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THE 

PHILOSOPHY OF ARITHMETIC 



AS DEVELOPED FROM THE 



THREE FUNDAMENTAL PROCESSES 



or 

SYNTHESIS, ANALYSIS AND COMPARISON 



CONTAINING ALSO 



A HISTORY OF ARITHMETIC 
REVISED EDITION. 



EDWARD BKOOKS, PH. D., 

Superintendent of the Public Schools of Philadelphia. 

LATE PRINCIPAL OF STATE NORMAL SCHOOL. PENNSYLVANIA, AND AUTHOR OF A 
NORMAL SERIES OF MATHEMATICS. 



"The highest Science is the greatest simplicity." 



PHILADELPHIA : 
NORMAL PUBLISHING COMPANY. 

1904. 




QA 



Entered according to Act of Congress, In the year 1876, by 

EDWARD BROOKS, 
In the Office of the Librarian of Congress, at Washington. 



COPYRIGHT, 1901, by EDWARD BROOKS 



LECTROTYPKI> A PRINTKD 

BY 

THE WICK F.aH*M PRINT-ISO CO., 
LANCASTER, PA. 




PREFACE. 



T"\ROGRESS in education is one of the most striking characteris- 
_L tics of this remarkable age. Never before was there so general 
an interest in the education of the people. The development of 
the intellectual resources of the nation has become an object of 
transcendent interest. Schools of all kinds and grades are multi- 
plying in every section of the country; improved methods of train- 
ing have been adopted ; dull routine has given way to a healthy 
intellectual activity ; instruction has become a science and teach- 
ing a profession. 

This advance is reflected in, and, to a certain extent, has been 
pioneered by, the improvements in the methods of teaching arith- 
metic. Fifty years ago, arithmetic was taught as a mere collection 
of rules to be committed to memory and applied mechanically to 
the solution of problems. No reasons for an operation were given, 
none were required ; and it was the privilege of only the favored 
few even to realize that there is any thought in the processes. 
Amidst this darkness a star arose in the East ; that star was the 
mental arithmetic of Warren Colburn. It caught the eyes of a few 
of the wise men of the schools, and led them to the adoption of 
methods of teaching that have lifted the mind from the slavery of 
dull routine to the freedom of independent thought. Through the 
influence of this little book, arithmetic was transformed from a dry 
collection of mechanical processes into a subject full of life and in- 
terest. The spirit of analysis, suggested and developed in it, runs 
to-day like a golden thread through the whole science, giving sim- 
plicity and beauty to all its various parts. 

(iii) 



IT PRKPACE. 

No one who did not in his earlier years learn arithmetic 
by the old mechanical methods, and who has not experienced 
the transition to the new analytic ones, can realize the com- 
pleteness of the revolution effected by this little work. But great 
as has been its influence, it should be remembered that it does 
not contain all that is essential to the science of numbers. Analysis 
in it.s minion, has done all that it was possible for it to accomplish, 
but it is not su ftiririit for the perfection of a science. There must be 
Hynthetic thought to build up, as well as analytic thought to separate 
and simplify. Comparison and generalization have an important 
work to perform in unfolding the relations of the various parts and 
in uniting them by the logical ties of thought, which should bind 
them together into an organic unity. What we now need for the 
perfection of the science of arithmetic and our methods of teaching 
it, is a more philosophical conception of its nature, and a logical 
relating of its parts which analysis leaves in a disconnected condition. 

It is worthy of remark that urithmetic,in respect to logical symme- 
try and completeness, differs widely from its sister branch geom- 
etry. The science of geometry came from the Greek mind almost as 
perfect as Minerva from the head of Jove. Beginning with definite 
ideas and self-evident truths, it traces its way, by the processes of 
deduction, to the profoundest theorem. For clearness of thought, 
closeness of reasoning.and exactness of truths, it is a model of excel- 
lence and beauty. It stands as a type of all that is best in the classi- 
cal culture of the thoughtful mind of Greece. Geometry is the per- 
fection of logic ; Euclid is as classic as Homer. 

The science of numbers, originating at the same time, seems to 
have presented less attractions or greater difficulties to the Greek 
mind. It is true that the great thinkers grew enthusiastic in the 
contemplation of numbers, and spent much time in fanciful specu- 
lations upon their properties, but this did comparatively little for 
the development of the science. The present system of arithmetic 
i mainly the product of the thought of the past three or four cen- 
tariefl. Developed by minds less logical than those of the old 



Greeks, and growing partly out of the necessities of business, it 
seems not to have acquired that scientific exactness and finish 
which belong to the science of geometry. That it has intrinsically 
as logical a basis and will admit of as logical a treatment, cannot 
be doubted. To endeavor to exhibit the true nature of the science, 
show the logical relation of its parts, and thus aid in placing it 
upon a logical foundation beside its sister branch, geometry, is the 
object of the present treatise. 

The work is divided into five parts, besides the Introduction. 
The Introduction contains a Logical Outline of Arithmetic, and a 
brief History of the science, including an account of the Origin of 
the Arabic system, the Origin of the Fundamental Operations, and 
an account of the Early Writers on the science. The facts pre- 
sented have been gathered from a variety of sources, and have 
been carefully compared, so far as was possible, with the originals, 
to secure entire accuracy in the statements. The principal author- 
ities followed are Leslie, Peacock, De Morgan, Fink, and Ball. 
As much is presented as it is supposed will be of interest to the 
teacher or general reader ; any who desire more detailed in- 
formation are referred to the writers mentioned. 

PART FIRST treats of the general nature of arithmetic, embracing 
the Nature of Number, the Nature of Arithmetical Language, and the 
Nature of Arithmetical Reasoning. The natu of Number is quite 
fully considered, especially in its relation to the idea of Time. 
Various definitions of Number are presented and examined, and 
the effort is made to ascertain that which may be regarded as the 
best for general use. 

The Nature of the Language of Arithmetic is discussed upon a 
broader basis than usual. The true relation of Numeration to 
Notation, which seems to have been overlooked by many authors, 
and which is frequently not understood by pupils, is explained. 
It is shown that Numeration is merely the oral and Notation the writ- 
ten language of Arithmetic. The philosophy of the Arabic system 
of notation, the objections to the decimal scale, and the advantages 
of a duodecimal system of arithmetic, are discussed. 



<r PRKFAC*. 

Considerable attention is given to the nature of Arithmetical 
Reasoning, a subject which seems not to have been very clearly 
understood by logicians and arithmeticians. The effort is made 
to put this matter upon a logical basis, and to ascertain and pre- 
sent the true nature of the logical processes by which the science 
of numbers is unfolded. The ground being almost entirely new, it 
is not to be supposed that the investigation is at all complete ; but 
it is hoped that what is given may induce some one to present a 
more thorough development of the subject. 

The fundamental idea of the work is that arithmetic has a triune 
hisit; that it is founded upon and grows out of the three logical 
processes, Analytic, Synthesis, and Comparison. This is a new gen- 
eralisation, and is believed to be correct. It has been previously 
maintained that all of Arithmetic is contained in the two processes, 
Addition and Subtraction; and that the whole science is a logical 
outgrowth of these two fundamental ones. In this work it is 
shown that Synthesis and Analysis are mechanical operations, giving 
rise to some of the divisions of the science, that the mechanical 
processes are directed by the thought process of Comparison, and 
that this itself gives rise to a larger part of the science. The old 
writers held that we can only unite and separate numbers ; in this 
work it is held that we can unite, $eparate, and compare numbers. 

Proceeding with this idea, it is shown that, regarding Addition, 
Subtraction, Multiplication, and Division, as the fundamental oper- 
ations of arithmetic, there will arise from them several other pro- 
cesses of a similar character, which I have called the Derivative 
Froceun of Synthesis and Analysis. It is then seen that for each 
analytical process there should be a corresponding synthetic pro- 
cess. There will thus arise a new process, the opposite of Factoring, 
to which I have given the name of Composition. This process, it 
will be seen, contains several interesting cases, which correlate with 
the different cases of Factoring. It is of especial interest in Alge- 
bra, as may be seen in my Elementary Algebra, 

Continuing this thought, it is shown that Ratio, Proportion, the 



PREFACE. Vli 

Progressions, etc., are not the outgrowth of either Synthesis or 
Analysis, but of the thought process Comparison. Attention is 
called to the nature of Ratio, a new definition is suggested, and the 
correctness of the prevailing method of finding the ratio of two 
numbers, which has been questioned, is vindicated. Suggestions 
are also made for improvements in some of the definitions and 
methods of treating Ratio, Proportion, Progressions, etc. The log- 
ical character of Percentage is exhibited, and the simplest and most 
practical method of treatment suggested. Several interesting 
chapters are also presented upon the Theory of Numbers. 

The subject of Fractions is quite fully discussed, the attempt be- 
ing made to exhibit their nature and their logical relation to inte- 
gers. The possible cases which may arise are considered, and a 
new case, called the Relation of Fractions, first given in one of my 
arithmetics, and already introduced into several other arithmetical 
works, is presented and explained. It is also shown that the sub- 
ject of Fractions admits of too methods of treatment, logically distinct 
in idea and form, and both treatments are presented. Especial at- 
tention is given to the treatment of Circulates, and the most impor- 
tant principles concerning them are collated. 

The nature of Denominate Numbers, which seems to have been 
imperfectly understood, is explained upon what is regarded as the 
correct basis. They are shown to be numerical expressions of con- 
tinuous quantity, in which some artificial unit is assumed as a meas- 
ure. This leads to the adoption of a new definition of Denominate 
Numbers, different from that which we usually find in our text- 
books. The origin of the measures in the various classes of 
Denominate Numbers is also stated, and many interesting facts 
concerning them are given. 

While the philosophical part of the work is that which will at- 
tract the most attention among thinkers, the historical part will be 
quite as interesting and instructive to the majority of younger 
readers. In the historical part; of course, no claims to original 
investigation are made ; but the best authorities have been con- 



Till 

tolled ; and, in many casea, their very language baa been used, their 
ezpremion being so clear and concise that I could not hope to im- 
prove it In thus combining with the philosophy of arithmetic its 
history, which in many case* aids in unfolding it* philosophy, I 
have aimed to present a work especially valuable to ttudtnlt and 
the younger teacher* of arithmetic. Such a work, I feel, would have 
been invaluable to me in my earlier years as a teacher. 

It is proper to remark that the work was mainly written ntx>nt 
twelve years ago. This might he regarded as an ad vantage; for, 
according to the recommendation of Horace, publication should not 
be hurried, but "a work thoiild be retained till the ninth year." Quin- 
tilian also remark* concerning his own great work on Oratory that 
he allowed time for reconsidering his ideas, " in order that when the 
ardor of invention had cooled I might judge of them on a more 
careful re-perusal, as a mere reader." In re-perusing the manuscript 
I see no reason for any change of opinion, in regard to any of the 
ideas presented, though I am conscious that the manner of pre- 
senting several subjects might, in some respects, be improved by 
being re-written; but I have decided to let them stand as originally 
conceived and expressed, thinking that they may thus gain in fresh- 
ness and vividness of conception what they may lack in elegance of 
style. 

Clirri-iliing ninny pleasant remembrances associated with the 
discussion of these ideas before my pupils in the class-room, to 
many of wlmm th-ir publication will prove a reminder of days 
gone l-y. I commit the work, with its merits and demerits, to an 
indulgent public, with the hope that it may be of assistance to 
the younger members of the profession, and contribute somewhat 
towards the fuller appreciation of the interesting and beautiful 
science of numbers. EDWARD BROOKS. 

Normal School. Millertrille, Pa., 
January 16. 1876. 

I revise the work after twenty-five years, giving the latest 
discoveries in the history of arithmetic. 

EDWARD BROOKS. 

Philadelphia, May 20, 1901. Supt. Public School*. . 



TABLE OF CONTENTS. 



INTRODUCTION. 

PA0K. 

CHAPTER I. Logical Outline of Arithmetic 9 

II. Origin and Development of Arithmetic 17 

III. Early Writers on Arithmetic 29 

" IV. Origin of Arithmetical Processes 44 

PART I. THE NATURE OP ARITHMETIC. 
SECTION I. The Nature of Number. 

CHAPTER I. Number, the Subject-matter of Arithmetic 67 

" II. Definition of Number 72 

" III. Classes of Numbers 76 

" IV. Numerical Ideas of the Ancients 81 

SECTION II. Arithmetical Language. 

CHAPTER I. Numeration, or the Naming of Numbers 93 

II. Notation, or the Writing of Numbers 101 

III. Origin of Arithmetical Symbols 108 

IV. The Basis of the Scale of Numeration 113 

V. Other Scales of Numeration . . . .' 121 

VI. A Duodecimal Scale 126 

VII. Greek Arithmetic 135 

VIII. Roman Arithmetic 141 

IX. Palpable Arithmetic 147 

SECTION III. Arithmetical Reasoning. 

CHAPTER I. There is Reasoning in Arithmetic 165 

II. Nature of Arithmetical Reasoning 171 

III. Reasoning in the Fundamental Operations 177 

IV. Arithmetical Analysis 185 

V. The Equation in Arithmetic 193 

VI. Induction in Arithmetic 197 

PART II. SYNTHESIS AND ANALYSIS. 
SECTION I. Fundamental Operations. 

CHAPTER I. Addition 207 

" II. Subtraction 213 

III. Multiplication 221 

IV. Division 227 

SECTION II. Derivative Operations. 

CHAPTER I. Introduction to Derivative Operations 237 

II. Composition 240 

III. Factoring 244 

IV. Greatest Common Divisor 249 

V. Least Common Multiple 257 

VI. Involution 261 

VII. Evolution 267 

1* fix) 



X CONTENTS. 

PART III. COMl'ARISOH. 

SECTION 1. Ratio and Proportion. PAaK 

CHAPTER I. Introduction to Comparison 291 

" II. Nature of Ratio 294 

" III. Nature of Proportion 805 

IV. Application of Simple Proportion . .810 

" V. Compound Proportion 818 

' VI. History of Proportion 826 

SECTION II. The Progressions. 

CHAPTER I. Arithmetical Progression 841 

" II. Geometrical Progression 845 

SECTION III. Percentage. 

CHAPTER I. Nature of Percentage 855 

" II. Nature of Interest 881 

SECTION IV. The Theory of Numbers. 

CHAPTER I. Nature of the Subject 871 

" II. Even and Odd Numbers 875 

" III. Prime and Composite Numbers 878 

" IV. Perfect, Imperfect, etc., Numbers 888 

" V. Divisibility of Numbers 889 

" VI. The Divisibility by Seven 897 

" VTL Properties of the Number Nine 404 

PART FV. FRACTIONS. 

SECTION I. Common Fractions. 

CHAPTER I. Nature of Fractions 418 

" II. Classes of Common Fractions 420 

" III. Treatment of Common Fractions 426 

" IV. Continued Fractions 484 

SECTION II. Decimal Fractions. 
CHAPTER I. Origin of Decimals 443 

II. Treatment of Decimals 455 

III. Nature of Circulates 460 

IV. Treatment of Circulates 464 

V. Principles of Circulates 470 

' VI. Complementary Repetends. 476 

VTI. A New Circulate Form 481 

PART V. DENOMINATE NUMBERS. 

CHAPTER I. Nature of Denominate Numbers 489 

II. Measures of Extension 497 

III. Measures of Weight 512 

IV. Measures of Value 521 

V. Measures of Time 541 

VI. The Metric System 555 



INTRODUCTION 



TO THE 



PHILOSOPHY OF ARITHMETIC 



I. LOGICAL OUTLINE OF ARITHMETIC. 



II. ORIGIN AND DEVELOPMENT OF ARITHMETIC. 



III. EARLY WRITERS ON ARITHMETIC. 



IV. ORIGIN OF ARITHMETICAL PROCESSES. 



INTRODUCTION 



CHAPTER I. 

A LOGICAL OUTLINE OF ARITHMETIC. 

rpHE Science of Arithmetic is one of the purest products of 
_L human thought. Based upon an idea among the ear- 
liest which spring up in the human mind, and so intimately 
associated with its commonest experience, it became in- 
terwoven with man's simplest thought and speech, and was 
gradually unfolded with the development of the race. The 
exactness of its ideas, and the simplicity and beauty of its re- 
lations, attracted the attention of reflective minds, and made 
it a familiar topic of thought ; and, receiving contributions from 
age to age, it continued to develop until it at last attained 
J,o the dignity of a science, eminent for the refinement of its 
principles and the certitude of its deductions. 

The science was aided in its growth by the rarest minds of 
antiquity, and enriched by the thought of the profoundqst 
thinkers. Over it Pythagoras mused with the deepest enthu- 
siasm; to it Plato gave the aid of his refined speculations; and 
in unfolding some of its mystic truths, Aristotle employed his 
peerless genius. In its processes and principles shines the 
thought of ancient and modern mind the subtle mind of the 
Hindoo, the classic mind of the Greek, the practical spirit of 
the Italian and English. It conies down to us adorned with 

(9) 



10 THE PHILOSOPHY OP ARITHMETIC. 

the offerings of a thousand intellects, and sparkling with the 
gems of thought received from the profouadest minds of nearly 
every age. 

