of reference, straight
lines should be drawn under the first, fifth and ninth multiples.
MULTIPLICATION AND DIVISION
31
'Example. Find the product of 74,853,169 and 2968457.
Multiple Table.
2968457
74853169
2968457
5936914
8905371
11873828
14842285
17810742
20779199
23747656
26716113
26716113
17810742
2968457
8905371
14842285
23747656
11873828
20779199
29684570
G5. Scholars' Division.
222198413490233
A table of multiples of the divisor
may be employed in the case of division in which both divisor and
dividend require many digits to express them,
Example. Divide 2808332109244 by 58679.
Multiple Table.
58679
47859236
58679)2808332109244
234716
461172
410753
504191
469432
347590
293395
541959
528111
138482
117358
211244
176037
352074
352074
66. Computers' Multiplication. In multiplying by a number
requiring several digits to express it, we may set down each
partial product as it is calculated, and then sum the whole of
them ; or, as each partial product after the first is calculated, we
may add to it the sum of alt the previously calculated partial
products.
32 ARITHMETIC.
Example. Multiply 56437 by 3852967.
The first line of products is simply 7
times the multiplicand. The next line is 56437
formed thus :— 3852967
6 times 7 and 5. the tens of the first line of 395059
S78127
products, =47. Write the 7 beneath the 5 _.i^.^
added in and carry 4. 6 times 3 and 4 167448
carried = 22. Write 2 on the left of the 7 last 298929
written and carry 2. 6 times 4 and 2 carried 481388
and 5 from the first line of products = 31. 217449898579
• Write 1 on the left of the 2 last written and '■
carry 3. 6 times 6 and 3 carried and 9 from
the first line of products = 48. Write 8 on
the left of the 1 last written and carry 4. 6 times 5 and 4 carried
and 3 from the first line of products = 37. Write 37 on the left of
the 8 last written. The partial product thus formed with the 9
brought from the line above is 3781275 which is 67 times 56437, the
multiplicand.
The third line of partial products is formed by multiplying the
multiplicand 56437 by 9 (hundred) and adding in successively the '
proper digits of the second partial product, thus : —
9 times 7 and 2 from the second partial product = 65. Write 5
beneath the 2 added in and carry 6. 9 times 3 and 6 carried and 1
from the second partial product =34. Write 4 on the left of the
5 last written and carry 3. 9 times 4 and 3 carried and 8 from the
second partial product =47. Write 7 on the left of the 4 last
written and carry 4. Proceeding in this way we obtain as third
partial product 5457457.9 (the 79 being brought down from the
lines above) which is 967 times 56437.
In like manner, multiplying by 2 (thousand) and adding in the
third partial product we obtain 2967 times 56437.
Next multiplying by 5 (ten thousand), then by 8 (hundred thousand)
and finally by 3 (million), each time adding in the last-obtained
partial product, we obtain 217449898579 which is the product of
56437 multiplied by 3852967. The six figures on the right in this
final product, viz. 898579, are the . right hand figures of the six
preceding partial products,
MULTIPLICATION AND DIVISION.
83
67. Computers' Division. In computers' multiplication the
product is built up by successive additions of multiples of the
multiplicand, these multiples being determined by the several digits
of the multiplier. In computers' division this process is reversed ;
the dividend is broken up or resolved by successive subtractions of
multiples of the divisor, these multiples determining the several
digits of the quotient.
Example 1. Divide 217,449,898,579 by 56,437.
)■ 3852967
^Here 56437 is contained 3 (million) 56437)217449898579
times in 217449 (million). Write 3 in 481386'
298929
the quotient and proceed to obtain the 1 8*74.4. <?
* remainder' by computers' subtraction, 545745
thus :— 378127
395059
written and carry
complement = 14.
written and carry
complement = 27.
3 (million) times 7 and 8 (million) complement =29 (million).
Write the 8 (million) complement under the 9 (million) in the
dividend and carry 2 from the 29. 3 times 3 and 2 carried and 3
complement = 14. Write the 3 complement on the left of the 8 last
1 from the 14. 3 times 4 and 1 carried and 1
Write the 1 complement on the left of the 3 last
1 from the 14. 3 times 6 and 1 carried and 8
Write the 8 complement on the left of the 1 last
written and carry 2 from the 27. 3 times 5 and 2 carried and 4
complement =21. Write the 4 complement on the left of the 8 last
written. This completes the subtraction of 3 (million) times 56437
from the dividend. To the right of the partial remainder 48138
just found, bring down 8, the * next figure ' of the dividend ; we
thus obtain 481388 as the second partial dividend giving 8 as the
* next ' figure of the quotient. Write 8 in the quotient on the right of
the 3 formerly written therein, and from 481388 take by computers'
subtraction 8 times 56437. There will remain 29892 to which
' bring down ' 9 from the dividend to obtain a new partial dividend.
Continue thus subtracting and * bringing down ' till the operation
is finished, or is carried to a sufficient degree of accuracy.
Comparing this example with the example given in the preceding
section, it will ba found, if the whole of the work be written out,
that tlie one process is the exact reverse of the other.
S4 AWTHMETIC.
68. A slightly more convenient arrangement of the work may be
obtained by ' carrying up ' the figures of the successive complements
or remainders, instead of ' bringing down ' the successive figures
of the dividend. This is merely a change in the arrangement,
not in the working of the division, 56437
the wording of the process will
remain the same with the exception of
the omission of "bring down the
next figure of the dividend, " Arranged
in this way the example just worked 3852967
out will appear as in the margin.
We give two other examples of this arrangement.
Example 2. Divide 372,956,483 by 7.
7x5+^=37. Write 5 in the quotient-line 71372956483
and 2 in the remainder-line. The next partial | 215dod54
dividend is thus 22. 7 x 3 4- 1 = 22. 53279497-f-
7x24-5=19. 7x74-6=55. 7x94-5=66. 7x44-6=34.
7x9-^5-68. 7x74-^=53.
Example 3. Divide 3,893,865,378 by 179.
217449898579
4813824425
29894710
167585
5479
33
3893865378
314518705
13967645
11
The first partial dividend is 389, giving 179
2 as the first figure of the quotient and
31 as the first remainder. The second
partial dividend is therefore 313 which — piHRCMQ^igs
gives 1 as the second figure of the quotient ^ ' ^
and 134 as the second remainder. ' The third partial dividend
is 1348, the third figure of the quotient is 7 and the third remainder
is 95. The fourth partial dividend is 956, the fourth figure of
the quotient is 5 and the fourth remainder is 61. The remaining
partial dividends are, in order, 615, 783, 677 and 1408. The final
remainder is 155.
69. Special Cases. In the case of certain multipliers and
certain divisors, special methods may be adopted with advantage.
The following are examples of these.
i. To multiply by 5, multiply mentally by 10 and divide the
product by 2. (5 = 104-2.)
ii. To multiply by 25, multiply mentally by 100 and divide the
product by 4. (25 = 100^4.)
p
MULTIPLICATION AND DIVISION. 35
If
iii. To multiply by 125, multiply mentally by 1000 and divide
the product by 8. (125 = 1000^8.)
iv. To multiply by 75, multiply by 300 and divide the product
by 4. (75 = 300-^4.)
V. To multiply by 375, multiply by 3000 and divide the product
by 8. (375 = 3000^8.)
vi. To multiply by 875, multiply by 7000 and divide the product
by 8. (875 = 7000^8.)
Til. To multiply by 11, add each figure of the multiplicand to
e figure on its right hand beginning from 0 mentally pictured as
written on both the right and the left of the multiplicand.
I Example. 35725876 x 11
392984636
Calculation.
.6=6. 7+6 = 13. 1 + 8 + 7 = 16. 1+5 + 8=14. 1 + 2+5=8.
2=9. 5 + 7 = 12. 1 + 3 + 5=9. 0+3=3.
lExpla/nation. 11 = 10 + 1, hence 35725876 x 11 = | ^ 035725876
392984636
viii. To multiply by 101, or 1001, or 10001, , employ
computers' multiplication, using the multiplicand as the first partial-
product line.
ix. To multiply by 13, 14, 17, 21, 31, 91,
102, 103, 109, 201, 301, 901, or other number
beginning with 1 or ending with 1, write the multiplier above the
multiplicand and use the multiplicand itself as the partial-product
line arising from the digit 1 in the multiplier.
Examples. 17 71
4372965 4372965
30610755 30610755
74340405 310480515
X. To multiply by 9, subtract the multiplicand from 10 times
itself by making up each figure to that on its right hand beginning
from 0 mentally pictured as written on both the right and the left
of the multiplicand.
Example. 7285634 x 9.
"65570706
36 ARITHMETIC.
Calculation. •4 + 6-10. 1 + 3 + 0-4. 6 + 7 = 13. 1 + 5+0 = 6.
8+7 = 15. 1 + 2 + 5 = 8. 7 + 5 = 12. 1 + 0+6=7.
Explanation. 9 = 10-1, hence 7285634 x 9 = | _ ^^285634
65570706
xi. To multiply by 99, 999, 9999, &c. subtract the multipliccand
from 100, 1000, 10000, &c. times itself by making up each figure to
the 2nd, 3rd, 4th, &c. on its right hand beginning from two,
three, four, &c., zeros mentally pictured as written on both the
right and the left of the multiplicand.
xii. To multiply by 97, 997, 9997, take 3 times
the multiplicand from 100, 1000, 10000, times the
multiplicand. To multiply by 974, 9974, 99974, take 26
times the multiplicand from 1000, 10000, 100000, times
the multiplicand. Similarly resolve other multipliers expresser'
by one or more 9's followed by one or more figures other than 9.
The subtractions should be made by computers' method, see the
Example following §49, p. 25.
Examples. 953784 =20 times. 114572 =400 times.
476892 X 19 28643 x 399
906094o = 19 times. 11428557 = 399 times.
xiii. If two or more consecutive figures in a multiplier constitute
a number which is a multiple of another figure of the multiplier
we may save a line of partial products.
Examples. {!.) 47289 •
_ ^67
331023= 7 times.
2648184 =7 times x 80=560 times.
26812863= ~567 times.
{2.) 2985643 {3.) 84629
36872 5397
23885144 800 592403 7
214966296 8x9= 72 4146821 490
107483148 72000+2 = 36000 4146821 4900
110086628696 36872. 456742713 5397,
MULTIPLICATION AND DIVISION. 37
H" xiv. If the multiplier is seen to be the product of two or more
small factors, multiply the given multiplicand by any one of these
factors, multiply the product so formed by a second factor, this
second product by a third factor, and do continue till all the factors
have been used. The final product is the product required.
I ^k XV. To divide by 5, multiply by 2 and divide the product by 10.
I Ht ^^^' ^^ divide by 25, multiply by 4 and divide the product by
"OO. (25x4 = 100.)
xvii. To divide by 125, nuiltiply by 8 and divide che product
by 1000. (125x8 = 1000.)
txviii. To divide by 75, or 175, or 225, or 275, , multiply
4 and divide the product by 300, or 700, or 900, or 1100,
. . . , as the case may be.
xix. To divide 375, or 875, or 1375, multiply by 8
and divide the product by 3000, or 7000, or 11000, #.s the
case may be.
I ^h XX. If the divisor is seen to be the product of two or more
■ ^ra,ctors each less than 13, divide by these factors *in succession,*
the final quotient is the quotient required.
(Example. 8765348 -r 462.
462 = 6 x7xlL 6
7
11
8765348_
14608911
2086983f
'I8972fft
The § might Imve been written i\ in which case f| would have
become hj and ^^^ become }r^'j. These are the forms in which the
fractions would have appeared had the divisor 462 been resolved
into 2x3x7x11 and these four factors used .as successive divisors.
xxi. Since 100 = 99x1 + 1
200 = 99x2 + 2
300 = 99x3 + 3
325 = 300 + 25 = 99x3 + 3 + 25
894 = 800 + 94 = 99x8 + 8 + 94
&c. = &c.
therefore when any number is divided by 99 the remainder increased
if necessary, by 99 or a nmltiple of 99, exceeds the remainder
38 ARITHMETIC.
when that number is divided by 100, by the quotient when 100 is
the divisor. Successive applications of this leads to a convenient
method ot dividing any number by. 99. Thus : —
297689 = 297600 + 89
= 2976x99 + 2976 + 89
= 2976x99 + 2900 + 76 + 89
= 2976x99 + 29x99 + 29 + 76 + 89
= 3005x99 + 194
= 3005x99 + 100 + 94
= 3005x99 + 1x99 + 1 + 94
= 3006x99 + 95
.-. 297689 + 99 = 3006«f.
This may be arrangjed for working as follows :—
2976 89
29 76
29
94
1
Quot.=3006 I 95 = Kern.
Similarly for any number expressed by 9's only.
70. Tests of Exact Divisibility. The following tests of
exact divisibility are often useful in a search for the factors of a
number.
i. A number is exactly divisible by 2 if its right-hand figure is zero
or a number exactly divisible by 2.
ii. A member is exactly divisible by 4 if its two right-hand
figures are zeros or express a number exactly divisible by 4.
Examples. 173528 is exactly divisible by 4, for 28 is exactly
divisible by 4 ; but 319378 is not a multiple of 4, for 78 is not
exactly divisible by 4.
iii. A number is exactly divisible by 8 if its three right-hand
figure? are zeros oi express a number exactly divisible by 8.
Examples. 536 is a multiple of 8, therefore 1397536 is exactly
divisible by 8 ; but 356 is not a multiple of 8, consequently 4679356
is not exactly divisible by 8.
iv. A number is exactly divisible by 5, 25, 125, if the
number expressed by the right-hand figure or the two, three,
right-hand figures is exactly divisible by 5, 25, 125,
MULTIPLICATION AND DIVISION. 39
V. A number is exactly divisible by 3 if the sum of its digits is
Exactly divisible by 3.
vi. A number is exactly divisible by 9 if the sum of its digits is
exactly divisible by 9.
Examples. Test whether 18637569 and 7385621 are divisible by 9.
1 + 8 + 6 + 3 + 7 + 5 + 6 + 9 = 45 = 9x5, .-. 18637569 is exactly
divisible by 9."
7 + 3 + 8 + 5 + 6 + 2 + 1 = 32 = 9x3 + 5, .-. 7385621 is not exactly
divisible by 9.
vii. A number is exactly divisible by 6 if it is exactly divisible by
both 2 and 3.
viii. A number is exactly divisible by 12 if it is exactly divisible
by both 4 and 3.
ix. A number is exactly divisible by 11 if the difference between
the sum of its 1st, 3rd, 5th, 7th, &c. figures and the sum of its
2nd, 4th, 6th, 8th, &c. figures is zero or a number exactly
divisible by 11.
Examples. Test whether 729583624 and 457983621 are exactly
divisible by 11.
4 + 6 + 8 + 9 + 7 = 34; 2 + 3+5 + 2 = 12; 34-12 = 22=11x2;
. •. 729583624 is exixctly divisible by 11.
1 + 6 + 8 + 7 + 4 = 26; 2 + 3 + 9 + 5 = 19; 26-19 = 7; .'. 457983621
is not exactly divisible by 11.
There are no easily applied tests for exact divisibility by 7 and by
13, but in the case of very large numbers the following may be
applied.
x. Point off the number into periods of three figures each,
beginning on the right ; if the difference between the sum of the
1st., 3rd., 5th., &c. periods and the sum of the 2nd., 4th., 6th.,
&c. periods is zero or is exactly divisible by 7, by 11, or by 13, the
number is exactly divisible by 7, by 11, or by 13, as the case may
be.
Example. Test 6,576,353,693 for 7, 11, and 13 as factors.
693 + 576-353-6 = 910=10x7x13; .'. 7 and 13 are factors
but 11 is not a factor.
xi. Any number less than 1000 will be exactly divisible by 7 if
the sum of the ones figure, thrice the tens figure and twice the
hundreds figure be exactly divisible by 7.
40 ARITHMETIC.
Examples. Is 7 a factor of 623 and of 685?
3 + 6 + 12 = 21 = 7x3, .-. 7 is a factor of 623.
5 + 24 + 12 = 41 = 7x5 + 6, .-. 7 is not a factor of 685.
xii. If a number is exactly divisible by each of two numbers
prime to each other, it is exactly divisible by their product ; and
conversely,
xiii. If a number is exactly divisible by the product of two
numbers, it is exactly divisible by each of the numbers.
71. Theorems xii and xiii follow immediately from Theorem
XIX, § 62. The truth of the other theorems may be sho^vn as
follows : —
i. 2 is a measure of 10 ;
. *. 2 is a measure of every multiple of 10 ;
. •. 2 is a measure of the part of any number consisting of the
tens, hundreds, thousands, &c. ;
. •. in testing any number for exact divisibility by 2, the tens,
hundreds, thousands, &c. may be neglected as certainly multiples.
ii. 4 is a measure of 100 ;
. •. 4 is a measure of every multiple of 100 ;
. •. 4 is a measure of the part of any number consisting of the
hundreds, thousands, ten-thousands, &c. ;
. •. in testing any number for exact divisibility by 4, the hundreds,
thousands, ten-thousands, &c., may be neglected as certamly
multiples.
iii. 8 is a measure of 1000 ;
. •. 8 a measure of every multiple of 1000 ;
. '. 8 is a measure of the part of any number consisting of the
thousands, ten-thousands, hundred-thousands, &c. ;
. '. in testing any number for exact divisibility by 8, the
thousands, ten-thousands, hundred-thousands, &c. , may be neglected
as certainly multiples.
iv. The demonstration is similar to those for 2, 4 and 8.
V. and vi. Bbth 3 and 9 are measures of 9, 99, 999, 9999, &c.,
that is, of 10-1, 100-1, 1000-1, 10000-1, &c.
. •. if from any number there be deducted the ones and 1 from
each 10, 1 from ea^h 100, 1 from each 1000, &c. , the remainders from
MULTIPLICATION AND DIVISION. 41
le tens, the hundreds, the thousands, S:c. , constitute a number
rhich will be a multiple of 9 and therefore also of 3 and which
may be neglected in testing the number for exact divisibility by
either 9 or 3. There will then remain to be tested the total of
the deductions ; these were the ones, the number of tens, the
immher oi hundreds, the number oi thousands, &c., and therefore
jtheir total is simply the sum of the digits of the given number.
Example. Is 78546 exactly divisible by either 9 or 3 ?
70000 = 7 times 10000 = 7 times 9999 and 7
8000=8 " 1000 = 8 " 999 " 8
500 = 5 " 100 = 5 " 99 " 5
40 = 4 " 10 = 4 " 9 " 4
6 = 6 " 1 = 6 " 0 " 6
Adding, 78546= a multiple of 9 and 30
30 is exactly divisible by 3 but not by 9,
•. 78546 is exactly divisible by 3 but not by 9.
It should be noticed that the theorem really proved is : —
TJie remxiiiider in dividing any number by 0 is the same as the
linder in dividing the sum of the digits of the number by 9.
vii and viii are special cases of xii.
ix. 11 is a measure of 11, 1111, 111111, 11111111, &c. ;
.-. 11 is a measure of 11, of 1111-110, of 111111-11110, of
millll- 1111110, &c.;
i. e.,11 is a measure of 11, 1001, 100001, 10000001, &c. ;
. •. 11 is a measure of 11, 99, 1(X)1, 9999, 100001, 999999, &c. ;
i. e. , of 10 + 1, 100 - 1, 1000 + 1, 10000 - 1, 100000 + 1, 1000000 - 1,
&c. ; . '. if fr(mi any number there be deducted the ones and 1 from
each 100, 1 from each 10000, 1 from each 1000000, &c. and there be
added 1 to each 10| 1 to each 1000, 1 to each 100000, Ac, the
resulting number will be a multiple of 11 and it may consequently be
neglected in testing the number for exact divisibility by 11. There
will then remain to be tested the difference between the deductions
and the additions which were made, i. e., the difference between
the sum of the digits in the odd places, (numbering from the right,)
and the sum of those in the even places.
42 ARITHMETIC.
Example. Is 8756439 exactly divisible by 11,
8000000-8 times 1000(X)0 = 8 times 999999 and 8
700000=7 " 100000=7 '' 100001 less 7
50000 = 5 " 10000 = 5 '' 9999 ami 5
6000 = 6 " 1000 = 6 " 1001 less 6
400 = 4 " 100 = 4 '' 99 ami 4
30 = 3 '' . 10 = 3 " 11 less 3
9 = 9 " r=9 " 0 ami 9
«
Adding, 8756439= a multiple of 11 and 26 less 16
26 less 16 = 10 which is not exactly divisible by 11
. •. 8756439 is not exactly divisible by 11.
Here also it should be noticed that the Theorem really proved is,
The remainder in dividing any number by 11 is the same as the
remainder in dividing the number obtained by subtracting the sum of
the digits in the even places, numberituj from the right, from the sum
of the digits in the odd places iiicreased if 'necessary by 11 or a
TThultiple of 11.
X. By substituting 1000 for 10 and 1001 for 11 and making
corresponding changes in the other numbers employed in the
preceding discussion of the test for exact divisibility by 11, it will
be proved that,
If any number be pointed off into periods of three figures each,
beginning from the right, the remainder in dividing the number by
1001 will be the same as the remainder in dividing the number
obtained by subtracting the sum of the periods in the even places,
numbering from the right, from the sum of the periods in the odd
places, increased if necessary by 1001 or a multiple of 1001.
Now 1001 = 13x11x7, hence if the remainder in dividing any
number by 1001 be exactly divisible by 7, by 11 or by 13, the
number itself will be exactly divisible by 7, by 11 or by 13 as the
case may be.
xi. From 10=7 + 3 and 100=98 + 2 = 14x7 + 2 comes at once
test or rule xi.
72. To cast the nines out of any number is to find the
remainder in dividing the number by 9. To do this, add together
the digits of the given number, omitting any nines there may be
among the digits, then add together the digits of that sum again
II
I
II
I
I
MULTIPLICATION AND DIVISION. 43
imitting all nines, and so continue until a number of one digit is
obtained. This last number, if it be less than 9, will be the
remainder in dividing the given number by 9 ; if it be 9, the
remainder will be zero. :
Example 1. Cast the nines out of 73856942.
7 + 3 + 8 + 5 + 6 + 4 + 2 = 35, 3 + 5 = 8, remainder.
Instead of adding all the digits-together and casting the nines out
of the sum, the nines may be cast out of the partial sums as fast as
they rise above 8. Adopting this method the preceding example
would appear
7 + 3 = 10, (1 + 0=1), l + 8 = 9;5 + 6 = ll, (1 + 1 = 2), 2 + 4 + 2 = 8.
Wording ; ten, one, nine, eleven, two, six, eight.
Example 2. Cast the nines out of 3587968594.
8, 16, (7), 14, (5), 11, (2), 10, (1), 6, 10, 1 remainder.
The applications of the operation of casting out the nines depend
upon two theorems : —
A. The sum of two numbers has the same remainder to 0 as the
sum of their remxihiders to 9.
B. The product of two numbers has the same remainder to 9 as the
product of their remainders to 9.
AU numbers may be regarded as multiples of 9 + their remainders
to 9. On adding or multiplying these numbers, all the multiples
of 9 will yield multiples of 9 and all these will disappear in
casting out the nines ; the result will therefore be the same as if
the numbers had been reduced from the first to their remainders to
9.
73. Proofs of Multiplication. Multiplication may be
proved, —
(1.) By repeating the calculation v/ith multiplier and multiplicand
interchanged.
(2. ) By dividing the product by the multiplicand ; the quotient
should be equal to the nmltiplier.
(3. ) By casting the nines out of the multiplier and the multiplicand,
then multiplying the remainders together and casting the nines
out of their product ; the remainder thus obtained should be the
same as the remainder ivom. casting the nines out of the product
of multiplier and multiplicand.
44 ARITHMETIC.
In arranging the several remainders it is usual to write the
remainder from the multiplicand on the left-hand of an oblitjue
cross, the remainder from the multiplier on the right-hand of the
cross, the remainder from the product below the cross and the
remainder from the product of the remainders above the cross.
Example 1. Apply the test of casting out the nines to
2968457 X 74853169 = 222198413490233. (See % 64. ;
Midtnd. b^Q Multr. 5x7-35 = 9x3-f8.
Prodiict.
Example 2. Prove 56437x3852967 = 217449898579 by casting
out the nines. (&ee § 66.)
%
74. Of these three proofs the second possesses the advantage of
locating any errors that may be detected but it doubles the labor
of calculation. The third proof is by far the easiest of application
but it is subject to the serious disadvantage of not pointing out an
error of 9 or a multiple of 9 in the product. Thus if 0 has been
written for 9 or 9 for 0, if a partial product has been set down in
the wrong place, if one or more noughts have been inserted or
omitted in any of the products, if two figures have been
interchanged or, generally, if one figure set down is as much too
great as another is too small, casting out the nines will fail to
declare the presence of error, for in each case the remainder to 9
will remain unafiected by the error.
75. Proofs of Division. Division may be prcwed, —
(1.) By repeating the calculation with the integral i)art of the
quotient for a divisor.
(2.) By multiplying the divisor by the complete (piotient ; or, as
it is generally stated, by multiplying the divisor by (the integral
part of) the quotient and adding the remainder to the product ; the
result should be equal to the dividend.
(3.) By casting the nines out of the divisor, the integral part of
the quotient and the 'remainder' in the division, multiplying the
MtfLTlPLICATlOK AND DIVISION.
45
it twcf of these remainders together and adding the third to their
'product and casting the nines out of this sum ; the remainder to 9
thus obtained should be the same as the remainder from casting the
nines out of the dividend.
Example. Prove 3893865378 -^179 = 21753437 Iff by casting out
the nines. (See § 68.)
■ Divisor ^^;^ 2 Quot 8x5 + 2 = 42-9x4 + 6.
^^^^p Dividend.
The proof of division by casting out the nines labors under
disadvantages corresponding to those to which the proof of
multiplication by casting out the nines is subject.
EXERCISE I.
MISCELLANEOUS PROBLEMS.
^m 1 . By what number must 2000 be divided that the quotient and
^»the * remainder ' may be the same as the quotient and the
* remainder ' in the division of 101 by 11 ?
2. "What number contains 13 '75 as often as 18*27 contains '0693 (
3. If a strip of carpet 27 in. wide and 50 yd. long make a roll
weighing 135 lb., what area could be covered by 4 T. of such
carpet ?
4. A rectangular block of granite measures 7' 1 ' x 2' 4" x 1' 3" ;
what must be the length of another rectangular block 2' 1" x 1\
(i), if it is to weigh the same as the first block,
(ii), if it is to have the same surface-area as the first block,
(a) exclusive of end-surfaces,
(6) inclusive of end-surfaces '(
5. A boat's crew rowed a, distance of 4 mi. 800 yd. in 36 min.
45 sec. What was the speed per hour ? What was the average
time per mile ?
6. A man who owns ^ J of a mill, sells f of his share ; what
fraction of the mill does he still own ? Had he sold f of the mill,
what fraction of the mill would he still have owned ?
46 ARITHMETIC.
7. A grocer drew off 4 gal. from a full barrel of vinegar and
filled the barrel up with water. Next day he drew off 4 gal. of
the mixture and then filled up the barrel with water. On the third
day he drew off 4 gal. of the mixture and filled up the barrel with
water. If the barrel held just 32 gallons, how many gallons of the
vinegar originally contained in the barrel remained in it after the
third drawing off?
8. G can do as much work in 4 days as H can do in 5 days, or as
much in 5 days as M can do in 9 days. The three undertake a
contract and G and H work together on it for 18 days, then M
takes (t's place and H and M work together on it for 26 days and
thus finish the contract. How long would it have taken G working
all the time alone to have executed the contract ?
9. A grocer buys two kinds of tea, one kind at 23ct. per lb. , the
other kind at 35ct. per lb. , and mixes them in the proportion of 51b.
of the cheaper to 3 lb. of the dearer kind. At what price per
pound (an integral number of cents) must he sell the mixture to
gain at least 30% on the buying price ?
10. Find the interest on $794-35 for 188 days at 5%.
11. The product of 25 and 25 is 625. By how much must this
product be increased to obtain the product of 26 and 25 ? By how
much must the product of 26 and 25 be diminished to obtain the
product of 26 and 24 % Hence by how much must the product of
25 and 25 be diminished to obtain the product of 26 and 24 % By
how much must the product of 26 and 24 be diminished to obtain
the product of 27 and 23 1 Hence by how much must the product
of 25 and 25 be diminished to obtain the product of 27 and 23, i.e.,
the product of 25 + 2 and 25 — 2 ? By how much must the product
of 25 and 25 be diminished to obtain the product of (i) 28 and 22,
(ii) 29 and 21, (iii) 30 and 20 ?
12. I take 344 steps in walking round a rectangular play-ground,
keeping 3 ft. within the boundary fence. Find the area of the
play-ground if 9 of my steps are equal to 7| yd. and a shorter side
of my walk is 68 steps in length.
13. Find the value of a rectangular field 330 yd. by 156 yd. ®
1^6 '50 per acre.
EXERCISES. 47
14. What must be the depth of a rectangular cistern to hold 350
gallons when filled to 6 in. from the top, if the horizontal section
of the cistern is to be 3' 6" square ?
15. How many miles will be travelled between 9,25 a.m. and
5,40 p.m. at an average of 22| mi. per hour for 3 hr. 35 min. and
of 28f mi. per hour for the remainder of the time ?
16. After drawing off 124 gal. of water from a cistern, -^ of the
water still remained. How many gallons did the cistern at first
contain ? How many gallons were left in it ?
17. A block of maple weighed 35 lb. and a block of red pine of
exactly the same size weighed 25 lb. Find the weight of a block of
maple of the same size as a block of red pine weighing 164 lb. and
the size of a block of red pine which will weigh the same as 23*75
cubic feet of maple, all three blocks of maple and likewise all three
of red pine being of the same quality.
18. A laborer was engaged @ $1 "12 and his board for each day
he worked, but was charged 38ct. for board for each day he was
idle. At the end of ^1 days he received $25*72. How many days
did he work ?
19. A tradesman bought goods for $1200 and sold one-third of
them at a loss of 10%. For how much must he sell the remainder
to gain 20% on the whole ?
20. Find the interest on $273*68 from 13th May to 7th Sept. at
ir —
^^^ 21. A rectangular lot 45 ft. front by 99 ft. deep was sold for
$3150. What was the price per foot frontage, and what the price
per acre at the rate of the selling-price of the lot ?
22. Find the area of a rectangle whose length is three times its
width and whose perimeter is 143*76 in.
23. What must be the depth of a cylindrical cistern 3' 6" in
diameter to hold 350 gal. when filled to 6 in. from the top ?
124. A man starts at 8,10 a.m. on a journey of 18 miles and
travels for 3j hours at the rate of 3j miles per hour. If he then
quicken his speed by § of a mile per hour, at what hour of the day
will he arrive at the end of his journey? How much sooner will
this be than would have been the hour of his arrival had he not
quickened his pace ?
48 - ARITHMETIC.
25. Herbert's age is just f of Maud's. Four years ago, his
father, who is now 36 years old, was just 5| times as old as Herbert
then was. How old is Maud ?
26. A man pays out ^^r of his income for rent and ^ for taxes.
What fraction of his income do these two sums form ? If the two
sums amount together to f 220 '90, what must be the amount of the
man's income ?
27. Two blocks of exactly the same size, the one of birch, the
other of willow, weighed 454 lb. and 249*7 lb. respectively. The
block of birch floated in water with only ^\ of its volume immersed.
How much of the volume of the willow-block would be immersed
were the block to float in water ?
28. A B and G can do a piece of work in 10 days, all three
working together. The three undertake the job and work on it
for 4 days, then G leaves off work, but A and J> continue and
finish the piece of work in 10 days. If A could have done the
whole work by himself in 30 days, in what time could jB, and in
what time could G have done it ? ♦
29. A tradesman sold |^ of a certain lot of goods at a loss of 10%,
at what per cent, advance on the cost must he sell the remainder of
the lot in order to gain 20% on the whole ?
30. To what sum would $87 '68 amount in 97 days @ 6|%
interest ?
31. Express the following distances in kilometres : — •
• (i). From Montreal to Toronto, 333 miles ; (ii), from Toronto to
Hamilton, 38*72 miles; (iii), from Toronto to Stratford, 88*34
miles; (iv), from Hamilton to London, 75*90 miles; (v), from
Stratford to London, 32 *68 miles ; (vi), from Montreal to London
via Hamilton ; (vii), from Montreal to London via Stratford.
32. In front and on one side of a rectangular lot 66 ft. by 132 ft.
and 2 ft. out from the line of the lot, a sidewalk 40 in. wide is laid.
How many square feet of ground does the side-walk cover ?
33. Into a rectangular cistern 3' 4" by 2' 9" in horizontal section,
water is flowing at the rate of 25 gal. per minute. How long will
it take at that rate of flow to increase the depth of the water in
the cistern by 4' 6" ?
EXERCISES. 49
34. Find the area of a rectangle of 6 •006 ft. perimeter, if its
ength is (i) equal to, (ii) double, (iii) thrice, (iv) four times, (v)
ive times, (vi) eight times, (vii) ten times its width.
35. A can run a mile in 5 min. 55 sec. and JB can run a mile in
> min. 2 sec. By how many yards would A win in a mile race run
.t these rates ?
36. Having paid an income-tax of 19 '2 mills on the $1, I have
.n income of $5735 "92 left. What amount of income-tax did I pay ?
37. Goods which cost $2756-13 for 17 T. 1335 lb. are sold at an
advance of j^j on cost. Find the selling price per cwt.
38. In a hundred-yard race, A can beat B by 17 yd. and 0 by
i yd. At these rates of running how many yards start ought G to
;ive B in a 200 yd. race that they may run a dead heat ?
39. (a), ^'s age is greater than B's by 12 yr. which is 25% of
I's age. Determine ^'s age.
(6). ilfs age which is 69 yr. is greater than JVs age by 15% of
Vs age. Determine iVs age.
(c). F's age is less than TF's age by 10% of W& age and the sum
»f their ages is 76 yr. Determine F's age.
40. At what rate per cent, per annum would $183*40 yield $2*78
nterest in 123 days 1
It
. What number is the same part of 95*9 that 18 '27 is of 29,
t,nd what number is the same multiple of *119 that 57057 is of
].5-96'?
42. The standard of fineness of British gold coins is j^g o^ alloy,
and 480 oz. Troy of standard gold is coined into 1869 sovereigns
equal in value to $4 "861 each. Find the value of (i) an oz. Troy,
(ii) an oz. avoirdupois, of pure gold.
43. Find the area of the outer surface of a cylindrical stove-drum
16" in diameter and 24" in height, deducting two circles, the
pipe-holes, of 7 " diameter each.
44. The depth of water in a rectangular cistern of 3' by 2' 9"
horizontal section increases at the rate of 5' 4" in 12 min. What
is the rate of inflow in gallons per minute ?
45. A horse trotted a mile in 2 min. 12 sec. Taking his stride
at 16 ft., how many times per second did his feet touch the ground ?
50 ARITHMETIC.
46. The municipal rates being reduced from 19§ mills to 17§
mills on the $1, my taxes are lowered by $4*05. For how much
am I assessed ?
47. A boat's crew that can row at the rate of 264 yd. per min. in
still water, rowed 3 miles down a stream in 16 min. Find the
velocity of the stream.
48. Sold 19 yd. of silk @ $1 '86 a yard, thus gaining the cost
price of 12 yd. Find the cost price per yard.
49. A's age which is 49 yr. is less than J5's age by 12^% of B's
age, and J5's age is less than (Js age by 12|% of Cs age. What is
(7s age ?
50. The interest on $270*25 for 93 days is $4-82 ; to what sum
would $725 amount in 125 days at the same rate 1
51. The sum of two numbers is 106 and one exceeds the other
by 28 •62. What fraction is the smaller of the larger number ?
52. Find the area of a circular field enclosed by a ring fence
440 yd. long.
53. A circular pond 17' 6" in diameter and 5' deep is to be filled
by means of a pipe which discharges 100 gal. per min. How long
will it take to fill the pond ?
54. A train runs the first 120 miles of a trip of 280 miles at a
speed of 32 miles per hour. At what speed must the remainder of
the trip be run, if the whole trip is to be accomplished in 8 hours ?
55. If during the day I pay out h, then ^, next ^^^ and lastly
yV of the money I had in the morning, what fraction of it have I
left ? If the sum left amounts to $1 "54 what sum had I at first ?
56. A train 220 ft. in length is running at the rate of 25 mi. per
hour. How long will it take to pass another train 330 ft. long if
the second train be (i), standing on a parallel track ; (ii), moving in
the opposite direction at the rate of 15 mi. per hour ; (iii), moving
in the same direction at the rate of 15 mi. per hour ?
.57. A has $480 which is less than what B has by 20% of what B
has, and the sum B has is greater than what C has by 20% of what
C has. What sum does C possess ?
58. A has more money than -B by 10% of JB's money. By what
per cent, of A' a money is jB's money less than ^'s ?
EXERCISES. 51
59. At what rate of interest would $379-45 amount to $396 in
245 days ?
60. The owner of a house offered an agent $500 commission if
the agent could sell the house for $10,500. What rate per cent,
commission was the owner offering ? Had the owner offered 5%
commission, what would have been the commission on $10,500?
II
61. A sum of money was divided between A and B, A receiving
$5 for every $4 received by By and it was found that A had
received $12 "GO less than double of what B had received. How
much did each receive ?
62. The area of Eurojie is 3,823,400 sq. mi. and its average
elevation above the level of the sea is 974 ft. Find the volume in
^fcbic miles of the portion of Europe above sea-level.
B63. If a cubic foot of gold weigh 1208 lb. , what must be the
Htickness of a gold ribbon 1^ in. wide and 10 ft. long, weighing
TiSO grains? {Avoirdupois Weight.)
64. If the telegraph poles beside a certain railway are placed at
intervals of 50 yd. , at what speed is a train running which
_ Jia verses two of these intervals in seven seconds ?
I ^p65. In a certain subscription list ^ of the number of subscriptions
" tSg for $5 each, ^ are for $4 each, ^ are for $2 each, ^ are for $1
each, and the remaining subscriptions, amounting to $10*50, are for
50ct. each. Find the whole number of subscribers and the total
amount of their subscriptions.
6Q. A train 80 yd. long crossed a bridge 140 yd. long in 22| sec.
_ J'ind the average speed of the train while crossing.
IB 67. Find the gain per $100 on a cargo of raw-sugar bought at
^3 per ton of 22401b., refined at a cost of $1-35 per 1001b. of
refined sugar and sold at 6^ct. per lb. , if 7 lb. of raw sugar yields
5 lb. of refined sugar.
68. A and B insure their houses against fire and A has to pay
$7 -50 more than B who pays $28-75. Find the value of their
houses, the rate of insurance being | %, and express the value of
B's house as a decimal of the value of A's house.
69. In what time would the interest on $182 "50 amount to $5 at
5%?
52 ARITHMETIC.
70. A man bought 50 shares in a company at $40 per share.
Next year the price was $45 per share but each year thereafter
there was a fall of $4 per share. Each year from the date of his
purchase he sold .out 10 shares and found at the end of five years
that including his dividends with the amounts realized by the sales
of his shares he had neither gained nor lost. What dividend per
share did the company pay 1
71. Prove that if 12 be added to the product of the first 11
integers, 13 will be a factor of the sum.
72. Prove that if 16 be added to the product of the first 15
integers, 17 will be a factor of the sum. (See Public School
Arithmetic, Ex. xxx, Probs. 26 and 27.)
73. Make out a bill dated Feb. 1st, 1889, for the following
transactions and receipt it on behalf of Messrs. Kent & Sons.
L. D. Walker bought of Messrs. Kent & Sons, Hamilton : —
Dec. 1st, 1888, Am't of Acc't rendered, $30-07 ; Dec. 14th,
IJyd. Lawn @ 28 ct., 2 Spools @ 5 ct., 1 Cloud $1-25, 2| yd. Lace
@ 80 ct., 1 Towel 27 ct., 2 yd. Ribbon @ 11 ct., IJ yd. Embroidery
@ 15'ct.; Dec. 22nd, SHk Handerchief 85 ct., do. do. $1'20,
6 Linen do. @ 22 ct.; Dec. 28th, 1 pr. Cashmere Hose 57ct., 4 Sk.
Wool @ 10 ct., I yd. Frilling @ 15 ct., |yd. do. @ 20 ct., 1 pr. Silk
Gloves 75 ct., 3|-yd. Pink Flannel @ 32 ct.; Jan'y 25th, 1889,
1\ yd. Lining @ 22 ct., 4 J yd. Silesia @ 13 ct., 4j yd. Jet Trimming
@22ct., 1 Spool Silk 15ct., 2 Twist @3ct., l^doz. Buttons® 10 ct.,
3 yd. Braid @ 2 ct.; Jan. 29th, J doz. Table-Nap. @ $2 '10, | doz.
do. @ $2-50 (per doz.), j yd. Veiling 25 ct., 1 yd. Frilling 20 ct.
Jan. 3rd, 1889, L. D. AValker paid Cash on Acc't., $25-00 ; Feb'y
4th, paid Acc't in full.
74. If the telegraph poles beside a certain railway are placed at
intervals of 66 yd. , at what speed can a train be running if it
traverse three of these intervals in between 11 and 12 seconds?
At what speed can the train be running if it traverse 10 intervals
in a time between 38 and 39 seconds in length 1
75. (a) How much must be added to the numerator of -/o that
the resulting fraction may be equal to | ?
(h) How much must be subtracted from the numerator of f that
the resulting fraction may be equal te ^^2 ^
II
EXERCISES. 53
76. A man distributed a bag of marbles among 4 classes
consisting of 7 boys each, giving the same number of marbles to
each boy in a class. Among the boys in the first class he distributed
half the marbles ; among those of the second class, ^ of them ;
among those of the third class, -^ of them ; and among those of the
fourth class, the remaining marbles which allowed the boys just
one apiece. How many did each boy in the other three classes
receive and how many marbles were there altogether 1
77. A train 54 yd. long running at the rate of 31 mi. per hr. ,
passed another train 78 yd. long running on a parallel track ; the
two trains completely clearing each other (i) in 5*4 sec. from the
time of meeting, (ii) in 22 '5 sec. from the time of the former
train overtaking the latter. Find the speed of the slower train ?
78. A house assessed at $2200 was rented for $23 a month, the
tenant to pay taxes and water-rates. The taxes were 17f mills on
the $1 and the water-rates were $5 per quarter year. How much
altogether did the tenant pay per year for the house. If the
property had cost the landlord $2500, what rate per cent, per
year was he receiving on his investment ?
79. In what time would $143 amount to $150 at 7 % interest ?
80. The manufacturer of an article makes a profit of 25 %, the
wholesale dealer makes a profit of 20 %, and the retail dealer makes
a profit of 30 %. What is the cost to the manufacturer of an
article that retails at $15*60.
81. Prove that if the number of integers less than 9 and prime
to it bo multiplied by the iiumber of integers less than 16 and
prime to it, the product will be the number of integers less than
144 ( = 16 X 9) and prime to it.
82. How many times must a man walk round a rectangular
play-ground 165 ft. by 132 ft. in order to travel 4^ miles ?
83. How many cubic feet of air will a rectangular room 27' 8"
X 18' 3" X 12' 4", contain, and how much will the air in the room
weigh if a cubic foot of the air weigh 565 grains ?
84. A can run a mile in 4 min. 56 sec. , B can run a mile in 5 min.
23 sec. If A give B 27 yd. start, in what distance will he overtake
him ?
54 ARITHMETIC.
85. Make out and receii)t for K. Dewar & Son the following
account : —
K. Dewar & Son of Stratford sold to Edwin Reesor on 2nd
Sept., 1885, 28 lb. Furnace Cement @ 20 ct., 7 ft. of 12-in. Hot-air
Pipe @ 46 ct. , 14 lengths of 8-in. Smoke-pipe @ 18 ct. , 1 Chimney
Ring, 25 ct., 1 8-in. Elbow, 50 ct., 2 8-in Rings for Lawson
Regulator® 35 ct., 21 1^-in. Bolts @2ct., 8 2-in. Bolts @ 3ct.,
16 2i-in. Bolts @ 4 ct. A man and an assistant from K. Dewar &
Son's worked 49 hours cleaning and repairing E. Reesor's furnace,
rate for the two together, 35 ct. per hour. E. Reesor paid $15 on
this account on the 3rd Oct. and the balance on the 28th Nov. ,
1885.
86. A man sold ^ of his farm, then | of the remainder, then
J of what remained, then | of what still remained, and he then
found that he had sold altogether 72 acres more than he had
remaining. How many acres had he at first ?
87. (a). How much must be added to the denominator of f
that the resulting fraction may be e(iual to /o ?
(h). How much must be subtracted from the denominator of
■^^ that the resulting fraction may be equal to f ?
88. A and B run a mile race ; A runS the whole course at a
uniform speed of 320 yd. per min. ; B runs the first half mile at a
speed of 300 yd. per min. and the second half mile at a speed
of 340 yd. per min. Which wins the race and by how many
yards ?
89. What principal would at 7% interest amount to $450 in
213 days ?
90. An agent receives $7850 to be invested. What sum should
he invest if he pay $12 '30 expenses and charge 1^% commission
on the amount of the investment ?
91. Two wheels in gear with one another have 30 and 128 teeth
respectively ; how many revolutions will the smaller wheel make
while the larger revolves 675 times ? If two marked teeth, one on
each wheel, are in contact at a certain moment, how many
revolutions will each wheel make before the same two teeth are in
contact again?
I
EXERCISES. 55
II.
92. Find the volume in cu. in. of 10 lb. of (a) lead, (h) cast-
^ron, (c) marble, (d) brickwork, fe^ oak, (f) birch, if a cubic foot of
Bead weigh 712 1b., of cast-iron 4441b., of marble 1721b., of
brickwork 112 lb., of oak 54 lb., and of birch 44*4 lb.
93. Taking the weight of a cubic foot of water to be 997 '7 oz. ,
what weight of water would fill a rectangular bath 35' 6" by 13' 3"
by 5' 7r ?
94. Make out an invoice of the following, supplying names and
dates : —
A. B. bought of a B. 15 doz. First Readers Pt. I @ $1-20, 18
doz. First Readers Pt. II @ $1'80, 27 doz. Second Readers @ $3 -OO^
24 doz. Third Readers @ $4-20, 9 doz. Fourth Readers @ $6-00,
30 doz. Public School Grammars @ $3-00, 30 doz. Public School
Arithmetics @ $3-00, 6 doz. Public School Geographies @ $9 -00 ;
the whole subject to 20 and 5 off.
95. A can run a mile in 4 min. 56 sec. , B can run a mile at the
te of 110 yards per 18 seconds. If they start together and run
uniformly at these rates which will be the first and by how many
yards (i) when A has run half a mile, (ii) when B has run a mile ?
96. If when -^^ of a certain time has elapsed, then 1 hr., and
then I of the remainder of the time, it is found that 16 min. of
the time still remain, what was the whole time ?
97. (a). What number added to both terms of -^^ will give a
fraction equal to f ?
(h). What number subtracted from both terms of f will give a
fraction equal to -^ ?
98. Find the length of a bridge which a train 100 yd. long
required 1 min. 15 sec. to cross, running at a speed of 15 mi. per
hour.
99. An account bearing interest at 6 % amounted at the end
of 93 days to $117 '45. What was the original amount of the
account ?
100. A grocer professes to retail a certain tea at 20 % profit but
mixes with it \ of its weight of an inferior tea which costs him
only I of the i)rice he pays for the better article. What rate -
per cent, of profit does he make 1
CHAPTER III.
<
APPROXIMATION.
CONVERGENT FRACTIONS. ,
76. In the calculations which arise in the ordinary course of
affairs, fractions with large terms are not unfrequently met with.
For these lare^e-termed fractions, other fractions nearly equal to
them in value but with smaller terms, may often be substituted
without impairing the accuracy of the result for practical purposes,
but with the advantage of very decidedly lessening the labor of
computation. If a small-termed fraction is in this way to replace
a fraction wdtli large terms, the difference in value between the two
fractions should be the least possible consistent with the condition
that the terms of the replacing fraction shall be small. This
requires us to be able to find all the fractions whose values approach
so near the value of any given fraction that it is impossible to
insert between the given fraction and any of the fractions found
another fraction intermediate in value but with terms less than
those of the fractions found. The fractions which fulfil this
condition are termed Convergents to the given fraction.
77. The following examples exhibit the simplest method of
computing the convergents to a given fraction.
Example 1. Find the convergents to ^f^.
They are
Q 1 S 3 4 13 _6JL^ 1,09 X^c
1) J> T> in» 15' 4 2' 197> .-Jo'J' **'^'
The method of calculation is as follows : —
Write J and below it ^ as initials.
From these initials treated as if they were fractions form another
fraction with the sum of their numerators as its numerator and the
sum of their denominators as its denominator; rXn^T' ^^^^^
newly formed fraction, \, being greater than the given fraction ^*^,
write it in the upper line, the line of J.
CONVERGENT FRACTIONS. 5*7
From ^ and -^ form a fraction with the sum of their numerators
as its numerator and the sum of their denominators as its
denominator; rTri="o"- This ^ being greater than ^*A, write it
in the upper line.
1 0 . 1+0 1
From - and - form the fraction ^> . -, =-77.
^ > j^/j, . •. write ^ in the upper line. .
1 0 1+0 1
From —and - form the intermediate fraction o. ^ ='7~'
i < 1%%, . '• write J in the lower line.
11 ,. . 1+1 2
From - and-- form the intermediate fraction tt~^>— TT-
4 3 4+3 7
f < j^/j, . '. write f in the lower line.
2 1 ... 2+1 3
From - - and . - form the intermediate fraction ^ir-rh — TTJ*
7 3 7+3 10
write -^jy in the lower line.
3 1 . ..3+14
From ^Yj and-^ form the intermediate fraction iai~q ~To«
A < -/s^s, . '. write ^% in the lower line.
4 1 .4+15
From 37. and -^ form the intermediate fraction ToXQ^Tp-
j^ > 1^/5, . •. write -^^.; in the upper line.
5 4 . 5+4 9
From r^ and -- form the intermediate fraction ToT^o — oq-
2^9 > 1^^, . '. write ^§ in the upper line.
9 4 . . . 9 + 4 13
From ^ and ^ form the intermediate fraction oq_i_io== To'
it ^ iV^o ' • '• write ^1 in the lower line.
13 9 " 13+9 22
From Tj^ and^Q form the intermediate fraction ZolToQ^'Ti'
f 1 > i^®5» • *• write f J in the upper line.
Continuing this process we arrive at length at the given fraction
^^ which may be placed in either line. If the calculation be
continued beyond this, the succeeding convergents will be all less
or all greater than the given fraction accoiding as the latter was
58 ARITHMETIC.
written in the upper or in the lower line. In the preceding list we
have placed ^f^ in the upper line and have given the next two
lower-line convergents, viz. yVr *^^^ 3 si- ^^^ ^^ written ^.- in
the lower line, we would have obtained as the next two upper-line
, 48 + 35 83 83 + 48 131
convergents jgg:p^3 = 2ggand2gg:p-gg = ^-23. Both lines bemg
endless may be continued indefinitely, but as the terms of all the
convergents following the given fraction are larger than the terms
of the latter, these succeeding convergents are useless for purposes
of approximation and need not here be considered.
78. The convergents f , ^, ^%, rKj and Jf are called Principal
Convergents to -^^j ; the others are named Secondary or
Intermediate Convergents.
Example 2. Find the convergents to y^-
Proceeding as in Example 1 we obtain
111111 -3- 14 ■ r
fl 1 J2_ _5_ 8 11 ^3**
1 6> 11> -iTJ ¥3) 59J
Computation,
0+11^ l+OJ. l+OJ- Wj^ 1+0 1
1+oT* 1+1~2- 2+l~"3" 3+l~4- 4+l~5'
l±?_i 1+1 2
5+1" 6" 6+5"~ir
2+13
11+5 16'
3J-2__5 5 + 3 8 8 + 3 _11
r6+ll~27* 27+16~43' 43-fl6~59-
11+ 3 14 14+11 25
59+16 ~ 75* 75+59 ~ 134'
The convergents following j%\ are omitted.
Here the Principal Convergents are ^, ^,
Example 3. Find convergents to f g.
Computing as before we obtain
.0123 1J5 2.:? 53 1'**
1' 1' 1' 1' 4 ' 7 » 10'
0+1 1 1+1 2 2+1 3
Computation. ^3^^ = -^. j— = --. j-p^=-
CONVERGENT FRACTIONS.
59
3+1
4
4+3 7
7+3 10
1+0"
1*
1+1 2'
■ 2+l~3-
10+3
13
13+10
23 23+10
33
33+10
43
3+r
TheP]
"4*
rinci
4 + 3"
nal Conver
'1' 7 + 3"
erents are ^ an
10*
d i^.
10+3 "
"13-
-. 79. From the method of formation of these convergents it is
apparent that ; —
(a) The difference between any two consecutive convergents,
whether in the same line or in different lines, is a fraction with 1
for numerator and with the product of the denominators of the
two convergents for denominator.
(h) Each convergent to a given fraction approaches nearer in
value to the given fraction than do any of the preceding convergents
in the same line.
Thus i-^/5>i
> _5_ _ JUL > _9_ .
•"^ 16 155 -^ 29
155 ^
and i^ft-i>M-t>T^A>i%>T¥5-T*5"> A-H, &e-
80. From these two fundamental laws, four others follow as
immediate consequences. These are : —
1°. All convergents are in their lowest terms.
2°. Between a given fraction and any convergent to it there
cannot be inserted a fraction of intermediate value with terms less
than those of the next succeeding convergent in the same line.
Thus, between \ and y^j there cannot be inserted a fraction < \
but > ^5^5, with terms less than those of ^\. Between J and ^f^
there cannot be inserted a fraction > j but < ^^^, with terms less
than those of f .
Cm^ollary. The terms of all fractions intermediate in value
between a given fraction and any pi'i'ncipal convergent to it are
greater than the terms of the next succeeding principal convergent.
3°. The difference between any two consecutive priiicipal
convergents is a fraction with 1 for numerator and with the product
of the denominators of the two convergents for denominator,
4°. The difference between a given fraction and any principal
convergent to it is less than the difference between the given fraction
and any fraction with terms smaller than those of the principal
convergent.
81. The Corollary to the Second Law applies to principal
convergents only and distinguishes them from intermediate
60 ARITHMETIC.
convergents, it being noted that the principal convergents to any
given fraction are alternately greater and less than the given
fraction.
82. The Fourth Law holds for principal convergents but does
not necessarily hold for intermediate convergents. Cases occur in
which a convergent in one line differs less from the given fraction
than does a succeeding and therefore larger-termed intermediate
fraction in the other line. Thus, in Example 1 page 56, y^A ~\>
\ - ^f,, so also y^, - f > i - T*A andj-s^ - ^f, > ^X - ^%. Hence both
^ and f are inferior to ^ and ^^ is inferior to j^y, if we consider
these fractions solely as approximations to j^A, regardless of
whether they are approximations in excess or in defect. This
Fourth Law, therefore, marks out the Principal Convergents to
any large-termed fraction as, in general, the best small-termed
substitutes for such large- termed fraction, in approximate
calculations. It is consequently important to have an expeditious
method of calculating the principal convergents to any given
fraction. Such a method is exhibited in the following examples.
Example 1. Find the principal convergents to ^f^.
A. Divide both terms of the given fraction by the numerator.
48 1 1
155 155-48 3 + 11
Now 3 < 3 + 11,
11 . 1 48
• 3"3Tii ^•^•'3"l55- ^'^
B. Divide both terms of \^ by the numerator.
11 1 1 48 1
48 48^11 4 + ,V ' ' 155~^
^^ 3 +
4 + A
Now 4<4-+T*T,
1 1
1 1 J_ 48 (ii)
3 + |^o . 1 i«^' 3 + 1^155-
3 +
4 + A
CONVERGENT FRACTIONS.
61
C. Divide both terms of ^j by the numerator.
4 11 48 1
ll~ll-h4~2 + |' • • 155-'i 1
3+-
4+-
Now 2<2 + f,
1 1
2 '2 + 1'
1 1
4 + i"^^ 1
" '+2+I
1
1
1 "
o . o ■
1
^+ 4 + i ^+ 1
2 + f
D. Divide both terms of
1 by tht
3 1 1
48
4~4+3"~l + ^' •
• 155 ~
2 + 1
3 +
48
155'
4 + J^
3 + -
4 + -
2 +
1 +
Now 1<1 + J,
1 _L
1 TTi'
1 1
2 + -- 2 + -
1 + i
4 +
3 +
1
2 + 1
1
4 +
4 +
2 +
1 + i
(iii)
(v)
1
-, I.e.
3 +
3 +
48^
i55
(iv)
2 + i
4 + -
4 +
2 + -
2 + 1
1 + i
62 ARITHMETIC.
We thus find that ^g-
1 1 (i)&(ii)
is less than ^ but greater than qTT>
1 1 (iii) & (iv)
is less than :, — but greater than ^ >
3+JL_ 3+—:':
2+i
and is equal to ^j—
3+ \
4+
(v)
1+i
83. That these fractions are the principal convergents to ^-^^
may be shown thus ; —
Reducing all to simple fractions they are J, y%, ^g, J|, y*/-.
1°. 48 _ 13^ 1
165 42 156X42*
.'. the terms of all fractions < //- but>|| are greater than the
terms of j% ;
xVk is t-he principal convergent to itself,
. *. ^1 is the principal convergent next preceding ^^.
2°. _9->^8.>13 and-9._i3=__i_,
29 156 42 29 42 29X42'
. '. the terms of all fractions > ^% but < -^rf are greater tlian the
terms of ^f ;
^f is a principal convergent to /A,
. •. -^g is the principal convergent next preceding ^|.
3°. Similarly it may be proved that j^^ and i^ are the other principal
convergents to ^*A.
84. The operations A, B, C and D may be summarized as
follows : —
Divide 155 by 48 ; divide 48 by 11, the remainder in the
preceding division ; divide 11, the first remainder, by 4, the second
remainder ; divide 4, the second remainder, by 3, the third
remainder ; divide 3, the third remainder, by 1, the fourth
remainder.
Now this is nothing else than the series of operations for finding
the G. C. M. of the two numbers 48 and 155. Arranging; the
CONVERGENT FRACTIONS.
63
work as in the Public School Arithmetic, page 100, it appears
thus ; —
Quotients.
3
4 2
1
3
155
144
48
44
11 4
8 3
3
3
1
11
4
3 1 1
The convergents may now be written down from the line of
quotients, thus ; —
3, 4, 2, 1, 3.
11 1
h
3+i'
3+
4+i
3+-
3-t-
4+
2+^
4+
2+
1+i
The simple fractions equivalent to these convergents may be
calculated by the ordinary method of reducing complex fractions
to simple forms, or otherwise thus ; —
Galcvlation. Convergents.
Quotients.
0
0
»
1
3
1+ 0x3 1
0+ lx3~ 3'
4
0-f- 1x4 4
1+ 3x4~13'
2 .
1+ 4x2_ 9
3 + 13x2 29'
1
4+9x1 13
13 + 29x1 42'
3
9 + 13x3_ 48
29 + 42x3 155
{Initial.)
(i)
(ii)
(iii)
(iv)
(v)
85. Limits for the errors arising from the substitution of ^, ^,
&c. for ^ may be obtained as follows :—
64
ARITHMETIC.
<-i---i
3 155 13' ' ■ 3 165 3 13 3X13»
i. e. the error arising from the use of J for y^A- is less than --L. .
_*8_ — Jl < _9. — ^ = _JL_
48
155
29'
13
i. e. the error arising from using -^ for -^^ is less than ■ ,^ ^^ ,
Similarly it may be shown that .^^-i — is a superior limit of error
in the substitution of ^^ for ^-^.
J^ is the error in the substitution of ^| for //g.
Example 2. Find the principal convergents to |m§,
1 2 3 8 1 3 13
33593
23478
10115
23478 I 10115 I 3248 I 371 280
20230 I 9744 | 2968 | 280 273
" 3248~i ^37rn'e8(rp9r| 1
91
91
7 = G. CM. of terms.
Quotients, 1, 2,
Convergents, 1, ?, \, §,
Limits of error, _i-, _J_
3,
8,
1,
A.
m
1
1
1x3 3XJ0 10X83 83X<j3 93X362
3, .
253
36 2 >
\
.02X4799'
13.
sac
i7"J
0.
3354
4 79 9-
86. If the given fraction be improper, reduce it to a mixed
number and use the integral part of the mixed number as
numerator in 23lace of 0 in the initial ^.
Example 3. Find a series of convergents to 3*14159265 which is
approximately the ratio of the circumference of a circle to its
diameter, i. e., approximately the measure of the circumference in
terms of the diameter as unit.
7
15 1
288
100000000
99114855
14159265
885145
885145
882090
882090
6110 ,
3055
885145
5307815
4425725
3055
27109
24440
882090
26690
24440
Quotients, 7, 15, 1.
Convergents, i, f, ^-, ^§-|, ff|.
Hence the circumference of a circle is longer than 3 diameters
of the circle, is shorter than ^^ diameters, is longer than f -gf
diameters and again is shorter than f ff diameters.
The limits of error are f, — 1 — , 1 , and
'' 7X106' 106X113
respectively.
113X32650
CONVERGENT FRACTIONS.
65
I
87 From this example it is evident that those cwivergents which
immediately precede large quotients are the best approximations to
employ as substitutes for exact values.
Example 4. Find a series of convergent comparisons of the
metre = 39 -370432 in. and the yard = 36 in.
The quotients of 36/39-370432 are
1, 10, 1, 2, 7, 2, 1, 5 ;
and the corresponding convergents, omitting initials, are
h \h \h ff> HI, nh uh mi
Hence 10 m. < 11 yd. but 11 m. > 12 yd. ;
32 m. <35 yd. but 235 m. >257 yd.
&c.
FIXBRCISB II.
Find the
principal convergents to ; —
1. le.
«• M%%'
11. 1-4142.
16.
•0498756.
2- m-
^- tVt^V
12. 1-73205.
ir.
•2439.
3- Ml
8. T^^V,
13. 2-44949.
18.
1-41844.
4. §?.
9. xmm-
14. -43589.
19.
2-71828.
5. ^M-
10. mm-
15. -55744.
20.
2-302585.
Find a series of convergent comparisons of : —
21. The kilometre = 1093^62311 yd. and the mile = 1760 yd.
22. The hectare and the acre.
23. The kilogramme and the pound.
24. The millier and the ton.
25. The kilolitre and the cubic yard.
26. The litre and the quart.
27. The Canadian standard metre = 39 -382 in. and the French
standard metre = 39 -37043 in.
2§. The earth's polar diameter = 41708954 ft. and its longest
equatorial diameter =41863258 ft.
20. The tenacity of steel and the tenacity of copper wire the
former being ||| times the latter.
30. The excess of the mean solar year of 365 da. 5 hr. 48 m.
47 "46 sec. over the ordinary civil year of 365 da. , and one day.
Hence show that if there were 8 leap-years in every 33 years, this
system would not be wrong by so nuich as 1 day in 4224 years, and
compare this with the Gregorian system of 97 leap years in every
400 years.
E
66 ARITHMETIC.
APPROXIMATE CALCULATIONS.
88. The greater part of the labor of computation in calculations
in which fractions occur arises in general from the several fractions
having different denominators. For example, if two or more
fractions are to be added together, they must all be brought to the
same denominator, if one fraction is to be divided by one or more
others all of different denominators, the terms of the quotient are
in most cases much larger than the terms of the dividend. The
labor of computation may be lessened by using convergents instead
of exact values ; it may often be lessened and the calculations may
always be simplified by replacing the fractions by approximately
equal decimal numbers. If we adopt either of these ways of
lessening the labor of computation, we deliberately incur an error
in calculation which we know will give a result sufficiently near the
truth for all practical purposes.
89. In calculations concerning quantities which presuppose
measurements, it should be remembered that these measurements
cannot be made with absolute accuracy. In the measurements of
every-day life we are satisfied if we do not err by more than one
part in a thousand ; in the most careful scientific work it is rarely
possible to reduce the error below one part in a million. The
results of calculations based on such measurements are necessarily
affected by the errors of measurement and it is therefore a mere
waste of time and labor to carry any calculation beyond the degree
of accuracy with which measurements can be made. It is moreover
misleading, for the results then present an appearance of exactness
where exactness does not and cannot exist.
90. The first significant figure in any number is the first digit, —
the first figure other than zero, — on the left of the number.
Examples. In 980*61 min., 9 is the first significant figure and in
•000122 da., 1 is the first significant figure.
91. A number is said to be correct to two, three, four,
significant figures if it does not differ from the number that would
express the exact value by more than 5 in the second, third, fourth,
place on the right of the first significant figure.
apphoximate calculations. 61
Example 1. If it is said that the length of a certain line is 3 "9 in.
correct to ttvo significant figures, it is meant that the actual length
is between 3 "85 in. and 3 "95 in.
If the length is given as 3*94 in. correct to three significant
figvires, it is meant that the actual length lies between 3 '935 in. and
3-945 in.
If the length is said to be 3*937 in. correct to fmir significant
figures, the actual length may be any between 3 "9365 in. and
3-9375 in.
Example 2. If the length of the greatest equatorial diameter of
the earth be given as 41,852,000 ft. and the length of the polar
diameter as 41,710,000 ft., correct in both cases to five significant
figures, it is meant that the actual length of that particular
equatorial diameter is not less than 41,851,500 ft. but is less than
41,852,500 ft., and that the actual length of the polar diameter is
not less than 41,709,500 ft. but is less than 41,710,500 ft.
92. The degree of any approximation is measured by the
fracti(jn which the total error is of the exact value, i. e., by
the quotient of the difierence between the exact and the
approximate value divided by the exact value. The degree of
approxirMitioii is therefore hulepoulent of the unit of measurement.
Example. If the length of the polar diameter of the earth is
41,710,000 ft. correct to five figures, the difference between this
length and the exact length is at most 500 ft. and the actual length
of the polar diameter is greater than 41,709,500 ft. Hence the
greatest possible rate of error is 500 ft. in 41,709,500 ft. =^1 part in
83,419. The degree of approximation is therefore at worst shItv ^^
the whole.
Had we used the mile instead of the foot, as the unit of
measurement in the foregoing, the degree of approximation would
have been found to be at least as close as -^^^ mi. in ^^|§|§^^ mi.
= 1 part in 83,419= yg^^^Tj^ of the whole.
93. Difterent degrees of approximation may be roughly compared
by comparing together the significant figures known to be correct
in each case.
Tlius if the first three significant figures are known to be correct
the approximation is about ten times as close as it would be if only
the first two were known to be correct. "Correct to six significant
68 ARITHMETIC.
figures " means an approximation about 1000 times as close as that
of " correct to three significant figures." *
94. In expressing mixed numbers and fractions by approximately
equal decimal numbers, it is in general sufficient if the calculations
are correct to four or at most to seven significant figures. Beymui
seven figures we very seldom need go.
So also if one approximate number is to be multiplied by another
or to be divided by another, the result need not be calculated to a
greater number of significant figures than are correct in the given
numbers.
Example 1. Find the product of 678-233 multiplied by 47-9583
correct to six significant figures.
ncontracte
d Form.
Contracted Form.
678-233
m?>'m
47-9583
47-9583
27129-32
27129-32
4747-63
1
4747-63 . .
(").
610-40
97
610-41 . .
(/,).
33-91
165
33-91 . .
{c).
5.42
5864
5-42 . .
(d).
•20
34699
•20 . .
32526-9
(e).
32526-90
16839
We begin by multiplying by 4, the first significant figure in the
multiplier. The product contains 7 significant figures ; this is one
more than the number required to be correct, but wfd retain all
seven that we may determine the ' carriage ' to the sixth significant
figure when adding together the partial products. We contract the
subsequently calculated partial products thus ; —
(a). Strike the right hand 3 from the multiplicand and multiply
by 7, carrying 2 from the 3x7 struck out.
(h). Strike the second 3 from the already contracted multiplicand,
and multiply by 9 carrying 3 from the 3x9 struck f>ut.
(c). Strike 2 from the multiplicand as contracted in (/>) and
multiply by 5 carrying 1 from 2x5 struck out.
(d). Strike 8 from the multiplicand as contracted in {<•) and
multiply by 8 carrying 6 from the 8x8 struck out.
(e). Strike 7 from the multiplicand as contracted in ((/) and
multiply by 3 carrying 2 from the 7x3 struck out.
APPROXIMATE CALCULATIONS.
69
The sum of the right-hand figures of the partial products is 9,
This woukl be the seventh significant figure of the product, but as
the product is to be correct to only six significant figures, we
change 9 to the nearest multiple of 10 which in this case is 10 itself.
We now complete the addition of the partial products as in the
ordinary uncontracted form.
The approximation in line (d) would have been closer had we
carried 7 from 8^ x 8 struck out instead oi carrying 6 from 8x8
struck out ; but as we are working to one figure more than the
number required to be correct in the result, the carried 6 is
practically as good an approximation as the carried 7 would be and is
more easily and quickly obtained requiring us to take account
of only one figure, the last figure struck out. In line (c), the
carriage should have been frcmi 8x3 instead of from 7x3, the 7^
struck out l^eing nearer 80 than 70.
[For the position of the multiplier and of the* decimal point in
the product, see Piiblic School Arithmetic p. 155 and the examples
on p. 156.]
Example 2. Find the product (jf 15876 multiplied by 15876
multiplied by 15876, correct to 5 significant figures.
1
15876
2
3
4
6
31752
47628
63504
79380
6
7
8
9
95256
111132
127008
142884
10 I 158760
15876
15876
15876
79380
12701
1111
95
252047(X)0 . .
. . Multiplier.
15876 . .
. . Multiplicand
31752
7938
318
6
1
4001500000000
[For a condensed notation applicable to examples like this, see
§ 122.]
Example 3. Divide 32526-9 by 678*233, obtaining the (iuotient
correct to 6 significant figures.
70
ARITHMETIC.
Uncontracted Form.
Contracted Form.
47-9583
47-9583
678233)32526900
2712932
678233)3252690^
2712932
539758
474763
0
1
9 0
9-7
539758
474763. . . .
64995
61041 . . .
3954
3391 . . .
563
542 .. .
21
20 . . .
{a)
64994
61040
. (h).
"3953
3391
562
542
9 30
165
76ll0
6 864
.(c)
.(./).
20
20
i 7860
3 4699
.(e).
—
1-6839
1
The sign — before 1 -6839 denotes that the quotient 47 '9583 is too
great ; it is however nearer the exact quotient than 47 -9582 would
be.
For the method of obtaining lines (a), (?>), (c), {d) and (e) see
Examjyle 1, page 68.
Computers' Contracted Form.
32526900-
539758
64995
3954
563
21
1
47-9583
Example 4. Find the weight (in Imperial tons of 2240 lb. each)
of the carbon in the carbonic acid gas in the atmosphere resting on a
square mile of land when the pressure of the atmosphere is 14*73 lb.
to the square inch, given (i) that each cubic foot of air contains
•00035 of a cubic foot of carbonic acid gas, correct to 2 significant
figures ; (ii) that the weight of any volume of carbonic acid gas is,
to 3 significant figures, 1 -52 times the weight of an equal volume
of air under the same pressure and at the same temperature ; (iii)
that -^Y ^y weight of all carbonic acid gas is carbon, correct to 4
significant figures. {See Huxley's Phijsiography, CJiap. VI. )
APPROXIMATE CALCULATIONS.
71
If
Our answer will be correct to only 2 significant figures, for
atum (ii) is correct to only 2 significant figures, and it is the
datum with the least number of figures correct that determines
the number of figures correct in tha result of any calculation. We
impute at first to 4 significant figures, reducing this number to 3
d finally to 2 as the number of operations* to be performed
ecome fewer.
Wt. of air on sq. in.
= 14-73 lb.
1 mi.
63360 in.
Wt. of air on sq. mi.
= 14-73 lb. X 63360x63360
= 59,130 W0,000 lb.
Wt. of carb. acid gas in this
air = 59,130, 000,000 lb. x -00035 x 1-52
= 31,500,000 lb.
Wt. of carbon in this
gas
= 31,500,000 lb. x^\
M = 3,800 T. Imperial.
95. These methods of contraction are easily adapted to
calculations in which the result is required to be correct to a given
number of decimal places.
Example. Find the interest on $79-27 for 93 days at 7|%.
-075
5-55
-40
5-95
5
6
3
5 53 -^p!
188
5
1
(a).
(6).
(c).
id).
$1-51
(a) The 7 in the multiplier stands above the 4th decimal place,
but only 2 decimal places are required in the result, therefore
strike '27, the two right-hand figures, out of the multiplicand, and
then multii>ly 79, the uncancelled part, by 7, carrying 2 from
-27 X 7 struck out.
(h) Strike 9 from the multiplicand as contracted in (a), and
multiply by 5 carrying 5 from 9x5 struck out.
{<■) $5-95 is the interest, to the nearest cent, on |79-27 for 1 year
at 7i%.
72 ARITHMETIC.
(d) To multiply 5*95 by 93, multiply 6 '95 by 7 and ' make up ' the
product, figure by figure as computed, to 595*00, i.e., to 5*95 x 100
setting down the ' making up ' numbers, thus, —
7 times 5 and 5 (set down) = 4,0'
7 " 9 and 4 (carried) "3 " " =7,0'
7 '' 5 " 7 " '* 3 '' " =4,5'
4 " "5 " " = 9'
5 *' *' = 5'
The accented figures are those of 595*00. {See §69, Case xii.)
96. In the preceding calculation, the sole influence of the 27
cents in the principal is the addition to the annncd interest, of the
2 cents ' carried ' in line (a). Even this small increment disappears
fVom the interest for 93 days, $1*51 being practically the interest
on $79 for 93 days at 7|%. The omission from the principal or the
addition to it of any number of cents less than 50, will not in
general change by more than one cent the computed amount of the
interest for a short-term loan, but the retention of the cents in the
calculation will considerably increase the labor of computation.
For this reason, business men compute on the nearest number of
dollars, when reckoning short-term interest and when determining
the equated time of an account. (See Public School Arith^netic,
p. 168.)
EXERCISE III.
1. Find the sum of 143*035472, 29*680037, '089173, 4*99876 and
2923*937958, correct to 4 decimal places.
2. Find the value of 379 '28056+29*68043+ 6 '8409207 - 44*398642
- 3 -7984061+ -2368592 - 300 '790797, correct to 6 significant figures.
3. Find the product of 478 '593 and 3*14159 correct to 3 decimal
places.
4. Find the value of 427*803 x *00749 correct to 5 decimal places.
5. Find the value of 3*1416x3*1416x3*1416 to the nearest
i nteger.
6. Find the product of 2*9957323 and -4342945 correct to 6 decimal
places.
7. Find the value of 5*7037825 x '4342945 correct to 6 decimal
places.^
8. Find the value of 3*14159265 x '96 x '995 x '9998 x -99992
X -999997, correct to 6 decimal places.
EXERCISES.
73
II
9. Find the value of 2-7182818 x '8 x '992 x -9993 x -99998
X -999993 X -9999994 correct to 7 decimal places.
10. Find the value of 2-3025851 x -9 x -97 x -995 x -99995
X -999997 correct to 7 decimal places.
11. Find the product of 1 •(XMX)127 x 1 -004 and -99898 x -99898
correct to 7 decimal places.
12. Find the value of 100(X)127 x 999987, correct to 8 significant
figures.
13. Find the value of 10-934 x 16-934 x 16-934 correct to 5
significant figures.
14. Find the value of 4-8784x4-8784x4-8784 correct to 5
significant figures.
16. Find the value of 9 0708324 x 9 0708324 x 9-0708324
X 9-0708324, correct to 6 significant figures.
16. Find the value of 2-0188223x2-0188223x2-0188223
X 2-0188223 x 2-0188223 x 2-0188223 correct to 6 significant figures.
Find the values of the following quotients, correct to 6
significant figures : —
23. l-^ 3 14159265. '
24. 1^-43429448. '
25. 11-^2 -22398 -=-2 -22398.
26. 4517 ^16 -5304 -^ 16 -5304.
27. 19 -5 -^ 2 -236068 ^6 -244998.
17. 100^1-414214. •»
18. 25000^3141593. 3
19. 07^2-64575. ^'
20. 1 -95 -^ 139 -6424. ^
21. -6931472 -r 2 -302585.,
22. 1-098612^2-302585.
Find the values of the following, to 5 significant figures
1 ■ L_^ L
1x2x3x4* 1x2x3x4x5
28. 1 + 1+1x2+1x2x3
-+&C.
^ 1 , 1 1 ■ 1 1
2^- l"^lx2 1x2x3+1x2x3x4 1x2x3x4x5+
1
-. + .
1_^ 1_
^ 1+1x3+1x3x5 ■ 1x3x5x7 ' 1x3x5x7x9
1:--^ + ^
+ &c.
0-. 1_u1j_1_l1_l1_l1.1.
^1- l"^2"^T + '8 +16+32+64+ *^^-
32. I+3-+9-+27+8i+243+ *^'
^' r+5:+25+125"^625'^ *^-
74 ARITHMETIC.
34. Prove that the answer to problem 29 is the reciprocal to 5
significant figures of the answer to problem 28.
35. Express 8749 yd. in metres correct to 4 significant figures.
36. Express 1760 metres in yards correct to 4 significant figures.
37. Express 4840 sq. yd. in centiares correct to 4 significant
figures.
38. Express 4840 centiares in sq. yd. correct to 4 significant figures.
39. Express 100 acres in hectares correct to 4 significant figures.
40. Express 100 hectares in acres correct to 4 significant figures.
41. Express 600 litres in gallons correct to 4 significant figures.
42. Express 132 gallons in litres correct to 4 significant figures.
43. The mean distance of the moon from the earth is 238800
miles ; express this in kilometres to 4 significant figures.
44. The mean distance of the sun from the earth is 91,430,000
miles ; express this in kilometres to 4 significant figures.
45. The mean distance of Saturn from the sun is 872,140,0(K)
miles, and of the earth from the sun 91,430,000 miles; form a
series of convergent comjmrisons of these distances.
46. Form a series of convergent comparisons of 346 "619 da. and
29*5306 da., and hence show that 19 times the former period is
nearly equal to 223 times the latter. ' Express these products in
terms of a year of 365 "25 days.
47. Taking the length of the sidereal year as 365 '25636 days and
that of the lunar month as 29*53059 days find a series of convergent
comparisons of the lunar month and the sidereal year.
48. Mars revolves about the sun in 686*9797 days and the earth
revolves about the sun in 365*2564 days ; find a series of convergent
comparisons of the length of the Martian year with that of the earth.
49. Jupiter rotates on its axis once every 9 hr. 55 min. 26 sec. ,
and the earth once every 23 hr. 56 min. 4 sec. ; find a series of
convergent comparisons of these times of rotation.
50. Mercury revolves about the sun in 87*9693 da. at a mean
distance of 35,392,000 miles ; and the earth revolves about the sun
in 365*2564 da. at a mean distance of 91,430,000 miles. Find
convergent comparisons of the speed of Mercury and the earth in
their orbits.
MISCELLANEOUS PROBLEMS. 75
EXERCISE IV.
MISCELLANEOUS PROBLEMS.
The mercury in a barometer rose •121 in, , '073 in. and '019 in.
Iri three successive clays, it fell '064 in. and '065 in. during the two
following days, rose "OSo in. on the sixth day and fell '028 in. on the
seventh day. If its height at the beginning of the first day was
30 '078 in., what was its height at the close of the seventh day ?
2. Find the weight of a rectangular beam of oak 18' x 13" x 13",
weighing 47 '375 lb. per cu. ft. How many cubic feet of water
would be of the same weight as the beam ?
3. A clock gains ]^ I of 3^ sec. in 2 hr. 30 min. If allowed to run
at this rate how much will the clock gain in 8 da. 8 hr. correct
time ? How much will it gain at this rate, if it run for 8 da. 8 hr.
by its own time ?
4. A man sold | of his wheat and then /^ of the remainder and
next ^J of what then remained and had 18 bushels more than '12 of
his wheat left. How many bushels had he at first ?
5. An india-rubber band 8" long §" wide and ^" thick is stretched
until it is 18" long and ^" wide. What must be its thickness, the
volume of the india-rubber remaining unchanged ?
6. If 1 lb. of brass consisting of 84 parts of copper and 16 of zinc,
be mixed with 2 lb. of brass consisting of 75 parts of copper and 25
( )f zinc, find the percentage of copi)er and of zinc in the mixture.
7. In 1881 the silver mines of Austria yielded 12,383 metric tons
of silver ore from which 31, 359 kilogrammes of silver were extracted.
What per(j^ntage of the ore was silver ? Express the weight of the
ore in Imperial tons and the weight of the silver in Troy ounces,
and employing these expressions of the weights, recalculate the
percentage which the silver constitutes of the ore.
8. A. B. bought goods amounting to $7400 subject to 25 and 5
off, $3730 subject to 30 off and $1492 subject to 20 and 10 off, find
the net cost of the goods. Were the invoice-clerk to bill A. B.
with goods amounting to $12682 subject to 30 off, what would be
the amount of the error in the net cost of the goods 1
9. Find the equated time of payment of a bill for $748 of which
$225 is at 30 days, $245 is at 60 days and the balance is at 90 days
all from 31st Aug. 1889.
76 ARITHMETIC.
10. The proceeds of a draft for $628*60 drawn at 90 days,
amounted to f 615 '79. What was the rate of discount ?
11. A train is due at a certain station at 42 niin. past 2 p.m. The
actual times of its arrival at the station for a certain week were : —
Monday, 2,38 p.m. ; Tuesday, 2,47 p.m. ; Wednesday, 3,07 p.m. ;
Thursday, 2,39 p.m. ; Friday, 2,42 p.m. ; Saturday, 3,11 p.m. By
how many minutes on an av^age was the train late that'week, (i)
not counting * minutes ahead of time, ' (ii) including ' minutes ahead
of time ' in the averaging ?
12. How often is the circumference of a circle 1' 9" radius con-
tained in the diameter of a circle whose circumference is 100 feet ?
13. What will be the weight of a rectangular sheet of glass 6' 3^'
long by 4' 4|" wide and ^ in. thick, the glass weighing 168 lb. per
cubic foot ?