And yet, rich as have been the contributions of the past, 
few of the great thinkers have endeavored to unfold its logical 
relations as a science, and discover and trace the philosophic 
thread of thought that binds together its parts into a complete 
and systematic whole. Unlike its sister branch geometry, 
which came from the Greek mind so perfect in its symmetry 
and classic in its logic, the science of arithmetic has been treated 
too much as a system of fragments, without the attempt to 
coordinate its parts and weave them together with the thread of 
logic into a complete unity. To remedy this defect is the special 
object of a work on the Philosophy of Arithmetic, and is the 
task which the author of the present work has with diffidence 
attempted. 

Like all science, which is an organic unity of truths and 
principles, the science of arithmetic has its fundamental ideas, 
out of which arise subordinate ones, which themselves give 
rise to others contained in them, and all so related as to give 
symmetry and proportion to the whole. What are these fun- 
damental and derivative ideas, what is the law of their evolu- 
tion, what is the philosophical character of each individual 
process, and what is the logical thread of thought that binds 
them all together into an organic unity ? These are the ques- 
tions that meet us at the threshold of the effort to unfold a 
philosophy of arithmetic; they are the foundation upon which 
such a superstructure must be erected ; and we begin the 
answer to these questions in the first chapter, under the head 
of A Logical Outline of Arithmetic, which exhibits the fun- 
damental operations and divisions of the science. 

To this Logical Outline the special attention of the reader 
is invited, as it is not only the foundation upon which the au- 
thor has builded, but also the frame-work of the system. In 



A LOGICAL OUTLINE OF ARITHMETIC. 11 

it the science is assumed to be based upon the three processes 
Synthesis, Analysis, and Comparison; general processes* in 
which each individual process must have its root, and from 
which it is developed. This generalization marks a new 
departure in the method of regarding the science, and the re- 
lation of its parts ; and shows the incorrectness of opinions 
around which has gathered the dust of centuries. Our first 
inquiry is, what is A Logical Outline of Arithmetic ? 

All numerical ideas begin with the Unit. It is the origin, 
the basis of arithmetic. From it, as a fundamental idea, 
originate all numbers and the science based upon them. Begin- 
ning, then, at the Unit, let us see how the science of arithmetic 
originates and is developed. 

The Unit can be multiplied or divided. This gives rise to 
two classes of numbers, Integers and Fractions. Integers 
originate in a process of synthesis, Fractions in a process of 
analysis. Each Integer is a synthetic product derived from a 
combination of units; each Fraction is an analytic product 
derived from the division of the unit. There are, therefore, 
two general classes of numbers, Integers and Fractions, 
treated of in the science of arithmetic. 

Having obtained numbers by a combination of units, we may 
unite two or more numbers and thus obtain a larger number 
by means of synthesis ; or we may reverse the operation and 
descend to a smaller number by means of analysis. Numbers, 
therefore, can be united together and taken apart; they can be 
synthetized and analyzed; hence Synthesis and Analysis are 
the two fundamental operations of arithmetic. These funda- 
mental operations give rise to others which are modifications 
or variations of them. Arithmetic, therefore, from its primary 
conception seems to consist of but two things, to increase and 
to diminish numbers, to unite and to separate them. Its pri- 
mary operations are Synthesis and Analysis. 

To determine when and how to unite, and when and how 



1J 1'HILOSOPHY OP ARITUMET1C. 

to separii a process of reasoning cat ;>ari- 

son. Ti. <-in|.;in -s numbers and determines their 

relations. Synthe.-i.s un<l Analysis are mechanical processes , 

parison is the thought process. Comparison directs the 
original processes, modilit -s them so as to product- from them 

, other p: "iitained 

in the original ones. It is, in other u this th<> , 

process working upon the idea of number, that tin- original 

esses of Synthesis and Analysis are directed and modified, 
that other processes are developed from them, and that new 
and independent processes arise, and the science of arithi; 
is developed. Comparison, therefore, in arithmetic as in geom- 
etry, is the process by which the science is constructed, or 
the key with which the learner unlocks its rich storehouse of 
interest and beauty. 

Arithmetic, it is thus seen, consists fundamentally of ti 
things; Synthesis, Analysis and Comparison. Synlfifsis and 

, : >/sis are fundamental mechanical operations, suggested in 

the formation of numbers; Comparison is the fundamental 

thought process which controls these operations, unfolds 

their potential ideas, and also gives rise to other divisions of 

the science jrn>\\ in.ir immediately out of itself. In other 

words, the science of arithmetic has a triune basis; it has its 

i ^rows out of, the three processes, Synthesis 

I, and Cui>ii>tiri*n. Let us examine these processes 

and see the number, nature, and relations of the divisions 

growing out of the fundamental operations, and thus deter- 

logical character of the science of arithmetic. 

lit. A general synthesis is called Aildttmn. A spe- 

nise of -iietic process of Addition, in which the 

nun .^d are all ei|iial, their .-um receiving the came of 

iiK't. is called forming of Com/ 

nberg by a synthesis of factors, which may be called 
Composition; Multiples, formed by a synthesis of particular 
factor? ; and i>y a synthesis of equal factors, are 



A LOGICAL OUTLINE OF ARITHMETIC. 13 

all included under Multiplication. Hence, since Involution, 
Multiples, and Composition, are special cases of Multiplication, 
and Multiplication is itself a special case of Addition, the pro- 
cess of Addition includes all the synthetic processes to which 
numbers can be subjected. 

ANALYSIS. A general analysis, the reverse of Addition, is 
called Subtraction. A special case of Subtraction, in which 
the same number or equal numbers are successively subtracted 
with the object of ascertaining how many times the number 
subtracted is contained in another, is called Division. Factor- 
ing is a special case of Division in which many or all of the 
factors of a number are required ; Evolution is a special case 
of factoring in which one of the several equal factors is re- 
quired ; and Common Divisor is a case of factoring in which 
some common factor of several numbers is required. The 
process of Division, therefore, includes the processes of Factor- 
ing, Common Divisor, and Evolution; and since Division is a 
special case of Subtraction, all of these processes are logically 
included under the general analytic process of Subtraction. 

COMPARISON. By comparison the general notion of relation 
is attained, out of which arise several distinct arithmetical 
processes. By comparing numbers, we perceive the relations 
of difference and quotient; and giving measures to these, we 
have Ratio. A comparison of equal ratios gives us Propor- 
tion. A comparison of several numbers differing by a common 
ratio gives us Arithmetical and Geometrical Progression. In 
comparing concrete numbers, when the unit is artificial, we 
perceive that they differ in regard to the value of the units, 
and also that we can change a number of units of one species 
into a number of another species of the same class ; and thus 
we have the process called Reduction. In comparing abstract 
numbers we notice certain relations and peculiarities which, 
investigated, give rise to the Properties or principles of num- 
bers. In comparing numbers, we may assume some number 
as a basis of reference and develop their relations in regard to 



14 THE PHILOSOPHY OF ARITHMETIC. 

this basis; when this basis is a hundred, we have the pr- 
een called Percentage. 

we obtain a complete outline of the science of nui 
ami more clearly the logical relations of tin- divisions 

of tin- .-. \ rithmetic is conceived as based upon 1 1.< 

. mien ml operations, synthesis and anai 
lions being controlled by which develop.- 

esses from these and also from itself. The who!- 

the outgrowth of this triune basis, E 
thesis, Analysis, and Comparison. The rest of aritln 
consists of the solution of problems, either real or tin < 
and may be included under the bead of Applications of Arith- 
metic. 

This conception of the subject is new and important. It has 
re held that addition and subtraction compre- 
ied the entire science of arithmetic; that all other pn>- 
cesses are contained in them, and are an outgrowth from them. 
This is a fallacy, which, among other things, has led logic 
to the absurd conclusion that there is no reasoning in arith- 
metic. Assuming that there is no reasoning in the prii. 

esses of synthesis and analysis, and that these primary 
processes contain the entire science, they naturally conclude 

there is no reasoning in the science itself. The ana! 
of ti t here given dispels this error and exhibits the 

subject in its true light. Synthesis and Analysi- Q to 

lie the primary mechanical processes; Comparison, the thought 
.,es them with her wand of magic, and they ger- 
minate and bring forth other processes, having their root- 
tbese primary ones. Comparison also becomes the foundation 
rocesses distinct from those of synthesis and anal\ 
-ses which cannot be conceived as growing out of syn- 
luit which have their root in the thought 
process of the science in Comparison. 

This outline of the science grows out of the pure idea of 
number, iudi ; f the language of arithmetic. These 



A LOGICAL OUTLINE OF ARITHMETIC. 



15 



fundamental processes are modified by the method of notation 
employed to express numbers. With the Roman or Greek 
methods of notation, the methods of operation would not be 
the same as with the Arabic system. The method of " carry- 
ing one for every ten," of "borrowing" in subtracting, the 
peculiar methods of multiplying and dividing, grow out of the 
Arabic system of notation. A portion of the treatment of 
common and decimal fractions arises from the notation adopted, 
and the principles and processes of repetends originate in the 
same manner. The methods of extracting square and cube 
root would be different if we employed a different method of 
expressing numbers. It is thus seen that the fundamental 
divisions of arithmetic arise from the pure idea of number, that 
the processes in these divisions are modified by the method of 
notation adopted, and also that some of the principles and pro- 
cesses of the science grow out of this notation. It may be 
remarked, also, that the power of arithmetic as a calculus 
depends upon the beautiful and ingenious system of notation 
adopted to express numbers. 

It is believed that the above view of arithmetic must tend 
to simplify the subject, and that much clearer notions of the 
science will be attained when these philosophical relations are 
apprehended. A general view of the subject is presented by 
the following analytical outline : 

T 9vnthpm's / Addition ' (Composition. 

8< I Multiplication. 4 ( Common Multiple. 

( (.Involution. 

II Analvsis / Subtraction. (Factoring. 
11. Analysis. -[ Division> J r Common Divisor. 

( \ Evolution. 

1. Ratio. 

2. Proportion. 
-3. Progression. 

4. Reduction. 

5. Percentage. 

6. Propertion of Numbers. 



Logical 
Outline 

of 
Arithmetic. 



III. Comparison. 






TUB PHILOSOPHY OF ARITHMETIC. 





D 
O 



CHAPTER II. 

ORIGIN AND DEVELOPMENT OP ARITHMETIC. 

A KNOWLEDGE of Arithmetic is coeval with the race. Every 
people, no matter how uncivilized, must have possessed some 
ideas of numbers, and employed .them in their transactions with 
one another. These ideas would be multiplied, and the methods 
of operation founded upon them gradually extended and improved 
as the nation advanced in civilization and intelligence. The his- 
tory of Arithmetic is, therefore, inseparably connected with the 
history of civilization and the race. The origin of its elementary 
processes must, of necessity, be involved in obscurity and uncer- 
tainty. History can speak positively only of some of the higher 
and more recent developments of the science. 

In presenting some of the principal facts concerning the history 
of arithmetic, we shall consider three things : the origin of our 
present system of arithmetic ; the early writers on the science ; and 
the origin of the fundamental operations. Other historical facts 
will be mentioned in connection with the particular subjects to 
which they belong. One of the most interesting inquiries is that 
which relates to the origin of the system of arithmetic now gen- 
erally adopted, which we shall consider in the present chapter. 

The basis of our system of arithmetic is the principle of place- 
value in writing numbers. All civilized nations, from the primi- 
tive habit of reckoning with the fingers, adopted a system of count- 
ing by groups of ten. Each group of ten is distinguished by a 
special name, and the names of the first nine numbers are used to 
number the groups and express the numbers between them. Thus 
all civilized peoples adopted the same general method of oral 
arithmetical language. In writing numbers, however, different 
2 (17) 



1" THE PHILOSOPHY OF ARITHMETIC. 

nations adopted widely different methods of notation. Our present 
simple and practical system of notation was reached by only a 
single nation of antiquity. The various methods of writing num- 
bers in use among the ancient nations and the origin of our present 
system will be briefly considered. 

The Egyptians represented numbers by written words, and also 
by symbols for each unit repeated as often as necessary. In one 
of the tombs near the pyramid of Gizeh, hieroglyphic numerals 
have been found in which 1 is represented by a vertical line ; 10 
by a kind of horse-shoe ; 100 by a short spiral ; 10,000 by a point- 
ing finger ; 100,000 by a frog, and 1,000,000 by the figure of a 
man in the attitude of wonder. In their hieratic writing they used 
symbols for numbers, but they did not combine them on the prin- 
ciple of place-value as in the modern system of notation. There 
are special characters for the nine units, and also for the tens, the 
hundreds, etc. The following are specimens of these symbols : 

I II III _ 1 A A ( A j. ~1 ill 3 

12 8 46 10 20 3040506070 

These are combined on the additive principle, the symbol of the 
larger value always being placed at the left of that for the smaller 
value. The papyrus of Ahmes in the British Museum indicates 
that the Egyptians at a very early period had considerable knowl- 
edge of the art of arithmetic. 

The ancient Babylonians used the wedge-shaped characters of 
their cuneiform system of writing in the representation of numbers. 
The mark for unity, a vertical arrow head, is repeated up to ten, 
whose symbol is a barbed sign pointing to the left. These symbols 
by mere repetition served to express numbers up to one hundred, 
for which a new sign was employed. The characters were written 
sometimes one beside another, and sometimes, to save space, one 
over another. The symbol for the smaller number written to the 
right of the symbol for a hundred denoted addition ; the same sym- 
bol written on the left denoted multiplication, or the number of 
hundreds. The Babylonians thus employed the principle of place 
value to some extent, but having no symbol for zero they were una- 



ORIGIN OP THE SCIENCE. 19 

ble to develop the modern system of notation and calculation. They 
used also along with the decimal system the sexagesimal system, that 
is one with a base of 60, and their operations with both integers 
and fractions show considerable mathematical facility and skill. 

The Chinese had a well developed number system, and seem to 
have come as near the present method of notation as any nation of 
antiquity, except the Hindoos. Of their early number symbols but 
little seems to be known. Later, as a result of foreign influence, 
there arose two new kinds of notation, whose figures are supposed 
to resemble the ancient symbols. Though the Chinese wrote their 
word-symbols in columns, yet their numbers were written from 
left to right, beginning with the highest order. The ordinal and 
cardinal numbers are usually arranged in two lines, one above 
another, with zeros in the form of small circles appearing as often 
as necessary. The following symbols will illustrate this system : 

II X u_ * fi o 

2 4 6 10 10,000 

" X 

/700 -*-_,_ 

20,046 

Their arithmetical calculations were made by means of the 
abacus or swan-pan, which is used at the present day among both 
their scholars and their merchants. 

The Phoenicians expressed numbers in words, and also by the 
use of special numerical symbols, using vertical marks for the 
units and horizontal marks for the tens. The Syrians somewhat 
later used the twenty-two letters of their alphabet to represent the 
numbers 1, 2, ... 9, 10, 20, ... 90, 100, ... 400 ; 500 was 
400+ 100, etc. The thousands were represented by the symbols 
for units with a subscript comma at the right. The notation of 
the Hebrews followed the same plan. None of these nations had 
a notation that could be used in making calculations as we do 
with the modern system. 

The early Greeks seem to have used the initial letters of their 



20 TIM: i-Hii.osoiMiY or AIMTIIMKTIC. 

number words to represent written numbers ; as (~) for 5 
A for 10 (<5a), and these letters were repeated as often as neces- 
sary. Soon after 500 B. C. two new systems appeared among the 
B. < )nc used the 24 letters of the Ionic alphabet in their 
nat iirnl order for the numbers from 1 to 24. The other arranged 
these letters, together with three other symbols, in an arbitrary 
order, thus aI, ft = 2, . . . < = 10, =20, . . . P=WO, <> = 
2( H, etc. The Greeks could perform the fundamental operations 
with these symbols with considerable facility, as may be seen in 
tin chapter on Greek arithmetic. The common method of calcu- 
lation, however, seems to have been with the abacus. The Greeks 
did not make use of the principle of place value, and they had no 
symbol for zero. 

The Romans also expressed numbers by means of letters. 
The characters are supposed to have been inherited from the 
Etruscans, and may originally have been symbolic, and subse- 
quently, on account of their resemblance to forms in their alpha- 
bet, they were replaced by letters. Mommsen says that the Roman 
numerals I, V, X represent the finger, the hand, and the double 
hand respectively. These characters were combined according to 
the additive principle, as in VI, VII, VIII, and also in accord- 
ance with the subtractive principle, as in IV, IX, XI,, XC. This 
Btihtractive principle is a distinctive characteristic of the Roman 
gystem of notation. The Romans could not use their notation for 
reckoning, but made their calculations with counters (calculi) or 
with the abacus. They seem to have had no conception of place 
value, or of a symbol for zero, as their system did not call for it. 

Our present number system, it is now known, had its origin 
among the Hindoos. They originated the modern position-system, 
Kinl introduced the zero to fill an unoccupied place. Their 
d books which have been in the hands of the priest- 
hood fur ci-nturies, contain the numerical characters. Their ear- 
liest symbols of the nine digits were after 3 merely abridged num- 
ber words, and tin- use of letters as figures is said to date from the 
second century B. C. The development of the system of place- 




ORIGI 

value seems to have 
writing numbers there is no indical 
value, though it appears in two other systems" 
prevailed in Southern India. Both of these methods 
tinguished by the fact that the same number can be made up in 
various ways. One method consisted in employing the alphabet, 
in groups of nine symbols, to denote the numbers from 1 to 9 re- 
peatedly, while certain vowels denote the zeros. A second 
method used type-words, and combined them according to the law 
of position. Thus abdhi (one of the 4 seas) = 4 ; surya (the sun 
with its 12 houses) = 12 ; a$vin (the two sons of the sun) = 2. 
The combination abdhisuryayvinas denoted the number 2124. 
These no doubt were stepping-stones to the present simple appli- 
cation of the position principle. The modern system of place 
value could not have been adopted before the invention of the 
zero, and there is no proof of its being introduced before 400 A. D. 
The first known use of the symbol on a document, Cantor says, 
dates from 738 A. D. 