14. How many days were there from 13th Nov. 1887 to 9th June
1888 ? Express the interval from noon on the former day to noon
on the latter day as a fraction of the year 1887 and also as a fraction
of the year 1888.
15. A watch is set right on Monday at 9,15 a.m. and it gains 3^
sec. per hour. On what day and at what hour will it have gained
exactly 5 min. and what time will it then indicate ? What will be
the correct time when the watch indicates 9,15 on the following
Monday morning ?
16. Out of a certain sum of money one-half was spent, then
one-third of the remainder, next one-twelfth of what still remained
and lastly one-fifteenth of what then remained, leaving 39ct. less than
one-half of what was spent. What was the original sum ?
17. A man buys milk at 5ct, a quart and having mixed it with
water, sells the mixture at 6ct. a quart. His profits are equal to 40%
of the cost of the milk. How much water is mixed with each quart
of milk ? What proportion of the mixture is water ?
- 18. If an investment of $7483-50 yield a net profit of $483-67,
what rate per cent, of profit is returned by the investment ? If
this profit is reinvested along with the original investment, and the
whole yield a second profit at the same rate per cent, as the first,
what will be the amount of this second profit ?
MISCELLANEOUS PROBLEMS. 77
19. James King & Co. of Brantford sold to Henry Adams of
Paris bills of merchandise as follows :— 12th Dec. 1888, $1174-80 at
90 da. ; 3rd Jan. 1889, $729-65 at 90 da. ; 21st Jan. 1889, $106-20
at 75 da. ; 12th Feb. 1889, $1485-45 at 60 da. j 7th March 1889,
$973 -28 at 30 da. Find the equated time and make out a statement
of account on the average date.
20. A note for $355 drawn on 3rd April 1889 was discounted on
11th April ; the proceeds amounted to $348*03. What was the rate
of discount, the rate of exchange being -1%, reckoned to nearest
cent.
II
21. A bicyclist rode 50 mi. in 3 hr. 6 min. 40 sec. ; what was his
rate in feet per second, in yards per minute, and in miles per hour ?
22. The leading wheels of a locomotive are 3' 2" in diameter and
the driving wheels 5' 6" ; how many revolutions will the former
make while the latter make 2166 ? What distance will have been
run ? If the distance is run in 20 min. at what rate in miles per
hour will the run be made ?
23. Find the weight of a slate blackboard measuring [19' 6'
X 3' 6" X §"] if a cubic foot of the slate weigh 178 lb.
24. In a certain gold mine, 11 tons of ore yielded 7j oz. (Troy) of
pure gold, what fraction of the ore was gold ? Express the proportion
< )f gold to ore in grammes per metric ton.
25. A clock which gains 9 sec. per 1 hr. 11 min. , is set right at
10 a.m. on 1st March, when will it denote correct time again ?
26. After drawing off 15 gal. of the contents of a certain cask and
then -j\ of what was left, the remainder sold at 5| ct. a pint brought
$3 -96. How many gallons were there originally in the cask ?
27. A mixture of coffee and chicory in the proportion of 8 parts
of coffee to 1 part of chicory is sold at 35 ct. a pound, being an advance
of 40% on the cost. The chicory cost 9 ct. a pound, find the cost
of the coffee per pound.
28. A man bought a house and lot for $4750. After spending
$1143 on repairs and improvc'ments and paying $128 for taxes and
other expenses, he sold the property for $6800. What -rate per
cent, of profit did his investment yield him ?
78 ARITHMETIC.
29. On 18th June 1888, a merchant purchased goods amounting
per catalogue prices to $647*80, subject to 25 and 5 off. He was
allowed 3 months credit after which he was charged interest at
8%. Find the amount of the account on 21st February 1889.
30. Find the difference between the discount taken off a draft
for $500 drawn at 90 days and discounted at 7% aaid the interest
on the proceeds for 93 days at 7%. Find the interest for 93 days
at 7% on the amount of the discount taken off the draft.
31. A man skated 10 miles in 36 min. 37 '2 sec. ; what was his
speed in yards per minute, in miles per hour, in metres per min. ,
in kilometres per hour ?
32. A circular race-track ^4 ft. wide encloses a circle of 50 yd.
radius. How long would it take a man to run round the outer edge
of the track at a speed which would take him round the inner edge
in one minute ?
33. Find the weight of slate per cubic foot if a rectangular slate
blackboard 16' 8" long, 3' 6" wide and f in. thick weigh 647 lb.
34. A road 44 ft. wide is made directly across a field 210 yd.
square. What fraction of the field does the road occupy ? What
would be the value of the part taken for the road, at $144 an acre ?
35. A cubic foot of pure water at 62° F. weighs 62 '356 lb. and a
cubic foot of sea- water at the same temperature weighs 64*05 lb, ;
find the weight of 25 gal, of sea-water.
36. A wheel makes 72 revolutions per minute. If its speed be
increased by jl^ of itself, how many revolutions will it make in 6
working days of 10 hours each ? Had the time of making a revolu-
tion been increased by y|-{) of itself, how many revolutions would
the wheel have made in 6 days of 10 hours each ?
37. A grocer buys 80 lb. of tea at 21 ct. a lb. and mixes it with
some dearer tea he has on hand. Selling the mixture for $43*75,
this being at the rate of 35ct. a lb., he clears $15 '25 on the whole.
How many pounds of the higher priced tea did he mix with the
other and how much per pound did this higher priced tea cost him?
38. The population of a certain city was 27,413 at the date of
taking one census and at £he time of taking the next census the
population had risen to 44,229 ; find the increase per cent, correct
to 4 significant figures. Express this as an increase per thousand.
L
MISCELLANEOUS PROBLEMS. 79
39. What rate of interest is e(iual to 8% discount for one year?
40. On 19th April 1889, a merchant purchased goods amounting
per catalogue prices to $1239-35, subject to 30 and 5 off; terms*
3 months credit or 5 off for cash, h% per month on accounts overdue.
Find the amount of this account on 19th Oct. 1889. What would
have been the amount had the account been paid on 19th April
1889? What rate of interest will the merchant be paying if he
settle on the 19th Oct. instead of on 19th April ?
Pi
41. Find to the nearest 100 sec. and also to the nearest minute
the time occupied by light in passing from the sun to the planet
Neptune, the velocity of light being 187,200 miles per second and
the distance of Neptune from the sun being 2,746,000,000 miles.
42. How many yards of carpet 27" wide will be required to
cari^et a room 25' 8" by 15' 8^' allowing 9" per width for matching ?
How many rolls of waU-paper and how many yards of bordering will
be required for the same room, allowing on the wall-paper a width
of 42" each for 3 windows and 2 doors ?
43. Find the value of a pile of cordwood 13' 4"4ong by 3' 9" high
at $4.50 the cord?
44. Find the weight of a circular copper plate f in. thick and 11"
in diameter, copper weighing 549 lb. per cubic foot.
45. If an express run at 30 mi. an hour and an accommodation
train at 22 miles an hour, what is a man's time worth if he would
Jose 45ct. in travelling a journey of 270 miles by accommodation
nstead of by express ?
46. Find the number of cubic inches which 10 lb. of (a) water,
(h) hard-coal, (c) silver, {d) oak will occupy if a cubic foot of water
weigh 62 lb. 6 '8 oz. and if hard-coal be 1*6 times and silver be 10*5
times heavier, volume for volume, than water, and if a cubic foot of
oak weigh ^ as much as a cubic foot of water,
47. A publisher sells a certain book at 78ct. per copy. He pays
the printer 172ct. , the binder 15ct. , and for other expenses 9ct. on
every copy printed. He also pays the author 12ict. on every copy sold.
Of one edition of 1000 copies he sells 879 and the rest are left on his
hands. Does he gain or does ho lose on the transaction ? How
much ? At what rate per cent. ?
80 ARITHMETIC.
48. A did 1^ of a piece of work, B did f of the remainder, C did
f of what was left undone by B, and T> then finished the work.
•How much should D get for his work if A receive $7 '00 for his?
49. Find the proceeds of the following joint note discounted in
St. Thomas on 18th Dec. 1888, at 7^%.
^347T^r^. St. Thomas 18th Dec, 1888.
Ninety days after date we jointly and severally promise to
pay to the order of Jno. Locke & Co. , Three hundred and forty-
seven ^^ dollars, at the Standard Bank here. Value received.
Isaac Harper.
H. H. Friedlaender.
60. What rate of discount is equal to 8% interest reckoning (a) for
a year, (h) for 93 days, (c) for 63 days ?
61. The British ship Egeria found a dfepth of ocean of 4430
fathoms at a certain place oflf the Friendly Islands and the U. S.
ship Tuscarora found a depth of 4655 fathoms off the north-east
coast of Japan. What must be the pressure per square inch due to
the superincumbent water at these depths, sea-water weighing
64 '05 lb. per cubic foot ? Express the pressure in kilogrammes per
square centimetre.
62. What will be the cost of 1000 yards of side-walk 8 ft. wide,
made of 3 in. plank laid on three lines of cedar stringers, if the planks
cost $12*00 per M., the cedars 4|^ct. per running-foot and preparing
and laying the sidewalk $3*50 per yard ?
63. Out of a circle 18" in diameter there is cut a circle 13*5'
in diameter. . What fraction of the original circle is left ?
64. Find the weight of a cast-iron cylinder 8' in length and 7 "
in diameter, if a cubic foot of cast-iron weigh 444 lb.
66. A vessel holds 2/^ qt., how many times can it be filled from
a barrel containing 31 1 gal. of oil ? After filling the vessel as often
as possible how much oil will remain in the barrel ? What fraction
of a vesself ul will this remaining quantity be ?
66. If 9 lb. of rice cost as much as 6| lb. of sugar and lOj lb.
of sugar cost as much as 1 lb. 10 oz. of tea and 1 '25 lb, of tea cost
as much as 2f lb. of coffee, find the cost of 100 lb. of cofibe if rice is
worth 7 ct. a pound.
MISCELLANEOUS PROBLEMS. 81
67. If a lamp burn '08 of a pint of oil per hour and 6 lamps are
used every night and 30 gal. of oil are consumed from 27th Sept. to
4th Jan. next following, both nights included, how many hours per
night are the lamps alight ?
58. In an examination A obtained 78% of the full number of marks
beating jB by 16% of the full number. If A received 975 marks,
how many did jB receive? What percentage of A'b number was JB's
number? What percentage of B's number was A's number? It
was afterwards decided to deduct 20% from the total number of
marks and also from the numbers obtained by A and B, what effect
would this change have on the answers to the preceding three
questions ?
59. Find the proceeds of the following note discounted in Toronto
on 17th Oct. 1888 at 7^%, exchange j% reckoned to nearest cent.
$211x«^. Hamilton, 12th Oct., 1888.
Three months after date I promise to pay to the order of
A. J. Wilson & Co. , Two hundred and eleven Dollars at the Bank
of Commerce here. Value received.
Henry Tomlinson.
60. For how much must a ninety-day note be drawn to realize
$190 when discounted at 6°/^ ?
61. If 8 metres of silk cost 76 francs what will be the price of
10 yd. at the same rate, reckoning 10 fjancs equal to $1 "93 ?
62. If 2 horses are worth as much as 7 oxen and 3 oxen as much
as 17 sheep, find the value of 5 horses that of 9 sheep being $60.
63. Find the price of a rectangular slate blackboard 23' 4" long
by 3' 6" wide @ 44 ct. per square foot.
64. Find the weight of a cast-iron pipe 7' 6" long and of 5j"
external and 4" internal diameter, a cubic foot of cast-iron weighing
4441b.
6b. A train is running at the rate of 20 miles per hour and a
second train starts after it at the rate of 27|^ miles per hour and
overtakes it in 3 hr. 25 min. How many miles an hour did the
second train gain on the first ? How far ahead was the first train
when the second traiii started ?
F
82 ARITHMETIC.
GQ. A man who has had his wages increased by tt^j is in receipt of
$12 "50 per week. What fraction of itself must be taken off this
weekly sum to reduce his wages to the original rate ?
67. How many boys each doing '6 of the work of a man must be
engaged with 51 men to do in 20 days as much work as 28 men could
do in 45 days ?
68. The average rainfall at Toronto is less than the average
rainfall at St. John, N. B. , by 45|°/^ of the latter, and the average
rainfall at Windsor, Ont. , which is 30 in. per annum, is greater than
the average rainfall at Toronto by 8'l°/„ of the latter. Find the
weight per acre of the average annual rainfall at St. John, N.B.
69. Find the proceeds of the following draft discounted on 15th
Feb., 1889 at 6 7„, exchange -^ 7, :
$791i%5o- GuELPH, 12th Feb., 1889.
Sixty days after date pay to the order of Henry Meadows & Co.
of Belleville, Seven hundred and ninety-one y%\j^ dollars. Value
received.
Stuart & Gee.
To J. J. Newcomb,
Belleville.
70. The proceeds of a note payable in 3 months from 1st Feb.
1889 and discounted on the 6th Feb. 1889, amounted to $847 '18.
For what sum was the note drawn ?
71. How many tiles 6" sqftiare would pave a hallway f the size
of a courtyard which required 9360 bricks to pave it, at the rate' of
8|' by 4^" per brick ? Find the length of the courtyard and the
width of the hallway given that the length of the hallway and the
width of the courtyard are each 42 ft. 6in.
72. Find the weight of 5 miles of steel wire of -147" diameter,
the steel weighing 492 lb. per cubic foot.
73. Sound travels at the rate of 1120 ft. per second more slowly
than light; at what distance is a lightning-flash the thunder of
which is heard 7^|- sec. after the lightning is seen ?
74. A by working on piece-work f as fast again as B is able t( >
earn $2 -09 per day. How much does B earn per day ?
L
MISCELLANEOUS PROBLEMS. 83
».
75. A man asks to have his working hours decreased from 10 hr.
to 8 hr. per day without any decrease in his daily pay. By what
fraction of his wages i)er hour does he ask them to be increased ?
76. A contractor undertakes a contract to be completed in 120
days. He employs 48 men and at the end of 25 days finds that he
has '2 of the work finished. How many additional men must he
now put on in order to have the contract completed 15 days sooner
than the time specified in his agreement ? (Only working-days are
cmiiited in this statenieiit. )
77. A can do a certain piece of work in 10 days working 8 hr.
er day. B can do the same work in 9 days working 12 hr. per
day. They decide to work' together and to finish the work in 6
days. How many hours a day must they work ?
78. A man travels 300 miles in 12 days travelling 8 hours per day.
If he increase his speed by 20 °/^, how many hours per day less than
before need he travel in order to accomplish 450 miles in 20 days ?
79. A market-woman bought a certain number of eggs @ 11 for
ct. and sold them, all but 3 which were broken and thrown away,
at 9 for 11 ct., thus clearing $2*63 on the transaction. How many
eggs did she buy and what rate per cent, of profit did she make ?
80. On 23rd July 1889, Messrs. Ingram, Hughes, Leighton &
Co. , of Toronto, take to the Bank of Commerce, to be discounted
and the proceeds placed to their credit, drafts as follows : — One at
60 days from date on S. Cassidy & Co. , Paris, for $372 '85 ; one at
90 days from date on Th. Moore & Co., Owen Sound, for $629 30 ;
one at 10 days from date on Gregg & Weir, Belleville, for $125; one
at 45 days from date on Brock & Eaton, St. Thomas, for $748 "50 ;
one at 4 mo. from date on Colby & Masson, Chatham, for $917 *60 ;
one at 2 mo. from date on Bowles & Co., Guelph, for $322*10.
Draw up and fill in a discount sheet for these drafts arranging
them in the order of maturing ; discount 7%, exchange (reckoned
to nearest cent on each bill) ^ % on drafts up to $400, }^ % on drafts
for more than $400.
81. A dealer bought eggs at 10 for 14ct. and sold them at 14 for
24ct. On a certain day ho received $6*60 ; how much of this was
profit ? Had he bought the eggs at 14 for 24ct. and sold them at
10 for 14ct. , how much would he have lost on the day's sales ?
84 ARITHMETIC.
82. Find the cost of painting the walls and ceiling of a hall
62' X 34' 6" X 15' 6" at 27ct. per square yard, — no deductions for
openings.
83. A rectangular box made of boards Ij" thick, measures on the
outside 3' 7" by 2' 5" by 1' 10". Find its internal content, (a) the
measurements, including the lid ; (h) the measurements being of
the box without the lid.
84. If a boat 3G ft. long travel f of its length at each stroke of
the oars, how many strokes will be required in rowing a distance
of 2| J miles ? How many strokes per minute will the rowers require
to make in order to row the distance in 26 min. 40 sec. ?
85. If a man earns ^ as much as 7 women and a boy earns | of f
of the wages of 2 women, what fraction of a man's wages does a boy
earn; the time of earning being in all cases the same ?
86. A town council was offered gravel unscreened at $4* 50 a cord,
screened at $5*50 a cord. Allowing 25ct. as the cost of screening a
cord of unscreened gravel ; at what fraction of the unscreened gravel
do the above prices estimate the loss by screening ?
87. Divide $40 '71 among 7 men, 16 women and 25 children, so
that 5 men may get as much as 6 women, and 5 women as much as
6 children.
88. By selling a certain book for $3 '96 I would lose 12 % of the
cost ; what advance on this proposed selling price would give a
profit of 12 % of the cost ? What rate per cent, on the proposed
selling price would this advance be 1
89. On 28th Aug. 1888, a merchant purchased goods amounting
per catalogue prices to $987 "50 subject to 20 and 5 off; terms 3
months credit or 5 % off for cash. To what rate of interest is this
5 % off for cash equal ? If the merchant were to discount at 7 % a
note drawn at 3 months for the credit amount of the above account,
by how much would the proceeds of the note exceed the cash amount
of the account ?
90. A merchant buys goods amounting per catalogue prices to
$1573-45, subject to 20 and 10 oft'; terms 90 days credit or 5% off
for cash. For how much must the merchant make a note payable
in 90 days, that the note discounted at 7 % may realize the cash
amount of the above bill ? For how much must the note be drawn
to allow I % off for exchange ?
V
MISCELLANEOUS PROBLEMS. 85
I
91. Four foremen A, B, 0, D, are placed over 260 men. For
every 4 men under A there are 5 under C, for every 9 under B
there are 10 under D, and for every 2 under A there are 3 under
B. How many are under each ?
92. Find the cost of plastering the walls and ceiling of a room
27' 8"^x 13' 4" X 9' 2" at 22ct. per square yard, there being 3 windows
6' 9" X 4' 3" and 2 doors 7' 3" x 4' 3". How many cubic feet of plaster
would be required to plaster the room, the average thickness of the
plaster being haK an inch ?
93. Find the surface-area and the volume of a rectangular block
3' 9"X2' 4"xl' 3". What fraction of the block would be cut away
and by what fraction of itself would its surface be diminished were
2" each to be taken oiF its length, its breadth and its thickness ?
94. How long will it take to travel 13 j^ kilometres at the rate
of 11*9 miles in 1 hr. 45 min. ? How long will it require to travel
13^ miles at the rate of 11*9 kilometres in 1 hr. 45 min. ?
95. A line A is half as long again as B and B is one quarter as
long again as G. What fraction of the length of A is equal to J of
the length of Gf
96. A merchant sold i of his stock for | of the cost of the whole
stock ; J of the remainder at a gain of $80 ; ^ of what still remained
for its cost, $150 ; and the rest at a reduction of § of the cost. What
was his total gain ?
97. If 7 men, 15 women and 9 boys earn $8701-40 in a year (313
working-days) and if a woman's earnings are '6 of a man's and a boy's
are f of a woman's, what are the weekly learnings of a man, of a
woman and of a boy respectively ?
98. Goods are sold at a loss of 15 ^ on the cost. By what per-
centage of itself should the selling price be advanced to yield a profit
of 15 % on the cost ?
99. What rate of discount is equal to 5% off for cash on a purchase
on 90 days credit, reckoning |% for exchange with the discount ?
100. What must a merchant charge for goods that cost, him
$976*50 cash, in order that after giving 6 months credit, thus
involving the discount @ 7% of a note drawn at 90 days to yield the
cash price of the goods and a renewal note also drawn at 90 days
and discounted at 7%, he may obtain a profit of 15% on the cash
price paid by him for the goods ?
CHAPTER IV.
THE THREE HIGHER OPERATIONS.
INVOLUTION,
97. An Integral Power of any number is the product or
the quotient resulting from successive multiplications or successive
divisions by the number, the initial multiplicand or initial dividend
being in every case one. The power is said to be positive, if
it be formed by multiplications; negative, if formed by divisions.
In naming positive powers, the term posrfire is usually omitted.
The second positive power of a number is conmionly called the
square of the number; the third positive power, its cube ; and
the initial 1, neither multiplied nor divided, the Zeroth power.
The first (positive) power of 5 is 1 x 5 =5
The second power or square of 5 is 1x5x5 = 5x5=- 25
The third power or cube of 5 is 1x5x5x5 =-25x5 = 125
The fourth power of 5 is 1x5x5x5x5 ^^ 125 x 5=625
The fifth power of 5 is 1x5x5x5x5x5 = 025x5 = 3125
The first negative power of 5 is 1-^5 =5
The second negative power of 5 is 1 -^ 5 -^ 5 = it -h 5 = J^
The third negative power of 5 is 1 -^ 5 -f 5 -^ 5 = oV ~=" ^ =" t i s
The zeroth power of 5 is 1 =1
98. The base of a power is the number used as multiplier ((^r
as divisor) in forming the power.
99. The Exponent or Index of a power is the number which
expresses how often the base occurs as factor (multiplier or divisor)
in forming the power. The figures of an exponent are usually
made somewhat smaller than those of its base and are placed on
the right of the base and a little above it. The sign minus is
employed as a negative sign and is written before the exponents of
negative powers.
Instead of 1 x 5 x 5 or 5 x 5 we write 5" which is read " 5 square."
Here 5 is the base and 2 is the exponent.
Instead of 1x7x7x7x7 or 7x7x7x7, we write 7* which is
read "7 to the fourth," power being understood after f mirth. In
this example, 7 is the base and 4 is the exponent.
INVOLUTION.
87
II
Similarly 3' which is read *' 3 to the seventh " (power), represents
1x3x3x3x3x3x3x3; and 3"% read "3 to the negative fifth,"
represents l-^3-=-3-=-3-^3-^-3. In 3^, the base is 3 and the exponent
is 7 ; in 3"^ the base is 3 and the exponent is -5.
The exponent 1 is not usually expressed, the first positive power
of any number being simply the number itself. Hence when no
exponent is expressed, the exponent 1 is to be utiderstood.
100. The Degree of a power is the number of successive
multiplications (or successive divisions) by the base. The exponent
of any power is, therefore, the Iiidex of the Degree of the power.
The greater the nmnber of multiplications, or the less the number
of divisions by the base, the higher is the degree of the power ; the
fewer the multiplications, or the more immerous the divisions, the
lower is the degree. The phrases * higher power, ' ' lower power '
are very frequently used instead of the full phrases * power of higher
degree, ' ' power of lower degree. '
101. Involution is the operation of raising a given base to a
power of given degree. In other words. Involution is the operation
of finding the product of a given
number of factors each equal to
a given number.
Example 1. Find the first six
positive powers of 1 "4678 correct
in each case to five figures.
1| 1-4678
2-9356
4-4034
5-8712
7-3390
• 1«<^ Voyrev. ^pg^^ ^^^^ ^^^^^^
8-8068
10-2746
11-7424
13-2102
1-4678
58712
8807
1027
117
2-15443 . . . 2nd power.
3-16227 ... 3rd power.
4-4034
14678
8807
294
29
10
3rd power.
5-8712
88068
5871
147
73
12^
6-81291
8-8068
1 17424
1468
294
132
1
9-99999 .
5th
power.
6th power.
88 ARITHMETIC.
We have made 1-4678 the multiplicand in each multiplication,
because by so doing, only a single table of multiples is required.
(See Example 2, page 69.) The computations have been carried to
six figures in order to ensure accuracy in the fifth. The six powers,
each correct to five figures, are 1-4678, 2-1544, 3-1623, 4-6416,
6-8129 and 10 respectively.
Example 2. Find the value of
^^^471 ^47^ -47^ ;47^
1 ^Ix2^1x2x3^1x2x3x4"^lx2x3x4x5-
correct to 4 decimal places. (Work to 5 decimals.)
1^L47
•47 (a)
•188
329
2) -2209
-11045 . . . . . (h)
"^047
47
19
2
2
94
3
1-41
4
1-88
5
2-35
6
2-82
7
3-29
8
3-76
9
4-23
3)^5191
•OlTSO" (c)
1)047~
329
14^
4) 00813 1
•00203 (d) '47 (a)
•1105 (h)
173 (c)
•00094
1
, 20 (d)
5) 00095 2 (e)
T00019 (e) re
We square '47 and divide by 2 and thus obtain {h). We next
multiply '47 by (6), this gives one-half of the cube of '47 ; we
divide by 3 and obtain (c) the sixth part of the cube. We then
multiply "47 by (c) and divide by 4 to obtain {d) ; and multiply -47
by (<^) and divide by 5 to obtain (e). Finally we find the sum of 1,
(a), (6), (c), {d) and (e) correct to the fourth decimal.
INVOLUTION.
89
EXERCISE V.
Write the following products as powers —
1. 2x2x2. 5. -IX-lX-lX-l.
a. 3X3. 6. 2-3x2-3x2-3x2-3.x2-3.
3. 5x5x5x5. 7. ^y^x|.
4. 10x10x10x10x10. 8. tx|xtX|XfX|.
Write the following powers as products : —
9. 3^.
12. 255.
15. (2)^.
10. 123.
13. 2-5^
16. (H)^
11. 152.
J4. -255.
17. 2-x32x5x72.
Find the value of : —
18. 26.
28. •02«
38. (i)s.
39. (1)^.
19. 62.
29. l-02«.
20. 5*.
30. 492.
--'^
21. 45.
31. 4-92.
41. 23x3*.
22. 23752.
32. -492.
42. 2* X 33x53.
23. 58733.
33. 23-63.
43. 22x33x72.
24. 273^
34. 2-363.
44. 2^x53x7x113x132.
25. 27^
35. -02363.
45. (72)3.
26. 1V\
36, (f)2.
46. 5«-^52.
27. 1^
37. #.
47. 27x3*x53-^2*-^32-^53
Resolve the following numbers into their prime factors, expressing
the repetition of a factor by an index : —
48. 2520. 49. 70200. 50. 1024. 51. 11368. 52. 530712.
Find the value, correct to four significant figures, of : —
53. ^ 1111 1 1 1 1
54.
55.
56.
57.
^+
5^ +
53
■^^+
T^ +
5
i+ 57+ 58
+
5"*
1
-6 +
1
1
63
1
1
1
0
1 1
+
1
C9-
1
9 ~
1
92 +
1
93
1
- 9^ +
1
9-^
1
9^
1
-+9-
1
102
3
1
10*
^1x2
1
10«
3
'1
x4x5 1
X.2 X 3 ^ 10«
1
2 ~
1
3^-
1
1 1
1
'~ 7
X -
1 1
2^+ 9
1
J~
1
11^
1
1
+ 3
1 1
-.3^-3^
1
1
3'
1
7
1
X 3,+
1
9^
1
3
)'
90 ARITHMETIC.
2 - 3 - 2^"^ 5 " 25" 7 '^ 2" ' 9 '^ 2'-^ 11 " 2^
99. rk ~ o X T»T: +,^X ct'^ ~ fr ^ 07 ' fl '^ 0<l 1 1 ^ oil
59.
+ 5r'"3"''"5^+5"'''5^"y 5-+9'' 5'-'
rii 1 1 1 1 1 1 i\
11111
/ 3 i^ 0 7"'
60. 4'', -~ - -- X -^ + ^ X — - -^ X -^+ -,- X -^ ^
"70"'" 3 ^70^"^ 99"" 3 ^99-'-
«, Jill 111 1\1
^5 3 5^ 5 o • 7 5' J 239
.7 .72 .73 .74 .7.-
1 1x2 1x2x3 1x2x3x4 ■ 1x2x3x4x5
Ix3x3x4x5x
.7 .72 .73 .74 ■ .7',
«a 1 — + — — _| . ^ /ire-
1 1x2 1x2x3^1x2x3x4 1x2x3x4x5^
64.
r fi ^j_ 1 _j_l
^V"" l3l'^3x313+5x315 + 7x31-J
'^^'^ l49"''3~^^4p+5^74pJ ^^"^ liGl"^3^nGpJ )
65. 2xi23x !„-;+„ »... + z-^TTT-+,
31 ' 3x31- ' 5x31^ ■ 7x31'
+ 1^^ [4^3-^^- + ^] +^^^ [lk+3^J }•
102. Horner's Method.— The simplest and easiest method of
raising a given base to a power of given j^ositive integral degree, is
that which Avas adopted in Exawjde i, p 87. In that system the
successive positive integral jjowers Avere calculated one after another,
all the figures of the base being used in the multiplicand in each
multiplication. The calculations may however be conducted on a
INVOLUTION.
91
different plan. We may begin with a single figure of the base, (by
preference the first on the left-hand) and having raised this single-
digit number to the assigned degree, we may then proceed to build
up the required power step by step as we add figure by figure to the
base. This way of computing the positive integral powers of
numbers is known as the Method of Differences, and the best
arrangement of the process, that exhibited in the following
examples, is named Horner's Method. In ordinary cases of
involution, Horner's Method is neither so easy nor so simple as that
employed in Example 1, p. 87, but it has the advantage of being
applicable to whole classes of problems for which the other method
is t)f little or no use.
Example 1. Find the square of 3472.
Mark off into columns
the space set apart for
the calculation, the
number of columns
being greater by one
than the exponent of
the required power. At
the top of the left-hand
column write 1, this 1 is to be understood as repeated in every line
down this initial column. The other columns contain the actual
calculations and may be called the working-columns and numbered
from the left. At the top of each of these write a zero. This forms
the first or initial line of the calculation.
Multiply 1 in the initial column by 3, the left-hand digit of the
base 3472, and add the product to the zero in the first working-
column. Set the result which is 3, in the first working-column.
Multiply the 3 just set in the first working-column by the base-digit
3 and adding the product to the 0 in the second working-column, set
the result which is 9, in the second working-column. Begin again
with the initial 1, multiply 1 by the base-digit 3 and add the product
to the 3 in the first working-column. Set the result which is 6,
in the first working-column.
We have now instead of the initial line 1, 0, 0, the new line
1, 6, 9 ; the 6 being double the base-digit 3, and the nine being the
square of this 3. Prepare this line for the next step by placing one
0
0
3
900
= 302,
60
64
115600 .
= 3402,
680
687
12040900
= 34702,
6940
6942
12054784
= 34722.
92 ARITHMETIC.
zero after the 6 and two zeros after the 0, thus converting them into
60, the double of 30, and 900 the square of 30.
We now repeat the system of operations just described using 4,
the next figure of the base after 3 instead of 3 and the line 1, 60, 900
instead of the line 1, 0, 0; thus : —
1 X 4 + 60 = 64, which is to be placed in first working-column.
64 X 4 -f- 900 = 1156, to be placed in the second working-column.
1 X 4+64 = 68 to be placed in the first working-column.
We thus obtain a third line of calculation, 1, 68, 1156, the 68
being double the base 34 and 1156 being 34 2. Prepare this line for
the next step by placing one zero after 68 and two zeros after 1156,
thus converting them into 680 = 340 x 2, and 115600 = 340^.
Repeat this system of operations using 7, the next figure of the
base, as multiplier and 1, 680, 115600 as line of calculation, thus : —
1 X 7+680 = 687, in 1st working-column ;
687 X 7 + 115600 = 120409, in 2nd working-column ;
1x7+687 = 694, in 1st working-column.
This gives as fourth line of calculation, 1, 694, 120409, which,
preparatory for the next step, is converted into 1, 6940, 12040900.
Repeat the first course in this system of operations using as
multiplier, 2, the last figure of the base, and as line of calculation,
1, 6940, 12040900.
1 X 2+6940 = 6942, in 1st working-column.
6942 X 2 + 12040900=12054784, in 2nd working-column.
This completes the calculation of 3472 2.
Example 2. Find the cube of 2574.
= 203
= 2503
= 25703
= 25743
I.
II.
III.
0
0
0
2
4
8000
4
1200
60
65
1525
15625000
70
187500
750
757'
192799
16974593000
764
19814700
7710
7714
19845556
17053975224
(7718)
(19876428)
(7722)
L
INVOLUTION. 93
Here we are required to find a third power, we must therefore
have three working-columns. In the first set of operations we take
2, the left-hand digit of the base 2574, as multiplier and we have
1, 0, 0, 0 as initial line,
(a) 1 X 2-t-0=2, in column I.
2x2+0=4, in col. 11.
4 X 2+0 = 8, in col. III. Change 8 to 8000.
(6) 1x2+2=4, in col. I.
4 X 2+4 = 12, in col. II. Change 12 to 1200.
(c) 1 X 2+4=6, in col. I. Change 6 to 60.
We have now a new line of calculation, 1, 60, 1200, 8000. In
lis line, 60=20x3, 1200=202 x 3 and 8000 =20^.
Repeat the system of operations starting from this new line of
dculation and using as multiplier 5, the second figure of the base.
(cO 1x5 + 60=65. Col.^I.
65x5 + 1200=1525. Col. II.
1525x5 + 8000=
15625. Col. III. 15625000.
(c) 1x5 + 65 = 70. Col. I.
70x5 + 1525 = 1875. Col. II. 187500.
(/) 1x5 + 70=75. Col. I. 750.
We thus obtain a third line of calculation, 1, 750, 187500,
15625000, in which 750=250x3, 187500 = 250^ x 3, 15625000 = 250^.
Repeat the system of operations, starting from the third line of
calculation and using the third figure of the base as multiplier.
f{<j) 1x7 + 750=757.
757 X 7 + 187500= 192799.
192799x7 + 15625000=
(h) 1x7 + 757 = 764.
764x7 + 192799=198147.
(k) 1x7 + 764=771.
We thus obtain a fourth line of calculation, 1, 7710, 19814700,
16974593000, in which 7710 = 2570x3, 19814700 = 2570^x3,
16974593000=25703.
Startmg with this fourth line of calculation, repeat the first course
of the system of operations, employing as multiplier the fourth
figure of the base.
(I) 1x4+7710 = 7714. Col. I.
7714 X 4 + 19814700= 19845556. Col. II.
19845556 x 4 + 16974593000 =
17053975224. Col. III.
Col. I.
Col. II.
16974593.
Col. III.
Col. I.
Col. II.
Col. I.
94
ARITHMETIC.
17053975224 being the cube of 2574, we need go no farther in
this system of operations unless we wish to prepare for another step
in advance. This we have done in the example, having calculated
and recorded (within parentheses) the lines marked (m) and (ti)
respectively.
Example 3. Find 1*584893193^ correct to 9 significant figures.
The required power being the fifth, five working-columns will be
needed. Nine figures are required to be correct, the computation
must therefore be carried to at least eleven figures in the fifth
working-column. The decimal jjoint is omitted as unnecessary,
except in the last working-column.
0
0
0
0
0
1
1
1
1
1-(MMM)0
,1
2
3
4
5(KM)()
3
6
10(X)0
4
1000
50
56
1275
16375
131875
7-59.37500000
•5
60
1575
24250
2531250(MM)
65
1900
33750000
70
225000
750
758
231064
35598512
2816038096
9-8465804768
,8
766
237192
37496048
3116006480
774
243384
39443120
»
782
249640
\ \
X^R
\ ^ \
\ y \ \
250
395431
396431
397431
313182372
314768096
9-97185342.56
,4
3976^'
31508618
9-9970603200
,8
3978
31540442
3980
^
40^^
3154404
9-9998992836
,9
1 1
3154764
9-9999939264
,3
\ \ \ \
9-9999970812
9-9999999207
10-O000000152
,1
,9
,3
Hence 1-5848931935 = 10*00000002, correct to the last figure.
In the first set of operations, we begin with 1, 0, 0, 0, 0, 0 as the
initial line of calculation and we take as multiplier 1, the left-hand
digit of the base, 1 -584893193. We obtain therefrom, the new line
of calculation 1, 5-0, 10-00, 10-000, 5-0000^ I'OOOOO.
I
INVOLUTION. 95
In the second set of operations, we begin with this new line of
calculation and we take as multiplier '5, the second figure of the
base. We obtain therefrom as third line of calculation,
1, 7-50, 22-5000, 33-750000, 25-31250000, 7-5937500000,
in which it is worthy of notice that
7-5 = 1-5 X 5
22-50 = 1-52x10,
33-750 = 1-5-^x10
25-3125 = 1-5* X 5,
and 7-59375 = 1-5^
In the third set of operations, we begin with the line of
calculation last obtained and we take as multiplier '08, the third
figure of the base. We obtain therefrom as fourth line of calculation
1, 7-900, 24-964000, 39-443120000, 31-160064800000,
9-846580476800000 ;
in which it should be noticed that
7-90 = 1-58 X 6,
24-9640 = 1-582x10,
39-443120=1-583 X 10,
31-16006480 = 1-58* X 5,
and 9-8465804768=1-585.
The contracting begins at the figure 4 of the base; the uncontracted
fifth working-column would on passing from 8 to 4 of the base,
receive an extension of Jim figures, these are all omitted and as a
consequence the other working-columns must also be contracted by
five figures each. Allowing for their "extensions" this will require
the cancelling of the right-hand figure in the fourth working-column,
of two figures on the right in the third, of three figures on the right
in the second and of four figures on the right in the first working-
column. In like manner, on proceeding from 4 of the base to the 8
following it, from 8 to 9, from 9 to 3, from 3 to 1, &c. , the first four
working-columns are contracted at each step by cancelling 1, 2, 3,
and 4 figures respectively.
Example If. Find the value of 3658" -f 2574-.
In ExampU 2 p. 92, we have the value of 2574-^ and the working-
columns of the calculations prepared for any addition to the base.
Now 3658-2574=1084, therefore we may take advantage of the
ARITHMETIC.
Qiilculation of 2574^ to obtain the value of SeSS-"* by giving the base
2574 the successive increments 4, 80 and 1000.
1
7722
19876428
17053975224=2574-^
7726
19907332
17133604552 -2578=^
7730
19938252
7734
7814
20563372
18778674312 = 2658«
7894
21194892
7974
8974
30168892
48947566:312=36583
. •. 36583 + 25743 = 66001541536.
Example 6. Find the value of 4-8773 - 116.
Instead of the initial line 1, 0, 0, 0 employed in finding the value
of 4*877^, use the initial line 1, 0, 0,-116, the sign - before 116
denoting that the difference is to be taken between 116 and the
number carried from the second working-column to the third.
,4
1 0
0
-116
4 .
16
- 52-000
8
4800
120
128
5824
-5-408(XM)
136
6912(K)
1440
1447
701329
-•498697(-HM)
1454
71150700
14610
14617
71253019
+ •0(KH)74133
ce 4-877=5 -116=
: -000074133.
,•8
,7
EXERCISE VI.
Find the value of : —
10:
22, 232, 2352, 23572, 235782, 2357812.
43, 433, 437s, 43753, 437593.
122, 123 . 1272, 1273 ; 12782, 12783 ; 127862, 127863.
51-4492, 51-4493, 51-449*.
-1362, -1363^ -1364^ -1365^ -1360.
205-389-5-93.
170-5-53913.
3-14159'^ -306.
8-2413-8-2412+8-241-500.
{Take 1, - 1, 1,-500 as initial Urn.)
11-483 + 11-482-1554.
{laJce 1, Ij 0, - 1554 as initial line.)
EVOLUTION. 97
EVOLUTION.
103. The square root of a given number is that number whose
square is the given number.
Examples. 4 is the square of 2, . '. 2 is the square root of 4 ; 9 is
the square of 3, . *. 3 is the square root of 9 ; 100 is the square of
10, . •. 10 is the square root of 100.
The cube root of a given number is that number vi^hose cube is the
given number.
Examples. 8 is the cube of 2, . *. 2 is the cube root of 8 ; 125 is
the cube of 5, . '. 6 is the cube root of 125 ; 1000 is the cube of
10, . ♦. 10 is the cube root of 1000,
The fourth root, fifth root, sixth root of a
given number is that number whose fourth power, fifth power,
sixth power is the given number.
Examples. 81 is the fourth power of 3, . ". 3 is the fourth root of
81 ; -00032 is the fifth power of "2, . '. -2 is the fifth root of '00032.
The square root, cube root, fourth root, fifth root, of
a given number is therefore the base whose square, cube, fourth
power, fifth power, is the given number.
104. Evolution is the operation of finding any root of a given
number. It is therefore the operation of finding the base of which
a given number is the power of given degree.
105. In Involution, the base and the exponent (the index of the
degree of the power) are given and the power is to be determined
therefrom. In Evolution, on the other hand, the base is to be
determined, the power itself being given and also the exponent or
index of its degree. Evolution is therefore an inverse of Involution.
106. There are two ways of denoting Evolution. In the first or
older notation, the square root of a given number is denoted by
prefixing the symbol a/ to the given number ; the cube root is
denoted by prefixing %[ , the fourth root by prefixing Xf , the fifth
root by prefixing %/ , and all other roots are similarly denoted, viz. ,
by prefixing to the given number the root-symbol J combined with
an index number indicating which root is to be taken.
Examples. ;y/64 denotes the square root of 64; ^64 denotes the
cube root of 64 ; V^l denotes the fourth root of 81 ; and %f^
denotes the fifth root of ^.
G
98 ARITHMETIC.
The second or modern notation for evolution employs fractional
exponents to denote the roots of numbers. The exponent of the
square root is |, that of the cube root is ^, that of the fourth root is
J, and, generally, the exponent of any root is the reciprocal of the
exponent of the corresponding power.
Examples. 49^ denotes the square root of 49 ; -125^ denotes the
. 1
cube root of "125 ; (ttjVj) denotes the tenth root of Y^^n 5 ^^^
81~* denotes the reciprocal of the fourth root of 81.
[107. The root-symbol ^ is merely a variant form of the letter
r. The employment of an index number with ^J is of -comparatively
recent date , the old notation was ,y/q for the square root, ijc for
the cube root, V^^ for the fourth root, ^^/cqfor the fifth root, /^/cc
for the sixth root and so on for other roots. After this came the
notation x/[6] for the sixth root, >/['7] ^o^ ^he seventh root,, and a
similar notation for other roots. Later still came the notation sj^^
fj"^ ^ &c. ; from this form our present notation is derived.
The exponential notation is as much superior to the root-symbol
notation as Arabic is to Roman notation and excels it very much in
the same respects. As a notation merely of record, the root-symbol
notation is perhaps quite equal to the exponential but the latter
notation by its very forms suggests calculation by exponents, (see
§143,) and the index laws and the many theorems following there-
from ; of these the root-symbol notation gives not the slightest hint,
tending rather to hide them from sight or make them obscure.]
EXERCISE VII.
Prove the following statements of equality : —
1.
252 = 5.
7.
6-25' =2-5.
13.
/16J_ 2
^81' 3 '
3.
125* = 5.
§.
49002 = 70.
14.
133 > 3 -6.
3.
4.
5.
16^ = 2.
81^=3.
1000» = 19.
9.
10.
11.
1-728-^ = 1-2.
-008^= -2.
0-001^^=0-1.
15.
16.
ir
IP < 2-224.
1-16^ > 1-05.
•41^ > -8.
6.
100000^ = 10.
13.
0-000015=0-l
1§.
25 < 1-1487.
II
EVOLUTION. 99
108. Evolution being an inverse of Involution a calculation in
the former will be merely the reversal or undoing of a calculation
in the latter. We require therefore a reversible process of
involution and such a reversible process we have in Horner's
Method. In it the required power is built up by successive
increments as additions are made to the base or as it is enlarged
figure by figure. To reverse this process we must withdraw the
successive increments of the direct process, and since the increments
may be added in any order (compare Example 2, p. 92 and Example
4, p. 95), they may also be withdrawn in any order. At the
beginning of the calculation, the only digit of the root, the unknown
base, of which we can be sure, is the first digit on the left,
therefore we commence by raising this digit to the degree of the
power which the given number is to be of the required root, and
subtracting this power from the given number.
In determining this first digit of the root, it must be remembered
that each figure subsequently added to the root or base gives two
additional figures in the square of that root, three in the cube, four
in the fourth power^ five in the fifth power, and
that for each figure to the right of the decimal point in the root there
will be two to the right in the square of that root, three to the
right in the cube, four to the right in the fourth power, five to the
right in the fifth power, Hence, in preparing to
extract any root of a number, we begin at the decimal point and
mark off the figures left and right in pairs in case of the square root,
in sets of three in the case of the cube root, in sets of four in the case of
the fourth root, in sets of five in the case of the fifth root,
This done, the set or period on the left will determine the first
figure on the left of the root.
109. The following Table will assist in determining the first
root-digit in cases of square root and cube root: —
3, 4, 5, 6, 7, 8, 9.
9, 16, 25, 36, 49, 64, 81.
27, 64, 125, 216, 343, 512, 729.
Root. -1, -2, -3, -4, -5, -6, '7, '8, "9.
Scpmre. 'OJ, '04, -09, -16, 25, -36, -49, '64, '81.
Ouhe, -001, -008, -027, '064, 125, '216, -343, '512, '729.
Root.
1,
2,
Square.
1,
4,
Cube.
1,
8,
100 ARITHMETIC.
Example 1. Find the square root of 5476.
The root is to be squared, hence two working columns will be
required. As the root is found its square is to be withdrawn or
subtracted from 6476, therefore we begin the second working
column with - 5476, the prefixed - indicating the subtraction of
the square of the root. The initial line will thus be 1, 0, - 5476.
1 0 - 5476(74 sg. r^.
7 49
1 140 -576
144 576
Mark off the figures of 5476 in pairs counting in this case from
the right-hand figure, there being no digits on the right of the
decimal point in the given number. The marking off may be done
by placing a point or dot over the right-hand figure of each period
except in the case of the period immediately on the left of the
decimal point, in which the decimal point serves as the marking ofi"
or distinguishing point. This period is named the zeroth period
and the others are numbered from it as jfirst, second, third, ,
positive or negative, (left or right,) as the case may be.
The first or left-hand period is 54. By the table of squares given
above 54>72 but <82,
. '. the square root of 54 > 7 but < 8,
. •. the square root of 5476 > 70 but < 80,
. '. the first figure of the root is 7.
Write 7 in the place set apart for the root and then proceed with the
calculation exactly as if the problem were to subtract 5476 from the
square of a given base whose first digit is 7. This gives as the
second line of calculation 1, 140, — 576.
To obtain a "trial digit " for the second figure of the root, divide
the 676 in the last column by the 140 in the next preceding column.
The ' quotient ' is 4. Write 4 as second figure in the root and
proceed as in involution to find the value of 74 ^ — 5476. There is
no 'remainder' therefore 74 is the square root of 5476.
Example 2. Find the square root of 12054784.
(Compare the calculation with that of Example i, page 91, noting
that there the square is built up, but that here the process is
virtually the opposite.)
EVOLUTION.
101
12054784(3472, sq. rt.
9
(0)
3
(60)
64
(680)
687
(6940) - 13884
6942 13884
The first period, 12, determines 3 as the first digit of the root.
From 12 substract 3^ and to the remainder, 3, 'bringing down' 06,
the next period o£ the given number, and complete the formation of
the second line of calculation 1, 60, -305. Divide 305 by 60. The
quotient 5 is found on trial to be too large but 4 on trial proves to be
the right digit. Proceed as in involution to form the third line of
calculation which will be found to be 1, 680, - 4947. Dividing 4947
by 680 gives 7 for trial as next digit of the root. On trial 7 is found to
be the right digit. Continuing this process it will be found that 2 is
the fourth digit of the root and that 3472 = 12054784^
In subsequent calculations we shall omit the initial column and,
in general, the minus signs in the last column and the lines in the
working-column corresponding to those enclosed in parentheses in
column two of the above example. If computers' subtraction be
employed the subtrahends need not be recorded in the last column.
Had all these omissions been made in the preceding example it
would have appeared thus : —
3472.
Example 3. Find the value of 204'081632 correct to six figures.
In the square of any number, if there be figures on the right of
the decimal point, the number of such figures is even, but in
204*08163 the number of figures on the right of the decimal point is
odd viz. 5. The number of ' decimal figures ' must be made even
and this is done by affixing a zero to the given number making it
204 081630.
3
12054784
64
305
687
4947
6942
13884
102
ARITHMETIC.
1
24
282
2848
204 -081630 2 = 14 -2857 +
1
104
96
808
564
24416
22784
28565
163230
* 142825
285707 2040500
1999949 .
40551
After 'bringing down ' all the periods in 204*081630, we find we
have only five figures in the root and six figures are required. To
obtain the additional root-figure we imagine a period of zeros, in
this case two zeros, affixed to the given number and ' bring them
down, ' we thus virtually extract the square root of 204 ■08163000.
Example 4- Find the value of 10^ correct to ten figures.
61
626
6322
63242
632447
63245^
10^
_9
100
61
=3-162277660,2
3900
3756
14400
12644
175600
126484
4911600
4427129
"484471
442715
41756
37950
3806
3792
14
EVOLUTION.
103
Having found six figures of the root by the ordinary uncontracted
process we may find four or five figures more by contracting the
process in exactly the same way as we contract in involution. In
this example we divide 484471 by 63245 by contracted division,
knowing that the figures rejected from the divisor will not affect
the quotient figures, here root-figures, till the divisor is reduced, to
one or at most to two figures. The root thus found is correct to
eleven figures.
The general rule is that when the number of figures obtained by
the uncontracted process is one more than half the number of
figures required in the square root, than a third of the number
required in the cube root, than a quarter of the number required
in the fourth root, than a fifth of the number required in the fifth
root, the rest of the figures may be obtained by
contracted operations.
Example 5. Find the cube root of 1*25 correct to ten figures.
0
1
2
307
314
3217
3224
0
1
1-260 1077217345
1
3
32149
•250000
225043
34347
3457219
3479787
3480433
348108'
24957000
24200533
756467
696087
60380
34811
25569
24368
1201
1044
157
139
18
■ Example 5 page 95 is virtually an example of the extraction of
the cube root of 116 correct to six figures and if each line in the
fifth working-column of Example 3 page 94 be subtracted from 10,
the example will exhibit the operation of extracting the fifth root
of 10 to ten figures.
104 ARITHMETIC.
EXERCISE VIII.
Find the square root of : —
1. 576. 3. 103041. 5. 2321-3124.
2. 1849. 4. 10-3041. 6. -0050367409.
Find the cube root of : —
7. 389017. 9. 700227072. 11. 6*199083253.
8. 814780504. 10. 700227*072. 12. -000160103007.
Find, correct to six significant figures, the value of : —
13. 22. 19. 402. 25. 123456*.
14. 202. 20. 4000^. 26. 123-456^.
15. 2002. 21. -42. 27. 2^.
16. 20002. 22. 74492. 28. ^oi
17. -22. 23. 10002. 29. 200*.
18. -022. 24. 609800-1922. 30. •2401*.
31. Find j^3 correct to six significant figures and hence prove
that 2- V3 is ihe reciprocal of 2+ v' 3 to six figures.
32. Prove to six significant figures that ^3x^^5 = ^/15 and that
the product of /^6+ ^3 and ^^5- V^ is 2.
110. The square of a given fraction has for numerator the square
of the numerator of the given fraction and for denominator the
square of the denominator of the given fraction. Hence inversely
the square root of a given fraction has for numerator the square
root of the numerator of the given fraction and for denominator
the square root of the denominator of the given fraction.
The cube, fourth power, fifth power of a given fraction
has for numerator the cube, f ourtl^power, fifth power of
the numerator of the given fraction and for denominator the cube,
fourth power, fifth power, of the denominator of the
cfiven power. Hence iilversely the cube root, fourth root, fifth root,'
, of a given fraction has for numerator the cube root,
fourth root, fifth root, of the numerator of the
given fraction, and for denominator the cube root, fourth root,
fifth root, of the denominator of the given fraction.
EVOLUTION.
105
I
Examples.
[I]
22
iiJ 112 121 l
49
121
2
492
1212
33_27_, .-. r 27 1 ^_ 27 ^_ 3
83 ~ 512 1 512 J ~g^23 ^
111. If in extracting any root of a given fraction, it is found
that the root of the denominator cannot be obtained exactly, the
fraction may be reduced to decimal form and the root extracted to
any required degree of accuracy. Another method is to multiply
both terms of the given fraction by any factor that will make the
denominator an exact power of degree the reciprocal of the degree
of the root ; the root of the resulting fraction is then extracted.
This process is called rationalizing the denominator. It can often
be used with advantage to obtain a rapid approximation to a
required root of a small number.
Example 1. Extract the square root of ^ correct to four figures.
[A]*=-
363636362 = -6030+-
2°
11x11
X 1
^442
~11
6-6333-
11
= -6030+.
Example 2. Find the value of Lr^
correct to six figures.
51
r2J
li2J
= •41666667 ='645497 +
= f 5 x3^ ^^152^
lr2x3J ~ 6 ~
3-872983 +
6
•645497 + .
3° -
f 51
1 12 J
2^ flS
1 36
r42-
1-6"^
11^
1 12 J
and
6x4x2
< error <
4
6 42x2-1
(i).
42x2-1
6x4x2'
31
6x8'
and
106 ARITHMETIC.
31 1 ,..,
< error < - — - x — — — ; (ii).
6x8x31x2 6x8 31^x2-1
1
.-. r 5 V 312x2-1 1921 ,
— < = , and
U2J 6x8x31x2 6x8x62
1921 1 .....
< error < - — - — — X :r;r^- ; ' \S^^h
6x8x62x1921x2 6x8x62 1921-^x2-1
19212x2-1 7380481
. 1
^^ 6 X 8 X 62 X 3842 x 7380481 x 2
7380481 1 .. .
6x8x62x3842 7380481^x2-1 ^ ^
In this 3° method we first rationalize the denominator and then
we try successively 15 xl^, 15x2^, 15x3^, 15x4^, till
we find a product that differs but little from a square number.
Such a product is 15 x 1^ which differs from 4^ by 1. We therefore
write — in the form from which we obtain
I36J I 62 J
at once — as a first approximation to the required root with an
6
error in excess somewhat greater than , as may be proved
6x4x2
4 1 4 142x2 — 1
by squaring — We next take =
^ ^ ^6 6x4x2 6 6x4x2 6x4x2
31
= as a second approximation in excess with an error
6x8
somewhat greater than — as may be proved by
^ 6x8x31x2 ^ ^
31 1 mu- • 31 1
squarmg • This gives
^ "'6x8 6x8x31x2 '' 6x8 6x8x31x2
312x2 — 1 1921
as a third approximation in excess
6x8x31x2 6x8x62
with an error somewhat greater than
6 X 8 X 62 X 1921 x 2
AX. ■ 1921 1
as may be proved by squaring _ _ - — — .
^ ^ ^ ^ 6x8x62 6x8x62x1921x2
ii
This gives
1921
EVOLUTION.
1
107
19212x2-1
6x8x62 6x8x62x 1921 x2 6x8x62x 1921 x 2
7380481
as a fourth approximation to the required root,
o x o X oZ X o842
The first approximation is correct to one decimal place, the second
I ■ is correct to three decimal places ; the third, to six decimal places ;
and the fourth, to fourteen decimal places.
It should be noticed that
• 15 42 — 1
(1.) comes from _ = ,
36 62 '
(ii.) comes from
(iii.) comes from
(iv.) comes from
1515x82312-1
36~36x82~62x82'
15 15 X 8-2 X 622 19212
and
36 36 X 82 X 622 ^2 ^ 82 x 622
15 15 X 82 X 622 X 38422 73804812 - 1
36 36 X 82 X 622 ^ 38422 62 x 82 x 622 x 3842-
r 2 ^ 2
Example 3. Find an approximate value of — ,'
22
11 11x11
f 212 5
112
riJ "fi'^St^-^
47
f 21 2 52x2-3
lilJ <
11x5x2 11x10
9
< ,rror <_x— ^33'
and
L^]
11 X 10 X 47 X 2
472x2-9
47 9
< error < ■- — — - x -— — - — - >
11x10 47^x2-9
4409
and
11x10x47x2 11x10x94
81 4409
< error < jj^iq -94 ^ 44092 x 2 - 81 '
81
11 X 10 X 94 X 4409 x 2
^2_y2- 44092x2-81
lllj '
38878481
11x10x94x4409x2 11x10x94x8818
The next correction would be
6561
11 X 10 X 94 X 8818 x 38878481 x 2*
the numerator 6561 being 81 2. From this example we may see
that if possible a numerator should be found that difiers from a
square number by but 1 or 2. This might easily have been done in
108 akithmj:tic.
this case by selecting 22 xS^ as numerator to work from. The
calculation would then have appeared as follows : —
2^_^_^ 32^198^142 + 2
il~ 112 "112^3^-332" 332-
f 2^^ 14
LnJ ^33'
and
2 1 14 2 14
> error > t^x—— — - — :r=7^x
33x14x2 33x14 33 142x2 + 2 33 I42 + I
rlV<y_^±l = _^, and
111 J 33x14
1
< error < ^--—-: x
33x14x197x2 33x14 1972x2-1
.-. f2^1^ 1972x2-1 ^ 77617
LllJ ^33x14x197x2 33x14x394 *^'
Here the first approximation is in defect, we therefore add the
first correction. This correction is in excess, hence the second
approximation is in defect, further since the numerator of the first
correction was reduced to 1, the numerators of all subsequent
corrections will also be 1. In fact the second approximation is
obtainable from the equality
2 ^198 198x142 1972-1
11-332 332 X 142-332 X 142 ■
Example 4- Find approximately the square root of 45.
45 = 72-4,
JL 2 2
. •. 452 < 7 and — < error < 7 x r— — - ;
7 7^-2
. .'1 72 - 2 47 ,2 ' 47 2
• • 45-<— — = -, and^^^< error <- x^^^-^;
.-. 45^<1?1Z_2^^207 nd^
7x47 7X47'
2 2207 2
< error < - — -- x
7x47 X 2207 7 x 47 22072 -2 '
45^<_2^Z!-Zl_=._^7^84^
7X47X2207 7x47x2207
2 4870847
< error <
7 X 47 x 2207 X 4870847 ' 7 X 47 X 2207 4870847 2 - 2
The terms of the errors and the corrections are reduced each
time by division by the common factor 2.
EVOI.UTION.
109
II
In any case in which only the result of the computation is
required, the limits of error need not be calculated for each
successive approximation ; it will be sufficient to examine the
superior limit to the error of the last approximation.
Example 5. Find the square root of 111 correct to six figures.
111 = 112-10, .'. the first three approximations to 111^ are
,., ,, .... 112-5 116 ,..., 1162x2-25 26887 ^•^'^^•7
(1), 11 ; (n), -^= -; (m), ^^^ =lTx-2^2 = '^ '''^ -
The error is less than 10*54 x
625
10 X
62^
26887^x2-625 25000^x2
= -000005. The error being thus less than 5 in the sixth decimal
place, the division of 26887 by 11 x 232 might have been carried
one step further ; the quotient is 10*535658 + , and allowing for the
error we obtain 1112 = 10*53565 + , correct to seven figures.
•7
Example 6. Find the cube root of _ correct to five figures.
r7^J
Li2J
= •583333333^*
83555
\l2) ( 2-2 X 3 J 1 23 X 33 S
126^
6
5 01330
6
3°
_7
12
= *83555-
7 _7x2x32_126_53 + l
22x3" 2-3x33 ~63 ~ 63
5,1
> error.
VkV
5 A ^
1_
12
7 126x753
12"" 63x76_3
i 12
1
53x3 + 1 376
_5
■^6 6x52x3 6x52x3 6x75
•835556
53156250 3763-1126
3 ^76
"^6x75
63 X 753
and
63x753
1126
12 J
6 X 75 X 3762 x 3
1126 3763x3-1126
376 ^
6x75 6x75x3762x3 6x75x3762x3
159471002
6x75x424128^
•83554965 +
(0-
(ii).
{a).
(iii).
110 ARITHMETIC.
(i) is correct to two figures ; (ii) is correct to five figures, the final 6
being rejected without augmenting the preceding 5, on account of
the sign < and the correction (a).
112. The process of forming a series of convergents to a given
fraction which was exemplified in §77 may be applied to obtain a
series of convergents to any root of a number.
Example 1. Find convergents to the square root of 6.
2 3
22 < 6 < 32, . •. we take — as the inferior and — as the superior
3 + 2 5
initial convergent of the series. The next convergent is tp -^ = -^ 5
j 5 ) 2 5 3
and since ) o" C ^ ^' "^ ^^ written in a line beside — higher than the
1-^2 5 + 2 7 . i 7 r . 7
line of — . ihe next convergent is - — 7 = ^ ' ) "^ ( ^^^ • "• o"
2
is written in the lower line, the line of - • The next convergent
. 7 + 5 12 < 12 ) 2 ^ . 12 . . . , ,
IS o~r^=~^ ; ) ~^ \ <"5 • -TT IS written m the lower line.
3 + 2 5 ' ( 5 ) 5
The process thus far followed is continued until there is obtained a
sufticiently close approximation to the required root.
1 A 27 49 267 485
1' 2 ll' 20' 109' 198'
1 7^12 17 22 71120 169218
T' 3' 5"' 7' "9' 29' 49"' 69' 89'
The principal convergents, as far as the series has been formed
^, , 2 5 22 49 218 485
are therefore-, -, -, -, -^, -,
these being alternately less and greater than 62, It is worthy of
3
notice that beginning with the superior initial — » there are
throughout the whole series two superior convergents followed by
four inferior convergents, followed in their turn by two superior
convergents. This enables us to form with great ease and rapidity
any required number of principal convergents, after the first two
are known. Thus, keeping to numerators alone, 5x4+2 = 22,
22x2+5 = 49, 49x4+22 = 218, 218x2+49 = 485. The denomina-
tors may be similarly computed, thus the denominator following
next after 198 is 198 X 4+89 = 881, The error committed in taking
EVOLUTION.
Ill
485
198^
2-449495
for 62
is less than
198x881 160000
•000007,
hence 62 = 2*44949 — , correct to six figures.
Example 2. Form a series of convergents to the cube root of 6.
13<6<2^, .". we take \ and f as initial convergents, and form
from them a series in the usual way, cubing each term to test whether
it is a superior or an inferior convergent.
2
1>
1 3 5 J S
T> ^' T?' 4» 5>
¥,n,
l£ &ft i(L9 129 149
;8> 49) 60 » 7T » 8^ >
1-69
93 )
m-
4f^ = 1-8171206 + , which is the cube root of 6 to eight figures.
The next two principal convergents to 6-^ are
467x508
149^^^467x509 + 149
I
257x508+ 82 257x509+ 82
113. This is the oldest and perhaps the simplest systematic
process for obtaining a series of approximations converging to the
value of any required r»ot of a given number. It is subject
however to the disadvantage of being extremely tedious and
laborious except where the law of immediate formation of the
successive principal convergents is known, in which case it becomes
an easy and rapid method of evolution. The following examples
exhibit one method of directly computing the successive principal
convergents to the square root of a given number.
Example 1. Find approximately the square root of 31.
31^=5 +
&c.
The first column always consists of 0, 1 and the greatest integer
whose square is less than the given number. In this example the
first column will therefore consist of 0, 1 and 5.
Let a, h and c denote the numbers in any column ; a denoting the
number in the 1st row ; 6, the number in the 2nd row ; and c, the
number in the 3rd row. Let A, B and 0 denote the corresponding
numbers in the next following column. The successive columns are
formed each from the column next before it, thus ; —
0
1
5
5
6
1
14 5
5 3 2
13 5
5
3
3
4 15
5 6 1
1 1 10
5
6
1
1 4 &c.
5 3 &c.
1 3 &c.
'Quotients.
Convergei
its.
5 1
T» 0> T» Tj
1
3 5
3
^1,
1 1 10
Iff, H^i Hm
112
ARITHMETIC.
A = be — a : B :
N~A'^
C = integral part of
I+A
b - ^ B
in which N denotes the number whose square root is required, in
this example 31, and I denotes the integral part of the scjuare root
of N, in this example 5.
Thus in the first column a = 0, b = l, c = 5;
. '. the second column is
In the second column a
the third column is
r A=
=
1x5-0
= 5.
B =
C =
-Int.
31-5
1
r5+5i
I 6 J
= 6.
= 1.
= 5,
rA=
b =
6 and c =
6x1-5
1;
=1.
B =
31 -l'
6
= 5.
C =
Int.
p + ll
I 6 J
= 1.
This process of forming each column from the preceding column
is continued until the second column occurs again, after which the
several columns are repeated in the same order.
The principal convergents to 31 ^ are obtained from the initials
^ and ^, by employing as ' quotients ' the numbers in the third row,
viz., 5, 1, 1, 3, 5, 3, 1, 1, 10, 1, 1, 3, 5, 3, 1, 1, 10, 1, 1, 3, &c.
Exam,ple 2. Find a series of principal convergents to 6".
The greatest integer whose square is less than 6 is 2, . '. the first
column is 0, 1, 2. The succeeding columns are formed each from
the immediately preceding column, thus : —
A = bc — a.
B:
6-A2
C = integral part of
2 + A
B
62 = 2 +
0
1
2
Quotients.
Convergents.
Sz.c.
4,
22
9'
2,
49
20'
4,
218
89'
2,
485
198'
4,
2158
881'
&c.-
Compare with Example 1, § 112, p. 110.
EVOLUTION.
113
EXERCISE IX.
Find, correct to six significant figures, the value of : —
f31*
1.
f 1 1 ^
2.
'4V^
3.
r256 1
1 2401 J
4.
5.
ri7v
I25J •
6.
f 9 ^ 2
I32.
7.
16^ 2-
§.
1 10 J
9.
8§.
rv V
30.
31.
32.
33.
10.
11.
12.
13.
14.
15.
16.
17.
1§.
8
ri2i 2
1175J •
r28] '
I45J •
19. 352.
20. 372.
21. 72.
3i
52.
152.
172.
242.
262.
22.
lli
23.
632.
24.
77I
25. 972.
given
f9lJ
i-20J '
t^24.
1
52,
frjven — = =
77 ^ 77x402
121 "112x402
45 45 X 242
26. 16012.
27. 24002.
3512-1
4402
1612-1
given
given 11 =
given 6 =
given 5 =
20 100 102x242 2402
13^ 78.^78x62 532-1
24~ 144~ 122 X 62 - 722 *
11x32 102-1
32 32
6 X 202 492 _ 1
202 202
5 X 42 92-1
42
42
114 ARITHMETIC.
34. 2*. given 2 = ^2^ Jl±l,
ako2 = 2_^ = 17-^\ also 2J~>'-^='-l^.
122 122 29- 292
I 343 ) I2I6J I 3 J
36. {1^]K 38. (^]\ 40.
U913J 1 8000 J
r 5 v^"
114. Two given quantities are commensurable if there be an
integral multiple of one of them which is also an integral multiple
of the other.
For example, let there be two lines A.
A and B of lengths such that a third B.
line which is five times the length of the line A is twelve times the
length of the line B. Divide this third line into 6 x 12 = 60 equal
parts, then the length of any one of these parts will be ^\j of five
times the length of the line A,, i. e., the sixtieth part of the third
line will be -^^ of the line A or be contained twelve times in the
line A. But the length of the same part will be -^^ of twelve
times the length of the line By i. e., the sixtieth part of the third
line will be | of the line JS or be contained five times in the line B.
Hence a sixtieth part of the third line will measure both the line A
and the line B, i. e. , the lines A and B have a common measure or
are commensurable.
Expressed in symbols the preceding example is : —
If 5A = 12B
5 A 12 B
5x12
A
12
5)
^2'
B
5
12
I 5".
5
fBl
l5J
and ^
. '. A and B are commensurable, -1 of B being a common measure
or common unit.
EVOLUTION. 115
115. If either of two commensurable quantities be expressed in
terms of the other as unit, the number expressing their ratio or
relative magnitude will be an integer, a fraction with integral
terms or with terms reducible to integers, or a mixed number
consisting in part of an integer and in part of an integral-termed
fraction. For this reason integers, integral-termed fractions and
integral-termed mixed numbers, whether decimally expressed or
otherwise, are called commensurable or rational numbers.
116. Two given quantities are incommensurable if no integral
multiple of one of them is an integral multiple of the other.
If either of two incommensurable quantities of the same kind be
expressed in terms of the other as unit, the number expressing
their ratio or relative magnitude will not be expressible exactly by
any integer, integral-termed fraction or integral-termed mixed
number whatever. For this reason a number which cannot be
expressed exactly by any integer or any fraction or mixed number
with integral terms is called an incommensurable or irrational
number.
If the length of the diagonal of a square be expressed in terms of
the length of a side of the square as unit, the number expressing
their ratio or relative magnitude will be the square root of 2.
Now, in extracting the square root of 2, whether as a decimal
number or as a fraction, there is always a remainder i. e., it is
impossible to find a rational or commensurable number of which
the square is exactly 2, Hence 2 ^ is an incommensurable number,
and the lengths of the diagonal and the side of the same square are
relatively incommensurable quantities.
Other examples of incommensurable numbers are 3'^, 5^, lO'-^, 2^
5*, 9% 100^, 2S 4^, 100*, sK
117. Every number formed by combining a definite number of
ones (or of integers) by means of the operations of addition,"
subtraction, multiplication and division, and of these only, is
reducible to an integer or to a fraction, proper or improper, with
integral terms, i. e., every number so formed is a commensurable
number, hence no incommensurable number can he expressed by
combining a definite number of commensurable numbers by additions,
subtractions^ midtiplications amd divisions and these only.
116 ARITHMETIC.
118. Incommensurable numbers which can be formed from a
definite number of commensurable numbers combined by means of
the operations of addition, subtraction, multiplication, division,
involution and evolution, are sometimes called surd numbers or
surds to distinguish them from incommensurable numbers which
cannot be so formed. The latter are called transcendental numbers.
Examples. 2 2 , 3*, 1 + 2^3 + 2^ - 4^, 5'^ x 6^ 8^ ^ 4^, are surds.
The ratio of the circumference of a circle to its diameter is a
transcendental number as also is the exponent which expresses the
degree of the power which 20 is of 10. (See Logarithms.)
119. Involution is the operation of raising a given base to a
power of given degree. In the examples of this operation hitherto
considered, the exponent or index of degree of the power has been
either an integer or, in the case of roots, the reciprocal of an
integer. But no such restriction need be laid on the values of
exponents ; these may be integral or fractional, commensurable or
incommensurable, positive or negative, provided that the terms
degree and power be interpreted in accordance with this extension
and provided that the laws laid down for operating upon and with
these generalized powers are consistent with each other and include
as particular or special cases, the laws governing operations upon
and with powers of integral degrees and their corresponding roots.
These laws which thus constitute the Fundamental Theorems of
Involution and Evolution are ; —
XXIV. If equals he raised to equal degrees, (have equal expotients),
the potvers are equcd.
(Equal-degreed roots of equals are equal. )
XXV. Equal powers of equals are of eqmd degree, (have equal
exponents. )
(Equal roots of equals are of equal degree.)
XXVI. Raising the base to any degree raises the pjower to the power
of itself of that degree.
(Extracting any root of the base extracts the equal-degreed root
of the power. )
XXVII. Midtiplying the exponent by any number raises the power
to a poimr of itself of degree denoted by the midtiplier.
(Dividing the exponent by any number reduces the power to it«
root of degree denoted by the reciprocal of the divisor. )
EVOLUTION.
117
XXVIII. The product of two or more powers of the same base is
that power of the base which has for exponeid the aggregate of the
exprnieitts of the factors.
(The quotient of two powers of the same base is that power of the
base which has for exponent, the remainder obtained by subtracting
the exponent of the divisor from the exponent of the dividend.)
^L XXIX. To multiply by a negative power of any base divide by the
^ ^p reciprocal of the power j i. e. , divide by the power of correspondi7ig
positive degree.
(To divide by a negative power of any base, multiply by the
reciprocal of the power.)
120. The Fundamental Theorem connecting the operations of
multiplication and division with the operations of involution and
evolution is, —
XXX. liaising the several factors of a product to any degree raises
the product to that degree.
(Reducing the several factors of a product to their roots of a
given degree reduces the product to its root of the same degree.)
Examples of Theorem XXVI.
3
1. Let 2 be the base and 2 be the power, and let the base be
I squared, then will
2 3 3 2
/2 ) =(2 ) ;
for (2 )■ =(2 X 2) X (2 X 2) X (2 X 2) = (2 X 2 X 2) X (2 X 2 X 2) = (2^)^
2. Let 729 be the base and 729^ be the power, and let the scpiare
root of the base be extracted, then will
I
for
and
3.
for
and
4.
for
and
5.
(7292)^ = (729^)^
729^ =27
27^ =3=
and
729^=9
(10^)1
(10^)^
10^ = 3-16228
(3-16228 -)K
L 2 2 1
(8-^) =(8)5
8^=2 and 82
2^=4=64^.
1 -3 -3 1
(124) = (12 )4.
- and 10' =1-58489 + ,
1-25893- =(1-58489 + )'.
64
118 ARITHMETIC.
6. (24 "5)"^ = (24"^)~3.
Examples of Theorem XXVII.
3
1. Let 2 be the i)ower and let the exponent be multiplied by 2,
then will
3X2 3 2 6 3 2
2 =(2 ) or 2 =(2 )
for "2'' =2 X 2 X 2 X 2 X 2 X 2 = (2 X 2 X 2) X (2 X 2 X 2) = (2^X •
3. Let the exponent of 729» be multiplied by |, then will
729^""^ - (729^)2 , or 729« = (7293)^
3. 10^'' 5^(102)^ or 101^5-^.. (io-2y.
4.. 8-i =(8«) or 8--^> = (8-0 .
_ov<L -3 1 _:i -3 1.
5. 6 ^''i = (6 )'^ or 6 + = (6 y.
Examples of Theorems XXVIII atul XXIX.
2 3 2+3 5
1. 7 x7 =7 =7 , ^
for 7 x7 =(7x7)x(7x7x7) = 7 .
2. 642 X 64^' = 642+» = 64«,
for 642=8and 64» = 4.
and 8 X 4 = 32 := 2 ^= (64«/ = 64^.
3. 7^ X 7^=7^^ = 71^-7 X 7T^.
5 2 5-2 3
4. 3 -^3 =3 =3 ,
for 3^ 4-3^ =(3 X 3 X 3 X 3 X 3)-^(3 x 3) = 3 x3 x 3 = 3" .
21 21 jr
5 -3-5 3 5-3
6. 5 x5 =5 4-5 =5 =52.
6 -3
for 5 x6 =(5x6x5 X 5x5) X (l-^5-^5-^5)
= (5x5x5x5 x5)-^(5x5x5) = 5 -^5 .
7. 642 X 64~3 = 64"2 -v-64-^ =642"» = 64«.
for 642 X 64~» =8 X | =8-f-4=642>64i
4 _2 4 2 4_2 2_
8. ll''Xll y = 115H-ll« = 115 •1=11T.\
2-525 3-3 2-5
9. 2 x2 =2 -^2 =l-=-2 =2 , =2 .
-3 —2 3 2 3 2 3+2 -5
10. 3 x3 =1-^3 -^3 =1^(3 x3 ) = 1^3 =3 .
3 -4 3 4 3+4 7
11. 3 ^3 =3 x3 =3 =3 .
-3 -4-344 3 4-3
12. 3 -^3 =3 x3 =3 -^3 =3 =3.
EVOLUTION.
119
1.
for
2.
for -
3.
id
4.
5.
Examples of Theorem XXX.
2 2 2 2
3 x5 =(3x5) =15
3 x5 =(3x3)x(5x5):
42x92= (4x9)2 = 362
42=2, 92=3 and 6 = 36^
42 x92— Qv5l^ft = ?{fi2
(3x5)x(3x5)=(3x5)'
2x3 = 6=36^
3^=(2x3)2 = 62;
22 = l-414214x, , 32 = 173205 + , 62=2-44949-
1 -414214 X 1 -73205 = 2 -44949 - .
.3 -3 .3 .3
7 xll =(7x11) =77 .
1
2 X
3 3 ^ 3 r ft -^
8 +27 =(8 + 27) = [|J
6. 83 +273 = (8+27)^ =
T. 7 +11
• LisJ
-2 2 2
= 11 +7 =
27
27 >•
111
J}
LnJ
112J
r8^5i«_ r2i
§
ir5''r2J
r9J
[121. The fundamental theorems of addition and subtraction set
forth in §42, those of multiplication and division set forth in §§62
and 63 and those of involution and evolution set forth in §§118
and 119, may by mere counting be proved to be true in every
instance in which the numbers to be combined are all commen-
surable, but they cannot be thus proved if the numbers to be
combined or operated upon are incommensurable. In the latter
case we practically assume or postulate the truth of these theorems
which thus contain implicitly, or rather actually become the
definitions of, the generalized operations of addition, subtraction,
multiplication, division, involution and evolution. For instance,
we may prove by mere counting that twice three is equal to thrice
two, that one-half of one-third is equal to one-third of one-half,
that the square root of four multiplied by the square root of nine is
equal to the square root of nine multiplied by the square root of
four, but we cannot by such method prove absolutely and completely
that the square root of two multiplied by the square rof>t of three
120 ARITHMETIC.
is equal to the square root of three multiplied by the square root of
two or even that twice the square root of three is equal to two
multiplied by the square root of three. So also we may prove
by mere counting that 2x3 = 6, that i x ^ = ^ and that
4^x9- =36'-^, but we cannot by counting and solely by counting
prove absolutely and completely that 22x3^ = 6"-, or that, if the
index of the power which 3 is of 10 be added to the index of the
power which 2 is of 10 the sum will be the index of the power
which 6 is of 10.]
EXERCISE X.
Prove the truth of the following statements: —
3 4 4 3 12 3 6 6 3 18
1. (2 ) =(2 ) =2 . 6. (7 ) =(7 ) =7 .
3443 12 37 73 21
2. (3 ) =(3 ) =3 . 7. (10 ) =(10 ) =10 .
3443 12 16612
3. (5 ) =(5 ) =5 .. 8. (8^) =(8 )-^=8 .
3563 16 11 111
4. (2 ) =(2 ) =2 . 9. (64"2)3 =(643)2 = 64«.
4 5 5 4 20
5. (2 ) =(2 ) =2 .
"• {[i]'}-{[h]y~-[k
/ 3\-4 /^-*\ 3 ^-12 / 12\-1 / -1\ 12
(2 ) =(2 ) =2 =(2 ) =(2 ) .
/ 3\ -4 /„-4\ 3 -12 ./„12\-1 /^-1\ 12
(3 ) =(3 ) =3 ^(3 ) =(3 ) .
(2-^)^ = (2^)-' = 2-^^=(2^^)-^ = (2-^)^\
15. (2^)-^ = (2-^)^ = 2-^^ = (2^^)-^ = (2-^r.
/ 3\ -6 /_-6\ 3 ^-18 /_ 18\ -1 / 18\-1
16. (7 ) =(7 ) =7 =(7 ) =(7 ) .
(io-^)'=(io'r=io-^'=(io")-'=(io-^)".
(64-4)?> = (64i) -i = (64i) -i = (64-.^)J=64-J.
I -3\ -4 / -3\-4 12 / -3\ -5 / -5\ -3 15
21. (2 ) =(2 ) =2 . 24. (2 ) ={2 ) =2 .
/ -3\ -4 / -4\ -3 12 / -4\-5 / -5\ -4 20
2a. (3 ) =(3 ) =3 . 25. (2 ) =(2 ) =2 .
/ -3\-4 / -4\ -3 12 / -3\-6 / -6\-3 18
23. (5 ) =(5 ) =5 . 26. (7 ) =(7 ) =7 .
11.
12.
13.
14.
17.
18.
19.
EVOLUTION.
121
27.
2S.
30.
31.
32.
33.
34.
35.
40.
41.
42.
43.
44.
45
50.
51.
52.
53.
54.
55.
60.
61.
62.
63.
64.
65.
70.
/ -3\-7 / -7\-3 5
(lO ) =(10 ) =10
/ _i\ -fi / -e\ _i 2
(8 a) =(8 ) « = 8 .
29.
(64~2 ) -h ^ (64"^ ) 2 = 646 .
WAYl
ire
-2 _1
"_ fi V:
3 4 3+4 7
2 X2 =2 =2 .
3 4 3+4 7
3 x3 =3 =3 .
3 4 3+4 7
5 x5 =5 -6 .
3 5 3+5 8
2 x2 =2 =2 .
4 5 4+5 9
2 x2 =2 =2 .
1 2
36.
37.
38.
39.
2i
16.
3 6 3+C y
7 x7 =7 =7 .
3 7 3+7 10
10 xlO =10 =10 .
83x8 =8
= 8«+^=«¥
642x64^ = 64^^3 = 64^
[^J'x[rel=[fJ =[
3 4 3-4 -1
2 H-2 =2 =2 . 46.
3 4 3-4 -1
3 -f3 =3 =3 . 47.
3 4 3-4 -1
5 -^5 =5 =5 . 4§.
5 3 5-3 2
2 ^2 =2 =2 . 49.
5 4 5-4
2. -J-2 =2 =2.
16J • UeJ Li6J
3 -4 3-4 -1
2 x2 =2 =2 . 56.
3 -4 3-4 -1
3 x3 =3 =3 . 57.
3 -4 3-4 -1
5 x5 =5 =5 . 58.
5 -3 5-3 2
2 x2 =2 =^2. 59.
5 -4 5-4
2 x2 =2 =2.
r 1 r' r 1 V _ r 1 1 "'""^
lieJ "^ lieJ " iFeJ
3-4347
2 ^2 =2 x2 =2 . 66.
3-4347
3 ^3 =3 x3 =3 . 67.
3-4347
5-^5 =5 x5 =5 . 68.
3-5358
2 -^2 =2 x2 =2 . 69.
4-5 4 5 9
2 -^2 =2 x2 =2 .
1 -2
3 6 3-6 -3
7 -7 -7 =7 .
7 3 7-3 4
10 -f-10 =10 =10 .
8«-^83 = 8""3 = 8^3\
642 -h 64^ = 642~^ = 64^
r 1 1 "^".
ireJ
3 -6 3-C -3
7 x7 =7 =7 .
7 -3 7-3 4
10 xlO =10 =10 .