The Arabs became acquainted with the Hindoo number-system 
and its figures, including zero, in the eighth century, and were 
instrumental in introducing the system into Europe. It was for 
many years thought that the present system of arithmetic origi- 
nated with the Arabians. The characters in general use were 
called Arabic characters, and the method of writing numbers was 
known as the Arabic system of notation. Further proof, it was 
thought, was found in the two words "cipher" and "zero," 
cipher being the Arabic as-sifr, meaning empty. This word, 
however, was derived no doubt from the Sanskrit name of the 
naught, sunya, the void. In Italy the character for naught was 
called " zephiro," which has, by rapid pronunciation, been changed 
to zero. The Arabs, however, it is now known, were not the 
authors of the system, but derived it from the Hindoos, and were 
only instrumental in introducing it into Europe. 

The Arabs from an early period had commercial relations with 
India, which brought them in contact with the Indian system of 



22 THE PHILOSOPHY OP ARITHMETIC. 

reckoning. It is known that they were acquainted with the 
Hindoo number system and its figures, including the zero, as 
early as the eighth century. The earliest definite date, says Ball, 
assigned for the use in Arabia of the decimal system of notation 
is 773. In that year some Indian astronomical tables were 
brought to Bagdad in which it is almost certain the Indian 
numerals, including the zero, were used. The Arabs no doubt 
developed the system somewhat slowly, as the custom of writing 
out number words continued among them until the beginning of 
the eleventh century. In the investigation we meet with the 
singular fact that the Arabs employed two kinds of figures : one 
used chiefly in the East called " Oriental ;" another used by the 
Western Arabs in Africa and Spain called the Gubar or dust 
numerals, so called because they were first introduced among the 
Arabs by an Indian who used a table covered with fine dust for the 
purpose of ciphering. These Gubar numerals are the ancestors 
of our modern numerals. They are said to be modifications of the 
initials of the Sanskrit word-numerals. 

It was through Spain, however, it is generally believed, rather 
than directly from Arabia, that the Arabic system was introduced 
into Europe. The Moors as early as 747 had conquered Spain, 
and established there their rule. They brought with them a taste 
for learning, and established schools and universities, so that by 
the tenth and eleventh centuries they had attained to a high degree 
of civilization. Though the political relations of the Arabs with 
the caliphs of Bagdad were not entirely cordial, yet they gave 
ready welcome to the works of the great Arabian mathematicians. 
The Arabs had studied with great avidity the Greek mathematics, 
and their translations of Euclid, Archimedes, Ptolemy, etc., along 
with the works of the Arabians themselves on arithmetic and 
algebra, were studied at the great Moorish universities of Gren- 
ada, Cordova and Seville. 

Thus while the Christian world was enveloped in ignorance, 
the Arabs were cultivating the learning and literature of Greece. 
Though not highly gifted with creative powers of mind by 



ORIGIN OF THE SCIENCE. 23 

which they made many valuable additions to what they thus 
acquired, a debt of gratitude is due them because they " preserved 
and fanned the holy fire." Their efforts at conquest had been 
crowned with brilliant success. Spain had yielded to their sway, 
and the Moors had become celebrated throughout Europe for the 
splendor of their institutions, the magnificence of their architecture, 
and the proficiency of their scholars. 

Disgusted with the trifling of their own schools, energetic and 
aspiring young men from England and France repaired to Spain 
to learn philosophy from the accomplished Moors. There they 
studied arithmetic, geometry and astronomy, and made themselves 
familiar with the Arabic method of notation and calculation. On 
their return they brought the characters and methods of the 
Arabic arithmetic with them and introduced them to the scholars 
of Northern Europe, and thus in time they gradually displaced the 
Roman system, which had been in use for many centuries. 

One of these tl pilgrims of science " was an obscure monk of 
Auvergne named Gerbert, who died in 1003. Returning to his 
native country he became widely celebrated for his genius and 
learning, and subsequently rose to the papal chair with the title 
of Sylvester II. His treatises on arithmetic and geometry were 
valuable, presenting many rules for abbreviating the operations in 
common use. He introduced an improvement in the use of the 
abacus by marking each of the nine beads in every column with a 
distinctive sign. These marks, called apices, are supposed to 
have been the same as the Gubar numerals, and thus Gerbert did 
much to introduce the old Hindoo numeral-forms into Western 
Europe. 

Efforts have been made to ascertain what persons were most 
conspicuous in the introduction of the Arabic characters into 
Northern Europe. There seems to have been some difficulty in 
obtaining access to the Moorish universities, as the Moors are said 
to have taken pains to conceal their learning from the Christian 
world. One of the earliest students from Christian Europe to 
acquire a knowledge of Moorish aud Arabian science was an Eng- 



24 TIIK rilll.o.M.l'HY OF ARITHMETIC. 

Bftb monk, Adelhard <>t liath, who, disguised as a Mahommedan 
Indent, got into Conlo\a :iliout 1120 and obtained a copy of 
Euclid's Elements. This copy translated into Latin is said to 
have been the foundation of all the editions of the work known in 
Europe until 1533. 

Another scholar who was influential in the introduction of 
Moorish learning into Northern Europe was Abraham Ben E/ra. 
a Jewish rabbi, born at Toledo in 1097 and died at Rome in 11C7. 
He wrote an arithmetic in which he explains the Arabic sy 
of notation with nine symbols and a zero, and gives the funda- 
mental processes of arithmetic and the rule of three. 

Another eminent scholar who aided in the introduction v/a> 
Gerard, born in 1114 and died in 1187. He translated the Arab 
edition of the Almage$t of Ptolemy in 1136, which seems to have 
been the earliest text-book among the Arab- that contained tin- 
Arabic notation, and which it is thought was instrumental in the 
introduction of the system to the Moors in Spain. A contempo- 
rary of Gerard, John Ilispalensi-, a Jewish rabbi converted to 
Christianity, translated several Arab and Moorish works, and also 
wrote a treatise on algorism, which is said to contain the earliest 
pies of the extraction of the square root of numbers by the 
aid of decimal numbers. 

The introduction of the Arabic system throughout Europe pro- 
ceeded slowly. The Roman system of calculation with the abacus 
had been in use many centuries, and it was difficult to lead the 
people to make the transition from it to the new system. The 
struggle between these two schools of arithmeticians, the old 
abacistic school and the new algoristic school, was long and no 
doubt often bitter. It was not easy for the mathematicians 
and business men who had been brought up on a system of calcu- 
lation with the abacus to drop it and adopt the new method of 
ronipiitincr with abstract symbols. 

One of the most influential men in bringing about the general 
OK of the new system was Leonardo Fibonacci, born at Pisa in 
1175. Educated in his youth at Bugia in Barbary, where his 



ORIGIN OF THE SCIENCE. 25 

father had charge of the custom house, he became acquainted with 
the Arabic system of notation and with the great work on algebra 
by Al Khowarazmi. He returned to Italy about 1200, and in 
1202 composed a treatise on mathematics known as Liber abaci, 
in which he explains the Arabic system of notation, and points 
out its great advantage over the Roman system. It begins thus : 
" The nine figures of the Hindoos are 9, 8, 7, 6, 5, 4, 3, 2, 1. 
With these nine figures and with this sign, 0, which in Arabic is 
called sifr, any number may be written." This work had a wide 
circulation, and practically introduced the use of the Arabic 
system throughout Christian Europe. It is supposed that the 
system was known before this time to the leading mathematicians 
who had read the works of Ben Ezra, Gerard and Hispalensis, 
and also by Christian merchants who had traded with the Mahom- 
edans, but the wide reputation of Leonardo gave a great impetus 
to its general adoption. 

The Arabic numerals were used at an early day by the astron- 
omers in composing calendars, and these calendars aided in dis- 
seminating a knowledge of the system. Shortly after the appear- 
ance of Leonardo's work, Alphonso of Castile, in 1252, published 
some astronomical tables founded on observations made in Arabia, 
which were computed by the Arabs and published in the Arabic 
notation. A frequent and free use of the zero in the 13th cen- 
tury is shown in the tables for the calculation of the tides at Lon- 
don and of the duration of moonlight. There is an almanac pre- 
served in one of the libraries of Cambridge University containing a 
table of eclipses for the period from 1330 to 1348. This almanac 
contains a brief explanation of the use of numerals and the prin- 
ciples of the denary notation, indicating that at that date the 
system was not generally understood. 

A little tract in the German language entitled De Algorismo, 
bearing the date of 1390, explains with great brevity the digital 
notation and the elementary rules of arithmetic. At the end of a 
short missal similar directions are given in verse, which from the 
form of the writing seems to belong to the same period. The 



THE PHILOSOPHY OF ARITHMETIC. 

characters, of uhi.-h tliuse in the margin are lac-similes, are in 
both manuscript* written tim 
right to left, the order whirl, the 
aus would naturally follow. 

Tin- ureat Italian poet, Petrarch, has the honor of leaving us 
one of the oldest authentic dates in the numeral characters. The 
.lati i- l;',7.~>, written upon a copy of St. Augustine. The college 
a.'.-i.uiKs in the English universities were generally kept in 
K.man numerals until the beginning of the sixteenth century. 
1 In- Arabic characters were not used in the parish registers of 
England before 1600. The oldest date met with in Scotland is 
that of 1490, which occurs in the rent-roll of the Diocese of St. 
Andrews. In Caxton's Mirrour of the World, issued in 1480, 
there is a wood-cut of an arithmetician sitting before a table on 
which there are tablets with Hindoo numerals upon them. 

According to Fink the Roman symbols were generally used in 
Germany with the abacus up to the year I.'HHI. From the Kith 
century on, these Hindoo numerals appear more frequently in 
Germany on monuments and in churches, but at that time they 
had not become common among the people. The oldest monu- 
ment in Germany with Arabic figures (in Katherein near Trop- 
pau) is said to date from 1007, and such monuments are found in 
Pforzheim (1371) and in Ulm (1888). In the year 1471 there 
appeared in Cologne a work of Petrarch with page numbers in 
the Arabic figures, and in 1482 the first German arithmetic with 
similar numbering was published at Bamberg. 

It mu>t have been somewhere from the year 1400 to 1450 that 
the Arabic system of arithmetic began to be generally dissemi- 
nated throughout Europe. Men of science and astronomers had 
become acquainted with the system by the middle of the 13th 
century. Tin- trail" of Europe during the 13th and 14th centu- 
ries was mostly in Italian hands, and the advantages of the alg.r- 
istic system led to its adoption in Italy for mercantile purposes. 
The change, however, was not made even among merchants with- 
out considerable opposition ; thus an edict was issued in Florence 



ORIGIN OF THE SCIENCE. 27 

in 1299 forbidding bankers to use the Arabic numerals, and the 
authorities of the University of Padua in 1348 directed that a list 
should be kept of books for sale with prices marked 4; non per 
cifras sed per literas claras." Most merchants seem to have con- 
tinued to keep their accounts in Roman numerals until about 1550, 
and monasteries and colleges until about 1650 ; though in both 
cases it is thought that the processes of arithmetic were performed 
by the Arabic system. It was not until the sixteenth century 
that the Hindoo position-arithmetic and its notation first found 
complete introduction among the civilized people of the West. 

The forms of several of the figures have undergone considerable 
change since their first introduction into Europe. In the oldest 
manuscripts the figures 4, 5 and 7 are most unlike the present 
characters. The 4 consisted of a loop with the ends pointing down 
thus 8; the 5 has some likeness to the figure 9, thus ^, and the 
7 is simply an inverted V, thus A. In the dates used by Caxton 
in the year 1480, the 4 has assumed its present shape, but the 5 
and 7 are still unlike the same characters of to-day. No reason is 
assigned for these changes ; they seem to have been gradual, and 
the result of chance rather than of intention. The forms of the 
figures at different periods may be seen in the table given on page 
forty-three. 

This explanation of the introduction of the Arabic characters 
and system of notation into Europe through Spain is the one now 
generally accepted as correct. M. Woepcke, an excellent Arabian 
scholar and mathematician, thinks that the Indian figures reached 
Europe through two different channels ; one passing through 
Encrypt about the third century ; another passing through Bagdad 
in the eighth century, and following the track of the victorious 
Islam. The first carried the earlier forms of the Indian figures 
from Alexandria to Rome, and as far as Spain ; the second carried 
the later forms from Bagdad to the principal countries conquered 
by the Kaliffs, with the exception of those where the earlier or 
Gubar figures had already taken firm root. The Gubar figures, 
he thinks, were adopted by the Neo- Pythagoreans, and introduced 



28 IMI run i>.-i>mt " AKI i H.MKTIC. 

Italy and n province*, Gaul and Spain, as early a* 

nth century, w> that the Mohammedans wh-n they reached 

Spain in the eighth century, found these figures already estab- 

niii adopted them. And so, likewise, when in the 

ninth ami tenth centuries the new Arabic treatises on arithmetic 

arri\ed in Spain from the East, they naturally adopted the mi, 

c,t system of ciphering carried on without the ahaen>, and 

kept the figures to which they as well as the Spaniard had 

been accustomed for centuries, and thus the Gubar figure* \\< -re 

retained by them. The only change produced in the ciphering 

Europe by the Arabs was, he claims, the suppression of the 

abacus, and the more extended use of the cipher required by the 

n. w >\>reni of reckoning. 

In the preparation of this and the following chapter, I have re- 
1 valuable assistance from Fink's History of Mathein-uio. 
translated from the German by Be man and Smith, and from Ball's 
lli>ti>ry of .Matheniatics, both valuable works to which the read* -r 
is referred for further information. I have al>o rec(i\ed many 
valuable suggestions from Dr. David Eugene Smith, I'rofe.-.-or of 
Mathematics in Teachers College, Columbia l'niver>ity, N 
The great authority on the history ot mathematics i- M-nit/ 
Cantor, whose works, however, have nut been tian-i.u. <i into 
English, 



CHAPTER III. 

EARLY WRITERS ON ARITHMETIC. 

ANE of the earliest known treatises on mathematics is the 
Ahmes papyrus of the British Museum. The manuscript was 
written by an Egyptian scribe named Ahmes sometime between 
2000 B. C. and 1700 B. C. The title of the work is " Directions 
for Obtaining the Knowledge of all Dark Things." It is believed 
to be a copy, with emendations, of a much older treatise, so that 
it probably represents the knowledge of the Egyptians on arith- 
metic many centuries earlier than its own date. Two other 
mathematical papyri have recently been found belonging to a 
much earlier period than that of Ahmes, which without entirely 
agreeing with the papyrus of Ahmes, exhibit many similarities to 
it, especially in the method of treating fractions. So that we have 
some knowledge of Egyptian arithmetic as early as the twelfth 
dynasty, or about 2oOO B. C. 

The treatise of Ahmes consists of the solution of problems on 
arithmetic and geometry ; the answers are given, but generally 
not the processes by which they were obtained. It deals with 
both whole numbers and fractions. The treatment of fractions is 
peculiar in that it is limited to those having unity for the numer- 
ator, except in the single case of . Fractions that cannot be 
expressed with a unit numerator are represented by the sum of 
two or more fractions whose numerators are each a unit ; thus for 
| Ahmes writes ^. A fraction is designated by writing the 
denominator with a certain symbol above it to indicate its nature. 
Special symbols were used for , ^, and . Ahmes treats 
also of numerical equations, ns when he says, " heap, its seventh, 
its whole, it makes nineteen ;" that ie, find a number such that the 

(20) 



"' Till PHILOSOPHY OP ARITHMETIC. 

sum of it and one-seventh of it shall equal 19 ; the answer given 
is 16 + ^ -f . The word hau or " heap " signifies the unknown 
quantity, or x, as seen again in the following: " heap, its $, 
its |, its }, its whole, gives 37 ; that is, $x + $x + \x + x 37." 
treatise contains examples in arithmetical and geometric-ill 
progression, and employs the method of " false position " so 
popular among the Hindoos, Arabs and modern Europeans. 

The Greeks obtained much of their mathematical knowl- 
originally from the early Phoenicians and Egyptians. They culti- 
vated the science of numbers to some considerable extent, but 
failed to invent a simple and convenient method of notation by 
which operations with numbers could be performed with any de- 
gree of facility. Like many other nations of antiquity, they 
depended upon the abacus in performing the operations of the 
fundamental rules, though in the time of Archimedes and Apol- 
lonius they could perform these operations to some extent by 
means of their notation. The science of arithmetic with the 
Greeks was speculative rather than practical. They did not 
to aim at the development of skill in computation, but delighted 
in investigating the properties of numbers and in the discovery <! 
fanciful analogies among them. It is a matter of surprise that 
while their works on geometry have been the models of later 
writers on that subject, the Greeks contributed but little of value 
to the science and art of numbers. 