8"3x8^ = 8''"^-8 3\
64^x64~3 = 64^~3" = 64^.
= rii"^
ll6J
3-6-369
7 -^7 =7 x7 =7 .
3-7 3 7 10
10 ^10 =10 xlO =10 .
6 _1 « 1 19
8 -!-8 3 = 8 x8''=8 3.
642 -^ 64"3 = 642 X 643 == 64« .
ll6J •
ireJ
ri V
Ire J
111 '_ nv.
lieJ ueJ
122 ARITHMETIC.
-3 -4 -3-4 -7 -3 -6 -3-6 -9
71. 2 x2 =2 =2 . 76. 7 x7 =7 =7 .
_-3 -4 -3-4 -7 -3 -7 -3-7 -10
72. 3 x3 =3 =3 . 77. 10 xlO =10 =10 .
_ -3 —4 -3-4 —7 -1 -G -1-6 -19
73.5 x5 =6 =5 . 7§. 8 3x8 =8 » =8 ^.
— 5 —3 —6—3 —8 —1 —1 — 1— J. — •*>
74. 2 X2 =2 =2 . 79. 64 ^x64 3=64 2 3=64 «.
—5 —4 —5—4 —9
7*. 2 x2 =2 =2 .
* • 1 16 J ^ ire J = ire J = I re J
-3 -4 -3 4 4-3
81. 2 ^2 =2 X2 =2 =2.
-3 -4 -3 4 4-3
82. 3 -^3 =3 x3 =3 =3.
-3 -4 -3 4 -3+4
83. 5 ^5 =5 X5 =6 =5.
-5 -3 -6 3 -5+3 -2
84. 2 ^2 =2 X2 =2 =2 .
-5-4-5445 -1
85. 2 -^2 =2 x2 =2 -^2 =2 .
3 -6-3 6 6-3 3
86. 7 ^7 =7 x7 =7 =7 .
-a -7 -3 7 7-3 4
87.10 -^10 =10 XlO =10 =10.
_1 -6 _1 6 6 1 U.
88. 8 3-^8 =8 ^xS =8 -f-83-8 3.
89. 64"2-i-64~3 = 64~^x643 = 643-^64^'"64~«
^ • ii6J "" lie J ~ ire J ^ ir6i = I16J
91. 3 x4 =(3x4) =12 . 95. 4 x5 =(4x5) =20 .
92. 3^x4^ = (3x4)■" = 12^ 96. s' x 6^ = (3x6)^ = 18^
5 5 / \5 5 10 10 / \10 10
93. 3 x4 =(3x4) =12 . 97. 7 x3 =(7x3) =21 .
22/ \2 2 / \8 8 / \8 8
94. 5 x3 =(5x3) =15 . 98. (1) x6 =(^x6) =2 .
i«o. 2;x(ir=(2xir=(|)'. ,«». (i)%(o'=(H4)=(i)'
101.-3%4^ = (3^)'' = (j)'. HO. 2'-r(i)' = (2-^i)' = 8'.
3 3/ \3 / \3 -2 -2 / \-2 -2
102. 3 ^4 =(3-=-4) =(|) . 111. 3 x4 =(3x4) =12 .
5 5/ \5 / \5 -3 -3 -3
103. 3 ^4 =(3^4) =(|) . 112. 3 x4 =12 .
2 2 / \2 -5 -5 -5
104. 5 -^3 =(i) . 113. 3 x4 =12 .
2 2 /' \2 -2 -2 / \-2
105. 4 ^5 =(i) . 114. 5 -^3 =(f) .
106. 3V6^ = (3-^6)^ = (i).' 115. 4~"-^5~^ = (|^^
10 10 / vlO " -7 -7 / \-7 -T
107. 7 -T-3 =(5) . 116. 3 x6 =(3x6) =18 .
/ \8 8 / \8 / \8 10 -10 10 10 10
108. (i) -6 =(i-6) =(i,) 117. 7-^3 =7x3 =21 .
EVOLUTION.
123
119. (J)^6-'=(J)^6^=(,g^=l8-^
i.». (i)%(ir=(4)%(j)==(j)'=(jr.
120.2-%(i)' = 2-'x(ir = (jr = (|)'=2'.
Find, correct to six figures, the square roots of
2, 3, 5, 6, 7, 8, 10, 12, 15, 18, 20, 24, 27, 30, 35, 50
and the cube roots of
2, 3, 4, 5, 6, 10, 12, 15, 16, 24, 25, 135, 256,
and employing these roots and actually performing the multipli-
cations indicated, prove that, (to five significant figures) : —
121. (2i)' = (2')i =2l
12a. (3ir = (3;)^, =3t.
123. {5*)* = (50i, =52 = 5^
124. (7^)^ = (7')i, E7* = 7^
125. (3Jr = (3^)J, =3«.
/ 1\2 / 2\1 • Z
126. (4-V =(4 )'\ =4».
127. (5^r = (5^K=5^.
10^.
129. 22x52:
130. 32x52 = 152.
131. 52x72 = 352.
132. 62x102 = 502.
133. 22x32x52=302.
134. 23x2^=4^ = 2^'.
135. 2^x3^ = 6^
12§. 2^x32 = 62.
136. 2^x43=8^, i.
137. 23x53 = 10«.
138. 3^x43 = 12«.
139. 3^x5^ = 15*.
140. 82=2^x42=2^x2,
144. 502=2^x252 = 2^5, =5^2.
e., 23x23 = 23 = 2.
141. 12^=32 X 42 =32 x2,=2V2
142. 18^ = 22x92 =22x3,= 3^2
143. 202=5^ X 4^=5^ x2,= 2V5
2/2.
24* = 6^x 4^=6^' x2,E2V6 147. 163 = (2*)3 = 23 x2, E 2^/2
27^=3^=3^x3,= 3 V3.
148. 243 = 33 X 83 = 33 X 2, = 23|/ 3
145.
146.
149. 1353=53x27-^ = 53x3, =33/5.
150. 2563 = (2'*)^ = 2''x23,= 4V4.
122. If a number which is correct to but a few significant
figures, be either very large or very small, it may in general be
most conveniently written as the product of two factors, one factor
being the number expressed by the significant figures with the
decimal point between the first and second of them, the other factor
124
ARITHMETIC.
being the power of 10 required to yield the proposed number as the
product of the two factors. The exponent of 10 in the second
factor is called the characteristic of the number to 10 as base.
Example 1. The sun's mass is 330,000 times that of the earth
and its distance from the earth is about 91,400,000 miles ; these
numbers might be written 3-3x10^ and 9-14x10^ respectively.
The characteristic of the first number is 5, that of the second is 7.
Example 2. The velocity of light is about 186,300 miles per
second and the wave-length of green light is about '0000208 of an
inch. These quantities may be written 1 '863 x 10^ miles per second
and 2-08 x 10~^ inch, respectively. The characteristic of the wave,
length number is oiegative.
Example 3, ^ind the cube of 15876 correct to five significant
figures.
1] 15876 1-5876 x 10*
1-5876 xlO*
31752
47628
63504
79380
95256
111132
127008
142884
158760
1-5876
79380
12701
1111
95
2-52047
1-5876
xl08
xlO^
31752
7938
318
6
1
4-0015 xl0i2
{^ee Example 2, %^4..)
Example Jf. Find the weight (in Imperial tons of 22401b. each)
of the carbon in the carbonic acid gas in the atmosphere resting on a
square mile of land when the pressure of the atmosphere is 14*73 lb.
to the square inch, given (i), that each cubic foot of air contains
•00035 of a cubic foot of carbonic acid gas, correct to 2 significant
figures ; (ii), that the weight of any volume of carbonic acid gas is,
to 3 significant figures, 1 -52 times the weight of an equal volume of
air under the same pressure and at the same temperature ; (iii),
that ^^ by weight of all carbonic acid gas is carbon, correct to 4
significant figures. (See Example 4^ § 94.)
EVOLUTION.
125
Wt. of air on sq. in.
Wfc. of air on sq. mi.
= 1-473 lb. X 10.
Imi. -6-33(>in. xlO^.
= l-4731b.x 6-3362x103
=5-9131b. xlOio.
Wt. of carb. acid gas in this air = 5. 913 lb. x lO^o x 3-5 x 10" ^ x 1-52
= 3-151b. xl07.
Wt. of carbon in this gas = 3 -15 lb. x 10' x 3 -^ 11.
= 8-61b. xlO«.
= 3,800 T. Imperial.
EXERCISE XL
What is the characteristic factor of
1. 33240. ' 4.
•0000335.
2. 7890000. 5.
•000000081.
3. 29986000000. 6.
12756-78
Write in ordinary notation :
r. 1-00074 xlo\ 10.
6xl0~^
8. 1-27418 xlo\ 11.
10832xlo''
9. 2-26xlO~\ 12.
3-04763x10
15. 10 ^1-2759.
Find to five significant figures the value of :-
13. 914 x 10^ X 1-60933 x 10^ .
14. 4-73 X 10^ X 1-0089 x 10^"
16. 3-98x10 -^(4.374xl0 ).
17. 1-863x10^ X 6-336 xloV(2-08xlO~').
18. 10%(981x 8 -837x10'^). 19. -33092 x
20. (l-27418x 10*)% 3-1416^6.
12
(6-37 xio'')'
X 283 x(l -22x10 0^
X
23. (2-37 xl0~*')2x 1-4707x10
24. (6-25x10
21. 1-96x10
22. (1-6 X lO*' X 6-3709 x 10^ x 2)^.
[. (6-25x10 '^)^-f(^
'. (1
3-1416x3-956x10^).
25. 13-003x10 ")«. 26. (4 xl0^)^-^(4x 10~'),
27. VI- 275678 X 10% 1 - 275584 X 10^ *< 1-271 278 X 10 ')^
LOGARITHMATION.
123. The Logarithm of a given number to a given base is the
exponent of the power which the given number is of the given base.
TJie terms logarithm attd exponent are therefore merely different
names for the same thing. Thus, instead of saying "the exponent
of 100 to base 10 is 2" we say *'the logarithm of 100 to base 10
is 2 ; " instead of saying *' the exponent of 32 to base 2 is 5 " we say
' ' the logarithm of 32 to base 2 is 5 j " and instead of writing 100 = 10^
and 32 = 2^ we may write log 100 = 2 and log 32 = 5. If the base is
. 10 . . . 2 _
10 it is usually omitted both hi writing and in reading logarithms.
Examples.
81 = 3\
log 81=4.
3
10 = 10\ loglO = l.
3
125 = 5 ,
10
1024 = 2 ,
loga25 = 3.
5
log 1024=10.
100 = 10^ logl00=2.
1000 = 10^ logl000=3.
2401 = 7*,
log 2401=4.
2= 8^, log^2 = ^.
1331 = ll^ log 1331 = 3.
11
log l-7<^ but log 1-71 >^-
5 5
4= 8^, log 4=§.
8
27= 9^'\ log 27 = 1-5
9
for 1-7
<53
but 1-71>53
i. e., l^"^
3
<5
but 1-71>5
3
for 1-7 =4-913 and 1 71 = 5 '000211.
EXERCISE XII.
Prove the truth of the following statements, and express them in
logarithmic notation : —
1. 128 = 2^
5.
3,125 = 5".
9. 2 = 16'^^
2. 256 = 4*.
6.
7,776 = 6'.
10. 4 = 16^V
3. 729 = 3'.
4. 729=9'.
7.
8.
14,641 = 11*.
1,000,000=10'.
11. 8 = 16'''.
. 1-2 5
12. 32 = 16
LOGARITHMATION.
127
13. 64 = 16''^^
1''. ^V = 2 ^.
21.
*=C
14. 1024=16''*^
1§. Jj = 4^ .
22.
^h=^ ' '<
15. 125 = 25''
19. ■ij=8\
23.
0.1 = 10 .
16. 279936 = 36^''
^ -1 .5
20. «\= 16 .
24.
0.0001 = 10
Prove the truth of the following statements and express them in
exponential notation : —
25. log 8 = 3.
33. log 1024 = 3^.
41. log 10 = 1.
26. log^64 = 3.
34. log^ 5= -5.
42. log 1000 = 3.
27. logV2 = 3.
35. log^^9 = i.
43. log 100000 = 5.
28. log^343 = 3.
36. log^(|)=-3.
4i. log 1 = 0.
29. log^2187=7.
^7.log{^,j)=-5.
45. log 0-1= -1.
30. log"^ 10077696 = 9.
38. log^(8V)=-4.
46. log 0.01= -2.
31. log'' 20736 = 4.
47. log 0.001 =-3.
32. log 16.777216 = 6.40. log (1^)= -3^.48. Iog0.00001= -5.
J '6 8
49. Prove that log 2 '154 < ^ but that log 2 "155 > J.
50. Prove that log 2 is somewhat greater than '3.
[124. The word logarithm means ratio-numhery and logarithms
were so named because they record the number of successive
multiplications (or successive divisions) by a fixed base, a common
ratio or rate of progression as it was at first called, the initial
multiplicand (or initial dividend) being in every case 1.
Thus, 2 is the fixed base, the common rate of progression by
multiplication, of the series of numbers
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
and 0, 1, 2, 3, 4, 6, 6, 7, 8, 9, 10
are the corresponding logarithms recording the number of successive
multiplications by the ratio 2.
The fixed base or common ratio of progression by multiplication is
3 in the series of numbers.
J. 1^ 1 1, 3, 9, 27, 81, 243, 729
27' 9 ' 3 '
and -3, -2, -1, 0, 1, 2, 3, 4, 5, 6
are the corresponding logarithms. The sign - preceding the first
128 -- ARITHMETIC.
three of these logarithms denotes that successive divisions, not
multiplications, are recorded.
The common rate of progression by multiplication is 10 in the series
00001, 0001, 001, 01, 1, 10, 100, 1000, 10000
and -4, -3, -2, -1, 0, 1, 2, 3, 4
are the corresponding logarithms.
If the fixed base or common rate of progression by multiplication
be 16, and if the series of numbers be
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096
then will 0, "25, '5, '75, 1, 1-25, 1'5, 1-75, 2, 2-25, 2*5, 2*75, 3
be the corresponding logarithms. In this example, the numbers
2, 4, 8 have been interpolated between 1 and 16 the zeroth and first
terms of the series to base 16, and the numbers 32, 64, 128 have
been interpolated between 16 and 256, the first and second terms of
the series to base 16.]
1 25. A logarithm being simply an exponent, the term logarithm
may be substituted for the term exponent in Theorems XXVII
and XXVIII, §119, which may then be expressed as follows : —
XXVIII (a). The logarithm of a product is the aggregate of the
logarithms of the factors.
(The logarithm, of a quotient is the remainder resulting from
suhtraA^ti')ig the logarithm of the divisor from the logarithm of the
dividoul.)
XXVII (a). The logarithm of a power (or of a root) of a number
is the product of the logarithm of the number and the expo^ient of the
power {or of the root. )
Example 1. log 8 = 3, log 32 = 5 ;
2 2
log (8x32) = log 256 = 8-3 + 5=log 8 + log 32.
2 2 2 2
i.e. 8 = 2^ 32 = 2 ;
8 3+5 3 5
8x32 = 256=2 =2 =2x2.
Example 2. log 4 = 2 ;
2
5 log 4 = 10=log 1024 = log (4'
2 22
2
4 = 2 ;
10 2X5
2 =1024 = 2 =4 .
LOGARITHMATION.
129
EXERCISE XIII.
Prove the truth of the foUowing statements : —
1.
log (16xl28)=log 16+log 128.
2.
log^(16xl28)=log^ 16 + log 128.
3.
log*(16 X 128) = log*16 + log^ 128.
4.
log^(512-^64)=log^512-log 64.
5.
log^(512-^64)=log''512 - log^64.
6.
log*(64^256)=log'^64-log 256.
'
7.
log^(2x32^4)=log 2+log*32-log
4.
8.
log^(27 X 243) = log 27 + log %AS.
8
9.
log^ (27 X 243)=log^ 27 +log^ 243.
10.
W (27-r243)=log^ 27 -log 243.
9 9 9.
11.
log 8^=3 log 8. 16.
2 2
log^8^ = ilog^8.
12.
log 8^ =3 log 8. 17.
log 3*^ ' = -25 log 3.
.9 9
13.
log 8^-= 3 log 8. 1§.
8 8
log 3'^ --2 log 3.
9 9
14.
log 8 ^3 log 8. 19.
log 27"^ =-4 log 27.
9 7 ^
15.
log 8^ = 1 log 8. 20.
lo^ -25 = -7 log -25.
8 8
126. Logarithmation is the operation of finding the logayrithm
of a given number to a given base. It is, therefore, an inverse both
of involution and of evolution ; for in involution a base and an
exponent are given and the power of the base denoted by the
exponent is required , and in evolution a power of an unknown base
and the exponent of that power are given and the unknown base is
to be found, but in logarithmation there are given a base and a
number considered as a power of that base and the exponent which
denotes that power is to be determined. For example, involution
and evolution would furnish answers to the questions * What is the
fourth power of 3 ? ', ' What is the cube of the tenth root of 10 ? ' ;
but logarithmation is required to answer the questions * What power
of 3 is 81 ? ', * What powBr of 10 is 2 ?' .
130
ARITHMETIC.
127. Thus of the seven fundamental operations of Arithmetic,
addition and subtraction are each the inverse of the other ; so also
are multiplication and division inverse to each other, but the three
remaining operations, viz. , involution, evolution and logarithmation,
are so related to one another that each has the other two operations
as its inverses.
128. There are several methods of computing logarithms, but
we shall give examples of only two of them. Of these, the first was
one of the methods proposed by Napier the inventor of logarithms,
and was the method by which the first published tables of logarithms
to base 10 were calculated.
129. First or Napier's Method. Extract the square-root of
the base correct to three figures more than the number of decimal
places to which the logarithms are to be correct. Next extract the
square-root of the root just found, then extract the square-root of
this last-found root, and so continue until there has been formed a
table similar to Table I which follows. In forming this table, 10
having been selected as the base, the roots were extracted to ten
decimal places and eight decimal places retained.
TABLE A.
.
10^
=io"
= 3-16227766.
3-162277662
• 25
= 10
= 1-77827941.
1-778279412
.125
= 10
= 1-33352143.
«^^*1 -333521432
.0625
= 10
= 1-15478198.
^^•^1-154781982
.03125
= 10
= 1-07460783.
1-074607832
.016625
= 10
= 1-03663293.
1-036632932
.0078125
= 10
= 1-01815172.
1-018151722
.00390625
= 10
= 100903505.
1-009035052
.001953125
= 10
= 1-00450736.
1-00450736^
.000976563
= 10
= 1-00225115.
1-00225115^
.000488281
= 10
= 1-00112494.
1-001124942
.000244141
= 100056231.
1-000562312
• 000122070
= 1-00028112.
1-00028112^
• 000061035
= 10
= 100014056.
LOGARITHMATION, 131
130. The exponents which would follow '000061035 in order in
the preceding Table, are obtained by taking |^, J, ^, •j'g, &c. of
•000061035, and the decimal parts of the corresponding powers,
correct to eight decimal places, by taking ^, |, ^, j^g, &c., of
•00014055, the decimal part of 1-00014055, the power of which
•000061035 is the exponent. Hence th8 logarithm to base 10 of any
number greater than 1 but less than 1 '00014055 is, to nine decimal
•000061035 , , . ,. -000061035 ^ ,,
I^^^""^» '00014055- ^"'*^^^^^^""^PP"^^"^^^'^"^<^ ^^ ^^'^
decimal or fractional part of the number.
'000061035
131. The fraction -—r-T-r—TT— which is equal to '434273+ is
'000140545
an approximation, correct to four decimal places, to a number called
the Modulus of logarithms to base 10. If any number other than
10 had been made the base in Table A, a different number would
have been obtained as the modulus ; e. g., had the base been
2-718281828459, the modulus would have been 1, i. e., the loga-
rithm to this base of any number greater than 1 but less than 1 '0001
is simply the decimal part of the number, correct to eight or more
decimal places. Had the roots in Table A been calculated to 32
decimal places, it would have been necessary to extend the columns
to fifty-five terms before the decimal parts of the roots would be
proportional to the exponents,"^ but in such case, the modulus would
have been obtained correct to some eighteen decimal places. It
has been computed to 136 decimal places t ; to twelve places it is
•434294481903.
132. If the powers in the third column of Table A be considered
as given numbers, the exponents in the second column of the Table
will be their logarithms to base 10. In Table B which follows, the
* Such a table was actually computed by Henry Briggs, Savilian Professor of
—54
Geometry at Oxford. The fifty-fifth exponent or 2 he found to be
o'ooo,ooo, 000,000, 000,055, 51 1, 151, 231, 257, 827
and the corresponding root, the result of fifty-four successive extractions of the
square root, to be
I •000,000,000,000,000, 127,819, 149,320,032,35.
Briggs was the first to compute and publish logarithms to the base 10.
+ By means of the series the earlier terms of which are given in problem 65,
page 90. The modulus is the reciprocal of that series.
132
ARITHMETIC.
powers are tabulated as numbers and the exponents as the loga-
rithms of tliese numbers.
TABLE B.
Number.
Logarithm.
Number.
Logarithm.
10 .
1
1^00903505
-00390625
3-16227766
-5
1-00450736
•001953125
1-77827941
-25
1-00225115
-000976563
1-33352143
-125
1 -00112494
-000488281
1-15478198
-0625
1-00056231
-000244141
1-07460783
-03125
1-00028112
-000122070
1-03663293
•015625
1^00014055
-000061035
1-01815172
•0078125
10001
-000043427
TABLE C.
Multiples of the Modulus {'i^f^g
1. -43427. 4. 1-73709.
2. -86855. 5. 2-17137.
3. 1-30282. 6. 2-60564.
7. 3-03992.
8. 3-47419.
9. 3-90846.
Example 1. Find the logarithm of 2 correct to eight decimal
places ; i. e. , find the exponent of the power to which 10 must be
raised so that the result may be 2.
From the columns of Numbers in Table B, (Col. Ill, the column
of powers in Table A, ) select the largest number less than the given
number 2, and divide 2 by the number thus selected. From the
columns of Numbers in Table B select the largest number less than
the quotient just obtained, and divide that quotient by this second
selected number. From the columns of Numbers in Table B select
the largest number less than the last obtained quotient and divide
LOGARITHMATION. 133
that quotient by this third selected number. Continue thus to
select and divide until there is obtained a quotient less than
1 •00014055. These operations resolve 2 into a series of factors all
of which, except the last, are numbers in Table B. Consequently
the logarithms of these factors, except that of the last factor, are
given in Table B and the logarithm of the last factor can be
obtained by multiplying the decimal part of the factor by the
modulus '43427. The logarithms of the factors being known, the
logarithm of 2, their product, may be found, being the sum of the
logarithms of the factors.
2 -^ 1-77827941 -112468265
tl 12468265 -=- 1 '07460783 = 1 '04659823
1 04659823 -^ 1 '03663293 = 1 '00961314
1 00961314 ^ 1 -00903505 = 1 '00057292
1 00057292 -^ 1 '00056231 = 1 '00001060
.-.2 = 1 '77827941 x 1 '07460783 x 1 03663293 x 1 '00903505
or
• 25 .03125 .015025 .00390625
= 10 xlO xlO xlO
• 000244141 .0000106X.4«427
X 10 X 10
.2 6 + .0;il2i>+.0156 2 5+.00390(52 5+-000244141 + .00 0004 604
= 10
• 3010 2999 5
= 10-
or log 2 = '30103000, correct to eight decimal places.
Written in logarithmic instead of in exponential notation, the
latter part of the preceding calculation would be
log 2 = log 1-77827941 + log 1 '07460783 + log 1 '03663293
+ log 1-00903505+ log 1 '00056231 + log 1-00001060
= '25+ 03125 + '015625+00390625+ '000244141
+ -0000106 X '43427
= '301029995,
.*. log 2=3'0103000, correct to eight decimal places.
[log 2= -301029995663981, correct to fifteen decimal places.]
Example 2. Find log 48847, correct to eight decimal places.
Write 48847 in the form 4'8847xl0* and resolve 4-8847 into
factors selected from the columns of Numbers in Table B.
4-8847 +3-16227766 = 1-54467777,
1 -54467777 + 1 -33352143 = 1 '15834491
134 ARITHMETIC.
1 -15834491 ^ 1 -15478198 = 1 00308537
1 -00308537 -r 1 -00225115 = 1 -00083235
1 -00083235 -^ 1 -00056231 = 1 -00026988
1 -00026988 -^ 1 -00014055 = 1 -00012931 ;
. •. 48847 = 10* X 3 -16227766 x 1 33352143 x 1 -15478198 x 1 -00225115
X 1-00056231 X 1-00014055 x 1-00012931 ;
. •. log 48847 = log 10* + log 3 -16227766 + log 1 -33352143
+ log 1-1 5478198 + log 1-00225115 + log 1-00056231
+ log 1 -00014055 + log 1 -00012931.
= 4+ -5+ -125+ -0625+ -000976563+ -000244141
+ -000061035+ -434273 x -00012931
= 4 -688837895, correct to within 1 in the last figure.
.*. log 48847 = 4-68883790, correct to eight places of decimals.
133, The logarithm of any number may be found by this method
indei3endently of finding the logarithm of any other number, but in
forming a table of logarithms, the logarithms of prime numbers alone
need be computed, the logarithm of any composite number being
the sum of the logarithms of the factors of such composite number
and the logarithm of a power being the product of the logarithm of the
base of the power and the exponent of the power. Thus knowing
log 2= -3010300, we obtain log 4 = log 2^=2 log 2= -6020600, log 8
= log 2 = 3 log 2 = -9030900, &c.
134. The knowledge of the logarithm of one number will often
greatly aid in computing the logarithm of another number which
differs but little from the number whose logarithm is known.
Example 3. Find log 81 correct to eight decimal places, given
log 80 = 1-903089987.
81=80+l = 80x(l + gV) = 80x 1-0125.
Resolve 1-0125 into factors selected from the columns of numbers
in Table B.
1 0125 + 1 -00903505 = 1 -00343393
1 00343393 + 1 -00225115 = 1 -0011801 2
1 00118012 + 1 -00112494 = 1 -00005512
. •. 81 = 80 X 1 -00903505 x 1 -00225115 x 1 -00112494 x 1 -00005512
. -. log 81 = log 80 + log 1 -00903505 + log 1 -00225115 + log 1 -00112494
+ log 1-00005512
= 1 -903089987 + -00390625 + "000976563 + -000488281
+ -43427 X -00005512
= 1-908485018, correct to within 1 in 'the last figure.
LOGARITHMATION.
135
3 -1=2x4:
2
3 +1
3^^- 1=8x10=80
log 81 = 1 "90848502, correct to eight places of decimals.
From log 81 we may obtain log 3 for
81 = 3*, . •. log 81 = log 3* = 4 log 3.
.-. 4 log 3 =1-90848502,
. *. log 3 = '47712125, correct to eight decimal places.
[log 3 = -477121254719662, correct to 15 decimal places.]
This problem is virtually, — Find log 3, given log 2. We
proceed thus ; —
3-1 = 2 and log 2 is given,
8+1=4 and log 4=2 log 2,
8 and log 8 = log 2 + log4 ;
= 10 and log 10 = 1,
and log 80 = log 8 + log 10;
3* =80 + l = 80x(l + ^o)=80x 1-0125.
The remainder of the calculation is that already given.
Example 4' Find log 7 given log 2 and log 3.
7-1 = 6 andlog6 = log2 + log3
7 + 1 = 8 and log 8 = 3 log 2.
2
and log 48 = log 6 + log 8
and log 50 = log 100 -log 2
and log 2400 =log 48 + log 50
: 2400 X (1 + 2 iVff) = 2400 X 1 -00041667.
1 00041667 + 1 -00028112 = 1 -00013551
. •. 7* = 2400 X 1-00028112 x 1-00013551
. •. log 7* = log 2400 + log 1-00028112 + log 1-00013551.
. -. 4 log 7 = 3-380211242 + "000122070+ -43427 x -00013551
= 3-380392160
.-. log 7= -845098040.
[log 7 = -845098040014257, correct to 15 decimal places.]
Example 5. Find log 11, given log 2, log 3 and log 7.
99 = 11 X 3^ . •. log 99 = log 11 + 2 log 3 = log 11 + -954242509.
99-1=98, log 98 = log 2 + log 49 = log 2 + 2 log 7,
99 + 1 = 100, log 100 = 2;
50
7 -1 = 48
2
7^ + 1^
7* -1 = 2400
7* = 2400 + 1:
136 ARITHMETIC.
.-. 99^-1 = 9800, Iog9800 = 2 + log2 + 2 1og7;
.-. 99^ = 9800+1 = 9800 x(H-9J^) = 9800x 1-00010204;
. •. log 99^ = log 9800 + log 1 -00010204
. •. 2 log 99 = 2 + log 2 + 2 log 7 + '434273 x -00010204
= 3-991270389.
.-. log 99 = 1-995635195
. •. log 11 + -954242509 = 1 -995635196
. % log 11 = 1 -995635195 - -954242509
= 1 -041392686, correct to eight places of decimals,
[log 11 = 1-041392685158225, correct to the 15th decimal.]
135. If the number to be resolved into factors selected from the
columns of Numbers in Table B or any quotient arising in the
course of its resolution be but very little less than one of the tabular
factors, it will in general be better to use such number or such
quotient as next divisor and the tabular factor next greater than it
as dividend. The tabular factor then becomes a divisor, not a
multiplier, in the resolved form of the given number.
Example 6. Find log 3*14159265, correct to eight decimal places.
3 - 16227766 + 3 -14159265 = 1 -00658424,
1 -00658424 + 1 -00450736 - 1 -00206756
1 -00225115 -r 1 -00206756 = 1 -00018321
1 -00018321 + 1 -00014055 = 1 -00004266
.•.3-14159265 = 3-16227766^-1 -00450736 +100225115 x 100014055
X 1-00004266
.-.log 3 -14159265 = log 3-16227766 -log 1 '00450736 - log 1-00225115
+ log 1-00014055 + log 1-00004266
= -5 - -001953125 - -000976563 + -000061035
+ -43427 X -00004266
= -497149873, correct to the last figure,
[log 3-14159265= -497149872694134-.]
Had 3-14159265 been resolved into & product of factors, as 2 was
resolved in Example 1 and 48847 in Example 2^ no less than nine
divisions would have been required to effect the resolution instead
of the four divisions required in the resolution just given.
LOGARITHMATION.
137
EXERCISE XIV.
Find, correct to 7 decimal places : —
1. log 1-00001. 4. log 1-00007.
2. log 1-00002. 5. log 1-000135.
3. log 1-00003. 6. log 1-0002497.
Find, correct to 4 decimal places : —
r. log 1-001. 10. log 3.
8. log 1-0012. 11. log 7.
9. log 1-0029. 12. log 2-718.
13. log 31, given log 32 = 1 -50515.
14. log 13, given log 7 and log 11 and that 7 x 11 X 13 = 1001.
15. log 17, given log 3 and log 7 and 3 x 7 = 1701.
Find, correct to 6 decimal places : —
16. log 7, given log 2 and log 3 and that 2 x 3 x 7^ =1000188.
17. log 17, given log 2 and log 7 and that *f x 17 = 2000033.
18. log 13, given log 2, log 3, log 7 and log 11 and that 123200
= 2^ X 7 X 11 X 10\nd 123201 =3^^ x 13 1^
19. log 19, given log 2 and log 3 and that 19" - 1 = 2 x 3" x 10.
20. log 19, given log 2 and log 3 and that 2 x 3^ x 19' = 10000422.
21. log 23, given log 2 and log 19 and that 23'^ = 190 • 109375 x 2 ^
22. log 29, given log 2, log 3, log 7, log 11 and log 13 and that
96059600 = 2^ x 7^ x 13^ x 10^ x 29, and 96059601 = 3^ x ll^
23. log 41, given log 2, log 3 and log 13 and that 2^ x 3 ' x 13^
= 410012928.
24. log 23, given log 2, log 3, log 7, log 11, log 13, and log 17 ^nd
1000000 2893400
*^^^^'^*^" "999999'' 2893401*
196 ''59
25. log 2, given that 2 =10
fl025"^
"" 1 1024 J
r 104857^1
I 1048575 J
f9801^ .
U80oJ '
f 65601 r 15624-)
U561J ^ 1 15625 J
136. Second or Taylor's Method. This is a method of
finding the convergent fractions to the logarithm of a gi\'en number.
The following examples which are self-explanatory will easily enable
one to understand the mode of procedure.
Example 1. Find log 2.
138 ' ARITHMETIC.
1 <10 A
2 > 1 B
1 x2 =2 =2 <10 AxB
2 x2 =2^ =4 <10 AxB
2^ x2 =2'^ =8 <10 C = AxB
2^ x2 =:2* = 16 >10 BxC
2* x2' =2 = 128 >10^ BxC""
7 3 10 3 3
2 x2 =2 = 1024 >10 D = BxC
10 3 13 4
2 x2 =2 = 8192 <10 CxD
2^^ x2^^ =2"'^ = 8388608 <10^ Cxd"
23 10 33 10 3
2 x2 =2 = 85898346.. <10 CxD
33 10 43 13 4
2 x 2 =2 = 87960930 < 10 CxD
4310 5 3 16 5
2 x2 =2 = 90071992 <10 CxD
53 10 0 3 19 6
2 x2 =2 = 91209720 <10 CxD
63 10 73 22 7
2 x2 -2 = 93398754 <10 CxD
2^ x2'^ =2^' = 96714066 <10^^ CxD^
83109 3 28 9
2 x2 =2 = 99035203 <10 E=CxD
9 3 10 103 31
2 x2 =2 = 101412048 >10 DxE
10393 19 6 59 2
2 x2 =2 =100433628 >10 r = DxE
196 93 289 _ 87
2 x2 =2 = 99464647 <10 ExF
289 196 485 146 2
2 x2 =2 = 99895954 <10 G = ExF
485 196 681 205
2 x2 =2 =100329130 >10 FxG
1651 485 2136 643 4
2 x2 =2 =100016289 >10 H = FxG
11165 2136 13301 4004 6
2 x2 =2 = 99993628 <10 J = GxH
15437 13301 28738 ^ 8651 2
2 x2 =2 = 100003544 >10 K = HxJ
28738 13301 42039 12655
2 x2 =2 = 99997172 <10 L = JxK
42029 28738 70777 21306
2 x2 =2 =100000716 >10 M = KxL
183593 70777 254370 76573 2
2 x2 =2 =99999320 < 10 N = LxM
254370 70777 325147 97879
2 x2 =2 =100000036 >10 P = MxN
LOGARITHMATION.
0 1 10
Writing A in the form 2 < 10 and B in the form 2 > 10 ,
"we have
I
I
A- =2"
< 10^ say
2<10o
or log 2 < ^
B ^ = 2^^
>io' .-.
2>10T
.. log2>^
C=Axb' = 2'
<10^ .-.
2<10*
.. log2<J
3 10
3
a.
D = BxC =2
>10
2 > 10^0
.. log2>35^
9 93
28
2<109§
E=CxD =2
<10
.. log 2 < II
2 196
59
2 > 10^9^
F-DxE =2
>10
•• log2>fVV
2 485
146
■ 146
G=ExF =2
<10
2 < 10* 8 5
•• log 2 < HI
4 2136
643
2 > 102T*^
H=FxG =2
>10
.. log2>JV%\
6 13301
4004
2 < 10T3 30T
J=GxH =2
<10
•• log2<^*ff^*T
2 28738
K = HxJ =2
86 51
>10
2 > 102¥7¥8
•• log2>,^5V
42039
12655
2<10^l^^«
L=JxK =2
<10
•• log 2 < net
70777
M = KxL=2
21306
>10
2>10^^^f
.• log2>f^f?f
3 254370 76673
2<102^I^^
N=LxM =2
<10
•• log2</^%7^
325147' 97879
P=Mx]S[=2 >10
2>10^^^
.. log2>^W^
We have obtained the first twelve principal convergents to log 2
by keeping a record of the exponents of the powers of 2 and of 10
which are of approximately equal values, but there is no absolute
necessity for the keeping of such a record. The convergents may be
computed by assuming— and — as initial convergents, the second
of these initials being the characteristic of 2 the given number to
10 the given base, and then taking as the convergent-quotients the
exponents of the multipliers B, C, D, &c. in the second column of
the above calculation. These exponents are, each of them, less by 1
than the number of successive multiplications required in the several
cases to pass from > through < to > again or vice versa ; thus they
record without repetitions the number of such multiplications.
Quotients, 3, 3, 9, 2, 2, 4, 6, 2, 1, 1, 3, 1 ;
Convergents,! 0 1 3 28 59 146 643 4004
0' 1' 3' 10' 93' 196'
8651 12655 21306 76573 97879
28738' 42039' 70777' 254370' 325147*
485' 2136' 13301'
140 ARITHMETIC.
The next quotient, the 13th. cannot be less than 1, and for 1 as
13th. quotient, the upper limit of error of the 12th. convergent is
1 1 -12
which is < „ .-.^^^ n^r^^r^r, < 6 X 10 » 5
325147 X (325147 + 254370) 300000 x 600000
97879
hence does not differ from log 2 by so much as 6 in the
325147
twelfth decimal place.
But the 11th. and 12th. convergents being close approximations
to log 2, the required number, it is not necessary, in order to
determine the 13th. quotient, to actually perform the multiplications
which that quotient records. Consider for example how the fourth
convergent quotient may be determined by the powers of 2 denoted
by D and ^, page 138. The fourth convergent-quotient is simply
the number of successive multiplications of 1024, the D-power of 2,
by 99035 , the E-power of 2, which are required to produce
3
the F-power of 2, and 1024 is approximately 10 , 99035
28
approximately 10 and the F-power of 2 approximately an integral
power of 10 ; the number of these multiplications will therefore be
less than the quotient of 1024 ^10^-1 divided by 1 - 99035. . . -4- lo! *
i. e., than •024-^*00965, but will be approximately equal to this
quotient. We may therefore use the integral part of "024-7- "00965
as a convergent quotient to form the fourth or F-convergent ; and
in point of fact the integral part of '024^ "00965 is 2, the fourth
convergent quotient. The correctness of the foregoing argument
may be seen at once, if the proper method of multiplying by
99035 .... be adopted, viz. , that described in § 69, xii, page 36.
It should however be noticed that if the terms of the division, here
"024-7- "00965, are not both very small the convergent-quotient
sought may be greater than the quotient arising from the division.
For example had we sought to determine the third convergent
from (1 - •8)-i-(l "024 — 1) we would have obtained 8 as the third
convergent-quotient instead of 9 the correct value.
In like manner from the powers of 2 and 10 yielding any two
consecutive convergents after the fourth, the quotient determining
the third consecutive convergent may be obtained, and consequently
the 13th. convergent may be computed from the powers of 2 and
10 yielding the 11th. and 12th. convergents. Thus
LOGARITHMATION.
the dividend obtained from N is 1- -99999320 . . . = -00000680
the divisor obtained from P is 1-00000036 . . . .- 1 - '00000036
.♦.the quotient is 680^36 = 18 +
^, ,^^, .. 1 o- 97879x18 + 76573 1838335
.•.the 13th. convergent to log 2 is ^- = — -
^ ^ 325147x18 + 254370 6107016
An upper limit of error for this convergent is
1
which is
•.log 2 =
6107016 X (6107016 + 325147)
<3xl0'
6000000 X 6000000
1838335
•3010299956634, correct to 13 decimal places.
6107016
Example 2. Find log 3.
Powers of 3.
1
3
3
9
27
243
2187
19683
177147 ,
1594323
1434891.
1291402..
1162261...
1046035...
941432...
984771...
1030105...
1014418...
998969...
1013...
The multiplier 3 occurs twice, 9 occurs ten times, the others twice,
twice and once respectively; hence the first five convergent quotients
to log 3 are 2, 10, 2, 2 and 1, and the sixth quotient will be the
integral part of (1 -014418 - 1) + (1 - -99§969) = 14418 + 1031 = 13 '9 + ,
which is 13. The characteristic of 3 to base 10 is 0, therefore the
initial convergents are — and — ; hence we have for log 3
Multipliers producing
the next power.
3
3
9
9
9
9
9
9
9
9
9
9
1046035,..
1046035...
984771...
984771...
1014418...
142 ARITHMETIC.
Quotients; 2 10 2 2 1 13
1 0 1 10 21 52 73 1001
Convergents; -^, T' ^' 21' 44' iM' 153' 2098'
or log 3= — —= '477121, correct to six decimal places, for the error
2098
1 1 ~^
of this convergent is < — < < 3 x 10
^ 2098 X (2098 + 153) 4000000
Had 13*9 been used instead of 13 as sixth convergent quotient,
the resulting convergent would have been
73x13-9+ 52 10667, .4..12126-
153x13-9 + 109 22357 ''^^^^"
which is a closer approximation than even — — .
2098
Examples. Find log 48847.
Powers of 48847. Multipliers producing
1 the next power.
48847
48847 48847
2386029 . . . 48847
1165504 . . . 48847
569314 . . . 1165504 • • •
663537 ...
773355 ...
901348 ...
1050525.... 901348- ••
946889 . . . 1050525 . . .
994730 ...
104
The first five convergent quotients are 1, 2, 4, 1, 2, and the sixth
is 9, the integral part of (1 -050525 - 1) +- (1 - -994730) = 50525 ^ 5270.
The characteristic of 48847 to base 10 is 4, and therefore the initial
convergents are — and— •
Quotients; 12 4 12 9
1 4 5 14 61 75 211 1974
Convergents; 0 T T 3 13 16 45" 42r-
The error of the last of these convergents is
< i <_Ji__<6xlO
421 X (421 + 45) 180000
1974
, *. log 48847 = -j^ =4-68884 - , correct to five decimal places.
LOGARITHMATION.
EXERCISE XV.
Obtain, correct to 4 decimal places : —
1. log 7. 3. log 31. 5. log 2-72.
^ 3. log 6. 4. log 6-6. 6. lo*gl-371.
^» 137. Many other methods of calculating logarithms have been
proposed, the greater number of them being merely variations of
one or other of the two processes already described, but all of these
methods are so tedious and involve so much labor in their applica-
tion that were it necessary to calculate a logarithm anew every time
it was required, computation by the aid of logarithms would be a
useless curiosity. To overcome this objection to their employment,
the logarithms of all integral numbers from 1 to 200,000 have been
calculated and recorded to seven places of decimals, once for all.
A small ]3art of this record, being a Table of Logarithms correct to six
decimal places, is given at the end of this volume. In Table I are
entered the logarithms to base 10 of all numbers from 1 to 100, in
Table II are given the logarithms to base 10 of all numbers from
1-000 to 9-999 by increments of '001, and Table III contains the
logarithms to bas& 10 of all numbers from 1 to 1'0999 by increments
of -0001. The logarithms entered in Tables II and III are all
decimals, but in printing the tables the decimal point has been
omitted as unnecessary. The decirnal part of a logarithm is
termed the mantissa of the logarithmy and the integral part, the
charaoteristio of the logarithm, (See § 122.)
138. The following examples wiU show how to use Tables II and
III either to find the logarithm of a given number or to find the
number corresponding to a given logarithm.
Example 1. Find log 4*884, log 48840 and log '04884.
We glance along the columns marked N^. until we find 488, the
first three digits of the given number ; we then pass horizontally
along the line of 488 to the column headed 4, the fourth digit of the
given number ; in that column we find 8776, these are the last four
figures of the mantissa of the required logarithm. The first two or
leading figures are 68, they will be found standing over the blank
space which appears in the line of 488 in the column headed 0,
144 ARITHMETIC.
Hence log 4 -884 = -688776. (A.)
48840 = 4-884x101
log 48840 = log 4-884 + log 10*
= •668776 + 4=4-688776. (B.)
-04884 = 4-884x10-2
. *. log -04884 = log 4- 884 + loglO-2 '
= -688776 -2 = 2 -688776. (0.)
It will be noticed that when the characteristic is negative, as it is
in (C), the minus sign is written above the characteristic, not in
front of it. The mantissa in (0) is positive, being the logarithm of
the factor 4*884.
Examples. Find log 4*076, log 407 '6, log 40760, and log -0004076.
We first find 407 in the columns marked N°. and then run
horizontally across to the column headed 6 in which we find "^0234,
the last four figures of the logarithm sought. The "^ in front of
these figures indicates that the two leading^ figures of the logarithm
are at the foot of the blank space in the column headed O. Looking
there we find the leading figures to be 61, hence
log 4-076= -610234.
407-6=4-076x102,
log 407 -6 = -610234 + 2 = 2*610234.
40760=4-076 xlOS
log 40760 = -610234 + 4 = 4 -610234.
•0004076 = 4-076 x 10-^,_
.-. log-0004076= -610234-4 = 4-610234.
139. It may be seen from these examples that changing the
position of the decimal point in a number changes the characteristic
but does not change the mantissa of the logarithm of the number.
The characteristic of the logarithm of a given number may and
should be written down before the mantissa is found in the Table
of Logs. , for the characteristic to base 10 is simply the number of
places which the first significant figure of the given number is from
the ones' figure of the number, the ones' figure itself not being
counted, i. e. , it is considered as standing in the zeroth place. If
the first significant figure of the given number stands to the left of
the decimal point, the characteristic will be positive or zero ; if it
stands to the right of the decimal point, the characteristic will be
negative.
LOGARITHMATION.
145
EXERCISE XVI.
^ind from Table I the logarithm of : —
1. 7. 2. 37. 3. 59. 4. 79.
Write down the characteristic of : —
6. 723. 7. -07^3. 8. 200000.
Find the losjarithm of : —
5. 97.
9. 0-00002. 10. 372-58.
11.
23.
16.
4-321.
21.
7246.
26.
1-178x101*.
12.
230.
17.
4321.
22.
•007246.
27.
1-63 xl0*2.
13.
2300.
IS.
•04321.
23.
676700.
28.
6-48 xlO-8.
14.
2-3.
19.
6789.
24.
67-67.
29.
4-496 xlO-«.
15.
•023.
20.
7810. .
25.
200-2.
30.
7-604x10-5.
y
140. Example 3. Find log 4-8847.
Log 4-8847 is not given in the Tables but evidently it lies in
value between log 4-884 and log 4-885 both of which are given and
it may be computed from these tabular logarithms as follows :
log 4-885
log -4 -884 =
log 4 -885 -log 4-884 =
•7 of -000089 =
log 4-8847 =
688865
688776
000089
000062
688776 + -000062 = -688838
We need not have actually subtracted log 4*884 from log 4-885 to
obtain the difference '000089, for this difference is recorded in the
right-hand column of the Table, the column headed D. Thus if we
glance along the horizontal line in which we find log 4 -884, we shall
find the ' difference ' 89 in the column D, and the computation of
log 4-8847 will then appear as follows : —
log 4-884 =-688776 D = 89-
7, 62,3
log 4-8847 = 688838
141. This method of computing the logarithm of a number
intermediate in value to two tabular numbers is merely a variation
of the method of calculating logarithms exhibited in Examples 3, ^,
134, for
4 -885 = 4 -884 + '001 = 4 -884 x (1 + f -.|||)
log 4-885 = log4-884 + log (l + ^sVi)
= -688776 + 4:3181 of '4343
= -688776+ -000089
= -688865, as given in Table II,
(A.)
146 ARITHMETIC.
4-8847-4-8840+ •0007 = 4-884 x(l + ^'.%"g^7)
log 4-8847 = log 4-884 + log (l + ^^W)
= •688776 + ^^1^ of -4343
= -688776+ -7 of jJ^t of "4343
= -688776 + -7 of -000089. (B. )
= •688776+ -000062
= -688838, as found in Example 3.
Now the * dijQference ' -000089 obtained above in {A) is given in
the Table of Logarithms, and knowing this difference and log 4-884,
we may at once write down the line marked {B), ,
Example 4. Find log 2-718282.
log 2 -718 = -434249 D = 160
2 32,0
8 12,80
2 ,320
log 2-718282= -434294.
rSee%ijJ
EXERCISE XVII.
Find the logarithm of : —
1.
7-3254.
6.
2.
59512.
y.
3.
47763.
8.
4.
•0049056.
9.
5.
295-947.
10.
142.
To find the i
tiumb
676767. 11. 6-37839x10.
• 186825. 12. 6 -35639 x 10** .
80008. 13. 1-0832 X 10^ \
•00457009. 14. 4-30725 xlO~^
30033000. 15. 3-04763 xlO~\
simply reverse the process of finding the logarithm of a given
number.
Example 1. Find the number of which -656769 is the logarithm.
We look in Table II along the columns headed O till we find 65,
the two leading figures of ^656769, the mantissa of the given
logarithm, and in the columns between the line led by 65 and that
led by 66 we look for 6769, the remaining figures of the given
mantissa. We find these four figures in the line of the number 453
and in the column headed 7, hence
LOGARITHMATION. 147
•656769=log 4-537.
Had the given logarithm been 3' 650769, we should have found
the number 4' 537 by means of the mantissa and then have moved
the decimal point three places farther to the right as indicated by
the characteristic 3, thus obtaining
3- 656769= log 4537.
In like manner may be found
5- 656769 = log 453700.
* 4 -656769 = log -0004537.
Example 2. Of what number is '497150 the logarithm ?
On looking for -497150 among the logarithms of Table II we
cannot find this mantissa, we therefore take out the logarithm next
smaller than '497150 the given mantissa, and also the number
corresponding to the logarithm taken out. This gives us
•497068 = log 3-141.
Then subtracting '497068, the tabular logarithm from '497150,
the given mantissa, we obtain '000082 as difierence. From the
column of differences we find that log 3 '142 -log 3-141= -000138,
i. e., a difference of '000138 in the logarithms makes a difference of
•001 in the corresponding numbers, hence a difference of '000082
in the logarithms will make a difference of of '001= of
-000138 138
•001 = '00059 in the corresponding numbers, hence '497068 + -000082
= log (3-141+ -00059), i.e., '497150= log 3-14159.
The actual calculation will appear as follows :
'497150
068_ =log 3-141
138)820 5
1300 9
58
•497150 = log 3 14159
The division is performed by the method exhibited in Example i, §, 67, page
33. It is not carried farther than the quotient 9 because the ' remainder' 58 is
practically within the limit of error of the tabular logarithm -497068, which is
correct to but 6 figures and which represents all logarithms from '49706750 to
•49706849 ; consequently the * remainder ' 58 may be too small by 50 or too
148 ARITHMETIC.
large by 49. In the former case, the figure next following 9 in 3-14159, would
be 6, in the latter case it would, to nearest approximation, be i ; it is therefore
indeterminate with the tables at our command and consequently we omit it,
ending our computation with 9.
143. The process of computing the logarithm of a number inter-
mediate in value to two tabular numbers or inversely of computing
the number corresponding to a logarithm intermediate in value to
two tabular logarithms is termed Interpolation of logarithms or
of numbers as the case may be.
144. In the early part of Table II, the differences between
consecutive logarithms are comparatively large and they change
rapidly ; as a consequence interpolation will not in this part of the
Table yield accurate results. This difficulty may however be
avoided by employing Table III for all numbers and logarithms
within its range. The method of using this Table is the same as
that of using Table II.
EXERCISE XVIII.
Find the numbers corresponding to the following logarithms : —
1. -480007. 6. -817342. 11. 2-830083. 16. -000000.
2. -734960. 7. 1-817342. 12. 4- 830457. 17. 2 000204.
3. -740047. §.5^-817342. 13. "3-900000. 1§. -301030.
4. 2-477121. 9. 1/817342. 14.. 3-301000. 19. 3 010300.
5. -937700. 10. 5-817342. 15. T'oOOOOo. 20. -030103.
Find the characteristic-factor and cofactor of the numbers cor-
responding to the following logarithms : —
21. 11716671. 22. '5-534626. 23. 10-817037. 24. 14 660000.
25. 100-000123.
COMPUTATION BY HELP OF LOGARITHMS.
145. The use of logarithms in lessening the labor of computation
depends on the theorems numbered xxvii (a) and xxviii (a) of § 125.
These may be restated as follows : — •
A. TJie logarithm of a product is the aggregate of the logarithms of
the factors of the product.
LOGARITHMATION.
149
B. TJie logarithm of a quotient is the remainder resulti^ig from the
subtraction of the logarithm of the divisor from the logarithm of the
dividend,
C. The logarithm of a power is the product of the exponent of the
power and the logarithm of the base of the power.
D. Tlie logarithm of a root of a given number is the quotient arising
from the division of the logarithm of the given number by the
root-index of the required root.
As the root-index is the reciprocal of the exponent of the root considered as
a fractional power, dividing by the root-index is equivalent to multiplying by the
exponent, hence D. is comprehended under O.
146. The following examples will show how these theorems are
applied to facilitate computation.
Example 1. Find the weight in tons of a rectangular block of
stone measuring 7 '413 x5'822 x 3-224 and weighing •097221b. per
cubic inch.
Volume of block = 7 -413 x 5 -822 x 3 '224 cu. ft.
= 7 -413 X 5 -822 X 3 '224 x 1728 cu. in.
•. Weight of block = 7 '413 x 5 -822 x 3 "224 x 1728 x . 09722 lb.
= (7'413'X 5-822 x 3-224 x 1728 x -09722^2000) T.
log 7 -413 = -869994
log 5-822 = -765072
log 3-224 = -508395
log 1728 =3-237544
log -09722 ==2-987756
4-368761
log 2000 3-301030
log 11-6877^
1067731
443
372)288
276
156
= log 11-
. •. Wt. of block = 11 -6877 Tons.
It will be observed that adding the 2 (the negative 2) of 2-987756 is equivalent
to subtracting 2. This is merely another way of saying " Multiplying hy 'oi
is equivalent to dividing by 100. " The mantissa '987756 is not negative and is
therefore added like the other mantissas.
150 ARITHMETIC.
Examples. Find the continued product of 3-9497, 3-9634 and
3 '1416, correct to five significant figures.
log 3-949 = -596487
7 77
log 3-963 = -598024
4 44
log 3-141 = -497068
6 83
log 49-1793 1-691783
700 -log 49-17
89)83 9
29 3
_ ??.
.-. 3-9497x3-9634x3-1416 = 49 179 + .
Example 3. Divide 3 "728 x '4837 by '02383 x -09769.
log 3-728= -571476 log -02385 = 2 -377124
log -4837 = 1-684576 log-09769 = 2 -989850
•256052 3-366974
3-366974
2 -889078= log 774-6
,\ 3-728X •4837+(-02383x -09769)=774-6.
jS7i476 + i-684576=-57i476+ -684576- 1=1*256052- i=-256o52.
2-377124 + 2-989850= I -366974 - 4 = -366974 -3 = 3 -366974.
•256052 - 3 -366974 = -256052 + 3 - -366974 = 3 -256052 - ^366974 = 2 -889078^
Notice that additig i is changed into subtracting i, and that subtracting'^ is
changed into adding 3.
Example 4. Find (a) the 77th. power of 74*13 and {h) the 110th.
power of f |?f.
(a) log 74-13 = 1-869994 (Muhiply log 74-13 by 77.)
77
13-089958
log 74-1377=:143-989538 = 1434-log 9-762,
■77 143
74 13 =9-762x10 .
(6) log 4205 = 3-623766
log 7413 = 3-869994
i -753772
110
28 -914920= log 8-2209 - 28
JARITHMATION.
151
r4502
17413
= 8-2209x10
8-2209^10
(log 4205 -log 7413) X iio=(753772 - 1) x 110=82-914920-110 = 28 914920.
Example 5. Find (a) the 11th. root of 35480, (&) the 7th. root
of -00075367, and (c) the 7th. power of the 11th. root of '7576.
(a) log 35480TT=^ of log 35480= Jj of 4-549984= -413635
= log 2-592,
35480^1=2-592.
(6) log -00075367 ' = 1 of log -00075367 = \ of T-877181
= i of (7 +3-877181)=T+-553883=l-553883=log-358,
•00075367^= -358.
(c) log •7576TT = /- of log -7576 = /j o^ ^ '879440 = /t of
(11 + 10-879440) ="7 +6-923280=3 -923280 = log -83807.
•75761^1 = -83807.
In {b) we do not at once divide the characteristic 4 by 7 the root-index, for
this would introduce a negative fraction into the quotient, in addition to the
(positive) fraction arising from the division of the mantissa "877181, and a
negative fraction in the quotient must be avoided if the required root is to be
expressed decimally. To overcome the difficulty of a negative fraction we add
3 to the characteristic 4, thus making the negative part of the dividend an exact
multiple of the divisor 7 and to counterbalance the addition of 3 we add 3 to
the mantissa •877181. The corresponding operation on the number '00075367
- 7*5367 . 75367
of which 4-877181 is the logarithm, is the change of -^^^^ "ito ^^^^^^'
Had we divided 4 and '877181 separately by 7, the required root would have
been obtained in the form of a fraction, the numerator being the 7th root of
7*5367 and the denominator the 7th root_of loooo.
In {c) we add 10 to the characteristic i, to make the negative part of the
logarithm exactly divisible by the root-index 11, and we counterbalance this
addition of 10 to the characteristic by adding 10 to the mantissa ; i.e., we change
7-576+10
/ mto \j'
10 ]
576 X 10 +10
t^-ir
7-576 x-io
")
=8-3807X10 +10 =^-3807+10 =-83807.
Example 6. What power of 1 -05 is 2 ?
log 1 -05 X exponent = log 2.
exponent = log 2-^log 1 -05 =
-301030-j- -021189 = 14 207
14 .207
1-05 =2.
152 ARITHMETIC.
EXERCISE XIX.
Apply logarithms to obtain approximate values of the •following
indicated products, quotients, powers and roots : —
1. 3-7485 X 42-396 x 3-14159. 2. 2*96374 x 4-83625 x -284639.
3. -372856 x -129745 x -386429 x -47638.
4. 43-8629 x -0048579 x -27846 x 1-49635.
5. 78549 X -0029638 x 43-7865 x -0085247.
6. 493764-^879-63. 9. 1-62964^-047285.
7. 2-98573-=-4-76845, 10. -029683-^- -0023867.
8. -379648-V-57-6483. 11. 39-6452 x -084763^-427859.
12. -27634 X -0028463 -^ -058496.
13. 4 -3785 -=-4986 -43 X -29739.
14. 8-976 xio'^ x2-8648xl0~%(7-293xl0^)
15. 1-4783 x 10~^ X 2-9653 x 10~^ -^-(3-4965 x 10~'*^).
16. 48-739^ 26. •002«. «^ ,,,1
17. 1-4786^! 27. -001^^-
1§. -4763". 28. 1-8476^^. „^ .,a
19. l-045^^ 29. •8643T''a.
20. -0999^^. 30. -008643^5 „^ / ^ VV^
21. 1010. 31. •1*\
22. 2-748612. 32. -02496'^
36. {l)K
38. (/^)'^\
/ \100
3». iUV'''
23. -087542. 33. -00478
24. -08754'^. 34. (3-954xl0"^)'7' .^ /aossV'''
L / -10\.3 5 '^V. \ijt^2 7/ *
25. -087545. 35. (4*658x10 ) .
41. 248-7%3-14159^ 44. -037624 ^-2785'^.
42. 248-7^x3-14159^ 45. -17458^^1 x -03965~*^^''^
43. -3762^x^02785^
From log 2, log 3, log 7, log 11, and log 13/ taken from Table I
obtain, —
46. log 32. 50. log 1024. 53. log 676-676.
47. log 48. 51. log 2401. 54. log Sfgi.
48. log 49. 52. log 1-701. 55. log ^fPgf.
49. log -625.
LOGARITHMATION.
153
log 12 X log -0478= log -0478.
10 12 10
56. Show that log 10 x log 7 = 1.
7 10
57. Express 10 as a power of 2, L e., from log 2 obtain log 10.
10 2
From log 347 to base 10 given in Table II obtain log 10 to the
base, —
5§. 3-47. 60. 3470. 62. '0347.
59. 34-7 61. -347.
Prove that, — •
63. log 12 X log 3 = log 3. 65.
64. log'l2xlog^ 1-37= log 1-37.
10 12 10
66. Hence show how, from a table of logarithms to base 10 a
table of logarithms to any other base may be computed.
67. Prove that log 23 x log 14 x log 9 -log 9.
10 23 14 10
6§. What power of 2 is 7 ?
69. What power of 7 is 2 ?
70. What power of 7 -386 is 94-853 ?
71. What power of 94-853 is 7-386?
72. What power of 29-84 is 4738?
73. What power of 4-768 is -04768 ?
74. What power of '02837 is 1-05 ?
75. What power of "0476 is '000476 ?
Find the exponent of the power which 2 is of, —
76. 1-035. 7§. 1-06. §0.
77. 104. 79. 1-07. §1.
Find the number of figures in the developed value of, —
1000 100 123 13
§2. 2 , §3. 3 . §4. 47 . 85. 2378 .
Find the number of figures in the integral part of the developed
value of, —
123 156 14T 48.32
§6.4-7 . 87.1-1 . 88. 23-78 TT-. 89.3-576 - .
How many zeros are there between the decimal point and the
left-hand digit in the developed value of, —
90. -104^ 92. •047^'''. 94. '000856*^^\
91. '2'^^ 93. -00976t\ 95. •0477''^'^
What is the decimal order of the first digit on the left in the
developed value of, —
54
ARITHMETIC.
96.
97.
—37
2-843 .
4768 -3"^^
9§. -47683"'.
99. -04862"'^'.
-•0473
100. -000674
147. It is now necessary to examine the degree of precision
attainable in calculations made by means of Tables I, II and III.
The mantissse entered in these tables are not absolutely correct,
they are merely the nearest representations of the correct values
attainable with six decimal places, and they are in some cases in
excess, in other cases in defect, but the excess or the defect is never
greater than 6 in the seventh decimal place, ^. e., the error in
a tabular logarithm, never exceeds '0000005. Now by
means of Table B p. 132 we find that "0000005 is the logarithm of
1-00000115, hence any logarithm actually entered in Table I,
Table II or Table III may be the logarithm of its corresponding
tabular number divided by 1-00000115, or of the tabular number
multiplied by 1-00000115, or of any number between these limits.
Hence the error in a tabular number never exceeds the
•000001 15 of the tabular number itself. If a logarithm be
obtained by interpolation, the operation of interpolation may itself
introduce an error not greater than '0000005, and this error may
be on the same side as the tabular error and consequently added to
it, so that a logarithm obtained by interpolation may be in error by
•000001.
If then we perform any calculation by help of logarithms, the
result is liable to an error of the "00000115 part (say the one nine-
hundred thousandth part) of itself for every logarithm employed and
for every interpolation made in the process of calculation. If a
logarithm be multiplied by any number, we must multiply the
possible error from that logarithm by the multiplier of the logarithm.
This is assuming that the errors lie all on one side, i. e. are all in
excess or all in defect, and that each error is nearly at its limit.
The cases in which this will occur will be comparatively rare, yet
rare as they may be, we must take them into account in estimating
the limit beyond which our result cannot err.
In ordinary computations by the help of 6-figure logarithms, we
may count on the result as almost certainly correct to 5 significant
figures and as probably correct to 6 figures. We exclude cases of
involution to high powers.
LOGARITHMS.
EXERCISE XX.
1 . The squares of the times of revolution of the planets round
the sun are as the cubes of their mean distances from the sun, i.e.,
A and B being two planets, if a fraction be formed having J.'s
time of revolution round the sun as numerator and B's time of
revolution as denominator and a second fraction be formed having
A 's mean distance from the sun as numerator and jB's mean distance
as denominator, the square of the former fraction will be equal to
the cube of the latter. Mercury performs a revolution about the
sun in 87 '969 days ; Venus performs a revolution in 224*701 days ;
the Earth, in 365 "256 days ; Mars, in 686*98 days ; Jupiter, in
4332*585 days, and Saturn, in 10759*22 days ; determine the mean
distance from the sun of Mercury, "Venus, Mars, Jupiter and
Saturn respectively, taking the mean distance of the earth from the
sun as the unit of length. Express these distances in miles,
assuming the mean distance of the earth from the sun to be (a)
91,430,000, (h) 92,780,000.
2. A pupil who was "strong at figures" undertook to multiply
15 by itself on the first day of his holidays, to multiply the product by
itself on the second day, to^ultiply the second product by itself on
the third day, to multiply the third product by itself on the fourth
day, and so to continue to do each day (Sundays and Saturdays
excepted) to the end of his holidays which were to last four weeks.
How many figures would there be in the twentieth product thus
formed ? Determine the first five and the last ten figures of this
product. How long would it take the boy to write down this
product at the rate of three figures per second? How many figures
would there be in the partial product formed in computing the
twentieth product from the nineteenth, assuming that in the
nineteenth product the several figures 0, 1, 2, .... 9, occur each
an equal number of times except that 5 occurs once oftener than
any of the others 1 Find to the nearest number of days how long
it would take 100 men to compute these partial products, working
at the rate of two figures per second for six hours per day for 313
days per year,
[Obtain log 15 to fifteen places of decimals from log 2 and log 3
156 ARITHMETIC.
which are given correct to fifteen places, the former on p. 133, the
latter on p. 135.]
3. In § 131, it is asserted that " had the base [in Table A] been
2-718281828459, the modulus would have been 1." Test the truth
of this assertion by forminaj a table with exponents the same as
those in Table A but with 2*71828 as base instead of 10, and with
the calculations carried to six decimal places instead of to eight.
(Extract the roots with the aid of Tables of Logarithms II and III. )
Show how to employ the table thus formed to calculate logarithms
to base 2-71828.
4. Form a six-decimal-place table similar to Table A but with 12
as base instead of 10 and show therefrom and from the Tables of
Logarithms that the modulus of logarithms to base 12 is equal to
log 2 -71828 -^ log 12 which is equal to log 2-71828.
12
Show how to employ this table to calculate logarithms to base 12.
6. A seven-figure table similar to Table A but with 2-71828 as
base instead of 10, having been formed, show that if the exponents
in the second column be all multiplied by -4342945, the modulus of
logarithms to base 10, the numbers in the first and third columns
remaining meanwhile unchanged, the common base of the second
column will be changed from 2-71828 to 10.
6. A seven-figure table similar to Table A but with 2-71828 as
base instead of 10, having been formed, show that if the pumbers
in the first and third columns be retained unchanged but the
exponents in the second column be all multiplied by the modulus
of logarithms to base 12, the common base of the second column
will be changed from 2-71828 to 12.
7. State and prove the general theorem of which the theorems
of Probs. 4 and 5 are particular cases, and thence show that the
modulus of the logarithms to any given base may be used as a
constant multiplier to convert logarithms to 2*71828 as base into
the corresponding logarithms to the given base.
8. Hence show that the modulus of the logarithms to a given
base is the logarithm of 2*71828 to the given base. (See Exercise
XIX, Frob. 66.) Example, "434294 = log 2 -71828.
9. If the difference between the logarithms of any two numbers
be divided by the difference between the numbers and the quotient
I
I
LOGARITHMS.
be multiplied by each of the two numbers, the products will be one
greater the other less than the modulus of the logarithms. Test
the accuracy of this theorem in the case of logarithms to base 10 by
applying it to numbers and tlieir logarithms selected from the
Tables of Logarithms I, II and III. This theorem seems to fail in
application to many pairs of numbers selected from Table III and
from the latter part of Table II, show that these may be cases of
seeming and not of real failure of the theorem. Example ; — Table
II gives log 7 '001 — log 7 = "000062 which is correct to six decimal
places, but to ten decimal places the difference is '0000620376 ; the
theorem fails if '000062 is taken as the difference between log 7 '001
and log 7, but it does not fail if '0000620376 is taken as the
difference.
10. Show that the theorem of Prob. 8 may be deduced from the
last theorem of § 130 in all cases in which the difference between the
numbers is less than the ten-thousandth part of the smaller number.
(In the case of six-figure logarithms, it will be sufficient if the
difference between the numbers is not greater than the thousandth
part of the smaller number as will at once appear if the numbers in
the third column of Table A be reduced to six decimals.)
1 1 . Show that the theorem of Problem 8 enables us to calculate
the differences of the logarithms in Tables II and III directly from
the modulus '4342945, without any previous calculation of the
logarithms themselves and that consequently a table of differences
having been thus computed, Tables II and III may be formed by
mere additions.
[This "method of differences " is the method which is nowemployed
whenever it is found desirable to extend a table of logarithms or,
for the purposes of verification, to recalculate any part of such a
table. In actual practice, the differences of the logarithms are not
obtained directly by division of the modulus as here proposed, but
are themselves computed from second differences. The number of
divisions which must be made, is thus greatly reduced.]
12. The modulus of logarithms to base 10 is "43429448 and
log 49 is 1 '69019608 each correct to eight decimals, determine
therefrom the logarithms of 4901, 4902, 4903, 4904, 4905, correct
in each case to six decimal places.
158' ARITHMETIC,
13. Show that the exponents of the powers to which the
bases I'Ol, 1-02, 1-03, 1-04, I'Oo, I'OB, 1-07 must severally be
raised to produce 2 are approximately equal to 70 divided by
1, 2, 3, 4, 5, 6 and 7 respectively, and to produce 3 the several
exponents are approximately the quotients of 110 divided by the
same seven numbers.
In the following problems, the values of the logarithms which are
stated to be ' given ' are to be taken from the Tables of Logarithms,
and the values of the logarithms to be computed are to be found
correct to six places of decimals.
14. Given log 2 and log 3 and 2^ x 3% 7^- 1000188, find log 7.
15. Given log3 and 3^^ = 177147, ll" =1771561, find log 11.
16. Given log 2, log 3, log 7 and log 11 and 2^x7x11 = 1232,
3^^ X 13^ = 123201, find log 13.
17. Given log 2 and log 7 and 7 ^ x 17 = 2000033, find log 17.
18. Given log 2 and log 3 and 2x3 x 19^ = 10000422, find
log 19.
19. Given log 2, log3 and log 11 and 2 x 3^ x if =71874,
5% 23 = 71875, find log 23.
." 2 2
20. Given log 2, log 3, log 7, log 11 and log 13 and 2x5x7
= 9800, 3* X 11^ =9801, 2 x 13% 29 = 9802, find log 29.
21. Given log 2, log 3, log 7, log 11 and log 13 and 2 x 3 x 7 x 11
X 13 = 6006, 6* x31^ =600625, find log 31.
22. Given log 3, log 7, log 11 and log 13 and s" x 7 x 11 x 13 x 37
=999999, find log 37.
23. Given log 17, log 19 and log 23 and 17 '^ x 19 " x 23?^
= 410006814589, find log 41.
24. Given log 2, log 3, log 7, log 11 and log 13 and 2" x 3" x 43
= 24768, 7x11^ xl3^ =2476803329, find log 43.
25. Given log3 and log 17 and 17 x 47*^ = 30004847, find log 47.
I
I
I
CHAPTER V.
MENSURATION OR METRICAL GEOMETRY.
148. To Measure any magnitude is to determine what
multiple or part or multiple of a part the magnitude is of a specified
magnitude of the same kind selected as a standard or unit of
measurement.
The number which expresses what multiple or part or multiple of
a part the measured magnitude is of the unit, is termed the
Measure of the magnitude.
The relation which is determined or smight to be determined by
such measurement is called the Ratio of the magnitude measured
to the unit of measurement.
149. If the first of two quantities of the same kind be divided
by the second, the quotient will be the measure of the first quantity
in terms of the second quantity as unit.
Thus 4 is the measure of 12 ft. in terms of 3 ft. as unit, for
12 ft. -^3 ft. =4 or, as it may otherwise be expressed, 12 ft. =4 (3 ft.)
The measure of 3 oz. in terms of 8 oz. as unit is | or '375 for
3oz. -j-8 oz. =§='375 or, otherwise expressed, 3 oz. =| (8 oz.)
= -375(8oz.)
150. Four ma2;nitudes are said to be proportional, to be in
proportion or to form a proportion, if the ratio of the first
magnitude to the second is the same as the ratio of the third
magnitude to the fourth.
151. Hence if four magnitudes he in proportion and if the first
magnitude he a multiple of the second, the third magnitude will he the
same midtiple of the fourth ; if the first magnitude he a part of the
second, the third magnitude vjUI he the same part of the fourth ; if the
first magnitude he a midtiple of a part of the second the third
magnitude will he the same midtiple of the same part of the fourth.
152. A, B, Oand D denoting four magnitudes of which A and
B are of the same kind and G and D also of the same kind, but not
necessarily of the same kind as A and B, the expression A:B::C:D,
read " J. is to 5 as 0 is to Z)," denotes that the magnitudes A, B,
C and D are in proportion in the order named, i. e. , that if A is a
160 ARITHMETIC.
multiple of By C is the same multiple of D; if ^ is a part of B, C is
the same part of D ; if ^ is a multiple of a part of B, C is the same
multiple of the same part of D ; and generally that the ratio of A to
B is the same as the ratio of G to D.
Thus 12 in. = 4 (3 in. ) and 20 lb. - 4 (5 lb. )
.-.12 in. :3in. :: 201b. :41b., read "12in. istoSin. as201b. is
to 41b."
So also, 15 gal. =f (35 gal.) and 1| min. =f (3| min.)
15 gal. : 35 gal. : : 1|^ min. : o\ min. ,
read " 15 gal. is to 35 gal. as 1^ min. is to 3| min.
163. The measure of the length of a line is the number which
expresses the ratio which the measured line bears to a line selected
as the unit of length.
The unit of length or linear unit is usually either
(a), a fundamental unit, or
(6), a multiple or a fraction of some fundamental linear unit.
The yard and the metre which are both defined by physical
standards (see pp. 12 and 17,) are examples of fundamental linear
units. The mile and the kilometre are examples of units which are
multiples of these fundamental units ; the inch, the foot and the
centimetre are examples of units which are definite parts or
determinate fractions of fundamental units.
Example 1. A certain rope is stated to be 37 yd. long. Here the
unit of measurement is the linear unit, a yard, and the measure
of the declared length of rope is the number 37.
Example 2. The length of the circumference of a certain circle is
found to be 47 '85 in. Here the number 47 '85 is the measure of the
length of the circumference, and the linear unit, an inch, is the
unit of measurement.
154. The measure of the area of a surface-figure is the number
which expresses the ratio which the measured figure bears to some
determinate surface-figure chosen as the unit of area.
The unit of area generally selected is either
(a), a square whose side is some specified unit of length, or
(6), a multiple of such a square.
Example 1. The area of the floor of a certain hall is 240 sq. yd.
Here the measure of the area of the floor is 240 and the unit of
MENSURATION. 161
measurement is the areal unit, a square yard, i. e., a square whose
sides are each a yard in length.
Example 2. The area of a certain field is found to be 7^ ac.
Here the measure of the area of the field is 7^ and the unit of
measurement is the areal unit an acre which is equal to 10 square
chains or 4840 square yards.
155. The measure of the volume of any solid or space-figure is
the number which expresses the ratio which the measured figure
bears to some determinate space-figure chosen as the unit of volume.
The unit of volume is either
(a), a cube whose edge is some specified unit of length, or
(6), the volume of a given mass of some specified substance under
stated conditions, or
(c), a multiple or a fraction of this volume.
Example 1. The volume of air in a certain school-room is 560
cu. yd. Here the measure of the volume of air is 560 and the unit
of measurement is the volume-unit a cubic yard, i. e. a cube whose
edges are each a yard long.
Example 2. A certain pitcher will hold f of a gallon of water.
Here the measure of the capacity of the pitcher is f and the unit of
measurement is a gallon, i.e., the volume often Dominion standard
pounds of distilled water weighed in air against brass weights with
the water and the air at the temperature of sixty-two degrees of
Fahrenheit's thermometer and with the barometer at thirty inches.
156. Two plane rectilineal figures are sjmilar if to every angle
in one of the figures there is a corresponding equal angle in the
other, and if also the sides about the angles in one figure are
proportional to the sides about the corresponding angles in the
other. The sides extending between corresponding angular points
are termed correspoQiding or homologmis sides.
157. Hence if the lengths of two sides of a triangle be given and
also the length of one of the corresponding sides of a triangle
similar to the former, the length of the second corresponding side
of the latter triangle can be determined.
158. If tivo triangles have two angles of the one equal to two
angles of the otJier, each to each, the triangles will be similar. (Euclid,
vi, 4.)
K
162 ARITHMETIC.
Example. Let the triangles ABC and K L M have the angle B
equal to the angle L and the angle C equal to the angle M ; also let
the sides A B, B C ahd C A be respectively 6, 8 and 9 sixteenths of
an inch in length and the side L M he 12 sixteenths of an inch long.
Find the lengths of the sides K L and K M.
K
k
B C L. M
By § 158 the triangles are similar and the corresponding sides
areABandKL, BCandLM, CAandMK;
KL :LM : : AB :BC.
The length of A B is 6 and that of B C is 8 sixteenths of an inch,
.-. - AB = fofBC
KL=fofLM
=f of 12 sixteenths of an inch
= 9 II M II II
Similarly, . • KM : ML : : AC : C B
and AC = |ofCB
KM = |ofML
= f of 12 sixteenths of an inch
= 13^ II M II II
EXERCISE XXI.
1. ABC and KLM are similar triangles, the angles A and K
being equal to one another and the angles B and L also equal to one
another ; the side A B is 9" long, the side B C is 10" long and the
side K L is 22 b" long, find the length of the side L M.
2. ABC and GHK are similar triangles, A and G being
corresponding angles and B and H also corresponding angles ; the
lengths of the sides AB, AC, GH and HK being 7", 15", 5-25"
and 15" respectively, find the lengths of B C and &IL 4/(^
3. A B C and GHK are similar triangles, the angles A and G
being equal to one another and the angles B and K also equal to
one another; the measures of the sides aTe AC = 25, GH=44,
HK = 35andK G=75. Find the measures of the sides ABand BC.
MENSURATION.
163
I
4. In A B, a side of the triangle A B C, a point D is taken, and
the straight line D E is drawn parallel to the side B C ; find the
length of D E, the length of A B being 35', that of B C 24' and that
of A D 11' 2 ".
5. The construction being the same as in problem 4, find the
measure of D E, given that the measures of A D, D B and B C are
7, 23 and 18 respectively.
6. The construction being the same as in problem 4, find the
measure of A D, given A B = 45, B C = 20 and D E - 8.
7. The construction being the same as in problem 4, determine
the length of B C, given that A D is 24 yd. long, D B, 30 yd. long,
and DE, 18 yd. long.
§. The construction being the same as in problem 4, determine
the length of B C, A B being 104 ft. long, B D being 44 ft. loijg, and
D E being 80 ft. long.
9. The construction being the same as in problem 4, what will
be the length of A D if B D be 36 chains long, B C, 36 chains long,
and D E, 15 chains long.
>/ 10. A stick 3' in length placed upright on the ground is found to
cast a shadow 2' 6" long, what must be the height of a flagpole
which casts a shadow 28' in length ?
v^ 11. A gas-jet is 12 ft. above the pavement, how far from the
ground-point directly beneath the jet must a man 5 ft. 8 in. in
height stand that his shadow may be 6 ft. long.
"^ 12. The vertical line through a gas-jet 9' 4" above the sidewalk
is 10*" 6" from a man 5' 10" in height, find the length of his shadow.
y 13. An electric light is 15 ft. above the pavement, what will be
the length of the shadow of a man 5 ft. 10 in. in height if he stand
30 ft. from the vertical line through the light ?
^14. The parallel sides of a trapezoid are respectively 27 ft. and
35 ft. in length and the non-parallel sides are respectively 18 ft.
7 in. and 23 ft. 11 in . long. The latter sides are produced to meet ;
find the respective lengths of the produced sides between the point
of meeting and the shorter of the parallel sides of the trapezoid.
. 15. The lengths of the parallel sides of a trapezoid are 10-75
and 12 "35 chains' respectively ; four straight lines are run across
164 ARITHMETIC.
the trapezoid parallel to these sides so that the six lines are at
equidistant intervals ; find the lengths of these four lines.
sj 16. The lengths of the parallel sides of a trapezoid are 15 and
28 inches respectively, and of the non-parallel sides 12 and 20
inches respectively ; through the intersection of the diagonals of
the trapezoid a straight line is drawn parallel to the parallel sides.
Find the lengths of the sections into which this line divides the
non-parallel sides.
1 7. Taking the diameter of the sun to be 880,000 miles and the
sun's distance from the earth to be 92 400,000 miles, what must be
the diameter of a circular disk that it may just hide the sun when
held between the eye and the sun and 21 inches in front of the eye ?
1 8. Three men A, B and G stand in a row on a level pavement,
^'s height is 5' 3|", ^'s is 5' 9" and O's is 6' l^; if A stand 10' to
the right of B, how far to the left of B must G stand that the tops
of the heads of the three men may range in a straight line ?
19. The lengths of the sides of a triangle are 7 yd., 11yd. and
12 yd. respectively and the perimeter of a similar triangle is 25 ft. ;
find the lengths of the sides of the latter.
V 20. The perimeters of two similar triangles are 26 ft. 6 in. and
56 yd. 2 ft. respectively. A side of the smaller triangle is 7 ft.
long and a non-corresponding side of the larger triangle is 17 yd.
1 ft. in length. Find the lengths of the other sides of Ihe triangles.
159. Any one of the sides of a parallelogram having been selected
as the base of the figure, the altitude of the parallelogram
is the perpe7idicular distance between the base and the side parallel
to the base.
One of the sides of a triangle having been selected as the base
of the figure, the opposite angle becomes the vertex, and the
altitude of the triangle is the length of the perpe7idicidar from
the vertex on the base, or the base produced.
160. A polyhedron is a solid-figure enclosed by plane
polygons.
A polyhedron enclosed by four polygons, (in this case, triangles)
is called a tetrahedron ; by six, a hexahedron ; by eight, an
octahedron ; by twelve, a dodecahedron ; by twenty, an icosa,hedron.
MENSURATION.
The faces of a polyhedron are the enclosing polygons. If the
faces are all equal and regular, the polyhedron is regular.
The edges of a polyhedron are the lines in which its faces
meet.
The summits of a polyhedron are the points in which its
edges meet.
A Prismatoid is a polyhedron two
of whose faces are polygons situated
in parallel planes and whose other faces
are triangles having the sides of the
polygons as bases and having their
vertices at the angular points of the
polygons. The polygons situated in
parallel planes are called the ends of '^i
the prismatoid, and if one of them be
taken as the base of the solid, the other
becomes the opposite 'parallel face. The other faces are called the
lateral faces and their common edges are named the lateral edges.