One of the earliest Greek writers on mathematics was Pytha- 
goras, an ancient geometer who is supposed to have lived from 
about 580 to about 500 B. C. He brought from the Kiist a pas- 
tor the mysterious properties of numbers, under the v 
which he probably concealed some of his secret and esoteric doc- 
trines. He regarded numbers as of divine origin the fountain of 
existence the model and archetype of things the essence of the 
universe. He divided them into classes, to each of which 
assigned distinct and peculiar properties. They wore Even and 
Prime and Composite, Plane and Solid, Triangular. Square, 
and Cubical. Even numbers were regarded as feminine ; odd 
numbers as masculine, partaking of celestial natures. 



EARLY WRITERS ON ARITHMETIC. 31 

Euclid, born about 330 B. C., was one of the early Greek 
writers upon arithmetic. His treatise is contained in the 7th, 8th, 
9th and 10th books of Euclid's Elements, in which he treats of 
the theory of numbers, including prime and composite numbers, 
greatest common divisor, least common multiple, continued pro- 
portion, geometrical progressions, etc. He develops the theory of 
prime numbers, shows that the number of primes is infinite, 
unfolds the properties of odd and even numbers, and shows how 
to construct a perfect number. These books of arithmetic are not 
included in the common editions of Euclid, but are found in an 
edition by the celebrated Dr. Barrow. It is supposed that Euclid 
was quite largely indebted to Thales and Pythagoras for his 
knowledge of the subject, though he undoubtedly added much to 
the science himself. The school at Alexandria in which he 
taught was highly celebrated, being attended by the Egyptian 
monarch Ptolemy Lagus. It was this pupil to whom Euclid, 
upon being asked if there was not an easier method of learning 
mathematics, is said to have replied, " There is no royal road to 
geometry ;" a statement, however, attributed to several other 
mathematicians of antiquity. 

Archimedes, born about 287 B. C., was one of the most eminent 
of the Greek mathematicians. He is especially celebrated for the 
discovery of the ratio of the cylinder to the inscribed sphere, in 
commemoration of which the figure of a sphere inscribed in a 
cylinder was engraved upon his tomb. He wrote two papers on 
arithmetic; the object of one, which is now lost, was to explain a 
convenient system of representing large numbers. The object of 
the second paper was to show that the method enabled one to 
write any number however large, in which he gave his celebrated 
illustration that the number of grains of sand required to fill the 
universe is less than 10 63 . 

Eratosthenes, who flourished about 250 years before the Christ- 
ian era, is said to have invented a method of determining prime 
numbers, known as Eratosthenes' sieve. He is also said to have 
suggested the calendar now known as the Julian Calendar, in 



Illl rilll.oMii'UV OK AKITIIMI 

which every fourth -.tain* 36G days. He determined tin- 

obliquity of ill-- rlijitu-, and measured a degree on the surfa 

.trtli \\Li.-li was subsequently found to be too long by about 
-. He U!M) describes an in-;. ..n.- nt fur the duplication 
ubi-. 

Nicoiiiat hns, whu is sup|H>M-d to have li\ed near the close of the 
first century of tin- Christian era, wrote an arithmetic \\ lndi in 
Latin translation remained lor a thousand years a standard 
authority U|>on that .-ubject. Hi- >|>< -cial aim wa- i be in\ -: Cation 
of the properties of numbers, and particularly of ratios. II 
gins with the explanation of even, odd, prime and perfect num- 
; then explain:- tractions in a tedious and elnni-y manner; 
then .ii-i-ussvs polygonal and solid numbers, and finally treat- <>t 
ratios, proportion, and the pro^reion-. He pi\e> the propo.-ition 
that all cubical numbers are equal to the >nm ot odd 

numbers; as 8 = 3 :. ; >'7=7 + 0+ll; 4 = 1 + l;'i+ 17-f-19. 
The work was translated by Boethius, and was the rem-M/, ,1 

-liook durinjr the Middle Ages. 

l'toh-m\ t. who died l,et\\e. n ]'2'> and \.'t\ A. 1)., was 

the author of numerous works on mathematics. C se on 

astronomy, called by the Aral's the A/nm>/rst. remaim-d a -tandard 
work on that subject until the time of Copernicus. In this work 
he treats of trigonometry, plane ami spherical, explains the 
obliquity of the ecliptic, uses 3-ffa as the approximate value of w, 
and em] 1 '1 >econds as i. 

Th- work exercised a strong influence in favor of sex a 21 simal 
iirithmetir, which uses the basis of sixty in the representation of 
numbers. 

a mathematician of Alexandria, who lived about 

tury, wrote a work railed Arithmetica, 

tfl of thirteen l>ooks>, only six of which have come down to 

It i< really a work on algebra, and before the discovery of 

the Ahmet papyrus was the oldest work extant on that su 

It treats of the properties of numbers, one of \\\* problems being 

to divide a number, as 18, which is the sum of two squares 4 and 



EARLY WRITERS ON ARITHMETIC. 33 

9, into two other square^ which he finds to be &^-- and -fa. It 
presents solutions of simple and quadratic equations, uses a symbol 
for the unknown quantity, and shows that " a number to be sub- 
tracted, multiplied by a number to be subtracted, gives a number 
to be added." The work is purely analytic in spirit, and is thus 
distinguished from the works of other Greek writers like Euclid. 
Diophantus originated the method of investigation known as Dio- 
phantine Analysis. 

Boethius, born at Rome between 480 and 482, wrote an Arith- 
metic based on that of Nicomachus. The arithmetic of Boethius 
was the classical work of the Middle Ages, and became the model 
of several subsequent writers even down to the fifteenth century. 
It was entirely theoretical, treating of the properties of numbers, 
particularly their ratios, and gave no rules of calculation, and wo 
have no means of telling whether the arithmeticians of thitf 
school reckoned on their fingers, or used an abacus. In the 
manuscript editions of this work, current during the llth century, 
there is a description of the Mensa Pythagorea, also called the 
abacus ; and mention is made of nine figures which are ascribed 
to the Pythagoreans or Neo Pythagoreans. This passage is by 
some considered spurious, and ascribed to a continuator ol 
Boethius. 

One of the earliest Hindoo writers upon the subject of mathe- 
matics was Aryabhatta, who was born in Pataliputra in 476 A. D. 
His work, entitled Aryabltattiyam, contains a number of rules and 
propositions written in verse. It consists of four parts, of which 
three are devoted to astronomy and the elements of spherical 
trigonometry ; the remaining part consists of thirty-three rules in 
arithmetic, algebra, and plane trigonometry. The algebra shows 
considerable knowledge of the subject, but there is no direct evi- 
dence that Aryabhatta was acquainted with the modern method 
of arithmetic. 

The next Hindoo writer of note is Brahmaguptn, born in 598, 
and was probably living up to 6GO. His work entitled Brahma- 
aphtito Siddhanta (i. e., the improved system of Brahma) is 
3 



34 



THE PHILOSOPHY OF ARITHMETIC. 



written in verse, and treats mainly of astronomy; though two 
chapters are devoted to arithmetic, algebra, and geometry. The 
arithmetic is entirely rhetorical ; most of the problems are worked 
out by the rule of three, and many of them are on the subject of 
interest. His algebra, which is also rhetorical, presents tin- 
fundamental cases of arithmetical progression, solves quadratic 
equations, and gives the method of solving indeterminate equa- 
tions of the second degree. 

The first known treatise among the Hindoos which contains a 
systematic exposition of the modern system of arithmetical nota- 
tion is that of Bhaskara, born 1114. This treatise was an 
astronomy, one chapter of which, called Lilawati, is an arithmetic 
written in verse, with explanatory notes in prose. After an intro- 
ductory preamble and colloquy of the gods, it begins with the 
expression of numbers by nine digits and the cipher or small 0. 
The characters are similar to those in present use, and the 
method of notation is the same. It contains the common rules of 
arithmetic and the extraction of the square and cube roots. The 
greater part of the work is taken up with the discussion of the 
" rule of three," which is used in solving numerous questions 
chiefly on interest and exchange. 

Another chapter of Bha-kara's work called Bjita-ganita (/. ., 
root computation) is a treatise in algebra. Abbreviations and 
initials are used for symbols ; subtraction is indicated by a dot, 
addition by juxtaposition merely, but no symbols are used for 
multiplication, equality, or inequality, these being written out at 
length. A product is indicated by the first syllable of the word 
subjoined to the factors, between which a dot is sometimes placed. 
In a quotient or a fraction, the divisor is written under the divi- 
dend without a line of separation. The two sides of an equation 
are written one under the other, confusion being prevented by the 
recital in words of all the steps which accompany the operation. 
Various symbols for the unknown quantity are used, but most of 
them nre the initials of the names of colors, and the word color is 
often used as synonymous with unknown quantity ; its Sanskrit 



EARLY WRITERS ON ARITHMETIC. 35 

equivalent also signifies a letter, and letters are sometimes used 
either from the alphabet or from the initial syllables of subjects 
of the problem. In one or two cases symbols are used for the 
given as well as the unknown quantities. The work contains also 
a treatise on trigonometry. 

The first Arabic arithmetic known to us is that of Al Kho- 
warazmi, written about the year 830. It begins with the words, 
" Spoken has Algoritmi. Let us give deserved praise to God, our 
leader and defender." Here the name of the author has passed 
into Algoritmi, from which comes our modern word algorism, 
meaning the art of computing in any particular way. The work 
treats of the fundamental rules by the Hindoo method, though the 
forms of operation are not so simple as those now used. Al Kho- 
warazmi also wrote a work on algebra in which the term " algebra," 
al-gebr, first occurs. This work holds an important place in the 
history of mathematics, as not only subsequent Arabian, but 
nearly all the early mediaeval works on algebra were based on it. 

It was from the writings of Al Khowarazmi that the Italians first 
obtained their ideas of algebra and of the modern method of arith- 
metic. This arithmetic was long known as algorism, or the art of 
Al Khowarazmi, in distinction from the arithmetic of BoethiuB, 
and this name was retained until the eighteenth century. The 
work had great influence in introducing the Arabic method of 
arithmetic to the scholars and mathematicians of Europe. 

Some of the early European writers on arithmetic were men- 
tioned in the previous chapter on the origin and development of 
arithmetic. These are Gerbert of the 10th century, and Leonardo, 
Ben Ezra and Gerard of the 12th century. From the 12th to the 
15th century there seem to have been few writers of note on the 
subject of mathematics, the most noted being Jordan us of Ger- 
many in the 13th century. One of the most distinguished mathe- 
maticians of the 15th century was Regiomontanus, who composed 
a work on trigonometry in 1464. This work contains the earliest 
known instances of the use of letters to denote known as well as 
unknown quantities. 



36 nil nm.osoi'iiY oi AKI i HMI.I K . 



In 1482 there appeared at Bamberg a small treatise on arith- 
metic which WHS attributed to Ulridi Wagner of Nuremburg. It 
was printed on parchment, and only fragments of a single copy of 
it are now extant. In 1483 the same Bamberg publishers brought 
out a second arithmetic, printed <>n paper, and covering seventy- 
seven pages. The work is anonymous, but I'lricli Wagi 
Mippo.-rd tn be its author. This Bamberger arithmetic of 1483, 
says Unger, bears no resemblance to previous Latin treatises, but 
aims especially at facility of computation in mercantile allairs. 
The method of solution, as in all the early books on arithmetic. 
was that of" the rule of three," known also as the " merchant.-' 
rule " or the " golden rule." 

An arithmetic by John Widmann was published in Leip/ig in 
1489, which is noted as being the earliest book in which the 
symbols -f- and have been found, though they had pre\ i 
appeared in a Vienna manuscript. They were not used, hov. 
as symbols of operation, but apparently merely as marks signify- 
ing excess or deficiency. It is supposed by some that they were 
originally warehouse marks to indicate more or less than the 
normal weights of boxes or chests containing goods. In Widmann 's 
book we find equation of payments treated according to the 
methods still in use. Problems of proportional parts and alliga- 
tion were solved by the use of as many proportions as there were 
groups to be separated. The work is obscure and deficient in 
rules for operations, and abounds in fanciful names of topics 
which Stifel in later years pronounced to be simply laughable. 

LII i, or Lucas di Borgo, an Italian monk, published 

}\\< great work entitled Xmnma de Arithmftica, Gcometria, Pi 

lif<i in Venice in 1 41) 4. The work consists of 
two p:irt. the fir.-t dealing with arithmetic and algebra, the second 
with geometry. This is one of the earliest printed treatises on 
arithmetic and algebra, and the earliest work presenting a >y>t in- 
atic exposition of nl^oristie arithmetic. It treats of the four 
fundamental rules, and present* methods of extracting the square 
root. In its practical application it deals largely with questions 



EARLY WRITERS ON ARITHMETIC. 37 

relating to mercantile transactions, including bills of exchange, 
working out numerous examples in these subjects. It also con- 
tains the first known treatise on double entry book-keeping. In 
this work the term " million " and also " nulla " or cero (zero) 
occurs for the first time in print. The work had a wide influence in 
the general introduction of the new arithmetic throughout Europe. 

Philip Calandri published a work on arithmetic at Florence in 
1491. It begins' with a picture of Pythagoras teaching, headed 
" Pictagoras Arithmetrice introductor." His notion of division 
is curious. When he divides by 8, he calls the divisor 7, demand- 
ing, as it were, that quotient which, with seven more like itself, 
will make the dividend. He describes the rules for fractions, and 
gives some geometrical and other applications. 

Jacob Kobel, in 1514, published, at Augsburg, a work on 
arithmetic in which the Arabic numerals are explained, but not 
used. The computation was by counters and Roman numerals. 
In the frontispiece is a cut representing the mistress settling 
accounts with her maid-servant by an abacus with counters. 

Cuthbert Tonstall, in 1522, published an arithmetic in Latin 
which had great influence on the development of the science in 
England. He gives the multiplication table in the form of a 
square, and also addition, subtraction, and division tables. For 
| of of J? he writes f ^ ^ ; and be gives a clear explanation of 
the multiplication of fractions. De Morgan says this book is 
" decidedly the most classical which ever was written on the sub- 
ject in Latin, both in purity of style and goodness of mutter." 

Jerome Cardan published, at Milan, in 1539. a work entitled 
Practica Arithmetica. It shows, as might have been expected 
from an Italian of that age, more power of computation than the 
French and German writers. It contains a chapter on the 
mystic properties of numbers, one use of which is in foretelling 
future events. These are mostly the numbers mentioned in the 
Old and New Testaments, but not altogether. In another treat- 
ise, Cardan expresses his opinion that it was Leonardo of Pisa 
who first introduced the Arabic numbers into Europe. 



;;.s mi. niii.o.-oriiY OF ARITHMETIC. 

iert Recorde publi.-ln-d hi.- r< Irbratcd work on arithmetic, 
called ' The Grounde of Art. -," about 1540. It was originally 
ated to Kdwanl VI. '!'!. work was subsequently revi.- 1 
and enlarged by John Dee, and published in 1 "ring tin* 

original dedication, which had b n omitted in the edition pre- 
pared during the reign of Mary. This work contains u number 
of the subjivts of modern text-books, including the rule of three, 
alligation, fellow.-hip, false position, and the method of testing 
operations by " casting out the 9's." He uses + and with tin- 
explanation, " + whyche betokeneth too muche, as this line, , 
pluine without a crosse line, betokeneth too little." It was sub- 
sequent lv revised by Mellis, who added a third part on practice 
and other things, and also by Hartwell. The last edition known 
in by Edward Hattoii, 10'J'J, which contains an additional book 
called " Decimals made easie." It is said to contain a large 
number of the principles and problems of modern text-books. 
Recorde introduced the sign of equality ( = ) in a work >: 
published in 1 ;">'><>. The work was called by the odd title, " The 
Whetstone of Witte," in which he gives his reason for the symbol 
in the following quaint language : " And to avoid the tedious 
repetition of these words, I will settle, as I doe often in worke use, 
a pair of parallel or Gemowe lines of one length, thus, =, because 
noe 2 thynges can be more equalle." 

Michael Stifel published, at Nuremberg, in 1">44, his celebrated 
work entitled Arittnnetica Integra. The first two books are on 
the properties of numbers, on surds and inconimi-n>urahles, 
learnedly treated, and with a full knowledge of what Km lid had 
done on the subject. The third book is on algebra, and did much 
for the introduction of algebra into Germany. Stifel acknowledges 
his obligations to Adam Riese, and professes to have tak'-n his 
examples from Christopher RudolflT. Stifel was the first to use the 
symbols + and to denote the operations of addition and sulv- 
traction. He introduced also the symbol of evolution, f / ', or- 
initial of radix or root, though Cantor says that 
Rudolff had previously used it. 



EARLY WRITERS ON ARITHMETIC. 39 

Nicholas Tartaglia, an eminent Italian mathematician, pub- 
lished a work on arithmetic, vols. 1 and 2 of which appeared in 
1556, and vol. 3 in 1560. The works are verbose, but give a 
clear account of the various arithmetical methods then in use, and 
present a large number of notes on the history of arithmetic. The 
work on arithmetic contains an immense number of questions on 
every kind of problem which would be likely to occur in mercan- 
tile arithmetic, and attempts are made to frame algebraic formulas 
applicable to particular problems. 'It contains also a large collec- 
tion of arithmetical puzzles and questions of an amusing character, 
among which is found the question, " What would 10 be if 4 were 
6 ?" and the problem of the three jealous husbands and their 
wives who were to cross a river with a single boat that would carry 
only two persons. The treatise on numbers was really an algebra, 
in which are found some interesting investigations. Tartaglia is 
believed to be the author of a method of solving cubic equations 
which Cardan obtained from him under a promise of secrecy, and 
afterward published under his own name in violation of his promise. 

Simon Stevinus published, at Leyden, in 1585, a work which 
was edited by Albert Girard in 1634. This work is character- 
ized by originality, accompanied by a great want of the respect 
for authority which prevailed in his time. For example, great 
names had made the point in geometry to correspond with the 
unit in arithmetic. Stevinus tells them that 0, and not 1, is the 
representative of the point. " And those who cannot see this," 
he adds, " may the Author of nature have pity upon their un- 
fortunate eyes; for the fault is not in the thing, but in the sight 
which we are not able to give them." A portion of this work 
contains " Les Tables d' Interest " and " La Disme," the latter 
of which exerted a great influence on the introduction of decimal 
fractions. 

John Mellis, in 1588, at London, published, " A briefe instruc- 
tion and manner how to keepe bookes of Accompts after the 
order of Debitor and Creditor," etc. This is the earliest English 
work on book-keeping by double entry. At the end of the book- 



40 THE PHILOSOPHY OP ARITHMETIC. 

keeping it * short tmitinc on arithmetic. Mellis says : Truly, 
1 am but the renuer and reviver of un uuncient old copie. printed 
here In London the 14 of August, 1543. Then collected, pub- 
liiihed, made and set forth by one Hugh Oldcastle, Scholemaster, 
who, as appcareth by his treatise then taught Arithmetike and 
this booke, in Saint Olluves parish in Marke Lane." 

In 1596, a work entitled, " The Pathway to Knowledge," was 
published in London, which was a translation from the Dutch, by 
W. P. The translator gives the following verses, of which he is 
supposed to be the author: 

Thirtic dales bath September, Aprill, June, and November, 
Februarie, eight and twentie alone ; all the rest thirtie and one. 

Mr. Davies, in his Key to Hutton's Course, quotes the follow- 
ing from a manuscript of the date of 1570, or near it : 

Multiplication is mie relation, 

And Division is quite as bad, 

The Golden Rule is mie stumbling stule. 

And Practice drives me mad. 

Cataldi, successively Professor of Mathematics at Florence, 
Perujiia and Bologna, published a work on the square root of 
numbers at Kolomna, in 1613. The rule for the square root is 
exhibited in the modern form, and he shows himself a most in- 
trepid calculator. The greatest novelty of the work is the intro- 
duction of continurd fractions, then, it seems, for the first time 
presented to the world. He reduces the square roots of even 
numlHTs to continued fractions, and then uses these fractions in 
approximation, but without the aid of the modern rule which 
tit-rives each approximation from the preceding two. 

Richard Witt, in 1613, published a work containing "Arith- 
metical questions " on annuities, rents, etc., ** briefly resolved by 
means of certain Breviats." These Breviats are tables, and this 
work is said to be the first English book containing tables of com- 
pound interest. Decimal fractions are really used. The tables 
being constructed for ten million pounds, seven figures have to be 



EARLY WRITERS ON ARITHMETIC. 41 

cut off; and the reduction to shillings and pence, with a temporary 
decimal separation, is introduced when wanted. The decimal 
separator used is a vertical line, and the tables are expressly 
stated to consist of numerators, with 100... for a denominator. 

John Napier, born 1550, died 1617, wrote a treatise on arith- 
metic which was published at Edinburgh in 1617, after the 
author's death. It contains a description of Napier's rods with 
applications. It is remarkable because it expressly attributes the 
use of decimal fractions to Stevinus. It also states that Napier 
invented the decimal point. De Morgan says this is not correct, 
since 1993.273 is written 19932'7"3'". Napier is illustrious as 
the inventor of logarithms. 

Robert Fludd, in 1617 and 1619, published a work on mathe- 
matics at Oppenheim. It contains two dedications, the first, 
signed Ego, homo, to his creator ; the second, on the opposite side 
of the leaf, to James I. of England, signed Robert Fludd. The 
first volume contains a treatise on arithmetic and algebra. The 
arithmetic is rich in the description of numbers, the Boethian 
divisions of ratios, the musical system, and all that has any con- 
nection with numerical mysteries of the sixteenth century. The 
algebra contains only four rules, referring for equations, etc., to 
Stifel and Recorde. The signs of addition and subtraction are P 
and M with strokes drawn through them. The second volume is 
strong upon the hidden theological force of numbers. 

Albert Girard published a treatise on algebra at Amsterdam 
in 1629, which contains a slight treatise on arithmetic. The 
arithmetic contains no examples in division by more than one 
figure. On one occasion the decimal point is used, though this 
was not the first time it had been employed. Girard introduced 
the parenthesis in place of the vinculum, which had been used by 
Recorde. 

Wm. Oughtred's Claris Mathematica, a work on arithmetic 
and algebra of great celebrity, was first published in 1631. It 
retains the old or scratch method of division which. Dr. Peacock 
observes, lasted nearly to the end of the seventeenth century. He 



42 THE PHILOeol'IIY OP AKITIIMtmc. 



not use the decimal point, but writes 12.3456 thus : 1213456. 
The symbol for multiplication, X, St. Andrew's cross, was intro- 
duced by Oughtred. lie seems to have first employed the 
symbol : : to denote the equality of ratio*. He wrote a treatise 
on trigonometry in 1657, in which abbreviations for tine, corine, 
etc., were employed. 

Nicholas Hunt published, in 1633, " The Hand-Maid to Arith- 
metick refined." The book is full on weights and measures, and 
commercial matters generally. It does not treat of decimal frac- 
tions, however. The author calls " dec i mull Arithmeticke, * 
a division of a pound into 10 primes of two shillings each ; each 
shilling into six primes of two pence each. It expresses the rule* 
in verse, of which the following is an example : 

Adde tbou upright, reserving every tonne. 
And write the digits downe all with tbj pen. 
Subtract the lesser from the greet noting the rest, 
Or ten to borrow you are ever prest 
To pay what borrowed was think it no paine. 
But honesty redounding to your gaiue 

Peter Herigone, in 1634, published at Paris" a work entitled 
44 Cursus Mathematici tomus secundus." It introduces the deci- 
mal fractions of Stevinus, having a chapter " des nombres de la 
dixme." The mark of the decimal is made by marking the 
place in which the last figure comes. Thus when 137 livres 16 
sous ift to be taken for 23 years 7 months, the product of 1378' 
and 23583'" is found to be 32497374"", or 3249 liv., 14 sous, 8 
denier*. 

William Webster published, in 1634, tables for simple and 
compound interest. This work treats decimal arithmetic as a 
thing known. No decimal point is recognized, only a partition 
lin<- to be used on occasion. It contains the first head-rule for 
turning a decimal fraction of a pound into shillings, pence and 
farthings. Many other interesting details will be found in the 
works of De Morgan, Unger, Fink, Ball, Gow and Cantor. 



EARLY WRITERS ON ARITHMETIC. 43 



1C 




N * O 1 

< ^ tf 



NOTE. This page of symbols is taken from Cajori's "History of 
Elementary Mathematics " by permission of the author and publisher. 






CHA1TKU IV. 

ORIGIN OF AK1TIIMKT1CA1. rKOCESSES. 

/~\NE of the most interesting |>oints connected with the hi.-tor\ 
^ of arithmetic, would be a full and complete account of the 
genesis of the different divisions and processes of the sci- 
This, ho u impossible. The origin of the elementary or 

fundamental processes dates back before the invention of printinjr, 
and can never be determined. Some of the principal facts, how- 
ever, upon this point, in addition to those already given, will be 
stated. 

ARITHMETICAL LANGUAGE. The notation of the nine dL'ii.- 
and zero, upon which the science of arithmetic is based and 
developed, originated, as we have already shown, among the 
Hindoos. This notation was adopted by the Arabians, and be- 
came general among Arabic writers on astronomy, as well as 
arithmetic and algebra, about the middle of the 10th century. 
From the Arabs, who, in the llth century, held jxissession of the 
southern provinces of Spain, the knowledge was communicated to 
the Spaniards and other nations of Europe. 

The Italians, from an early period, adopted tin- method of dis- 
tributing the digits of a number into groups or period- of six, and 
consequently proceeding by millions. This is the method of 
numeration given by I'acioli, 1494. The method of reckoning by 
three places, as used in this country and on the Continent, seem- 
to have originated with the Spanish. In a work on arithmetic by 
Juan de Ortega, 1.030, we find the following method of numera- 
tion ; 10, dc/.ena ; HIM. centena ; 1000, millar ; 10000, dezena de 
millar; 100000, centena de millar; 1000000, cuento. The term 
ii//in, however, had not yet been introduced, and it has not been 
fully ascertained at what time this introduction took place. 

(44) 






ORIGIN OF ARITHMETICAL PROCESSES. 45 

Cantor says that the term millione occurs the first time in print in 
the summa do arithmetica of Paciola. Bishop Tonstall, 1522, 
speaks of the term million as in common use, but rejects it as bar- 
barous, being used only by the vulgar. 

Stevinus divided numbers into periods of three places, called 
each period membres, aad distinguished them as le premier membre, 
le seconds membre, etc. Instead of million he says mille mille ; for 
a thousand million he uses mille mille mille; and for a million- 
million he uses mille mille mille mille. It would appear from the 
practice of Stevinus, and from the observation of his contempor- 
ary, Clavius, that the term million was not at this time in general 
use amongst mathematicians. Albert Girard divides numbers 
into periods of six places, which he terms premiere masse, seconde 
masse, troisieme masse, etc., the first of which only is divided into 
periods of three places each ; but he does not use the word million. 
Ducange of Rymer mentions the word million in 1514, and in 
1540 it occurs once in the arithmetic of Christopher Rudolff. The 
term was introduced into Recorde's arithmetic, 1540, and subse- 
quently appeared in all succeeding authors. It appears to have 
been admitted into German works much later than into the French 
and English. The terms billion, trillion, etc., so far as known, 
appeared first in a manuscript work on arithmetic by Nicolas 
Chuquet, a gifted French physician of Lyons, and appear in 
1520 in a printed work of La Roche. 

FUNDAMENTAL OPERATIONS The fundamental operations 
of arithmetic were, without doubt, invented by the Hindoos at 
a very early period. The work from which our knowledge of 
Hindoo arithmetic has been mainly derived, is the Lilawati of 
Bhaskara, who lived about the middle of the 12th century. 
The work is named after the author's daughter, Lilawati, who, 
it appeared, was destined to pass her life unmarried and re- 
main without children. The father, however, having ascer- 
tained a lucky hour for contracting her in marriage, left an 
hour- cup on a vessel of water, intending that when the cup 
should subside, the marriage should take place. It happened, 



46 TIIK PIIII.060PHY OF ARITHMETIC. 

however, that the girl, from a curiosity natural to children, 
looked into the cup to see the water coming in at the hole, 
when, by chance, a pearl separated from her bridal dress, fell 
into the cup, nnd rolling down to the hole, stopped the influx 
of water. When the operation of the cup had thus been de- 
layed, the father was in consternation ; and, examining, he 
found that a small pearl had stopped the flow of the water, 
'and the long expected hour was passed. Thus disappointed, 
the father said to his unfortunate daughter, " I will write a book 
of your name, which shall remain to the latest times, for a good 
name is a second life, nnd the groundwork of eternal existence." 

This work frequently quotes Brahmagupta, an author who is 
known to have lived in the early part of the 7ih century, and 
portions of whose works, containing treatises on arithmetic and 
mensuration, are still extant. Brahmagupta also refers to an 
earlier author, Arabhatta, who wrote an algebra and arithmetic 
as early as the 6th century, and who is considered one of the old- 
est writers among the Hindoos. In tracing the history of the 
operations of arithmetic, we must therefore begin with the Lilu- 
wati of Bhaskara. 

The fundamental operations of arithmetic, as given in the 
Lil-nnitl, are eight in number ; namely, addition, subtraction, 
multiplication, division, square, square root, cube, cube root. To 
the first of these the Arabs added two, namely, duplation and 
mediation or halving, considering them as operations distinct from 
multiplication and division, in consequence of the readiness with 
which they were performed ; and they appear as such in many of 
the arithmetical books in the 16th century. 

Addition. The rule given in the Lilawati for addition is as 
follows : ' The sum of the figures, according to their places, is 
to be taken in the direct or inverse order," which is interpreted 
to mean, * from the first on the right towards the left, or from 
the last on the left towards the right." In other words, they 
commenced indifferently with the figures in the highest or low- 
est places, a practice which would not lead to much incon 



ORIGIN OF ARITHMETICAL PROCESSES. 47 

venience in their mode of working. Thus, to add 2, 5, 32, 193, 

18, 10, 100, they proceed as follows: 

Sum of the units, 2, 5, 2, 3, 8, 0, 0, 20 

Sum of the tens, 3, 9, 1, 1, 0, 14 

Sum of the hundreds, 1, 0, 0, 1, 2 

Sum of the sums, 360 

Subtraction. The process of subtraction was also com- 
menced either at the right or the left, but much more commonly 
at the latter ; and it is remarkable that this method of begin- 
ning to subtract at the highest place, which is subject to 
considerable inconvenience, should have been so general. It 
is found in Arabic writers, in Maximus Planudes, a Byzantine 
writer of about the middle of the 13th century, and in many 
European writers as late as the end of the 16th century. 

In Planudes, numbers to be added or subtracted are placed 
one underneath another, as in modern works on arithmetic ; 
and the sum or difference is written above these numbers. 
When a term in the subtrahend is greater than the correspond- 
ing one in the minuend, a unit is written beneath them, as in 
the example in the margin. 

In performing the operation, 3 is increased 18769 rem. 
by the unit in the next place to the right, and 54612 rain, 
also 5, 8, 4, and the terms thus increased are !??, 
subtracted from the terms above, increased by 
10, to find the remainder. 

In other cases, the numbers are arranged, as 06779 rem. 
in the margin, the digits 3, 0, 0, 2 in the minuend ^9( 
being replaced by 2, 9, 9, 1, and then 5 is 2 3245 Tub 
subtracted from 4, 4 from 1, 2 from 9, 3 from 
9, and 2 from 2, in order to get the remainder. It is obvious, 
that when such a preparation is made, it is indifferent where 
we commence the operation. 

Bishop Tonstall attributes the invention of the modern 
practice of subtraction to an English arithmetician of the name 
of Garth. This method he has illustrated with great detail, 



48 THE PHILOSOPHY OF ARITHMETIC. 

and added, for the assistance of the learner, a subtraction table, 
giving the successive remainders of the nine digits when sub- 
traeted from the series of natural numbers from 11 to 19 inclu- 

tbe only cases which can occur in practice. 
In speaking of the methods of preceding writers, 2 91010 
he has presented tin- example in the margin, in 3 ]. 
whi-h it will be seen that the numbers from . 

which the subtraction is actually made, are 
placed above the terms of the minuend. 

In the arithmetic of Ramus, which was published in 1584, 
though written at an earlier period, we find the operation 
performed from left to right, and this method is followed 
by some other writers. Thus, in subtracting 345 from 7 
432 the terms to be subtracted and the remainder are 
written as in the margin. When 3 is subtracted from 
4, the remainder should be 1 ; but it is replaced by zero, 
since the next term in the subtrahend is greater than the corres- 
ponding term of the minuend ; in the second term the remainder, 
which should be 9, is reduced to 8, since 5, the next term of 
the subtrahend, is greater than 2, the term above it, but the 
last remainder 7, is not changed. 

Orontius Fineus, the predecessor of Ramus in the professor- 
ship of Mathematics at Paris, in his De Arithrtielica Practica, 
1555, subtracts according to the method now used ; and it is 
difficult to account for the adoption by Ramus of so inconven- 
ient a method as he employed, when the method of Fineus 
must have been familiar to him, unless we attribute it to that 
love of singularity which led him to aspire to the honor of 
founding a school of his own. 

Multiplication. The author of Lilawati has noticed six 
different methods of multiplying numbers, and two others are 
mentioned by his commentators. These may be illustrated by 
their application to the following example: "Beautiful and 
dear Lilawati, whose eyes are like a fawn's, tell me what are 
the numbers resulting from one hundred and thirty-five taken 



ORIGIN OF ARITHMETICAL PROCESSES. 49 

into twelve ? If thou be skilled in multiplication, by whole or 
by parts, whether by division or separation of digits, tell me, 
auspicious woman, what is the quotient of the product divided 
by the same multiplier ?" 

Here the multiplicand is 135, and the multi- 135 
plier 12; and the first method, which consists of 12 12 12 
multiplying the terms of the multiplicand sue- 12 60 
oessively by the multiplier, is indicated in the 3 6 

margin. 16 20 

The second method, which consists in sub- 
dividing the multiplier into parts, as 8 and 4, 135 8 1080 
and severally multiplying the multiplicand by 135 4 540 
them, is also indicated in the margin. 1620 

The third method, which con- 
sists in separating the multiplier 

12, into its two factors, 3 and 4, and 1354 205403 120 
multiplying successively by these 
factors, the last product being the 
result, is also represented in the 
margin. 

The fourth method consists in taking the 
digits as parts, viz., 1 and 2, the multiplicand 135 135 
being multiplied by them severally, and the 



products being added together according to the 

places of the figures, as is represented in the 

1620 
margin. 