(A B C D E F G is a prismatoid on the quadrilateral base A B C D,
the opposite parallel face is the triangle E F G.)
The midcross-section of a prismatoid is its section by a plane
parallel to the planes in which are situated the end polygons and
midway between these planes. Tlie midcross-section therefore bisects
all the lateral edges of the prismatoid. (HKLMNPQ is the
midcross-section of the prismatoid A B C D E F G. The angular
points H, K, L, M, N, P, Q are the mid-points of A F, B F, B G,
C G, D G, D E, and A E respectively. )
If the bases of two adjacent lateral faces of a prismatoid are
parallel the two faces lie in one plane and together form a
trapezoid.
A Prismoid is a prismatoid whose
lateral faces are all trapezoids. The end
polygons must therefore have the same
number of sides and each corresponding
pair must be co-parallel. (ABCDEFG
II K L is a prismoid with pentagonal
ends ABODE andFGHKL.)
166 ARITHMETIC.
A Wedge is a solid enclosed by five plane
figures, the base is a trapezoid, two of the
lateral faces are trapezoids and the other two
lateral faces are triangles, A wedge is therefore
a prism atoid on a trapezoidal base, in which the
face opposite the base has become reduced to a b^ -~*(?
straight line parallel to the two co-parallel sides of the base.
(ABCDEF is a wedge ; the base ABCD is a trapezoid, the sides
BC and AD being parallel to each other ; EF is parallel to both
BC and AD, hence BCEF and ADEF are both trapezoids.)
A Prism is a polyhedron two of whose faces are parallel
polygons, and the other faces, parallelograms.
The bases or ends of a prism are the parallel polygons.
The altitude of a prism is the perpe'ndicvlar distance between
the planes of its bases.
A right prism, is one whose lateral edges are perpendicular to
its bases.
A parallelepiped is a prism whose bases are parallelograms.
A parallelepiped is therefore a solid contained by six parallelograms
of which every opposite pair are parallel.
A quad or quadrate solid is a right parallelepiped with
rectangular bases. It is therefore contained by six rectangles. A
cube is a quad whose faces are all squares.
161. A cylindric surface is a surface generated by a straight
line so moving that it is always parallel to a fixed straight line.
A cylinder is a solid enclosed by a cylindric surface and two
parallel planes.
The bases of a cylinder are the parallel plane faces.
The altitude of a cylinder is the perpetidicvlar distance
between the planes of its bases.
A right cylinder is one in which the generating lines of the
cylindric surfaces are perpendicular to the bases of the cylinder.
A right circular cylinder is a right cylinder whose bases are
circles,
MENSURATION.
167
I
A Cylindroid is a solid bounded by two
parallel planes and the surface described
by a straight line which simultaneously
describes two closed curves, one in each
of the parallel planes. The plane figures
enclosed by the curves in the parallel
planes are called the ends of the cylindroid.
A Sphenoid is a prismatoid or a
cylindroid, one of whose ends has become
reduced to a line.
162. A pyramid is a polyhedron one
of whose faces, called the base, is a
polygon and whose other faces are triangles whose bases form the
sides of the polygon and whose vertices meet in a point called
the vertex of the pyramid.
A pyramid is therefore a prismatoid one of whose parallel ends
has become reduced to a point.
A regular pyramid is one whose base is a regular polygon and
whose other faces are equal isosceles triangles
The altitude of a pyramid is the length of the ijer'pzndk.vlar
let fall from the vertex on the plane of the base.
163. A conical surface is a surface generated by a straight
line which so moves that it always passes through a fixed point
called the vertex of the surface.
A cone is a solid enclosed by a conical surface and a plane. It
is therefore a cylindroid one of whose parallel ends has become
reduced to a point.
The base of a cone is the plane face opposite the vertex.
The altitude of a cone is the length of the perpendicular let
fall from the vertex on the plane of the base.
A right circular cone has a circle for its base, and the straight
line joining the vertex of the cone and the centre of the base is
perpendicular to the plane of the base.
The frustum of a pyramid or of a cone is the portion
included between the base and a plane cutting the pyramid or the
cone parallel to the base.
168 ARITHMETIC.
164. Two polyhedra are similar if to every solid angle in one
of them there is a corresponding equal solid angle in the other, and
to every face of one of them there is a corresponding similar face in
the other.
The corresponding edges of similar polyhedra are those
which are corresponding sides of corresponding faces.
165. Similar surface-figures need not be rectilinea., they need
not even be plane surfaces. Thus all circles are similar to one
another, parallel plane sections of a cone are similar figures, all
spherical surfaces are similar to one another, and generally the
complete surfaces of similar solids are themselves similar. If two
plane surface-figures are similar, they are, or they may be so placed
as to be, parallel plane sections of a pyramid or else of a cone.
Similar solid-figures are not necessarily bounded by plane
surfaces ; e. </. , all spheres are similar to one another, so also are the
spheroids described by similar ellipses rotating about corresponding
axes.
[Similar figures, whether surface or solid, may be described as
figures which are alike in form but which are not necessarily equal
in size.]
166. In the theorems which immediately follow, the areal unit
is the square and the volume unit is the cube described on the
linear unit as side and edge respectively.
167. The measure of the area of a square is the square of the
measure of the length of a side of the square.
The measure of the length of a side of a square is the square root of
the measure of the area of the square.
168. The measure of the volume of a cube is the cube of the measure
of the length of an edge of the cube.
The measure of the length of an edge of a cube is the cube root of the
measure of the volume of the cube.
Examples. If the length of a side of a square be 5 ft. , the area of
the square will be 5^ sq. ft. If the area of a square be 4840 sq. yd.,
1
the length of a side of the square will be 4840^ yd.
MENSURATION.
If the length of an edge of a cube be 7 "3 in. , the volume of the
cube will be 7*3^ cu. in. If the volume of a cube be 10 cu. ft., the
length of an edge of the cube will be 10^ ft., and the area of a face
of the cube will be 10^ sq. ft.
169. In the formulae which follow S, a and b denote the measures
of the area, the altitude and the length of the base respectively and
r, I, t and z subscribed^ to S are to be severally read rectangle,
parallelogram, triangle and trapezoid.
i. The measure of the area of a rectangle is the product of the
mea^^wres of the lengths of two adjacent sidesj or
S, = ab.
(Special case, — square.)
ii. The measure of the area of a parallelogram is the product of
the measures of the altitude and the length . — y
of the base, or
8i = ab.
(Special case, — rectangle.)
iii. The memure of the area of a triangle is ONE-HALF of the
product of the measures of the altitude avid the
length of the base, or
St = ^ab.
(Special case, — sector of a circle, including
circle itself)
iv. The measure of the area of a trapezoid is ONE-HALF of the
product of the measure of the altitude and the suth of the measures
of the parallel sides, or
(Special cases, — parallelogram, tri-
angle and sector of an annulusj,ncluding
antrndus itself.)
Example 1. The width of a, rectangular building-lot is to its
170 ARITHMETIC.
LENGTH
length as 3 to 5, and if the length
of the lot be increased by 8 yd. and
the width by 5 yd. , its area will be
increased by 481 sq. yd. Find the
length of the lot.
The increment to the lot may be
considered to consist of three parts,
viz.,
1° A rectangle 8 yd. by the width of the lot ;
2° A rectangle 6 yd. by the length of the lot ;
3° A rectangle 8 yd. by 5 yd.
The area of these three rectangles taken together is 481 sq. yd.
and the area of the 3° rectangle is 40 sq. yd. Subtracting 40 sq. yd.
from 481 sq. yd. , there will remain 441 sq. yd. as the area of the 1°
and 2° rectangles taken together.
The 1° rectangle is 8 yd. by the width of the lot.
The width of the lot is f of its length.
Therefore the 1° rectangle is 8 yd. by f of the length of the lot,
which is equivalent to a rectangle ^^ yd. by the length of the lot,
—^ yd. by the length of the lot.
The 2° rectangle is 5 yd. by the length of the lot.
Therefore the two rectangles are together equivalent to a
rectangle (^^* + 5) yd. by the length of the lot,
=^/ yd. by the length of the lot.
The sum of the areas of these two rectangles is 441 sq. yd. ;
. '. ^- of the measure of the length of the lot = 441 ;
the measure of the length of the lot = 441 -^ ^^,
=45;
. •. the length of the lot is 45 yd.
Example 2. A rectangular park is 400 yd. by 660 yd. It is
surrounded by a road of uniform width the whole area of which fs
one-sixth of the area of the park. Determine the width of the road.
The area of the park is (400 x 660)sq. yd. - 264000 sq. yd.
The area of the road is J of 264000 sq. yd. = 44000 sq. yd.
Therefore the area of the rectangle composed of both road and
park = 264000 sq. yd. + 44000 sq. yd. = 308000 sq . yd".
The park is a rectangle 260 yd. longer than it is wide.
MENSURATION.
171
W'DTH + 2S0 YD
,
-^: WIDTH -^
i
no YD
130 YO.
len the road is included with the park, both the length and
the width of the rectangle is increased by double the width of the
road ; the resulting block of land is therefore still a rectangle
260 yd. longer than it is wide.
Hence if thp length of the block
be reduced by 130 yd. and the
width of the block when thus
shortened be increased by 130 yd. ,
the resulting rectangle will be a
SQUARE whose sides will be each
130 yd. longer than the width of
the original block.
Reducing the length of the block
by 130 yd takes from the block a
rectangle 130 yd. by the width of the block.
Increasing the width of the shortened block adds to this block a
rectangle 130 yd. by 130 yd. more than the width of the original
block, i. e. , it adds a rectangle 130 yd. by the width of the original
block and a square 130 yd. square.
Hence the two operations of reducing the length of the original
bjpck and increasing the width of this shortened block increase the
area of the resulting figure as compared with the area of the original
block, by the area of the ' completing square ' of 130 yd, square,
i. e., by an area of 130 ^ sq. yd. =16900 sq. yd.
The area of the original block was found to be 308000 sq. yd.
The area of the completing square has been found to be
16900 sq. yd.
Therefore the area of the completed square or square block will
be 308000 sq. yd. +16900 sq. yd. = 324900 sq. yd.
Therefore the length of the side of the square block = (324900)^yd.
= 570 yd.
Therefore the length of the rectangular block = 570 yd. + 130 yd.
= 700 yd.
The length of the park = 660 yd.
Therefore double the width of the road = 700 yd. - 660 yd. = 40 yd.
Therefore the width of the roftd = 20yd.
172 ARITHMETIC.
EXERCISE XXII.
* 1. Find to the nearest inch the length of the side of a square
whose area is an acre ?
2. A square field contains exactly 8 acres. Determine the length
of a side of the field, correct to the nearest link.
3. The area of a chess-board marked in 8 rows of 8 squares each,
is 100 sq. in. Find the length of a side of a square.
4. On a certain map it is found that an area of 16000 acres is
represented by an area of 6 '25 sq. in. Give the scale of the map in
mile* to the inch and also in the form of a ratio.
5. A rectangle measures 18' by 30' ; find the difierence between
its area and that of a square of equal perimeter.
6. Six sheets of paper measuring 8 in. by 10 in. weigh an ounce ;
find the weight of 120 sheets of the same kind of paper, each sheet
measuring 11 in. by 17 in.
7. Two rectangular fields are of equal area, one field measures
15 chains by 20 chains, the other is square. Find the length of a
side of the latter field, correct to the nearest link.
8. How many stalks of wheat could grow on an acre of ground,
allowing each stalk a rectangular space of 2" by 3" 1
9. How many pieces of turf 3' 6" by 1' 3" will be required to sod
a rectangular lawn 28' by 60' ?
10. Sidewalks 12 ft. wide are laid on both sides of a street
440 yd. long. Find the cost of the sidewalks at $1'35 per square
yard for the pavement and 75 cents per lineal yard for curbing ;
deducting three crossings of 64 ft. each on both sides of the street.
11. The area of a rectangular field is 15 acres ; the length of the
field is double the width, find the length of the field.
12. How many yards of fencing- wire will be required to enclose
a rectangular field thrice as long as it is wide, if the field contain
10 acres and the fence be made 5 wires high ?
13. The lengths of the sides of a rectangular piece of land are as
3 to 8, and its area is 60 acres. Find the lengths of the sides.
14. The perimeter of a rectangle is 154 in. , and the difierence in
MENSURATION.
173
length of two adjacent sides is 11 in. Find the area of the
rectangle.
15. The length of a rectangle is 88 ft, ; if the width were
increased by 8ft., the area of the rectangle would in such case be
616 sq. yd. Find the width of the original rectangle.
16. The area of a certain rectangle is 1980 sq. yd. If the length
of the rectangle were increased by 12 ft. , the area would be
2100 sq. yd. Determine the lengths of the sides of the rectangle.
17. Find the difference between the perimeter of a square field
containing 22*5 acres and the perimeter of a rectangular field of
equal area, the length of the latter field being to its width as 5 to 2.
18. A rectangular block of building-lots is 660 ft. long by 198 ft.
wide. Find the area covered by an eight-foot sidewalk around the
block just outside of it.
19. A six-foot sidewalk of Sin. planks is to be laid around a
rectangle 266 ft. 8 in. by 480 ft. , the inner edge of the sidewalk to
be twelve inches out from the sides of the rectangle. Find the
value at $14 the M, board-measure, of the planking for the sidewalk.
y 20. Find the areas of the outer and the inner surface of a hollow
iron cube measuring Sin. on the outside edge, the iron being fin.
thick.
SI. Find the area of the inside surface of a hollow quad
measuring 3' 2" by 2' 8" by 2' 1" externally, the enclosing walls
being Ij" thick.
\/ 22. The length of the base of a parallelogram is 45 ft. ; the
length of the perpendicular on the base from the opposite side is
28 ft. ; the length of a side adjacent to the base is 35 ft. ; find the
length of the perpendicular on this side from the side opposite to it.
^ 23. The adjacent sides of a parallelogram measure 132 ft. and
84 ft. respectively and the area of the parallelogram is two-thirds of
that of a square of equal perimeter. Find the perpendicular
distance between each pair of parallel sides.
V 24. Find the cost of painting the gable-end of a house @ 22 ct.
per sq. yd. , the width of the house being 32 ft. ; the height of the
eaves above the ground, 36 ft. ; and the perpendicular height of
the ridge of the roof above the eaves, 16 ft. ;, ^ c/ , .: ^
25. Find the area of a field in the form of an isosceles right-angled
174 ARITHMETIC.
triangle, the length of the perpendicular on the hypothenuse being
7*50 chains. ^^r ^ tl'" CUt^
V 26. The length of one of the diagonals of a quadrilateral is
27 '7 ft. and the lengths of the perpendiculars on this diagonal from
opposite angles of the quadrilateral are 18 '5 ft. and 11 '3 ft.
respectively. Find the area of the quadrilateral, 1°, if the diagonal
lies wholly within the quadrilateral ; 2°, if the diagonal lies wholly
without the quadrilateral. //; ///2> '/iS^/^ *^ f^'ftr^f^^t
• 27. The lengths of the diagonals of a courtyard in the form of'^ a
rhombus are 40 ft. and 25 ft. How many bricks 9" by 4|" will be
required to pave the courtyard ; adding 5 % to the area to allow for
broken bricks and for waste at the sides of the courtyard ? /^Cl
V 28. One of the diagonals of a parallelogram measures 819 ft. and
the perpendicular on it from an opposite angle of the parallelogram
measures 237 ft. Find the area of the parallelogram,
i 29. The area of a quadrilateral is 7956 sq. yd. , the length of one
/iO .li" of the diagonals is 416 ft. and the length of the perpendicular on
^ this diagonal from an opposite angle of the quadrilateral is 192 ft.
^ ''3i» \^ Find the length of the perpendicular from the other opposite angle,
1°, if the diagonal is internal ; 2°, if it is external.
30. The area of a quadrilateral is 12*48 acres and the length of
one of the internal diagonals is 19 '50 chains. Find the sum of the
lengths of the perpendiculars on this diagonal from the two opposite
angles. / 2 • ^^ <2^.
31. The area of a quadrilateral is 906 "5 sq. yd. ; the length of one
of the internal diagonals is 147 ft. ; and the difference between the
lengths of the perpendiculars on this diagonal from the opposite
angles of the quadrilateral is 33 ft. Find the lengths of these
perpendiculars.
32. ABCD is a quadrilateral, A B = 400 ft., B C = 203 ft.,
CD = 396 ft., and DA = 195ft. ; the angles at A and C are right
angles. Find the area of the quadrilateral.
\' 33. Find the area of a trapezoid whose parallel sides measure
12' *l" and 19' 3'' respectively, the perpendicular distance between
them being 8' 6".
34. ABCD is a quadrilateral; A B =37 '48 chains, BC = 21-85
chains and CD = 29-64 chains. AB is parallel to DC and the
MENSURATION. 175
angle at C is a right angle. Determine the areas of the triangles
A B D and A C D and of the quadrilateral.
r 35. Find the area of a quadrilateral one of whose sides measures
23 '29 chains and the perpendiculars on this side from the opposite
angles of the quadrilateral 17 '75 chains and 13-45 chains respectively,
the distances of the feet of these perpendiculars from the adjacent
angles being 3 "64 chains and 2 "40 chains respectively.
' 36. The area of a trapezoidal field is 3^ acres and the sum of the
lengths of the parallel sides is 440 yd. Find the perpendicular
distance between these sides. The lengths of the sides being in
the ratio of 6 to 6, find these lengths.
V 37. The area of a trapezoid is 9750 sq. yd. and the perpendicular
distance between the parallel sides is 234 ft. If the length of one
of the parallel sides be 410 ft., what will be the length of the other
parallel side %
V 38. The area of a trapezoid is 47*142 acres. One of the parallel
sides is 6-12 chains longer than the other and the perpendicular
distance between the parallel sides is 11 '64 chains. Determine the
lengths of the two parallel sides.
39. The lengths of the parallel sides of a trapezoid are 12 ft. and
17 ft. and the perpendicular distance between these sides is 8 ft.
A straight line is drawn across the trapezoid parallel to the parallel
sides and midway between them. Find the areas of the two parts
into which the trapezoid is thus divided.
I^^V 40. The lengths of the parallel sides of a trapezoidal field are
^K 15 '80 chains and 18*70 chains respectively and the perpendicular
^^ distance between these parallel sides is 14*40 chains. Four straight
lines are drawn across the field parallel to the two parallel sides and
dividing the distance between these sides into five equal parts.
Find the areas of these five parts of the field.
^ 41. The area of a triangle is 551 sq. yd. and the length of its base
is 95 ft. Two straight lines are drawn across the triangle parallel
to the base and dividing into three equal parts the perpendicular
from the vertex on the base. Find the areas of the parts into
which the triangle is divided by these lines.
V 42. A trapezoid with parallel sides whose lengths are to be as 4
to 3 is to be cut from a rectangular board 14 ft. lontr. Find the
176 ARITHMETIC.
lengths of the parallel sides that the trapezoid may be one-third of
the board, the trapezoid to be of the same width as the board.
f 43. The length of a rectangle is to its width as 7 to 4, and if its
length be diminished by 3 ft. while its width is increased by 3 ft. ,
its area will be increased by 198 sq. ft. Find the length of the
rectangle.
44. The length of a rectangular piece of land is to its breadth as
9 to 5 ; if its length be increased by 4 ft. and its breadth be
diminished by 3 ft. , its area will be diminished by 355 sq. ft. Find
the length and the breadth of the piece of land.
45. The length of a rectangle is to its width as 16 to 9 ; if its
length be diminished by 2 ft. and its width diminished by 3 ft. , its
area will be diminished by 720 sq.ft. Find the area of the
rectangle.
46. A rectangular field 200 yd. long is surrounded by a road of
the uniform width of 60 ft. The total area of both field and road is
9 A. 1240 sq. yd. Find the width of the field.
4T. A rectangular field 780 ft. in length is surrounded by a road
of the uniform width of 50 ft. , the area of the whole road being
15000 sq. yd. Find the area of the field.
48. A rectangular field 180 yd. by 150yd. is surrounded by
a walk of uniform width, the whole area of the walk being
10000 sq. ft. Find the width of the walk.
49. Around a rectangular park runs a path of uniform width ;
paths of the same width cross the park dividing it into four equal
rectangles. The total length of the park, including paths is
330 yd. ; its area, including paths, is 15 A. ; exclusive of paths the
area is 13-775 A. Find the width of the paths.
60. The areas of two squares differ by 64 sq. yd. and the lengths
of their sides difier by 2 yd. Find their areas.
51.~The sum of the perimeters of two squares is 200 ft. and the
difierence of their areas is 400 sq. ft. Find their areas.
52. The area of a rectangle is 945 sq. ft. and that of a square of
equal perimeter is 961 sq. ft. Find the lengths of the sides of the
rectangle.
MENSURATION.
177
IP
53. The area of a rectangle is 37249 sq. ft. and its length exceeds
its breadth by 40 ft. Find its length.
54. The area of a triangle is 2 A. 2152 sq. yd. and the length of
the base exceeds the altitude of the triangle by 38 yd. Find the
length of the base.
55. A certain rectangular field of area 3§ A. is surrounded by a
road of the uniform width of 55 ft. , the total area of the road being
2 J A. Find the length and the width of the field.
170. In the formulae which follow V, a, B and M denote the
measures of the volume, the altitude, the area of the base and the
. ^^ area of the midcross-section respectively, and q, p^ c, i/, A;, /", d and w
I H| subscribed to V are to be read severally quad, prism, cylinder,
' ^" pyramid, cone, frustum of pyramid or of cone, prismatoid (or
prismoid) and wedge.
t. The measure of tlie volume of a quad (rectangular parallelepiped , )
I is the product of the measures of the lengths of three adjacent edges,
i. e. , of three edges meeting in a summit, or
Fq = ahi ^2 = <^^^»
in which 6^ and 62 denote the measures of adjacent edges of the
base and consequently bib^ = B.
(Special case, — cube.)
II. The tneaswre of the volume of a prism is the product of the
m^aswres of the altitude a/rtd the areoj of the base, or
V^ = aB.
(Special cases. — quad, and cylinder.)
V. This theorem is true of the oblique parallelepiped for every
oblique parallelepiped can be transformed into a rectangular
parallelepiped with base equal to and altitude the same as that
of the oblique parallelepiped. ,. a m
For example, let ABCDa^cci /\
be an oblique parallelepiped. / \„
Through e, a point in the edge ■ ""
Art, pass a plane at right angles
to the edges A/y, B6, Cc, T)d and
cutting these edges in the points
I
178
ARITHMETIC.
/
Hi
"i\
e, /, gr and /«/ respectively. Transfer the solid e/gf 7ia6cf? from end
to end of ABCDa6cfZ thus transforming this parallelepiped into
the parallelepiped EFGHe/(//fc on the rectangular base EH/ie
which is equal to the base KDda.
Through /c, a point in the edge e/i,
pass a plane at right angles to the edges
e/i, /(/, EH, EG, and cutting these edges
in the points /c, ?, K, and L respectively.
Transfer the solid EFLKe/Zfc from side pt.
to side of EFGHe/(//i, thus transform- \
ing this parallelepiped into the rectangu-
lar parallelepiped KLMNfcZmn. ^
The measure of the volume of AC 6f? is the same as the measure
of the volume of KM^n, the two parallelepipeds being made up of
the same parts diflferently arranged. The measure of the volume of
K M Z n is, by Theorem I, the product of the measures of its altitude
and its base-area. Hence the measure of the volume of KQhd
is the product of the measures of the altitude and base-area of
K M I tiy which is the same as the product of the measures of the
altitude and base-area of AGhd itself, for the altitude of the
parallelepipeds remains uAchanged during the transfers and the
base K N 71 jt is merely the base AD da with its parts transposed.
The theorem is therefore true of parallelepipeds.
2°. The theorem is true of a prism on a triangular base, for two
similar and equal prisms on triangular bases
may be so joined together as to form a
parallelepiped with both volume and base
double the volume and base of either prism.
Hence double the measure of the volume of a
prism on a triangular base is the product of
the measure of the altitude and the measure
of double the area of the base, and therefore
the measure of the volume is the product of
the measures of the altitude and the area of the base.
3^. The theorem is true of prisms whose bases have five or more
sides for by passing planes through any one lateral edge and
through all the other lateral edges except the two immediately
adjacent to the first edge, any such prism will be resolved into an
^__,
MENSURATION.
179
all
the
same
A
1 \^ /
\
[
\
/'
1
\
aggregate of triangular based prisms which have
altitude as the resolved prism and whose triangular
bases together make up the base of the resolved
prism.
All prisms are included under one or other of 1°, 2°
or 3°, therefore the theorem is true generally.
[The student should make models of the solid-
figures here considered and also of those considered
under theorems III and IV which follow. Solid-
figures can very easily be cut out of potatoes or turnips.]
III. The measure of the volume of a pyramid is ONE-THIRD of
the product of the measures of the altitude and the area of the hase^ or
Y,=^aB.
(Special cases, — coiie and sector' of a sphere^ including sphere itsdf.
This theorem is true of tetrahedra (triangular pyramids) for any
triangular prism, e.g., ABCDe/, can be divided
into three tetrahedra of which two, A B C D and
D e/C, will be of the same altitude as the prism
and will have the triangular faces of the prism as
their respective bases ; the third tetrahedron
BCDe, may be seen to have an altitude and a
base equal to each of the other two by resting
the prism first on the face Ae and next on the
face B/.
The theorem is true of pyramids with bases which
have four or more sides ; for, by passing planes
through any one lateral edge and all the other
lateral edges except the two adjacent to the first
edge any such pyramid will be resolved into an
aggregate of tetrahedra which have all the same
altitude as the pyramid and whose bases together
make the base of the pyramid.
Hence the theorem is true of all pyramids.
[The preceding proof assumes that two tetrahedra on equal
and similar bases and of the same altitude are of equal volulhe, a
proposition which is a particular case of Euclid xii, 5. The
proposition may also be proved as follows : —
180
ARITHMETIC.
Divide one of the lateral edges of each tetrahedron into any
number of equal parts, the same number in both tetrahedra, and
through the points of division pass planes parallel to the bases. All
the sections of the first tetrahedron are triangles equal and similar
to the corresponding sections of the second tetrahedron.
Beginning with the base of the first tetrahedron, construct on the
base and on each section as
base a prism with lateral edges
parallel to one of the edges of the
tetrahedron and with altitude
equal to the perpendicular
distance between the sections.
Beginning with the first section
above the base of the second
tetrahedron, construct on each
section as anti-base or upper
triangular surface, a prism with
lateral edges parallel to one of
the edges of the tetrahedron and with altitude equal to the
perpendicular distance between the sections.
The aggregate of the first-constructed series of prisms is greater
than the first tetrahedron and the aggregate of the second series is
less than the second tetrahedron, therefore the difierence in volume
between the tetrahedra is less than the difference in volume
between the prism-aggregates.
But, by II p. 178, each prism in the second tetrahedron is equal in
volume to the prism in the first tetrahedron next above it in- order
numbering from the prisms on the bases of the tetrahedra.
Therefore the difference between the prism-aggregates is the basal
prism in the first aggregate.
Now the volume of this basal prism may be made less than any
assignable volume, for the measure of its volume is the product of
the measures of the altitude and the area of the base. The base is
constant being the base of the tetrahedron, but the altitude being
the perpendicular distance between the sections may be increased
or diminished by changing the number of the sections. By
doubling the number of the sections the altitude and with it the
volume of the prism will be diminished by one-half of itself. If wa
MENSURATION.
181
again double the number of sections, we shall again diminish the
volume of the prism by one-half of itself. Repeating the doubling
we repeat the subdividing, and the process may be continued till
the volume of the basal i)rism is less than that of any assigned solid
however small.
Hence the tetrahedra can have no assignable difference of volume,
and, both being constants, they cannot have a variable difference ;
therefore they are of equal volume.]
IV. The measv/re of the volume of a prismatoid is ONE-SIXTH of
the product obtained hy midtiplying the measure of the altitude by the
sum f Mined by adding the measures of the areas of the parallel faces to
four times the measure of the area of the midcross-sectio7i, or
(Special cases, — prismoid and cylindroid, wedge and sphenoid,
prism and cylinder, pyramid and cone and frusta of pyramid and
cone, ellipsoid and frustum of ellipsoid, by planes perpendicular to an
axis.)
Let ABCDEFG be a prismatoid and denote the measure of its
altitude by a, the measure of the area of the base ABCD by ^i, and
the measure of the area of the face EFG, the face parallel to the
base, by Bq,
Bisect the lateral edge r
BG in the point H and
through H pass a plane
parallel to the base AB
C D, cutting, and therefore
bisecting, the other lateral
edges in K, L, M, N, P
and Q respectively. The
polygon HKLMNPQ
is the midcross- section
of the prismatoid and
its perpendicular distance
both from the base ABCD
and from the parallel face
EFG is one-half of the altitude of the prismatoid. The measure of
that distance is therefore ^a. Let M denote the measure of the
area of the midcross-section.
182 ARITHMETIC.
In the plane of the midcross-section take any point R and pas&
planes through R and each edge of the prismatoid thus resolving
that solid into the nine pyramids RABCD, REFG, RBCG,
RGCE, RODE, RDAE, REPA, RABF, RFGB, a pyramid
on each face of the prismatoid.
The measure of the volume of RABCD is one-third of the
product of icfr and B^ i.e., ^aB^.
The measure of the volume of R EFG is one-third of the product
of ia a.ndB.2, i.e., ^aB^.
To determine the measure of the volume of the other pyramids
join RH, RK, RL, RM, RN, RP and RQ, also join BK. Let
the measures of the areas of the triangles RHK, RKL, RLM,
RMN, RNP, RPQ, RQH be denoted respectively hy m^ mg,
mg, m4, mg, mg and rrij, therefore
Because CG is bisected in K, the triangle BCG is double the
triangle BKG. Because BG is bisected in H, the triangle BKG
is double of the triangle HKG. Therefore the triangle BCG is
four times the triangle HKG. Therefore the pyramid RBCG is
four times the pyramid RHKG. But taking G as the apex and
RHK as the base of RHKG, the altitude of this pyramid is
one-half that of the prismatoid therefore the measure of the volume
of RHKG is one-third of the product of ^ a and m^., i. e., ^am^.
Therefore
the measure of the volume of RBCG is fa Wi.
In like manner it may be shown that
the measure of the volume of RGCE is ^rtmg,
II II II M RCDE is ^arriQ,
M II II II RDAE is ^am4,
II II II II REFA is famg,
II ,1 I, I. RABF is ^aniQj
,1 „ I, I, RFGBisf«m7.
Hence the sum of the measures of the volumes of the pyramids
on the lateral faces of the prismatoid is
ia(m-^ -H mg + m.3 + m^ -}- W5 -f m^ -f- m- ) ^ f a M.
Adding to this sum the measure of the volume of the basal
MENSUKATION.
183
pyramids RABCD and REFG, the measure of the volume of the
whole prismatoid is found to be
or V^=^B^i-4:M+B.).
This is known as the Prismoidal Formula. It is of the
very highest importance, nearly all the elementary formulae in
stereometry being but special cases of it.
IV, a. The measure of the volume of a frusUmi of a pyramid is
ONE-THIRD of the prodiwt foi-med by midtiplyin^ the measure of
the altitude by the sum obtahied by adding the measures of the areas
of the tivo parallel faces to the square root of the product of these ttvo
measu/res; or
Y,= ^a{BMB^B,)i +B,}.
This theorem may easily be deduced from the Prismoidal Formula,
but it may also be proved, independently as follows : —
All cases of frusta on bases having four or more sides can be
reduced to the case of frusta on triangular bases ; for, by passing
planes through any one lateral edge and all the other lateral edges
except the two adjacent to the first edge, any frustum whose base
has more than three sides will be resolved by these planes into an
aggregate of triangular-based frusta which have all the same altitude
as the given frustum and whose triangular bases together make up
the base of the given frustum. Hence it will be sufficient to
consider only frusta on triangular bases.
Let AB CDEFbethe frustum of a tetrahedron
Let a denote the measure of its altitude ; B^^
the measure of the area of the base ABC ; and
JBg, the measure of the area of the face D EF.
Pass a plane through AC and F and another
plane through EF and C, thus resolving the
frustum into the three tetrahedra ABCF,
DEFC, ACEF. Let Fj, V^ and F3 denote (
the measures of the volumes of these several
tetrahedra, therefore
184 ARITHMETIC.
Complete the pyramid of which ABCDEF is a frustum by
producing the lateral faces, — and thereby producing the lateral
edges, — to meet in a common point G.
Taking the triangle ABC as the base of ABCF, the tetrahedron
and the frustum have the same altitude ; therefore the measure of
the volume of ABCF is laB^, or
Taking the triangle DEF as the base of DEFC, the tetrahedron
and the frustum have the same altitude ; therefore the measure of
the volume of DEFC is la Boy or
V,^ = laBo.
Taking C as the common summit of the tetrahedra ABFC and
AEFC, these two pyramids will have the same altitude, and
therefore in determining the ratio of their volumes their common
altitude may be omitted as being merely a common factor. The
volumes of the tetrahedra will therefore liave the same ratio as the
areas of their bases have, or
ABFCABF .^.
AEFC AEF* ^
Taking F as the common summit of tlie tetrahedra CAEF and
CDEF, these two pyramids will have the same altitude, and
therefore in determining the ratio of their volumes, their common
altitude may be omitted. The volumes of the tetrahedra will
therefore have the same ratio as the areas of their bases have, or
CAEF CAE
CDEF CDE
(2)
AB and EF being parallel, the triangles ABF and AEF have
the same altitude, viz., the perpendicular distance of EF from AB ;
the areas of these triangles will therefore have the same ratio as
the lengths of their bases AB and EF have, or
ABF^AB
AEF EF* ^ ^
Ai AB AG ,,.
Also = . (4)
EF EG ^ ^
MENSURATION.
185
CA and DE being parallel, the triangles CAE and CDE have
'the same altitude ; the areas of these triangles will therefore have
the same ratio as the lengths of their bases CA and DE have, or
CAE CA
Also
CDE
DE
CA
DE"
AG
EG
(5)
(6)
I
Collecting the equalities numbered (1), (3), (4), (6), (5) and (2)
and arranging them in the order here indicated, we obtain
• ABFCABFABAGCACAE^CAEF
AEFC~AEF~EF~EG~DE~CDE~CDEF'
^, , ABFC AEFC /hx
*^^^^^^^'^ AEFC^DE-FC' ^^^
For the volumes of the three tetrahedra ABFC, DEFC, AEFC
substitute the measures of these volumes in terms of a common
Sand (7) becomes
F, F.
^3 = ^1^2
V,= laB^+laB^ + la{B^B^Y
IV, h. The measure of the vohime of a wedge is ONE-SIXTH of
the cmitimied product of the mea»\ire of the altitiide of the ivedge, the
measure of the width of the base and the sum of the measures of the
lengths of the three parallel edges, or
in which a denotes the measure of the altitude of the wedge, h
denotes the measure of the width of the base and 6j, 62 ^^^ ^s
denote the measures of the respective lengths of the three co-parallel
edges.
186 ARITHMETIC.
Let ABCDEF be a wedge on the base ABCD. It may be
treated as a prismatoid whose base is a EN F
trapezoid and whose face parallel to the yj fi ~^
base has become reduced to the straight y/\'r 7''/ — "^.J
line EF. / A J\^"h
Let N P Q be a plane section of the wedge *\j ^ \j \/
at right angles to the edge EF and therefore 0 Q c
also at right angles to the edges AB and CD which are parallel to
EF. The length of the line PQ is the width of the base and the
length of the perpendicular from N on PQ is the altitude of the
wedge. Hence the measure of the length of PQ is 6, and the
measure of the length of the perpendicular from N on PQ is a.
Let GHKL be the midcross-section of the prismatoid and let it cut
the triangle NPQ in the straight line RS which will therefore be
parallel to PQ. R is the mid-point of NP and S is the mid-point of
NQ, therefore RS = JPQ, and therefore the measure of RS is hh.
The measure of the length of AB is fcj, that of the length of EF
is 63, and G and H are the respective mid-points of AE and BF,
therefore the measure of the length of GH is ^{hj +63).
The measure of the length of CD is 62, that of the length of
F E is 63, and K and L are the respective mid-points of C F and
DE, therefore the measure of the length of KL isK62 + ^3)-
Applying the Prismoidal Formula, the measure of the volume of
the wedge is
ia{B^+^M+B^)' (1)
Bi is the measure of the area of the trapezoid ABCD, The
measures of the lengths of the parallel sides of this trapezoid are
bi and b^ respectively and the measure of its width is b.
.'. B^=ib(b,+b,). (2)
M is the measure of the area of the trapezoid GHKL. The
measures of the lengths of the parallel sides of this trapQzoid are
^(61+63) and 5(^2 + ^3) respectively and the measure of its
width is ^ 6,
.-. M=kb[^(b,+b.,)+i(b,+b,)]
.-. 4.M=h{ib^ + ib. + b.,). (3)
MENSURATION.
187
J^2 is the measure of the area of the line EF,
.-. J5,=0. . (4)
Substitute in (1) the vahies of JB^, 4ilf and B^ given in (2), (3)
md(4),
In the case of the common wedge or wedge on a rectangular base,
b and b^ are the measures of the lengths of adjacent basal edges
and b2 = b-^
V^^ab{2b^-\-b,).
rV, c. The measure of the volume of a tetrahedron is T WO-THIRDS
of the product of the measure of the perpendicidar distance between
any two opposite edges and the mAiasure of the area of the parallelogram
whose aiujular points are the mid-points of the other four edges of the
tetrahedron^ or
V,= ?.aM.
The tetrahedron is a prismatoid whose
parallel faces are reduced to two straight
lines, and the mid-parallelogram is its mid-
cross-section, therefore by the Prismoidal
Formula,
Each side of the mid-parallelogram is
equal to half of the edge of the tetrahedron parallel to the side,
therefore, if the midcross-section of the tetrahedron be a rectangle
and Vt=|aCiC2,
in which Cj and Ca denote half the measures of the lengths of
the two edges parallel to the midcross-section. In this case
the midcross-section divides the tetrahedron into two wedges
(hemitetrahedra) whose altitudes are equal as also are their volumes.
It is worthy of notice that if a prism, a hemitetrahedron and a
pyramid are on equal bases and are of the same altitude, the
volume of the prism is thrice and the volume of the hemitetrahedron
is twice that of the pyramid, or
F„ = aB, K, = 2 a R F, - X aB.
188
ARITHMETIC.
Example 1. An iron tank in the form of a hollow cube whose
sides, bojitom and top are all and everyAvhere of the same thickness,
has a capacity of 381 gallons. The length of an outside edge of the
tank is 4 ft. Find the thickness of the sides.
The capacity of the tank is 381 gal.
= 277-118CU. in. x381.
=(277-118 X 381) cu. in.
The length of the edge of a cube of this capacity is
(277 -118x381)=^ in!
The cube root of 277*118x381 may be obtained directly by
multiplication and evolution, or it may be computed by the aid of
logarithms thus : —
log (277 -118 X 381)-^ = i(log 277 '118 + log 381)
= 1(2 -442665 + 2 -580925)
= 1-674530= log 47 -264.
.-. {277 -118x381)^^=47 -264 in.
. •. the length of an inside edge of the tank is 47*264 in.
The M M If outside n n w tt ti 48 in.
The difference between the lengths of an outside and an inside
edge is double the thickness of the sides ;
. •. double the thickness of the sides is 48 in. - 47 *264 in.
= •736 in.
. *. the thickness of the sides is ^ of -736 in. = '368 in. which is very
nearly three-eighths of an hich.
Example 2. Find the ^, __»
air-capacity of an attic
given the accompany-
ing plan and the follow-
ing dimensions : —
The floor of the attic
is a hexagon ABCDEF ;
the ceiling is a trapezoid
GHKL; AB,MC,ND,
rE,GHandLKareall
parallel to each other,
AF and GL are also
parallel to each other,
AF is at right angles
MENSURATION. 189
to AB and therefore also to MC, ND and FE, and GL is at right
angles to GH and LK. AB = 28 ft., MC = 32 ft., ND = 34 ft.,
FE = 30ft. ; GH = 22ft., LK = 23ft. ; AM=6ft., MN = 16ft.,
NF = 8 f t. ; GL = 12 ft. ; and the vertical height of the ceiling above
the floor is 10 ft. 6 in.
The area of the floor is the sum of the areas of the three trapezoids
ABCM, MCDN, NDEF.
The area of ABCM is h of 6(28 + 32) sq. ft. = 180 sq. ft.
u I, M MCDNis I of 15(32 + 34)sq. ft. =495sq. ft.
ti .1 .1 NDEF is I of 8(34 + 30) sq. ft. = 256 sq. ft.
. •. the area of the floor is (180 + 495 + 256) sq. ft. = 931 sq. ft. (1)
The area of the ceiling is ^ of 12(22 + 23) sq . ft. = 270 sq. ft. (2)
The area of the midcross-section is the sum of the areas of the
foiir trapezoids PQRW, WRSX, XSTZ, ZTUV. To determhie the
areas of these trapezoids, the lengths of their parallel sides and of
the normal distances between these sides must first be found.
PQ=HAB + GH)=H28 + 22)ft.=25ft.
WR = J (MC + GH) - 1(32 + 22) ft. = 27 ft.
XS-|(ND + GH) = i(34 + 22)ft.=28ft.
ZT=|(ND + LK)=|(34 + 23)ft.=28ift.
VU=^(FE + LK) = J(30+23)ft.=26|ft.
PW = JAM = Jof 6ft.=3ft.
\VX = JM:N=iof 15ft.=7ifl.
XZ -JGL =^ of 12ft. =6^.
ZV=i]SrF = |of 8ft.=4ft.
The area of PQRW is ^ of 3(25 + 27) sq. ft. =78 sq. ft.
.n ., n WRSX is I of 71(27 + 28) sq. ft. =206^ sq.ft.
„ M ., XSTZ is 1 of 6(284-28i)sq. ft.=169^sq.ft.
M ., M ZTUV is I of 4(28^ + 26i)sq. ft. =110 sq.ft.
The area of the midcross-section is (78 + 206| + 169f + 110) sq. ft.
= 563| sq.ft. (3)
.-. the capacity of the attic is ^ of 10i<; 931 + 4(563-]) + 270 ^cu. ft.
= 6048cu. ft;
EXERCISE XXIII.
1. Find the number of cubic inches in the volume of a quad
measuring a foot by a yard by a metre.
190 ARITHMETIC.
2, Find bo the nearest gallon the volume of a quad measuring
75 in. by 87 '5 in. by 126-875 in.
Jl. Find, correct to four significant figures, the length of the
inside edge of a cubical vessel which will just hold 10 gallons.
4. Find, correct to four significant figures, the length of the
inside edge of a cubical vessel which will just hold 100 gallons.
5. One acre of a certain wheat-field yielded 21001b. of wheat
weighing 7 lb. lOA oz. per measiired gallon. At this rate what was
the yield in cubic inches per square yard of the field, and what
would be the length of the edge of a cube equal to the yield of a
square inch of the acre ?
6. A quadrate reservoir is 147 ft. 8 in. long, 103 ft. 6 in. wide and
11 ft. 9 in. deep. When the reservoir is nearly full of water, how
many cubic feet of water must be drawn ofi" that the water-surface
may sink 4 ft. 4 in. ?
7. Find to the nearest gallon the capacity of an open quadrate
tank measuring 7' 6" by & 4" by 5' 8" externally ; the material
of which the tank is made being 1 j inches in thickness.
§. Three cubes of lead measuring respectively |, ^, and f of an
inch on the edge were melted tdgether and cast into a single cube.
Find the length of the edge of the cube thus formed, neglecting
loss of lead in melting and casting.
9. Four cubes of lead n^easuring respectively 6, 7, 8 and 9
inches on the edge were melted together and cast into a single
cube. Find the length of the edge of the cube thus formed, if
4 per cent, of the lead was lost in the melting and casting.
10. Three cubes of lead measuring respectively 3*1, 3*6 and
37 inches on the edge were melted and cast into a single quadrate
lump 5 in. long by 4-1 in. wide. Find the height of the quad,
neglecting loss of lead in melting and casting.
11. A cube of lead measuring 64-1 mm. on the edge was melted
and cast in the form of a quad with square ends and with length
three times the width . Find the dimensions of the quad, neglecting
loss of lead in melting and casting.
12. The length of a quad is thrice its width and the width is
double the height. Find the length of the quad, its volume being
a cubic yard.
MENSURATION. 191
13. The three adjacent edges of a quad are to one another as
2:3:5 and its volume is a cubic metre . Find the length of the
edges and the areas of the faces of the quad.
14. Find the volume of a cube the area of whose surface is
100-86 sq. in.
15. The surface of a cube measures 30 sq. in. Find the area of
the surface of a cube of five times the volume of the former.
16. The volume of a cube is 30 cu. in. Find the volume of a
cube whose- surface has an area five times the area of the surface of
the former cube.
17. A cube measures 5 in. on the edge. A second cube is of
thrice the volume of the first. By how much does the length of an
edge of the second cube exceed that of an edge of the first cube ?
1§. A cube measures 5 in. on the edge. Find the volume of a
cube whose surface-area is thrice that of the former cube.
19. A quadrate cistern is 5 ft. wide by 6 ft. long by 4 ft. 2 in.
deep. Its width and its length are each increased by 6 inches.
How much deeper must it be made that the total increase of its
capacity may be 250 gallons ?
20. If a quad has its length, its breadth and its height respectively
a twelfth, a thirteenth and a fourteenth as long again as the
corresponding dimensions of another quad ; show that the volume
of the first quad will be a quarter as large again as the volume of
the second quad.
21. By raising the temperature of a cube of iron, the length of
each of its edges was increased by '5 per cent. Find correct to four
decimals the ratio of increase in the volume of the cube.
22. Each edge of a cube is diminished by a tenth of its length.
By what fraction of itself is the volume diminished ? By what
fraction of itself is the area of the surface diminished ?
23. By taking the decimeter as equal to 4 in. what percentage
of error is introduced into (a), linear measurements ; (6), areal
measurements ; (c), volume measurements ?
21. The height of a solid six-inch cube of India-rubber is
diminished by pressure to 5 85 in. If the volume of the solid
remain the same and the lateral expansion be uniform throughout,
what will be the dimensions of the new base ?
192 ARITHMETIC.
25. The length of a quad is 7 in., its height is 3 in., and the
total area of its surface is a square foot. Find the volume of the
quad.
26. The length of a quad is 13 '3 in., its width is 8 '4 in., and the
total area of its surface is 466 '5 sq. in. Find its volume.
27. The width of a quad is 7-05 ft., its height is 3 -13 ft. and the
total area of its surface is 30 sq. yd. Find its volume.
2§. The width of a quad is 371mm., its height is 284 mm. , its
volume is a cubic metre. Find the area of its base.
29. Find the area of the surface of a quad 31 '62 cm. wide by
38 '73 cm. long and of '03 cubic metre volume.
30. The area of the base of a quad is 71 '288 sq. in., that of a side
of the quad is 56*868 sq. in. and that of an end is 52 "65 sq. in.
Find the volume of the quad.
ft 1 . The length of the perimeter of the base of a quad is 20 in. ;
the area of the base is 20*16 sq. in. ; and the total area of the surface
of the quad is 90*32 sq. in. Find the volume of the quad.
32. The perimeter of the base of a quad measures 25 '4 in. , the
area of one end of the quad is 23*1 sq. in., and the volume of the
quad is 166 "32 cu. in. Find the lengths of the edges of the quad.
33. Find the measure of the length of the edge of a cube the
measure of whose volume is equal to the measure of the area of its
surface.
34. The measure of the volume of a quad two of whose edges
measure 3 in. and 4 in. respectively, is the same as the measure of
the area of the whole surface of the quad. Find the length of the
third edge.
35. The area of the surface of a quad on a square base is
192 sq. in. The area of the base is equal to the sum of the areas of
the two sides and two ends. Find the volume of the quad.
36. The volume of a quad on a square base is 6572 cu. in. , the
height of the quad is 11 '8 in. Find the length of an edge of the
base.
37. Find the weight of the air in a rectangular roctm measuring
27' 8" by 23' 5" by 12' 4", the weight of the air being the '001295 of
the weight of an equal volume of water.
MENSURATION. 193
38. If a cubic foot of gold weigh 1200 lb. , find the thickness of
gold-leaf of which 1200 leaves 3^ inches square weigh an ounce
troy.
39. A quadrate block of stone measuring 5*297" by 7*472" by
9 '57" weighs 38 '14 lb. Compare the weight of any volume of the
stone with the weight of an equal volume of water at 62° F.
40. What will be the weight of 36 iron rods each 14 ft. long and
of cross-section | of an inch square, if the specific gravity of the
iron be 7 '7 ?
41. What length of a bar of iron will weigh 10 lb. , the cross-section
of the bar being a rectangle measuring | in. by 1^ in. ?
42. Wliat weight will just keep under water a stick of square-timber
measuring 36 ft. by 10 inches square, the specific gravity of the
wood being "725 ?
43. What sized cube of iron placed on a quad of dry pine
measuring 8 ft. by 5 '6 ft. by 4 in. will just sink the quad in water,
the specific gravity of the pine being '472 and that of the iron 7 *7 'i
44. An open quadrate tank is 5 ft. 6 in, long, 4 ft. 3 in. wide and
3 ft. 8 in. high, the sides and bottom of the tank are | in. thick.
Find the number of cubic inches of material in the vessel.
45. Find the weight of a hollow iron cube measuring 2*735
inches on the outer edge, the thickness of the iron being '167 of an
inch and its specific gravity 7 "7.
46. Find the thickness of the sides of an iron box in the form of
a hollow cube, which weighs 266 lb. when empty and 566 lb. when
filled with water ; the sides, bottom and top being all of the same
thickness and the specific gravity of the iron 7 *7.
47. The sides, bottom and lid of a quadrate box have a uniform
thickness of § in. The outside measurements of the box are 8 in.
by 12 -5 hi. by 16 '25 in. How many cubes each f of an inch on the
edge, will the box hold.
4§. Find the thickness of the material of which a closed hollow
iron cube is constructed, if the cube weigh 33 lb. 4 oz. and measure
10*5 in. on an outside edge, the specific gravity of iron being 7*7.
49. An iron cube is coated with a unifonn thickness of gold.
Find the thickness of the gold if the coated-cube is 3 inches long
194 ARITHMETIC.
and weighs 7 '525 lb., the specific gravity of the gold being 19 '26
and that of the iron 7*7.
V 50. Find the volume of a right triangular prism 8 in. long, the
terminal triangles being right-angled and the lengths of the sides
containing the right angle being l"2in. and 2'lin. respectively.
51. The normal length of a triangular prism is 79 mm., the
length of an edge of one of the terminal triangles is 43 mm. , and
the length of the perpendicular on that edge from the opposite
superficial angle is 29 mm. Find the volume of the prism in cubic
centimetres.
(The normal length is the length measured at right angles to the
parallel ends. If one of these ends be taken as the base of the
prism, the normal length will be the altitude of the solid.)
y 52. The altitude of a prism is 17 '3 in. and its base is a
parallelogram of length 25 '75 in. and normal width 9 "7 in. Find
the volume of the prism.
53. The normal length of a trapezoidal prism is 97 ft. 6 in. , the
lengths of the parallel edges of the trapezoidal ends are 37 ft. 5 in.
and 23 ft. 4 in. respectively and the perpendicular distance between
these edges is 9 ft. 6 in. Find the volume of the prism.
54. The length of a prism is 9 ft. 4 in. A right cross-section of
the prism is a quadrilateral, one of whose diagonals measures 3 ft.
7 in. and the perpendiculars on that diagonal from the opposite
angles of the section are respectively 1 ft. 5 in. and 1 ft. 7 in. long.
Find the volume of the prism.
55. The fabled wall of China was said to be 26 ft. wide at the
bottom, 15 ft. wide at the top, 20 ft. high and 1500 miles long.
How many cubic yards of material would such a wall contain.
56. H'ow many cubic yards of earth must be removed in the
digging of a ditch 147 ft. long, 8 ft. wide at the top, 6 ft. wide at
the bottom and 4 ft. 6 in. deep, the ends of the ditch being vertical?
57. How many gallons of water will fill a horse trough 7 ft. 6 in.
long, 10 in. deep, 14 in. wide at the top and 11 in. wide at the
bottom ; the ends of the trough being at right angles to the bottom
and sides %
58. How many prismatic bars of lead each 10 '5 in, long must be
melted down to make a cube 6*26 in. on the edge, a right cross-
MENSURATION. 195
section of each bar being a trapezoid measuring 1 "3 in. and "6 in.
respectively on the parallel sides and "5 in. in perpendicular distance
between these sides ; "5 per cent, of the lead being lost in the
melting ?
59. 7843 cu. yd. of earth were removed in digging a ditch 2 ft.
9 in. deep, 4 ft. 6 in. wide at the top and 3 ft. wide at the bottom.
Find the length of the ditch assuming that the ends were vertical,
60. The cross-section of a canal is 36 ft. wide at the surface of
the water and 20 ft. wide at the bottom ; what must be the depth of
the water if 100 yd. in length of the canal contain 689315 gallons
of water ?
61. A ditch 125 yd. long is filled to a depth of 1ft. 9 in. by
10571 gal. of water. What must be the width of the ditch at the
bottom if the width at the surface of the water be 3 ft. and the ends
of the ditch be vertical ?
62. A stream flows at the rate of 3 miles per hour through a
trough whose cross-section is a trapezoid. The width of the bottom
of the trough is 21 inches, the depth of the water is 4 "5 inches and
the width of the surface of the water in the trough is 25 inches.
How many gallons flows through the trough per minute ?
63. A prism of 21 inches altitude weighs one ton. Find the
area of the base, the material of the prism weighing 5241b. per
cubic foot.
64. The volume of a prism is 6cu. ft. ; its height is 9 in., and its
base is an isosceles right-angled triangle. Find the lengths of the
edges of the base.
65. A shed with a single sloping roof is 22 ft. long by 12 ft.
wide ; the heigl t of the roof above the floor is 12 ft. at the front
and 8 ft. at the back. Find the total capacity of the shed.
66. A school-room with attic ceiling is 32 ft. long by 28 ft. wide.
The ceiling at the side walls is 10 ft. above the floor and slopes
upward until it attains a height of 14 ft. 6 in. and then becomes
level, the width of the level part being 12 ft. The ceiling meets
the end walls at right angles. Find the air-capacity of the
school-room.
67. A parallelepiped is cut by two .planes which neither meet
the ends nor intersect. The area of a right cross-section is 96 sq. in.
and the lengths between the cutting planes of the four parallel
196 ARITHMETIC.
edges are respectively 6m. 7 "Sin. lOin. and 8*5 in. Find the volume
of the portion of the parallelepiped between the cutting planes.
>/ 6§. The base of a pyramid is a triangle one side of which measures
15 '3 in. ; the length of the perpendicular on that side from Ihe
opposite angle of the base is 9 6 in. and the altitude of the pyramid
is 12-5 in. Find the volume of the pyramid.
69. Find the volume of a tetrahedron whose base is a right-
angled triangle, the sides of the base containing the right angle
measuring 17 in. and 19 in. respectively and the altitude of the
tetrahedron being 18 in.
TO. Find the volume of a tetrahedron whose base is a right-angled
isosceles triangle, the altitude of the tetrahedron being 7 ft. 5 in.
and the length of the hypothenuse of the base being 5 ft. 7 in.
71, Find the weight of the pyramid formed by cutting off a
comer of a cube of lead by a plane passing through three adjacent
corners, the length of an edge of the cube being 2 '5 in. and the
specific gravity of lead being 11*4.
7fi, One of the corners of a quad of gold is cut off by a plane
which meets the three conterminous edges, 2 '7 inches, 4*3 inches
and 3 '6 inches respectively from their common point. Find the
value of the piece cut off, the specific gravity of the gold being
17 '66 and its value $18*95 per ounce troy.
73. The base of a pyramid is a square whose side is 3 "45 ft. long.
The altitude of the pyramid is 4 "75 ft. Find the volume of the
pyramid.
74. Find the volume of a pyramid whose altitude is 4 ft. 5 in.
and whose base is a rectangle measuring 3 ft. 4 in. by 3 ft. 9 in.
75. The altitude of a pyramid is 2 ft. 3 in., its base is a trapezoid
whose parallel sides measure 1 ft. 9 in. and 1 ft. 3 in. respectively,
the perpendicular distance between these sides being 1 ft. 4 in.
Find the volume of the pyramid.
76. The base of a pyramid is a square 2 ft. 7 in. long and its
volume is 3-2cu. ft. Find the altitude of tlie pyramid.
77. The volume of a pyramid on a rectangular base is half a
cubic yard. The length of the base is 3 ft. 9 in. and the altitude of
the pyramid is 3 '2 ft. Fin4 the width of the base.
78. The volume of a pyramid on a square base is a cubic yard
and its altitude is a yard. Find tlie length of an edge of the base.
MENSURATION. 197
>• 79. The volume of a pyramid on a square base is 30'87cu. in.
and the altitude of the pyramid is equal to the length of an edge of
the base. Find the altitude.
yf 80. The volume of a pyramid is 77 cu. in. The base of the
pyramid is a quadrilateral ; the length of one of the diagonals of the
base is 15 in, and the lengths of the perpendiculars on this diagonal
• from the opposite angles of the base are 10*6 in. and 9 in. Find the
altitude of th e pyramid.
V §1. The base of a pyramid is a square 15 in, long and the
altitude of the pyramid is 16 in. The base of another pyramid is a
rectangle 16 in. long by 12*5 in. wide. Find the altitude of the
second pyramid, the volumes of the two pyramids being equal.
82. The Great Pyramid of Egypt when complete was 480 ft. 9 in.
in height, and its base was a square 764 ft, in length ; in its present
condition the pyramid is 450 ft. 9 in. high and its base is a square
746 ft. long and wide. Find to the nearest cubic yard the volume
of the pyramid in its complete and also in its present state.
83. The representative gold pyramid in the International
Exhibition of 1862 was 10 ft. square at the base and 44 ft. 9j in.
high. Find the volume of the pyramid, and the weight and the
value of the gold represented by it, taking the specific gravity of the
gold at 19 "25 and its value at $20 '67 X)er ounce troy.
84. Since the construction of the pyramid mentioned in problem
83, about 25,000,000 ounces troy of gold have been mined ; how
much higher would the pyramid require to be made to include this
quantity ?
85. The base of a pyramid is a square whose sides are 25 in. long.
The altitude is 16 in. A plane parallel to the base divides the
pyramid into parts of equal volume. Find the perpendicular height
of the plane above the base.
86. The base of a pyramid is a trapezoid whose parallel sides
measure 19 5 in. and 13*7 in., the perpendicular distance between
them being 12 '6 in. The altitude of the pyramid is 14 in. At
what height above the base must a plane parallel t the base be
drawn, that it may bisect the pyramid ?
87. The base of a pyramid is a square whose area is 7 sq, ft.
The altitude of the pyramid is one yard. A plane parallel to the
base so divides the pyramid that the volume of the frustum between
198 ARITHMETIC.
the base and the plane is double the volume of the pyramid abov
the plane. Find the height of the frustum.
88. The altitude of a pyramid is loin. A plane parallel to the
base divides the pyramid into two parts whose volumes are such
that thrice the volume of the frustum between the plane and the
base is equal to five times the volume of the pyramid above the
plane. Find the height of the frustum.
§9. Find the volume of a prismoid whose top and bottom are
rectangles the corresponding dimensions of which are 3 ft. by 2 ft.
and 5ft. by 3 '5 ft., the altitude of the prismoid being 3*5 ft.
90. Find the volume of a prismoid whose top and bottom are
rectangles the corresponding dimensions of which are 3 ft. by 2 ft.
and 3 '6 ft. by 5 ft., the altitude of the prismoid being 3 "5 ft.
91. Find the capacity of a cart the top of which measures 4' 3"
by 3' 8" ; the bottom, 3' 9" by 3' 2" ; and the depth, 2' 3".
92. How many gallons of water will fill a ditch 2 ft. deep, the
top and bottom of the ditch being rectangles whose corresponding
dimensions are 148 ft. by 3 ft. 4 in. and 146 ft. 6 in. by 2 ft. 3 in. 1
93. Find the weight of an iron shaft whose ends are rectangles,
one end measuring 10 '5 in. by 17 in., the other end measuring 7 in.
by 12 in., the length of the shaft being 13 ft. 6 in. and the specific
gravity of the iron, 7 •7.
94. What weight will just sink a scow in the form of a hollow
prismoid with rectangular base, the length of the scow over all being
14 ft. 1 in. ; its width, 3 ft. 8 in. ; its full depth, 2 ft. 11 in. ; the
length of the bottom outside, 12 ft. ; the width of the bottom 3 ft.
and its weight 920 lb.
95. Find the volume of a pile of broken stones, the base of the
pile being a rectangle measuring 13 ft. 6 in. by 7 ft. 5 in. ; the top
of the pile a rectangle measuring 12 ft. 2 in. by 6 ft. ; and the height
of the pile being 2 ft. 10 in.
96. It is usual to take as the measure of the volume of a pile of
broken stones the product of the measure of the altitude of the pile
and the measure of the area of its midcross-section. By how much
would the volume thus calculated be in defect of the actual volume
in the case of the pile described in the problem immediately
I^receding.
MENSURATION. 199
97. Find the number of cubic yards in a railway cutting in the
form of a prismoid with trapezoidal ends ; the lengths of the parallel
Sides at one end being 124 ft. and 33 ft., and the distance between
them 28 ft. ; the corresponding dimensions of the other end being
104ft., 33ft. and 21ft. respectively; and the distance between
the ends being 236*5 yd.
98. A straight ditch with a fall of 1ft. in 300 yd. is to be dug in
level ground. The sides are to slope 1 in 1, the bottom is to be
4 ft. wide, and the depth at the upper end is to be 3 ft. 6 in. Find
the number of cubic yards of earth that will require to be removed
in digging the first 1000 yards of the ditch.
99. How many cubic yards of earth will be excavated in making
a railway cutting through ground whose surface is an inclined plane
rising in the same direction as the rails, the length of the cutting
being 123 yd. ; the width at the bottom 33 ft. ; the width at the
top at one end, 66 ft. ; at the other end, 100 ft. ; and the depths of
these ends, 22 ft. and 48 ft. respectively ?
100. A railway-embankment is made on ground which falls at
20 ft. per mile in the same direction of the rails, which themselves
fall 1 in 800. The length of the embankment is 2100 yd. ; its width
at the top is 33 ft., the slope of the sides is 1 in 1 and the height at
the upper end is 1 ft. 8 in. * Find the number of cubic yards of
earth in the embankment.
101. The ends of a prismoid are rectangles whose corresponding
dimensions are 17 '3 in. by 11 "4 in. and 9*6 in. by 6'6in. ; the
altitude of the prismoid is 21 '6 in. The prismoid is divided in two
parts by a plane parallel to the ends and midway between them.
Find the volume of each part.
102. The ends of a prismoid are rectangles whose corresponding
dimensions are 7 ft. by 5 ft. and 3 ft. by 2 ft. The prismoid is
divided by planes parallel to the ends, into three prismoids each
1ft. Sin. in altitude. Find the volume of each of. these three
prismoids.
103. A prismoid, one of whose ends is a rectangle measuring
15 in. by 12*5 in., the opposite end measuring 9*6 in. by 8*4 in.,
and whose altitude is 2 ft. is cut into two wedges by a plane which
passes through the longer edge of one end and the opposite longer
edge of the other end. Find the volumes of the wedges.
200 ARITHMETIC.
104. The height of a wedge is 18 in., the length of the edge is
16 in., and the dimensions of the base which is a rectangle, are
12 in. by 8 in. The wedge is divided into two parts by a plane
parallel to the base and midway between the base and the edge.
Find the volume of each part.
105. Had the wedge described in problem 104 been bisected by
the plane j3arallel to the base, what would have been the height of
the plane above the base ?
106. The length of the edge of a wedge is 8*5 in., the length of
the base which is a parallelogram is 6*3 in. and its normal width is
4*5 in., the height of the wedge is 15 in. The wedge is divided
into three parts of equal height by planes parallel to the base.
Find the volume of each part.
107. The ends of a prismoid are rectangles whose corresponding
dimensions are 18 in. by 15 in. , and 10 in. by 18 in. ; the height of
the prismoid is 7 ft. 4 in. The prismoid is cut by a plane parallel
to the ends and at a distance of 2 ft. from the larger end. Show
that the section is a square, and find the volumes of the parts into
which the plane divides the jirismoid.
108. Find the number of cubic yards of earth in an embankment
from the accompanying plan and following data : —
The base ABCD is a ^
quadrilateral and the top
EFGHK is a pentagon. Tne
edges AB, EF, KH and DC
are all parallel to each other,
AD and EK are in a plane
at right-angles to AB, and
LG is parallel to EF. AB = 96yd., DC = 124yd. ; EF = 84yd.,
LG = 98yd., KH = 90yd. ; AE = 18ft., EL = 16ft., LK = 18ft.,
KD = 12 ft., the last four measurements being * in plan ', i.e. being
the horizontal distances between verticals through the points
A, E, L, K and D. The height of the embankment is 18 ft.
109. The lengths of two opposite edges of a tetrahedron are
7 '2 in. and 5 '6 in. respectively and the perpendicular distance
between them is 6 -4 in. The midcross-section is a rectangle. Find
the volume of the tetrahedron.
MENSURATION. 201
1 10. A recbiiigular tank 3 ft. long by 2 ft. 4 in. wide by 2 ft. 6 in.
deep, rested on props 3 in. high, a prop at each comer. By accident
one of the props was knocked out of its place and the cistern w^is
tilted on the adjacent two until the unsupported corner touched
the ground. How much less water would the tank hold in that
position than it would hold when level ?
111. The base of a wedge is a rectangle measuring 3 "6 in. by
2 '4 in., the length of the opposite edge is 3 in., the height of the
wedge is Sin. Find the volume 1° if the three-inch edge is parallel
to ,the longer side of the base ; 2° if it is parallel to the shorter side
of the base.
112. The base of a sphenoid is a square whose sides are 10 in,
long ; the opposite edge is parallel to the diagonal of the base, and
of the same length as the diagonal ; the altitude of the sphenoid is
15 in. Find the volume of the sphenoid.
113. The lengths of the three parallel edges of a wedge are
7 '5 in., 5'7in. and 6'9in. respectively and the area of a section at
right angles to these edges 76 sq. in. Find the volume of the wedge.
114. The base of a wedge is a rectangle measuring 13 '5 in. by
11 "2 in., the length of the opposite edge is 5 "4 in., this edge being
parallel to the longer side of the base ; the perpendicular distance
of this edge from the plane of the base is 18 in. The wedge is
divided into two pieces by a plane which intersects the edge opposite
the base at a point distant 7 '5 in. from one end and which cuts the
two edges parallel to this edge at points distant 5 "25 in. and 7*5 in.
from the ends corresponding to that from which the 7 '5 inches was
measured. Find the volume of each part.
115. The base of a wedge is a rectangle measuring 18 in. by
15 in. ; the opposite edge is parallel to the longer side of the base
and is 10 in. long ; the length of the perpendicular from this edge
on the base is 21 in. Find the volume of the parts into which the
wedge is cut by a plane passing through one end of the edge
opposite the base and which is parallel to the triangular face at the
other end.
116. The base of a wedge is a trapezoid whose parallel edges are
3 ft. and 1ft. 9 in. long respectively and whose width at right
angles to these sides is 15 in. , the length of the edge opposite the
202 ARITHMETIC.
base is 18 in., and the volume of the wedge is 2cu. ft. Find the
altitude of the wedge.
117. The length of a side of the base of a frustum of a square
pyramid is 3' 9", that of a side of the top is 1' 8", the altitude of the
frustum is 2' 6". Find the volume of the frustum.
118. Find the number of cubic feet in a stick of square timber
18" square at one end, 14" square at the other end and 36'«4ong.
119. Find the weight of a frustum of a square pyramid of
marble, the height of the frustum being 6 ft. 6 in. ; the length of
an edge of the base, 4 ft. 4 in. , and of an edge of the top, 2 ft. 8 in. ,
the weight of a cubic foot of marble 172 lb.
120. In the frustum of a square pyramid whose base-area is
2 sq. yd. and whose altitude is 4 ft. 6 in. , the lengths of the basal
edges are to those of the top edges as 3 to 2. Find the volume of
the frustum.
121. The areas of the base and top of a frustum of an iron
pyramid are 1 sq. ft. 48 sq. in. and 1 sq. ft. 3 sq. in. respectively and
the weight of the pyramid is 8881b. Find the height of the
pyramid, the specific gravity of the iron being 7 "11.
122. The altitude of a frustum of a square pyramid is equal to
the length of a side of the base and is double of the length of a side of
the top. Find the altitude, the volume of the frustum being 4 cu. ft.
123. A frustum of a pyramid has the area of its base nine times
the area of its top. Compare its volume with that of a prism whose
altitude and base-area are respectively the same as the altitude and
the base-area of the frustum.
1 24. A frustum of a pyramid has the area of its base four times
the area of its top. Compare its volume with that of a pyramid
whose altitude and base-area are respectively the same as the
altitude and the base-area of the frustum.
125. Find the volume of the frustum of a pyramid on a
rectangular base measuring 4 ft. by 2 ft. 8 in. , the height of the
frustum being 5 ft. 8 in. and the length of the top, 3 ft. 6 in.
126. The volume of the frustum of a pyramid on a rectangular
base is 3*6 cu. ft. The length of one side of the base is 1 ft. 6 in.,
the length of the corresponding side of the top is 10 in., the height
of the frustum is 1 ft. 4 in. Find the lengths of the other sides of
the base and top.
W MENSURATION. 203
139*. The base of the frustum of a pyramid is a rectangle whose
length is double its width ; the area of the top is half the area of
the base ; the height of the frustum is 3ft. 8 in. and its volume is
a cubic yard. Find the length of the base.
128. The base of a frustum of a pyramid is a trapezoid the lengths
of whose parallel sides are 275 cm. and 225 cm. respectively, the .
distance between them being 192 cm. The height of the frustum
is 2375 mm. and the width of its top is 148 cm. Find the volume
of the frustum in cubic metres.
129. Find the area of the surface of a square pyramid whose basal
edges are each 3' 4" long, the slant height of each side being 3' 6".
130. Find the area of the surface of a frustum of a square
pyramid, the length of a side of the base being 18 in. ; that of a side
of the top, 6 in. ; and the slant height of each of the lateral faces
being 27 in.
131. Find the height and the width of a quad whose length is
3 ft. whose volume is 9 cu. ft. and the area of whose surface is
28 sq. ft. 108sq.in.
132. A horse-trough 9 ft. long, 15 in. wide at the top and 10 in.
wide at the bottom, and 12 in. deep, is full of water. If 30 gallons
of water be drawn oft* by how many inches will the surface of the
water in the trough sink ? (The ends of the trough are vertical ;
the calculation is to be made to the nearest tenth of an inch.)
133. The cross-section of a canal is 33 ft. wide at the bottom and
58 ft. wide at a height of 10 ft. from the bottom. At what depth
must the water in the canal stand that 1000 yd. in length of the
canal may contain 4,545,725 gallons ?
134. The volume of the frustum of a square pyramid is
172 cu. ft., the height of the frustum is 36ft. and the length of a
side of the base is 2 ft. 8 in. Find the length of each side of the
top**
1 35. A covered rectangular tank whose dimensions are 3' 6" by
2' 11" by 1' 9" will hold just 81 gallons. What must be the thickness
of the material of which the tank is made, the bottom, sides and
top being all of the same thickness ? (This problem is of a type
which is the inverse of the type to which problems 9 and 10 of Exercise
VI belong. The calciolations for these three problems will therefore
follow parallel lines. )
204 ARITHMETIC.
171. In any right-angled triangle, the squares on the sides
containing the right angle are together equal to the square on the
hypothenuse or side opposite the right angle. (Euclid I, 47.)
Let ABC be a triangle right-angled at C, and let a, h and c be the
MEASURES of the lengths of the sides opposite the
angles A, B and C respectively ; . '. a'^, h"^ and c^ are
the measures of the areas of the squares on these
sides and it follows from this and the preceding
proposition that
«2+6^'=c2, (1)
and .-. , c = {a^+h^f, (A) ^
that is ; — hi any rifjht-aiigled triangle, the measure of the length of
the hypothenuse is the square root of the sum of the sqiuxres of the
measures of the leicgths of the sides containing the right angle.
From(l) a^ = c^-b^ = {c + h)(c-h), ' (2)
.-. ^ a=^(e + h)(c-h)y\ (B)
that is ; — In any right-angled triangle, the measure of the length of
either of the sides containing the right ayigle is the square root of the
product of the sum and the difference of the measures of the lengths of
the other two sides of the triangle.
Example 1. The lengths of the sides containing the right angle
of gr right-angled triangle are 336 ft. and 627 ft. respectively ; find
the length of the hypothenuse and of the perpendicular from the
right angle on the hypothenuse.
Length of hypothenuse = (336 2 -f- 527 - f ft.
= (112896-1- 277729)^" ft.
= 390625^ ft.
= 625 ft.
Measure of length of perpendicular on hypothenuse x 625
= double of the measure of the area of the triangle
= 336x527
=177072.
. *. the measure of the length of the perpendicular on the hypothenuse
= 177072 -f 625
= 283-3152;
. •. the length of the perpendicular on the hyi)othenuse = 283 "3152 ft.
MENSURATION.
205
Example 2. The lengths of two of the sides of a triangle are
1590 mm. and 1037 mm. respectively, and the length of the
perpendicular from the opposite angle on the longer of these sides
is 988 mm. Find the length of the third side of the triangle.
Let ABC be the triangle, AB = 1590 mm. , C
and BC = 1037 mm. From C let fall on AB
the perpendicular CD then CD = 988 mm.
(TJie accompanyinci fifjure is drawn mi a
scale of 1 : 64. )
DB = -{ (1037 + 988) x (1037
= (2025x49pmm.
= 315 mm.
. • . AD = 1590 mm . - 315 mm .
= 1275 mm.
. •. AC = (12752 +9882)^ mm.
= (1625625 + 976144)^ mm.
= 2601769^ mm.
= 1613 mm.
172. If m and n be any two whole numbers then shall
m^—'ii^f 2tnn and m^+^t^
be the measures of the lengths of the sides of a right-angled
triangle .
For {7n--n-Y=m^-2m'^n.'^ +n^,
{2mny = 4:111^)1^
and (m2 + h2)2 ^m^ +2n'^u'^ +u*.
EXERCISE XXIV.
1. Find the length of the hypothenuse of a right-angled triangle
whose other sides are 65 in. and 72 in. long respectively.
2. Find the length of the hypothenuse of a right-angled triangle,
the lengths of the other sides being 777 mm. and 464 mm.
3. Find the length of the diagonal of a square whose side is one
foot long.
4. Find the length of the diagonal of a cube whose edge is one
foot long.
200 ARITHMETIC.
5. What is the length of the side of a square whose diagonal is
one foot long ?
O. What is the length of the edge of a cube whose diagonal is one
foot long ?
7. What is the length of the edge of the largest cube that can be
cut out of a sphere 6 inches in diameter ?
8. What is the length of the diagonal of a cube if the length of a
diagonal of one of the faces of the cube is 3 ft. ?
9. What will be the length of the diagonals of the faces of the
largest cube that can be cut out of a sphere 3 inches in diameter ?
10. A quad measures 24 in. by 11 '7 in. by 4-4 in. Find the
lengths of the diagonals of its faces.
1 1 . A quad is 14 ft. long by 5 ft. wide by 2 ft. thick. Find the
length of its diagonal.
12. A quad measures 6*325 m. by 5*796 m. by "528 m. Determine
the length of its diagonal and the lengths of the diagonals of its faces.
13. The lengths of the diagonals of the faces of a quad are 22 ft.,
8 ft. and 3 ft. respectively. Find the length of the diagonal of the
quad.
14. The diagonals of the faces of a quad are respectively 25 in.,
23*79 in. and 9*79 in. long. Determine the lengths of the edges of
the quad.
15. The lengths of the sides of the base of a triangular pyramid
are 38 '83 in. , 30*92 in. and 25*95 in. respectively. The lateral edges
meet at right angles at the vertex. Find the volume of the pyramid.
16. The lateral edges of a pyramid are all equal to one another.
The base is a rectangle 4 ft. 8 in. long by 4 ft. wide. The height
of the pyramid is 3 ft. 9 in. Find the area of the surface.
17. The hypothenuse of a right-angled triangle is 16*13 in. in
length and one of the other sides is 12 '75 in. long. Determine the
length of the third side.
1§. Two sides of a triangle are 218 ft. and 241 ft. in length and
the perpendicular from the included angle on the third side is 120
ft. long. Find the length of the third side.
19. A ladder 25 ft. long stands vertically against a wall. How
far must the foot of the ladder be drawn out liorizontally from the
wall that the top of the ladder may be drawn down one foot %
MENSURATION.
207
20. A rope hanging loose from a hook 26 ft. above level ground,
just reaches the ground. How high above the ground will the
lower end of the rope be when it is drawn 10 ft. aside from the
vertical ?
21. Two of the sides of a triangle are 1450 ft. and 1021ft. long
respectively. From the contained angle a perpendicular is let fall
on the third side, and the segment of that third side between the
foot of the perpendicular and the shorter of the first mentioned
two sides is 779 ft. in length. Find the area of the triangle.
22. The base of a pyramid is a rectangle 12 in. long by 10 in.
wide. The lateral edges are each 31 in. long. Find the volume of
the pyramid and the area of its surface.
23. A flagpole 53 ft. 4 in. in height is broken by the wind and
the top falling over strikes the ground 14 ft. 8 in. from the foot of
the pole before the pieces part at the place of breaking. Find the
length of the piece broken off, the ground being level.
24. A B and C are three houses standing at the angles of a
right-angled triangle. A is 80 ch. east of 0, and B is north of C
and 51"20ch. nearer to it than to A. Find the distance from A
to B.
25. The lengths of the four sides of a trapezoid taken in order
are 608 ft. , 554 ft. , 250 ft. and 520 ft. Find its area and the lengths
of its diagonals.
173. In any obtuse-angled triangle, the squares on the sides
containing the obtuse angle are together less than the square on the
third side or side opposite the obtuse angle by twice the rectangle
under either of the two sides containing the obtuse angle and the
IDrojection on it of the other of these two sides. (Euclid II, 12. )
Let ABC be a triangle obtuse- B
angled at C, and let a, h and c be |
the MEASURES of the lengths of I
the sides opposite the angles A, B [
and C respectively, and let ^-„ be •
L-
the measure of the length of CD, d ' " C b
the projection of CB on AC produced, tnen will «-, h^ and e^ be
the measures of the areas of the squares on the sides and 6/_a will
208
ARITHMETIC.
be the measure of the area of the rectangle contained by CA and
CD. It follows from this and the preceding proposition that
a2 + 6- + 26/_, = c2 (3)
and .-. (a2 + 62 + 2?>/_,)2 = c, (C)
that is ; — In any ohtuse-angled triangle, if to the sum of the squares
of the measures of the lengths of the sides containing the obtuse angle
there be added tivice the product of the measures of the lengths of either
of these sides and the projection on it of the other of these sides, the
square root of the sum will be the measure of the length of the side
opposite the obtuse angle.
174. In any triangle, the squares on the sides containing an
acute angle are together greater than the square on the third side,
or side opposite the acute angle, by twice the rectangle under either
of the two sides containing the acute angle and the projection on it
of the other of these two sides. (Euclid II, 13. )
Let ABC be a triangle acute-angled
at C and let a, b and c be the measures
of the lengths of the sides opposite the
angles A B and C respectively, and
let y^a be the measure of the length of
CD, the projection of CB on CA
(produced if necessary) ; then will a- ^
b^ and c^ be the measures of the areas of the squares on the sides
and bd^ will be the meas^ire of the area of the rectangle contained
by CA and CD. It follows from this and the preceding proposition
that
a2 + 62-26< = c2 (4)
and.-. (a2 + 62_264)V=:c, (D)
that is ; — In any triangle, if from the sum of the squares of the
measures of the lengths of the sides containing an acute angle there be
subtracted tivice the product of the Tueasures of the lengths of either of
these sides and the projection on it of the other of these sides, the square
root of the remainder will be the measure of the length of the side
opposite the acute angle.
175. If the angle BCD be one-third of two right angles, i.e., if
it be equal to the angle of an equilateral triangle, the line CD will
MENSURATION. 209
be equal to half of the side C B ; therefore twice the rectangle
under CD and CA will be equal to the rectangle under CB and
CA and consequently (C) of § 173 will become
c = (a^ + b^+ab)^ (Cc)
and (D) of § 174 will become
c = (a2 + 62-a6)2 (Dd)
It should be noticed that in the case of (Cc) the angle BCD is
an external angle of the triangle ABC and the internal obtuse
angle is two-thirds of two right angles.
EXERCISE XXV.
1 . The lengths of the sides of a triangle are 125 ft. , 244 ft. and
267 ft. respectively. Find the area of the rectangle under each
side and the projection on it of either of the other sides. Find
also the lengths of the sides of three squares equal in area to the
three rectangles thus obtained.
2. The lengths of the sides of a triangle are 84ft. 1 in., 158 ft.
2 in. and 188 ft. 3 in. respectively. Find the length of the
projection of the shortest side on the longest.
3. The lengths of the sides of a triangle are 595 mm. , 769 mm.
and 965 mm. respectively. Find the length of the projection of
the shortest side on each of the others.
4. The lengths of the sides of a triangle are 25 in., 39 in., and
40 in. respectively. Find the lengths of the projection of the
shortest side on each of the others and the lengths of the perpen-
diculars on these sides from the opposite angles.
5. In a right-angled triangle, the lengths of the sides containing
the right angle are 30 ft. 4 in. and 52 ft. 3 in. Find the lengths
of the segments into which the hypothenuse is divided by the
perpendicular on it from the right-angle, and also the length of that
perpendicular, and prove that the product of the measures of the
lengths of the segments of the hypothenuse is equal to the square
of the measure of the length of the perpendicular.
6. The lengths of the sides of a triangle are 13 yd. , 14 yd. and
15 yd. respectively. Find the lengths of the perpendiculars on the
sides from the opposite angles,
N
210 ARITHMETIC.
7. Show that the triangle whose sides are respectively 25 ft.,
39 ft. and 66 ft. long, is obtuse angled and find the lengths of the
projections of each side on the other two.
8. From a point O, three lines OA, OB and OC whose lengths
are respectively 196 ft., 264 ft. and 326 ft. are drawn making equal
angles with one another in the same plane. Find the lengths of
the lines AB, BC and CA.
9. From a point O, three lines OD, OB and OC whose lengths
are respectively 440 ft. , 264 ft. and 325 ft. are drawn making equal
angles with one another in the same plane. Find the lengths of
the sides of the triangle BCD.
10. The lengths of the sides of a triangle are 21ft. 2 in., 21ft.
10 in. and 26 ft. 4 in. respectively. Find the lengths of the medians
of the triangle. (See Mackay's Euclid, Ap. II, Prop. 1.)
176. If the lengths of the sides of a triangle are known, the
propositions of §§ 173 and 174 will enable us to determine the length
of the perpendicular on any side from the opposite angle and
consequently to find the area of the triangle. It is not however
necessary to compute the length of the perpendicular on a side in
order to find the area of the triangle, this may be determined
directly from the lengths of the sides as follows : —
iii, a. From the measure of the length of the semiperimeter of the
triangle svbtract the measure of the length of each side separately,
multiply together the three remainders and the commmi minuend, the
square root of the product will he the measure of the area of the triangle ;
or, j^
^.= ^ s{s-a){s-h){s-c) ^2
in which >S^t is the measure of the area of the triangle, a, h and c are
the measures of the lengths of the sides and s is the measure of the
semiperimeter ; i.e.,
2s — a + h + c.
Let X, y and h denote the measures of the lengths of AD, CD
and BD respectively, see Figs, of §§ 173 and 174, then will
Fig. of § 173. Fig. of § 174.
x — y = h x + y==b
y2 =a2+7i,2 yi = a^ ^h^
211
and
MENSURATION.
jg2 _^2_g2 _^^2
£C2 -1/2=^2 -a2
y)(x + y) = c^-a^
(a; + i/)(a;-i/) = c2-a2
b{x + y) = c^-a^
6(x-i/) = c2-a2
h(x-y) = b^
6(x + ]/) = 62
2hx =62+c2-
-0,2
26x =62^c2
/l2 = c2 — iC^ = (C + ic) (C — £C)
462/^2 = (26c + 26x) (26c - 26x0
=26c + 62 + c2-a2)(26c-62-c2+a2)
= <!(6 + c)2-a2 y ^ a2-(6-c)2 ^
= (6 + c + a)(6 + c-a)(a + 6-c)(a-6 + c)
= 2s(2s - 2a) (26- - 2c) (2s - 26)
= 16s(s-a)(s-6)(s-c).
i62;t2=s(,>_ct)(s-6)(s-c);
^6/i =
-is(s-
-a)(s-
-a){s-
-c)y^.
S.-
= lbh
Sr-
= -{s(s-
-a)(s-
-b)(s-
-c)P.
But
An important advantage of this method of computing the area of
a triangle is that it can be adapted to calculation by logarithms for
it yields at once
log >S^t = Y"^ log 8 + log {s — a) + log (s - 6) + log {s -c) )-.
Example. Find the area of a triangle the lengths of whose sides
are 13'14m., 14'15 m. and 15'13m. respectively.
26- -42 -42
s = 21-21 .-. log s =1-326541
s-a= 8-07 log(s-a)= -906874
s-6= 7-06 log(s-6)= -848805
s-c== 6-08 log(s-c)= -783904
2)3-866124
.-. logS = 1-933662 = log 85 -716
the area of the triangle is 85-716 square metres.
177. If the measures of the lengths of the sides of a triangle be
kl (m2 + »fc2), mn {k^ + ^2), (kn + Im) (km - In),
the measure of the area of the triangle will be
klmn (hi + Im) Qcm, - In)
fc, I, m and n denoting any numbers whatsoever.
212 ARITHMETIC.
The triangle can be resolved into two right-angled triangles, the
measures of the lengths of the sides of the first being
M (m^ + n-), hi (m^— n^), 2klmn,
and the measures of the lengths of the sides of the second being
mn (k^ +1^), mn (k^—P), 2khnn ;
and kl{m'^-n'^) + m7i{k^ — l^) = (hi + Im) {km - In).
EXERCISE XXVI.
Find the areas of the triangles the lengths of whose sides are
respectively
- 1. 13yd., 10yd. and 13 yd. 6. 13 in., 21 in. and 20 in.
^ 2. 13 yd., 24yd. and 13 yd. 7. 13 m., 37 m. and 40 m.
» 3. 13 ft., 4 ft. and 15 ft. . 8. 13 m., 45 m. and 40 m.
' 4.13 ft., 14 ft. and 15 ft. , 9. 1.23 ch., 5-95 ch. and 6-76 ch.
* 5. 13 in., 11 in. and 20 in. 10. 73'2ch., 45-5 ch. and 87 -6 ch.
11. What will be the value at $73 per acre of a triangular piece
of land the lengths <^f whose sides are 478 '5 chains, 329 '6 chains and
237 *4 chains respectively ?
12. A triangular piece of land the lengths of whose sides were
•1234 miles, '2315 miles and '2086 miles respectively was sold for
$975. What was the price per acre ?
! 3. The lengths of the sides of a triangle are respectively 212 ft..
225 ft. and 247 ft. A straight line is drawn across the triangle
joining the mid-points of two of the sides. Find the area of the
trapezoid thus formed.
14. The lengths of the sides of a triangle are 126 m., 269 m. and
325 m. respectively. Straight lines are drawn across the triangle
parallel to one of the sides and joining points of trisection of the
other two sides. Find the areas of the parts into which the triangle
is thus divided.
15. The length of the side ot a square is 44 ft. A point is taken
within the square distant 12 '9 ft. and 37 '7 ft. respectively from the
ends of one side. Find the are#s of the triangles formed by joining
the point to the four corners of the square.
MENSURATION.
213
^' 16. The lengths of two adjacent sides of a rectangle are 349 ft.
and 247 ft. A point is taken within the rectangle distant 225 ft.
and 164 ft. respectively from the ends of the longer side of the
rectangle. Find the areas of the triangles into which the rectangle
is divided by lines joining its angular points to the given point,
17. The lengths of two of the sides of a triangle are 55 ch. and
39 ch. respectively and the angle contained between these sides is
two-thirds of a right-angle. Find the area of the triangle.
18. Find the area of the gable end of a barn 66 '2 ft. wide, the
height of the eaves being 19 ft. at the front of the barn and 8 ft. at
the back, and the lengths of the rafters being 29 ft. on the front
and 56 "2 on the back, the bam standing on level ground.
19. The lengths of the sides of the triangle ABC are 6983 mm. ,
17079 mm. and 18574 mm. Find the area of a triangle whose sides
are equal to the medians of the triangle ABC.
20. The lengths of the medians of a triangle are 16"45ch.,
47 '77 ch. and 60*52 chains respectively; find the area of the triangle.
I
178. Given the le'}igth of the radius of a circle and the length of the
chord of any arc of the circle, to find the length of the chord of half
the arc.
Let r, k and fcg denote the measures of the lengths of the radius,,
the chord of the arc and the chord of half the arc respectively, then
^ill h, = -{2r^-,i4r^.-kl)^yk
Let ABK be the circle, C its centre, ADB the arc and D the
mid-point of the arc. Join AD,
AB, AC and CD ; the radius
CD will bisect the chord AB
at right angles, say in E. The
measures of the lengths of AC,
AB and AD are respectively r,
fci and fcg. Let q denote the
measure of the length of CE.
CE2 = CA2-AE-
r
4(^'
4 7-2-fc^
214
by(i)
%
ARITHMETIC.
2r-
= (4r2
-K)^
.D2:
= AC2
+ CD2-
-2CDCE
=2 CD
2-2CDCE
fc^
= 2r2-
-2r(i
= 2r2-
-r(4r2
-1^1)^
h,:
= ^2r
2-r(4^
r--l\YY
(1>
(2)
(3)
Example 1. The side of a regular hexagon inscribed in a circle is
equal to the radius of the circle, find the length of a side of the
inscribed regular convex dodecagon.
In this case we are given fc i = r,
h2 = ^ 2r2-^-(4r2-r2)i[.,^
=(2r2-3^r2)^
= (2-l-73205081)2r
= •267949192r
= -siyessoQr.
Therefore the length of a side of a recpdar convex dodecagon
inscribed in a circle is '51763809 of the length of the radius of the
circle.
The length of the semiperimeter of the dodecagon is six times the
length of a side and -51763809 r x 6 = 3-1058285 r, therefore
The length of the semijyeri meter of a regular convex dodecagon
inscribed in a circle is 3*1058285 times the length of the radius of the
circle. /
Example 2. Fiiid the length of a side of a regular convex polygon
of 24 sides, inscribed in a circle.
In this case fc^ is the measure of the length of a side of regular
convex polygon of twelve sides, inscribed in the circle, . '. by
Example 1, A; i = -51763809 r,
A- 2= -i 2 r2 - r (4r^ - '517638092 /•2)2 j. h
= (2r2-3-73205081^r2)2
= -26105238 n
MENSURATION.
215
Therefme the hiigth of a side of a regidar convex 24-gon inscribed in
a circle is '26105238 of the length of the radius of the circle.
The length of the semiperimeter of the 24-gon is 12 times the
length of a side and -26105238 r x 12 -3 '1326286 r, therefore
The length of the semiperimeter of a regular convex 24-gon inscribed
in a circle is 3*1326286 times the length of the radius of the circle.
1 79. Given the lengths of the radius of a circle, of the chord of any
arc of the circle aiid of the chord of half the arc, to find the sum of the
lengths of the tangents from the ends of the half -arc to their point of
intersection.
Let r, fci and kz denote the measures of the lengths of the radius,
the chord of the arc and the chord of half the arc respectively, and
let t denote the sum of the measures of the lengths of the tangents
from the ends of the half -arc to their point of intersection, then
will
Let ABK be the circle, ADB
the arc and D the mid-point of
this arc. Join AD and AB and
draw tangents to the circle at
A and D and let them meet in
G. Draw the diameter D K
bisecting the chord AB in E.
Join KA and produce KA and
DG to meet in F. Then because
GA and GD are equal, being
tangents from the point G, and
DAF is a right angle, therefore
the angle GAF is equal to the
angle GFA, therefore GA is equal to GF and consequently the
tangents AG and DG are together equal to DF. The measures of
the lengths of AB and AD are fc^ and h^, and the sum of the
measures of the lengths of AG and DG which is equal to the
measure of the length of DF, is t.
EA is parallel to DF, both being at right angles to DK, therefore
the angle EAD is equal to the angle ADF, also the angle AED is
216 ARITHMETIC.
equal to the angle DAF, both being right angles, therefore the
triangle AED is similar to the triangle DAF
FD:DA::DA:AE
t 2^2
Example. Find the length of a side of a regular convex dodecagon
circumscribed about a circle.
Let T be the measure of the length of the radius of the circle,
then will fcj =r and h^ = '51763809 r. (See Example 1, p. 214.)
i = 2(-5l763809r)2^r
= •53589838 r.
Therefore the le^igth of a side of a regular convex dodecagmt
circumscribed about a circle is '53589838 of the length of the radius oj^
the circle.
The length of the semiperimeter of the dodecagon is six times
the length of a side, and '53589838 r x 6=3-2153903 r ; therefore
The length of the semiperimeter of a regular convex dodecagon
circumscribed about a circle is 3 '2153903 times the length of the radius
of the circle.
EXERCISE XXVII.
Find the length of a side and also the length of the semiperimeter
of a regular convex polygon inscribed in a circle, the unit of
measurement being the radius of the circle and the number of the
sides of the polygon being
1. 48. 2. 96.
Find the length of a side and also the length of the semiperimeter
of a regular convex polygon circumscribed about a circle, the unit
of measurement being the radius of the circle and the number of
the sides of the polygon being
3. 24. 4. 48. 5. 96r
MENSURATION.
217
180. RECTIFICATION OF THE
CIRCLE. Let ABK be a circle,
C its centre, ADB an arc of the
circle and D the mid-point of
the arc. Join AD, AB, AC and
CB. Draw the diameter DCK
bisecting the chord AB at right
angles in E. Draw AG and
DG, tangents to the circle at A
and D. Produce CA and DG
to meet in H, and CB and GD
to meet in M. Draw KA and
produce it to meet DH in F.
If the chord AB be a side of
a regular convex polygon of
n sides, say a regular n-gon,
inscribed in the circle ABK, the
chord AD will be a side of a
regular 2M-gon inscribed in the
circle, HM will be a side of a regular n-gon circumscribed about the
circle and FD which is equal to AG + GD, will be equal to a side
of a regular 2n-gon circumscribed about the circle.
The perimeter of the inscribed n-gon will be n times AB which is
equal to 2n times AE. Let 2n (AE) denote and be read " 2n times
AE."
The perimeter of the inscribed 2 7/-gon will be 2n (AD).
The perimeter of the circumscribed n-gon will be n (HM) which
is equal to 2n (HD).
The perimeter of the circumscribed 29i-gon will be 2n (FD).
Now AED being a right angle, AD is greater than AE,
2>t(AD)>2n(AE),
i.e., the perimeter of the inscribed regular 2ii-gon is greater than
the perimeter of the inscribed regular n-gan.
Because FD is less than HD
2n(FD)<2vi(HD),
i.e., the perimeter of the circumscribed regular 2;(,-gon is less than
the perimeter of the circumscribed regular u-gon.
218 ARITHMETIC.
The angle DAF being a right angle, FD is greater than AD
2n(FD)>2n(AD)
i.e., the perimeter of the circumscribed regular 2?i-gon is greater
than the perimeter of the inscribed regular 2?t-gon.
If then a regular hexagon be inscribed in a circle, and a similar
hexagon be circumscribed about the circle, the perimeter of the
circumscribed hexagon will be greater than the perimeter of the
inscribed hexagon.
If next a regular convex dodecagon be inscribed in the circle
in which the hexagon was inscribed and a similar dodecagon be
circumscribed about the same circle, the perimeter of the inscribed
dodecagon will be greater than the perimeter of the inscribed
hexagon, and the perimeter of the circumscribed dodecagon will be
less than the perimeter of the circumscribed hexagon but will be
greater than the perimeter of the inscribed dodecagon. Hence
the difference in length between the circumscribed and inscribed
dodecagons is less than the difference in length between the
circumscribed and inscribed hexagons.
If next a regular 24-gon be inscribed in the circle and a similar
24-gon be circumscribed about the circle, the perimeter of the
inscribed 24-gon will be greater than the perimeter of the inscribed
12-gon, and the perimeter of the circumscribed 24-gon will be less
than the perimeter of the circumscribed 12-gon but greater than
the perimeter of the inscribed 24-gon. Hence the difference in
length between the perimeters of the circumscribed and inscribed
24-gons is less than the difference in length between the perimeters
of the circumscribed and inscribed 12-gons.
If next a regular 48-gon be inscribed in the circle and a similar
48-gon be circumscribed about the circle, the difference in length
between their perimeters will be less than the difference in length
between the perimeters of the circumscribed and inscribed 24-gons.
By continuing this process we shall obtain a series of pairs of
polygons whose perimeters become more and more nearly equal at
each doubling of the number of their sides.
Now as the circumference of a circle is greater than the perimeter
of any regular convex polygon inscribed in the circle but is less than
the perimeter of any similar polygon circumscribed about the
circle, the lengths of the perimeters of these polygons may be taken
MENSURATION. 219
as limits between which the length of the circumference must lie.
But it has been shown that, beginning with a regular hexagon, as
the number of sides of the inscribed and circumscribed polygons is
successively doubled the difference between the lengths of their
perimeters becomes less and less. In other words, by repeatedly
doubling the number of the sides of similar inscribed and
circumscribed polygons, the limits between which the length of the
circumference lies, are made continually to approach each other *
and therefore a nearer and nearer approach may be made to the
exact length of the circumference.
As an example let us take the measures of the lengths of the
semiperimeters of the inscribed and circumscribed regular convex
polygons of 12, 24, 48, and 96 sides respectively, which are given
in the Examples of §§ 178 and 179 and in the answers to the
problems in Exercise xxvii, and, 7t denoting the measure of the
length of the semicircumference
measurement, we shall obtain
from the 12-gons
from the 24-gons
from the 48-gons
from the 96-gons
Since 3fo< 3 141 and 3 "1428 ■
3if<;r<3ia.
These are known as Archimedes' Limits of the ratio of the
semicircumference of a circle to its radius, or of the circumference
to its diameter.
[181. Had we carried our calculations beyond the 96-gons to
the 12,288-gons we should have obtained
3 1415926 <7t< 3-1415927.
Vieta, doubling the number of sides 16 times successively
computed to 10 places of decimals the lengths of the perimeters of
the inscribed and circumscribed regular 393,216-gons and found
that
31415926535 <7t< 3-1415926537.
* We do not here enquire whether the Hmits thus found may be made
approach each other indefinitely, nor is it necessary to ascertain whether they
do so, for we seek not an exact but only an approximate rectification of the
circle.
ce when the
radius is the
unit of
3-10 <
1C
< 3-22
313 <
It
< 3 16
3 139 <
Tt
< 3 1461
3-141 <
It
< 3 1428.
<3ig
220 ARITHMETIC.
Ludolph van Ceulen starting from squares and successively
doubling the number of sides 60 times, determined it to 35 decimal
places.
182. The method which has been described of approximating to
the value of tc depends on the proposition that the arc AD (see
Fig. on p. 215) is greater than the chord AD but is less than the
sum of the tangents AG and GD, i. e. , less than FD. This method
is extremely tedious if 7t is to be computed to more than three or
four decimal places, but the following theorems afford a means of
greatly reducing the labor of calculation.
r. Thearc AD > chord AD + ^^ (AD- AE);
2\ The arc AD < chord AD + i (DF - AD).
If for these magnitudes we substitute the measures of their
lengths in terms of the radius as unit of measurement, and multiply
throughout by 2 n, we shall obtain
2 TT > 2 nfeg + ^ (271^2 - ^^^i\
2 TT < 2 nk^ + J {2nt - 2 nk^).
If Pn and P^n denote the measures of the lengths of the perimeters
of the inscribed regular 7i-gon and 2 »t-gon respectively and Q.^n
denote the measure of the length of the perimeter of the
circumscribed regular 2 -jt-gon, in terms of the radius as unit of
measurement, then will
P„==7tfc|, F 2^ = 2)1^2 and ^2^ = 2 n^
and the preceding limits may be written
2^>P2n+i3(^2n-P»),
27r<P2„+^(g2„-P2„).
As an example of the closeness of these limits let us take the
case in which Pgn is the measure of the length of the perimeter of
the inscribed regular 96-gon, then will
^P„ = 3-1393502,
JP2n = 3-1410319, and ^(;)2„ = 3 1427146
and .-. :;r>3-1410319 + i(31410319-3-1393502)
but 7t < 3-1410319 + 1 (3 1427146- 3-1410319)
i.e., 7r> 3-1415925
but ;r< 3 -1415928,
MENSURATION. 221
1 83. Numerous other geometrical constructions of the approximate
length of an arc have been proposed for the evaluation of it, the best
being one which yields
30;r<8^4„ + 8P2a-Pa
This was published in 1670 by James Gregory who at the same
time laid the foundation of the modern methods of computing 7t by
proving that
i^=i-HI-H5-T\+ ,
the series to be continued endlessly.
Twenty-nine years later, Machin announced that
, _ /1_^^ 1 1
^5 3x5s 5x55 7x5-
^239 .3x239-^"^5x2395~7'^"23"97'^ /
and computed tt thereby to 100 places of decimals. Recently W.
Shanks, employing this series, has calculated 7t to 707 places of
decimals.
The rapidity of convergence of Machin's series gives it a great
advantage over Gregory's for purposes of calculation, but it and the
many other series which have been proposed and used for the
evoluation of tt, can be easily deduced from Gregory's series.
It may here be mentioned that it has been proved that tt is a
transcendental number, i.e., Tt cannot he exactly expressed by a
definite nwinber of integers combined by the operations of addition,
subtraction, midtiplication, division, involution and evolution.]
184. Let Pc be the length of the circumference of a circle and R
be the length of the radius, therefore, since 27t is the measure of Po
in terms of R as unit of measurement, Pc will be equal to 27? radii,
which we shall denote by P^ — 27r(R). If we now adopt any other
unit than R, say U, and if p^ be the measure of Pc and )• be the
measure of R both in terms of U, then will
Pc=i?c(U)andR = r(U),
and.-. p.(U)=^27t^r(U)y .
= (2^r)(U),
2)c = 27rr.
222 ARITHMETIC.
that is, the measure of the length of the circumference of a cirde is the
prodiict of Tt and twice the measure of the length of the radius^
r -2^- correct to 3 significant figures,
Tt being < 3 '1416 n n 5 n it
185. If 2 a. 2 6 and 'p^ denote the measures of the lengths of the
major and minor axes and of the perimeter of an ellipse and if
a^ -6^ be small compared to a^, then will
a2+3 62
^ ii^«'+36M
but 'p,<Tt\ 2(a2 + 62) j3 .
[186. If a conical spiral beginning with a radius of r^ units,
advance in n revolutions through a distance of h units measured
on the axis of the cone, and have then a radius of r^ units, the
measure of the length of the spiral will be roughly approximate to
If ro=rn, the curve is a cylindric spiral or helix, (the edge of the
thread of a screw) and the rectification is exact.
If /i=o, the curve is the common spiral or spiral of Archimedes.]
EXERCISE XXVIII.
1. The inner diameter of a circular drive is 210 ft. in length and
the width of the drive is 28 ft. Find the length of the inner and
of the outer edge of the drive.
2. What will be the cost of the wire at $1.25 per 100 yd. for a
barbed-wire fence five wires high around a circular fish-pond 60 ft.
in diameter %
3. The minute hand of a clock measures 1ft. 3f in. from the
centre of its arbor to the tip of the hand. Find the distance travelled
by the tip of the hand during the course of 365 days.
4. Find the length of the .l-a^ius of a wheel which made 1600
revolutions in rolling 3*25 miles. •
5. A circular path is 400 yd. in length on its inner edge. What
will be its length 6 ft out from that edge all around ?
MENSURATION. 223
■^ 6. The length of the hypothenuse of a right-angled triangle is
2 "9 in. and that of one of the other sides is 2'lin. Find the length
of the radius of a cisrcle whose circumference is equal to the sum of
the lengths of the circumferences of circles described on the three
sides of the triangle as diameter.
7. The difference in length between the diameter and the
circumference of a circle is 2 ft. 6 in. ; find the length of the diameter.
8. If Mercury describe round the sun in 87 '97 days a circle whose
radius is 35,700,000 miles in length and Saturn describe in 10759-22
days a circle whose radius is 882,000,000 miles long, what will be
the orbital speed in miles per minute of each of these planets ?
V 9. Find the length of the arc which subtends an angle of 60° at
the centre of a circle of 10 in. radius.
10. Find the length of the arc which subtends an angle of 36" at
the centre of a circle of 25 in. radius.
11. Of how many degrees will the angle be which an arc whose
length is 1 ft. , subtends at the centre of a circle of 2 ft. radius.
' 12. How many degrees will there be in the angle subtended at
the centre of a circle of 1 ft. radius, by an arc whose length is 2 ft. ?
V 13. How many degrees will there be in the angle subtended at
the centre of a circle by an arc whose length is equal to the length
of the radius, if the length of the radius be (a) 1 ft., (6) 2 ft., (c)
3 ft., (ci) 7 ft., (e) 27-3 in.
14. The length of the ladius of a circle is 17*5 in. ; find the
length of the perimeter of a sector of which the angle is (a) 90% (6)
270°.
"^ 15. What will be the length of the perimeter of the segment of
a circle of 18 in. radius, if the arc of the segment subtend an angle
of 45° at the centre of the circle ?
V 16. The length of the perimeter of a semicircle is 5 ft. ; find the
length of the diameter.
17. The length of the perimeter of a sector of a circle if 7 '2 ft. ;
find the length of the radius the angle of the sector being 30°.
"" 1§. The leng^i of the perimeter of the segment of a circle is
7 -2 ft. ; find the length of the radius if the arc of the segment
subtend an angle of 30° at the centre of the circle.
224 ARITHMETIC.
19. Find the length of the perimeter of an ellipse the lengths of
whose axes are 12 in. and 10 in. respectively.
20. Find the length of the quadrantal arc of an ellipse whose
semiaxes measure 11 '9 in. and 7 "9 in. respectively.
21. Find the length of the quadrantal arc of an ellipse whose
semiaxes are 10 '199 m. and 9*799 m. respectively in length.
22. Find the length of the radius of a circle whose circumference
is of the same length as the perimeter of an ellipse whose semiaxes
are 40*399 yd. and 39*599 yd. long respectively.
23. Find the length of the equator ; 1°, assuming it to be an
ellipse the lengths of whose semiaxes are 20,926,629 feet and
20,925,105 feet respectively; 2°, assuming it to be a circle of
20,926, 202 feet radius.
24. The French metre was originally defined to be the
10,000,000th part of the length of a meridian quadrant taken from
the equator to the pole. Had this definition been retained what
would be the length of a metre in inches, if the length of the polar
axis of the earth be 41,709,790 ft. and the length of the equatorial
diameter be 41,852,404 ft.
25. Mars revolves around the sun in an ellipse, the centre of the
sun being one of the foci of the ellipse. Find the lengths of the
semiaxes and of the perimeter of Mars' orbit if the greatest and the
least distance of the planet from the sun be respectively 154,000,000
miles and 128,000,000 miles.
20. Find the average speed in miles per minute of Mars in his
orbit, given the data in problem 25 and that his periodic time is
687 days.
si 27. Assuming the earth to be a sphere of 7913 miles diameter,
find the length of a degree of longitude in latitude 60\
N/ 2§. Assuming the earth to be a sphere 7913 miles in diameter,
find the length of a degree of longitude in 45° north latitude.
29. The length of the perimeter of an ellipse is 383 in. and the
length of the axes are as 10 to 7 ; find the lengths of the axes.
V 30. The difference between the lengths of the radii of a front
and a hind wheel of a carriage is 7 in. What must be the lengths
of these radii if the front wheel make 70 revolutions more than the
hind wheel makes in rolling a mile.
MENSURATION.
225
187. Let ABK be a circle ; C, its centre ; AF, half of a side of
a regular /t-gon cir-
cumscribed about
the circle ; B F,
half of an adjoining
side of the 7(-gon.
Join AB and CF.
CF \vill bisect the
clkord AB at right
angles, say at E,
and will bisect the
arc A B, say at
D. Draw GDH,
tangent to the circle at D and meeting AF in G and BF in H.
Then GH is equal to a side of a regular 2n-gon circumscribed about
the circle ABK. Also AG = GD = DH = HB. Bisect AF in M and
draw GL and MN i)arallel to FC and meeting AE in L and N
respectively.
Because the angle GDF is a right angle
GD<GF,
AG<GF
. •. M lies between G and F,
and
But
But
2AG+^GM = AF.
2AG=2GD=2LE=2LN+2NE
= 2LN + AE.
4AG-f2GM = AF+2LN-f-AE.
GM>LN
4AG<AF+AE
AE<arcAD
AF-f AE<AF + arcAD
4AG<AF-l-arcAD,
and?. •. 4 AG - 2 arc AD < AF - arc AD.
.-. (AG + GH + HB)-arcAB<i(AF^FB)-|arcAB,
(AG4-GH + HB)-arcAB<i^ (AF-(-FB)-arcAB 1- ;
i.e.', the excess of the length of the broken line AGHB over the
length of the arc AB is less than half the excess of 'the length of
the broken line AFB over the length of the arc AB.
226 ARITHMETIC.
Applying this theorem to all the other pairs of adjoining half-sides
of the regular n-gon circumscribed to ABK, and taking thi
aggregates of the excesses, we find that
The excess of the length of the perimeter of a regular ^n-gai
circumscribed about a circle over the length of the circumference of the
circle is less than half of the excess of the length of the perimeter of the
regular n-gon circumscribed about the same circle over the length oj
the circumference of the circle.
188. Join AD, AC and CB in the figure in the preceding section.
AG<GF,
. •. the triangle AGD < the triangle GFD,
. •. double the triangle AGD < the triangle AFD,
. •. 2 (triangle AGD) - 2 (segment AD) < triangle AFD - segment AD ;
. •. triangle AGD - segment AD < ^ (triangle AFD - segment AD);
. -. figure AGDC - sector ADC < | (triangle AFC - sector ADC ;)
. •. figure AGHBC - sector ADBC < ^figure AFBC - sector ADBC) ;
i.e., the excess of a sector of the circumscribed regular 2n-gon over
the corresponding sector of the circle is less than half of the excess
of the corresponding sector of tin circumscribed regular ?t-gon over
the sector of the circle.
Applying this theorem to all the sectors of the circumscribed
polygons and taking the aggregates of the excesses, we find that
The excess of the area of a regular 2n-gon circumscribed about a
circle over the area of the circle is less than half of the excess of the
area of the regidar n-gon circumscribed about the same circle over the
area of the circle.
189. Every n-gon circumscribed about a circle is the aggregate
of the triangles whose bases are the sides of the 7i-gon and whose
vertices all meet at the centre of the circle. Now the radius of the
circle is the altitude of each of these triangles, therefore the area of
the ^t-gon is the sum of the areas of the triangles whose bases arc
the sides of the ?i-gon and whose common altitude is the radius <jf
the circle about which the n-gow is circumscribed. In terms of the
measures of the areas,
in which r is the measure of the length of the radius, and </„ is the
MENSURATION. 227
measure of the length of the perimeter and aS„ the measure of the
area of the ^t-goii.
190. Quadrature of the Circle. Describe a circle and
circumscribe a regular hexagon and a regular convex dodecagon
about it. The excess of the length of the perimeter of the dodecagon
over the length of the circumference of the circle is less than half
of the excess of the length of the perimeter of the hexagon over the
length of the circumference of the circle. Circumscribe a regular
24-gon about the circle. The excess of the length of the perimeter
of the 24-gon over the length of the circumference of the circle is
less than half of the excess of the length of the perimeter of the
dodecagon over the length of the circumference of the circle. So
also the excess of the length of the perimeter of a circumscribed
regular 48-gon, over the length of the circumference of the circle is
less than half of the excess of the length of the perimeter of the
circumscribed regular 24-gon over the length of the circumference
of the circle.
Thus, eveiy time the number of the sides of the circumscribed
regular convex polygon is doubled, the excess of the length of the
perimeter of the polygon over the length of the circumference of the
circle is reduced to less than half of what it was before the doubling
took place.
Hence by repeating the doubling a sufficient number of times, the
excess of the length of the perimeter of the circumscribed regular
n-gon over the length of the circumference of the circle can be made
less than any explicitly assigned length however small.
Had we begvm with any other circumscribed regular n-gon than
the hexagon, the reasoning would have advanced step by step with
the preceding reasoning, and we should have arrived at the same
result.
Expressing that result in terms of the measures of the lengths of
the perimeters and circumference, it becomes
qn-Pe can, by sufficiently increasing /(, be made less than any
proposed number however small.
Therefore, r being constant, ^r(q^-p^) can, by sufficiently
increasing /i, be made less than any proposed number howevei
smaJl.
228 ARITHMETIC.
But I r (g„ - 2h) = h'^qn- h W'
and aS'„ = I rq^
. '. S^—^rpt can, by sufficiently increasing n, be made loss than any
proposed number however small.
191. By a train of reasoning similar to that in the preceding
section, but applied to areas instead of to lengths of perimeters, it
may be proved that by doubling the number of the sides of a
regular n-gon circumscribed about a circle the excess of the area of
the n-gon over the area of the circle is reduced to less than half of
what it was before the doubling took place, and that by repeating
the doubling a siifficient number of times, the excess of the area of
the circumscribed 7(,-gon over the area of the circle can be made
less than any explicitly assigned area however small. This result
expressed in terms of the measures of the areas instead of in terms
of the areas themselves is, —
>Sf„ - So can, by sufficiently increasing n, be made less than any
proposed number however small.
But it was shown in the preceding section that
/Si„ - 1 rpa can, by sufficiently increasing x, be made less than any
proposed number however small,
.'. (Sr,-^rpe)-(Sn~So) Can, by sufficiently increasing /(, be made
less than any proposed number however small,
But(S^-hrpS-iS^-S,) = S,-hrj), ^
Now >Sfc, T and p^ are constants, and increasing n can liave no effect
on them,
. '. Sc — \Tpa must be less than any proposed number however
imall ; and it cannot be variable,
bhat is; — The measure of the area of a circle is one-half of the
nrodiict of the measures of the letigths of the radins ami the circmnferenee
:>/ the circle.
192. Hence, by Euclid, VI, 33,
iii, h. The measure of the area of a sector of a circle Is one-half
yf the prod'iict of the measures of the lengths of the radius and tite (trr
jf the sector.
MENSURATIOJN. 229
193. Substitvite Jiir for p^ in the ecjuation
and it becomes
J'. The measure of the area of a circle is the irroduct of it and the
square of the measure of the leiujth of the radms of the circle.
194. Let 2a denote the measure of the length of the major axis
and 26 denote the measure of the length of the minor axis of an
ellipse and let S^ denote the area of the ellipse.
The ratio of the area of an ellipse to the area of the circle
described on the major axis as diameter is the same as the ratio of
the length of the minor axis to the length of the major axis. But
the length of the minor axis is h/a of the length of the major axis,
therefore the area of the ellipse is h/a of the area of the circle
described on the major axis as diameter,
^; = -of TTa-,
a '
S, — 7tab.
vi. The measure of the area of an ellipse is the continued product
of It aud tlie measures of the len<jt1ui of the semiaxes of the ellipse.
EXERCISE XXIX.
[In the following problems tt may be taken equal to 3 '1416 and
log ;r= -497150.]
1. Find the area of a circle the length of whose radius is 3 '75 in.
2. Find the area of a circle of 7 ft. diameter.
3. Find the area of a circle whose circumference is 13*09 cm. in
length.
4. Find the length of the radius of a circle whose area is an acre.
5. Find the length of the diameter of a circle whose area is a
square mile.
6. Find the length of the circumference of a circle whose area is
18 "7 acres.
7. How much will it cost to gravel a circular piece of ground
51 ft. in diameter, at 7 cents per square yard ?
230 ARITHMETIC.
8. Find the length of the radius of a circle whose area is equal to
the sum of the areas of four circles of 10 in. , 15 in. , 18 in. and 24 in.
radius respectively,
9. Find the total pressure on a plate 25 inches in diameter, the
pressure per square inch being 65 lb.
10. The circumference of the circular basin of a fountain
measures 117 '81 ft. on the outside of the masonry and the thickness
of the masonry is 30 in. Find the area of the surface of the water
within the basin.
11. A circular hole is cut in a circular metal plate of 7 in. radius,
so that the weight of the plate is reduced by 40 per cent. Find the
length of the radius of the hole.
12. A rectangular room, 27' 6" by 13' 6", has a semicircular
bow-window 8' 4" in diameter, thrown out at the side. Find the
area of the floor of the whole room.
13. The area of a semicircle is 13*1 sq. in. Find the length of
its perimeter.
14. The lengths of the sides' of a triangle are 13 ft. 14 ft. and
15 ft. respectively. Find the difierence between the area of the
triangle and that of a circle of equal perimeter.
15. The perimeters of a circle, a square and an equilateral
triangle are each 6 ft. in length. Find by how much the area of
the circle exceeds the area of each of the other figures.
16. Find the difference between the area of a circle of 5 m.
radius and that of a regular hexagon of equal perimeter.
17. Find the length of the diameter of a circle whose area is
equal to that of a square whose sides are each 12 ft. long.
1§. The length of the diameter of a circle is 187 yd. Find the
length of the side of a square whose area is ec^ual to that of the
circle.
19. A circle is inscribed in a square wh(jse sides are each 17 in.
long. Find the area between the sides of the square and the
circumference of the circle.
20. A square is inscribed in a circle of 11 ft. radius. Find the
area between the circumference of the circle and the sides of the
square.
21. Find the difference between the area of a circle of 7*7 m.
radius and that of a regular inscribed hexagon.
MENSURATION. 231
22. Find the areji of the semicircle described on the hypothenuse
©f a right-angled triangle as diameter, the lengths of the other sides
of the triangle being 7 ft. and 17 ft. respectively.
23. Show that in any right-angled triangle, the area of the
semicircle described on the hypothenuse as diameter is equal to the
sum of the areas of the semicircles described on the other two sides
as diameters.
24. The lengths of the radii of an annulus or plane ring are
23 '4 cm. and 36 "6 cm. respectively. Find the area of the annulus.
25. Out of a circle of radius 3 ft. is taken a circle of radius 2 ft.
Find the area of the remainder.
26. The length of the radius of the inner boundary of an annulus
is 25 ft. and the area of the annulus is lOOsq.yd. Find the length
of the outer boundary.
27. The length of the chord touching the inner boundary of an
annulus is 6 ft. Find the area of the annulus.
28. A circular fish-pond whose area is 2 '5 acres is surrounded by
a walk 3 yd. wide. Find the cost at 9 ct. per square yard of
gravelling the walk from its outer boundary to within one foot of
the edge of the pond.
29. Around a circular lawn containing 2*36 acres runs a walk of
uniform width containing a quarter of an acre. Find the width of
the walk.
30. A circular lawn 98 yards in diameter has a drive of uniform
width around it. Find that width, if the area of the drive is just
half that of the lawn.
31. What will it cost to pave a circular courtyard 55 ft. in
diameter, at 60c. per square foot, leaving in the centre unpaved a
hexagonal space whose sides are each 3 ft. long.
32. A circle of 54 in. radius is divided into three equal parts by
two concentric circles. Find the lengths of the radii of these circles.
33. Find the area of a sector of 45°, the length of the radius
being 10-5 in.
34. Find the area of a sector of 36° the length of the circumference
of the whole circle being 1309 mm.
35. The area of a sector is 11 "9 sq. ft. and the angle of the sector
is 30°. Find the length of the radius.
232 ARITHMETIC.
36. The area of a sector is equal to the area of the square on the
radius of the sector. Find the number of degrees in the angle of
the sector.
37. A sector of an annulus is 12 inches broad and the lengths of
its bounding arcs are 35 in. and 28 in. respectively. Find the area
of the sector, its angle and the lengths of its radii.
The length of the radius of a circle being one foot tind the area of
a segment which subtends at the centre of its circle an angle of
38. 60°. 39. 120° 40. 90°.
41. The length of the radius of a circle is 24 in. Two parallel
chords are drawn both on the same side of the centre, one subtending
/ an angle of 60° at the centre, the other subtending there an angle
of 90°. Find the area of the zone between the chords. ' ." . 7 "'
42. Show that the chord of a quadrant divides the circle into
parts whose areas are very nearly in the same ratio of 10 to 1.
-/ 43. Three circles so intersect that the circumference of each
passes through the centres of the other two. Find the area of the
figure common to the ^hree circles, the length of the radius of each
circle being 15 in. J/ 'h -^2-1 ^ ^ Culm^o. ^ />i^S ' 7/ ^-^ «-
44. Three circles bf 2 ft. ramus each, touch each other. Tind
the area of the figure enclosed by them.
45. The length of the chord of a sector is 5*73 in. and the length
of the radius is 10 in. Find the area of the sector. (Apply
1° theorem, § 182, p. 220.)
46. An elliptic flower-bed is described by means of a string 16 ft.
long passing over two pegs 6 ft. apart. What is the area of the
bed?
47. The area of the circle circumscribed about an ellipse is
12 sq. ft. , and that of the circle inscribed in the ellipse is 7 '5 sq. ft.
Find the area of the ellipse.
48. In a rectangular plot of land measuring 100 yd. by 70 yd.
there is dug a fish-pond in the shape of an ellipse the lengths of
whose axes are 90 yd. by 60 yd. Find the cost of gravelling the
remainder of the plot at 7*5 ct. per square yard.
49. A lawn in the shape of an ellipse the lengths of whose axes
are 98 ft. and 58 ft. is surrounded by a v/alk 2 yards wide. Find
the area of the walk.
MENSURATIOIS. 233
50. The clear span of h seinielliptic arch is 72 ft. and the clear
height is 24 ft. The thickness of the arch at the crov/n is 6 ft. and
at the sprinsfing it is 7 ft. 6 in, Find the area of the face.
195. The mantel of a cylinder or of a cone is the lateral or
curved surface (jf the cylinder or the cone.
S denoting the measure of the area of a surface, cij, men, '"^j inrk,
s and £ subscribed to S, are to be read of a cylinder, of the mantel
of a cylinder, of a right circular cone, of the mantel of a right
circular cone, of a sphere and of a zone of a sphere, respectively.
196. vii. The measure of the area of the mantel of a cylinder
is the prochict of the measure of the length of the WAintd aud the
measure of the length of the perimeter of a right cross-section of the
cylinder^ or
The truth of this tlieorem will appear at once on developing or
unwrapping the mantel by rolling the cylinder on a plane surface.
The developed mantel can by a single transposition of parts be
transformed into a parallelogram whose base is a generating line of
the mantel and wh(^se width at right angles to the base is the length
of the perimeter of a right cross-section of the cylinder. In the
case of the right cylinder the mantel develops into a rectangle.
vii, a. In the case of the right circular cylinder
2) = 27tr,
^mcy = 27tra
in which r is the measure of the length of the radius of the base and
a is the measure of the altitude of the cylinder.
vii, b. Adding the areas of the ends to the area of the mantel,
gives for the area of the whole surface of a right circular cylinder
S^y = 27tr{a + r).
197. viii. The measure of the area of the mantel of a right
CIRCULAR CONE is the co7itinued product of it the measure of the slant
heiglit of the cone and the measure of the length of the radius of the
base, or
S^rM^'^rl.
If the cone be rolled on a plane surface, the mantel will develop
into the sector of a circle whose radius is the slant height of the
234
ARITHMETIC.
cone and whose arc is equal in length to the circumference of the
base of the cone ; and the measure of the length of that circumference
is 2 Ttr.
viii, a. In the case of a frustum of a right circular cone, the
mantel develops into the sector of an annulus, therefore
= 2;r?r
in which
2 and r^ are the measures of the lengths of the radii
of the ends and of the midcross-section of the frustum.
viii, h. Adding the area of the base to the area of the mantel
8,^ = Ttr{l + r).
198. ix. The measure of the area of the surface of a sphere
in four times the product of it ay id the square of the measure of the
length of the radius of the sphere^ or,
B,^4:Ttr'\
ix, a. The measure of the area of a CAP oru ZONE of a sphere is
tivice tlie continued product of it the meastire of the length of the radius
of the sphere and, the measure of the altitude of the segment whose
curved surface is the cap or the zo'iie to he measured, or,
/S', = 2nrh.
199. Let ABC be the
quadrant of the circle, C
being its centre. Draw the
tangents AD and BD. Divide
the arc AB into any number
of equal parts, "^sayAEjEF,
FG, GH and HB, and draw
KEL, LFM, MGN and
NHP tangents to the arc
AB at the points of division
which will therefore be the
mid-points of the tangents.
The broken line AKLMNPB
is the quarter of the perimeter of a regular convex polygon which
would circumscribe the circle of which ABC is the quadrant.
*In the figure as drawn, the arc AB is divided into five equal arcs, but it
might have been divided into any other number and the method of proof would
have applied equally well.
MENSURATION. 235
If now the whole figure revolve about AC as axis, the arc AB
will generate the surface of a hemisphere, the tangents KL, LM,
MN, NP will generate the mantels of a series of frusta of right
circular cones, and BD will generate the mantel of a right circular
cylinder circumscribed about the hemisphere. We proceed to prove
that each mantel generated by a tangent is equal in area to the
mantel generated by the projection of that tangent on BD, the
generating line of the mantel of the cylinder.
Consider the mantel generated by the tangent MGN join GC and
let fall GQ perpendicular to AC, and MR and JSTS perpendicular to
BD, and NT perpendicular to MR.
By viii a, p. 284, the mantel generated by MN revolving round
AC as axis is equal to 2Tt rectangles each equal to the rectangle
under GQ and MN, which quantity we shall denote by 2it (GQ.MN).
The triangle MNT is similar to the triangle CGQ,
CG : GQ : : MN : NT
GQ. MN = CG. NT = CB. SR
27C (GQ. MN) = 27r(CB. SR)
. *. the mantel generated by MN rotating about AC as axis is ecjual
to27r(CB.SR).
But 27C (CB. SR) is equal to the mantel generated by SR rotating
about AC as axis, and SR is the projection of NM on BD
. •. the mantel generated by MN rotating about AC is equal to the
mantel generated by the projection of MN on BD, rotating about
AC.
In a similar manner it may be proved that the mantels generated
by the other tangents, KL, LM, NP are each equal in area to the
mantels generated by their projections on BD. Hence the aggregate
of the mantels generated by the tangents will be equal to the mantel
generated by the aggregate of the projections of the tangents on
BD, i.e., the mantel generated by PD.
Therefore, the aggregate of the mantels generated by the broken
line KLMNPB revolving around AC will be equal to the mantel
generated by the line BD revolving around AC.
If now the number of equal parts into which the arc AB is
divided, be doubled, the number of mantels of frusta will be doubled
(including in each case the mantel generated by the * final'
236 ARITHMETIC.
tangent, PB), but each mantel generated by a tangent being still
equal to the mantel generated by tire projection of that tangent on
BD, and the aggregate of the projections being still BD, the
aggregate of the mantels generated by the tangents will still be the
mantel generated by BD. We .may therefore double the number
of tangents as often as we please and the aggregate of the mantels
generated by them will remain equal to the mantel generated by BD,
By doubling often enough the number of equal parts into which
the arc AB is divided, the point K can be brought as near to A a»
we please, and therefore the aggregate length of the tangents, KL,
LM, MN, can be made differ from the length of the
broken line AKL B by less than any explicitly assigned
length however small. Hence the surface generated by the broken
line revolving about AC as axis can be made to differ in area from
the mantel generated by BD revolving about AC as axis, by less
than any explicitly assigned area however small.
But, ^ 187, by doubling the number of tangents often enough, the
broken line AKL B can be made differ in length from the
arc AB by less than any explicitly assigned length however small.
Hence the surface generated by the broken line AKL !B
revolving about AC as axis can be made differ in area from the
hemisphere-surface generated by the arc AB revolving about AC as
axis, by less than any explicitly assigned area howevet small.
Hence the hemisphere-surface generated by the arc AB and the
cylindric mantel generated by BD differ in area by less than any
explicitly assigned area. Therefore the diflference in area of these
surfaces cannot be constant.
Neither can their difference in area be variable, for the surfaces
themselves are constant and two constants cannot have a variable
difference.
Therefore if the figure ACBD revolve about AC as axis, the area
of the curved surface of the hemisphere generated by the quadrant
ABC will be equal to the area of the mantel of the cylinder generated
by the square ADBC, i.e., the mantel of the cylinder circumscribing
the hemisphere.
But by vii «, p. 233, the mantel of this circumscribing cylinder is
equal to
27r (BC.BD) - 27t (sq. on BC^
MENSURATION. 237
Hence the surface of the sphere whose radius is BC is equal to
47r (sq. on BC),
200. From the preceding investigation, it is evident that if a
right circular cylinder be circumscribed about a sphere and two
planes parallel to the ends of the cylinder cut both sphere and
cylinder, the area of the zone between the planes is equal to the
area of the cylindric mantel between the planes. If one of the
planes coincide with an end of the cylinder, the zone will become a
spherical cap. But r being the measure of the length of the radius
of "the sphere and /*. being the measure of the normal distance
between the planes of section, the measure of the area of the mantel
between these planes will be 27trh,
JS, = 27Crh.
201. If 21 and 2k denote the measures of the lengths respectively
of the polar axis and of an equatorial diameter of a spheroid, and if
/ and k are very nearly equal, the measure of the area of the surface
df the spheroid will be nearly
27tk^\k^-\-V^).
In the case of the oblate spheroid, in which A>^, the measure of
the area of the surface will be
>^^k- —— — -.7" I
but <2Tt}fi (k^ + fi).
EXERCISE XXX.
1. Find the area of the mantel of a right cylinder of 3 ft. altitude
and 15 in. perimeter of base.
2. The slant height of a cylinder is 39 in. and the length of tlie
perimeter of a right cross-section is 40 in. Find the area of the
mantel.
3. The length of a cylinder is 22 ft. and its least girth is 22 in.
Find tlie area of the mantel. •
4. Find the area of a right circular cylinder of 25 in. altitude and
12 in. radius of base.
238 ARITHMETIC.
5. The axes cf the base of a right elliptic cylinder are 15 in. and
12 in. long respectively and the length of the cylinder is 7 ft. 6 in.
Find the area of the mantel.
6. Find the area of the whole surface of a right circular cylinder
of 15 in. radius and 5 ft. altitude.
7. Find the area of the ivhole surface of a cylindric pipe 8 ft. 6 in.
long and an inch and a quarter thick, the length of the internal
diameter being lOi in.
§. Find tlie area of the whole surface of a right elliptic cylinder
6 ft. long, the lengths of the axes of the base being 12 in. and 10 in.
respectively.
9. The area of the mantel of a cylinder is 8 sq. ft. and the length
of the perimeter of a right cross-section is 3 ft. Find the length of
the cylinder.
1 0. The area of the mantel of a right circular cylinder is 2 sq. ft.
117 sq. in. and the length of the radius of the base is 6 '75 in. Find
the length of the cylinder.
11. The area of the whole surface of a right circular cylinder is
21 sq. ft. and the height of the cylinder is equal to the length of the
diameter of the base. Find the length of the diameter. ; .^ ; ,
1 2. The area of the whole surface of a right circular cylinder is
27 sq. ft. and the length of the cylinder is thrice the length of the
radius. Find the length of the radius.
Find the area of the mantel of a right circular cone whose
dimensions are
13. Slant height 3ft. 6 in., length of circumference of base 4ft,
9 in.
14. Slant height 4ft. 6 in., length of radius of base 1 ft. 3 in.
15. Altitude 3 ft. 9 in., length of radius of base 2ft. 4 in.
16. Altitude 8 ft. 3 in,, length of circumference of base 5 ft. 3 in.
Find the area of the whole surface of a right circular cone whose
dimensions are
IT. Slant height 2ft. 6 in., length of radius of base lO^n.
18. Slant height 7 ft. 5 in., length of circumference of base 7 ft.
lin.
19. Altitude 2 ft., length of radius )f base 10 in.
MENSURATION. 239
20. Altitude 5 ft., length of circumference of base 9 ft. 11 in.
21. The area of the mantel of a right circular cone is 5 sq. ft. and
the length of the circumference of the hase is 45 in. Find the
slant height of the cone.
22. The area of th6 mantel of a right circular cone is 7 sq. ft.
72 sq. in. and the length of the circumference of the base is 5 ft.
Find the altitude of the cone.
23. Find the slant height of a right circular cone whose mantel
has an area of 15 sq. in. and whose base-radius has a length of 1*5 in.
24. Find the altitude of a right circular cone, given that the area
of its mantel is 5 sq. ft. and the length of the radius of its base is
6 in.
25. The area of the mantel of a right circular cone is 2*5 sq. ft.
and its slant height is 25 in. Find the length of the circumference
of the base.
26. The area of the mantel of a right circular cone is 15 sq. ft.
and the slant height is 2 ft. Find the length of the radius of the
base.
27. The area of the whole surface of a right circular cone is
2 sq. yd. and the slant height is twice the length of the diameter of
the base. Find the length of the diameter of the base.
2§. How many yards of canvas 45 in. wide will be required to
make a conical tent 10 ft. wide and 9 ft. high ?
29. How many yards of canvas 32 in. wide will be required to
make a conical tent 15 ft. wide and 10 ft. high, if 10 % of the canvass
is cut away or turned in, in the making of the tent.
JIO. The area of the mantel of a right circular cone is twice the
area of the base. Find the vertical angle.
31. A right circular cylinder and a right circular cone stand on
equal bases and are of the same altitude, the altitude being equal to
the length of a diameter of either base. Find the ratio of («) the
mantels, (h) the whole surfaces of the cone and cylinder.
32. Find the area of the mantel of the frustum of a right circular
cone whose slant height is 7 in., the lengths of the circumferences
of the ends of the frustum being 15 in. and 2 ft. respectively.
33. The radii of the ends of the frustum of a right circular cone
are 15 in. and 5 in. long respectively and the slant height of Uie
frustum is 12 in. Find the area of its mantel.
240 ARITHMETIC.
V 34. The altitude of ^. fmstum of a right circular cone is 12 in,
and the lengths of the end-radii are 9 in. and 16 in. respectively.
Find the area of the mantel.
35. Find the area of the whole surface of the frustum of a right
circular cone, the lengths, of the circumferences of the ends being
11 in. and 17 in. respectively and the slant height of the frustum
being 7 in.
36. The lengths of the end-radii of a frustum of a right circular
cone are 3 3 ft. and 1 '7 ft. respectively, and the slant height of the
frustum is 27 in. Find the area of the whole surface of the frustum.
37. The altitude of a frustum of a right circular cone is 7 '7 in.
and the lengths of the end-radii are 6-4 in. and 10 in. respectively..
Find the area of the whole surface of the frustum.
38. The altitude of a frustum of a right circular cone is 20*8 in.
and the lengths of the end-radii are 7 '5 in. and 18 in. respectively.
If the frustum be divided into two frusta whose mantels are of
equal area, wlxat will be the altitude of each ?
39. The lengths of the sides containing the right angle of a
right-angled triangle are 1 '248 and 1 '265 metres respectively. If the
triangle revolve about an axis parallel to and 1"25 metres distant
from its shortest side, what will be the- area of the whole surface
described by the sides of the triangle ?
40. Find the area of the surface of a sphere of 3 in. radius.
41. Find the area of the surface of a sphere 12 inches in
circumference.
42. The area of the surface of a sphere is a square foot. Find
the length of the radius to the nearest hundredth of an inch.
43. A cylindric tube 8 ft. long and 2 ft. 6 in. in diameter is
slosed at each end by a hemisphere. Find the area of the whole
external surface.
44. The length of the radius of a sphere is 15 in. Find the area
Df a cap on the sphere, 5 inches in height. ^ _ '
45. Find the area of the whole surface of a segment of a sphere
Df 21 inches radius, the height of the segment being 10 inches, and
:he distance of its base from the centre of the sphere, 11 inches,
46. Find the area of the whole surface of a zonal segment of a
sphere of 12 in. radius, the distances from the centre of the sphere
MENSURATION. 241
of the terminal circles of the zone being 5 in. and 9 in. both on the
same side of the centre.
^ 47. The length of the diameter of a sphere is 30 in. and the
length of the radius of the base of a cap-segment of the sphere is
5 in. Find the height of the cap at right angles to its base.
4§. A sphere is 30 inches in diameter. What fraction of the
whole surface will be visible to an eye placed at a distance of 10 ft.
from the centre of the sphere %
V 49. At what distance from the centre of a sphere of 9 in. radius
must a luminous point be placed to light up one-third of the
surface of the sphere %
50. Find in square miles the area of the surface of the earth
assuming it to be practically an oblate spheroid the lengths of whose
semiaxes are 20,926,202 feet and 20,854,895 feet respectively.
202. If two solids on equal hoses and of equal altitudes are such
that all plane sections of the solids pandlel to and at equal distatwes
from their bases are equal to one aiiother, the section of one solid at
each and every distance from its base equal to the section of the other
solid at the same distance from its base, then will the solids be equal in
volume.
This proposition may be shown to follow from Theorem II
p. 177, by applying a method of demonstration similar to that
employed on pp. 180 and 181 to prove that tetrahedra on equal
and similar bases and of the same altitude are of equal volumes, and
on pp. 225 to 228 to obtain the quadrature of the circle.
203. II, a. The measure of the volume of a cylinder is the product
of the measures of the altitude of the cylinder and the area of its base, or
V,y = aB.
This proposition follows immediately from Theorem II, p. 177,
and § 202.
In the case of the right circular cylinder,
by §193 B=itr^,
and . '. F'rcy = Ttar-^.
204. Ill, a. The measure of the volume of a cone is ONE-THIRD
of the product of the measures of the altitude of the cone and the area
of its base, or
. V^ = ^aB.
P
242
ARITHMETIC.
This i)roposition follows iininecliately from Theorem III, p. 179,
and § 202.
In the case of the right circular cone, *"
by § 193 B=7tT'^
and.". Vri=^'n:ar^.
For the measure of the volume of a frustum of a right circular
cone, IV a, p. 183, gives
205. V. The measure of the volum,e of an ellipsoid is FOUR-
THIRDS of the continued product of it and the measures of the
lengths of the semiaxes oj the ellipsoid, or
V, = ^7rahc.
V, a. The measure of the volume of the sphere is four-thirds of the
product of Tt and the cube of the measure of the length of the radius of
the sphere, or
V, = ^7tr'\
These theorems may be obtained at once from the Prismoidal
Formula but they may also be proved independently as follows :—
Let AFDB be a sphere, C its centre and ACB a diameter.
Let MNPQ be a right circular cylinder whose diameter and
altitude are both equal to the diameter of the sphere. Let there
be hollowed out of the cylinder two right circular cones MGN and
PGQ whose bases are the ends of the cylinder and whose vertices
meet at G the mid-point of RS the axis of the cylinder.
MENSURATION. 243
In CA take any point E, draw EF at right angles to CA and
meeting the surface of the sphere in F, and join CF. EF is the
radius of the small circle which is the plane section of the sphere
at the distance CE from the centre.
Let the measures of the lengths of CF and CE be r and x
respectively then will r'^ — x- be the square of the measure of thei
length of EF, and therefore the measure of the area of the small
circle at the distance CE from C will be Tti^r"^ —x"^.)
In GR take GH equal to CE and draw HLK at right angles to
GR and cutting GM in L and PM in K. The section of the
hollowed cylinder by a plane through H parallel to the base of the
cylinder is the annulus whose centre is H and whose radii are HL
and HK.
Because RM is equal to GR, therefore HL is equal to GH. But
GH is equal to CE and HK is equal to CF therefore the measure of
the length of HL is x and that of the length of HK is r. Therefore,
the measure of the area of the annulus whose centre is H and radii
HL and HK, is 7t{r^ -x*^). But this is the measure of the area of
the small circle which is the plane section oi the sphere at distance
CE, equal to GH, from the centre.
Hence the area of a plane section of the sphere at any distance
from its centre, C, is equal to the area of the right cross-section of
the hollowed cylinder at the same distance from its centre, G.
Hence by § 202, the volume of the sphere is equal to the volume
of the hollowed cylinder, and if the cylinder be constituted between
planes tangent to the sphere (as it is in the figure), the volume
of the spherical segment between any two planes parallel to the
tangent planes is equal to the volume of the part of the hollowed
cylinder between the two parallel planes.
By § 203, the measure of the volume of a right circular cylinder
is Ttar^ and here a = 2r, therefore the volume of the unhollowed
cylinder is 27rr^.
But by § 204, the measure of the volume of each of the two cones
hollowed out of the cylinder is ^Ttr^,
Therefore the measure of the volume of the hollowed cylinder is
244 ARITHMETIC.
But the measure of the volume of the sphere is equal to the
measure of the volume of the hollowed cylinder,
For the hollowed cylinder in the preceding proof, there may be
substituted a tetrahedron whose altitude (distance between a pair
of opposite edges) is equal to a diameter of the sphere and whose
midcross-section is equal in area to a midcross-section of the sphere.
206. The proof of V follows step by step the preceding proof of
V,a, employing, however, an ellipsoid instead of a sphere, a right
elliptic hollowed cylinder instead of a right circular hollowed
cylinder and ellipses instead of circles. CA and CD of the figure
should be semiaxes of the ellipsoid.
207. It should be noticed that if the sphere be inscribed in the
hollowed cylinder and two planes parallel to the ends of the cylinder
be drawn cutting the figures, not only will the volumes of the
sphere-segment and hollowed cylinder between the cutting planes
be equal, the areas of the zone of the sphere and the mantel of the
cylinder between the cutting planes will also be equal.
208. If a right circular cylinder, a hemisphere and a right
circular cone be on equal bases and of the same altitude, the
volume of the cylinder will be thrice and the volume of the
hemisphere will be twice the volume of the cone, or
Compare these relations in volume with those of the prism, the
hemitetrahedron and the pyramid, given on page 187.
209. X. The measure of the area of a tore or ring is the product
of the measures of the length of the perimeter of a right cross-section
and the levigth of the axis of the tore.
VI. The measure of the volume of a tore is the product of the
'measure of the area of a right cross-section and the measure of the
length of the axis of the tore.
210. TTie areas o/ SIMILAR plane figures or of similar surfaces
are to one another as the squares of the measures of the lengths of their
corresponding linear dimensions.
The volumes of SIMIL4R solids are to one another as the cubes of
the measures of the loigths of their corresponding linear dimensio7is.
MENSURATION. 245
EXERCISE XXXI.
1. The length of the radius of the base of a right circular
cylinder is 5 in. and the altitude of the cylinder is 8 in. Find its
volume.
2. The lengths of the axes of the base of an elliptic cylinder are
6 in. and 4 in. respectively and the altitude of the cylinder is 12 in.
Find its volume.
3. Find the length of the radius of the base of a cylinder whose
volume is a cubic foot and whose altitude is a linear foot.
4. The area of the mantel of a right circular cylinder is 6 sq. ft.
and the volume of the cylinder is 6 cu. ft. Find the length of the
radius of the base.
5. The area of the mantel of a right circular cylinder is a square
yard and the volume of the cylinder is a cubic foot. Find the
length of the cylinder.
6. The area of the base of a right circular cylinder is 5 sq. ft. and
the volume of the cylinder is 5 cu. ft. Find the area of the mantel.
yl A vessel, in the form of a right circular cylinder is to have a
capacity of one gallon and the depth of the vessel is to be equal to
the length of the diameter of a right cross-section of it. Find the
depth and the whole internal area, the vessel being without a lid.
8. The French and German liquid measures are right circular
cylinders whose depth is in each case. equal to twice the length of
its diameter. Find the diameter of a measure holding 10 litres.
9. The French dry measures are right circular cylinders whose
depth is in each case equal to the length of its diameter. Find the
depth of the hectolitre.
10. The German dry measures are right circular cylinders whose
depth is in each case equal to two-thirds of the length of its
diameter. Find the depth of the hectolitre.
11. A cubic foot of brass is drawn into wire the twentieth of an
inch in diameter. Find the length of the wire.
13. Mr. C. V. Boys has drawn quartz fibres which have been
estimated to be only the millionth of an inch in diameter. How
246 ARITHMETIC.
many miles of such a fibre would a grain of sand make, the grain
being a right circular cylinder one-hundredth of an inch long by
one-hundredth of an inch in diameter ?
V 13. Find the volume of a hollow right circular cylinder, the
length of the radius of the inner surface being 3 '5 in; ; of the radius
of the outer surface, 4*125 in. ; and of the cylinder, 7 ft. 6 in.
"^ 14. Find the thickness of the lead in a pipe of three-quarter
inch bore, if 10 ft. of the pipe weigh 211b. and a cubic foot of lead
weigh 7121b.
"^ 15, A hollow right circular cylinder of cast iron 15 feet in
length and 4 feet in diameter of outer surface, is set upright and
bears on the top a weight of 250 tons. Determine the thickness of
the metal so that the pressure on the base may be 1500 lb. per
square inch, the weight ol a cubic foot of cast iron being 4441b.
16. Find the volume of a hollow-elliptic cylinder 75 ft. in length,
the lengths of the axes of the inner surface being 5 ft. and 3 ft.
respectively and the thickness of the ua.lls being 8 in.
17. Find the volume of a cone whose altitude is 15 in. and whose
base is a circle 10 in. in diameter.
1§. The volume of a cone is 3*5 cubit feet and its altitude is
5 feet. Find the length of the radius of the base which is a circle.
19. Find the volume of a cone whose slant height is 05 in. and
whose base is a circle 32 in. in diameter.
20. Find the volume of a cone whose altitude is 35 in. and whose
slant height all round is 37 in.
21. Find the volume of a cone on a circular base of 5 in. radius,
the area of the mantel of the cone being a square foot.
22. Find the volume of a cone on a circular base, the altitude of
the cone being 10 in. and the area of the mantel being a square foot.
23. Find the volume of the frustum of a cone on a circular base,
the height of the frustum being 10*5 in. and the lengths of the radii
of the ends being 5 in. and 2 in.
21. The slant height of a^frustuin of a right circular cone is
/ I 10 in. and the lengths of the ^§fc6rthe ends are 16 in. and 10 in.
respectively. Find the volume of the frustum. ^., ' v •■^(■'(yi"^ :
25. Find the volume of the cone from which the frustum in
problem 24 was cut.
MEJ^SURATICNn 247
ae. The lengths of the radii of the ends of a frustum of a right
circular cone are 6 ft. and 9 ft. respectively and the altitude of the
frustum is 4 ft. Find the volumes of the two frusta formed by
cutting the frustum by a plane parallel to the ends and midway
between them.
Q7. The lengths of the radii of the ends of a frustum of a right
circular cone are 4 ft and 6 ft. respectively and the altitude of the
frustum is 3 ft. Find the volumes of the three pieces produced by
cutting the frustum by two planes parallel to the ends and trisecting
the height of the frustum.
28. A pyramid 15 inches in altitude is divided into three parts
of e<pial volumes by planes parallel to the base. Find the altitudes
of the three parts.
29. The lower portion of a haystack is in the form of a frustum
of a right circular cone with the end of shorter diameter below, the
upper part of the stack is in the form of a cone. The total height
of the stack is 25 ft., the length of its greatest circumference is
54 ft., the height of the frustum is 15 ft. and the length of the
diameter of the base is 15 ft. How many cubic yards are there in
the stack ?
30. The area of the whole surface of a right circular cone is
25 sq^t. Find the volume of the cone, the slant height being five
times the length of the radius of the base.
31. The volume of a right circular cone is 7854 cubic inches.
Find the area of the whole surface of the cone, the altitude being-
thrice the length of the radius of the base.
3^. A vessel in the form of q, right circular cone whose slant depth
is equal to the length of the diameter of its mouth, just holds a
gallon. Find the slant depth.
33. Find the volume of a sphere 12 inches in diametei .
34. Find the volume of a sphere a great circle of which is 33 in.
in circumference.
35. The area of the surface of a sphere is a square yard. Find
the volume of the sphere.
36. How many gallons will a hemispherical bowl 18 inches in
diameter hold?
24^ ARITHMETIC.
37. What will be the weight of a spherical shot of cast iron
5*5 inches in diameter if a cubic foot of iron weigh 4441b. ?
3§. Find the weight of a sphere of lead 3*75 inches in diameter,
the lead weighing 712 lb. per cubic foot.
39. What weight of gunpowder will fill a spherical shell of 7 in.
internal diameter, if 30 cubic inches of the gunpowder weigh a
pound ?
40. Pind the volume 1° of the greatest sphere, 2° of the greatest
hemisphere, that can be cut out of a cube of wood measuring
7 *5 inches on the edge.
41. The largest possible cube is cut out of a sphere one foot in
diameter. Find the length of an edge of the cube and the volume
of material cut away in making the cube.
42. Find the weight of a spherical shell l'75in. thick and of 8
inches external radius, the material composing the shell weighing
490 lb. per cubic foot.
43. The length of the greatest circumference of a spherical shell
is 25 in. and the length of the internal diameter is 5' 75 in. Find
the weight of the shell, the substance of which it is composed
weighing 500 lb. per cubic foot.
44. A spherical shell weighs 13 lb. and the lengths of the external
and internal diameters are 6 in. and 4 in. respectively. Find the
weight of a shell of the same substance but of 8 in. external and
6 in. internal diameter.
45. Find the volume of a solid in the form of a right circular
cylinder with hemispherical ends, the length of the diameter of the
cylinder being 3 ft. 6 in. and the extreme length of the solid being
25 feet.
46. A cylindrical pontoon with hemispherical ends is constructed
of sheet-iron "125 in. thick, the extreme length of the pontoon is
22 ft. and the length of its outside diameter is 2 ft. 6 in. Find the
weight which the pontoon will support when half immersed and also
the greatest load it will bear assuming the specific gravity of
sheet-iron to be 7 '75 and taking the weight of water at 62 '5 lb. per
cubic foot.
47. Find the thickness of an 8-inch shell if it weigh half a?
much as a solid ball of the same diameter and of like material.
:nsuration.
4§. A spherical shell 10 in. in diameter weighs '9 as much as a
solid ball of the same diameter and substance. Find the length of
the internal diameter.
49. A cast iron shell Sin. in diameter, is filled with gunpowder and
plugged with iron ; the whole then weighs 75 "5 lb. Find the
thickness of the shell, supposing the iron to weigh 444 lb. per cubic
foot and the gunpowder to weigh 57 '6 per cubic foot.
50. If the nature of the earth's crust be known to an average
depth of 5 miles, what proportion of the whole volume of the earth
is known, assuming the earth to be a sphere 7912 miles in diameter .^
ftl. If the ocean cover 73*5 jier cent, of the earth's surface and
its average depth be 2 miles, what proportion will its volume bear to
the volume of the whole earth considered as a sphere 7912 miles in
diameter ?
52. If the atmosphere extend to a height of 45 miles above the
earth's surface what proportion will its volume bear to that of the
earth assumed to be a sphere of 7912 miles diameter ?
53. The radius of the base of a right circular cone is 2 inches
and the volume of the cone is equal to that of a spherical shell of
4 in. external and 2 in. internal diameter. Find the altitude of the
cone.
54. A Stilton cheese is in the form of a cylinder, a Dutch cheese
is in the form of a sphere. Find the length of the diameter of a
Dutch cheese weighing 9 lb. , a Stilton cheese 8 inches in diameter
and 7 inches high weighing 6 lb.
55. The length of the radius of the base of a segmeYit of a sphere
is 2 in. and the length of the radius of the sphere is 6 in. Find the
volume of the segment.
56. The height of a segment of a sphere is 6 in. and the length
of the radius of the base is 8 in. Find the volume of the segment.
57. The lengths of the radii of the ends of a zonal segment of a
sphere are 5 in. and 8 in. respectively, and the height of the segment
is 3 in. Find the volume of the segment.
5§. Find the volume of a zonal segment of a sphere, the ends of
the segment being on opposite sides of the centre of the sphere and
distant from it 10 in. and 15 in. respectively, the length of the
radius of the sphere being 20 inches.
260 ARITHMETIC.
•S9. A section parallel to the base of a hemisphere bisects its
altitude. Find the ratio of the volumes of the segments.
60. A sphere whose volume is a cubic yard is divided by a plane
into segments whose altitudes are as 2 to 3. Find the volumes of
the segments.
61. How much water will run over if a heavy globe of 2 in.
diameter be dropped into a conical glass full of water, the diameter
of the mouth of the glass being 2 '5 in. and its depth 3 in. ?
62. Find the volume of .the prolate spheroid generated by an
ellipse of 12 in. major and 10 in. minor axis.
63. Find the volume of the earth assuming it to be an oblate
spheroid of 41,709,790 ft. polar axis and 41,852,404 ft. equatorial
diameter.
6 1. Find the volume of the earth assuming it to be an ellipsoid
the lengths of whose semiaxes are 20,926,029 ft., 20,92.5,105 ft.
and 20,854,477 ft. respectively. Find also the length of the
mean-radius or radius of a sphere of the same volume as thq earth.
65. Find the length of 100 complete C(jils of a wire one-tenth of
an inch in diameter coiled closely upon a cylinder of 5 in. radius.
66. On examining and taking the dimensions of a steep cistern,
which was supposed to be perfectly cylindrical, I found the bottom
cross diameters to be 70 inches each, but the toj) diameters were
68 and 72 inches respectively. The depth of the vessel was 65
inches. What is the difference in the capacities of the true
cylinder at 70 inches diameter and the one examined ?
6T. A steep cistern in the form of a frustum of an elliptic cone,
the cross diameters at the bottom being 84 and 64 inches,. and the
diameters at the top 72 and 57 inches, is 50 inches deep, and is,
filled to the depth of 25 inches with dry barley. How many cubic
inches does it contain ?
68. A cylindrical iron tank, 20 feet long and 4 feet 6 inches in
diameter, was jjlaced horizontally on a fiat car and filled with oil
at Petrolia. When it arrived at Toronto, it was found upon being
dipped from the top, to be 10 inches to the surface of the oil.
What was the wantage in gallons ?
CHAPTER VI.
PROPORTIONAL AND IRREGULAR DISTRIBUTION AND
PARTNERSHIP.
211. If four magnitudes be in proportion and if the first
magnitude be a multiple of the second, the third magnitude will bu
the same multiple of the fourth ; if the first magnitude be a part of
the second, the third magnitude will be the same part of the
fourth ; if the first magnitude be a multiple of a part of the second,
tlie third magnitude will be the same multiple of the same part of
the fourth ; and, gienerally^ according as the first magnitude is
greater than, e<iual to or less than any multiple or part or multiple
of a part of the second, the third magnitude is also greater than,
ecfual to or less than the same multiple or the same part or the same
multiple of the same, part of the fourth ; and, conversely ; If these
(■onditu/}is are satvified the four magnitudes are in proportion. (See
.:^§ 148 to 152, pp. 159 and 160.)
212. Hence if four quantities be in proportion the first and
second quantities will also be i)roportional to any equimultiples of
the third and fourth (quantities or to any equifractional parts of
tliese quantities, i.e. the third and ff)urth (juantities may both be
multiplied or both divided by the same number without affecting
the proportion.
Example. $12 = | of $18 and 641b. = .| of 961b.,
$12: $18: :641b :961b.
Dividing both 641b. and 96 lb. by 7 will not affect the § in the
statement 641b. =§ of 961b., nor will multiplying the two quotients
by 4 affect the |, .
$12: $18: :9ilb. :13flb.
and $12 : $18 : : 364 lb. : 54^ lb.
So also if four quantities be in proportion, the first and second
quantities may both be multiplied or both divided by the same
number without affecting the proportion.
Thus in the preceding example, multiplying both $12 and $18
251
252 ARITHMETIC.
by 2 and dividing the products by 5 will not
statement $12 -| of |18,
I of $12 : I of $18 : : 641b. : 961b.,
i.e. $4•80^ $7 -20: :641b. :961b.,
and . •. $4 -80 : $7 '20 : : 36^ lb : 54f lb.
Hence, generally, if four quantities be in proportion any
equimultiples or equif ractional parts of the first and second quantities
will also be proportional to any equimultiples or equif ractional
parts of the third and fourth quantities.
213. If four quantities be in proportion and if any equimultiples
or equifractional parts of the first and third quantities be taken
and also any equimultiples or equifractional parts of the second and
fourth quantities, these multiples or fractional parts taken in the
order of the quantities are in proportion.
Example. 15 in. = | of 25 in. and 57 gal. = f of 95 gal.
15 in. : 25 in. : : 57 gal. : 95 gal.
Multiplying both 15 in. and 57 gal. by 6 will multiply the | by 6
in both the statements,
15 in. = f of 25 in. and 57 gal. = f of 95 gal.
which thus become
15 in. X 6 = 1 X 6 of 25 in. and 57 gal. x 6 = f x 6 of 95 gal.
Multiplying both 25 in. and 95 gal.' by 7 will divide the | x 6 by 7
in both these statements which thus oecome
15 in. x 6 = I x f of (25 in. x 7) and 57 gal. x 6 = | x f of (95 gal. x 7)
(15 in. X 6) : (25 in. x 7) : : (57 gal. x 6) : (95 gal. x 7),
i.e. 90 in. : 175 in. : : 342 gal. : 665 gal.
214. If four quantities be in proportion and if the first and
second quantities be expressed in terms of one and the same unit
and the third and fourth quantities be also expressed in terms of
one and the same unit, the unit of the first and second quantities
not being necessarily the same as the unit of the third and fourth
quantities, it follows from the preceding section that the product of
the measures of the first and fourth quantities is equal to the product
of the measures of the second and third quantities. For, if the
first and third quantities both be multiplied by the measure of the
fourth quantity, and the second and fourth quantities both be
multiplied by the measure of the third cj[uantity, in the proportion
PROPORTIONAL AND IRREGULAR DISTRIBUTION. 253
formed by these multiples the third and fourth quantities will be
equal to one another and therefore the first and second quantities
will be equal to one another. But the measure of the first quantity
in this proportion formed by the multiples, is the product of the
measures of the first and fourth quantities of the original proportion,
and the measure of the second quantity in the new proportion is
the product of the measures of the second and third quantities of
the original proportion. Hence the product of the measures of the
first and fourth quantities of the original proportion is equal to the
product of the measures of the second and third quantities of the
original proportion.
Example. $35 = f of $56 and 55 yd. = § of 88 yd.
$35: $56: :55 yd. :88 yd.
Multiply $35 and 55 yd. both by 88, the measure of 88 yd., the
f<jurth quantity or term of the proportion.
Also multiply $56 and 88 yd. both by 55, the measure of 55 yd. ,
the third quantity or term of the proportion. Then by § 213
$35 X 88 : $56 X 55 : : 55 yd. x 88 : 88 yd. x 55
But 55 yd. x 88=88 yd. x 55
$35x88 = $56x55.
215. If four quantities form a proportion, the quantities are
called the terms of the proportion ; the first and fourth quantities
are called the extreme terms or the extremes of the proportion
and the second and third quantities are . called the mean terms or
the means of the proportion.
Employing this phraseology and with the implication of the
conditions regarding the units of the terms, the theorem of § 2] 4
may be briefly stated under the form
The prochict of the measures of the extremes of a proportion is equal
to the product of the measures of the means of the proportion.
th e mean terms of a proportion be equal to one another, i. e. ,
if the first of three quantities of the same kind be to the second
as the second is to the third, the third quantity is said to be a
third proportional to the first and second quantities, and the second
quantity is said to be a mean proportional between the first
and third quantities.
254 ARITHMETIC.
216. If the nrst of four quantities be to the second as the third
is to the fourth ;
i. The second quantity will he to the first as the fourth quantity is
to the third;
ii. The sum of the first and second quantities tmll he to the second as
the sum of the third and fourth quantities is to the fourth ; and
iii. TJie difference between the first and the second quantity will he
to the second qumitity as the difference between the third and the fourth
quantity is to the fourth quantity.
These theorems follow immediately from § 211 but in the case of
a proportion with commensurable terms ii and iii are merely special
cases of the theorem of § 213.
Examples. If A's money : B's money : : $3 : $5
then will B's money : A's money : : $5 : $3,
A's money + B's money : B's money : : $8 : $5,
and B's money : A's money + B's money : : $5 : $8.
So also if N's weight : M's weight H-N's weight : : 41b. : 11 lb.
then will M's weighty- N's weight : N's weight : : 11 lb. : 41b.
and M's weight : N's weight c : 7 lb. : 4 lb.
217. If either the first and second quantities or the third and
fourth quantities of a projiortion be replaced by their measures in
terms of a common unit, the other pair of quantities are then said
to be proportional to the numbers which constitute these measures.
Thus, if A's money is to B's money as $3 to $5, we may say that
A's money is to B's money as 3 to 5.
EXERCISE XXXII.
Prove that
1. $12 :$18: :42 yd. :63 yd.
2. 26537 gal. : 56865 gal. : : 54992 min. : 117840 min.
3. 2-6A. : 26-6 A : : 27f bu. : 285 bu.
4. 1yd. 8 in. : Imi. 256 yd. 2 ft. : : 36 min. : 5wk. 6 da. 6hr.
5. 8 ft. :12^ft. ::48^:3
6. 2^ sec. : 3^ sec. : : 648^ mi. : 3 mi.
PROPORTIONAL AND IRREGULAR DISTRIBUTION. 255.
Supply the missing term in
7. $12: $15: :20gaL : ( ). ,
§. 1yd. :2yd. : :3da. :( ).
9. 3^ yd. :3|yd. : :( ):2awk.
10. ( );7-5A: : 3332 T. : 5236 T.
11. ( ):ioz.-ioz. ::i + ^:h-h
12. 2^ + 3^ :( )::3 + 108^:3.
13. What sum is to $1*25 as 25 ft. to 4 ft. ?
14. One waterpipe discharges 141 gal, per hour, another
discharges 235 gal. pt r hour. Compare their rates of discharge
[a), per hour ; (6), per minute ; (c), per second ; (d), per day ;
^e), per seventh of a day. Also compare the times in which the
pipes would each discharge (a), 705 gal, ; (6), 705 qt. ; (c), 705 pt. ;
(^), 1000 gal.; (e), Igal.
15. Two taps when both open discharge water at the rate of
481 gal. per hour ; the discharge of the smaller of the two being at
the rate of 148 gal. per hour. Compare the volume discharged by
the larger tap in any given time with the volume discharged by the
smaller tap in the same time. Compare also the time in which the
larger tap will discharge a given number of gallons with the time
required by the smaller to discharge the same number of gallons.
16. One train travels 8h mi. in 20min., and a second train 9 mi.
in 15 min , ; compare their rates per hour.
17. A person walks from his house to his office at the rate of
4 mi. per hr. ; but iSnding he has forgotten something returns at the
rate of 5 mi. per hour. ; compare the time spent in going with that
spent in returning.
1§. A man can row 6 mi. an hour in still water ; compare his
rate of rowing down a stream Which flows at the rate of 2^ mi. an
hour with his rate of rowing up.
19. A greyhound pursuing a hare takes 3. leaps to every 4 the
hare takes ; bub 2 leaps of the hound are equal in length to 3
leai)s of the hare ; compare the speed of the hound with that of
the hare.
20. .^'s money is to B'» as 3 to 4, and B's, money to O's as 4 to 5.
How much money has A compared to C 'i
256 ARITHMETIC. -
21. A grocer has 841b. of a mixture of green and black teas, the
weight of green tea in the mixture being to the weight of black tea
in it as 5 to 1 ; how many pounds of black tea must be added to
make the weight of green to that of black as 4 to 1 ?
22. Milk is worth 20 cents a gallon, but by watering it the value
is reduced to 15 cents a gallon. Find the proportion of water to
milk in the mixture.
23. Divide $4500 between two persons in proportion to their
ages which are 21 and 24 years.