The fifth method consists in multiplying the 
multiplicand by the multiplier less 2, namely, 135 10 1350 
10, and adding the result to twice the multipli- 135 2 270 
eand, as may be seen in the margin. 1620 

The sixth method consists in multiplying the 
multiplicand by the multiplier increased by 
8, namely, 20, and subtracting 8 times the 
multiplicand, as represented in the margin. 



50 



THE PHILOSOPHY OF ARITHMETIC. 




135 

12 

~To 
11 

5 



The other two methods are given in the Commentary of 
Ganesa. The first of these, which is 
represented in the margin, appears to 
have been very popular in the East, 
and was adopted by the Arabs, who 
termed it shabacah, or net-work, from 
the reticulated appearance of the figure 
which it formed, and also by the Per- * 
sians under a slight alteration of form. It is found likewise in 
the works of the early Italian writers on algebra, and the same 
principle may be recognized in the process of multipli- 
cation by Napier's rods. 

The second of these two methods of multiplica- 
tion, as represented in the margin, is described in 
full by Ganesa. He, however, considers this method 
difficult, and not to be learned by dull scholars with- l 
out oral instruction. 1620 

The number and variety of these methods would 
seem to show that the operation of multiplication was regarded 
as difficult, and it is remarkable that the method now used is not 
found amongst them. We find no notice of the multiplication 
table among either them or the Arabs. At all events, it did 
not form a part of their elementary system of instruction, a 
circumstance which would account for some of the expedients 
which they appear to have made use of, for the purpose of 
relieving the memory from the labor of forming the products 
of the higher digits with each other. 

The Arabs adopted most of the Hindoo methods of multi- 
plication, and added some others of their own ; among which 
are some peculiar contrivances for the multiplication of small 
numbers. They may also be considered as the authors of the 
method of quarter squares, or of finding the product of two num- 
bers by subtracting the square of half their difference from the 
square of half their sum. The Arabs were most probably the in- 
ventors of the method of proof by casting out 9's, which is as yet 
unknown to the Hindoos ; they called it tarazit, or the balance. 



ORIGIN OF ARITHMETICAL PROCESSES. 51 

The work of Planudes was chiefly collected from the Arabic 
writers, as appears from his being acquainted with the 
method of casting out 9's. In multiplication he has g,Q 
chiefly followed the method of multiplying crosswise or 35 
Kara TOV xiaapav, from the figure x, which is employed to x 
connect the digits to be multiplied together. Thus, in 24 
multiplying 24 into 35, we should write the factors as in 
the margin ; and then multiply 4 into 5 (/wwaArf^ write down and 
retain 2 for the next place ; multiply 4 into 3, and 3 into 5, the 
sum is 22, which added to 2, makes 24 (<*<*<*<%), write down 4 
and retain 2 ; lastly, multiply 2 into 3, add 2, which makes 
8 (mzTwraJef) ( and the product is 840. He also gives another 
method which he acknowledges to be very difficult to per- 
form with ink upon paper, but very commodious on a board 
strewed with sand, where the digits may be readily 
effaced and replaced by others. Thus, taking the same 
example, we multiply 2 into 3, write 6 above the 3 ; * , , 
multiply 2 into 5, the result is 10; add 1 to 6, and 35 
replace it by 7, or write 7 above it; multiply 4 into 3, 24 
the product is 12; write 2 above 5, and add 1 to 7, 
which is replaced by 8, or 8 written above it ; lastly, multiply 
4 into 5, the result is 20; add 2 to 2, place 4 above it and after 
it the cipher ; the last figures, or those which remain without 
accents, will express the product required. 

Division. The extreme brevity with which the rules of 
division are stated in the Lilawati renders it difficult to 
describe the Hindoo method of dividing numbers. We are 
directed to abridge the dividend and divisor by an equal 
number, whenever that is- practicable; that is, to divide them 
both by any common measure; thus, instead of dividing I' 1 *-* 1 
by 12, we may divide 540 by 4, or 405 by 3. We find, how 
over, in one of the commentators on this work, a description 
of the process of long division, which, if exhibited in a schomr. 
would exactly agree with the modern rule 

ITALIAN METHODS. The Italians, who cultivated arithmetic 



52 THE PHILOSOPHY OF ARITHMETIC. 

with so much zeal and success, from a very early period 
adopted from their Oriental masters many of their processes 
for the multiplication and division of numbers ; adding, how- 
ever, many of their own, and particularly those which are 
practiced at the present time. In the Summa de Arithmetical, 
of Lucas di Borgo, we find eight different methods of multi- 
plication, some of which are designated by quaint and fanciful 
names. We shall mention them in their order. 

1. Multiplicatio : bericuocoli e schacherii. The second of 
these names is derived from the resemblance of the written 
process to the squares of a chess-board ; 
the first from its resemblance to the check- 456 

ers on a species of sweetmeat or cake, 3 7 

made chiefly from the paste of bacochi or 



apricots, which were commonly used at j 3 1 1 



648 



festivals. The process is exhibited in the 
margin. This method is presented by 



92 



Tartaglia and later Italian writers with- 172368 
out the squares; and it thus became the 
method which is now universally used, and which was adopted 
from the beginning of the 16th century by all writers on arith- 
metic, nearly to the exclusion of every other method. 

2. Castelluccio ; by the little castle. This 
method, as indicated in the margin, uses the |876 
upper number as the multiplier, and begins with 
the higher terms. This method was much prac- 



ticed by the Florentines, by whom it was some- ' 4175230 
times called alV indietro, from the operation 40734 

beginning with the highest places, more Arabum, 67048164 
according to the statement of Pacioli. 

3. Columna, o per tavoletta ; by the column, or by the tablets. 
These were tables of multiplication, arranged in columns, the 
first containing the squares of the digits, the second the pro- 
ducts of 2 into all digits above 2 ; the third, of 3 into all digits 
above 3 ; and so on, extending in some cases as far as the pro- 



ORIGIN OF ARITHMETICAL PROCESSES. 



53 




ducts of all numbers less than 100 into each other. Pacioli 
says that these tablets were learned by the Florentines, and 
their familiarity with them was considered by him as a princi- 
pal cause of their superior dexterity in arithmetical operations. 
This method is used in multiplying any number, however large, 
into another which is within the limits of the table. Thus, to 
multiply 4685 by 13, the terms of the multiplicand are multiplied 
successively by 13, and the results formed 
in the ordinary manner. 

4. Crocetta sive casella ; by cross multi- 
plication. This method is said to require 
more mental exertion than any other, par- 
ticularly when many figures are to be 

combined together. Pacioli expresses his 

admiration of this method, and then takes 20 7 9 3 6 
the opportunity of enlarging on the great difficulty of attaining 
excellence, whether in morals or in science, without labor. 

5. Quadrilatero ; by the square. This is 
a method which has been characterized as 
elegant, and as not requiring the operator 
to attend to the places of the figures when 
performing the multiplications. The method 
is represented in the margin, and will be 
readily understood. 

6. Gelosia sive graticola ; latticed multiplication. 
called," says Pacioli, " because the dispo- 
sition of the operation resembles the form 

of a lattice, a term by which we designate 

the blinds or gratings which are placed in 

the windows of houses inhabited by ladies 

so that they may not easily be seen, as well 

as by other nuns, in which the lofty city of 

Venice greatly .abounds." The method will "9 7 4 

be readily understood by the example given in the margin, 

which multiplies 987 by 987. It is the same as one previously 



5 4 
5 4 


3 

3 


1 


G 


i 


9 


2 


1 


7 


9 


2 


7 


1 


5 


2 9 
on. " It 


4 

is > 




54 THE P1IILOSOPHY OF ARITHMETIC. 

noticed, which was in common use among the Hindoos, Ara- 
bians, and Persians. 

7. Ripiego ; multiplication by the unfolding or resolution 
of the multiplier into its component factors. Thus, to multiply 
157 by 42, resolve 42 into its ripieghi or factors, 6 and 7, and 
multiply successively by them. 

8. Scapezzo ; multiplication by cutting up, "* 5, 6 

or separating the multiplier into a number of '. '. . 

oarts, which compose it by addition. Thus, 81624 30 

1 1 A 1 9 A I Q A I f* A 

to multiply 2093 by 17, we separate 17 into ' g.jy 

10 and 7, multiply by each, and take the sum 

_ iii. bU yU IoU 

of the products. In some cases both multi- 
plicand and multiplier were separated into parts. Thus, the 
multiplication of 15 by 12 was performed as in the margin. 

In another Italian arithmetic, published in 1567, by Pietro 
Cataneo Sienese, we find the same distinctions preserved, and 
the same names, or nearly so, attached to them ; the method of 
cross multiplication is expressly attributed to Leonardo ,. 
of Pisa, who derived it, in common with Maximus >^ 
Planudes, from the Hindoos, through the Arabians. 4 7 
It is not impossible tliat St. Andrew's cross, which ~~^~ 
is the sign of multiplication, was derived from the 
custom of uniting the numbers to be multiplied together by lines 
which crossed each other, as in the example given in the margin. 

Both Lucas di Borgo and Tartaglia mention other methods 
of multiplication which were made use of in their time. An 
extraordinary passion seems to have prevailed in that age for 
'the invention of new forms of multiplication, and every pro- 
fessional practitioner of arithmetic considered it as an important 
triumph of his art if he could produce a figure more elegant 
and more refined in its composition and arrangement than 
those which were used by others. They are, all of them, how- 
ever, characterized by Pacioli as inconvenient, at least com- 
pared with those which he had given ; and Tartaglia treats them 
as trifling and superfluous, such as any one may invent who is 
acquainted with the 2d proposition in the 2d Book of Euclid. 



ORIGIN OF ARITHMETICAL PROCESSES. 55 

The Hindoos, as has been stated, had no proper knowledge 
of the multiplication table, and the Arabs do not appear to 
have made use of the table of Pythagoras as the basis of their 
arithmetical education ; the credit of introducing it, therefore, 
is due to the early Italian writers on the science, who probably 
found it in the writings of Boethius, and adopted it thence. 
Even after the Italian arithmeticians were familiar with this 
table, many writers of other countries considered it important 
to relieve the memory from the labor of retaining it for the 
products of all digits exceeding 5, by giving rules for their 
formation. The principal rule for this purpose, called regula 
ignavi, or the sluggard's rule, was adapted from the Arabians, 
and is found in Orontius Fineus, liecorde, Laurenberg, and 
most other writers between the middle of the 16th 
and Itth centuries. The rule is as follows: Sub- 73 82 91 
tract each digit from 10, and write down the XXX 
difference ; multiply these differences together, "^ *j __ 
and add as many tens to their product as the ^ 56 72 
first digit exceeds the second difference, or the 
second digit the first difference. The Arabians made use of 
this and other similar rules which applied to numbers of two 
places of figures, a practice which may be accounted for by 
their very general use of sexagesimals, and the consequent 
importance of being able to form the products which are found 
in a sexagesimal table. 

Many other expedients were proposed to relieve the mem- 
ory, in the process of multiplication, from the labor 514*) 
of carrying the tens. An interesting one is pre- 43 

sented by Laurenberg, an author who endeavored lou 

to elevate the character of the common study of 1532 
arithmetic by collecting all his examples from clas- 
sical authors, and by making them illustrative of 
the geography, chronology, weights and measures 110fl 
of antiquity. It will be understood from the example given, 
without explanation. 



66 THE PHILOSOPHY OF ARITHMETIC. 

Division. Neither Planudes nor the early Arabic writers 
seem to have presented any methods of dividing that merit the 
special notice of the writers on the history of arithmetic. 
Lucas di Borgo gives four distinct methods which we proceed 
to explain. These methods had particular names, as in mul- 
tiplication. 

1. Partire a regolo, sometimes called also partire per testa 
or division by the head, was used when the divisor was a 
single digit, or a number of two places, such as 12, 
13, etc., included in the librettine or Italian tables 6 

of multiplication. The method will be readily 3478 
understood from the example given. Di Borgo says: 579|- 
" This method of division is called by the vulgar, the 
rule, from the similitude of the figure to the carpenter's rule 
which is made use of in the making of dining-tables, boxes, 
and other articles, which rules are long and narrow." 

2. Per ripiego ; which consists in resolving 
the divisor into its simple factors, or ripieghi. 



It will be readily understood from the example g 35721 
given, and be recognized as a common method of 3969 

modern arithmetics. 

3. A danda ; which the author says is thus called for rea- 
sons which will be readily seen in the opera- 

tion itself, which represents the division of D ^ s 7 r ' ^g^ 1 *' 
230265 by 357, giving a quotient of 645. 230 265 

The process is the same as our common 2142 
method of long division, only the numbers ~1606~ 
are not so conveniently written. It was 1428 
called a danda, or by giving, because after 1785 

every subtraction we give or add one or 1785 

more figures on the right hand. The author, 
however, prefers the next method. 

4. Galea vel galera vel batello ; so called from the process 
resembling a galley, " the vessel of all others most foared on 



ORIGIN OF ARITHMETICAL PROCESSES. 57 

the sea by those who have good knowledge 

of it ; the most secure and swiftest ; the most 

975 
rapid and lightest of the boats that pass on T$9Ai 

the water. " The method may be illustrated 9/535399(9 

by dividing 97535399 by 9876. We first 98,70 
write the dividend, and underneath it the 

divisor, and commence with the second figure of the dividend, 

since the divisor is not contained in the first four terms of the divi- 
dend. Multiplying the divisor by the first term 

of the quotient, 9 times 9 are 81, which sub- 86 

tracted from 97 leaves 16, which is written $%& 

above 97 ; then cancel 97 and 9 in the divi- 97-35399/98 

sor ; 9 times 8 are 72, which taken from 165, 98766 

leaves 93 ; write 9 above 16 and 3 above 5 987 
in the dividend, and cancel 165, and 8 in 

divisor ; 9 times 7 are 63, which subtracted from 933 leaves 
870; cancel 933 in remainder, and 7 in divisor; 9 times 6 are 
54, which subtracted from 705 leaves 651 ; 
cancelling 705, and 6 in the divisor, we have 
as a remainder 8651399. For multiplying 
by the second quotient figure, we arrange 

the divisor as in the margin, and proceed as 8(31022 

before. The complete operation is repre- 9/55$5 

sented by the last work in the margin, and ^$5^5/3 

is so apparent that it needs no further expla- 9/75^5^99(9876 

mm> 

nauon. 98777 

Tartaglia states that it was the custom in 988 

Venice for masters to propose to their pupils 9 

as the last proof of their proficiency in this 
process of division, examples which would produce the com- 
plete form of the galley, with its masts and pendant. The 
last addition to the work was supplied by the scheme for the 
proof of the accuracy of the operation by casting out the 
9's. Dr. Peacock gives an example showing the numbers 



58 THE PHILOSOPHY OF ARITHMETIC. 

thus arranged, which is very curious, but too long for insertion 
here. 

The same process is illustrated by an example ^ ^ , 

from the numerous calculations by Regiomon- 3134 
tanus, in his tract on the quadrature of the 154750 [4 
circle, written as early as 1464, though not 276548 
published until 1532. The question proposed 
is to divide 18190735 by 415. The divisor is 4HH 
placed under the dividend and repeated at 444 

every step backward, and all the figures erased 43333 
in succession. The quotient, 43833 is placed 
down the side and along the bottom, the remainder 40 being the 
only digits left on the board. 

It is amusing to observe the enthusiastic admiration of Di 
Borgo for this method of division. When describing the pre- 
ceding method he seems impatient, and looks forward with 
pleasure to the description of the method a la galea, as pos- 
sessing a certain charm and solace, remarking that it is a 
noble thing to see in any species and scheme of numbers, a 
galley perfectly exhibited, so as to be able to observe its mast, 
its sail, its yards and its oars, launched in the spacious ocean 
of arithmetic. This method, we are surprised to learn, 
appears to have been preferred by nearly every writer on 
arithmetic as late as the end of the 17th century. It was 
adopted by the Spaniards, French, Germans, and English; and 
it is the only method which they have thought necessary to 
notice. It is found almost universally in the works of Tonstall, 
Recorde, Stifel, Ramus, Stevinus, and Wallis ; and it was 
only at the beginning of the 18th century that this method of 
division, called by the English arithmeticians the scratch 
method of division, from the scratches used in cancelling the 
figures, was superseded by the method now in common use, 
which was specifically called Italian division, from the country 
whence it was derived. 




ORIGIN OF ARITHMETICAL PROCESSES. 59 

Recorde noticed the Italian method of 
division, which, he says, " I first learned 
of, and is practiced by my ancient and espe- 
cial loving friend, Master Henry Bridges, 
wherein not any one figure is cancelled or 
defaced. He illustrates the method by an ^ 

example which we subjoin ; though, as before 
stated, he preferred the scratch method of dividing. 

POWEUS AND ROOTS. The author of the Lilawati has given 
rules for the formation of squares and cubes, as well 
as for the extraction of the corresponding roots. 
The rule for the formation of the square, which is gj 
very ingenious, is as follows: Place the square of 28 
the last digit over the number, and the rest of the 126 

digits doubled and multiplied by the last are to be 1? 

placed above them respectively ; then repeating the _J^ 
number with the omission of the last digit, perform 88209 
the same operation. This is illustrated in squaring the num- 
ber 297. 

In performing the converse operation, every uneven place is 
marked by a vertical line, and the intermediate digits by a 
horizontal one ; but if the place be even, it is joined 
with the contiguous odd digit. It may be illus- ' ' 
trated by extracting the square root of 88209, 
enough of the work being indicated to show the 48209 

nature of the method. We subtract from the last i 

uneven place, 8, the square 4, and there remains 12209 
48209, represented as in the margin. Double the ' 

root 2, making 4, and divide 48, the number de- 
noted l>y the next two terms, by the result, 
obtaining '.) (10 would be too large), and subtracting 9 times 4 
or '.',('}, \v<- have 12209. From the uneven place, with the resi- 
due, 122, subtract the square of 9, or 81; the remainder is 
4109 Double 9, giving 18, and unite the result with 4, giving 
58, and divide 410 by it, and we have 7, and the remainder, 



60 



THE PHILOSOPHY OF ARITHMETIC. 



8 



49, to which the square of the quotient 7, or 49, answers with- 
out a residue. The double of the quotient, 14, is put in a line 
with the preceding double number, 58, making 594, the half 
of which is the root sought, 297. 

This account of the Hindoo method of extracting square 
root, is taken from the commentators on the Lilawati, and does 
not differ essentially from the method now used ; and the same 
may be said of the method of extracting the cube root, the 
principal difference from the present method being found in 
their peculiar methods of multiplying and dividing. 

The method of extracting the square root used by the Ara- 
bians resembled their method of division ; and it is prob- 
able that they are both founded on 
the Greek methods of performing these 
operations with sexagesimals. The 
example given will show the form of 
operation. Vertical lines being drawn 
and the numbers distinguished into 
periods of two figures, the nearest root 
of 10 is 3, which is placed both below 
and above, and its square, 9, subtracted ; 
the 3 is now doubled, and 6 being writ- 
ten in the next column, is contained 
twice in 17, or the remainder with the 
first figure of the next period ; the 2 is 
therefore set down both above and 
below, and being multiplied into 6 
gives 12, which is subtracted from 17, 
leaving 5 ; the square of 2, or 4, is now 
subtracted from 55, and 518, the re- 
mainder, with the succeeding figure, is 
divided by 64, or the double of 32, giving 8 for the quotient ; 
then 8 times 64 are 512, which, subtracted from 618 leaves 6-, 
and 64 is exhausted by taking from it the square of 8. It is 
said that this mode was adopted from the Arabs by the Hindoos. 



1 





7 


5 


8 


4 




9 












1 


7 










1 


2 












5 


5 












4 










5 


1 


8 








5 


1 


2 












6 


4 










6 


4 










4 


8 






6 


6 








3 




2 







ORIGIN OF ARITHMETICAL PROCESSES. 



61 




^2416 

604 
4(304 

2416 



00 

m 



The earlier mathematicians of Europe employed a similar 
method of extracting the square root, though perhaps not quite so 
systematic and regular. In proof of the rule which they followed, 
they constantly refer to the 4th proposition of the 2d book of 
Euclid. I will give several examples illustrating their methods 

The first is from the arithmetic of Pelletier, 
the first edition of which was published in 
1550. It represents his method of extract- 
ing the square root of 92416, and is so sim- 
ple it needs no explanation. It will be seen 
that the dots marking the periods into which 

the number is separated are placed under the number, instead 

of above it as is now the custom. 

The second example is from the work of 
Lucas di Borgo, and is in the form of the 

process which was most commonly adopted. 

The example, as will be seen, is the extrac- 
tion of the square root of 99980001. The 

scheme will require no explanation, but will 

be readily understood by those who are fam- 
iliar with the galley form of division. 

We present another illustration taken from the tract, already 

mentioned, of Regiomontanus. The question is to find the 

square root of the number 5261216896. 

Now the nearest square to 52 is 49, leaving 

3 to be set above the 2, while 7, the root, is 

placed in the vertical line ; then double of 

1, or 14, being set under the 36, is contained 

twice, and 2 is accordingly placed under the 

7 ; but twice 1 is 2, which taken from 3 

leaves 1, and twice 4 are 8, which taken 

from 6, or 16, leaves 8, and extinguishes the 

1 before it ; and twice 2 are 4, which taken 

from 1, or 11, leaves 7, and converts the pre- 
ceding 8 into 7. In this way the process advances till the 






123 
2465 
1757174 
38796595 
5261216896 
14406 
430 
145 
14 
1 



72534 



02 



THE PHILOSOPHY OF ARITHMETIC. 



figures become successively effaced. The root, 72534, is placed 
both at the right hand side and also immediately below the 
work. The divisors do not appear to be right, but we do not 
feel sufficiently acquainted with the subject to change them, and 
do not possess the original work by which we can verify them. 
The method of extracting cube root used by the Arabians and 
Persians, and by them communicated to the 
Hindoos, resembles likewise their method 5 

of performing division. We will illus- 
trate it by extracting the cube root of 
91125. Having drawn the vertical lines 
as indicated, the several digits of the num- 
ber are inscribed between them, and dots 
set over the first, fourth, seventh, etc., 
reckoning from the right. The nearest 
cube to 91 is 64, which is set down and 
subtracted, leaving 27. To obtain the 
next term of the root, 3 times 18, which 
is 3 times the square of the root found, is 
written below, and being contained 5 
times in 271, the divisor is completed by 
adding 3 times the product of 4 and 5, or 
60, and then the square of 5, or 25, mak- 
ing f in all 5425, each term of which is 
multiplied by 5, and the products sub- 
tracted in succession. 
The ancient mode of extracting the cube root practiced in 



9 


1 


1 


a 


5 


6 


4 








2 


7 








2 


5 










2 










2 













1 










1 













'2 
2 


5 




4 


8 










6 


8 










2 


5 


1 


5 


4 


2 


5 



Europe was similar to the process 
just explained, but not so regular 
and formal. The annexed example 
is taken from the Ars Supputandi 
of the famous Cuthbert Tonstall, 
Bishop of Durham, the earliest 
treatise on arithmetic published in 
England, and a work of no common 
merit. The number 250523582464, 



4' 7'6' 
3'" 4'0'" 
2'5'0'5'2'3'5'8'2'4'6'4' 



6 







3'4'l / 8 / 7'8'9 / 8 7 9 7 0'4 ' 
1'0'2'1'5'9' 
1'2'2'S'l' 0' 
4'9'fi' 8'2'4'6' 

7' 



ORIGIN OF ARITHMETICAL PROCESSES. 63 

whose root is to be extracted, is placed above two parallel 
lines, between which the root 6304 is inserted ; the successive 
divisors and the corresponding remainders being written alter- 
nately below and above, and the figures erased as fast, as 
che operation advances, the operation of erasure being here 
denoted by accents. 

M. Stifel, who usually sought to generalize the methods of 
his predecessors, has considered the process of extracting the 
square root in connection with those of higher powers. By 
observing the formation of the powers themselves, he discovered 
certain schemes, or pictures as he calls them, for extracting the 
square, cube, biquadrate, etc., roots. If we indicate the terms 
of a binomial root by a and b, his scheme for the square root 
would consist of a-20-6 and 6 2 written under the b to denote 
addition. The meaning of the scheme is $ 
that in extracting the square root, the first 070r)2p / T(2601 
term, a, must be multiplied by 20 to get 2 - 20-6. 

the divisor from which we determine the 36-276 

26 - 20 0-0 
second term, o; after which the sum of 2-60-20 1 

the product of a, 20, and b, and b 1 must 1-5201 

be subtracted from the first remainder. 

His method is illustrated by the extraction of the square root 
of 6765201, as here given. 

The history of the origin of these arithmetical processes is 
derived from Prof. Leslie and Dr. Peacock, much of it having 
been copied word for word from the originals. The origin of 
methods in Fractions, Decimals, Rule of Three, Continued 
Fractions, etc., will be given in connection with those subjects ; 
and such other historical information as it is thought will be 
of interest to the reader will be presented in its appropriate 
place. Occasionally the same fact is repeated, in order to give 
a completeness to tho particular subject discussed. 



PART I. 

THE NATURE OF ARITHMETIC. 



SECTION I. 
THE NATURE OF NUMBER. 

SECTION II. 
ARITHMETICAL LANGUAGE. 

SECTION III. 
ARITHMETICAL REASONING. 



SECTION 1. 

THE NATURE OF NUMBER. 



I. SUIUECT MATTER OP ARITHMETIC 



II. DEFINITION OF NUMI;ER 



III. CLASSES OF NUMBERS. 



IV. NUMERICAL IDEAS OF THE ANCIENT* 



CHAPTER I. 

NUMBER, THE SUBJECT MATTER OF ARITHMETIC. 

\TUMBER was primarily a thought in the mind of Deity. 
-L i He put forth His creative hand, and number became a fact 
of the universe. It was projected everywhere, in all things, 
and through all things. The flower numbered its petals, the 
crystal counted its faces, the insect its eyes, the evening its 
stars, aud ihc moon, time's golden horologe, marked the months 
and the seasons. 

Man was created to apprehend the numerical idea. Finding 
it embodied in the material world, he exclaimed, with the enthu- 
siasm of Pythagoras, " Number is the essence of the universe, 
the archetype of creation." He meditated upon it with enthu- 
siasm, followed its combinations, traced its relations, unfolded 
its mystic laws, and created with it a science the beautiful 
science of Arithmetic. Let us consider the origin and nature 
of the idea out of which man has created this science of exact 
relations and interesting principles. 

Origin. The conception of number begins with the contem- 
plation of material objects. Objects are found in combinations 
or collections, and the inquiry, how many of such a collection, 
gives rise to the idea of number. The young mind looks out 
upon nature, communes with its material forms, sees unity and 
plurality, the one and the many, all around it, and awakens to 
the numerical idea. Strange law of spiritual development! 
the material thing calls into being the immaterial thought. 
The unity and plurality, as it dwelt in the God-mind and was 

( 07 ) 



68 THE PHILOSOPHY OF ARITHMETIC. 

embodied in the material world, passes over to the mind of 
man, and appears as an idea of the immaterial spirit. 

The idea of definite numbers is developed by a mental act 
called counting. We ascertain the how-many of a collection, 
by counting tb*> objects in the collection. The act of counting, 
(one, two, three, etc.), is the foundation of all our knowledge 
of number. In counting, we pass in succession from one 
object to another. Succession implies time, and is only possi- 
ble in time. The idea of number, therefore, has its origin in 
the fact of time, and is possible only in this great fact. A brief 
consideration of this relation will not be uninteresting. 

Time is one of the two great infinitudes of nature. Space 
and Time are the conditions of all existence. Time enables 
us to ask the question, when ; Space, the question, where. 
Space is the condition of matter regarded as extended, and is 
thus the condition of extension. Extension has three dimen- 
sions, length, breadth, and thickness. The science of extension 
is geometry. Space is thus seen to be the basis or condition 
of the science of geometry. 

Time is the condition of events, as Space is of objects. 
Every event exists in Time, as every object must exist in Space. 
Time has somewhat the same relation to the world of mind, 
that Space has to the world of matter. Matter extends in 
Space, as mind protends in Time. This intimate relation of 
Number and Time leads me to present a few thoughts concern- 
ing the nature of Time, and the development of the idea of 
Number from it. 

Time is not a mere abstraction. It is not a quality per- 
ceived in an object and drawn away from it by the power of 
abstract thought, and conceived as an abstract notion. Neither 
is it a general idea, or a concept. We do not first get partic- 
ular notions of Time, and then, by putting these together, form 
a general idea of it. No summation of particular times can 
give the grand, unlimited idea of Time that the mind possesses. 
Indeed, we do not consider particular times as examples of 



NUMBER, THE SUBJECT MATTER OF ARITHMETIC. 69 

Time in general ; but we conceive all particular times to be 
parts of a single endless Time. This continually flowing 
and endless time is what offers itself to us when we contem- 
plate any series of occurrences. All actual and possible 
times exist as parts of this original and genera. Time. There- 
fore, since all particular times are considered as derivable from 
time in general, it is manifest that the notion of time in general, 
cannot be derived from the notions of particular times. 

Time is a grand intuition. It is an idea which is formed in 
the mind when the proper occasion of sensible experience is 
presented. Sensible experience is not the cause, but the occa- 
sion upon which the mind conceives or originates this idea. 
It is the product of the higher intuitive power known as the 
Reason. But Time is not only an idea, it is a great reality. It 
has a real objective existence, independent of the mind which 
conceives it. Were there no minds to conceive it, time would 
still exist as the condition of events. Were all events blotted 
out of existence, time would remain an endless on-going. 

Time is infinite. No mind can conceive its beginning ; no 
mind can conceive its end. All limited times merely divide, 
but do not terminate the extent of absolute time. In it every 
event begins and ends, while it never begins and never ends. 
It is, in its very nature, like Him who inhabiteth eternity, with- 
out beginning and without end. 

Time gives rise to succession, as space does to extension. 
Out of succession grows the idea of Number, and the science of 
Number is Arithmetic. Arithmetic, therefore, has somewhat the 
same relation to time, that geometry has to space. In view of 
this fact, some philosophers have called geometry the science 
of space, and arithmetic the science of time. This view of 
ju-ithmetic, however, has not been adopted by all writers, since 
t here are other ideas growing out of time than that of number. 
VVhewell, in writing of the Pure Sciences, speaks of the three 
great ideas Space, Time, and Number ; thus distinguishing 
between Number and Time. Several efforts have been made 



70 THE PHILOSOPHY OF ARITHMETIC. 

to construct a science of Time; the most remarkable is that of 
Sir William Rowan Hamilton, which resulted in the invention 
of the wonderful Calculus of Quaternions. 

Time is considered as having but one dimension. In this 
respect it differs from Space, which has three dimensions, 
length, breadth, and thickness. Time may be regarded as 
analogous to a line, but it has no analogy to a surface or a vol- 
ume. Time exists as a series of instants which are before and 
after one another ; and they have no other relation than this 
of before and after. This analogy between Time and a line- 
is so close, that the same terms are applied to both ideas, and 
it is difficult to say to which they originally belonged. Time 
and lines are called long and short; we speak of the beginning 
and the end of a line, of & point of time, and of the limits of a 
portion of duration. 

There being nothing in Time which corresponds to more 
than one dimension of extension, there is nothing which bears 
any analogy with figure. Time resembles a line extending 
indefinitely both ways ; all partial times are portions of this 
line ; and no mode of conceiving time suggests to us a line 
making an angle with the original line, or any other combina- 
tion which might give rise to figures of any kind. The anal- 
ogy between time and space, which in many circumstances is 
so clear, here disappears altogether. Spaces of two and of 
three dimensions, surfaces and volumes, have nothing to which 
we can compare them in the conceptions arising out of time. 

The conception which peculiarly belongs to thrre, as figure 
does to space, is that of the recurrence of times similarly 
marked. This may be called rhythm, using the word in a 
general sense. The forms of such recurrence are noticed in 
the versification of poetry and the melodies of music. All 
kinds of versification, and the still more varied forms of recur- 
rence of notes of different lengths, which are hoard in all the 
varied strains of melodies, are only examples of such modifica- 
tions or configurations, as we may call them, of time. They 



HUXBEB, THE SUBJECT MATTER 07 ABITHMETIC. 71 

involve relations of various portions of time, as figures involve 
relations of various portions of space. But jet the analogy 
between rhythm and figure is by no means very elose ; for in 
rhythm we hare relations of quantity alone in parts of time, 
whereas in figure we hare relations not only of quantity, but 
of a kind altogether different namely, of position. On the 
other band, a repetition of similar elements, which does not 
necessarily occur in figures, is quite essential in order to 
impress upon us that measured progress of time of which we 
here speak. And thus the ideas of time and space bare each 
their peculiar and exclusive relations; position and figure 
belonging only to space,- while repetition and rhythm are ap- 
propriate only to time. 

One of the simplest forms of recurrence is alternation, as 
we have alternate accented and unaccented syllables For 
example: 

" Come one', cotne all', this rock' shall fly V 
Or without any subordination, as when we reckon numbers, 
and call them in succession, odd, even, odd, even, etc. 

But the simplest of all forms of recurrence is that which 
has no variety, in which a series of units, each considered as 
exactly similar to the rest, succeed one another; as one, one, 
one, and so on. In this case, however, we are led to consider 
each unit with reference to all that have preceded ; and thus 
the series one, one, one, and so forth, becomes one, two, three, 
four, five, and so on; a series with which all are familiar, 
and which may be continued without limit. We thus collect 
from that repetition of which time admits, the conception of 
Number. 

This view of the origin of the idea of Number is now accepted 
by a large number of thinkers, bat there are those wbo bold other 
theories. Toe most objectionable view is tbat number is a cense 
perception, the absurdity of which is seen in the fact that number 
IMS no color or form or any attribute of a percept. Number is not 
a percept ; it is an intuition. 



CHAPTER II. 

DEFINITION OF NUMBER. 

THE idea of number is so elementary that it is difficult to 
define it scientifically. Various definitions have been pre- 
sented by different writers upon the subject, though no one 
has hitherto given one which is, in all respects, satisfactory. 
The two most celebrated definitions are those of Newton and 
Euclid, both of which will be briefly considered. 

Newton defined number as " the abstract ratio of one quan- 
tity to another quantity of the same species." This definition 
is philosophical and accurate. It shows number to be a pure 
abstraction derived from a comparison of things. In discrete 
quantity, it regards one of the individual things as the unit of 
comparison ; while in continuous quantity the unit is assumed 
to be some definite portion of the quantity considered. 