24. Two men receive $15 for doing a certain piece of work.
Now one man had worked but 3 days while the other had
worked 5 days on the job. If the money is to be divided in
proportion to the lengths of time the men worked, how much should
each receive ?
25. A farm is divided into two parts whose areas are as 9 to 13,
and the area of the larger part exceeds that of the smaller by 18 A.
880 sq. yd. Find the area of the farm.
218. Let there be any number of quantities, say A, B, C, D,
, all of one kind and an equal number of quantities, say
a, b, C, d, ....... , also all of one kind but not necessarily of the
same kind as the quantities of the first set, then if
A : B : : a : b,
B : C : : b : c,
C : D : : c : d,
and so on throughout the two sets, the quantities A, B, C, D,
are said to be proportional to the quantities a, b, C, d,
A and a, B and b, C and c, D and d, are called
corresponding or homologous terms^
If the quantities of either set be replaced by their measures in
terms of a common unit, the quantities of the other set are then
said to be proportional to the numbers which constitute these
measures.
The expression
A:B:C:D: :a:b:c:d
denotes that A, B, C and D are proportional to a, b, C and d.
PROPORTIONAL AND IRREGULAR DISTRIBUTION. 257
Eoaample 1. Divide $720 into parts proportional to 4, 5 and 6.
4 + 5 + 6 = 15,
. •. if 15 be divided into parts proportional to 4, 5 and 6, these parts
will be 4, 5 and 6 ;
. *. if 1 be divided into parts proportional to 4. 5 and 6, these parts
will be jf., ^, ^V,
.*. if $720 be divided into parts proportional to 4, 5 and 6, these
parts will be jK .,f $720, f^^ of $720, and ^^^ of $720.
j\ of $720 = $192.
T'Cof $720 = $240
j^Cof $720 = $288
Proof. $192 + $240 + $288 = $720,
Also $192 = t of $240 i.e. , $192 : $240 : : 4 : 5
and $240 = f of $288 i.e., $240 : $288 : : 5 : 0,
$192 : $240 : $288 : : 4j 5 : 6.
J^xample^. Divide 3161b. into parts proportional to ^, l, J.
:v + 5 + 8 - ^20 "^ 120 "^ 120 ~ 120'
.*. if //g be divided into parts proportional to }, 1 and i, these
parts will be j%, ^Vo and ^V^,
.'. if 79 be divided into parts proportional to I, 3 and J these parts
will be 40, 24 and 15 ;
.'. if 1 be divided into parts proportional to ;\, I and ^ these parts
will be 4tt, ^ and Ig.
.*. if 3161b. be divided into parts proportional to I, I and ^ these
parts will be i?^ of 316 lb. , f * of 316 lb. and 1 j] . .f 316 lb.
fa of 316 lb. =160 lb.,
' ^A of 316 lb. = 961b.,
15 of 316 lb. = 601b.
Proof. 160 lb. + 96 lb. + 60 lb . = 316 lb.
1601b. -r-96 lb. =5+3 = ^ + 1, i.e., 1601b. :961b. : : ^ : i
9r> lb. -+ 60 lb. = 8 + 5 = i + i , /. e. , 96 lb. : 60 lb. ::l:^
1601b. :961b. :601b. : :1 :1:^
Q
258 AllTTHMETia
EXERCISE XXXIII.
Divide —
1 . 1331 into parts proportional to 2, 4, 5.
2. 19 T. 11201b. into parts proportional to |, I, |.
3. $57 into parts proportional to |, ?, ^^.
4. $169'65 into parts proj)<»rtionai to 1, 2, 3, 3, 4.
5. $1064 into parts proportional to 2, 2^, 2?.
6. $1720 into parts proportional to 10, 2h, 1, i h
7. 18011), into parts jiroportional to 3 '3, '7, '5.
§. $253 in the proportion of 6, 7, and 10.
9. $6336 in the i)roportion of |, |, and '7.
10. 15223 in the proportion of f, ^, 1%, ^%, f..
11. Sugar is composed of 49*856 parts oxygen, 43 '265 carbon,
and 6-879 hydrogen ; how many pounds of each is there in 1300 lb.
of sugar ?
12. Gunpowder is composed of nitre, charcoal and sul])hur in the
proportion of 33, 7 and 5.
(1.) How many lb. of sulphur are tliere in 180. lb. of powder ?
(2.) How many lb. of powder can ])e made with 301b. of sulphur ?
(3.) How much nitre and sulphur nuist V)e mixed with 1121b. of
charcoal to form gunpowder ?
13. ■ A man divides $3300 amongst his three sons, whose ages are
16, 19, and 25 years, in sums j)roportional to their ages : two years
afterwards he similarly divides an equal sum, and again after three
years more ; how much does each receive in all ?
14. Two sums of money are to be divided among three persons,
one sum equally and the other in the proportion of 3, 5, and 8. The
shares of the first two amount to $64*56 and $81*36 respectively.
Determine the sums.
15. I want an alloy consisting of 19 parts by weight of nickel,
17 of lead, and 41 of tin. The only nickel I can obtain is 101b. of
an alloy containing 11 parts of nickel to 7 parts of tin and 5 of
lead. How nmch lead and tin must I add to make up the alloy I
want?
PROPORTIONAL AND IRREGULAR DISTRIBTTTION. 259
16. Two persons travelling together agree to pay expenses in the
ratio of $7 to $5. The first (who contributes the.greater sum) pays
on the whole $103 '20, the second $63 '40. What must one pay the
other to settle their expenses according to agreement ?
17. Capital originally invested so as to yield an income of $22500,
at the rate of 9%, is reinvested at 10%, and then divided among
three persons in the i)roportion of 4, 7 and 9. Find the yearly
income of each.
1 §. Three persons, A, B, C, agree to pay their hotel bill in the
proportion of 4, 5, 6. A pays the first day's bill which amounts to
$6*10 ; B the second, which amounts to $8*66 ; and C the third,
which amounts to $9 '24. How must they settle accounts ?
19. A founder is required to supply a ton (22401b.) of fusible
metal consisting of 8 parts by weight of bismuth, 5 of lead, and 3
of tin. The only bismuth he has in stock is in an alloy consisting
of 9 parts bismuth, 4 lead and 3 tin. How much of the alloy
must he take, and how much lead and tin must he add to make up
the order 1
Example 3. Divide 53 "5 A. among three men so that the first man
may receive 7 A. as often as the second receives 8 A., and the second
may receive 5 A. as often as the third receives 4 A.
Share of 1st : share of 2nd : : 7 A. : 8A.
share <jf lst = ^ of share of 2nd
Share of 2nd : share of 3rd : : 5A. • 4A.
share of 2nd = | of share of 3rd,
and . '. share of lst = | of f of share of 3rd.
share of 1st + share of 2nd + plus share of 3rd
= {l «f 4+1 + 1) share of 3rd,
= (II + M + i I) share of 3rd,
= ^?fi/ of share of 3rd.
53 -5 A = -i:f/ of share of 3rd
T\h ^^ 53 "5 A = share of 3rd
share of 3rd =^16 A.
and share of 2nd=| of share of 3rd = 20 A.
and share of lst = f of | of share of 3rd = 17 "5 A.
260 ARTTHMETIC.
SXERCISB XXXIV.
1 . Divide $1050 among A, B, C and D so that A's share may be
to B's as 2 to 3, B's share to C's as 4 to 5, and C's to D's as 6 to 7.
2. Divide £28. 13s. 8d. among A, B and C, so that for every
shilling given to A, B gets 10s., and C a half -guinea. (2l8. = 1 guinea.)
3. Divide 32 gal. 3 qt. 1^ pt. into four measures so that the first
shall be to the second as 9 to 14, the second to the third as 21 to
25, the third to the fourth as 20 to 23.
4. An assemblage of 700 persons consists of 5 men for every
2 children, and 3 children for every 7 women. How many of each ?
5. The joint capital of four partners, A, B, C, D, is $12600 ; A's
investment is $10 for every $17 of B's, C's is $34 for every $65 of
D's, and B's is half as much again as C's. Required the amount of
the investment of each
6. Divide $3274-70 among A, B and C, giving A five per cent
more than B, and six per cent, less than C.
7. A's rate of working is to B's as 7 to 5, B's to C's as 4 to 3, C's
to D's as 5 to 6 ; time A works per day is to time B works per day
as 9 to 10, time B works to that C works at 10 to 11, that of C
to that of D as 10 to 7 ; number of days A works to number B
works as 15 to 7, number B works to number C works as 11 to 20,
and number C works to number D works as 7 to 5. How should
$1220, the sum paid for the work, be divided among them ?
Example 4- Divide the number 429 into three parts such that five
times the first part may be equal to seven times the second and t;)
nine times the third.
First X 5 = second x 7 = third x 9
and first + secf )nd + th ird --= 429
first X 7 X 9 + second x 7 x 9 + third x 7 x 9 = 429 x 7 x 9 =27027
first X 7 X 9-l-first X 5 X 9-l-first X 7 X 5 = 27027
first X 143 = 27027
first -27027-^-143 = 189, -
and second = first x 5 -^ 7 = 135,
and third = first x 5 ^ 9 = 105.
PROPORTIONAL AND IRREGULAR DLSTRIBUTION. 261
EXERCISE XXXV.
1. Divide $9*60 between A and B so that 3 times A's share may
be equal to 5 times B's.
2. A, B, and C liave together $1740 ; if ^ of A's-/o '^^ B's =
l^Q of C's, find the share of each.
3. A pound of tea, a pound of coffee, and a pound of sugar
together cost $1"37 ; find the price of each having given that 71b.
of tea cost as much as 16 lb. of coffee, and 31b. of coflee as much as
11 lb. of sugar.
4. Divide $1650 into two parts, such that the simple interest on
one of them at 4:h % for 3 years would be equal to the simple interest
on the other at 5 % for 2j years.
5. Divide $1560 "50 into three such parts that the amount of the
first for 2^ years at 5 % may be equal to the amount of the second
for 2^ years at 3^ % and also to the amount of the third for 4 years
at 4 %, simple interest.
6. A father leaves $15000 to be divided among his three sons, aged
respectively 16, 18, and 20 years so that if their respective shares
be put to simple interest at 6 %, they may have equal shares on
coming of age. How is the money to be divided ?
7. Divide 365 into three parts, such that twice the first, 5 times
the second, and 24 % of the third, may be equal to one another.
§. Three coal wagons contain 195 cwt. of coal in such proportions
that 10 times the load in the first, 12 times that in the second, and
15 times that in the third, are equal quantities. What weight does
each wagon carry ?
9. A man, a woman, and a boy finish in a day a piece of work for
which $4-65 is paid. Find the share of each on the supp(^sition
that 2 men do as much as 3 women or 5 boys, and that the pay is
proportional to the work done by each.
10. Divide the number 80 into four such i)arts that the first
increased by 3 the second diminished by 3, the third multiplied by
3 and the f(jurth divided by 3, may give equal results.
262 ARITHMETIC.
Example 5. The daily wages of 9 men, 11 women and 12 boys is
$53*40. Find the daily wages of each man, on the supposition that
3 men do as much work as 5 women, and 4 women as much as 5 boys.
Assume the work done by one woman in one day as the unit of
work. Then
the 11 women do 11 units of work
the 9 men do § of 5 = 15 units of work
the 12 boys do ^f- of 4 = 9f units of work
Hence the money must be divided in the proportion of 15, 11,
and 9|,
which is in the proportion of 75, 55 and 48.
the 9 men's daily wages = yyg of $53*40
each man's daily wages = ^ of 3^ of $53 "40 = $2 "50.
EXERCISE XXXVI.
1. Divide $490 among 2 men, 8 women, and 10 children for work
done, on the supposition that 1 man does as much as 3 women or
5 children.
2. A, B, C, rent a pasture for $92 ; A puts in 6 horses for 8 weeks,
B, 12 oxen for 10 weeks, C, 50 cows for 12 weeks. If 5 cows are
reckoned equivalent to 3 oxen, and 4 oxen to 3 horses, what shall
each pay ?
3. Three workmen, A, B, C, did a certain piece of work and were
paid daily wages according to their several degrees of skill. A's
efficiency was to B's as 4 to 3, and B's to C's as 6 to 5 ; A worked
5 days, B, 6 days, and C, 8 days. The whole amount paid for
the work was $36*25. Find each man's daily wages.
4. Three men, working respectively 8, 9, 10 hours a day, receive
the same daily wages. After working thus for 3 days, each works
one hour a day longer, and the work is finished in 3 days more. li
$114*05 is paid for the work, how much should each man receive <
5. Three mechanics. A, B, C, are to divide among them the
proceeds of a job valued at $125*50, and finished in 9 weeks, the
share of each being proportional to the work done by him. B car
do half as much again in the same time as C, and A twice as much.
PROPORTIONAL AND IRREGULAR DISTRIBUTION. 263
C works steadily 8 hours a day ; B works 7 hours a day for the first
2 weeks, 5 for the next 2, 3 for the next 4, and 11 for the last.
During the first 7 weeks, A works only 2 hours a day for 4 days of
the week, and during the last 2 he works 14 hours a day, but finds
that in the last 4 hours of each day he can get through no more work
than C could. How much should each receive ?
Example 6. A drover bought oxen at $40, cows at $30, and slieej)
at $10 a head, paying for all $1440. There were 2h times as many
cows as oxen, and 5 times as many sheep as cows, how many did he
buy of each 1
No. oxen : No. cows : : 1 : 2J : : 2 : 5,
No, cows : No. sheep : : 1 : 5 : : 5 : 25,
No. oxen : No. cows : No, sheep : : 2 : 5 : 25 ;
. •, as often as he expends $80 in purchasing oxen he will expend
$150 in i)urchasing cows, and $250 in sheep ;
Hence the money nmst be divided hi the i)roportion of 80, 150,
250, which is in tlie proportion of 8, 15, 25 ;
cost of oxen = f^ of ^1440 = $240
No, oxen = $240- $40 = 6.
EXERCISE XXXVII.
1. A person bought wheat at 80c, barley at 75c, and oats at 40c
a bushel, expending for barley half as much again as for wheat, and
for oats twice as much as for wheat. He sold the wheat at a gain
of 5 %, the barley at a gain of 8 %, and the oats at a gain of 10 %,
and received altogether $9740, How many bushels of each did he
buy 1
ft. Suppose that $95 '10 is to l)e divided among a certain number
of men, women and boys ; that there are 10 boys for every 3 men,
and 16 men for every 39 women, that each boy receives 5 cents,
each woman 10 cents, and each man 25 cents ; find the number of
men, of women, and of boys.
JJ. A debt of $176 is paid in $5 bills, $2 bills, and $1 bills, the
number of each denomination being proportional to 4, 7 and 10 ;
how many were there of each ?
264 ARITHMETIC.
4. A debt of $350 is paid in $10 bills, $5 bills, and $2 bills, there
are | as many ten's as five's and 2h times as many two's as five's.
How many were there of each denomination ?
5. A merchant paid $84 for 100 yd. of cloth of three different
kinds. For every 4 yd. of the first kind he had 3.\ of the second
and for every l^yd. of the second he had I4 yd. of the third ; if
2 yd. of the first cost as much as 3 yd. of the second, and 5 yd. of
the second as much as 4 yd. of the third ; find the price per yard of
each kind of cloth .
Example 7. Divide $7840 70 among A,|B, C and D, giving A $77 '74
more than 40 % of what B and D receive ; B $88 less than f of
what C and D receive ; and C $99 more than 33| % of what D
receives.
Assume D's share as the tmit, that is, express the shares of the
others in terms of D's share and known (Quantities. Then, since
D's share = D's share,
C's „ =:lD's „ +$99,
B'sshare = ^(C's + D's)-$88 = | D's „ -$28*60,
A's n =|(B's + D's) + $77.74 = .ViD's n +$66-30.
sum of shares = -^^.M-D's share + $136 '70.
-27V D's share + $136 '70 =$7840 '70 ;
D's share = ($7840 -70 -$136 -70) x ^^
= $2700.
EXERCISE XXXVIII.
1. Divide $3000 among A, B, C and D so that A. may receive
$40 more than 33^ % of what B, C and D receive ; B $50 less than
60 % of the united shares of C and D ; and C f of D's share and $30
besides.
SJ. Two men A and B, make a bet on the result of a walking
match, the total sum staked being $105. A's stake is to B's as B's
original money is to A's. If A win he will have 2h times as niuch
money as B will have left, but if he lose he will have left f| of the
sum B will then have : how iiuich had each at first?
PROPORTIONAL AND IRREGULAR DISTRIBUTION. 265
3. Four men own a timber limit, which they sell for $7200 ; the
first receives $900 more than f of what the other three get ; the
second $600 less than 70 % of the joint shares of the third and
fourth ; and the third $400 more than f of a sum which exceeds
the share of the fourth by $2300. How much do each receive,
after paying their proportionate share of the expenses of the sale
which amount to $360 ?
4. Divide $52' 50 among A, B and C so that B's share may be
half as nmch again as A's, and C's one-third as much again as A's
and B's together.
5. Divide $252*50 among A, B, C and D so that the sum of the
shares of A and B may be f of the sum of the shares of C and D,
and that B's share may be ^^ of A's, and C's ^^ of B's.
6. In a certain factory the number of men is ^^ the number of
boys, and the number of women 36 % of the whole number of
persons employed. If to give each boy 6d. , each woman Is. , and
each man 2s. 6d. requires £47. lis., find the number of men,
wcmien, and boys.
7. A, B and C engage to hoe an acre of corn for $4*68. A alone
could hoe it in 48 hours ; B, in 36 hours ; and C, in 24 hours. A
begins first and works alone 10 hours ; then B commences and A
ami B work together 6 hours, when C begins and all work together
till the job is finished. How much should each receive ?
§. Two men, A and B, hired a span of horses and a carriage for
$7 to go from M to R, a distance of 42 miles. At N, 12 miles from
M, tliey took in C, agreeing to carry him to R and back to N for
iiis proportionate share of the expenses. At P, 24 miles from M,
they took in D, agreeing to take him to R and back to P for his
proportionate share of the expenses. What should each person pay ?
(Give briefly the arguments for and those against each of the two
commonly presented solutions of this problem.)
9. $1200 is to be distributed among A, B and C. From part of
it they are to receive equal amounts, and of the rest B's shares is to
be 10 % more than A's, and C's 10 % more than B's. Altogether
B's share is 8-^^^),^ % more than A's and 7§f % less than C's. Find
the part of the $1200 that was equally divided.
266 ARITHMETIC.
PARTNERSHIP.
219. A Partnership is a voluntary association of two or more
persons who combine their money, goods or other jjroperty, their
labor or their skill, any or all of these, for the transaction of business
or the joint prosecution of any occupation or calling, such as the
carrying on of any manufacture or trade or the practice of any
profession, upon an agreement that all gains and losses shall bo
shared in certain specified proportions among the persons constituting
the partnership.
Such an association is styled a Firm, a Company, or a Honse and
the persons uniting to constitute the association are called the
Partners of the Firm.
The Investment of a partner in a firm is the money or property
contributed by him to the firm.
The Capital of a firm is the total of the investments of the
partners.
The Net Gain within a certain period is the excess of the total
gains of a firm over its total losses within the period.
The Net Loss within a certain period is the excess of the total
losses of a firm over its total gains within the period.
A Dividend is the share of the net gain or of any sum divided
among the members of a firm or a company, which belongs to any
partner. The dividends to the several partners are generally in
proportion to their investments.
220. In a partnership in which the gains and losses are to be
divided among the partners in i)roportion to their investments, to
find each partner's share of any net gain or net loss : —
i. If the investments are contributed for equal times, divide the net
gain or the net loss in proportion to the investments.
ii. If the investments are contributed for unequal times, midtiply
ea^h investment by the measure of 4he length of tim,e during whi<^h if
was invested and divide the net gain or the net loss in proportion to the
products.
JHIP. ^^^^^ 267
EXERCISE XXXIX.
1 . R. Stuart and G, Armstrong enter into partnership and agree
to share all gains and losses in proportion to their investments.
Stuart contributes $4500 to the partnership and Armstrong
contributes $7500. Their net gain at the end of the year is $1750.
How much of this sum should each partner receive ?
2. Three partners inVest respectively $7800, $5750 and $9450
in business. At the end of the first year they find their net gain
to be $3156. What is the amount of each partner's share of this
gain ?
3. Two contractors, G . Rose and W. Crerar, undertake to build
a bridge for the sum of $31,500. Crerar supplies the material at a
cost of $11,727 and Rose pays the wages of the mechanics and
laborers and all other expenses connected with the contract,
amounting altogether to $15,645-80. If the profit on the contract
is to be divided in proportion to investment, how much of the
$31,500 should each partner receive ?
4. A. Jones and D. Smith enter into partnership, the former
investing $13,500 and the latter investing $22,800, and they agree
that Jones shall receive a salary of $2000 for managing the business,
and that all gains over and above this sum and all losses shall be
shared in proportion to their respective investments. At the end
of a year their resources are $74,850 and their liabilities are
$17,943-86. Find the amount of the interest of each partner at
the end of the year.
5. Th. Sinclair, C. Harvey and H. .Stevens enter into partnership,
Sinclair investing $37,500, Harvey $28,600, and Stevens $24,000,
and they agree to share all gains and all losses in proportion to
their investments. At the end of the year the resources of the firm
are $124,368-50 and the liabilities are $37,429-50. Stevens now
wishes to withdraw from the firm and sells to his partners his
interest in the business in shares proportional to their interests in
it. How much should he receive from each ?
O. T. Allan and E. Jamieson engage in business with a joint
capital of $19,200 and agree to share gains and losses in proportion
268 ARITHMETIC.
to their investments. At the end of a year Allan receives a dividend
of fllOO and Jainieson a dividend of $1300. What was the amount
of the investment of each ?
7. D. Rowan, F. Galbraith and J. Munro enter into partnership
and agree to share all gains and all losses in proportion to their
several investments. They gain $7500 of which Rowan receives
$2100, Galbraith $3100, and Munro the balance. How much did
Rowan and Galbraith respectively invest if the amount of Munro's
investment was $18,000 ?
^. Three merchants enter into partnership, the first invests
$1855 for 7 months, the second invests $887 "50 for 10 months and
the third invests $770 for 11 months ; and they gain $434. What
should be each partner's share of the gain ?
9. L, M and N entered into partnership and invested respectively
$19,200, $22,500 and $28,300. At the end of 5 months L invested
$3800 additional ; M, $2500 ; and N, $3700. At the end of a
year the net gain of the firm was found to be $7850. What was each
partner's share of this, if all gains and all losses v^-ere shared among
the partners in proportion to their average investments ?
10. Graves and Barr form a partnership, Graves investing $7000
and Barr $8000. At the end of 3 months Graves increases his
investment to $9000 but at the end of 5 months more he withdraws
$4000 from the business. Barr, 4 months after the formation of
the partnership, withdraws $2000 of his investment but 5 months
later increases it by $4000. At the end of the year the resources
of the firm are $27,850 and its liabilities are $8460. What is the
amount of each partner's interest in the business now, the net gain
being divided between the partners in proportion to their average
investments ?
1 1 . Stuart and Moss enter into partnership, Stuart contributing
$5000 more capital than Moss. At the end of 5 months Stuart
withdraws $2500 of his capital and 2 months later Moss increases his
investment by $2500. At the end of their first year of partnership,
their assets exceed their liabilities by $24,800 and on dividing their
net gain in the ratio of their average investments, Stuart's interest
in the business is found to exceed that of Moss by $461 '54. Find
the amount of the original investment of c:ich.
CHAPTER VII.
I. PERCENTAGE.
221. The phrase per cent, which is a shortened form of the
Latin pe/- centum, is equivalent to the English word hundredths.
Hence a rate per cent, is a rate or ratio per hundred and a
number expressing a rate per cent of any quantity expresses simply
so many hundredths of the quantity. Thus 5 per cent, of any sum
of money is 5 hundredths of the sum ; 7h per cent, of a given length
is 7^ hundredths of the length ; and 225 per cent, is 225 hundredths.
222. The symbol % is frequently employed to denote the words
per cent., and may therefore be read either percent, or hundredths.
Thus 5% = -05, 25% = -25, |% = -005, 133|%-1-33|,
7^ % of 840 = -075 of 840 = 63, 145 % of f 640 = 1 "45 of $640 = $928.
EXERCISE XL.
1. A lawyer collected $287*50 and charged 5% for his services;
how much did he retain, and how much did he pay over ? What
per cent, is the amount paid over of the amount collected ?
2. On Jan. 10, a merchant buys goods, invoiced at $876 '40 on
the following terms : 4 mos. , or less 6 % if paid in 10 days. What
sum will pay the debt on Jan. 15 ?
3. A house is sold for $16,400, and 25% of the purchase money
is paid down, the balance to remain on mortgage. How much
remains on mortgage ?
4. A man invests 42 % of his capital in real estate and has
$53,070 left ; what is his capital ?
5. A horse was sold for $658 which was 16§ % more than its cost ;
how much did it cost ?
6. A bankrupt's assets are $23,625, and he pays 40% of his
liabilities ; what are his liabilities ?
269
270 ARITHMETIC.
7. A paymaster receives $150,000 from the treasury but fails to
account for f 2250 ; what is the percentage of loss to the government ?
§. $640 increased by a certain per cent, of itself ecpials $720 ;
required the rate per cert.
9. A tea merchant mixes 401b. <)f tea at 45ct. per lb. with 501b.
at 27ct. per lb. and sells the mixture at 42ct. per 11). What per
cent, profit does he make ?
10. A merchant buys a bill of dry goods, Apl. 16, amounting to
$6377'84, on the following terms : 4mos., or less 5% if paid within
30 days. How much would settle the account on May 16? The
amount paid May 16 is what % of the full amount of the bill ?
11. On Aug. 16, a merchant buys a bill of goods amounting to
$2475 on the following terms : 4mos., or less 5% if paid in 30 days.
Sept. 15, he makes a payment of $1000, with the understanding that
he is to have the benefit of the discount of 5 %. With what amount
should he be credited on the books of the seller ? How nmch
would be due at the expiration of the 4mos. ?
12. Paid $664 '25 for transportation on an invoice of goods
amounting to $8866. What per cent, must be added to the invoice
price to make a profit of 20% on the full cost ?
13. A business firm's resources consist of notes, merchandise,
personal accounts, &c., to the amount of $9117*61, and a l)alance,
which is 44% of their entire capital, on deposit in bank. How
nmch is on deposit ?
14. At a forced sale a bankrupt's house was sold for $8000, which
was 20 % less than its real value . If the house had been sold for
$12,000 what per cent, of its real value would it have brought ?
1»>. The population of a town of 64,000 inhabitants increases at
the rate of 2|% in each year, find its population (i) 1, (ii) 2, (iii) 3
years hence.
16. The population of a city increases at the rate of 2% yearly.
It now has 132,651 inhabitants ; how many had it (i) 1, (ii) 2, and
(iii) 3 years ago ?
17. A ship depreciates in value each year at the rate of 10 % of
its value at the beginning of the year, and its value at the end of
3 years is $14,580 ; v/hat was its original value ?
PEUCENTAfilE. 271
18. A man in Inisiness loses in his first year 5 % of his capital,
but in his second year he gains 0% of what he had at the end of the
first year, and his cai)ital is now $14 more than at first ; what was
his original capital ?
19. Wine which contains 7h% of spirit is frozen, and the ice
which contains no spirit being removed, the proportion of spirit in
the wine is increased to S'^%. How much water in the state of ice
was removed from 504 gal. of the original wine ?
20. The stuff out of a lead mine contains at first 15*9% of lead,
^fter washing, by which process the amount of lead ore is not
diminished, the stuff crmtains 87 '45% of lead. How nuich rock
was washed away out of 210 tons 6 cwt. of the original stufi"?
21. The money deposited in a savings bank during the year 1885
was 5% greater than that deposited in 1884. In 1886 the deposits
were 33^% greater than in 1885, while the amount deposited in
1887 e iceeded the average of the three previous years by 20%.
The aggregate of the four years was $150,937*50. Find the amount
deposited in each year.
22. In 1871 the populations of Toronto, Hamilton and St, Thomas
were severally 56091, 26716 and 2197. In the next ten years they
increased 54%, 34 '6%, and 280*8% respectively. Determine the
increase per cent, of their united population,
23. The cattle on a stock-farm increase at the rate of 18| % per
annum. In 1889 there were 6859 head of cattle on the farm ; how
many were there in 1886 ?
24. In a certain election A polled 88% of the votes promised
him, and B polled 90% of those promised him, and B was elected
by a majority of 3 votes. Had each candidate received the full
number of votes ])romised him, A would have been elected by a
majority of 25. How many votes did each candidate receive ?
25. The delivery of letters in a certain town is carried on by four
postmen, two of whom deliver on 14 streets and two on 17 streets,
but the work of the latter two is 20 % less per street than that of
the former two. A fifth man is put on to help them. In what
ratio should he help the two pairs of men so that all five shall have
equal work ?
272 ARITHMETIC.
II. PROFIT AND LOSS.
223. The Prime Cost of merchandise or other property is the
net sum paid by the purchaser thereof to the seller thereof.
The Gross Cost of merchandise or other property is the sum of
the prime cost, all charges for purchasing, and all expenses for
freight, storage, handling, and such like.
224. ProJSlt is the amount by which the selling price exceeds
the cost price. Net Projit or Gahi is tl e amount by which the
selling price exceeds the gross cost.
The Rate of Profit is usually expressed as a percentage of the
prime cost.
225. Loss is the amount by which the selling price falls short
of the cost price. Net loss is the amount l)y which the selling price
falls short of the gross cost.
The Rate of Loss is usually expressed as a percentage f)f the
prime cost.
EXERCISE XLI.
1. A lot of dry goods was sold at an advance of 18 %. If the gain
was $436 '50, what was the cost ?
2. I made a mixture of wine consisting of one gallon at 50 cents,
3 at 90 cents, 4 at $1'20, and 12 at 40 cents. I sell the mixture at
$1*60 a gallon ; find my gain %.
3. A merchant's price is 25% above cost ; if he allow a customer
a discount of 12 % on his bill, what % profit does he make ?
4. If cloth, when sold at a. loss of 25%, brings $5 a yard, what
would be the gain or loss % if sold at f 6 "40 a yard ?
5. Eggs are bought at 27 cents a dozen, and sold at the rate of
8 for 25 cents ; find rate of profit.
6. A merchant sells goods to a customer at a profit of 60%, but
the buyer becomes bankrupt and i)ays only 70 cents on the dollar ;
what % does the merchant gain or lose on the sale ?
PROFIT AND LOSS. 273
7. A man sells an article at 5 % profit ; if he had bought it at 5 %
less and sold it for $12 less he would have gained 10%. Find cost
price.
§. A man bought a horse which he sold again at a loss of 10%.
If he had received $45 more for him he would have gained 12^ % ;
find cost of horse.
9. A merchant buys wine at 16s. a gal. ; 20% of it is wasted ; at
what price per gal. must he sell the remainder to gain 20 % on his
outlay'^
10. A tradesman proposes to retail his goods at 10 % profit ; but
adulterates them by adding J of their weight of an inferior article
which costs him i of the price of the better ; what % profit does he
make ?
11. I purchase 2276 lb. of coffee at 21ct. per lb. and mix it with
chicory at 4^ct per lb. in the ratio of 3 parts by weight of the former
to 2 of the latter ; at what price per lb. must I sell it to gain 25 % ?
IJ8. I buy oranges at the rate of 3 for 2d., and a third as many
at the rate of 2 for Id. ; at what rate per doz. must I sell them to
gain 20 % on my outlay ? Supposing my total profit to be 5s. 4d. ,
how many did I buy ?
13. A merchant buys 3150 yd. of cloth. He sells I of it at a gain
of 6 %, ^ at a gain of 8 %, } at a gain of 12 %, and the remainder at a
loss of 3 %. Had he sold the whole at a gain of 5 % he would have
received $28*98 more than he did. Find the prime cost of one yard.
14. Sold steel at $25*44 a ton, making thereby a profit of 6%,
and a total profit of $103*32. Find the quantity sold.
15. A baker's outlay for flour is 70% of his gross receipts, and
his other trade expenses amount to i of his receipts. The price of
flour falls 50% and the other trade expenses are thereby reduced
25 % ; to make the same amount of profit, by how much should he
now reduce the price of the 5 cent loaf 1
16. A man havjng bought a certain quantity of goods for $150,
sells ^ of them at a loss of 4 %, by what increase % must he raise
that selling price that by selling th6 whole at that increased rate he
may gain 4 % on his entire outlay ?
274 ARITHMETIC.
17. 4 liorses and 7 cows cost $390 ; but, if the price of the horses
were to rise 25 % and that of the cows 15 % they would cost $466*50 ;
find the cost of a horse and of a cow,
1§. The cost of freight and insurance on a certain (quantity of
goods was 15% and that of duty 10% on the original outlay. The
goods were sold at a h^ss of 5 %, but had they brought $3 more there
would have been a gain of 1 %.• How much did they cost ?
19. A bookseller sold a book at 17% below cost, but had he
charged 50 cents more for it, he would have gained 7 %. Find the
cost of the book to the bookseller, and the price at which he sold it.
20. A man buys pears at 35ct. a score, and after selling 7 dozen
at 45ct. a dozen (giving 13 to the dozen) he finds he has cleared his
original outlay. If he then sell the remainder at the rate of 2 for
a cent, what will he gain % on the whole transaction ?
21. I buy two cows for $55 ; if I sell the first at a loss of 5% and
the second at a sain of 5%, I should gain y\ % ; what was the price
of each cow ?
22. I bought a lot of coffee at 12ct. per lb. Allowing that the
coffee will fall short about 5 % in roasting and weighing it out, and
that 10% of the sales will be bad debts, for how much per pound
must I sell it so as to gain 14 % on the cost ?
23. A grocer mixed together two kinds of tea and sold the
mixture, 144 lb., at an advance of 20% on cost, receiving for it
$62 "lO, Had he sold each kind of tea at the same price per pound
as he sold the mixture he would have gained 15 % on the one and
25% on the other. How many pounds of each were there in the
mixture, and what was the cost of each per pound ?
24. The manufacturer of an article charged 20% profit, the
wholesale dealer charged 25 % of an advance on the manufacturer's
price and the retail dealer charged 30 % of an advance on the
wholesale price. Find the cost to the manufacturer of an article
for which the retail dealer charged $23 •40.
25. I sold for $296, two horses which had cost me $280. The
gain per $100 on one of them was ec^ual t(j the loss ])er $100 on the
other and also equal to the difference in cost of the two horses.
Find the cost of each.
INSURANCE. 275
III. INSURANCE.
226. Insurance is a contract bj which one party, the insurer,
in consideration of a sum of money received from another party, the
insured^ engages to pay a stipulated sum on the happening of a
particular event or undertakes to indemnify the insured or his
representatives for loss or damage arising from certain specified
causes, if sustained within a stated time.
The instrument or document setting forth the contract is termed
an InsuFance Policy.
The sum paid by the insured to the insurer is styled the
Premiuni. It is generally a fixed ])ercentage of the amount
insured.
The Term, (jf an insurance is the period for which the contract is
made and the risk assumed.
227. The ordinary kinds of insurance are Fire Insurance,
Marine Insurance and Life Insurance.
228. In Fire Insurance', the insurer undertakes to indemnify
the insured up to a specified sum, for loss or damage that may occur
to certain property described in the policy, if caused by fire, within
a stated time, generally one, two or three years.
229. In Marine Insurance, the insurers contract to indemnify
the insured up to a stipulated sum for any loss or damage that may
occur to a certain ship, cargo or freight, any or all of them, by
storms or other perils of navigation during a particular voyage or
within a specified period not usually exceeding twelve months,
230. In -Life Insurarice, the insurer engages to pay on the death
of the insured, a sum specified in the policy. In an Etidowmeid
Policy, the stipulated sum is payable to the insured if he should
survive a specified number of years, but should he die before the
expiration of the period named, the sum assured is to be paid to
the representatives of the insured or to a person named in the
policy.
231. Fire and life insurances are usually undertaken by
companies or corporations organized to carry on such business ;
276 ARITHMETIC.
marine insurance is undertaken both by companies and by private
persons. A marine insurance by private individuals is generally
undertaken by several parties and each of them writes his name
under or at the foot of the policy, and engages on his own account
to indemnify the insured to the amount set opposite his name : on
this account individual marine insurers are called iindertvriters.
232. In an ordinary fire policy, if the loss is only partial, the
insurer undertakes to pay the full value of the property destroyed
or the full amount of the depreciation of the property damaged,
provided it does not exceed the sum covered by the insurajice. In
marine policies there is commonly an average clause which declares
that the indemnity for a partial loss of property not insured to its
full value will be the same part of the loss as the sum covered by
the insurance is of the full value of the property.
233. If a property is insured in two or more companies or by
two or more underwriters, the insurers are liable for the indemnity
for a partial loss, in sums proportionate to the amounts of the risks
severally assumed by them.
EXERCISE XLII.
1. A factory valued at $35,000 was insured for | of its value, the
rate of insurance being § % for one year. What was the amount of
the premium ? .
2. A warehouse valued at $62,500 was insured for f of its value,
the rate of insurance w^as 1 j % for three years, and the cost of the
policy and the agent's expenses were $2*50. What was the amount
paid for the insurance ?
3. What will be the cost of insuring a cargo of 24,000 bushels of
wheat valued at $1*05 per bushel, the insurance covering f of the
value of the cargo, the premium rate being 1\% and the other
expenses of the insurance being 2| % of the premium ?
4. A merchant's stock was insured for $42,000, | of this amount
being at |^%, § of the remainder at |% and the remainder at ^%.
Find the total amount of premium paid.
INSURANCE. 277
5. A building and contents are insured as follows : — $12,000 in the
Imperial, $8000 in the National and $5000 in the Lancashire
Insurance Company. Were a loss to the extent of $3500 to occur
through lire, what portion of the loss should each company bear ?
6. Merchandise valued at $63,000 was insured in the Phoenix
Insurance Co. for $15,000, in the North British and Mercantile
Insurance Co. for $12,000 and in the Norwich Union Fire Insurance
Society for $8000 ; if the merchandise is damaged Jay fire to the
extent of $10,500, how much of the damage should each company
pay?
7. A merchant insured his stock for $33,000 for one year at 1%.
Six UKHiths thereafter the policy was cancelled at the request of the
insured. Find the amount of premium returned, the short rate for
six months being §%.
§. A factory and the machinery therein is insured for $65, OCX) ;
f of this sum is at |% premium and the remainder is at ^%.
What is the average rate per cent, of premium paid <m the whole ?
9. A fire insurance company insured a building for $60,000 at | %
premium and reinsured one-half of the risk in another company at
§% and one-third of the risk in a third company at |%. What
amount and what rate of premium did the company net on the
remainder of their risk ?
10. A steamboat worth $60,000 is insured in three companies, in
two to the amount of $15,000 each and in the third to the amount of
$20,000. For what sum would each company be liable if the vessel
were to sustain damage to the extent of $6600 ?
11. A ship worth $56,000 was insured for $15,000 in one insurance
company at ^% premium and for $32,000 in another company at
I %. The vessel received damage in a storm to the extent of $7500.
What amount had each company to pay to the owners of the vessel
and by how much did each amount exceed the premium received by
the company paying that amount ?
12. A fire insurance company charged $196*88 for insuring a
house for $17,500. What was the rate per cent, of insurance ?
13. A merchant's stock was worth $120,000 ; he insured it at f
its value ]>aying $700 premium. What was the rate per cent, of
insurance ? What was the rate in cents per $100 ?
278 ARITHMETIC.
14. A shipment of goods is insured for $7500 and $18*75 is paid
as premium. At that rate, what would be the amount of the
premium on $18,750 ?
15. The sum of $285 was paid for the insurance at | of its value
of a ship worth $50,000. What was the rate per cent, of premium,
if $3*75 was charged for the policy and the preliminary survey ?
16. For what sum was a house insured if the premium paid was
$17 '50 and the rate of insurance ^ % ?
1 7. For what sum was a shop insured if the rate of insurance
was 65 cents per $100 and the premium paid was $81 '25 ?
I *, A fire insurance company received $350 for insuring a factory
at 1|% premium, and charged |% for insuring a less hazardous
property of the same valuation as the factory. What was the
amount of the premium paid on the second property ?
19. A merchant owns § of ,a steamship and insures f of his interest
at § %, paying $337 *50 premium. What was the value of his interest
in the steamer ? If during the continuance of the policy, the vessel
be damaged in a collision to the extent of $35,000, what sum will the
merchant be entitled to receive from the insurance company ?
20. The invoice price of a shipment of goods is $1845. The
shipper wishes to insure the goods for such a sum as will, in case of
loss, cover both invoice price and amount of premium. For what
sum should the shipment be insured if the rate of insurance is f % ?
21. The value of a consignment is $4250. For what sum should
it be insured that the owner may receive both the value of the
consignment and the amount of the premium in case of total loss,
the rate of insurance being 55 cents per $100 ?
22. For what sum should a cargo worth $18,750 be insured to
cover the value of the cargo, the cost of insurance at ^% and $2-50
for the policy and broker's charges ?
23. A cargo of wheat invoiced at $9930 is insured for $10,000
which sum covers not only the invoice value of the wheat but also
the premium paid and $5 for expenses. What was the rate per
cent, of the insurance ?
24. A shipment of goods is insured for $6000, which sum covers
the value of the goods, the premium at 1^ % and $2*50 for expenses.
What was the value of the goods ?
I
COMMISSION AND BROKERAGE. 279
IV. COMMISSION AND BROKERAGE.
234. An Agent is a person authot-ized to transact business for
another. The person for whom the agent transacts business is
called his Principal.
235. A Commission Merchant is one who buys or sells
goods for other ])ersons by their authority. Commission merchants
are usually i)laced in possession of the goods bought or sold.
236. A Broker is a person who, in the name of his principal,
' effects contracts to buy or to sell. The broker is not in general
placed in possession of the goods bought or sold.
The title Broker is also applied to ])ersons who deal in stocks,
bonds, bills of exchange, promissory notes, &c., and to mercantile
agents who transact the business for a ship when in port.
237. Commission is the charge made by anagentfor transacting
business.
238. The Gross Proceeds of a sale or of a collection is the
total amount received by an agent for his principal.
239. The Net Proceeds of a sale or of a collection is the sum
due the principal from the agent, after deducting his commission
and all other charges. These charges include freight, handling,
storage, advertising, and such like.
240. The Prime Cost of a purchase is the net sum paid by an
agent for merchandise or other property and does not include his
commission or other charges.
241. Commission is nisuaUy reckoned at a rate per cent, on the
gross proceeds of sales and collections, on the prime cost of purchases,
and on the net amount of investrtients.
EXERCISE XLIII.
I. A commission merchant sold 270 barrels of flour at $6 a
barrel, and received 5 % commission. What was his commission ?
How much did he remit to his employer
280 ARITHMETIC.
2. A commission of $242 "58 was charged for selling $.3772 worth
of goods. What was the rate of commission 'i
3. A grain-dealer charged 3i % for selling a quantity of wheat,
and received for his commission $218*40 ; for how much did he sell
the wheat ?
4. A real-estate broker sold a house on 6| % ccmimission, and sent
to the owner $3060. What was the broker's commission, and what
sum did he receive for the house ?
5. A merchant sent $3238*30 to New Orleans to be expended in
cotton. The broker in New Orleans charged 6% connnission.
What sum was paid for the cotton ?
6. If $512*50 include the price paid for certain goods and 2i %
commission to the agent, how much money does the agent expend
in purchasing the goods ?
7. An agent sold 210 bush, of oats at 60ct. a bush, and charged
$3*78 for doing so. Find his rate of commission.
§. How many yards of cloth at 90ct. a yd, can an agent l)uy witli
the commission received from the sale of 360 bush, of potatoes at
oOct. a bush., his rate of commission being 1| % ?
9. A man bought a horse and carriage for $450, which sum was
his commission at 2| % on the sale of a farm. For how nuich was
the farm S( )ld ?
10. A broker is oflFered a commission of 5| % for selling wool and
guaranteeing payment, or a commission of 3| % without guaranteeing
payment. He accepts the 5| % and guarantees ])ayment. The
sales amount to $17,000, and the bad debts to $295.50. How much
did he gain by choosing the 5^ % ?
11. Sent to a commission merchant in Guelph $2080 "80 to invest
in flour, his commission being 2 % on the amount expended ; how
many barrels of flour could be purchased at $4*25 a barrel ?
1 2. An agent sold 6 mowing-machines at $120 each, and 12 at
$140 each. He paid for transportation $72, and, after deducting
his commission, remitted $2208 to his employer. What was the
rate of commission ?
13. A man allows his agent 5 % of his gross rentals, and receives
a net rental of $3488*40. If the gross rental is 6% of the value of
the property, what is the value of the property ?
COMMISSION AND BROKERAGE. 281
14. On a debt of $1725 a creditor receives a dividend of 60%, on
which he allows his attorney 5 % . He receives a further dividend
of 25%, on which he allows his attorney 6%. What is the net
amount that he receives ?
15. An agent sold a quantity of cotton amounting to $7317 '83,
and charged a commission of 2| %. He was instructed to invest the
proceeds in dry goods, after deducting a commission of 1| % on the
amount so expended. What was his total commission ?
16. An agent sold 300 bales of cotton, averaging 4621b. to the
bale, at 15"7ct. per lb., his commission being 25ct. per bale, and the
charges being $161. He purchased for the consignor dry goods
amounting to $2576*37, charging a commission of Ih % . How much
was still due the consignor ?
17. A commission merchant sold a consignment of bacon at
11| ct.. per pound and invested the proceeds, less his commission,
in tea at 38 ct. per pound. His commission on the two transactions
at the rate of 5 % on the sale <jf the bacon and 2 % on the purchase
of the tea amounted altogether to $52 '50. How many jumnds of
bacon did he sell and how many pounds of tea did he buy 1
1§. A miller sends 4000 bbl. of flour to a commission merchant
with instructions to sell the flour and remit the net proceeds by
draft. The consignee pays $462" 40 for freight and other exj^enses,
sells the flour at $6 '75 per barrel, charges 3 % connnission and pays
J % premium for draft. Find the amount of the draft.
19. The owner of certain property pays his agent 2 J % for
collecting his rents, insurance and repairs cost him 6| % of his ')iet
income but on this sum he pays no income tax, his income tax at
17A mills on the dollar amounts to $153*73. Find the gross rents
from his property.
20. An agent sold a consignment of boots and shoes 'for $3825 and
invested the proceeds, less his commission, in leather. His total
commission on the two transactions amounted to $150. What rate
did he charge, the rates on both sale and purchase being the same ?
91 . An agent sold a consignment of fish for $2460 and invested
the proceeds, less his commission, in flour. The commission on the
sale exceeded the commission on the purchase by $3. What rate
did he charge, the rates being the same on the two transactions ?
282 ARITHMETIC.
V. DISCOUNT.
242. Discount is an abatement or reduction from the nominal
price or value of anything ; as, for example, from the catalogue or
list price of an article, from the amount of a bill or invoice of goods
or of a debt, or from the face value of a i)romissory note .
243. The Rate of Discount is usually stated as a rate per
cent, of the amount fro7H which tlie discount is tnade.
244. Trade Discounts are reductions made from the catalogue
or list prices of goods.
In some branches of business the manufacturers and the wholesale
dealers catalogue their goods at fixed prices, usually the retail
selling price, and then allow retail dealers reductions or discounts
from these catalogue prices. These discounts generally depend on
the amount of the purchase and the terms of payment, whether
cash or credit. By varying the rate of discount, the manufacturer
can raise or lower the price of his goods without issuing a new
catalogue.
245. Very often two or even more successive trade discounts are
to be deducted. In such cases the^r^^ rate denotes a i)ercentage of
the catalogue jwice ; the second rate denotes a percentage of the
remainder after the first discount hns been made ; the third rate, a
percentage of tJie remainder after the second discount has been made ;
and so on.
Thus, discounts of 20% and 5% in succession off any amount, or,
as it is generally expressed in business, SO and 5 off, means that "20
of the amount is to be deducted from it, and then from the remainder
•05 f)f that remainder is to be taken.
EXERCISE XLIV.
1. What is the difference between discounting a bill of $30rO at
40%, and then taking a discount off the remainder of 5% iov cash,
and discounting the whole at 45 % ?
2. An invoice of crockery, amounting to $1473 "20, was sold
Jan. 3, at 90 days, subject to 40% and 10% discount, with an
I
DISCOUNT. ^^^V 283
additional discount of 6% if paid within 20 days. How much will
be required to pay the bill on Jan. 21 ?
3. What must be the marking price so that a merchant, in closing
out a sale, may sell broadcloth costing $3"()0ayard at 10% below
cost, and yet be able to allow 40 % off the marking price ?
4. A cabinet dealer directed his salesman to mark a set of
furniture so that, by allowing 20 % off the marked price he may
realize a gain of 25 % . The salesman marked the set by mistake at
$200, or at a loss to the dealer of 20 % of the sale. How much less
tlfan the re(piired marking price was the set marked ?
5. "What single discount is equivalent to successive discounts of
20% and 10%?
6. A merchant buys goods at 40 and 20 off the list price and sells
them at 30 and 10 off the list price. What is his gain percent.?
7. A manufacturer sells certain goods at 30 and 10 off, and gains
thereby 12|%. What is the list price, if the goods cost $28?
8. I purchase bf)oks at $2 each, less 33^5%, and 5% for cash.
What is the net cost ? What % disccjuiit may be given off the list
price so that I may sell them at a net profit of 10 % ?
O, Show that successive discounts of specified rates may be taken
off a list price in any order without affecting the net price. Thus
20 and 10 off is equivalent to 10 and 20 off, so also 30 and 10 and 5
off, 10 and 30 and 5 off, and 5 and 30 and 10 off are all equivalent.
10. 20 and what rate off are equivalent to 40% off?
,11. 25 and what rate off are equivalent to 40% off?
12. 30 and what rate off are equivalent to 40 % off?
13. 20 and what rate off are equivalent to 33_^ % off ?
14. What rate taken off twice in succession is equivalent to 36 %
off?
15. What rate taken off twice in succession is equivalent to 44%
off?
16. What rate taken off thrice in succession is equivalent to 48 "8 %
off?
17. What rate taken off thrice in succession is ecjuivalent to 34 %
off?
1 §. What rate put on a list price and then taken off the increased
price is equivalent to 4 % off the list price ?
284 AlllTHMETIC.
246. A Promissory Note (often called briefly a Note) is a
written promise to pay, unconditionally, on demand or at a fixed or
a determinable future time, a specified sum of money, to a particular
person named in the note, or to a person named or his order, or to
bearer.
A note which is, or on the face of it purports to be, both made
and i^ayable within Canada, is an inland note : any other note is a
foreign note.
247. The Maker of a note is the person who signs the promise.
The Payee is the person to whom or to whose order the note, is
made jiayable.
The Holder or Bearer of a note is the person who lawfully
possesses it.
The Pace Value (or simply the Face) of a note is the sum of
money (exclusive of interest) which the maker promises to pay.
248. A promissory note may be made by two or more makers,
and they may be liable thereon jointly, or jointly and severally
according to its tenor. If a note runs " I promise to pay," and is
signed by two or more persons, it is deemed to be their joint and
several note.
249. An Indorser of a note is a person who writes his name on
the back of the note. By so doing he guarantees its payment and
becomes responsible therefor, unless when indorsing he writes above
his signature the words "without recourse." A note payable to
orde7' must be indorsed by the payee when transferred to anyone
else, but a note payable to hearer need not be indorsed.
A special indorsement specifies the person, called the indorsee^ to
whom, or to whose order, the note is to be payable.
An indorsement in blank specifies no indorsee, and a note so
indorsed becomes payable to bearer. When a note has been
indorsed in blank, any holder may convert the blank indorsement
into a special indorsement by writing above the indorser's signature
a direction to pay the note to or to the order of himself or some
other person.
An indorsement is restrictive which prohibits the further
negotiation of the note or which ex})resses that it is a mere
authority to deal with the note as thereby directed, and not a
DISCOUNT. 285
transfer of the ownership thereof, as, for example, if a note be
indorsed ''PayD only," or "'Pay D for the account of X," or
"Pay D or order for collection." A restrictive indorsement gives
the indorsee the right to receive payment of the note and to sue
any party thereto that his indorser could have sued, but gives him
no power to transfer his rights as indorsee unless it expressly
authorise him to do so. Where a restrictive indorsement authorises
further transfer, all subsequent indorsees take the note with the
same rights and subject to the same liabilities as the first indorsee
under the restrictive indorsement.
250. A Negotiable Note is one which may be sold or
transferred by the payee to anyone else ; and a note is negotiated
when it is transferred from one person to another in such a manner
as to constitute the transferee the holder of the note. A negotiable
note may be payable either to order or to bearer. A note is payable
to bearer which is expressed to be so payable, or on which the only
or last indorsement is an indorsement in blank. A note is payable
to order which is expressed to be so payable, or which is expressed
to be payable to a particular person, and does not contain words
prohibiting transfer or indicating an intention that it should not be
transferable. Where a note either originally or by indorsement, is
expressed to be payable to the order of a specified person, and not
to him or his order, . it is nevertheless payable to him or his order,
at his option.
A note payable to bearer is negotiated by delivery. A note
payable to order is negotiated by the endorsement of the holder
completed by delivery.
Where the holder of a note payable to bearer negotiates it by
delivery without indorsing it, he is called a transferor by delivery.
A transferor by delivery is not liable on the instrument. A
transferor by delivery who negotiates a note thereby warrants to his
immediate transferee, being a holder for value, that the note is what
it purports ^o be, that he has a right to transfer it, and that at the
time of transfer he is not aware of any fact which renders it valueless.
When a ncjte contains words prohibiting transfer, or indicating
an intention that it should not be transferable, it is valid as between
the parties thereto, but it is not negotiable.
286 ARITHMETIC.
251. Where a i)roiiiissory note is in the body of it made payable
at a particular place, it must be presented for payment at that
place in order to render the maker liable : in any other case,
presentment for payment is not necessary in order to render the
maker liable. Presentment for payment is necessary in order to
render the indorser of a note liable. Where a note is in the body
of it made payable at a particular place, presentment at that place
is necessary in order to render an indorser liable ; but when
a i)lace of payment is indicated by way of memorandum only,
presentment at that place is sufficient to render the indorser liable,
but a presentment to the maker elsewhere, if sufficient in other
respects, will also suffice.
252. Maturity (properly Date of Maturity) is the day on
which the note becomes legally due. Where a note is not payable
on demand, the day on which it falls due is determined as
follows : —
Three days called days of grace, are, in every case where the note"*
itself does not otherwise provide, added to the time of payment as
fixed by the note, and the note is due and payable on the last day
of grace. Whenever the last day of grace falls on a legal holiday
or non-juridical day in the Province where any such note is payable,
then the day next following, not being a legal holiday or non-juridical
day in such Province, is the last day of grace.
A note is payable on demand, which is expressed to be payable on
demand, or on presentation, or in which no time for payment is
expressed.
253. Where a bill is payable at a fixed period after date, after
sight, or after the happening of a specified event, the time of
payment is determined by excluding the day from which the time
is to begin t j run and by including the day of payment. The term
" Month " in a note means the calendar month. Every note which
is made payable at a month or months after date becomes due on
the same numbered day of the month in which it is made payable
as the day on which it is dated — unless there is no such day in the
month in which it is made payable, in which case it becomes due on
the last day of that month — with the addition, in all cases, of the
days of grace.
DISCOUNT. 287
254. A note is not invalid by reason only that it is antedated or
post-dated, or that it bears date on a Sunday.
255. A Draft or Bill of Exchange is a written order by one
person (called the Drawer) directing a second person (called the
Drawee) to pay, unconditionally, on demand or at a fixed or j,
determinable future time, a specified sum of money (called the Face
or Par) to a third person (called the Payee) or to the payee's order,
or to bearer.
Sections 249 to 254 apply to bills of exchange as well as to
promissory notes.
256. Bank Discount is a dedndion made from the face ndue
of a note or a draft for cashing it or buying it before maturity.
257. The Term of Discount is the time between the date of
the discounting and the date of maturity.
258. The Rate of Discount is the percentage of the face
VALUE which would be deducted if the term of discount were one
YEAR.
259. Exchange is a charge made for collection in cases in
which the place of payment of the note or the draft is nt>G the place
of discount. The rate of exchange is generally from -^ to I of 1 %
of the face value, but if the face value is less than SlOO, the full
exchange on |100 is usually charged.
260. The Proceeds of a note is the sum of money received for
it on discounting it. It is equal to the sum due at maturity less
the discount and the exchange.
Example. A note for $572 80 drawn on 13th June and payable
4 months after date, was discounted at 7 % on 27th June. Find
the proceeds.
Maturity is 4 mo. 3 da. from 13th June = 16th Oct.
Term of discount is from 27th June to 16th Oct. = lllda. = ;^^-^ yr.
Face of note = $572 SO. Rate cf discount - "07.
Discount = $572 '80 -x -07 x 111 4- 365 - $12 20.
Proceeds = $572 -80- $12 -20 = $560 60.
[Calcidation of the disconnt.
log 572 -8 + log -07 4- log 111 - log 365
= 2 -758003 + -845098 -2 + 2 045323 - 2 '562293
= 1-086131 = log 12-194.]
Date of
Rate of
Date of Note.
Time.
Discount.
Discount
13th May, 1890.
90 da.
13th May.
6 %.
5tli Sept. 1892.
3 mo.
16th Sept.
7 %.
28th Aug. 1891.
60 da.
4th Sept.
7 %,
17th Dec. 1889.
2 mo.
23rd Dec.
7i%.
28th Dec. 1891.
4 mo.
15th Jan.
8 %.
288 ARITHMETIC.
EXERCISE XLV.
Find the date of maturity, the term of discount and the proceeds
in the following cases.
Face of
Note.
I. $312-80.
a. 1975-65.
3. $450.
4. $79-50.
5. $586-67.
6. Find the proceeds of the following note discounted in Toronto
on 1st May 1890, at 7%, exchange ^%.
$390t%o^ Ottawa, 1st May, 1890.
Three months after date I promise to pay to the order of
Thomas A Stuart, Three Hundred and Ninety y%% Dollars, at the
Bank of Commerce here. Value received.
James Henderson.
7. A note for $250 was discounted 40 days before maturity and
the proceeds were $247 '80. What was the rate of discount, there
being no exchange ?
§. A note for $742-76 was discounted 93 days before maturity
and the proceeds were $730-47. What was the rate of discount,
the rate of exchange being g % ?
9. For what sum nmst a note be drawn in order that if discounted
89 days before maturity, the proceeds may be $425 ; the rate of
discount being 7 % and there being no exchange ?
I O. For what sum must a draft payable thirty days after sight be
drawn in order that if discounted on day of drawing the proceeds
ill ay be $745-25; the rate of discount being 7o% and that of
exchange | % ?
II. A promissory note for $385 -20 was discounted on 1st March,
1890, at 7 % discount and ^ % exchange and the proce.eds were
$377 "70. Determine the date of maturity of the note.
S
INTEREST. 289
VI. INTEREST.
261. Interest is the sum which the lender of money charges
the borrower for the use of the sum borrowed, or which a creditor
charges a debtor for allowing his debt to remain unpaid after it has
become due,
262. The Principal is the sum borrowed or due.
263. The Amount is the sum total of principal and interest.
264. The Rate of Interest is always expressed as the rate
per cent, of the principal which would be charged for its use for one
FEAR.
265. The Time is the period, expressed in years, for which
interest is reckoned. ^
266. Simple Interest is interest reckoned on the original
principal and on it alone for the whole term during which that
principal bears interest.
267. Compound Interest is interest which is reckoned for
stated periods and added at the end of each period to the principal
on which it was reckoned, the amount or sum total of principal and
interest at the end of each period becoming the principal for the
succeeding period.
It is as if the original principal had been loaned at simple interest
for the first period, then the amount from that period loaned for
the next period, the second amount loaned for the third period and
3o on, a nfw loan of the sum total of principal and all accrued
interest being entered upon at the beginning of each period.
Thus compound interest reckons interest upon interest.
268. The names Annual, Semiannual, Quarterly and
Monthly Interest are applied to interest which is payable at
the end of each year, half-year, quarter-year or month, as the
case may be, throughout the time during which the principal bears
interest.
269. Annual or other periodically payable interest differs from
simple interest in that it is to be paid at stated intervals while
simple interest is not due and collectible until the principal matures.
290 ARITHMETIC.
270. Annual or other periodically payable interest differs from
compound interest in that it like simple interest is reckoned on the
original principal and on it alone while compound interest is
computed for each period on the original principal increased by all
accrued interest. In eflect, however, periodically paid interest is
equivalent to compound interest, for the borrower loses the use of
the money he pays as interest at the end of each period and the
lender gains the use of it, and the value of this use is assumed to be
interest at the rate paid on the principal. Hence, in calcidations
concerning periodic payments, the tnethods, not of simple hnt of
compound interest, should be emploijed. For example it is usual with
savings banks which pay annual interest to credit each dept)sitor at
the end of every interest year with all interest on his deposit
accrued but undrawn, treating such interest as a new deposit, the
net result being that the banks pay compound interest.
271. If interest which is by agreement to be paid at specified
intervals, is not so paid, and the lender has to coUect it by process
of law, the courts have authority to grant at their discretion simple
interest on the accrued periodic interest. The interest up(jn
interest, if thus granted, is styled damages and its maximum rate is
the legal rate of six per cent, per annum.
Simple Interest.
272. Problems in simple interest involve the consideration of
principal, rate, time, interest and amount ; and any three of these
being known the other two may be determined, for by definition ; —
1°. The interest is the contim led prod \ict of the principaU the rate
'per unit and the measure of the time.
2°. Tlie amount is the siim of the prinrApal and the interest.
273. Expressed in general symbols these statements are
1°. I=Frt,
2°. A=P+I,
the letters J, P, t arid A denoting severally the measures of the
interest, principal, time and amount, and r denoting the rate per
unit.
INTEREST. 291
EXERCISE XLVL
Find the simple interest on and the amount of
1. $473-28 for 3 years at 6 %.
2. $385-35 for U years at 5 %.
3. $628-25 for 185 days at 4|%.
4. $935-68 for 66 days at 6| %.
5. $147 -50 for 3 years 93 days at 7 %.
6. $250 from 9th July to 18th Aug. at 8%.
7. What principal will yield $43*25 interest in 2J years at 5^% ?
§. What principal will in 95 days yield $9-20 interest at 7 % ?
9. What principal will yield $10 as interest at 6 % from 1st. May
to 31st, Oct. of the same year 1
10. What principal will amount to $1000 in 4| years at 4| % ?
11. What principal will amount to $73 '56 in 66 days at 8% ?
12. A debt due on 3rd March was not paid and interest at 6j%
was charged on it from that date. On 6th June following, the debt
amounted to $100. What was the sum due on 3rd March ?
13. At what rate will $375*50 amount at simple interest to
$441-21 in 2| years?
14. At what rate will $222 '66 yield $21 simple interest in 1 year
and 94 days ?
15. At what rate will $438*88 borrowed on 17th Ap. amount at
simple interest to $446*93 on 29th July next following ?
16. At what rate will a sum of money at simple interest double
itself in 20 years ?
1 T. At what rate will a sum of money at simple interest quadruple
itself in 50 years ?
i§. In what time will $273*85 yield $28*86 simple interest a.t
6%?
19. In how many days will $733*65 amount to $743*70 at 5%
simple interest ?
20. A debt of $175 became due on 13th June after which date
interest was charged at the rate of 7%. When the debt was paid
the interest accrued on it was $4*10. When was the debt paid ?
292 ARITHMETIC.
21. In what time will a sum of money double itself at 5% simple
interest ?
22. In what time will a sum of money triple itself at 8 % simple
interest ?
23. The proceeds of a note for $137 '50 discounted 40 days before
maturity, were $136*30. What was the rate of discount charged
on the face of the note and what was the rate of interest paid on
the proceeds?
24. Find the discount off $385-77 due 86 days hence, (i) at 8 %
discount, (ii) at 8 % interest. Show that the difference between
amounts (i) and (ii) is the interest at 8 % on (i) or the discount at
8%off(ii).
25. What rate of interest is equivalent to 10% discount, the
term of discount being one year ?
26. What rate of interest is equivalent to 10% discount, the
term of discount being 95 days ?
[274. The Present Worth at a specified rate of interest of a
bill or a promissory note is the sum of money which jiut out at
interest at the specified rate will when the bill is due or the note
matures amount to the sum due on bill or note.
The difference between the present worth at a specified rate of
interest of a bill or a promissory note and the amount of the bill or
the note when due, is by some writers termed the True Discount,
and the specified rate of interest is called the JRate of Discount.
Considered as an abatement or deduction made from the amount of
the bill or the note, the so-called True Discount is certainly a
discount, hut so woidd be any other abatement, but to call the rate
at which the present worth increases by interest, a rate of discount,
i.e., a rate of counting off, is a perversion of the term which is
not sanctioned by commercial usage and which leads to needless
confusion when pupils go from the class-room to the counting-house.
The problems which are commonly given under the head of True
Discount are properly problems on Interest and were they correctly
worded and proposed as problems on Interest they would be perfectly
legitimate and unexceptionable. Thus Prob. 10, Ex. xlvi, p. 291,
may be put under the form : — What is the present worth at 4 "5 %
interest of $1000 due 4*5 years hence ?]
INTEREST. 293
Averaging Accounts.
275. When one person owes another several amounts due at
different times, the date on which all these debts may be discharged
by payment of their sum, without loss of interest to either the
debtor or the creditor is called the Average Date or Equated
Time.
Example. A bought goods of B as follows : — May 17, |200 at
30 days' credit ; June 3, $250 at 60 days' credit ; June 12, $210 at
90 days' credit. On July 5, A paid B $300 on account. Find the
equated time for paying the balance.
Had A paid B $200 + $250 + $210 = $660 on May 17, B would have
gained the interest on $200 for 30 days, the interest on $250 for
77 days and the interest on $210 for 116 days.
But if A delay from May 17 to July 5 to pay $300 of the $660,
B'a gains will be reduced by the interest on the $300 for 49 days,
the number of days of delay.
And if A defer the payment of the $360, balance of the $660,
until the equated date, B will lose the balance of the interest he
would have gained had all the payments been made on May 17.
Interest on $200 for. 30 da. =Int. on ($200 x 30 = $ 6000) for 1 da.
„ „ 250 M 77 M = „ M ( 250 X 77= 19250) „
H 210 n 116 M = M „ ( 210x117= 24360) ,, „
60 $49610 for Ida.
Interest on $300 for 49 da. = Int. on ($300 x 49= 14700) „ „
$360 $360 )$34910( 97
Interest on $34910 for Ida. =Ini. on $360 for (34910-^360) days
- Int. on $360 for 97 days.
Equated time = 97 days after May 17 = Aug, 22.
276. Should any of the items include cents, omit the cents in
the calculation, and take the nearest number of dollars to the
amounts of the items.
277. The method of determining the equated time of an account,
which is exhibited in the preceding solution, is based on the
assumption that what the debtor gains by retaining certain sums
294 ARITHMETIC.
after they become due he loses by paying other sums before these
become due, but as both gains and losses are computed on the full
amounts of the items, while the actual gain is the interest on the
amounts of the deferred payments and the actual loss is the interest
on the present worth of the anticipated payments, it is evident that
the solution is not absolutely exact. However, in ordinary
business transactions, the error is too small to materially affect the
result.
EXEBCIBE XLVII.
Find the equated date of payment or
1. Sep. 3, $350 @ 60 da. 2. Aug. 27, $325 @ 60 da.
„ $520 @ 90 da. Sept. 20, $280 @ 30 da.
,. $175 @ 30 da. Oct. 31, $785 @ 90 da.
3. On May 2, goods amounting to $1250 were purchased on the
following term's ; $400 payable in 30 days, $500 payable in 60 days
and the balance payable in 90 days. Find the equated date for the
payment of the whole bill.
4. On Sep. 19, a commission merchant received a consignment
of 600 barrels of apples. He sold 120 barrels at $2*25 on Sept. 24 ;
75 barrels at $2-30 on Sep. 27 ; 150 barrels at $2-40 on Oct. 7th ;
150 barrels at $2" 35 on Oct. 22 ; and the balance at $2*20 on Nov. 18.
Find the equated date of the total sales.
5. Henry Simpson sold A. Thomson & Co. merchandise as
follows : Sep. 1, 225 bbl. flour @ $6, on 30 days' credit ; Sep. 9,
180 bbl. of pork averaging 2081b. @ ll|ct., on 60 days' credit ;
Sep. 17, 150 doz. eggs @ 16 ct. per dozen on 2 months' credit ;
Oct. 7, 5721b. bacon @ 13| ct. on 3 months' credit; Nov. 10,
460 lb. butter @ 21| ct. on 90 days' credit. Find the equated date
for the payment of the sum-total of the several bills.
6. A holds three promissory notes made by B, one is for$245'60
payable in 3 months from Feb. 13, 1889 ; another is for $425
payable 60 days after date of Mar. 5, 1889 ; and the third is
for $186-25 and is dated Ap. 3, 1889, and payable 90 days after
date. On Ap. 17, 1889, B offers to pay $500 on the notes, and give
in exchange for them a single note for the balance on them unpaid.
When should the single note be payable ?
PARTIAL PAYMENTS. 295
Partial Payments.
278. A Partial Payment is a payment of only a part of a
debt and its accrued interest.
279. A Receipt Indorsement is an acknowledgment of the
receipt of a partial payment written on the back of a note, mortgage
or other documentary evidence of debt, stating the amount and the
date of the payment.
280. When partial payments have been made on an interest-
bearing note or other obligation, the balance unpaid and due at any
given date may be found as follows : —
Find the interest on the pi'imipal from the date of the note or other
obligation to the date of the first partial payment.
(«) If the first partial payment is equal to or exceeds the interest
thus found, subtract the first payment from, the sum of the principal
and its accrued interest, and consider the reinainder as a new principal.
(6) If the first partial payment is less than the interest thus found,
find the interest on the principal to the date of the next or of the
earliest subsequent partial 'payment at which the sum of the payments
equals or exceeds the interest due at such date, and subtract the sum of
the payments to that date from the sum of the principal and its accrued
interest to that date, avid cansider the remainder as a new principal.
Similarly find the interest on the new principal to the date of the
next partial paymoit. If that payment be equal to the interest thus
found or if it be greater than the interest, proceed as in (a); but, if
the payment be less than the interest, proceed a^ in (6). So continue
to the date of settlement.
281. A partial i)ayment in excess of the accrued interest will
have the effect of reducing the principal, since, after discharging
such accrued interest, there will remain a surplus to be so applied.
A partial payment less than the accrued interest will not reduce
the principal since such payment is not sufficient to discharge the
accrued interest which must first be paid. No new principal should
exceed the preceding principal, for such excess could arise only by
the addition of interest to that preceding principal, and the effect
would be to compute interest on interest, in computing the interest
on the new principal.
296 ARITHMETIC.
282. In opon accounts merchants generally charge interest upon
all debts from the time they become due to the time of balancing
accounts and allow interest to the same time upon all partial
payments from the time they are made ; they then deduct the sum
of the partial payments and their accrued interest from the sum of
the debts and their accrued interest, the remainder being the
balance due. Upon this balance, if the account be not meanwhile
paid or closed by note, interest is charged to the time when the
accounts are again balanced, and is allowed to the same time upon
all j^artial payments from the time they are made ; and this process
is continued until the account is either paid or closed by note.
EXERCISE XLVIII.
1. On a note for |620 on demand, dated Oct 18, 1888, and
drawing 6 % interest are indorsed the following payments : Nov.
26, 1888, $47-50 ; Dec. 28, 1888, $108 -93 ; Feb. 11, 1889, $216-18 ;
June 6, 1889, $60-10 ; Sep. 2, 1889, $183-25. How much was due
on the note on Nov. 11, 1889 ?
2. On a mortgage for $3750 dated May 16, 1887, and bearing
interest at 6 %, there were paid May 16, 1888, $350 ; Sept. 18,
1888, $280; Jan. 22, 1889, $750; May 16, 1889, $925; Oct. 31,
1889, $500. What sum was due on the mortgage on Jan. 2, 1890 ?
3. How much was due on the following note, on Oct. 31, 1889?
$850. Toronto, Oct. 31, 1887.
For value received, I promise to pay Alex. Thompson or order,
on demand. Eight hundred and fifty Dollars, with interest from
date at six per centum. John Stuart.
On this note the following payments were indorsed.
April 20, 1888, $125. Jan. 21, 1889, $75.
Nov. 20, 1888, $125. July 20, 1889, $425.
COMPOUND INTEREST. 297
Compound Interest.
283. Compound Interest is interest which is computed for
stated jjeriods and added at the end of each period to the principal
on which it was computed, the sum-total of principal and accrued
interest at the end of each period becoming a new principal on
which interest is computed for the next succeeding period.
284. The interest is said to be compounded annually, semi-
annually, quarterly, monthly, according as the addition of
interest to principal is made every year, half-year, quarter-year,
month, or other interval.
285. In stating the rate of interest, one year is taken as the
unit of time but is not expressed, and the rate is reduced to an
annual rate as if it were for simple interest. Thus 4 % compounded
semi-annually does not mean 4 % per half-year but 2 % per six
months, the full phrase being, — * 4 % per annum but compounded
semi-annually. ' A rate expressed in this way as if it were a simple
interest rate is called a nominal rate to distinguish it from the
actual or effective rate. A nominal rate of 6 % compounded
quarterly is an actual rate of 1| % per quarter-year, and a nominal
rate of 12 % compounded monthly is an actual rate of 1 % per month.
Example. If $1250 deposited in a savings-bank, draw interest at
4 % payable semi-annually, the interest accrued and due at the end
of the first half-year will be '02 of $1250 which is $25 00. If this
$25 "00 be not drawn it will be placed to the credit of' the depositor,
making his deposit $1275.
The interest for the second half-year will be computed on the
increased deposit and will therefore be '02 of $1275 which is $25-50.
If this $25-50 be not drawn it will be plg-ced to the credit of the
depositor, making his deposit $1300-50 at the beginning of the third
period of six months.
The interest for the third half year will be computed on the
$1300-50 deposit and will therefore be '02 of $1300*50 which is
$26 -01. This sum, if it be not drawn, will be added to the $1300-50
making a total of $1326*51 at the credit of the depositor at the end
of 18 months.
298 ' ARITHMETIC.
Thus $1250 at 4 % interest compounded semi-annually will in
a year and a half aniount to $1326 "51 ; and the compound interest at
the specified rate and for the stated time will be $1326 '51 — $1250
-$76-51,
Computation. $1250 = original amount or principal.
l"02 = mie of increase in amount.
~~2500
1250 _
$1275 = amount at end of 1st i)eriod.
^1;02
25^
1275
$1300 "50 =: amount at end of 2nd i)eriod.
102
26-0100
1300-50
$1326 "51 = amount at end of 3rd period.
EXERCISE XLIX.
Find the amount and the compound interest of : —
1 . $800 for 3 years at ^ % compounded annually.
2. $425 for 4 years at 4 % compounded annually.
3. $250 for 2 years at 6 % compounded semi-annually.
4. $366-67 for 2^ years at 4 % compounded semi-annually.
5. $722*50 for 1^ years at 4 % compounded quarterly.
Find correct to six significant figures the amount of $1 at
compound interest at 6 % for one year, interest compounded.
^. annually. 7. semi-annually. §. quarterly,
Find correct to six significant figures the rate of increase in the
amount of $1 at 5 % interest compounded annually for
• 9. three years. 10. five years. 1 1. seven years.
Find correct to six significant figures the rate of increase in the
amount of $1 at 4 % interest compounded quarterly for
12. one year. 13. two years. 14. three years.
COMPOUND INTEREST. 299
286. Problems in Compound Interest involve the consideration
of 07'igi7ial amount or principal, rate, number of compouiidiugs,
final amount and interest, and any three of these being known the
other two may be determined.
Let r denote the nominal rate of interest per unit ; t the
measure in years of the length of time between two successive
compoundings ; n the number of compoundings ; Ao the measure
of the original amount, the principal ; A^ the measure of the amount
after ?«, compoundings ; and /„ the measure of the interest after n
compoundings ; then will
^1 = ^0(1 + ^*^),
A^^A, (l + H) = Ao(l + rtr
A^ = A2{l + rt)^Ao{l + rtY
A^ = A^(l+rt) = A^(l+Hy
^n = -4„_i (1 + H) = ^0 (1 + rty, (A . )
and . •. log A^ = log Jq + u log (1 + rt) ; (Aa. )
and J„ = ^„-^o* (-5.)
287. If there should occur a broke^i period whose measure in
years is t^, t^ being < i, the rate of increase for ti is by commercial
usage taken to be l + rt^.
Example 1. What will be the amount of $437*50 in 10 years at
5 % payable and compounded half-yearly ?
The nominal rate of interest is '05 per unit and the periods or
terms are \ yr. each,
. '. the actual rate of interest is \ of "05 per unit= "025 per unit.
. •. the rate of increase by compounding is 1 "025 per half year ;
. •. the rate of increase for 10 years- is 1 '025^0
.-. the amount sought to be known is $437 "50 x 1 '0252 0.
logl-025F= -010724
20
20 log 1-025= -21448
log 437 -5 = 2 -640978
2^-855458 = log 716 -9
amount at end of 10 years = $7 16 -90.
300 ARITHMETIC.
Example 2. What will be the discount off $100 for 6 years at 6 %
interest, compounded quarterly ?
The actual rate of interest is ( "06 x ^) per unit = '015 per unit.
The interest is compounded (6-f^) times = 24 times.
$100 present value will in 6 yr. amount to $100 x 1 '0152 *
.-. ($100^1-0152*) „ „ „ „ „ „ „ $100.
log 100 - 24 log 1 -015 = 2 - -155184 = -844816 = log 69 -95
.-. $69-95 present value will in 6yr. amount to $100
.-. discount = $100 -$69 -95 = $30 -05.
[The student should note the distinctuni between discounting at 6 %
INTEREST {whether simple or compound) and discounting at 6% of
DISCOUNT. See § 274, p. 292.]
EXERCISE L.
Find the amount and the compound interest of : —
1. $750 for 15 years at 5 % compounded annually.
2. $365 for 10 years at 6 % compounded semi-annually.
3. $1250 for 20 years at 4 % compounded quarterly.
4. $36-25 for 5 years at 6 % compounded monthly.
5. $427 "50 for 15 years at 5 % compounded triennially.
6. $125 for 100 years at 4 % compounded quinquennially.
y. Find, correct to five significant figures, the sum to which one
cent would amount in 1890 years at (a) 1 %, (6) 2 %, (c) 3 % interest
compounded annually, given log 1 "01 = "0043213738,
log 1-02= -0086001718 log 1-03= -0128372247.
Find the present worth of : —
8. $1000 payable 20 yr. hence, at 4 % interest compounded
annually.
9. $372-50 payable 7^yr. hence, at 5 % interest compounded
semi-annually. '
10. $372-50 payable 7|yr. hence, at 5 % interest compounded
quarterly.
Find the discount off $125 payable 10 years hence at
11. 5 % discount. 12. 5 % simple interest.
13. 5 % interest compounded annually.
14. 5 % interest compounded semi-annually.
15. 5 % interest compounded quarterly.
COMPOUND INTEREST. 301
16. In what time will $300 amount to $426 -63 at ^ % compounded
annually ?
17. In what time will $250 amount to $376-20 at 6 % compounded
quarterly ?
18. Show that a sum of money will about double itself in (70-^2)
compoundings at 2 %, (70-^3) compoundings at 3 /^, (70 ^3|)
compoundings at 3.| %, and in correspondingly obtained numbers of
compoundings for 4, 4i, 5, 5^, 6, 7, 8, 9 and 10% respectively.
(See problem 13, Exercise xx, p. 158).
In what time will a sum of money drawing 8 % interest increase
to 10 times the original sum
19. if the interest be compounded annually ?
20. if the interest be compounded semi-annually ?
21. if the interest be compounded quarterly ?
22. At what rate will $225 amount to $302' 60 in 12 years, interest
compounded annually ?
23. At what rate will $133 amount to $456-15 in 14 years, interest
compounded semi-annually ?
24. In 1871 the population of a certain city was 27512, in 1881
it was 44653 ; what was the annual rate of increase of the city's
population ?
25. Two equal sums of money are placed at interest, one sum
at 6 % the other sum at 1^ %, the interest in both case being
compounded annually. In what time will the amount at the higher
rate be 10 times that at the lower rate ?
26. Two equal sums of money are placed at interest, both at a
nominal rate of 12 %, but in one case the interest is compounded
monthly while in the other case it is compounded annually. In
what time will the amount at the higher effective rate be double
that at the lower ?
27. What will be the effective rate per annum if the nominal
rate be 6 % and the interest be compounded (a), monthly ; (h),
daily ; (c), hourly ; (d), per minute ?
2§. At what rate will a sum of money treble itself in 30 years,
interest compounded quarterly ? To what multiple of the original
sum will it amount in 100 years at this rate ?
302 ARITHMETIC.
VI. Stocks and Bonds.
288. A Corporation or Incorporated Company is an
association of persons authorized by law to transact business as a
single individual. The powers, rights, duties and obligations of a
corporation, as such, are distinct from those of the members
forming it.
289. The capital of a corporation or of a public company is
usually divided into a definite number of equal parts called Shares.
A share commonly represents $100 (or £100) of the original capital
of the corporation, but in some cases it represents as low as f 1 (or
£1) of it and in other cases as high as $1000 (or £1000) of it.
290. Any number of shares in a corporation or any amount of its
capital is called Stock, but in the United States this term is also
used distinctively for shares of $100 each, shares of $50 and of $25
being called half-stock and quarter-stock respectively.
291. The proprietors of shares in a corporation or in a public
company are called shareholders or stockholders. Each
owner of stock may sell his shares or otherwise transfer them to
another person without the consent of the other shareholders.
292. A Stock Certificate is an instrument issued by a
corporation, certifying that the holder thereof owns a stated
number of shares of the capital stock of the corporation.
293. The Par Value of a share is the value which is specified
upon the face of the certificate for the share, and represents the
amount of capital stock for which it was originally issued.
294. The Market Value of a share is the sum for which it can
be sold.
Stock is said to be above par or at a premium when the market
value of the shares is greater thg,n their par value ; it is said to be
below par or at a discount when the market value of the shares is
less than their par value.
295. A Dividend is the part of the net earnings or profits of a
corporation or a public company, which is divided among the
stockholders thereof. Dividends are usually declared annually.
STOCKS AND 150NDS. ^0^
semi-annually, or quarterly at a specified rate per cent, of the par
value of the stock.
296. Preferred Stock is that part of the capital stock of a
corporation on which a specified percentage is payable annually out
of the net earnings, before any dividend can be declared on the
ordinary stock.
297. A Bond or Debenture is a written obligation to pay the
holder thereof a certain sum of money at the expiry of a certain
term of years, and interest thereon at a specified rate per cent, at
stated intervals. Bonds and debentures are issued for money
borrowed by the General and Local Governments and by municipal
and business corporations. Debentures frequently charge certain
specified property with the repayment of the money borrowed on
them ; in such cases the debentures are practically mortgages on the
property.
298. An Interest Coupon is an interest certificate payable to
bearer, printed at the bottom of bonds and debentures given for a
term of years. There are as many coupons attached to each bond as
there are instalments of interest to be paid on it, a coupon for each
instalment. Each coup(jn is cut off and presented for payment
when the interest for the period mentioned in it becomes due.
299. Consols, i.e.. Consolidated Annuities are British Govern-
ment securities bearing 3 % interest. These with the other British
Government securities for which permanent provision has been
made, the most important of which are the Reduced Annuities and
New three per cent. Annuities, are in England termed the Public
Funds.
300. Rentes (i.e. Annuities) are French Government securities
bearing various rates of interest.
301. Stock Brokers are persons who deal in stocks, bonds
and similar securities. When a stock broker buys or sells for a
principal he charges a conunission, technically termed brokerage,
which ranges, according to circumstances and previous agreement,
from ^ of 1 % to 1^ of 1 % of the par value of the securities bought
or sold, the most common rate being ^ of 1 %. Occasionally special
rates are agreed upon and paid.
304
ARITHMETIC.
In England, stock brokers do not deal directly with each other
but sell to or buy from stock-jobbers who act for themselves and
make their profits out of the turn of the market.
302. In Canada and the United States, stock quotations usually
state only the rates per cent, which the market values of the stocks
and bonds quoted bear to their par values ; but in England quotations
of other than government securities generally give the price per
share or per bond.
The following is an illustrative example of a stock report and
quotations : —
The closing prices on the Toronto Stock Exchange to-day (6 Dec. ,
1889), were as follows :
1 P. M.
Stocks.
^
o
o
■Ji
A
4 P. M.
5*ft
Banks.
Montreal .
Ontario . . .
Molsons . . ,
Toronto . .
Merchants'
Commerce
Imperial . .
Dominion .
Standard . ,
Hamilton .
200
100
50
200
100
50
100
50
50
i 100
5
3J
31
4"
5
225|
132
158
219
142
121^
152|
2221
138"
224i
213
139
121
150
221
137^
146
225^
131^
156
220
140
121i
153
223
138
224|
131
212*
139
121
150|
222
1371
147
It will be seen from this report that in Toronto, on 6 Dec, 1889,
sellers of Bank of Montreal stock were offering it at the rate of
$225 '25 cash per $100 stock and as each share represents $200 of
stock, sellers were really asking $450 '50 per share. The report
also shows that buyers of Bank of Montreal stock were offering
for it 224^ % to 224| % of its par value, i.e., were offering $449 to
$449*50 per share for it.
STOCKS AND BONDS. 305
Other forms of report may be seen in the ' Financial Columns '
of the Toronto and Montreal daily newspapers and in the ' Share
Lists ' of any Stock Exchange.
Example 1. Find the price at 140| of 40 shares (^40 each) of
Western Assurance Co. stock, brokerage | %.
Par value oi stock = |40 x 40 = $1600.
Rate paid = 140| % + 1 % - 1 '401.
Cost of stock = $1600 X 1 -401 =$2254.
Example 2. I sold 500 shares of Bank of Montreal stock at 224|
and invested the proceeds in Bank of Commerce stock at 124j,
paying i % brokerage on each transaction. Find the increase in
my annual income, the Bank of Montreal paying a half-yearly
dividend of 5 %, the Bank of Commerce a half-yearly dividend of
^%•
Par value of B. of M. stock = $200 x 500 = $100,000.
Rate received = 224| % - 1 % = 2 -24^.
Amount to be invested =$100, 000 x 2 •24f = $224,625.
Rate paid for B. of C. stock = 124| % + i % = l-24f.
Price of 1 share of B. of C. stock=$50x 1 -241 = $62-1875.
Number of shares bought is the integral part of
$224625 -^ $62 -1875
which is 3612
ind there is $3*75 of cash over.
Par value of 3612 shares of B. of C. stock =$180,600.
2 dividends at 5 % each on $100,000 of B. of M. stock
= $100000 X •10=$10000.
2 dividends at ^ % each on $180600 of B. of C. stock
= $180600 X -07 = $12642.
Increase of annual income = $12642 - $10000 =
306 ARITHMETIC.
EXERCISE LI.
Find the cash value of
1. 25 shares Ontario Bank at 131.
2. 18 M Standard Bank at 137^.
3. 75 n Bank of Toronto at 218.
4.250 M (|50) Dominion Telegraph Co. at 83|.
5. 950 „ ($100) Canadian Pacific R.R. at 72|-.
6. 350 M ($24-331) North West Land Co. at 79|.
7. Sold through a broker 1500 shares ($100) of Jersey Central
R.R. stock at 121J, brokerage | %. What were the net proceeds of
the sale ?
8. Bought through a broker 1600 shares ($100) St. Paul R.R.
stock at 69 J, brokerage | % . What was the gross cost of the stock 1
9. A speculator bought 36500 shares ($100) Reading R.R. stock
at 39| and sold them at 40§. What was his gain on the transaction ?
10. A man bought through a broker 1900 shares ($100) Canada
Southern R. R. stock at 54| and sold them at 55§. What was his
net profit on the transaction, brokerage each way |- % ?
11. A man bought through a broker 7600 shares ($100) of Lake
Shore R.R. stock at 107^ and sold 2400 shares at 107| and the
remainder at 107|. What was the amount of his losses on the
transactions, brokerage being |- % each way ?
1J8. A bank declared a dividend of Sh%. How much should a
stockholder owning 120 shares ($50) receive ?
13. An insurance company declared a dividend of 6 %. What
rate is that on the market value of the shares which are at 185
14. Compare the rates on the cash values of 6 % on stock at 216
and 3^ % on stock at 125.
15. Sold 37 shares ($25) B. and L. Association stock, receiving
therefor $1019 •81. At what rate was the stock sold ?
16. Bought through a broker 750 shares ($50) in the Farmers'
Loan and Savings Society paying therefor $43968*75. At what
quotation were they bought, brokerage | % ?
17. Sold through a broker 215 shares ($50) in the Dominion
Savings and Loan Society receiving from him for them $9728*75.
At what quotation did the broker sell them, brokerage | % ?
STOCKS AND BONDS. 307
18. Bought stock at 197| and sold it at 194|, having meanwhile
received a dividend of 6 % on it. My net gain by the transaction
after paying | % brokerage each way, is $336. How many shares
($40) did I buy ?
19. A man received $495 as dividend at 4|^ % on his bank stock.
He sold 40 shares ($100) at 143j and the remainder at 144^, paying
^ % brokerage. What were the net proceeds of the sale ?
20. A capitalist had$20000 to invest. He purchased $8700, par
value, of Canadian 4 % bonds at 103 and $7300, par value, of
Canadian 3| % bonds at 93| and invested the balance as far as he
could in bank stock (shares $100) at 149j, paying half-yearly
dividends of 4 % each. What was the gross amount of his investment
he paying ^ % brokerage for buying each class of securities ? What
was his annual income from these investments ? What average rate
per cent, per annum did he receive on these investments ?
21. The difference between the annual income derived from a
certain sum invested in 7 % stock at 150 and that from an equal
sum invested in 9 % stock at 202|^, is $40. What is the amount
invested in the 7 % stock and what is the annual income therefrom 1
22. A shareholder receives a dividend of 6 % on his stock and
pays thereon an income-tax of 16| mills on the dollar. Next year
he receives a dividend of 6| % and pays an income-tax of 12^ mills
on the dollar. He finds that his income is $830 more in the latter
year than it was in the former. How much stock does he hold ?
23. A man invests a certain sum in 3 % stock at 90 and an equal
sum in 4 % at 95. Each stock rises 5 % in price ; the investor then
sells out and invests the proceeds of each stock in the other. The
stocks fall to their former value and he again sells out at a total loss
of $1943.90. Find the sum he originally invested.
24. What sum invested in the three per cents at 95 will in 17|
years amount to £10000, the price of the funds having risen
meanwhile to 100^ ; interest to be payable and compounded half
yearly ?
25. If money be worth 5 %, what should be the price of 6 %
bonds which are to be paid off at par 3 years after the date of
purchase, the interest on the bonds being payable half-yearly.
CHAPTER VIII.
EXCHANGE.
303. Exchange is the system by which accounts between
persona in distant places are settled without the necessity of
sending large sums of money or large quantities of gold or silver
from one place to the other, thus avoiding the risk and expense of
transportation.
For example, suppose that A of Halifax owes B of Toronto $7500
for wheat and that X of Toronto owes Y of Halifax ^7500 for dried
fish. In such case, B in Toronto can draw on A in Halifax for
$7500 and sell the draft to X who transmits it to Y who in turn
presents it to A wlio thereupon pays Y. Thus instead of A
sending $7500 from Halifax to Toronto to pay B, and X sending
$7500 from Toronto to Halifax to pay Y, X of Toronto pays B in
Toronto and A of Halifax pays Y in Halifax, the debts being as it
were exchanged.
Domestic or Inland Exchange is exchange carried on between two
cities in the same country.
Foreign Exchange is exchange carried on between two cities in
different countries.
304. A Draft or Bill of Exchange is a written order by
one person, called the drawer, directing a second person, called the
drawee, to pay a specified sum of money to a third person, called
the payee, or to the payee's order.
A Domestic or Inland Bill of Exchange, usually called a Draft, is
one of which drawer and drawee reside in the same country.
A Foreign Bill of Exchange is one of which drawer and drawee
reside in different countries. Foreign bills of exchange are usually
drawn in sets of three, called respectively the First, the Second and
the Third of Exchange, and are of the same tenor and date and so
worded that when one of the set is paid, the others become void.
The object of thus drawing the bills in sets of three is to provide
against loss in transmission. The bills or two of them are sent
either by different routes or by the same route at different dates.
EXCHANGE. 309
305. An Acceptance is an agreement by the drawee to pay
the sum specified in the draft or bill of exchange. The usual mode
3f accepting a bill of exchange is for the drawee to sign his name
ander the word " accepted" written across the face of the bill. If
bhe bill be payable a specified number of days after sight, the date
of acceptance should be inserted.
306. If the drawee of a bill refuses acceptance or if, having
accepted, he fails to make payment when it is due, the bill is
immediately protested^ i.e., a, written declaration is made by a public
oflicer called a Notary Public, at the request of the holder or person
in legal possession of the bill, notifying the drawer and the. indorsers
of its non-acceptance or non-payment.
307. Bills of exchange are negotiable or non-negotiable upon the
same conditions and are subject to the same indorsements as
promissory notes. The date of maturity of bills of exchange is
ascertained in the same manner as that of notes ; see § 252, p. 286,
308. The Pace or Par of a bill of exchange is the sum specified
in the bill, exclusive of interest, premiums, discount, or commission.
When bills of exchange on a given place sell for more than their
par value, exchange on that place is said to be above par or at a
premium ; when they sell for less than their face value, exchange
on that place is said to be below par or at a discount.
309. Exchange is usually conducted through bankers or brokers
who buy commercial bills on distant cities and mail them for collection
to their correspondents or agents in those cities. Drafts or bills of
exchange are then drawn on the correspondents for the whole or
for any required part of the sums thus placed to the credit of the
principals and sold to persons who wish to use money in those cities.
Bankers and their correspondents also draw on each other for sums
required by persons dealing with them and at stated periods strike
a balance of the sums thus drawn.
310. The Par of Exchange between two countries is the value
of the monetary unit of one of the countries expressed in terms of
the currency of the other.
The intrinsic par of exchange is the real or intrinsic value of coins
estimated by the weight and purity of the metals of which they are
composed.
X
310 ARITHMETIC.
The legal par of exchange is the par established under authority of
statute.
The dollar of Canada is defined by statute to be of such value
that four dollars and eighty-six cents and two-thirds of a cent shall
be equal in value to one pound sterling; thus $4*86| per £1 is the
legal par of exchange between Canada and Great Britain. There
being no Canadian gold coinage and the silver and bronze coins of
Canada being only a token coinage, there is no intrinsic par of
exchange between Canada and Great Britain.
The intrinsic value of the sovereign, the coin which determines
the value of the pound sterling of Great Britain, in terms of the
gold dollar, the monetary unit of the United States of North
America, is $4 "866564— ; for, 1869 sovereigns contain 211200 grains
of pure gold and the United States gold eagle contains 232 '2 grains
of pure gold and 211200 -^ 1869 -^ 23 -22 = 4 '866564-. The value
determined at the United States Mint and proclaimed by the
Secretary of the Treasury is $4*8665, a sum which approaches the
intrinsic value far within the * remedy ' allowed on the sovereign.
The intrinsic value of the ten-franc gold pieces of France, Belgium
and Switzerland is $1*93. The intrinsic value of the ten-mark gold
piece of the German Empire is $2-88.
311. The Rate of Foreign Exchange is the market or
commercial value of the monetary unit of one country expressed in
terms of the currency of another.
The following quotations were given by the New York Argents of
the Canadian Bank of Commerce as indicating the rates for actual
business in sterling exchange on 7 Dec. , 1889.
Prime Bankers, 60 days 4 '80^
do. Demand 4 -841
do. Cables 4'84|
Commercial 60 days 4 791-^
Documentary do. 4 -781 @ 4-79.
Prime Bankers' Sterling Bills are those drawn by first-class
banking houses in New York on first class banking-houses in
London, England.
Commercial Bills are those drawn by merchants or commercial
houses of good standing in America on their correspondents abroad.
'EXCHANGE.
A Documentary Commercial Bill is a bill drawn by a shipper upon
his consignee for merchandise shipped. It is accompanied by a Bill
of Lading and a Letter of Hypothecation giving control of the
merchandise to the holder of the bill, with recourse to the drawer
for the deficiency, if any should arise.
312. The New York quotations for bills on London are always
given in dollars per pound sterling.
The quotations for bills on Paris, Antwerp or Geneva are given
in francs per dollar. The quotations for bills on Hamburg, Bremen,
Berlin and Frankfort are given in cents per four marks.
In Canada the legal par of sterling exchange was formerly $4 '441
per £1 and Canadian quotations are still usually given as a percentage
premium on this old par. Thus when sterling exchange is quoted
at 9^ it is meant that the rate of exchange is $4 '441 x 1"095 per £1,
i.e., $4'86| per £1 which is the new par. So also sterling exchange
at 9 means $4 '441 x 1-09 per £1, i.e., $4'84A per £1.
313. The usage of Canadian bankers is to draw bills of exchange
on London payable either at 60 days after sight or on demand, but
as the greater part of the business is done in the former class of
bills, quotations are assumed to be for sixty-day bills unless it is
specifically stated to be otherwise at the time of making them.
314. A Circular Letter of Credit is a letter issued by a
banking-house to a person who purposes to travel abroad and
addressed to bankers generally and to the agents and correspondents
of the banking-house in particular in the several countries which the
traveller is about to visit, requesting them to supply the traveller
with money as he requires it until a total amount has been paid him
not exceeding the sum specified in the letter. The sums paid to the
traveller from time to time are indorsed on the letter. A letter of
credit is not transferable from one person to another.
Example 1 . What will a bill of exchange on London for £6000
realise in Toronto exchange at 8^ ?
The S\ here means 8^ % premium on the old par of exchange of
^•44| which gives $*^-x 1*085 as the rate of exchange for the
transaction.
. •. £6000 is equivalent to %^f x 1 '085 x 6000 = $28933 -33.
312 ARITHMETIC.
Example 2. Exchange at New York on London is 4 '841, and at
London on Paris it is 25*25 francs per £1. What sum remitted
from New York through London to Paris will pay a debt in Paris
of 12500 francs ?
25-25fr.=£l = |4-84|
12500 fr.= |4-84|xl2500H-25-25 = |2399-75.
EXERCISE LIT.
1 . What will a bill on London for £75 cost, exchange at 9^ ?
a. What will a bill on London for £225 cost at 9| ?
3. What will be the value of a bill for £60 at 8 1
4. What must be paid for a bill on London for £15 7s. fxl. at 10 ?
5. What sum sterling will be e(j[ual to $100 Canadian, exchange
9i?
6. What sum sterling should I receive for $5500 Canadian,
exchange 9| ?
7. The Government of Canada purchased the following sterling
exchanges : For transmission to Messrs. Glyn, Mills & Co. ,
£50,000 at 8^ and £10,000 at 8| ; for transmission to Messrs.
Baring Brothers &Co., £20,000 at 9 and £40,000 at $4-846 per
£1 stg. ; and for transmission to the Bank of Montreal, London,
£20,000 at 8f| , £20,000 at 8^|, £20,000 at 8||, and £20,000 at 8§|.
Calculate the cost of each of the eight purchases and find what
amount in dollars and cents should be charged to Glyn, Barings and
the Bank of Montreal, London, respectively, that they may be
charged at the par value 9^, in their accounts.
8. Find the cost of a bill of exchange on Paris for 2400 francs at
5-16ifr. per$l.
9. A merchant wishes to transmit 2400 francs from Toronto to
Paris, through London. For what sum (sterling) should the bill
on London be drawn and how much will the merchant have to pay
for it, sterling exchange being 9| and exchange between London and
Paris 25-20 francs per £1 ?
10. What will be the cost of a bill of exchange on Berlin for
2400 marks, rate of exchange 95 1 cents per 4 marks I
EXCHANGE. ^^^B^ ^l^
11. I bought in Ottawa a bill of exchange on London, England,
for £G0 at 9f and forwarded it to Calvary & Co. of Berlin who sold
it for 1224 i)iarks and gave me credit for the proceeds. What rate
of exchange on Berlin did I thus obtain ?
12. Immediate payments to the extent of £200,000 stg. are
required to be made in England on behalf of the Canadian
Government, and in response to calls the following tenders have
been received ;— for 60 days sight drafts £200,000 at 8|f , and for
demand drafts the same sum at 9|. Which tender would be the
more profitable to the Government, taking the rate of discount in
England at 3^ per cent, and the time 63 days ? How much would
the Government gain by accepting the more profitable tender 1
13. I purchased through a broker in New York a bill of exchange
on London for £432 12s. 6d. at 4-84§. Wliat was the total cost,
brokerage ^ % ?
1 4. I sold through a New Y( )rk broker a bill of exchange on
Hamburg for 1260 marks at 95^. What were the net proceeds due
me, brokerage j % ?
15. I bought through a brf»ker in Boston a bill of exchange on
Liverpool for £300 paying the broker $1457-64 for it. At what
quotation was the bill purchased, allowing | % for brokerage ?
16. I paid a broker $1511-00 for a bill of exchange on Bremen
for 6400 marks. At what (juotation was the bill purchased allowing
1 % for brokerage ?
17. I sold through a broker a bill of exchange on Manchester for
£600 and received $2912 '35 as the net proceeds. At what rate of
exchange was the bill sold allowing i % for brokerage ?
1 §. I sold a bill of exchange on Paris for 8330 francs and received
$1606*10 as the net i)roceeds. What was the rate of exchange on
Paris, a brokerage of |^ % having been charged me for selling the
bill?
19. I paid $2*40 as brokerage at |- % on a bill of exchange on
Hamburg for 8040 marks. What was the rate of exchange ?
20. I sold through a broker a bill of exchange on London at
4*85 and received $4773-37 as net proceeds. What was the face of
the bill, brokerage j % ?
314 ARITHMETIC.
21. The cost including brokerage at | %, of a bill of exchange on
Genevabought at 5-20 was $3764-70. What was the face of the
bill ?
22. Find the cost of 120 marks paid in Berlin on a letter of
credit, the rate of exchange being 95^^ and 28 cents being charged
for commission and interest.
23. Complete the following : —
New York, December 10, 1889.
i^anar/icin (^Suhk o/%omtnelce,
I 10 Wall Street.
Acct. Letter of Credit 7520, paid Berlin, Nov. 25, to F. O. 16r
marks, receipt enclosed ;
@ 95^
Com. \ % - -
Int. 30 days (S? 6 %. —
24. The Government of Canada procured silver coinage to tlie
extent of $200,000, for which the following (juantities of bar silvei
were purchased, viz. :
50, 341 '80 ounces Troy at 51|d. per oz.
50,046-27 M n 51}fd. „
49,055-26 M ^^ 52 d. „
On the value of the silver so purchased brokerage was charged at ^
per cent. ; the carriage and insurance from England to Canada,
calculated at the par of 9 J, was, on $60,000 at 18s. 6d. per £100 ;
on $80,000 at 16s. per £100 ; and on $60,000 at 13s. per £100, and
the cost of coinage £2,166 17s. 6d. What profit accrued to the
Government in dollars and cents, taking the rate at 9^ per cent, on
the transaction ; and what weight of silver in grains is contained in
a dollar ?
^iPiiPEisriDix:.
CIRCULATING DECIMALS.
By an extension of the ordinary or Arabic system of notation
the decimal fractions /q, j%, tWo are severally written 7, "89 and
•541, and conversely '3, '47 and -293 denote -^% i%^(j and t%%V
respectively. Still further extending this system to the expression
71 89?^ 541^
of complex decimal fractions jx » jtJ and ~--~ may be written
'TJ, '89| and •541f respectively, and conversely 'Si^, •27f and '001^
will severally denote t^-, 7-^ and Y^y?u\J with analogous expres-
sions for all other fractions whose numerators are mixed numbers
and denominators powers of 10.
A system of notation similar to this notation for decimal fractions
is sometimes employed in the writing of fractions whose denominators
are one less than a power of 10. For example ^ is written "7, f f
is written "85 and f^f^ is written '3069, with corresponding
expressions for all other fractions whose denominators are expressed
by 9's only. Conversely, when this system is employed '5 denotes
I, -216 denotes |^| and "230769 denotes if §9 |y-, with a corresponding
interpretation of all similar expressions.
This notation may be combined with that for decimal fractions ;
e. g., -41, -351 and 4*27?,^^ may be written '47, '352 and 4-27583
respectively, and 4 '866, '385 and 70 '02437 will severally denote
4-86f , -311 and 70'02^ff.
When it is necessary to reduce such complex fractions as "47,
•352 and "27583 to simple form advantage should be taken of the
relations 9 = 10 - 1, 99 - 100 - 1 , 999 = 1000 - 1, &c. Thus,—
*/ - *9 - 9Q 90 90 '
■o,A .o,. 35(10 -l) + 2_352-35_317.
•27583- -gr^^-^ 27(1000 - 1) + 583 _ 27583-27,27556
316 APPENDIX.
Example 1. Prove that "5= -55= '555= "5555^ "55= "555= "5555.
i. e. -5 = -55 = -555 = '5555 = - -
Also, 1= If = -511= -55^1 = - -
{. e. -5 = -55 = '555 = "5555 = - -
Example 2. Prove thab -237 = -2372= -23723= 237237 = -237237
m-=;2e||= -23511= •2372M=-2372fB =
•237 = -2372 = -23723 = -237237 = -2372372 :
Also 3^3 I =2 3 7237 — . 03123123= .0312.3 12 3— _ . _ 23 7 2 3 7 23 T _ .
J^iau, yyy 9 99 99 9 ~ "^999999 '^'^9911999— "" 999999999~
i. e. , -237 = -237237 = -2372372 = -23723723 = - - = -237237237 = ■
These two examples exhibit a property of fractions whose
denominators are expressed by 9's only or by one or more 9's
followed by one or more O's, which has led to this class of fractions
receiving the name of repeating or circulating decimals.
If the circle of recurring figures includes all the figures to the right
of the decimal point, the fraction is termed a pure circulating
decimal. Examples; -74, '853, 14-3257. If there are one or more
figures between the decimal point and the circle of recurring figures,
the fraction is called a mixed circidating decimal. Examples ; -574,
-853, 14-3257, 3 002.
Example 3. Express j\- in decimal notation.
A = '6 A = '63/1, Work.
-AxlOO=63A,
1^x99 = 63,
117 00
47
t\ = ?J = -63.
•63
Example ^. Express \} in decimal notation.
^H-3^f=-35;^=-351i?,
Work.
ifxl000 = 351if,
-13x999 = 351,
37
13 000
1953
1
- ^f=fM = -35i.
-35i
CIRCULATING DECIMALS. 317
The work of division in problems such as Examples 3 and ^ is to
be continued until a remainder occurs which is the same as either
the original dividend or a preceding remainder ; if the division be
carried beyond the second of these equal remainders the quotient-
figures from the former remainder to the latter will all recur in the
same order thus showing that they form a ' circle. ' As the
remainders must all be less than the divisor, the number of different
remainders and therefore the number of figures in the circle cannot
exceed 7i-l, in which n is the divisor, i. e. , the denominator of the
fraction to be expressed in decimal notation.
Example 5. Express || in decimal notation. • ^
Worl. 56 47 000000000
2 226820846
51434021
•839285714 = ^7
Explanation. il= -83911 = -83928571411
839i| X 1000000 = 839285714if
839,^1 X 999999 = 839285714 - 839
839i|= ^^^^^^^P&3o =.839|f|^ii = 839-285714
•8391^ = -839285714,
f^= -839285714.
TABLBS
OF
LOGARITHMS OF NUMBERS.
TABLE I.
N.
Log.
N.
Log.
N.
Log.
N.
Log.
1
0-000000
26
1-414973
51
1-707570
76
1-880814
2
0-301030
27
1-431364
52
1-716003
77
1-886491
3
0-477121
28
1-447158
53
1-724276
78
1-892095
4
0-602060
29
1-462398
54
1-732394
79
1-897627
5
0-698970
30
1-477121
55
1-740363
80
1-903090
6
0-778151
31
1-491362
56
1-748188
81
1-908485
7
0-845098
32
1-505150
57
1-755875
82
1-913814
8
0-903090
33
1-518514
58
1-763428
83
1-9] 9078
9
0-954243
34
1-531479
59
1-770852
84
1-924279
10
1-000000
35
1 -544068
60
1.778151
85
1-929419
11
1041393
36
1-556303
61
1-785330
86
1-934498
12
1-079181
37
1-568202
62
1-792392
87
1-939519
13
1-113943
38
1-579784
63
1-799341
83
1-944483
14
1-146128
39
1-591065
64
1-806180
89
1-949390
15
1-176091
40
1-602060
65
1-812913
90
1-954243
16
1-204120
41
1-612784
66
1-819544
91
1-959041
17
1-230449
42
1-623249
67
1-826075
92
1-963788
18
1-255273
43
1-633468
68
1-832509
93
1-968483
19
1-278754
44
1-643453
69
1-838849
94
1-973128
20
1-301030
45
1-653213
70
1-845098
95
1-977724
21
1-322219
46
1-662758
71
1-851258
96
1-982271
22
1-342423
47
1-672098
72
1-857333
97
1-986772
23
1-361728
48
1-681241
78
1-863323
98
1-991226
24
1-380211
49
1-690196
74
1-869232
99
1-995635
25
1-397940
50
1-698970
75
1-875061
100
2-000000
LOGARITHMS. — TABLE IL
319
No
0
1
2
3
4
1734
5
2166
6
7
8
9
D.
100
000000
0434
0868
1301
2598
3029
3461
3891
432
101
4321
4751
5181
5609
6038
6466
6894
7321
7748
8174
428
102
8600
9026
9451
9876
*0300
*0724
*1147
*1 70
*1993
*2415
424
103
012837
3259
3680
4100
4521
4940
5360
5779
6197
6616
419
104
7033
7451
7868
8284
8700
9116
9532
9947
*0361
*0775
416
105
021189
1603
2016
2428
2841
3252
3664
4075
4486
4896
412
106
5306
5715
6125
6533
6942
7350
7757
8164
8571
8978
408
107
9384
9789
*0195
*0600
*1004
n408
n8i2
*2216
*2619
*3021
404
108
033424
3826
4227
4628
5029
5430
5830
6230
6629
7028
400
109
7426
7825
8223
8620
9017
9414
9811
*0207
*0602
*0998
396
110
041393
1787
2182
2576
2969
3362
3755
4148
4540
4932
393
111
5323
5714
0105
6495
6885
7275
7664
8053
8442
8830
389
112
9218
9606
9993
*0380
*0766
ni53
n538
*1924
*2309
*2694
386
113
053078
3463
3846
4230
4613
4996
5378
5760
6142
6524
382
114
6905
7286
7666
8046
8426
8805
9185
9563
9942
0320
379
115
060698
1075
1452
1829
2206
2r.82
2958
3333
3709
4083
376
116
.4458
4832
5206
5580
5953
6326
6699
7071
7443
7815
372
117
8186
8557
8928
9298
9668
*0038
*0407
*0776
*1145
*1514
369
118
071882
2250
2617
2985
3352
3718
4085
4451
4816
5182
366
119
5547
5912
6276
6640
7004
7368
7731
8094
8457
8819
363
120
9181
9543
9904
*0266
*0626
*0987
n347
*1707
*2067
*2426
360
121
082785
3144
3503
3861
4219
4576
4934
5291
5647
6004
357
122
6360
6716
7071
7426
7781
8136
8490
8845
9198
9552
355
123
9905
*0258
*0611
*0963
*1315
n667
*2018
*2370
*2721
*3071
351
124
093422
3772
4122
4471
4820
5169
5518
5866
6215
6562
349
125
6910
7257
7604
7951
8298
8644
8990
9335
9681
*0026
346
126
100371
0715
1059
1403
1747
2091
2434
2777
3119
3462
343
127
3804
4146
4487
4828
5169
5510
5851
6191
6531
6871
340
128
7210
7549
7888
8227
8565
8903
9241
9579
9916
*0253
338
129
110590
0926
1263
1599
1934
2270
2605
2940
3275
3609
335
130
3943
4277
4611
4944
5278
5611
5943
6276
6(;08
6940
333
131
7271
7603
7934
8265
8595
8926
9256
9586
9915
*0245
330
132
120574
0903
1231
1560
1888
2216
2544
2871
3198
3525
328
133
3852
4178
4504
4830
5156
5481
6806
6131
6456
6781
325
134
7105
7429
7753
8076
8399
8722
9045
9368
9690
*0012
323
135
130334
0655
0977
1298
1619
1939
2260
2580
2900
3219
321
136
3539
3858
4177
4496
4814
5133
5451
5709
6086
6403
318
137
6721
7037
7354
7671
7987
8303
8618
8934
9249
9564
315
138
9879
*0194
•^0508
*0822
»1136
n450
*1763
*2076
*2389
*2702
314
139
143015
3327
3639
3951
4263
4574
4885
5196
5507
5818
311
No
0
1
2
3
4
5
6
7
8
9
D.
320
LOGARITHMS. — TABLE II.
No
0
146128
1
6438
2
6748
3
7058
4
7367
5
7676
6
7
8
9
D.
140
7985
8294
8603
8911
309
141
9219
9527
9835 *0142
*0449
*0756
*1063
*1370
*1676
*1982
307
142
152288
2594
2800
3205
3510
3815
4120
4424
4728
5032
305
143
5336
5640
5943
6246
6549
6852
7154
7457
7759
8C61
303
144
8362
8664
8965
9266
9567
9868
*0168
*0469
*0769
*1068
301
145
161368
1667
1967
2266
2564
2863
3161
3460
3758
4055
299
146
43r,3
4650
4947
5244
5541
6838
6134
6430
6726
7022
297
147
7317
7613
7908
8203
8497
8792
9686
9380
9674
9968
295
148
1702P2
0555
0848
1141
1434
1726
2019
2311
2603
2895
293
149
3186
3478
3769
4060
4351
4641
4932
5222
6512
58C2
291
150
176091
6381
6670
6959
7248
7536
7825
8113
8401
8689
289
151
8977
9264
9652
9839
*0126
*0413
*0699
*0985
*1272
*1558
287
152
181844
2129
2415
27t0
2985
3270
3555
3839
4123
4407
285
153
4691
4975
5259
5542
5825
6108
6391
6674
6956
7239
283
154
7521
7803
8(;84
8366
8647
8928
9209
9490
9771
*C051
281
155
190332
0612
0892
1171
1451
1730
2010
2289
2567
2846
279
156
3125
3403
3681
3959
4237
4514
4792
5069
6346
5623
278
157
5899
6176
6453
6729
7(05
7281
7^.56
7832
8107
8382
276
158
8657
8^*32
9206
9481
9755
*0029
*0303
*0577
*0850
*1124
274
159
201397
1670
1943
2216
2488
2761
3033
3305
3577
3848
272
160
4120
4391
4663
4934
5204
5475
5746
6016
6286
6556
271
161
6826
7096
7365
7634
7904
8173
8441
8710
8979
9247
269
162
9515
9783
*0051
*0319
*0586
*C853
ni2i
*1388
*1654
*1921
267
163
212188
24;-;4
2720
2986
3252
3518
3783
4049
4314
4579
266
164
4844
5109
5373
6638
5902
6166
6430
6694
6957
7221
264
165
7484
7747
8010
8273
8536
8798
9060
9323
9585
9846
262
166
220108
0370
0631
0892
1153
1414
1675
1936
2196
2456
261
167
2716
2976
3236
3496
3755
4015
4274
4533
4792
5051
259
168
5309
5568
5826
6084
6342
6600
6858
7115
7372
7630
258
169
7887
8144
8400
8657
8913
9170
9426
9682
9938
*0193
256
170
230449
0704
09C0
1215
1470
1724
1979
2234
2488
2742
254
171
2996
3250
3504
3757
4011
4264
4517
4770
5023
5276
253
172
5528
5781
6033
6:^85
6537
6789
7041
7292
7544
7795
252
173
8046
8297
85J8
8799
9049
9299
9550
9800
*0050
*0300
250
174
240549
1799
1048
1297
1546
1795
2044
2293
2541
2790
249
175
3038
3286
3534
3782
4030
4277
4525
4772
5019
5266
248
176
5513
5759
6CC6
6252
6499
6745
6991
7237
7482
7728
246
177
7973
8219
8464
8709
8954
9198
9443
9687
9932
*0176
245
178
250420
0664
0908
1151
1395
1638
1881
2125
2368
2610
243
179
2853
3096
3338
3580
3822
4064
4306
4548
4790
5031
242
No
O
1
2
3
4
6
6
7
8
9
D.
LOGARITHMS. — TABLE II.
321
Ko
0
1
2
3
4
6
6
7
8
9
D.
180
255273
5514
5755
5996
6237
6477
6718
6958
7198
7439
241
181
7679
7918
8158
8398
8637
8877
9116
9366
9594
9833
239
182
260071
0310
o:48
0787
1025
1263
1501
1739
1976
2214
238
183
2451
2688
2925
3162
3399
3636
3873
4109
4346
4582
237
184
4818
6054
5290
5525
6761
5996
6232
6467
6702
6937
235
185
7172
7406
7641
7875
8110
8344
8578
8812
9046
9279
234
186
9513
9746
9980 *0213 *0446
*0079
^0912
*1144
*1377
*1609
233
187
271842
2074
2306
2538
2770
3001
3233
3464
3696
3927
232
188
4158
4389
4620
4850
5081
5311
5542
5772
6002
6232
230
189
6462
6692
6921
7151
7380
7609
7838
8067
8296
8525
229
19C
8754
8982
9211
9439
9667
9895
*0123
*0351
*0578
*0806
228
191
281033
1261
1488
1716
1942
2169
2396
2622
2849
3075
227
192
3301
3527
3753
3979
4205
4431
4656
4882
5107
6332
226
193
5557
6782
6007
6232
6456
6681
6905
7130
7364
7578
225
194
7802
8026
8249
8473
8C96
8920
9143
9366
9589
9812
223
195
290035
0257
0480
C702
0925
1147
1369
1691
1813
2034
222
196
2256
2478
2699
2920
3141
3363
3584
3804
4025
4246
221
197
4466
4687
4907
5127
5347
5567
5787
6C07
6226
6446
220
198
6065
6884
7104
7323
7.^42
7761
7979
8198
8416
8635
219
199
8853
9071
9289
9:07
9726
9943
•^0161
*0378
*0595
*0813
218
200
301 0"0
1247
1464
1681
1898
2114
2331
2647
2764
2980
217
201
3196
3412
3628
3844
4069
4275
4491
4706
4921
5136
216
202
5351
5566
5781
5996
6211
6425
6639
6854
7068
7282
215
203
7496
7710
7924
8137
8351
8564
8778
8991
9204
9417
213
204
9030
9843 *0056
*0268
•^0481
*0693 *0906
nii8
*1330
*1542
212
205
311754
1966
2177
2389
2600
2812
3023
3234
3446
3656
211
206
3867
4078
4289
4499
4710
4920
5130
5340
5551
5760
210
207
5970
6180
6390
6599
6809
7018
7227
7436
7646
7854
209
208
8063
8272
8481
8689
8898
9106
9314
9522
9730
9938
208
209
320146
0354
0562
0769
0977
1184
1391
1598
1805
2012
207
210
2219
2426
2633
2839
3046
3252
3458
3665
3871
4077
206
211
4282
4488
4694
4899
5106
5310
5516
5721
5926
6131
205
212
6336
6541
6745
6950
7155
7359
7563
7767
7972
8176
204
213
8380
8583
8787
8991
9194
9398
9601
9805
*0008
*0211
203
214
330414 '
0617
0819
1022
1225
1427
1630
1832
2034
2236
202
215
2438
2640
2842
3044
3246
3447
3649
3850
4051
4253
202
216
4454
4665
4856
6057
5267
5453
5658
58" 9
6059
6260
201
217
6460
C600
6860
7060
7260
7459
7659
7858
8058
8257
200
218
8456
8656
8855
9054
9253
9451
9650
9849
*0047
*0246
199
239
340444
0642
0841
1039
1237
1435
1632
1830
2028
2226
198
No
0
1
2
3
4
5
6
7
8 .
9
I>.
322 LOGARITHMS. — TABLE II.
No
0
1
2
3
4
6
6
7
8
9
D.
220
342423
2620
2817
3014
3212
3409
3606
3802
3999
4196
197
221
4392
4589
4785
4981
5178
5374
5570
5766
5962
6157
196
222
6353
5549
6?44
6939
7135
7330
7525
7720
7915
8110
195
223
8305
8500
8694
8889
9083
9278
9472
9666
9860
*0054
194
224
350248
0442
0636
0829
1023
1216
1410
1603
1796
1989
193
225
2183
2375
2568
2761
2954
3147
3339
3532
3724
3916
193
226
4108
4301
4493
4685
4876
5068
5260
5452
5643
5834
192
227
6026
6217
6408
6599
6790
6981
7172
7363
7554
7744
191
228
7935
8125
8316
8506
8696
8886
9076
9266
9456
9646
190
229
9835
*0025
*0215
*0404
*0593
*0783
*0972
*1161
*1350
*1539
189
230
361728
1917
2105
2294
2482
2671
2859
3048
3236
3424
188
231
3612
3800
3988
4176
4363
4551
4739
4926
5113
5301
188
2d2
5488
5675
5862
6049
6236
6423
6610
6796
6983
7169
187
233
7356
7542
7729
7915
8101
8287
8473
8659
8845
9030
186
234
9216
9401
9587
9772
9958
*0143
*0328
*0513
*0698
*0883
185
235
371068
1253
1437
1622
1806
1991
2175
2360
2544
2728
184
236
2912
3096
3280
3464
3647
3831
4015
4198
4382
4565
184
237
4748
4932
5115
5298
5481
5664
5846
6029
6212
6394
183
238
6577
6759
6942
7124
7306
7488
7670
7852
80.^.4
8216
182
239
8398
8580
8761
8943
9124
9306
9487
9668
9849
*0030
181
240
380211
0392
0573
0754
0934
1115
1296
1476
1656
1837
181
241
2017
2197
2377
2557
2737
2917
3097
3277
3453
3636
180
242
3815
3995
4174
4353
4r)33
4712
4891
5070
5249
5428
179
243
5606
5785
5964
6142
6321
6499
6677
6856
7034
7212
178
244
7390
7568
7746
7923
8101
8279
8456
8634
8811
8989
178
245
9166
9343
9520
9698
9875
*0051
*0228
*0405
*0582
*0759
177
246 390935 1112 1288 1464 1641 1817 1993 2169 2345 2521
247! 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277
248 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025
249 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766
250
251
252
253
254
255
256
257
258
259
No
7940 8114 8287 8461 8634 8808 8981 9154 9328 9501
9674 9847*0020*0192*0365*0538*0711 *0883 *1056 *r228
401401 1573 1745 1917 2089 2261 2433 2605 2777 2949
3121 3292 3464 3635 3807 3978 4149 4320 4492 4663
4834 5005 5176 5346 5517 5688 5858 6029 6199 6370
6540 6710 6881 7051 7221 7391 7561 7731 7901 8070
8240 8410 8579 8749 8918 9087 9257 9426 9595 9764
9933 *0102 *0271 *U440 *0609 *0777 *0946 »1114 *1283 *1451
411620 1788 1956 2124 2293 2461 2629 2796 2964 3132
3300 3467 3635 3803 3970 4137 4305 4472 4639 4806
ARITHMETIC.
323
Ko
0
1
2
3
4
5
6
7
8
9
D.
260
414973
5140
5307
5474
5641
5808
5974
6141
6308
6474
167
261
6641
681.7
6973
7139
7306
7472
7638
7804
7970
8135
166
262
8301
8467
8633
8798
8964
6129
9295
9460
9625
9791
165
263
9956
*0121
*0286
*0451
*0616
*078L
*0945
*1110
*1275
*1439
165
264
421604
1768
1933
2097
2261
2426
2590
2754
2918
3082
164
265
3246
3410
3574
3737
3901
4065
4228
4392
4555
4718
164
266
4882
5045
5208
5371
5534
5697
5860
6023
6186
6349
163
267
6511
6674
6836
6999
7161
7324
7486
7648
7811
7973
162
268
8135
8297
8459
8621
8783
8944
9106
9268
9429
9591
162
269
9752
9914 *0075
*0236
*0398
*0559 *0720
*0881
*1042
*1203
,161
270
431364
1525
1685
1846
2007
21C7
2328
2488
2649
2809
161
271
2969
3130
3290
3450
3610
3770
3930
4090
4249
4409
360
272
4569
4729
4888
5048
5207
5367
5526
5685
5844
6004
159
273
6163
6322
6481
6640
6799
6957
7116
7275
7433
7592
159
274
7751
7909
8067
8226
8:^4
8542
8701
8859
9017
9175
158
275
9333
9491
9648
9806
9964 *0122
"0279
*0437
*0594
*0752
158
276
440909
1066
1224
1381
1538
1695
1852
2009
2166
2323
157
277
2480
2637
2793
2950
3106
3263
3419
3576
3732
3889
157
278
4045
4201
4357
4513
4669
4825
4981
5137
5293
5449
If 6
279
5604
5760
5915
6071
6226
6382
6537
6692
6848
7003
155
280
7158
7313
7468
7623
7778
7933
8088
8242
8397
8552
155
281
8706
8861
9j15
9170
9324
9478
9633
9787
9941
*0095
154
282
450249
0403
0557
0711
0865
1018
1172
1326
1479
1633
154
283
1786
1940
2093
2247
2400
25.^3
2706
2859
3012
3165
153
284
3318
3471
3624
3777
3930
4082
4235
4387
4540
4692
153
285
4845
4997
5150
5302
5454
5606
5758
5910
6062
6214
152
286
6366
6518
6670
6821
6973
7125
7276
7428
7579
7731
152
287
7882
8033
8184
8336
8487
8638
8789
8940
9091
9242
151
288
9392
9543
9694
9845
9995
*0146
*0296
*0447
*0597
*0743
151
289
460898
1048
1198
1348
1499
1649
1799
1948
2098
2248
150
290
2398
2548
2697
2847
2997
3146
3296
3445
3594
3744
150
291
3893
4042
4191
4340
4490
4639
4788
•1936
5085
5234
149
292
5383
5532
5680
5829
5977
6126
6274
6423
6571
6719
149
293
6868
7016
7164
7312
7460
7608
7756
7904
8052
8200
148
294
8347
8495
8643
8790
8938
9085
9233
9380
9527
9675
148
295
9822
9969
*0116
*0263 *0410
*0557
*0704
*0851
*0998
*1145
147
296
471292
1438
1585
1732
1878
2025
2171
2318
2464
2610
146
297
2756
2903
3049
3195
3341
3487
3633
3779
3925
4071
146
298
4216
4362
4508
4653
4799
4914
5090
5235
5381
5526
146
299
5671
5816
5962
6107
6252
6397
6542
6687
6832
6976
145
No
0
1
2
3
4
5
6
7
8
9*
D.
324
LOGARITHMS. — TABLE II.
No
0
1
2
3
4
5 6
7
8
9
D.
300
477121
7266
7411
7555
7700
7844 7989
8133
8278
8422
145
301
8566
8711
8855
8999
9143
9287 9431
9575
9719
9863
144
302
480007
0151
0294
0438
0582
0725 0869
1012
1156
1299
144
303
1443
1586
1729
1872
2016
2159 2302
2445
2588
2731
143
304
2874
3016
3159
3302
3445
3587 3730
3872
4015
4157
143
305
4300
4442
4585
4727
4869
5011 5153
5295
5437
5579
142
306
5721
5863
6005
6147
6289
6430 6572
6714
6855
6997
142
307
7138
7280
7421
7563
7704
7845 7986
8127
8269
8410
141
308
8551
8692
8833
8974
9114
9255 9396
9537
9677
9818
141
309
9958
*0099
*0239 *0380
*0520
^0661 *0801
*0941
*1081
*1222
140
310
491362
1502
1642
1782
1922
2062 2201
2341
2481
2621
140
311
2760
2900
3040
3179
3319
3458 3597
3737
3876
4015
139
312
4155
4294
4433
4572
4711
4850 49fc9
5128
5267
5406
139
313
5544
5683
5822
5960
6099
6238 6376
6515
6653
6791
139
314
6930
7068
7206
7344
7483
7621 7759
7897
8035
8173
138
315
8311
8448
8586
8724
8862
8999 9137
9275
9412
9550
138
316
9687
9824
9962
*0U99 *0236
'^0374 *0oll
*0648
*0785
*0922
137
317
501059
1196
1333
1470
1607
1744 1880
2017
2154'
2291
137
318
2427
2564
2700
2837
2973
3109 3246
3382
3518
3655
136
319
3791
3927
4063
4199
4335
4471 4607
4743
4878
5014
136
320
5150
5286
5421
5557
5693
5828 5964
6099
6234
6370
136
321
6505
6640
6776
6911
7046
7181 7316
7451
7586
7721
135
322
7856
7991
8126
8260
8395
8530 8664
8799
8934
9063
135
323
9203
9337
9471
9606
9740
9874 *00i)9
*0143
*0277
*0411
134
324
510545
0679
0813
0947
1081
1215 1349
1482
1616
1750
134
325
1883
2017
2151
2284
2418
2551 2684
2818
2951
3084
133
326
3218
3351
3484
3617
3750
3883 4016
4149
4282
4414
133
327
4548
4681
4813
4946
5079
5211 5344
5476
5609
5741
133
328
5874
6006
6139
6271
6403
6535 6668
6800
6932
7064
132
329
7196
7328
7460
7592
7724
7855 7987
8119
8251
8382
132
330
8514
8646
8777
8909
9040
9171 9303
9434
9566
9697
131
331
9828
9959
*0090
*0221
*U353 *0484 *0615
*0745
*0876
*1007
131
332
521138
1269
1400
1530
1661
1792 1922
2053
2183
2314
131
333
2144
2575
2705
2835
2966
3096 3226
3356
3486
3616
130
334
3746
3876
4006
4136
4266
4396 4526
4656
4785
4915
130
335
5045
5174
5304
5434
5563
5693 5822
5951
6081
6210
129
336
6339
6469
6598
6727
6856
6985 7114
7243
7372
7501
129
337
7630
7759
7888
8016
8145
8274 8402
8531
8660
8783
129
338
8917
9045
9174
9302
9430
9559 9687
9815
9943
*0072
128
339
530200
0328
0456
0584
0712
1840 0968
1096
1223
1351
128
No
O
1
2
3
4
5 6
7
8
9
D.
ARITHMETIC.
325
No
O
1
2
3
4
1990
5
2117
6
7
8
9
D.
MO
531479
1607
1734
1862
2245
2372
2500
2627
128
341
2754
2882
3009
3136
3204
3391
3518
3645
3772
3899
127
342
4026
4153
4280
4407
4034
4661
4787
4914
5041
5167
127
343
5294
5421
5547
5674
5800
5927
6053
6180
6306
6432
126
344
6558
6685
6811
6937
7063
7189
7315
7441
7567
7693
126
345
7819
7945
8071
8197
8322
8448
8574
8699
8825
8951
126
346
9076
92U2
9327
9462
9578
9703
9829
9954
*0079
■»0204
125
347
540329
0455
0580
0705
0830
0955
1080
1205
1330
1454
125
348
1579
1704
1829
1953
2078
2203
2327
2452
2576
2701
125
349
2820
2950
3074
3199
♦-:323
3447
3571
3696
3820
3944
124
350
4068
4192
4316
4440
4564
4688
4812
4936
5060
5183
124
351
5307
5431
5555
5078
5802
5925
6049
6172
6296
6419
124
352
6543
6666
6789
6913
7036
7159
7282
7405
7529
7652
123
353
7775
7898
8021
8144
8267
8389
8512
8635
8758
8^81
123
354
9003
9126
9249
9371
9494
9616
9739
9861
9984
0106
123
355
550228
0351
0473
0595
0717
0840
0962
1084
1206
1328
122
350
1450
1572
1694
1816
1938
2060
2181
2303
2425
2547
122
357
2668
2790
2911
3033
3155
3276
3398
3519
3640
3762
121
358
3883
4004
4126
4247
4368
4489
4610
4731
4852
4973
121
359
6094
5215
5336
5457
5578
5699
5820
5940
6061
6182
121
360
6303
6423
6544
6664
6785
6905
7026
7146
7267
7387
120
361
7507
7627
7748
7868
7988
8108
8228
8349
8469
8589
120
362
8709
8829
8948
9068
9188
9308
0428
9548
9667
9787
120
363
9907
*0026
*0146
*0265
*0385 *0504
*0624
*0743
*0863
*0982
119
364
561101
1221
1340
1459
1578
1698
1817
1936
2055
2174
119
365
2293
2412
2531
C350
270D
2887
300G
3125
3244
3362
119
366
3481
3600
3718
3837
3950
4074
4192
4311
4429
4548
119
367
4fJ66
4784
4903
5021
5130
5257
5376
5494
5612
5730
118
368
5848
5966
6084
6202
0320
6437
6555
6673
6791
6909
118
369
7026
7144
7262
7379
7407
7614
7732
7849
7967
8084
118
370
8202
8319
8436
S554
8G71
8788
8905
9023
9140
9257
117
371
9374
9491
9608
9725
9842
9959
*0076
*0193
*0309
*0426
117
372
570543
0660
0776
0893
1010
1126
1243
1309
1476
1592
117
373
1709
1825
1942
2058
2174
2291
2407
2523
2639
2755
116
374
2872
2988
3104
3220
3336
3402
3568
3684
3800
3015
116
375
4031
4147
4263
4379
4494
4610
4726
4841
4957
5072
116
376
5188
5303
5419
5534
5600
5765
5880
5996
6111
6226
115
377
6341
6457
0572
6687
6-!02
6917
7032
7147
7262
7377
115
378
7492
7607
7722
7836
7951
8066
8181
8295
8410
8525
115
379
8639
8754
8868
8983
9097
4
9212
9326
9441
9555
9669
114
No
0
1
2
3
5
6
7
8
9
D.
326
LOGARITHMS. — TABLE II.
No
O
1
9898
2
**12
3
•^0126
4
•^0241
6
•^0355
6
7
8
9
D.
380
579784
•^0469
*0583
*0697
*0811
114
381
580925
1039
1153
1267
1381
1495
1608
1722
1836
1950
114
382
2063
2177
2291
2404
2518
2631
2745
2858
2972
3085
114
383
3199
3312
3426
3539
3652
3765
3879
3992
4105
4218
113
394
4331
4444
4557
4670
4783
4896
5009
5122
5235
5348
113
385
5461
5574
5686
5799
5912
6024
6137
6250
6362
6475
113
286
6587
6700
6812
6925
7037
7149
7262
7374
7486
7599
112
387
7711
7823
7935
8047
8160
8272
8384
8496
8608
8720
112
388
8832
8944
9056
9107
9279
9391
9503
9615
9726
9838
112
389
9950
**61
*0173
*0284
*0396
^^0507
*0619
*0730
*0842
*0953
112
390
591065
1176
1287
1399
1510
1621
1732
1843
1955
2066
111
391
2177
2288
2399
2610
2621
2732
2843
2954
3064
3175
111
392
3286
3397
3508
3618
3729
3840
3950
4061
4171
4282
111
393
4393
4603
4614
4724
4834
4945
5055
5165
5276
5386
110
-394
5496
5606
5717
5827
5937
6047
6157
6267
6377
6487
110
395
6597
6707
6817
6927
7037
7146
7256
7366
7476
7586
110
396
7695
7805
7914
8024
8134
8243
8353
8462
8572
8681
110
397
8791
8900
9009
9119
9228
9337
9446
9556
9G65
9774
109
398
9883
9992 *0101 *0210
*0319
*0428
*0537
*0646
*0755
*0864
109
399
600973
1082
1191
1299
1408
1517
1625
1734
•1843
1951
109
400
2060
2169
2277
2386
2494
2603
2711
2819
2928
3036
108
401
3144
3253
3361
3469
3577
3686
3794
3902
4010
4118
108
-402
4226
4334
4442
4550
4658
4766
4874
4982
5089
5197
108
403
5305
5413
5521
5628
5736
5844
5951
6059
6166
6274
108
404
6381
6489
6596
6704
6811
6919
7026
7133
7241
7348
107
405
7455
7562
7669
7777
7884
7991
8098
8205
8312
8419
107
406
8526
8633
8740
8847
8954
9061
9167
9274
9381
9488
107
407
9594
9701
9808
9914
**21
*0128
*0234
*0341
*0447
*0554
107
408
610660
0767
0873
0979
1086
1192
1298
1405
1511
1617
106
409
1723
1829
1936
2042
2148
2254
2360
2466
2572
2678
106
410
2784
2890
2996
3102
3207
3313
3419
3525
3630
3736
106
411
3842
3947
4053
4159
4264
4370
4475
4581
4686
4792
106
412
4897
5003
5108
5213
5319
5424
5529
5634
5740
5845
105
413
5950
6055
6160
6265
6370
6476
6581
6686
6790
6895
105
414
7000
7105
7210
7315
7420
7525
7629
7734
7839
7943
105
415
8048
8153
8257
8362
8466
8571
8676
8780
8884
8989
105
416
9093
9198
9302
9406
9511
9615
9719
9824
9928
0032
104
417
620136
0240
0344
0448
0552
0656
0760
0864
0968
1072
104
418
1176
1280
1384
1488
1592
1695
1799
1903
2007
2110
104
419
2214
2318
2421
2
2525
2628
2732
2835
2939
3042
3146
104
No
0
1
3
4
6
6
7
8
9
D.
ARITHMETIC. 327
No
420 623249 3353 3456 3559 3663 3766 3869 3973 4076 4179
421 4282 4385 4488 4591 4695 4798 4901 5004 5107 5210
422 6312 5415 5518 5621 5724 5827 5929 6032 6135 6238
423 6340 6443 6546 6648 6751 6853 6956 7058 7161 7263
424 7366 7468 7571 7673 7775 7878 7980 8082 8185 8287
425 8389 8491 8593 8695 8797 8900 9002 9104 9206 9308
426 9410 9512 9613 9715 9817 9919 **21 *0123 *0224 *0326
427 630428 0530 0631 0733 0835 0936 1038 1139 1241 1342
428 1444 1545 1647 1748 1849 1951 2052 2153 2255 2366
429 2457 2559 2660 2761 2862 2963 3064 3165 3266 3367
430 3468 3569 3670 3771 3872 3973 4074 4175 4276 4376
431 4477 4578 4679 4779 4880 4981 5081 5182 5283 5383
432 5484 5584 5085 5785 588() 598<> 0087 6187 6287 6388
433 6488 6588 G688 6789 6889 6989 7089 7189 7290 7390
434 7490 7590 7690 7790 7890 7990 8090 8190 8290 8389
435 8489 8589 86S9 8789 8888 8988 9088 9188 9287 9387
436 9486 9586 mSG 9785 9885 9984 **84 *0183 *0283 *0382
437 640481 05S1 0680 0779 0879 0978 1077 1177 1276 1375
438 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366
2465 2563 2662 2761 2860 2959 3058 3156 3255 3354
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
No
4143
4242
4340
5127
5226
5324
6110
6208
6306
7089
7187
7285
8067
8165
8262
3453 3551 3650 3749 3847 3946 4044
4439 4537 4636 4734 4832 4931 5029
5422 5521 5619 5717 5815 5913 6011
6404 6502 6600 6698 6796 6894 6992
7383 7481 7579 7676 7774 7872 7969
8360 8458 8555 8653 8750 8848 8945 9043 9140 9237
9335 9432 9530 9627 9724 9821 9919 **16 *0113 *0210
650308 0405 0502 0599 0696 0793 0890 0987 1084 1181
1278 1375 1472 1569 1666 1762 1859 1956 2053 2150
2246 2343 2440 2536 2633 2730 2826 2923 3019 3116
3213 3309 3405 3502 3598 3695 3791 3888 3984 4080
4177 4273 4369 4465 4562 4658 4754 4850 4946 5042
5138 5235 5331 5427 5523 5619 5715 5810 5906 6002
6098 6194 6290 6386 6482 6577 6673 6769 6864 6S60
7056 7152 7247 7343 7438 7534 7629 7725 7820 7916
8011 8107 8202 8298 8393 8488 8584 8679 8774 8870
8965 9060 9155 925G 9346 9441 9536 . 9631 9726 9821
9916 **11 *0106 *0201 *0296 *0391 *0486 *0581 *0676 *0771
660865 0960 1055 1150 1245 1339 1434 1529 1623 1718
1813 1907 2002 2096 2191 2286 2380 2475 2569 2663
328
LOGARITHMS. — TABLE IL
No
0
1
2
3
4
5
6
7
8
9
460
662758
2852
2947
3041
3135
3230
3324
3418
3512
3607
94
461
37.»1
3795
3889
3983
4078
4172
4266
4360
4454
4548
94
462
4642
4736
4830
4924
5018
5112
5206
5299
5393
5487
94
46:}
5581
5675
5769
5862
5956
6050
6143
6237
6331
6424
94
464
6518
6612
6705
6799
6892
6986
7079
7173
7266
7360
94
465
7453
7546
7640
7733
7826
7920
8013
8106
8199
8293
93
466
8386
8479
8572
8665
8759
8852
8945
9038
9131
9224
93
467
9317
9410
9503
9596
9689
9782
9875
9967
**60
*0153
93
468
670246
0339
0431
0524
0617
0710
0802
0895
0988
1080
93
469
1173
1266
1358
1451
1543
1636
1728
1821
1913
2005
93
470
2098
2190
2283
2375
2467
2560
2652
2744
2836
2929
92
471
3021
3113
3205
3297
3390
3482
3574
3666
3758
3850
92
472
3942
4034
4126
4218
4310
4402
4494
4586
4677
4769
92
473
4861
4953
50-15
5137
5228
5320
5412
5503
5595
5687
92
474
5778
5870
5962
6053
6145
6236
6328
6419
6511
6602
9J
475
6694
6785
6876
6968
7059
7151
7242
7333
7424
7516
91
476
7607
7698
7789
7881
7972
8063
8154
8245
8336
8427
91
477
8518
8609
8700
8791
8882
8973
9064
9155
9246
9337
91
478
9428
9519
9610
9700
9791
9882
9973
**63
*0154
*0245
91
479
680336
0426
0517
0607
0698
0789
0879
0970
1060
1151
91
480
1241
1332
1422
1513
1603
1693
1784
1874
1964
2055
90
481
2145
2235
2326
2416
2506
2596
2686
2777
2867
2957
90
482
3047
3137
3227
3317
3407
3497
3587
3677
3767
3857
90
483
3947
4037
4127
4217
4307
4396
4486
4576
4666
4756
90
484
4845
4935
5025
5114
5204
6294
5383
5473
5563
5652
90
485
5742
5831
5921
6010
6100
6189
6279
6368
6458
6547
89
486
6636
6726
6815
6904
6994
7083
7172
7261
7351
7440
89
487
7529
7618
770r
7796
7886
7975
8064
8153
8242
8331
89
488
8420
8509
8598
8(j87
8776
8865
8953
9042
9131
9220
89
489
9309
9398
9486
9575
9664
9753
9841
9930
**19
*oior
89
490
690196
0285
0373
0462
0550
0639
0728
0816
0905
0993
89
491
1081
1170
1258
1347
1435
1524
1612
1706
1789
1877
88
49iJ
1965
2053
2142
2230
2318
2406
2494
2583
2671
2759
88
493
2847
2935
3023
3111
3199
3287
3375
3463
3551
3639
88
494
3727
3815
3903
3991
4078
4166
4254
4342
4430
4517
88
495
4605
4693
4781
4868
4956
5044
5131
5219
5307
5394
88
496
5482
5569
5«i57.
5744
5832
5919
6007
6094
6182
6269
87
497
6356
6444
6531
6618
6706
6793
6880
6968
7055
7142
87
498
7229
7317
7404
7491
7n78
7665
7752
7839
7926
8014
87
499
8101
8188
8275
8362
8449
.8535
8622
8709
8796
8883
87
No
O
1
2
3
4
5
6
7
8
9
D.
ARITHMETIC.
329
No
O
1
2
3
4
5
6
7
8
0
D.
50f
698970
9057
9144
9231
9317
9404
9i91
9578
9664
9751
87
501
1)838
9924
**11
**98
*0184
*0271
*0358
*0444
*0531
*0617
87
50:^
700704
0790
0877
0963
1050
1136
1222
1309
1395
1482
86
503
1568
1654
1741
1827
1913
1999
2086
2172
2258
2344
86
504
2431
2517
2603
2689
2775
2861
2947
3033
3119
3205
86
505
3291
3377
3463
3549
3635
3721
3807
3893
3979
4065
86
500
4151
423r>
4322
4408
4494
4579
4665
4751
4837
4922
86
507
5008
5094
5179
5265
5350
5436
5522
5607
5693
5778
80
508
5864
5949
6035
6120
6206
6291
6376
0462
6547
6632
85
509
6718
6803
6888
6974
7050
7144
7229
7315
7400
7485
85
510
7570
7655
7740
7826
7911
7996
8081
8166
8251
8336
85
511
8421
8506
8591
8676
8761
8846
8931
9015
9100
9185
85
512
9270
9355
9440
9524
9609
9694
9779
9863
9948
f033
85
513
710117
0202
0287
0371
0456
0540
0625
0710
0794
0879
85
514
0963
1048
1132
1217
1301
1385
1470
1554
1639
1723
84
515
1807
1892
1976
2060
2144
2229
2313
2397
2481
2565
84
516
2650
2734
2818
2902
2986
3<i70
3154
3238
3323
3407
84
517
3491
3575
3659
3742
3826
3910
3994
4078
4102
4246
84
518
4330
4414
4497
458 L
4665
4749
4833
4916
5000
5084
84
519
6167
5251
5335
5418
5502
5586
5669
5753
5836
5920
84
620
6003
6087
6170
6254
6337
6421
6504
6588
6671
6754
83
521
6838
6921
7004
7088
7171
7254
7338
7421
7504
7587
83
522
7671
7754
7837
7920
8003
8086
8169
8253
8336
8419
83
523
8502
8585
8668
8751
8834
8917
9000
9083
9165
9248
83
524
9331
9414
9497
9580
9663
9745
9828
9911
9994
**77
83
525
720159
0242
0325
0407
0490
0573
0655
0738
0821
0903
83
526
t98G
]068
1151
1233
1316
1398
1481
1563
1646
1728
82
527
1811
1893
1975
2058
2140
2222
2305
2387
2469
2552
82
528
2634
2716
2798
2881
2963
3045
3127
3209
3291
3.>74
82
529
3456
3538
3620
3702
3784
3866
3948
4030
4112
4194
82
530
4276
4358
4440
4522
4604
4685
4767
4849
4931
5013
82
531
5095
5176
5258
5340
5422
5503
5585
5667
5748
5830
82
532
5912
5993
6075
6156
6238
6320
6401
6483
6564
0646
82
533
6727
6809
6890
6972
7053
7134
7216
7297
7379
7460
81
634
7541
7623
7704
7785
7866
7948
8029
8110
8191
8273
81
635
8354
8435
8516
8597
8fi78
8759
8841
8922
9003
9084
81
636
9165
9246
9327
9408
9489
9570
9651
9732
9813
9893
81
537
9974
**55
'^0136
•^0217 *0298
"^0378
*0459
*0540
*0621
*0702
81
538
730782
0863
0944
1024
1105
1186
1266
1347
1428
1508
81
639
1589
1669
1750
1830
1911
1991
2072
2152
2233
2313
81
No
O
1
2
3
4
5
6
7
8
9
D.
330
LOGARITHMS. — TABLE II.
No
O
1
2
3
4
5
6
7
8
9
D.
510
732394
2474
2555
2635
2715
2796
2876
2956
3037
3117
80
541
3197
3278
3358
3438
3518
3598
3679
3759
3839
3919
80
542
3999
4079
4160
4240
4320
4400
4480
4560
4640
4720
80
543
4800
4880
4960
5040
5120
5200
5279
5359
5439
5519
80
544
5599
6679
5759
5838
5918
5998
6078
6157
6237
6317
80
545
6397
6476
6556
6635
6715
6795
6874
6954
7034
7113
80
546
7193
7272
7352
7431
7511
7590
7670
7749
7829
7908
79
547
7987
8067
8146
8225
8305
8384
8463
8543
8622
8701
79
548
8781
8860
8939
9018
9097
9177
9256
9335
9414
9493
79
549
9572
9651
9731
9810
9889
9968
**47
*0126
*0205
*0284
79
550
740363
0442
0521
0600
0678
0757
0836
0915
0994
1073
79
551
1152
1230
1309
1388
1467
1H6
1624
1703
1782
1860
79
552
1939
2018
2096
2175
2254
2332
2411
2489
2568
2647
79
553
2725
2804
2882
2961
3039
3118
3196
3275
3353
3431
78
554
3510
3588
3667
3745
3823
3902
3980
4058
4136
4215
78
555
4293
4371
4449
4528
4606
4684
4762
4840
4919
4997
78
556
5075
5153
5231
5309
5387
5465
5543
5621
5699
5777
78
557
5855
5933
6011
6089
6167
6245
6323
6401
6479
6556
78
558
6634
6712
6790
6868
6945
7023
7101
7179
7256
7334
78
559
7412
7489
7567
7645
7722
7800
7878
7955
8033
8110
78
560
8188
8266
8343
8421
8498
8576
8653
8731
8808
8885
77
561
8963
9040
9118
9195
9272
9350
9427
9504
9582
9659
77
562
9736
9814
9891
9968
**45
*0123
^^0200
*0277
*0354
*0431
77
563
750508
0586
0663
0740
0817
0894
0971
1U48
1125
1202
77
564
1279
1356
1433
1510
1587
1664
1741
1818
1895
1972
77
565
2048
2125
2202
2279
2356
2433
2509
2586
2663
2740
77
566
2816
2893
2970
3047
3123
3200
3277
3353
3430
3506
77
567
3583
3660
3736
3813
3889
3966
4042
4119
4195
4272
77
568
4348
4425
4501
4578
4654
4730
4807
4883
4960
5036
76
569
5112
5189
5265
5341
5417
5494
5570
5646
5722
5799
76
570
5875
5951
6027
6103
6180
6256
6332
6408
6484
6560
76
571
6636
6712
6788
6864
6940
7016
7092
7168
7244
7320
. 76
572
7396
7472
7548
7024
7700
7775
7851
7927
8003
8079
76
573
8155
8230
8306
8382
8458
8533
8609
8685
8761
8836
76
574
8912
8988
9063
9139
9214
9290
9366
9441
9517
9592
76
575
9668
9743
9819
9894
9970
**45 *0121
*0196
*0272
*0347
75
576
760422
0498
0573
0649
0724
0799
0875
0950
1025
1101
75
577
1176
1251
1326
1402
1477
1552
1627
1702
1778
18.3
75
578
1928
2003
2078
2153
2228
2303
2378
2453
2529
2604
75
579
2679
2754
2829
2904
2978
3053
3128
3203
3278
3353
75
No
0
1
2
3
4
5
6
7
8
9
D.
ARITHMETIO.
331
615
616
617
618
619
763428 3503 3578 3653 3727 3802 3877 3952 4027 4101
4176 4251 4326 4400 4475 4550 4624 4699 4774 4848
4923 4998 5072 5147 6221 5296 5370 5445 5520 5594
5669 5743 5818 5892 5966 6041 6115 6190 6264 6338
6413 6487 6562 6636 6710 6785 6859 6933 7007 7082
7156 7230 7304 7379 7453 7527 7601 7675 7749 7823
7898 7972 8046 8120 8194 8268 8342 8416 8490 8564
86S8 8712 8786 8860 8934 9008 9082 9156 9230 9303
9377 9451 9525 9599 9673 9746 9820 9894 9968 0042
770115 0189 0263 0336 0410 0484 0557 0631 0705 0778
0852 0926 0999 1073 1146 1220 1293 1367 1440 1514
1587 1661 1734 1808 1881 1955 2028 2102 2175 2248
2322 2395 2468 2542 2615 2688 2762 2835 2908 2981
3055 3128 3201 3274 3348 3421 3494 3567 3640 3713
3786 3860 3933 4006 4079 4152 4225 4298 4371 4444
4517 4590 4663 4736 4809 4882 4955 5028 5100 5173
5246 6319 5392 5465 5538 5610 5683 5756 5829 6902
5974 6047 6120 6193 6265 6338 6411 6183 6556 6629
6701 6774 6846 6919 6992 7064 7137 72fi9 7282 7354
7427 7499 7572 7644 7717 7789 7862 7934 8005 8079
8151 8224 8296 8368 8441 8513 8585 8658 8730 8802
8874 8947 9019 9091 9163 9236 9308 9380 9452 9524
9596 9669 9741 9813 9885 9957*0029 *0101 *0173 *0245
780317 0389 0461 0533 0605 0677 0749 0821 0893 -0965
1037 1109 1181 1253 1324 1396 1468 1540 1612 1684
1765 1827 1899 1971 2042 2114 2186 2258 2329 2401
2473 2544 2616 2688 2759 2831 2902 2974 3046 3117
3189 3260 3332 3403 3175 3546 3618 3689 3761 3832
3904 3975 4046 4118 4189 4261 4332 4403 4475 4546
4617 4689 4760 4831 4902 4974 6045 5116 5187 5259
5330 5401 6472 5543 5615 5686 5757 5828 5899 5970
6041 6112 6183 6254 6325 6396 6467 6538 6609 6680
6751 6822 6893 6964 7035 7106 7177 7248 7319 7390
7460 7531 7602 7673 7744 7815 7885 7956 8027 8098
8168 8239 8310 8381 8451 8522 8593 8663 8734 8804
8875 8946 9016 9087 9157 9228 9299 9369 9440 9510
9581 9651 9722 9792 9863 9933 ***4 **74 *0144 *0215
790285 0356 0426 0496 0567 0637 0707 0778 0848 0918
0988 1059 1129 1199 1269 1340 1410 1480 1550 1620
1691 1761 1831 1901 1971 2041 2111 2181 2252 2322
332 LOGARITHMS. — TABLE 11.
620
621
622
623
624
625
626
627
628
629
630
631
792392 2462 2532 2602 2672 2742 2812 2882 2952 3022
3092 3162 3231 3301 3371 3441 3511 3581 3651 3721
3790 3860 3930 4000 4070 4139 4209 4279 4349 4418
4488 4558 4627 4697 4767 4x36 4916 4976 5045 5115
5185 5254 5324 5393 5163 5532 5602 5672 5741 6811
5880 5949 6019 6088 6158 6227 6297 6366 6436 6505
6574 6644 6713 6782 6852 6921 6990 7060 7129 7198
7268 7337 7406 7475 7545 7614 7683 7752 7821 7890
7960 8029 8098 8167 8236 8305 8374 8443 8513 8582
8651 8720 8789 885^ 8i)27 8996 9065 9134 9203 9272
9341 9409 9478 9547 9616 9685 9754 9823 9892 9961
800029 0098 0167 0236 0305 0373 0442 0511 0580 0648
632 0717 0786 0851 0923 0992 1061 1129 1198 1266 1335
633
634
635
636
637
638
639
640
641
642
643
644
645
646
•947
1404 1472 1541 1609 1678 1747 1815 1884 1952 2021
2089 2168 2226 2295 2363 2432 2500 2568 2637 2705
2774 2842 2910 2979 3047 3116 3184 3252 3321 3389
3457 3525 3594 3662 3730 3798 3867 3935 4003 4071
4139 4208 4276 4344 4412 4480 4548 4616 4685 4753
4821 4889 4957 5025 5093 5161 5229 5297 5365 5433
5501 5569 5637 57u5 5773 5841 5908 5976 6044 6112
6180 6248 6316 6384 6451 6519 6587
6858 6926 6994 7061 7129 7197 7264
7535 7603 7670 7738 780^> 7873 7941
8211 8279 8346 8414 8481 8549 8616
8886 8953 9021 9088 9156 9223 9290
9560 9627 9694 9762 9829 9896 9964 **31 **98 *0165
810233 0300 0367 0434 Ooul 0569 0636 0703 0770 0837
0904 0971 1039 1106 1173 1240 1307 1374 1441 1508
1575 1642 1709 1776 1843 1910 1977 2044 2111 2178
6655
6723
6790
7332
7400
7467
8008
8076
8143
8684
8751
8818
9358
9425
9492
649( 2245 2312 2379 2445 2512 2579 2646 2713 2780 2847
653
651
652
653
654
655
656
657
658
859
2913 2980 3047 3114 3181 3247 3314 3381 3448 3514
3581 3648 3714 3781 3848 3914 3981 4048 4114 4181
4248 4314 4381 4447 4514 4581 4«47 4714 4780 4847
4913 4980 5 146 5113 5179 5246 5312 5378 5445 5511
5578 -5644 5711 5777 5843 5910 5976 6042 6109 6175
6241 6308 6374 6440 6506 6573 6639 67C5 6771 6838
6904 6970 7036 7102 7169 7235 7301 7367 7433 7499
7565 7631 7698 77G4 7830 7806 7962 8028 8094 8160
8226 8292 8358 8424 8490 8556 8622 8688 8754 8820
8885 8951 9017 9083 9149 9215 9281 9346 9412 9478
Nol O 12 3 4 5 6
ARITHMETIC.
333
No
O
1
2
3
9741
4 5
6
7
8
9
D.
660
819544
9610
9676
9807 9873
9939
***4
**70
*0136
66
661
820201
0267
0333
0399
0464 0530
0595
0661
0727
0792
66
662
0858
0924
0989
1055
1120 1186
1251
1317
1382
1448
66
663
1514
1579
1645
1710
1775 1841
1906
1972
2037
2103
65
664
2168
2233
2299
23(54
2430 2495
2560
2626
2691
2756
65
665
2822
2887
2952
3018
3083 3148
3213
3279
3344
3409
65
666
3474
3539
3605
3670
3735 3800
3865
3930
3996
4061
65
667
4126
4191
4256
4321
4386 4451
4516
4581
4646
4711
65
668
4776
4841
4906
497 L
5036 5101
5166
5231
5296
5361
65
66^
6426
5491
5556
5621
5686 5751
5815
5880
5945
6010
65
670
6075
6140
6204
6269
6334 6399
6464
6528
6593
6658
65
671
6723
6787
6852
6917
6981 7046
7111
7175
7240
7365
65
672
7369
7434
7499
7563
7628 7692
7757
7821
7886
7951
65
673
8015
8080
8144
8209
8273 8333
8402
8467
8531
8595
64
674
8660
8724
8789
8853
8918 8982
9046
9111
9175
9239
64
675
9304
9368
9432
9497
9561 9625
9690
9754
9818
9882
64
676
9947
**11
**75
*0139
■^0204 *0268 *0332
*0396
*0460
*0525
64
677
830589
0653
0717
0781
0845 0909
0973
1037
1102
1166
64
67«
1230
1294
1358
1422
1486 1550
1614
1678
1742
1806
64
679
1870
1934
1998
2062
2126 2189
2253
2317
2381
2445
.64
680
2509
2573
2637
2700
2764 2828
2892
2956
3020
3083
64
681
3147
3211
3275
3338
3402 3466
353)
3593
3657
3721
64
662
3784
3818
3912
3975
4039 4103
4166
4230
4294
4357
64
683
4421
4484
4548
4611
4675 4739
4802
4866
4929
4993
64
684
5056
5120
5183
5247
5310 5373
5437
5500
5564
6627
63
685
5691
5754
5817
6881
5944 6007
6071
6134
6197
6261
63
68G
6324
6387
6451
65.4
6577 6641
6704
6767
6830
6894
63
687
6957
7020
7083
7146
7210 7273
7336
7399
7462
7525
63
688
7588
7652
7715
7778
7841 7904
7967
8030
8093
8156
63
689
8219
8282
8345
8408
8471 8534
8597
8660
8723
8786
6S
690
8849
8912
8975
9038
9101 9164
9227
9289
9352
9415
63
691
9478
9541
9604
9667
9729 9792
9855
9918
9981
**43
63
692
840106
0169
0232
0294
0357 0420
0482
0545
0608
0671
63
693
0733
0796
08o9
0921
0984 1046
1109
1172
1234
1297
63
694
1359
1422
1485
1547
1610 1672
1735
1797
1860
1922
63
695
1985
2047
2110
2172
2235 2297
2360
2422
2484
2547
62
696
2609
2672
2734
2796
2859 2921
2983
3046
3108
3170
62
697
3233
3295
3357
3420
3482 3544
3606
3669
3731
3793
62
698
3855
3918
3980
4042
4104 4166
4229
4291
4353
4415
62
699
4477
4539
4601
4664
4726 4783
4^50
4912
4974
5036
62
No
O
1
2
3
4 5
6
7
8
9
n
334 LOGARITHMS. — ^TABLE II.
No
O
1
2
52 .'2
3
5284
4
5346
6
6
7
8
9
D.
700
845098
5160
5408
5470
5532
5594
6656
62
701
5718
5780
5842
5904
5966
6028
6090
6151
6213
6275
62
702
6337
6399
6461
6523
6585
6646
6708
6770
6832
6894
62
703
6955
7017
7079
7141
7202
7264
7326
7388
7449
7511
62
704
7573
7634
7696
7758
7819
7881
7943
8004
8066
8128
62
705
8189
8251
8312
8374
8435
8497
8559
8620
8682
8743
62
706
8805
8866
8928
8989
9051
9112
9174
9235
9297
9358
61
707
9419
9481
9542
9604
9665
9726
9788
9849
9911
9972
61
708
850033
0095
0156
0217
0279
0340
0401
0462
0524
0585
61
709
0646
0707
0769
0830
0891
0952
1014
1075
1136
1197
61
710
1258
1320
1381
1442
1503
1564
1625
1686
1747
1809
61
711
1870
1931
1992
2053
2114
2175
2236
2297
2358
2419
61
712
2480
2541
2602
2663
2724
2785
2846
2907
2968
3029
61
713
3090
3150
3211
3272
3333
3394
3455
3516
3577
3637
61
714
3698
3759
3820
3881
3941
4002
4063
4124
4185
4245
61
715
4306
4367
4428
4488
4549
4610
4670
4731
4792
4852
61
716
4913
4974
5034
5095
5156
5216
5277
5337
5398
5459
61
717
5519
5580
5640
5701
5761
5822
5882
5943
6003
6064
61
718
6124
6185
6245
6306
6366
6427
6487
6548
6608
6668
60
719
. 6729
6789
6850
6910
6970
7031
7091
7152
7212
7272
60
720
7332
7393
7453
7513
7574
7634
7694
7755
7815
7875
60
721
7935
7995
8056
8116
8176
8236
8297
8357
8417
8477
60
722
8537
8597
8657
8718
8778
8838
8898
8958
9018
9078
60
723
9138
9198
9258
9318
9379
9439
9499
9559
9619
9579
60
724
9739
9799
9859
9918
9978
**38
**98
*0158
*0218
*0278
60
725
860338
0398
0458
0518
0578
0637
0697
0757
0817
0877
60
726
0937
0996
1056
1116
1176
1236
1295
1355
1415
1475
60
727
1534
1594
1654
1714
1773
1833
1893
1952
2012
2072
60
728
2131
2191
2251
2310
2370
2430
2489
2549
2608
2668
60
729
2728
2787
2847
2906
2966
3025
3085
3144
3204
3263
60
730
3323
3382
3442
3501
3561
3620
3680
3739
3799
3858
59
731
3917
3977
4036
41/96
4155
4214
4274
4333
4392
4452
59
732
4511
4570
4630
4689
4748
4808
4867
4926
4985
5045
69
733
5104
5163
5222
5282
5341
5400
5459
5519
5578
5637
69
734
5696
6755
5814
5874
5933
5992
6051
6110
6169
6228
59
735
6287
6346
6405
6465
6524
6583
6642
6701
6760
6819
59
736
6878
6937
6996
7055
7114
7173
7232
7291
7350
7409
59
737
7467
752.)
7585
7644
7703
7762
7821
7880
7939
7998
69
738
8056
8115
8174
8233
8292
8350
8409
8168
8527
8586
59
739
8644^
8703
8762
8821
8879
8938
8997
S056
9114
9173
59
No
O
1
2
3
4
6
6
7
8
9
D.
ARITHMETIC.
335
No
O
1
2
3
4
6
6
7
8
9
D.
740
869232
9290
9349
9408
9466
9525
9584
9642
9701
9760
59
741
9818
9877
9935
9994
**53
'^Olll
*0170
*0228
*0287
*0345
59
742
870404
0462
0521
0579
0638
0696
0755
0813
0872
0930
58
743
0989
1047
1106
1164
1223
1281
1339
1398
1456
1515
58
744
1573
1631
1690
1748
1806
1865
1923
1981
2040
2098
68
745
2156
2215
2273
2331
23^9
2448
2506
2564
2622
2681
68
746
2739
2797
2855
2913
2972
3030
3088
3146
3204
3262
58
747
3321
3379
3437
3195
3553
3611
3669
3727
3785
3844
58
748
3902
3960
4018
4076
4134
4192
4250
4308
4366
4424
58
749
4482
4540
4598
4656
4714
4772
4830
4888
4945
5003
58
750
5061
5119
5177
6235
5293
5351
5409
5466
5524
5582
58
751
5640
5698
5756
5813
5871
5929
5987
6045
6102
6160
58
752
6218
6276
6333
6391
6449
6507
6564
6622
6680
6737
58
753
6795
6853
6910
6968
7026
7083
7141
7199
7256
7314
58
754
7371
7429
7487
7644
7602
7659
7717
7774
7832
7889
58
755
7947
8004
8062
8119
8177
8234
8292
.8349
8407
8464
67
756
8522
8579
8637
8694
8752
8809
8866
8924
8981
9039
67
757
9096
9153
9211
9208
9325
9383
9440
9497
9555
9612
57
758
9669
9726
9784
9841
9898
9956
**13
**70
*0127
*0185
57
759
880242
0299
03&6
0413
0171
0528
0585
0642
0699
0756
67
760
0814
0871
0928
0985
1042
1099
1156
1213
1271
1328
57
761
1385
1442
1499
1556
1613
1670
1727
1784
1841
1898
67
762
1955
2012
2069
2126
2183
2240
2297
2354
2411
246S
67
763
2525
2581
2638
2695
2752
28u9
2866
2923
2980
3037
67
764
3093
3150
3207
3264
3321
3377
3434
3491
3548
3605
67
765
3661
3718
3775
3832
3888
3945
4002
4059
4115
4172
67
766
4229
4285
4342
4399
4455
4512
4569
4625
4682
4739
67
767
4795
4852
49U9
4965
5022
5078
5135
5193
5248
5305
67
768
5361
5418
5474
5531
55h7
5644
5700
5757
5813
5870
57
769
6926
5983
6039
6096
6152
6209
6265
6321
6378
6434
66
770
6491
6547
6604
6660
6716
6773
6829
6885
6942
6998
66
771
7054
7111
7167
7223
7280
7336
7392
7449
7505
7561
66
772
7617
7674
7730
7786
7842
7898
7955
8011
8067
8123
66
773
8179
8236
8292
8348
8404
8460
8516-
8573
8629
8685
66
774
8741
8797
8853
8909
8965
9021
9077
9134
9190
9246
56
775
9302
9358
9414
9470
9526
9582
9638
9694
9750
9806
66
776
9862
9918
9974
**30
**86
*0141
^^0197
*0253
*0309
*0365
66
777
890421
0477
0533
0589
0645
0700
0756
0812
0868
0924
56
778
0980
1035
1091
1147
1203
1259
1314
1370
1426
1482
66
779
1537
1593
1649
1705
1760
1816
1872
1928
1983
2039
66
No
0
1
2
3
4
5
6
7
8
9
D.
336 LOGARITHMS. — TABLE II.
780
781
782
788
784
785
786
787
788
789
790
791
792
793
794
796
797
798
799
800
801
802
803
b04
892095 2150 2206 2202 2317 2373 2429 2484 2540 2595
2651 2707 2762 2818 2873 2929 2985 3040 3096 3151
3207 3262 3318 3373 3429 3484 3540 3595 3651 3706
3762 3817 3873 3928 3984 4039 4094 4150 4205 426L
4316 4371 4427 4482 4538 4593 4648 4704 4759 4814
4870 4925 498J 5036 5091 5146 5201 5257 5312 5367
5423 5178 5533 5588 5644 5699 5754 5809 5864 5920
5975 6030 6085 6140 6195 6251 6306 6361 6416 6471
6526 6581 6636 6692 6747 6802 6857 6912 6967 7022
70/7 7132 7187 7242 7297 7352 7407 7462 7517 7572
7627 7682 7737 7792 7847 7902 7957 8012 8067 8122
8176 8231 8286 8341 8396 8451 8506 8561 8615 8670
8725 8780 8835 8890 8914 8999 9054 9109 9164 9218
9273 9328 9383 9437 9492 9547 9602 9656 9711 9766
9821 9875 9930 9985 **39 **94*0149 *0203 *0258 *0312
795 900367 0422 0476 0531 0586 0640 0695 0749 0804 0859
0913 0968 1022 1077 1131 1186 1240 1295 1349 1404
1458 1513 1567 1622 1676 1731 1785 1840 1894 1948
2003 2057 2112 2166 2221 2275 2329 2384 2438 2492
2547 2601 2655 2710 2764 2818 2873 2927 2981 3036
3090 3144 3199 3253 3307 3361 3416 3470 3524 3578
3633 3687 3741 3795 3849 8904 3958 4012 4066 4120
4174 4229 4283 4337 4391 4445 4499 4553 4607 4661
4716 4770 4824 4878 4932 4986 5040 5094 5148 5202
6256 5310 5364 5418 5472 5526 5580 5634 5688 5742
805
5796
5850
5904
5958
6012
6066
6119
6173
6227
6281
54
806
6335
6389
6443
6497
6551
6604
6658
6712
6766
6820
54
807
6874
6927
6981
7035
7089
7143
7196
72»0
7304
735S
54
808
7411
7465
7519
7573
7626
7680
7731
7787
7841
7895
54
809
7949
8002
8056
8110
8163
8217
8270
8324
8378
8431
54
810
8485
8539
8592
8646
8699
8753
8807
8860
8914
8967
54
«11
9021
9074
9128
9U1
9235
9289
9342
9396
9449
9503
54
81^
9556
9610
9663
9716
9770
9823 ^877
9930
9984
**37
53
813
910091
0144
0197
0251
om
0358
0411
0464
0518
0571
53
814
0624
0678
0731
0784
0838
0891
0944
0998
1051
1104
53
815
1158
1211
1264
1317
1371
1424
1477
1530
1584
1637
53
816
1690
1743
1797
1850
1903
1956
2009
2063
2116
2169
53
817
2222
2275
2328
2381
2435
2488
2541
2594
2647
2700
53
818
2753
2806
2859
2913
2966
3019
3072
3125
3178
3231
53
819
3284
3337
3390
3143
3496
3549
3602
3655
3708
3761
53
No
O
1
2
3
4
5
6
7
8
9
D.
■
H
■
r
—J
tllTHMETIC.
w
337
No
o
1
2
3
4
6
6
7
8
9
D.
820
913814
3867
3920
3973
4026
4079
4132
4184
4237
4290
53
821
4343
4396
4449
4502
4555
4608
4660
4713
4766
4819
53
822
4872
4925
4977
5030
5083
5136
5189
5241
5294
6347
53
823
5400
54 i3
5505
5558
5611
5664
5716
5769
5822
6875
53
824
5927
6980
6033
6U85
6J38
6191
6243
6296
6349
6401
53
825
6454
6507
6559
6612
6664
6717
6770
6822
6875
6927
53
826
6980
7033
7085
7138
7190
7243
7295
7348
7400
7453
53
827
7506
7558
7611
7663
7710
7768
7820
7873
7925
7978
52
828
8030
8083
8135
8188
8240
8293
8345
8397
8450
8502
52
829
8555
8607
8659
8712
8764
8816
8869
8921
8973
9026
52
830
9078
9130
9183
9235
9287
9340
9392
9444
9496
9549
62
831
9601
9653
9706
9758
9810
9862
9914
9967
**19
**71
52
832
920123
0176
0228
0280
0332
0384
0436
0489
0541
0593
52
833
0615
0697
0749
0801
0853
0906
0958
1010
1062
1114
52
834
1166
1218
1270
1322
1374
1426
1478
1530
1582
1634
52
835
1686
1738
1790
1842
1894
1946
1998
2050
2102
2154
52
836
2206
2258
2310
2362
2414
2466
2518
2570
2622
2674
62
837
2725
2777
2829
2881
2933
2985
3037
3089
3140
3192
52
838
3244
3296
3348
3399
3451
3503
3555
3607
3658
3710
52
839
3762
3814
3865
3917
3969
4021
4072
4124
4176
4228
52
840
4279
4331
4383
4434
4486
4538
4589
4641
4693
4744
52
841
4796
4848
4899
4951
50O3
5054
5106
5157
5209
5261
52
842
5312
5364
5415
5467
5518
5570
5621
5673
5725
5776
62
843
5828
5879
5931
5982
6034
6085
6137
6188
6240
6291
61
844
6342
6394
6445
6497
6548
6600
6651
6702
6754
6805
51
845
6857
6908
6959
7011
7062
7114
7165
7216
7268
7319
51
846
7370
7422
7473
7524
7576
7627
7678
7730
7781
7*832
61
847
7883
793 >
7986
8037
8088
8140
8191
8242
8293
8345
61
848
8396
8447
8498
8549
8601
8652
8703
8754
8805
8857
61
849
8908
8959
9010
9061
9112
9163
9215
9266
9317
9368
51
850
9419
9470
9521
9572
9623
9674
9725
9776
9827
9879
51
851
9930
9981
**32
**83
^0134
*0185 *0236
*0287
*0338
*0389
51
852
930440
0491
0542
0592
0643
0694
0745
0796
0847
0898
61
853
0949
1000
1051
1102
1153
1204
1254
1305
1356
1407
51
854
1458
1509
1560
1610
1661
1712
1763
1814
1865
1915
51
855
1966
2017
2068
2118
2169
2220
2271
2322
2372
2423
51
856
2474
2524
2575
2626
26^7
2727
2778
2829
2879
2930
61
857
2981
3031
3082
3133
3183
3234
3285
3335
3386
3437
61
iJo8
3487
3538
3589
3S39
3690
3740
3791
3841
3892
3943
51
859
3993
4044
4094
4145
4195
4246
4296
4347
4397
4448
51
No
0
1
2
3
4
6
6
7
8
9
D.
338
LOGARITHMS. — TABLE II.
No
O
1
2
3
4
6
6
7
8
9
D.
860
934498
4549
4599
4610
4700
4751
4801
4852
4902
4953
50
861
5003
5054
5104
5154
5205
5255
5306
5356
5406
5157
50
862
5507
5558
5r)08
5658
5709
5759
5809
5860
5910
5960
50
863
6011
6061
6111
6162
6212
6262
6313
6363
6413
6463
50
864
6514
6564
6614
6665
6715
6765
6815
6865
6916
6966
50
865
7016
7066
7117
7167
7217
7267
7317
7367
7418
7468
50
866
7518
7568
7618
7668
7718
7769
7819
7869
7919
7969
50
867
.8019
8069
8119
8169
8219
8269
8320
8370
8420
8470
50
868
8520
8570
8rt20
8670
8720
8770
8820
8870
8920
8970
50
869
9(»20
9070
9120
9170
9220
9270
9320
9369
9419
9469
50
870
9519
9509
9619
9669
9719
9769
9819
9869
9918
9968
50
871
940018
0068
0118
0168
0218
0267
0317
0367
0U7
04fi7
50
872
0516
0566
0616
0666
0716
0765
0815
0865
0915
0964
50
873
1014
1064
1114
1163
1213
1263
1313
1362
1412
1462
50
874
1511
1561
1611
1660
1710
1760
1809
1859
1909
1958
50
875
2008
2058
2107
2157
2207
2256
2306
2355
2405
2455
50
876
2504
2554
2603
2653
2702
2752
2801
2851
2901
2950
50
877
3000
3049
3099
3148
3 198
3247
3297
3346
3396
3445
49
878
3495
3514
3593
3643
3692
3742
3791
3841
3890
3939
49
879
3989
40.J8
4088
41J7
4186
4236
4285
4335
4384
4433
49
880
4483
4532
4581
4631
4680
4729
4779
4828
4877
4927
49
88.1
4976
5025
5074
5124
5173
5222
5272
5321
5370
5419
49
882
5469
5518
5567
5616
5665
5715
5764
5813
58b2
5912
49
883
5961
6010
6059
6108
6157
6207
6256
6305
6354
6403
49
884
6452
6501
6551
6600
6649
6698
6747
6796
6845
6894
49
885
6943
6992
7041
7090
7140
7189
7238
7287
7336
7385
49
886
7434
74u3
7532
7581
7630
76?9
7728
7777
7626
7875
49
887
7924
7973
8022
8070
8119
8168
8217
8266
8315
8364
49
888
8413
8462
8511
8560
8609
8657
8706
8755
8804
8853
49
889
8902
8951
8999
9048
9097
9146
9195
9244
9292
9341
49
890
9390
9439
9488
9536
95^5
9634
9683
9731
9780
9829
49
891
9878
9926
9975
**24
**73
•^0121 *017()
*0219
*0267
*0316
49
892
9503u5
0414
0462
0511
0560
0608
0637
0706
0754
0803
49
893
0851
0900
C949
0997
1046
1095
1L43
1192
1240
1289
49
894
1338
1386
1435
1483
1532
1580
1629
1677
1726
1775
49
895
1823
1872
1920
1969
2017
2066
2114
2163
2211
2260
48
896
2308
2356
2405
2453
2502
2550
2599
2647
2696
2744
48
897
2792
2841
2889
2038
2986
3034
3083
31:^1
31fcO
3228
48
898
3276
3325
3373
3421
3470
3518
3566
3615
3663
3711
48
899
3760
b8L8
3856
3905
3953
4001
4049
4098
4146
4194
48
No
O
1
2
3
4
5
6
7
8
9
D.
]
■
t
ARITHMETIC.
339
No
0
1
2
3
4
6 6
7
8
9
D.
900
954243
4291
4339
4387
4435
4484 4532
4580
4628
4677
48
901
4725
4773
4821
4869
4918
4966 5014
5062
5110
5158
48
902
5207
5255
5303
5351
5399
5447 5495
5543.
5592
5640
48
903
5688
5736
5784
5832
5880
5928 5976
6024
6072
6120
48
904
6168
6216
6265
6313
6361
6409 6457
6505
6553
6601
48
905
6649
6697
6745
6793
6840
6888 6936
6984
7032
7080
48
906
7128
7176
7224
7272
7320
7368 7416
7464
7512
7559
48
907
7607
7655
7703
7751
7799
7847 7894
7942
7990
8038
48
908
8086
8134
8181
8229
8277
8325 8373
8421
8468
8516
48
909
8564
8612
8659
8707
8755
8803 8850
8898
8946
8994
48
910
9041
9089
9137
9185
9232
9280 9328
9375
9423
9471
48
911
9518
9566
9614
9661
9709
9757 9864
9852
9900
9947
48
912
9995
**42
**90
*0138
*0185
*0233 *0280
*0b28
*0376
*0423
48
918
960471
0518
0566
0613
0661
0709 (.756
0804
0851
0899
4S
914
0946
0994
1041
1089
1136
1184 1231
1279
1326
1374
47
915
1421
1469
1516
1563
1611
1658 1706
1753
1801
1848
47
916
1895
1943
1990
2038
2085
2132 2180
2227
2275
2322
47
917
2369
2417
2464
2511
2559
2606 2653
2701
2748
271'5
47
918
2843
2890
2937
2985
3032
3079 3126
3174
3221
3268
47
919
3316
3363
3410
3457
3504
3552 3599
3646
3693
3741
47
920
3788
3835
3882
3929
3977
4024 4071
4118
4165
4212
47
921
4260
4307
4354
4401
4448
4495 4542
4590
4637
4684
47
922
4731
4778
4825
4872
4919
4966 5013
5061
5108
5155
47
923
5202
5249
5296
5343
5390
5437 5484
5531
5578
5625
47
924
5672
5719
5766
5813
6860
5907 5954
6001
6048
6095
47
925
6142
6189
6236
6283
6329
6376 6423
6470
6517
6564
47
926
6611
6658
6705
6752
6799
6845 6892
6939
6986
7033
47
927
7080
7127
7173
7220
7267
7314 7361
7408
7454
7501
47
928
7548
7595
7642
7688
7735
7782 7829
7875
7922
7969
47
929
8016
8062
8109
8156
8203
8249 8296
8343
8390
8436
47
930
8483
8530
8576
8623
8670
8716 8763
8810
8856
8903
47
931
8950
8996
9043
9090
9136
9183 9229
9276
9323
9369
47
932
9416
9463
9509
9556
9t;02
9649 9695
9742
9^89
9835
47
933
9882
9928
9975
**21
**o8-
^0114 *0161
*0207
*0254
*0300
47
934
970347
0393
0440
0486
0533
0579 0626
0672
0719
0765
46
935
0812
0858
0904
0951
0997
1044 1090
1137
1183
1229
46
936
1276
1322
1369
1415
1481
1508 1554
1601
1647
1693
46
937
1740
1786
1832
1879
19^5
1971 2018
2064
2110
2157
46
938
2203
2249
2295
2342
2338
2434 2481
2527
2573
2619
46
939
2666
2712
2758
2804
2851
2897 2943
2989
3035
3082
46
No
0
1
2
3
4
5 6
7
8
9
D.
340
LOGARITHMS — TABLE IL
No
0
1
2
3
4
3313
5
3359
6
7
8
9
3543
D.
940
973128
3174
3220
3266
3405
3451
3497
46
941
3590
3636
3682
3728
3774
3820
3866
3913
3959
4005
46
942
4051
4097
4143
4189
4235
4281
4327
4374
4420
4466
46
94S
4512
4558
4604
4650
4696
4742
4788
4834
4880
49:^6
46
944
4972
6018
5064
5110
5156
5202
5248
5294
6340
5386
46
945
5432
5478
5524
6570
6616
5662
5707
5763
5799
5845
46
946
6891
5937
5983
6029
6075
6121
6167
6212
6258
6304
46
947
6350
6396
6442
6488
6533
6579
6625
6671
6717
6763
46
948
6808
6854
6900
6946
6992
7037
7083
7129
7175
7220
46
949
7266
7312
7358
7403
7449
7495
7541
7586
7632
7678
46
950
7724
7769
7815 7861
7906
7952
7998
8043
8089
8136
46
951
8181
8226
8272
8317
8363
8409
8454
8500
8546
8591
46
952
8637
8683
8728
8774
8819
8865
8911
8956
9002
9047
46
953
9093
9138
9184
9230
9275
9321
9366
9412
9457
9503
46
954
9548
9594
9639
9685
9730
9776
9821
9867
9912
9958
46
955
980003
0049
0094
0140
0185
0231
0276
0322
0367
0412
45
957
0458
0503
0549
0594
0640
0685
0730
0776
0821
0867
45
957
0912
0957
1003
10^8
1093
1139
1184
1229
1275
1320
45
958
1366
1411
1456
loOl
1547
1592
1637
1683
1728
1773
45
959
1819
1864
1909
1954
2000
2045
2090
2136
2181
2226
45
960
2271
2316
2362
2407
2452
2497
2543
2688
2633
2678
46
961
2723
2769
2814
2859
2904
2949
2994
3040
3086
3130
45
962
3175
3220
3265
b310
3356
3401
3446
3491
3536
3581
45
963
3626
8671
3716
3762
3807
3852
3897
3942
3987
4032
45
964
4077
4122
4167
4212
4257
4302
4347
4392
4437
4482
45
965
4527
4672
4617
4662
4707
4762
4797
4842
4887
4932
45
966
4977
5022
6067
5112
5157
6202
5247
6292
5337
5382
46
967
5426
5471
6516
5561
5606
5651
5696
5741
6786
5830
45
968
5875
5920
5966
6010
6055
6100
6144
6189
6234
6279
45
969
6324
6369
6413
6458
6503
6548
6593
6637
6682
6727
45
970
6772
6817
6861
6906
6951
6996
7040
7085
7130
7175
45
971
7219
7264
7309
7353
7398
7443
7488
7532
7577
7622
45
972
7666
7711
7756
7800
7846
7890
7934
7979
8024
8068
45
973
8113
8157
8202
8247
8291
8336
8381
8425
8470
8514
45
974
8559
8604
8648
8693
8737
8782
8826
8871
8916
8960
46
975
9005
9049
9094
9138
9183
9227
9272
9316
9361
9405
46
976
9450
9494
9539
9583
9628
9672
9717
9761
9806
9850
44
977
9895
9939
9983
**28
**72
^0117 *0161
*0206
*02n0
*0294
44
978
990339
0383
0428
0472
0516
0561
0605
0650
0694
0738
44
979
0783
0827
0871
0916
0960
1004
1049
1093
1137
1182
44
No
0
1
2
3
4
5
6
7
8
9
D.
c
m
E
^A
RITHMEtl
TT"
^3¥r
No
O
1
2
3
1359
4
5
6
7
8
9
D.
980
991226
1270
1315
1403
1448
1492
1536
1580
1625
44
981
1669
1713
1758
1802
1846
1890
1935
1979
2023
2067
44
982
2111
2156
2200
2244
2288
2333
2377
2421
2465
2509
44
983
2554
2598
2642
2686
2730
2774
2819
2863
2907
2951
44
984
2995
3039
3083
3127
3172
3216
3260
3304
3348
3392
44
985
3436
3480
3524
3568
3613
3657
3701
3745
3789
3833
44
986
3877
3921
3965
4009
4053
4097
4141
4185
4229
4273
44
987
4317
4361
4405
4449
4493
4537
4581
4625
4669
4713
44
988
4757
4801
4845
4889
4933
4977
5021
5065
5108
5152
44
989
6196
5240
6284
5328
5372
5416
6460
5504
5547
5591
44
990
5635
5679
5723
6767
5811
6864
6898
6942
6986
6030
44
991
6074
6117
6161
6205
6249
6293
6337
6380
6424
6468
44
992
6512
6555
6599
6643
6687
6731
6774
6818
6862
6906
44
993
6919
6993
7037
7080
7124
7168
7212
7255
7299
7343
44
994
7385
7430
7474
7517
7561
7605
7648
7692
7736
7779
44
995
7823
7867
7910
7954
7998
8041
8085
8129
8172
8216
44
996
8259
8303
8347
8390
8434
8477
8521
8564
8608
8652
44
997
8695
8739
8782
8826
8869
8bl3
8956
9000
9043
9087
44
998
9131
9174
9218
9261
9305
9348
9392
9435
9479
9522
44
999
9565
9609
9652
9696
9739
97S3
9826
9870
9913
9957
43
1000
000000
0043
0087
0130
0174
0217
0260
0304
0347
0391
43
1001
0434
0477
0521
0564
0608
0651
0694
0738
0781
0824
43
1002
0868
0911
0954
0998
1041
1084
1128
1171
1214
1258
43
1003
1301
1344
1388
1431
1474
1517
1561
1604
1647
1690
43
1004
1734
1777
1820
1863
1907
1950
1993
2036
2080
2123
43
1005
2166
2209
2252
2296
2339
2382
2425
2468
2512
2555
43
1006
2598
2641
2684
2727
2771
2814
2857
2900
2943
2980
43
1007
3029
3073
3116
3159
3202
3245
3288
3331
3374
3417
43
1008
3461
3504
3547
3590
3633
3676
3719
3762
3805
3848
43
1009
3891
3934
3977
4020
4063
4106
4149
4192
4235
4278
43
1010
4321
4364
4407
4450
4493
4536
4579
4622
4666
4708
43
1011
4751
4794
4837
4880
4923
4966
5009
5052
5095
5138
43
1012
6181
5223
5266
5309
5352
5395
5438
5481
6524
6567
43
1013
5609
5652
5695
5738
5781
5824
5867
5909
5952
6995
43
1014
6038
6081
6124
6166
6209
6252
6295
6338
6380
6423
43
1015
6466
6509
6552
6594
6637
6680
6723
6765
6808
6851
43
1016
6894
6936
6979
7022
7065
7107
7150
7193
7236
7278
43
1017
7321
7364
7406
7449
7492
7534
7577
7620
7662
7705
43
1018
7748
7790
7833
7876
7918
7961
8004
8046
8089
8132
43
1019
8174
8217
8259
8302
8345
8387
8430
8472
8515
8558
43
No
O
1
2
3
4
5
6
7
8
9
D.
342 LOGARITHMS. — TABLE III
No
0
008600
1
8643
2
3
4
6
6
7
8
9
D.
1020
8685
8728
8770
8813
8850
8898
8941
8983
43
1021
9026
90ii8
9111
9153
9196
9238
9281
9323
9366
9408
42
1022
9451
9493
9536
9578
9621
96()3
9706
9748
9791
9833
42
102.S
9876
9918
9961
*0003
*0045
*0088
*0130
*0173
*0215
*0258
42
1024
010300
0342
0..85
0427
0470
0512
0554
0597
0639
0681
42
1025
0721
0706
0809
0851
0893
0936
0978
1020
1063
1105
42
1026
1147
1190
1232
1274
1317
1359
1401
1444
1486
1528
42
1027
1570
1613
1655
1697
1740
1782
1824
1866
1909
1951
42
1028
1993
2035
2078
21_0
2162
2204
2247
2289
2331
2373
42
1029
2415
2458
2500
2542
2584
2626
2669
2711
2753
2795
42
1030
2837
2879
2922
2964
3006
3048
3090
3132
3174
3217
42
1031
3259
3301
3343
3385
3427
3469
3511
3553
3596
3638
42
1032
3680
3722
3764
3806
3848
3890
3932
3974
4016
4058
42
1033
4100
4112
4184
4226
4268
4310
4353
4395
4437
4479
42
1034
4521
4563
4605
4647
4689
4730
4772
4814
4856
4898
42
1035
4940
4982
5024
5066
5108
5150
5192
5234
5276
6318
42
1035
5360
5402
5444
5485
5527
5569
5611
5653
5695
6737
42
1037
5779
582 L
5863
5904
5946
5988
6030
6072
6114
6156
42
1038
6197
6239
6281
6323
6365
6407
6448
6490
6532
6574
42
1039
6616
6657
6699
6741
6783
6824
6866
6908
G950
6992
42
1040
7033
7075
7117
7159
7200
7242
7284
7326
7367
7409
42
1041
7451
7492
7534
7576
7618
7659
7701
7743
7784
2826
42
1042
7868
7909
7951
7993
8u34
8076
8118
8159
8201
8243
42
1043
8284
8326
8368
8409
8451
8492
8334
8576
8617
8659
42
1044
870C
8742
8784
8825
8867
8908
8950
8992
9033
9076
42
1045
9116
9158
9199
9241
9282
9324
9366
9407
9449
9490
42
1046
9532
9573
9615
9656
9698
9739
9781
9822
9864
9905
42
1047
9947
9988
•^0030
•^0071
'^0113
•^0154
■^0195
*0237
*0278
*0320
41
1048
020361
O403
0444
0486
0527
0568
0610
0651
0693
0734
41
1049
0775
0817
0858
0900
0941
0982
1024
1066
1107
1148
41
1050
1189
1231
1272
1313
1355
1396
1437
1479
1520
1561
41
1051
1603
1644
1685
1727
1768
1809
1851
1892
1933
1974
41
1(52
2016
2057
2098
2140
2181
2222
2263
2306
2346
2387
41
1053
2428
2470
2511
2552
2593
2635
2676
2717
2768
2799
41
1054
2841
2882
2923
2904
3005
3047
3088
3129
S170
3211
41
1055
3252
3294
3335
3376
3417
3458
3499
3541
3682
3623
41
1056
3664
3705
3746
3787
3828
3870
3911
3952
3993
4034
41
1057
4075
4116
4157
4198
4239
4280
4321
43H3
4404
4445
41
1058
4486
4527
4568
4609
4650
4691
4732
4773
4814
4855
41
1059
4896
4937
4978
5019
6060
6101
5142
6183
6224
6265
41
No
O
1
2
3
4
6
6
7
8
9
D.
ARITHMETIC.
843
No
O
1
2
3
4
5
6
7
8
9
D.
1060
025306
5347
5388
5429
5470
5511
5552
5593
5634
5674
41
lOGl
5715
5756
5797
5838
5879
5920
5961
6002
6043
6084
41
1002
6125
6165
6206
6247
6288
6329
6370
6411
6452
6492
41
10G3
6533
6574
6615
6656
6697
6737
6778
6819
6860
6901
41
1064
6942
6982
7023
7064
7105
7146
7186
7227
7268
7309
41
1065
7350
7390
7431
7472
7513
7553
7594
7635
7676
7716
41
1066
7757
7798
7839
7879
7920
7961
8002
8042
fc083
8124
41
1067
8164
8205
8246
8287
8327
83(38
840i)
8449
8490
{=531
41
1068
8571
8612
8653
8693
8734
8775
8815
8856
8896
8937
41
1069
8978
9018
9059
9100
9140
9181
9221
9262
9303
9343
41
1070
9384
9424
9465
9506
9546
9587
9027
9668
9708
9749
41
1071
9789
9830
9871
9911
9952
9992 '
^0033
*0073
*0114
*0154
41
1072
030195
0235
0276
0316
0357
0397
0438
0478
0519
0559
40
1073
0600
0640
0u81
0721
0762
0802
0843
0883
0923
0964
40
1074
1004
1045
1085
1126
1166
1206
1247
1287
1328
i3(;8 j
40
1075
1408
1449
1489
1530
1570
1610
1651
1691
1732
1772 '•
40
1076
1812
1853
1893
1933
1974
2014
2054
2095
2135
2175
40
1077
2216
2256
2296
2337
2377
2417
2458
2498
2538
2578
40
1078
2619
2659
2699
2740
2780
2820
2860
2901
2941
2981
40
1079
3021
3062
3102
3142
3182
3223
3263
3303
3343
3384
40
1080
3424
3464
3504
3544
3585
3625
3665
3705
3745
3786
40
1081
3826
3866
3906
3.!46
3986
4027
4067
4107
4147
4187
40
1082
4227
4267
4308
4318
4388
4428
4468
4508
4548
4588
40
1083
4628
4669
4709
4749
4789
4829
4C69
4909
4949
4989
40
1U84
5029
5069
5109
5149
5190
5230
5270
5310
5350
5390
40
1085
5430
5470
5510
5550
5590
5630
5670
5710
5750
5790
40
1086
5830
5870
5910
5950
5990
6030
6070
6110
6150
6190
40
1087
6230
6269
6309
6349
6389
6429
6469
6509
6549
6589
40
1088
6629
6669
6709
6749
6789
6828
6868
6908
6948
6988
40
1089
7028
7068
7108
7148
7187
7227
7267
7307
7347
7387
40
1090
7426
7466
7506
7546
7586
7626
7665
7705
7745
7785
40
1091
7825
7865
7904
7944
7984
8024
8064
8103
8143
8183
40
1092
8223
8262
8302
8342
8382
8421
8461
8501
8541
8580
40
1093
8620
8660
8700
8739
8779
8819
8859
8898
8938
8978
40
1094
9017
9057
9097
9136
9176
9216
9255
9295
9335
9374
40
1095
9414
9454
9493
9533
9573
,9612
^0009
9652
9692
9731
9771
40
1096
9811
9850
9890
9929
9969
*0048
*0088
*0127
*0167
40
1097
040207
0246
0286
0325
0365
0405
0444
0484
(523
0563
40
109£
0602
0642
0681
0721
0761
0800
0840
0879
0919
0958
40
109£
0998
1037
1077
1116
1156
1195
1235
1274
1314
1353
39
Nc
1 »>
1
1
2
3
4
6
6
7
8
9
D.
^1TS"V7-E]I?.S.
(The ansivers of Exercises I, II and IV are due to Mr. Thomai
Mc Janet of Ottawa ; those of Exercises XXXII to XLVI, to Mr.
Thomas Kirkconndl, Mathematical Master of Port Hope High
School ; tJie latter (jentleman also tested the answers of Exercises I
and III.)
Exercise I. 1. 222. 2. 3625. 3. 2222 sq. yd. 2 sq. ft.
4. (i), 9' 11" ; (ii), (tf) 8' 22f' , (b) 8' H^K"- 5- 7t\ mi. per lir. ;
8i min. per mile. 6. (i), f ; (ii), rHs- 7. 21^5"^ gal. 8. 67|| da.
9. 36 ct. 10. $20-46. 12. 1 A. 361 sq. yd. 7 sq. ft. 13. $388-23.
14. 5' r. 15. 213,Vo- i»^i- 16. (i\ 170| gal. ; (ii), 46| gal.
17. 229-6 lb. ; 33-25 c. ft. 18. 25 da. 19. $1080. 20. $658.
21. $70 ; $30800. 22. 968-7627 sq. in. 23. 6' 4". 24. (i),
1:28^V P-m. ; (ii), MgVV min. 25. 15 years. 26. ^4^^ ; $1974.
27. 0-4. 28. £in37ida., Oin25da. 29. 35%. 30. $89-20.
31. (i), 535-90 Km. ; (ii), 62-31 Km. ; (iii), 142-17 Km. ; (iv),
122-15 Km. ; (v), 52-59 Km. ; (vi), 720-36 Km. ; (vii), 730-66' Km.
32. 684* sq. ft. 33. 10i% min. 34. (i), 2-2545 sq. ft. ; (ii),
2-004 sq. ft. ; (iii), 1-6908 sq. ft. ; (iv), 1-4428 sq. ft ; (v),
1-2525 sq. ft. ; (vi), 0-8906 sq. ft. ; (vii), 0-7452 sq. ft. 35. Uj^j yd.
36. $112-29. 37. $9-60. 38. 19i:3yd. 39. («), 36yr. ; (6), 60yr.;
(c), 36 yr. 40. 4| %. 41. 60-417 and 425-425. 42. (i), $20-67 ;
(ii), $18-84. 43. 10 sq. ft. 92} sq. in. 44. 22-86 gal. 45. 5 times
l)er2sec. 46. $2700. 47. 66 yd. per min. 48. $114. 49. 64yr.
50. $742-38. 51. rW 52. 3^^ A. 53. 1 hr. 15 min. 54. 37H
mi. pex hr. 53. j^^ ; $92-40. 56. (i), 15 sec. ; (ii), 9f sec. ;
(iii), 37isec. 57. $500. 58. 9^^%. 59.6^/0- 60. (i),4if%;
(ii), $525. 61. A. $21, B, $16-80. 62. 705301f§ cubic miles.
63. 0 000545 in. 64. 29}f mi. per hour. 65. 120 subscribers,
$253.50. 66. 20 mi. per hour. 67. $34 05. 68. .4's $5800;
346 ARITHMETIC.
B's $4600 ; 0-7931. 69. 200 da. 70. $100. 73, Feb'y 4th, paid
$20-88. 74. Between 33 and 37 miles per hour. Between 34 and
36 miles per. hour. 75. (a) If. (6) |i 76. i, 12 ; ii, 8 ; iii, 3.
168 marbles. 77. 19 mi. per hour. 78. $335-05 ; 11-04 %.
79. 255 days. 80. $8-00. 82. 40 times. 83. 6227^^ c. ft. :
502 lb. 4427|f gr. 84. 323 yd. 85. Nov. 28th, paid $16-24.
86. 120 acres. 87. (a) If (6) 2^ 88. ^, by 7 J yd. 89. $432-34.
90. $7750-51. 91. 2880 revolutions, 64 and 15 revolutions.
92. (a) 24 -27 c. in., (6) 38 -92 0. in., (c) 100 -465 0. in., ((^) 154 -286c. in.,
(«) 320 c. in. , (/) 389-2 c. in. 93. 164985 lb. 13}i|-oz. 94. $372 -55.
95. (i), B, by 24f yd., (ii), B, by 47§i yd. 96. 4 hr. 97. (a) 5.|.
(6) 2|. 98. 450 yd. 99. $11568. 100. 28f %.
Exercise II. 1. 1 § ,^ |«. 2. i J A A M iU- 3. 1 1 ^
f^ If ill- 4. I V- Sf 5. 2 V H iU m VM- 6. I #, T%\
^§% mi 7. 1 i ^ 3% Ii Ml m i^ i§§f rwyn. s. ^ ^
f% t¥9 m ii% imv 9. i f ?* i^tV^ ft? ^fr?i iim nm
iHih WM^. 10. ^ ? u m Mf if If im mm- n. 1 1
17 41fi0 14(12 3 0 7071 10 9ilX 19 2fi 71 97 2f55 3fi3 9 £9 3340
T]J '29 1^ 99 T69 5(J?J0- -'•'^- ^34 11 T5 4T 56 T53 109 sTl 135T
3329 166_5J5 34G41 1 Q 1 2_2 49 2_l_a 4_8A 2JJ>S 2643 i444 lj_r>.3_l
19^^ 9(T39 20000- ■'■'-'• 2 9 20 89 T98 881 T0T9 3^39 7157
24975 422-Q& 'JXiSl 24.4949 1 A L 3 _^7_ 10 17 1489 1 5Q6 7_513_
TOT 9 6 17353 275 49 1 00 00 0' ^^' 2 7 1« 23 39 3 4TB^ 34 5 5 17 236
JIPIJL 4 3 5_8 9 15 11 45. 29 34 131 296 7 2^^ 17 42 1« 1
20ff91 T^OO^^- ■*■*-'• -■■ 5 T 9 52^ 61 235 531 1^9T 3125- ■•■"- ^S
4^A 8Wt sVtV THh TlUh Tf-Ml5 T^m^T 5¥9¥^^Y IT^O^^^V^
ii<M^o- 17. i 1^ i? T^y^. 18. f I V^ II B f^? MUh'
19. f I ^ ¥ §1 ^i^ W ^^ ¥5^ ¥i^^ 1/^^ UUh
20. 5 M H n w ¥A fHf fii^ m\ ^tm nm mw
llli! ¥rm^- IBB**^. 21. 1 i I f § if If T%1. fl-1 f§J§f
MiM T'^o^wiv im^^ ^ttNUi \%mkii Mffii?-f \%m\u
1QQ3 6 23.11 • 00 5 4 2 2J.7 2£a 855 5 6729 171042 227 771 1082126
1T6000000- -^'^- ^ TT Toi T2t 346" 229.57 692T7 9^171 4^7915
^^V^Mf. 23. I V- If fl m Mi fill IfM ^sWify^ WVW^
f§IMI§ IMfflf W^^^5¥ ffMeift 4!Mlfli ^-i^mm
flflffM L^MiMM- 24. J^- H II Ii III IM mk lilt
7.3 0 ail 2^0 16 8 a ^13 AI8 7 43212.184 31684555 226644294 771617.437
ff¥J98T T9ff7359 465765? 392^586rT 83T74&8r 2^05608373 700000000-
25 * II 42^^ ^^^ A^^^ 48334 1981^5 . 222X81_a 15792868 6.5.3.911291
^^' 3 lIT ^50 lf63 3669 36955 T5148r' 1703244 T2 0 7 4 1 8 9 5000000U*
96 1 I i4 -^2, 37 2Q7 411 6 58 1109 106 3£ 3 3.02.6 1717 6a 2Q8I9 5
^O. X § ^Y ^. ^^ ^35 gj.^ 74Y T259 T^078 37493 199543 237036
AifSWERS. 347
Hmu mmi umu- 27. m^i m^ mn mm mm
mim MMm- 28. 1 m tm um urn mm mhu
mm mi>m imm^ num imm iuum- 29. 2 n
W W V¥- m- 30. i q^ is ^\ x¥t M A%\ m\ f tM Mf^l
AV.^^ Bm¥^ iiiflf i-myoV
Exercise III. 1. 3101-7414. 2. 67 0509. 3. 1503-543.
4. 3-20424. 5. 31. 6. 1 301030. 7. 2*477121. 8. 3. 9. 2-1556589.
10. 2. 11. 1-0019656. 12. 99999970. ^Jj3^f856. 14. 1161.
15. 6770. 16. 67-7. 17. 70-7107. 18. 795^7^5! ^ 19. 00264575.
20. 0-013964. 21. 0*301030. 22. 0-477121. 23. 0-318310.
24.2-30259. 25.2-22398. 26.16-5304. 27.1-39642. 28.2 7183.
29.0-36788. 30.14107. 31.2. 32 15. 33.125. 35.8000 m.
36. 1925 yd. 37. 4047 centiares. 38. 5789 sq. yd. 39. 40-47
hectares. 40. 247 1 A. 41. 132 1 gal. 42. 599-4 litres.
43. 384,300 Km. 44. 147,100,000 Km. 45. f , Y, i^, ^A', tV-
46. -V, -V^ 4,^ -Y, W, W-, h'h'- 18-03 yr. 47. ^^,\\, #7, ,%,
ih. i\%' 48. 1, f, ^^, ff, If, H, fit- 49. i I, t\, hi if
50. 1, f , I- f , f, 1^, M-
Exercise IV. 1. 30-197 in. 2. 1000796875 lb. 16-051 c. ft.
3. 3min. 25^ sec. ; 3 min. 25 2W0V7 sec. 4. 1100 bu. 5. ^Vi^i-
6. 78 % of copper, 22 % of zinc. 7. 0-2532 %. 8. $9000-49, $123-09.
9. 2nd Nov., 1889. 10. 8 %. 11. (i), 9| min. ; (ii), 8f min.
12. 2-89 times. 13. 120-4264 lb. 14. 209 da. §§| ; §fl|.
15. Friday at 3,15 a.m. ; 3,20 a.m. 9,05 {j^j a.m. 16. $5-40.
17. Jqt. ; f 18. 6^%%\%. %514:-93. 19. 2nd Ap., 1889.
20. 7%. 21. 23f ft. per sec. ; 4715- yd. per min. ; 16,1^ mi. per hr.
22. 3762 revolutions ; 7 mi. 160f yd. ; 21if ^ mi. per hr.
23. 632f 7 lb. 24. ^s 5%do ; 23f f grammes per millier. 25. Oct. 24
at 2 a.m. 26. 31^ gal. 27. 27 ct. 28. 125-§|f %. 29. $477-34.
30. 16_ct. ; 16 ct. 31. 480^ I §^ yd. per min. ; I6/3V1 mi. per hr. ;
439-467 m. per min. ; 26- 368 Km. per hr. 32. 1 min. 9 6 sec.
33. 180-041 lb. 34. -^^^ ; $91 64. 35. 256791 lb. 36. 271296 ;
247643/^^. 37. 45 lb. @ 26 ct. 38. 61-35 %. 613-5 per 1000.
39. 8^g %. 40. $836-53 ; $78309 ; 1365 %. 41. 4 hr. 5 min. ;
348 ARITHMETIC.
4 hr. 4 niin. 42. 61| yd. 16 roUs 28 yd. 43. $7 '04. 44. 8 "878 lb.
45. 13-75ct. per hr. 46. (a) 276-812 c. in. ; (&) 173 lb. ; (c) 26-363 lb. ;
(d) 316-357 lb. 47. Gains $160-75 ; 30-63 %. 48. $2-34.
49. $340-86. 50. (a) 7^^ % ; (6) 7 -834 % ; (c) 7 -91 %. 51. 5-911 T. ,
6-2115 T., per sq. in. 831-2 kilog. ; 873-4 kilog., per sq. cm.
52. $4769. 53. xV 54. 949| lb. 55. 54 times ; 1^^^ qt. ; f .
56. $28-66. 57. 5 hr. 58. 775 marks ; 79^| % ; 125§f %. It
Vvould reduce the 775 marks to 620 marks but would have no effect
on the percentages. 59. $206-56. 60. $192-95. 61. $16-77.
62. $661-12. 63. $35-94. 64. 210-07 lb. 65. 7^ mi. per. hr.
25f mi. 66. 3j2- 67. 20 boys. 68. 5789-658 T. 69. $783*04.
70. $863-65. 71. 2210 tiles; 58' G" ; 13'. 72. 1531*46 lb.
73. 1-596 mi. 74. $1*71. 75. i- 76. 12 men. 77. 7ff hr.
78. 3hr. 79. 55 doz. 48^f %. 80. Net proceeds $3061*71.
81. $1-21 ; $1-48. 82. $153-92. 83. (a) 12 c. ft. 1534 c. in. ;
(b) 12 c. ft. 192-4 c. in. 84. 667 strokes ; 25 strokes. 85. ^^%.
86. \% 87. $7-56; $14-40; $1875. 88. $1*08; 27,^1%.
89. 21yV % ; $23-84. 90. $1095-78 ; $1097-97. 91. A, 48 men ;
B, 72 men ; 0, 60 men ; D, 80 men. 92. $25-59 ; 40-536 c. ft.
93. 32 sq. ft. 102 sq. in. ; 10 c. ft. 1620 c. in. ; |^§ ; j^j\.
94. 1 hr. 11 min. 41-84 sec. ; 3 hr. 5 min. 41*4 sec. 95. fj.
96. $511-25. 97. $8-40; $5-04; $3-60. 98. 35*3%. 99.19-264 %.
100. $116414.
Exercise V. 1. 23. 2. 3^. 3. 5*. 4. 10^. 6. 01*.
6. 2-35. .7. {^y. 8. ay. 9. 3x3x3x3. 10. 12x12x12.
11.15x15. 12.25x25x25x25x25. 13.2-5x2-5x2-5x2-5x2-6
14.0-25x0-25x0-25x0-25x0-25. 15. fx|x|x§. 16. J^x
HxH- 17.2x2x2x3x3x5x7x7. 18.64. 19.36. 20.625.
21. 1024. 22. 5640625. 23. 202572273617. 24. 5554571841.
25. 14348907. 26. 1-331. 27. 000001. 28. 0*000000000064.
29. 1-126162419264. 30. 2401. 31. 24 01. 32. 0 2401.
33. 13144-256. 34. 13-144256. 35. 0000013144256. 36. f
37. U' 38. ^\. 39. Ml- 40. Iff. 41. 648. 42. 54000.
43. 5292. 44. 629829200a 45. 117649. 46. 625. 47. 72.
ANSWERS.
349
48. 2-^ X 32 X 5 X 7. 49. 2^ x 3^ x 52 x 13.
52. 2-x3«x7xl3. 53.0-25. 54.0-2.
57 to 61. 0-7854. 62. ,2-014. 63.
65. 2-303.
50.210. 51.23x72x29,
55. 01. 56. 0009706.
0-4966. 64. 0'6931.
Exercise VI. 1. 4, 529, 55225, 5555449, 555922084, 55592679961.
2. 64, 79507, 83453453, 83740234375, 83791924694479. 3. 144,1728;
16129, 2048383 ; 1633284, 2087336952 ; 163481796, 2090278243656.
4. 2646-999601, 136185-482471849, 7
5. 0 018496, 0 002515456, 0 000342102016
0 000006327518887936. 6. O'Ol. 7. 0-05139.
10. 90-744.
Exercise VIII. 1. 24. 2. 43. 3. 321
6. 0 07097. 7. 73. 8. 934. 9. 8^
12.00543. 13.1-41421. 14.4-47214
17. 0-447214. 18. 0-141421. 19.
21. 0-632456. 22. 863076. 23.
25. 49-7933. 26. 4 97933. 27.
29. 5-84804. 30. 0 621455.
006606-887694149201.
, 0-000046525874176,
8. 0-02. 9. 0 00686.
4. 3-21. 5. 48-18.
, 10. 88-8. 11. 1-837.
15. 141421. 16. 44-7214,
6-32456. 20. 63-2456.
31-6228. 24. 780-897.
1-25992. 28. 2 71442.
Exercise IX. 1.
i 2.1 3
i|.
4. 0 745356
. 5.
0-824621
6. 0-530330.
7.
0-769800.
8.
0-547723.
9.
0-845154
10. 0-612372.
11.
0-261861.
12.
0-788811.
13.
1-73205.
14. 2-23607.
15.
3-87298.
16.
412311.
17.
4-89898.
18. 5-09902.
19.
5-91608.
20.
608276.
21.
2-64575.
22. 3-31662.
23.
7-28011.
24.
8 -77496.
25.
9-84886.
26. 40 0125.
27.
48-9898.
28.
0-797724.
29.
0-670820
30. 0-73598.
31.
3-31662.
32.
2-44949.
33.
2.23607.
34. 1-41421.
35.
f . 36. A.
37
0-609884.
38.
0-471957.
39. 0-87358.
40.
0-893904.
Exercise XI. 1. 10*. 2. 10«.
7. 10,007,400. 8. 12741-8. I
3. 1010; 4. 10-5. 5. 10-8. 6. 10^
). 0-000226. 10. 0- 000, 000, 006.
11.1, 083, 200, 000, 000, 000, 000, 000.
13. 1-4709x1011.
16. 90992x103.
14. 4-8721x10-3.
17. 5-875 X 101*.
12. 0-000,030,476,3.
15. 7-8376x108.
18. 1-1535x107.
350
ARITHMETIC.
19. 8-5534 X 1019. 20. 10832 x 10^2 21. 37267 x 10*.
22. 4-5152 xl0% 23. 2-2641x104. 24. 6-3611 x lO-io.
25. 6-6966x10-*. 26. 8-5499x10-1. 27. 127418x10'.
Exercise XIV. 1. 00000043. 2. 0-0000087. 3. 00000130.
4. 0-0000304. 6. 0-0000586. 6. 0-0001084.
7. 0-0004341.
11. 0-8450980.
15. 1-2304489.
19. 1-278754.
23. 1-612784.
8. 00005208. 9. 0-0012576. 10. 04771213.
12.0-4342495. 13. 1-4913617. 14. 1-1139434.
16. 0-845098. 17. 1230449. 18. 1113943.
20. 1-278754. 21. 1361728. 22. 1462398.
24.1-361728. 25.0 301030.
Exercise XV. 1. 0 8451. 2. 0 7782. 3. 04914. 4. 0 8195
6. 0-4346. 6. 0-1370.
Exercise XVI. 6. 10^. 7.
11. 1-361728. 12. 2-361728
10-2. 8. 105. 9. 10-5. 10. 102.
13. 3-361728. 14. 0-361728.
15. 2-361728. 16. 0635584. 17. 3635584. 18. 2635584.
19. 3-831806. 20. 3-892651. 21. 3-860098. 22. 3860098.
23. 5-830396. 24. 1830396. 25. 2 301464. 26. 14 071145.
27. 42-212188. 28. 8811575. 29. 6652826. 30. 5881042.
Exercise XVII. 1. 0864831. 2. 2774604. 3- 4679092.
4. 3-690692. 6. 2471214. 6. 5-830439. 7. 1271435.
8. 4-903133. 9. 3659925. 10. 7477599. 11. 8804711.
12. 8-803211. 13. 27-034709. 14. 7*634200. 15. 5-483962.
Exercise XVIII. 1. 3-02. 2. 5-432. 3. 5-496. 4. 300.
5. 8-6636. 6. 6-5666. 7. 65 666. 8. 656660. 9. 0-65666.
10. 0-000065666. 11. 67621. 12. 67680. 13. 00079433.
14. 1999-9. 15. 0-31623. 16. 1. 17. 100 047. 18. 2.
19. 1024. 20. 1-0718. 21. 5 208 xlO^. 22. 3-4247x10-5.
23.6-562x10-10. 24. 4-5709x10^*. 25. 1 -00028 xl">-^oo.
Exercise XIX. 1. 499-27. 2. 4-0798. 3- 0-0089054.
4.0-088785. 5.86-898. 6.561-33. 7.0-62614. 8. 0 0065856.
9. 34 464. 10. 12-4368. 11. 78541. 12. 0013446.
13.0-00026113. 14.0-0035259. 15. 12537 x lO-'^. 16.115779.
ANSWERS. 351
17. 1730-6. 18. 0024513. 19. 4-66735. 20. 9-8019x10-21.
21.1-2589.22.1-08791. 23.0-29587. 24.0-44402. 25.0-61439.
26. 0-45986. 27. 0 74989. 28. 1-24732. 29. 0-96691.
30. 0-1J8921. 31. 0-79433. 32. 0-076522. 33. 0-1423.
34.5-4143x10-24. 35.5 4184x10-*. 36.0-80274. 37.0-5848.
38. 0-491515. 39. 099718. 40. 1*0000025. 41. 1917791.
42. 23-097. 43. 0-168792. 44. 0-034278. 45. 2-2339.
46. 51og2. 47. 4 log 2 + log 3. 48. 2 log 7. 49. (2-41og2)- 1.
60. 10 log 2. 51. 4 log 7. 62. 61og3+log7-3. 53. 21og2 +
log 7 + log 11 + 3 log 13 -3. 64. 41og3 + 21ogll-log2-21og7-2.
65. (41og2 + log7 + logll + 3-61og3-21ogl3)-l. 57. 2^ -^^^ =10;
l-flog2. 68.1-8507. 59.0-64921. 60.0*28246. 61.-2-1755.
62.-0-68512. 68. 2 80736. 69. 0*35621. 70. 2 2766.
71. 0-43924. 72. 2-4923. 73. -1-94843. 74. -0*0136958.
75.2-5124. 76.20-149. 77.17.673. 78.11*8956. 79.10-2448.
80. 9-58435. 81. 27267. 82. 302. 83. 48. 84. 206. 85. 44.
86. 83. 87. 7. 88. 46. 89. 27. 90. 6. 91. 69. 92. 164.
93. None. 94. None. 96. 32. 96. -17. 97. -1. 98. 1.
99. 10. 100. 0.
Exercise XX. 1 . Mer. , 0 *3871 ; Yen. , 0*72333 ; Mars, 1 -52369 ;
Jup., 5-2012; Sat., 9-538. (a), 35,915,000; 66,134,000; 139,310,000;
475,540,000; 872,060,000. (6), 35,915,000; 67,111,000; 141,370,000 ;
482,560,000 ; 884,930,000. 2. 1233222 figures ; 1,169,649
18,212,890,625; 114 hr. 11 min. 14 sec. 342,188,706,078 figures;
253 yr. 21 da.
Exercise XXI. 1. 25". 2. 20" ; 11*25". 3. 19*886" ; 42-614".
4. 7' 7-886". 6.4-2. 6.18. 7. 40-5 yd. 8. 138' 8". 9. 25-714 ch.
10. 33' 7*2". 11. 6' 8-5". 12. 17' 6". 13. 19' 1091".
14. 62' 8-625", SO' 8-625". 15. 11-07 ch., 11-39 ch., 11-71 ch.,
1203 ch. 16. 4-186", 7 814"' ; 6 977', 13-023"'. 17. 0-2".
18. 8' 2*182"'. 19. 5' 10", 9' 2", 10'. 20. 7 ft. 9 -6 in., 10 ft.
^-4in. ; 15yd. 1ft. 8in., 23yd. 2ft. 4in.
Exercise XXII. 1 . 69 yd. 1 ft. 9 in. 2. 8 -944 ch. 3. 1 *25 in.
4. 2 mi. to 1 in. 1 : 3520. 6. 36 sq. ft. 6. 46*75 oz, 7. 17*32 ch.
852 ARITHMETIC.
a 1045440 stalks. 9. 384 pieces. 10. $4747-80. 11. 17 '32 ch
12. 5081yd. 13. 15ch., 40ch. 14. 1452 sq. in. 15. 55 ft,
16. 30yd. by66yd. 17. 6-41 ch. 18. 13984 sq. ft. 19. $384*38,
20. 315 -375 sq. in. 21. 37 sq.ft. 91-5 sq. in. 22.36ft. 23. 58 "9 ft.
92-57 ft. 24. $34-03. 25. 5-625 A. 26. F 402*73 sq. ft.
2°, 99 -72 sq.ft. 27. 1867 bricks. 28. 4 A. 2207 sq. yd. 29. 1°,
152-25ft. ; 2°, 536-25ft. 30. 12-80 ch. 31. 127-5ft., 94-5ft.
32. 79194 sq.ft. 33. 133 sq.ft. 139 sq. in. 34. 40-9469 A, ;
32-3817 A. 35. 31 7495 A. 36. 200 yd., 240 yd. 37. 340ft.
38. 43-56 ch., 37*44 ch. 39. 63 sq. ft., 63 sq. ft. 40. 4-63392 A.,
4-80096 A., 4-968 A., 5-13504 A., 5-30208 A. 41. 551 sq. ft.,
1653 sq. ft., 2755 sq. ft. 42. 5' 4", 4'. 43. 161 ft. 44. 441 ft.,
245 ft. 45. 17424 sq.ft. 46. 146 yd. 2 ft. 47. 413600 sq.ft.
48. 5 ft. 49. 11 ft. 50. 289 sq. yd. , 225 sq. yd. 51 . 841 sq ft. ,
441 sq.ft.. 62. 35 ft., 27 ft. 53. 194*0335 ft. 54. 174 yd.
55. 484ft., 330ft.
Exercise XXIIT. 1. 17008 c. in. 2. 3005 gal. 3. 14 05 in.
4. 30-26 in. 5. 15 -704c. in. ; 0-2297 in. 6. 66228- 5 eft. 7. 1549 gal.
8. lin. 9. 12 in. 10. 2 014 in. 11. 133 '3334 mm. by 44 '4445 mm.
by 44-4445 mm. 12. 22894 yd. 13. 643*66 mm. by 965-49mm. by
1609*15 mm. ; 0*6214, 1 -0357 and 1 -5536 centiares. 14. 68*921 c. in.
15. 87-72sq.in. 16. 335 41 c. in. 17. 2*211in. 18.649-52c.in.
19. 5*4 in. 20. 0*0151 to 1. 21. (a^ 0*271; (6)0*19. 22. In
reductions from metric expressions a ' calculated length ' will be in
excess by 1 '599 % of the actual length and should therefore be
decreased by 1*.576% of itself; a 'calculated area' will be in
excess by 3 *224 % of the actual area and should be decreased by
3*123% itself; and a 'calculated' volume will be in excess by
4*874 % of the actual volume and should be decreased by 4*648 % of
itself. In reductions to metric expressions a ' calculated length '
will be in ^defect by 1 '574 % of the actual length and should be
increased by 1 "599 % of itself ; &c. 23. 6 *8457 in. square.
24. 107*lc.in. 25. 625*683 c. in. 26. 244 "798 eft. 27. 3*5211 ca.
28. 0*3447 ca. 29. 461*468c. in. 30. 50*4 c. in. 31. 7*2in. ;
ANSWERS.
•5 in. ; 4-2 in. 32. 6. 33. 12 in. 34. 128 c. in. 35. 23 6 in.
36. 645-2251b. 37.0-0000067. 38. 27904 to 10000. 39.236-321b.
40. 76-777 in. 41. 584 -59 lb. 42. 12095 in. 43. 5066 c. in.
44. l-841b. 45. yLin. 46. 24149 cubes. 4:7,j%in. 48. 0-00102 in.
ya ^. 10-08 c. in. ba^'/ 49 -2565 millilitres. ^l. 4321-1 c. in.
52. 28134c. ft., 1458c. in. 53. 50c. ft., 288 c. ii^.''"54. 117,333,-
333 i c. yd. 55. 159-25 c. yd. 56. 406 gal. 5^ 49-2 bars.
58. 6844 -8 yd. 59. lift. 3 in. 60. 2 ft. 2 in. Sf. 394 4 gal.
62. 2-181 sq.ft. -^4 ft. by 4 ft. by 5 -6578 ft. 64. 2640 eft.
65. 11840 c. ft. 6Bf 768 c. in. i99V 306 c. in. 68. 969 c. in.
69. 33293-4 c. in. "TO- 1-07 lb. 71^^235-31. 72. 18-8456 c. ft.
73. 18 c. ft., 696 c.^. 74. 1-5 eft. 75. 16-846 in. 76.3 ft.
4-5in. 77. 5-196ft. "i^. 4 5243 in. 7^1-57in. m- 18in.
81. 3464344 c. yd. ; 309/1908 c. yd. 82^" 1492-36 c. ft. i 895 T.
13631b. 1722 gr. ; $539984058-47. 83. 42 ft. 10-^ in. 84. 3-3 in.
85. 2 -888 in. 36. 11 04 in. 87 2 175 in. 88. 39 375 eft.
89. 40-25c.ft. 90. 30-484c.ft. 91. 5128*27 gal. 92. 5776 -711^
93.70081b. 94. 244-369c.ft. 95. 771c. in. 96. 47272 -264c. yd.
97. 5365-226 c. yd. 98. 28246 722 c. yd. 99. 101138 343 c. yd.
100. 1699-38 c. in. ; 972-972c.in. 101. 47 '6850. ft. ;-29-352c.ft. ;
15 c. ft. 102. 165 c. in. ; 95 76 c. in. 103. 696 c. in. : 264 c. in.
104. 5-654 in. 105. 124 542 c. in. ; 82-791c. in. ; 30-042 c. in.
106. 3-618 c. ft. ; 8-045 c. ft. 107. 10138 c. yd. 6 c. ft.
108. 43008 c. in. 109. 11 gal. 110. 32 64 c. in. ; 37*44 c. in.
111. 1000 c. in. 112. 509-2 c. in. 113. 647 234 c. in. ;
441-406 c. in. 114. 1575 c. in. ; 840 c. in. 115. 18 432 in.
116. 19c. ft. 418c. in. 117.64ic.ft. 118. 13954 -31b. 119.57c.ft.
120. 20-48 in. 121. 1-8998 ft. 122. 13 to 9. 123. 7 to 4.
124. 53c. ft. 352c. in. 125. 2 -9 ft. 126. 4 -474 ft. 127. 8987 litres.
128. 34 sq.ft. 64sq.in. 129. 11 5 sq.ft. 130. 2' 8" ; 1' IJ".
131. 5-6 in. 132. 6 ft. 133. 1'8". 134. l^in.
Exercise XXIV. 1. 97 in. 2. 905 mm. 3. 1ft. 5 in.
4. 1ft. 8| in. 5. 81 in. 6. 6 -928 in. 7. 3 -464 in. 8. 3-674 ft.
9. 2-45 in. 10. 267 in., 24-4in., 12-5in. 11. 15ft. 12. 8-595m. ;
354 ARITHMETIC.
8-579m., 6-347m., 5-82m. 13. 14. 4'29in., S'Sin.,
23-4in. 15. 962-676 c. in. 16. 56sq. ft. 40sq. in. 17. 9-88 in,
18. 391ft. 19. 7ft. 20. 2ft. 21. 683128 sq.ft. or 168988 sq.ft.
22. 1200 c. in. ; 790-9 sq. in. 23. 28ft. 8-2 in. 24. 36-9 ch.
25. 231 66 sq.ft. ; 696ft., 630ft.
Exercise XXV. 1. 1936 sq.ft., 57600 sq.ft., 13689 sq. ft. ;
44ft., 240ft., 117 ft. 2. 46ft. 5-494in. 3. 9-2 mm., 359-5 mm.
4. 7 in., 8-8 in. ; 24in., 23 '4 in. 5. 15 ft. 2-753 in., 45ft. 2-247 in. ;
26ft. 2-797 in. 6. 12 -923 yd., 12yd., 11-2 yd. 7. 19 -8 ft.,
12-692ft., 20ft., 44-8 ft., 36ft., 51-692ft. 8. 399ft., 455ft.,
511ft. 9. 616ft., 665 ft., 511ft. 10. 17 ft., 21ft. 3. in., 21ft.
9in.
Exercise XX VI. 1. 60sq.yd. 2. 60sq.yd. 3. 24 sq.ft.
4. 84sq.ft. 5. 66sq.ft. 6. 126sq.in. 7.240sq.in. 8. 252 sq. in.
9. 2-9274sq.ch. 10. 166-417 A. 11. $260653. 12. $11868.
13. 16672-5 sq. ft. 14. 18*2 Ares, 54-6 Ares, 91 Ares.
15. 227-04 sq. ft., 804-32 sq. ft., 740 ' 96 sq. f t. , 163-68 sq.ft.
16. 14760sq.ft., 17352-28 sq.ft., 28341 '5 sq. ft., 25749 -22 sq. ft.
17. 92-8812 A. 18. 1698-8 sq.ft. 19. 44-7154 sq. metres.
20. 37-0843 A.
Exercise XXVII. 1. 0-130806; 3*13935. 2. 0-065438;
3-14103. 3. 0-263305; 3*15966. 4. 0*131087; 3-146086.
6. 0-0654732 ; 3-142715.
Exercise XXVIII. 1. 659 '734 ft. ; 835 -664 ft. 2. $3-93.
a 2mi. 1710yd. 2ft. 3in. 4. 1ft. 8Lin. 5. 410yd. l|ft. 5in. 6. 3|in.
7. 14 in. 8. 1770-7 mi. per min. ; 357.7 mi. per min. 9. 10-472 in.
10. 15-708 in. 11. 28° 38' 52-4". 12. 114° 35' 29-6'.
13. 57° 17' 44-8". 14. 62 -489 in. ; 82-467 in. 15. 27-914in.
16. 1-945 ft. 17. 2-853ft. 18. 6-915ft. 19. 34-6 in. 20.15-8in.
21. 15-7 m. 22. 40yd. 23. 43,827,033yd.; 43,827,735 yd.
24. 1-093827 yd. 25. 141, 000, 000 mi. and 140 , 400 , 000 mi.
26. 899 mi. per min. 27. 34 '527 mi. 28.48-83mi. 29. 141 '244 in.
and 98-87 in. 30. 28-45 in. and 35-45 in.
ANSWERS. 355
Exercise XXIX. 1. 44 18 sq. in. 2. 153-94 sq.ft.
a 13-636 sq. cm. 4. 117 '75 ft. 5. 595784 ft. 6. 3199-41ft.
7. $15-89. 8. 35in. 9. 319071b. 10. 829-58sq.ft. 11. 4-427 in.
12. 386-146 sq. ft. 13. 14-848 in. 14. 56-37 sq. ft. 15. 0*615 sq. ft. ;
l-133sq.ft. 16. 7-31 sq.m. 17. 1354 ft. 18. 16572 yd.
19.62-02sq.in. 20. 138-13sq.ft. 21. 32 -225 sq.m. 22. 93-46sq. ft.
24. 2488-14sq.cm. 25.15-708sq.ft. 26. 189-69ft. 27. 28-274sq.ft.
28. $96-10. 29. 3-1 yd. 30.11yd. 31. $1411-47. 32. 44 1 in.
33. 40-15sq. in. 34. 136-35sq.cm. 35. 6742 ft. 36. 57° 17' 44-8".
37. 33°25'2r. 38. 0-0906 sq.ft. 39. 0-6142 sq.ft. 40. 0-2854sq.ft.
41. 112-2 sq. in. 42. 158 -57 sq. in. 44. 92-88 sq. in. 45. 29-05 sq. in.
46. 62-832 sq.ft. 47. 9-487 sq.ft. 48. $206-91. 49. 1269-21 sq. ft.
60. 692 -72 sq.ft.
Exercise XXX. 1. 3sq. ft. lOSsq. in. 2. lOsq. ft. 120sq. in.
3. 40 sq. ft. 48 sq. in. 4. 19-37 sq. in. 5. 384 sq in. 6. 49 sq. ft.
120-6 sq. in. 7. 52 sq.ft. 134*7 sq. in. 8. 18 sq.ft. 94-89 sq. in.
9. 3ft.8in. 10. 9-55in. 11. 25 -33 in. 12. 12 44 in. 13. 8sq.ft.
45 sq. in. 14. 17 sq.ft. 96 -69 sq. in. 15. 64 sq.ft. 104-66 sq. in.
16. 21 sq.ft. 111-2 sq. in. 17. 9-2775 sq.ft. 18. 30/26 sq.ft.
19. 7sq. ft. 122-97 sq. in. 20. 33 sq. ft. 118-55 sq. in. 21. 2ft. Sin.
22. 2 ft. 10-7 in. 23. 3-183 in. 24. 3 ft. 1-72 in. 25. 2 ft. 4-8 in.
26. 2-387 ft. 27. 25-69 in. 28. 14fyd. 29. 41yd. 30.60°.
31. 2 to 1. 32. 136-5 sq. in. 33. 753 -98 sq. in. 34. 1021 -02 sq. in.
35. 130 -627 sq. in. 36. 78 -63 sq.ft. 37. 880 '78 sq. in. 38 8*343 in.
and 12 -457 in. 39. 45 -783 sq. m. 40. 113 -1 sq. in. 41. 45-837 sq. in.
42. 3-39 in. 43. 82-467 sq. ft. 44. 47i»l'e4 sq. in. 45. 16-144sq. ft.
46. 6 -065 sq.ft. 47. 0 858 in. or 29 -142 in. 48. yV 49. 27 in.
60. 196,940,000 sq. miles.
Exercise XXXI. 1. 628 -32 c. in. 2. 226-194 c. in. 3. 6-77 in.
4. 2 ft. 6. 2fin. 6. 7 -927 ft. 7. 7 "0663 in. ; 196 '0844 sq. in.
8. 185-336mm. 9. 503-08mm. 10. 383-92mm. 11. 13mi.
1566-2yd. 12. 15 -783 mi. 13. 1347 45 c. in. 14. 0*1502 in.
15. 2-4 in. 16. 733 037 c. ft. IT 392-7 c. in. 18. 981 in.
19. 16889-24 c. in. 20. 5579 47 c. in. 21. 201* 16 c. in.
356 ARITHMETIC.
22. 186 -7 c. in. 23. 428 -83 c. in. 24. 4322 '840. in. 26. 5719-108c.in
26. 428-828 eft. 27. 101 c. ft. ; 78-66c. ft. ; 59 'left.
28. 1 -8963 in. ; 2*7033 in. ; 10-4004 in. 29. 141-87 c. yd.
30.7-836c.ft. 31.2408-66sq.in. 32. 10-8573in. 33. 904 '780. in.
34. 606 -8630. in. 35. 4387* 14c. in. 36. •5096gal. 37. 22 -283 lb.
38. 11 •3771b. 39. 5-98651b. 40. 110-446c. in. 41. 6-928in. ;
574-226 c. in. 42. 101-274 lb. 43. 47*545 lb. 44. 33-1 lb.
45. 229-303 c. ft. 46. 23789 lb. ; 5625-9 lb. 47. 0-825 in.
48. 4 -64 in. 49. 1*42 m. 50.0 003787. 51. 1 to 900. 52. 103 to
1000. 53. 7 in. 54. 10-01 in. 55. 20-123 c. in. 56. 51-662 c. in.
57. 433-541 c. in. 58. 25525 4 c. in. 59. 13 to 6. 60. 0-36c. yd, ;
0-64c.yd. 61. 3-28392c.in. 62. 928-32c.in. 63. 2 '5988170. mi. x 10^ J
64. 2-598682c.mi. x lO^i ; 20902046ft. 65. 3173 in. 66. 68-068c.in.
67. 99041 c. in. 68. 253 gdl.
Exercise XXXII. 7. 25 gal. 8. 6 da. 9. 2f|wk. 10. 4 A.
3740 sq. yd. 11. |foz. 12. 3^ 13. $7-8125. 14.3:5.5:3.
15. 9:4. 4:9. 16. 17:24. 17. 5:4. 18. 17:7. 19. 9:8. 20. 5:8.
21. 3^ lb. 22. 1 to 3. 23. $2100, $2400. 24. $5 -621, $9 -37^.
25. 100 A.
Exercise XXXIII. 1. 242, 484, 605. 2. 18055-41b., 12036 -91b.,
9027-7 lb. 3. $17-45, $27 92, $11-43. 4. $13*05, $26-10, $39*15,
$39-15, $52-20. 5. $320, $360, $384. 6. $1200, $300, $120, $60,
$40. 7. 1321b., 281b., 201b. 8. $66, $77, $110. 9. $1860,
$2112, $2464. 10. 2925, 3640, 4212, 1950, 2496. 11. 648-1281b.
of oxygen, 562 '445 lb. of carbon, 89*427 of hydrogen. 12. 201b.
2701b. 5281b. of nitre and 801b. of sulphur. 13. $2704, $3151,
$4045. 14. $134-40, $118-08. 15. 2//vlb. of lead, 17H^lb. of tin.
16. $6-02. 17. $5000, $8750, $11250. 18. A to C, 30ct. ;
B to C, 36 ct. 19. 199111b. ; 202|lb., 46.|lb.
Exercise XXXIV. 1. $160 ,$240, $30% $350. 2. £1 6s. SJd.,
£13 6s. 9|d., £14 Os. 2d. 3. 40-3088 pt., 62-7025 pt., 74-6459 pt.,
85 -8428 pt. 4. 300 m., 120 ch., 280 w. 5. $2100, $3570, $2380,
$4550. 6. A, $1085-70; B, $1034; C, $1155. 7. A, $540:
B, $200 ; C, $300 ; D,
ANSWERS
357
Exercise XXXV. 1. $6, $3-60. 2. $600, $840, $300.
3. 88ct., 38|ct.,10^ct. 4. $900, $750. 5. $522, $536, $502-50.
6. $4507-06 ; $4965-41 ; $5527 53. 7. 37-5, 15, 312-5. 8. 78001b.,
66001b., 52001b. 9. $2-25, $1-50, 90 ct. 10. 12, 18, 5, 45.
Exercise XXXVI. 1. $147, $196, $147. 2. $10-82, $20-29,
$60-89. 3. $2 -.50, $1-87^ $l-56i. 4. $38*25, $38, $37-80.
5. $40, $42-30, $43-20.
1. 2500 bu., 4000 bu., 10000 bu.
16, 28, 40. 4. 15, 20, 50. 6. 99-8 ct.,
2.
4.
Exercise XXXVII.
2. U4:m.3bliv.,480b. 3.
66-53 ct., 83-17 ct.
Exercise XXXVIIT.
2. A, $240 ; B, $180. 3.
$13-50, $30. 5. $79-98,
480 6., 351m 7. $2-10, $1-50, $1-08.
95 ct. or $2-42, $2-42, $1-41, 75 ct. 9.
Exercise XXXIX. 1. $656 25, $1093-75.
$789, $1296-70. 3. $18004-83, $13495-17.
$34486-50. 5. $13138-05, $10019-95. 6.
7. $16434-78, $24260-87. 8. $185-81, $126-99,
9. $2216-98, $2480-08, $3152*94. 10. $9368-20,
11. $13000, $8000.
Exercise XL. 1. $14-38. 2. $823-82. 3. $12300. 4. $91500.
6. $564. 6. $59062-50. 7. 1^ %. 8. m %. 9. 20 %.
10. $6058-95, 95%. 11. $105263, $1422-37. 12. 21-5%.
13. $7163-84. 14. 120 %. 15. 65600, 67240, 68921. 16. 130050,
127500, 125000. 17. $20000. 18. $2000. 19. 72 gallons.
20.3538641b. 21.$31250, $32812-50, $43750, $43125. 22.53-7%.
23. 4096. 24. 1122:1125. 25. 37 -.32 in work.
Exercise XLI. 1. $2425. 2. 150%. 3. 10%. 4. 4% loss.
5. 38f%. 6. 12% gain. 7. $2400. 8. $210. 9.24 s. 10.14/2%.
11. 18ct. 12. 9d. ; 512. 13. $140. 14. 71-75T. 15. 2ct.
16. 12i%. 17. $45, $30. 18. $40. 19. $208. 20. 1413%.
21. $25, $30. 22. 16 ct. 23. 751b. at37^ct., 691b. at 34^ ct
24. $12. 25. $160, $120.
1. $780, $801-25, $426-79, $991-96.
$2565, $1425, $1710, $1140. 4. $9,
$1714, $11-99, $143-38. 6. 144 m.,
8. $2-23, $2-23, $1*59,
$1070-30,
$20419-64,
$10400.
$121-40.
$10021-80.
S58 ARITHMETIC.
Exercise XLII. 1. $131-25. 2. |471-25. 3. $232-47.
4. $332-50. 5. $1680, $1120, $700. 6. $4500, $3600, $2400.
7. $82-50. 8. 0-8%. 9. $187 '50; 1^%. 10. • $1980, $2640.
11. $2393-62, $5106-38. 12. U %. 13. i%; 87ict. per $100.
14. $46-87i 16. I %. 16. $2000. 17. $12500. 18. $245.
19. $88333-33, $14000. 20. $1851*94. 21. $4273-50. 22. $18918
23. 65 ct. per $100. 24. $5930.
Exercise XLIII. 1. $81, $1539. 2. $6-43. 3. $6240.
4. $3264. 6. $3055. 6. $500. 7. 3%. 8. 3 yd. 9. $18000.
10. $2. 11. $480. 12. 5%. 13. $61200. 14. $1388-62.
15. $288-39. 16. $18909 18. 17. 66521b., 19011b. 18. $25663 44.
19. $9653-38. 20. 2%. 21. 2i %.
Exercise XLIV. 1. $60. 2. $747*80. 3. $5*40. 4. $175.
6. 28 %. 6. 31i %. 7. $50. 8. $1-27 ; 30^ %. 10. 25 %•
11. 20%. 12. 14^%. 13. 16|%. 14. 20%. 16. 25-17%
16. 20%. 17. 13%. 18. 20%.
Exercise XLV. 1. $308-02. 2. $960-12. 3. $445-17.
4. $78-54. 6. $573-04. 6. $382-90. 7. 8%. 8. 6%. 9. $432-38.
10. $751-28. 11. June 4.
Exercise XLVI. 1. $85-19. 2. $28-90. 3. $14-33. 4. $11.
6. $33-61. 6. $2-19. 7. $349*49. 8. $504-96. 9. $333-33.
10. $831-60. 11. $72-51. 12. $98-40. 13.7%. 14. 7^%^
15. 6|%. 16. 5%, 17. 6%. 18. lyr. 276 da. 19. lOOda.
20. Oct. 13. 21. 20 yr. 22. 25 yr. 23. 8%. 8-034%.
24. $7-27; $7-14. 26. ll^j%. 26.10-267%.
Exercise XLVII. 1. 12 Nov. 2. 17 Dec. 3. 30 June.
4. 10 Oct. 5. 2 Nov. 6. 4 July or 8 July, 1889.
Exercise XL VIII. 1. $22-58. 2, $2364-33. 3. $71*41.
Exercise XLIX. 1. $92610; $12610. 2. $497-19; $72-19.
3. $281-38; $3138. 4. $404 83; $38-16. 5. $766-95; $44-45.
6. $106. 7. $1-0609. 8. $1061364. 9.1-357625. 10.1-276281.
11. 1-4071. 12. 1040604. 13. 1082857. 14. 1-126825.
ANSWERS. 859
Exercise L. 1. $1559-20; $809-20. 2.. $659-23; $294-23.
3. $2770-89; $1520-89. 4. $48-90; $12-65. 5. $650-17; $222-67.
6. $4792 -20 ; $4667 "20. 7. $1470268 ; $1470268 ; $1 -796076 x 10^ * •
$1-829594x102 2. 8. $456 39. 9. $257*20. 10. $256-61.
11. $62-50. 12. $41-67. 13. $4826. 14. $4872. 15. $48-95.
16. 8yr. 17. 6yr. 314 da. 19. 29 yr. 325 da. 20. 29 yr. 129 da.
21. 29 yr. 25 da. 22. 2^%. 23. 9%. 24. Nearly 5%.
25. 53 yr. 29 da. 26. 114 yr. 34 da. 27. 6-167%; 6183%;
6-184%. 28. 3-6^%. 38-94.
Exercise LI. 1. $3275. 2. $1237-50. 3. $32700. 4. $10468-75.
5. $69231-25. 6. $679204. 7. $182062-50. 8. $111J00.
9. $22812-50. 10. $71250. 11. $4200. 12. $210. 13. 324 %.
14. 125:126. 15. 110^. 16. 117^ 17. 90f. 18. . 280.
19. $15807-63. 20. $19989; $827-50; 4-14%. 21. $18000; 840.
22. $160000. 23. $35938-44. 24. £5625. 25. 102j.
Exercise LII. 1. $365-00. 2. $109750. 3. $288. 4. $7517.
5. £20 10s. ll^d. 6. £1127 Us. 3d. 7. $241111*11, $48388-89;
$96888-89, $193840; $96833*33, $96722*22, $96750, $96805-56,
$292000, $292000, $389333 -33. 8. $464*89. 9. $464-02. 10. $570-75.
11. 95f . 12. The drafts at 60 days' sight. $648-51. 13. $2098*05.
14. $298*89. 15. $4*84§. 16. 94§. 17. 9|. 18. 518.
19. 95|. 20. £986 13s. 4d. 21. 19552 fr. 22. $28*86.
23. $38*48. 24. $30538*93; 358-664 gr.
360 ARITHMETIC.
CORRECTIONS.
12, line 5 up ; after length insert is the yard which.
Page 77, Prob. 20 ; after 1889 insert and payable 9 July, 1889.
82, Prob. 70 ; after discounted insert at 8 % .
83, Prob. 81 ; /or |6 '60 read $29*40. The answers will then
he $5-39 and |6-60.
Page 84, Prob. 90 ; for payable in read drawn at.
Page 98, line 7 up ; omit of equality.
Page 123, Prob. 141 ; /or ^2 read ^3.
Page 137, Prob. 24 ; hiseH x 3^ » ^ 73 x 11 x 13 -=- 2« -^ 5« -f- 17.
Page- 151, line 13 up ; m second denominator, for 6 read 0.
Page 151, line 5 up ; /or -^10-l read X 10"!.
Page 155, line 9 up ; for partial product read partial products.
Page 162, line 5 up ; for HK read GK.
Page 187, line 10 ; fo'r 06(261+63) read 106(261+63).
Page 192, Prob. 34 ; /or 3 in. read 6 in.
Page 200, Prob. 108, figure ; join FB and HC.
Page 206, Prob. 13 ; for 22 ft., 6ft. and 3 ft. read 53 ft., 48 ft. and
43 ft.
Page 213, line 15 up ; for k read k-^.
Page 223, Prob. 6 ; for diameter read diameters.
Page 249, Prob. 49 ; /or 7 5 "5 lb. reac^ 45 '5 lb. a7idinsert\h.afterb*J'(S.
Page 307, Prob. 23 ; for at a total loss of $1943-90 read at a loss
of $3514*75 on the amount realized by his former sales.
The answer will then he $64980.
Page 316, line 12 up ; for "574 read. -bji.
m
QA Glashan, J. C.
103 Arithmetic for high schools
G53 and collegiate institutes
Physical &
Applied Sci»
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