This definition was no doubt primarily intended to apply to 
extended quantity, in which there is no natural unit, but in 
which some definite portion of the quantity is assumed as a 
unit of measure, and the quantity estimated by comparing it 
with this unit as a standard. Such comparison gives rise to 
three kinds of numbers; integral, fractional, and surd numbers. 
When the quantity measured contains the unit a definite 
number of times, the number is integral ; when it is only a 
definite part of the measure, the number is fractional ; when 
there is no common measure between the unit and th quan- 
tity measured, the number is a surd or radical. 

The definition of Newton, though admirable in many 
respects, is not suitable for popular use. It is too abstract and 

(72) 



DEFINITION OF NUMBER. 73 

difficult to be understood by young pupils ; and cannot, there- 
fore, be recommended for our elementary text-books. It may 
be said, also, that it does not express clearly the process of 
thought by which we attain the idea of number. It is more 
appropriate as applied to continuous than to discrete quantity, 
while the idea of number begins with discrete rather than con- 
tinuous quantity. In this latter respect it may possibly be 
improved by changing the form of expression, while retaining 
its spirit: thus, A number is the relation of a collection to the 
single thing. This is simpler than the original form, and is in 
many respects a very satisfactory definition. 

Euclid defined number to be "an assemblage or collection of 
units or things of the same species." This definition, slightly 
modified, has been generally adopted by mathematicians. In 
its original form it excluded the number one, since one thing is 
not an assemblage or collection, and hence it has been changed 
to read A number is a unit or a collection of units. This is 
the definition which is now found in a large number of text- 
books. 

This definition, 'however, is not strictly correct. A number 
is not precisely the same as a collection of units, and a collec- 
tion of units is not necessarily a number. In other words, 
there is a difference between a collection of things and a num- 
ber of things. This may be more clearly seen by the use of 
the corresponding verbs. To collect and to number -are two 
different things. We may collect without numbering, and we 
may number without collecting ; I may collect & number of 
things, and I may number a collection of things. If a basket 
of apples were strewn over the floor and I were told to collect 
them, I might do so without numbering them ; or, if told to 
number them, I might do so without collecting them. In the 
latter case I would have a number of apples without having a 
collection of apples, except the mental collection, from which it 
appears that a number is not precisely the same as a collection. 
Number is more definite than collection. A collection is an 



74 THE PHILOSOPHY OF ARITHMETIC. 

indefinite thing, numerically considered ; number is that which 
makes it definite. Number and collection are not, therefore, 
identical. Number is rather the how many of the collection. 
It is thus seen that Euclid's definition, as modified and now 
introduced into most of our text-books, is not without scien- 
tific objections. It must be admitted, however, that there is 
no other one word which so nearly expresses the idea of the 
word number as collection; and, for ordinary purposes, they 
may be used interchangeably. Thus we may say, in analysis, 
we pass from the collection to the single thing ; from a number 
to one. It is, therefore, regarded as the best definition for the 
ordinary text-book, that has hitherto been presented. 

From this discussion it will appear, as above stated, that it 
is difficult to present a good definition of Number. This diffi- 
culty is due to the fact that Number is a simple term express- 
ing a simple idea, for which we have no other word of 
precisely the same signification. Simple terms are always 
difficult to define, from the very fact that they define themselves. 
Indeed, perhaps there is nothing in the way of a definition of 
number clearer than the identity "A Number is a Number." 
The following, though liable to a verbal objection, seems to me 
to come as near the truth as anything that has yet been pre- 
sented: A Number is the how-many of a collection of units; 
or, A Number is how many times a single thing is reckoned, or 
is contained in a collection 

The first excludes the number one, unless, as some writers 
propose, we give a special signification to collection. The 
second provides for the number one, but is not, in other respects, 
eo satisfactory as the first. These definitions express pre- 
cisely the idea of a number, but the use of the expression hoiv 
many as a noun, is not elegant in the English language. The 
simplest and most satisfactory definition for a text-book is, 
"A Number is a unit or a collection of units." 

The definitions of a number, as given in some of our text- 
books, are very objectionable. One author says : " Numbers 



DEFINITION OF NUMBER. 76 

are repetitions of units." This may answer as a popular state- 
ment, but is very far from meeting the requirements of a sci- 
entific definition. Another author says: "A number is a 
definite expression of quantity." So is a triangle or a circle, 
each of which should be a number if this definition is correct 
Another says: "A number is an expression that tells how 
many." The two errors are, first, that a number is not an 
expression; and, second, that a number does not tell anything. 
The following definitions have also been given by different 
writers: "Number is a term signifying one or more units;" 
"A number is an expression of one or more things of a kind;" 
" A number is an expression of quantity by a unit, or by its repe- 
tition, or by its parts;" "Number consists of a repetition of 
units;" "A number is either a unit or composed of an assem- 
blage of units;" "A number is a term expressing a particular 
sameness of repetition." Other definitions, equally incorrect, 
may be found by leafing over text-books upon the subject 
A very simple definition, and especially suitable for a primary 
text-book is, "A number is one or more units." It may bo 
remarked that authors seem to be adopting the definition of 
Euclid, with the modification presented above, so that the 
standard definition in our text-books is becoming, " A number 
is a unit or a collection of units." 

To give a perfect definition of Number is exceedingly diffi- 
cult, if not impossible. Stevinus defines it as "that by which 
the quantity of anything is expressed," but mathematicians 
have not adopted it. Euler's definition, " number is nothing 
else than the ratio of one quantity to another quantity taken 
as a unit," has been highly commended. "Number is a do fi- 
nite expression of quantity," has its advocates. "Number is 
quantity conceived as made up of parts, and answers to the 
question, How man}'?" has the authority of a very c-nrcful 
writer. The world, however, still waits for a simple and uc- 
carate definition, which may be generally adopted. 



CHAPTER III. 

CLASSES OF NUMBERS. 

N" UMBERS have been variously classified with respect to 
different properties, or by regarding them from different 
points of view. The fundamental classes to which attention 
is here called, are Integers, Fractions, and Denominate Num- 
bers. These three classes are practically and philosophically 
distinguished, and constitute the basis of three principal 
divisions of the science of arithmetic. Logically, the distinc- 
tion is not without exception, for a Fraction may be denomi- 
nate, and a Denominate Number may be integral; but the 
division is regarded as philosophical, since they are not only 
different in character, but require distinct methods of treat- 
ment, and give rise to distinct rules and processes. The 
philosophical character and relation of these three classes of 
numbers, will appear from the following considerations : 

Integers. The Unit is the basis or beginning of numbers. 
A number is a synthesis of units; it is the how -many of a 
collection of units. These units, as they exist in nature, are 
whole things, undivided ; hence the first numbers of which a 
knowledge is acquired, are whole numbers, that is, collections 
of entire or undivided units. Such units, being entire, are 
called integral units, and the numbers composed of them are 
Called integral numbers, or Integers. An Integer is, therefore, 
a collection of integral units, or, as popularly defined, it is a 
whole number. It is a product of pure synthesis. 

Fractions. The Unit, as the basis of arithmetic, may be 
multiplied or divided. A synthesis of units, as we have seen, 

(76) 



CLASSES OF NUMBERS. 77 

gives rise to Integers ; a division of the unit gives rise to 
Fractions. Dividing the unit into a number of equal parts, we 
see that these parts bear a definite relation to the unit divided, 
and by taking one or more of these parts, we have a Fraction. 
It is thus seen that the conception of a fraction implies three 
things: first, a division of the unit; second, a comparison of 
the part to the unit; and third, a collection of the fractional 
parts. In other words it is the product of three operations, 
division, comparison, and collection ; or, like the logical nature 
of the science of arithmetic itself, a fraction is a triune product, 
consisting of analysis, comparison, and synthesis. 

Denominate Numbers. The unit of a simple integral num- 
ber exists in nature. A Denominate Number is a collection of 
units not found in nature ; it is a collection of artificial units 
adopted to measure quantity of magnitude. The philosophical 
character of a denominate number is indicated in the following 
statement: Nature, regarded as how many and how much, 
gives rise to two distinct forms of quantity ; quantity of 
multitude, and quantity of magnitude. Quantity of multitude 
is primarily expressed by numbers, since it exists in the form 
of individuals, or units ; quantity of magnitude does not admit, 
primarily, of being expressed in numerical form. To estimate 
quantity of magnitude, we must fix upon some definite part of 
the quantity considered as a unit of measure, by which we can 
give it a numerical form of expression. 

A Denominate Number may, therefore, be defined as a 
numerical expression of quantity of magnitude. Or, since 
the unit is a measure by which the quantity is estimated, we 
may define it to be a number whose unit is a measure. 
Again, since tho unit is not natural but artificial, we may de- 
fine it to be a number whose unit is artificial. Either of these 
definitions suffices to distinguish it from the other two classrj- 
of numbers. It differs from them in respect of the nature of 
the quantity to which it refers, and also in its origin and com- 
position. In the simple integral numbers, the units, as found 



78 THE PHILOSOPHY OP ARITHMETIC. 

in nature, are collected ; in the denominate number, the unit 
is assumed, the quantity compared with the unit, and the 
result expressed numerically. The same kind of quantity may 
be measured by different units, bearing a definite relation to 
each other, which gives rise to a scale of units. Taking our 
scales as they now exist, we have a series of units definitely 
related to each other, forming a Compound Number, which 
does not appear in the other classes of numbers. This, how- 
ever, is rather incidental than essential, as it partially vanishes 
when we apply the decimal scale to quantity of magnitude, as 
in the metric system of weights and measures. 

It is thus seen that there are three distinct classes of num- 
bers; and, since they require different methods of treatment, 
they will be considered independently. The remainder of this 
chapter will be devoted to the discussion of some of the pecu- 
liarities of integral numbers. 

Classes of Integers. Simple Integral Numbers, being 
learned before Fractions and Denominate Numbers, are the 
first class to which the term number was applied; they have 
consequently appropriated to themselves the almost exclusive 
use of the word number. Thus, it is the general custom to 
speak of Numbers, Fractions, and Denominate Numbers, appar- 
ently forgetful that they are all numbers. This custom being 
o common, the word Integer being somewhat inconvenient, 
and some of the properties which belong to integral numbers 
applying also to the other two classes, I will also use the word 
number in place of integral number in considering this part 
of the subject. 

Numbers are of two general classes, Concrete and Abstract. 
A Concrete Number is a number in which the kind of unit is 
named. An Abstract Number is a number in which the kind 
of unit is not named. A concrete number may also be defined 
as a number associated with something which it numbers. 
This is seen in the etymology of the term, con and cresco, a 
growing together. An abstract number may also be defined 



CLASSES OF NUMBERS. 79 

as a number not associated with anything numbered. This 
is indicated by the etymology of the term, ab and traho, a 
drawing from. It is not true, therefore, as has been asserted, 
that " all numbers are concrete." Number is never concrete, in 
the popular sense of material. When I think of four apples, 
the apples are concrete, but the four is purely numerical and 
in no sense material. It would be much nearer the truth to 
say that all numbers are abstract; for the number itself is 
always a pure abstraction. The distinction between an abstract 
and a concrete number is not a difference in the numbers them- 
selves, but a distinction founded upon the fact of their being 
associated or not associated with something numbered. 

This distinction is clearly seen in the origin of the idea of 
number. The idea of number is awakened by the contem- 
plation of material objects. The mind takes the thought of 
the how-many, abstracts it from the material things with which 
it was at first associated, lifts it up into the region of the ideal, 
and conceives it as pure number. Though the idea was pri- 
marily awakened by the objects of the material world as the 
occasion, yet so distinct is number from matter, that if all 
material things were destroyed, we could still have a science 
of number as complete as that which now exists. 

There is still another method of conceiving the distinction 
between concrete and abstract numbers. All numbers are 
composed of units. The unit gives character and value to the 
number of which it is the basis. A number is clearly appre- 
hended only as we have a clear apprehension of the unit: thus, 
6 pounds or 6 tons are only clear and definite ideas to us as 
we have clear and definite ideas of the units, pound and 
Ion. Hence, also, the nature of numbers depends upon the 
nature of the units which compose them. Fundamentally, 
units are of two classes, concrete and abstract. A concrete 
unit is some object in nature or art, as, an apple, a book ; or 
some definite quantity agreed upon to measure quantity of 
magnitude; as, a yard, a pound, etc. An abstract unit is 



80 THE PHILOSOPHY OF ARITHMETIC. 

merely one without any reference to any particular thing. The 
concrete unit is not a number, it is only one of the things num- 
bered ; the abstract unit is the number one. A collection of 
abstract units gives us an Abstract Number; a collection of 
concrete units gives us what is called a Concrete Number. 
An Abstract Number is thus merely a number of abstract 
units ; a Concrete Number is a number of concrete units. The 
number itself and the things numbered, considered together, 
constitute what is called the Concrete Number. This is the 
usual method of conceiving the distinction between an abstract 
and a concrete number ; but it is not as simple as the one pre- 
viously presented. 

From either method of conceiving the difference between 
these two classes of numbers, it will be seen that the Concrete 
Number is dual in its nature, consisting of two classes of units. 
Thus, in the concrete number, four apples, the concrete unit 
is one apple; while the basis of the number four itself is the 
abstract unit, one. Both of these classes of units must be 
clearly apprehended in order to have a clear and adequate idea 
of any concrete number. 



CHAPTER IV. 

NUMERICAL IDEAS OF THE ANCIENTS. 

AMONG the ancients, much time was spent in discussing 
the properties of numbers. The science, with them, was 
mainly speculative, abounding in fanciful analogies. Pythag- 
oras, the greatest mathematician of his age, was deeply 
imbued with this passion for the mysterious properties of 
numbers. He regarded number as of Divine origin, the foun- 
dation of existence, the model and archetype of things, the 
essence of the universe. 

Plato ascribed the invention of numbers to Theuth, as may 
be seen in the following passage in the Phsedrus: " I have 
heard, then, that at Naucratis, in Egypt, there was one of the 
ancient gods of that country, to whom was consecrated the 
bird which they call Ibis ; but the name of the deity himself 
was Theuth. He was the first to invent numbers, and arith- 
metic, and geometry, and astronomy, and moreover draughts 
and dice, and especially letters." In the Timseus, he presents 
the conception of the relation of numbers to time, with great 
beauty of expression. " Hence, God ventured to form a cer- 
tain movable image of eternity; and thus, while he was 
disposing the parts of the universe, he, out of that eternity 
which rests in unity, formed an eternal image on the principle 
of numbers, and to this we give the appellation of Time." 

Aristotle, in speaking of the Pythagoreans, says, "They 
supposed the elements of numbers to be the elements of all 
entities, and the whole heaven to be an harmony and number.' 1 
6 (81 ) 



82 THE PHILOSOPHY OF ARITHMETIC. 

And again he says, "Plato affirmed the existence of numbers 
independent of sensibles; whereas, the Pythagoreans say that 
numbers constitute the things themselves, and they do not set 
down mathematical entities as intermediate between these." 

The views of Pythagoras are so curious and interesting that 
they may be stated somewhat in detail. He regarded Numbers 
as of Divine origin, as above stated, and divided them into 
various classes, to each of which were assigned distinct proper- 
ties. Even numbers he regarded as feminine, and allied to 
the earth ; odd numbers were supposed to be endued with 
masculine virtues, and partook of the celestial nature. 

One, or the monad, was held as the most eminently sacred, 
as the parent of scientific numbers. Two, or the duad, was 
viewed as the associate of the monad, and the mother of the 
elements, and the recipient of all things material ; and three, 
or the triad, was regarded as perfect, being the first of the mas 
culine numbers, comprehending the beginning, middle, and end, 
and hence fitted to regulate by its combinations the repetition 
of prayers and libations. It was the source of love and sym- 
phony, the fountain of energy and intelligence, the director of 
music, geometry, and astronomy. As the monad represented 
the Divinity, or Creative Power, so the duad was the image 
of matter ; and the triad, resulting from their mutual con- 
junction, became the emblem of ideal forms. 

Four, or the tetrad, was the number which Pythagoras 
affected to venerate the most. It is a square, and contains 
within itself all the musical proportions, and exhibits by sum- 
mation (1 + 2+3-f 4) all the digits as far as ten, the root of 
the universal scale of numeration. It marks the seasons, the 
elements, and the successive ages of man ; and also represents 
the cardinal virtues, and the opposite vices. It marked the 
ancient fourfold division of science into arithmetic, geometry, 
astronomy, and music, which was termed tetractys, or quater- 
nion. Hence, Dr. Barrow explains the oath familiar to the 



NUMERICAL IDEAS OF THE ANCIENTS. 83 

disciples of Pythagoras: " I swear by him who communicated 
the Tetractys." Five, or the pentad, being composed of the 
first male and female numbers, was styled the number of the 
world. Repeated in any manner by an odd multiple, it always 
reappeared ; and it marked the animal senses and the zones of 
the globe. 

Six, or the hexad, composed of the sum of its several fac- 
tors (1 + 2+3), was reckoned perfect and analogical. It wa. 
likewise valued as indicating the faces of the cube, and as 
entering into the composition of other important numbers. It 
was deemed harmonious, kind, and nuptial. The third power 
of 6, or 216, was conceived to indicate the number of years 
that constitute the period of metempsychosis. 

Seven, or the heptad, formed from the junction of the triad 
and tetrad, has been celebrated in every age. Being unpro- 
ductive, it was dedicated to the virgin Minerva, though pos- 
sessed of a masculine character. It marked the series of the 
lunar phases, the number of the planets, and seemed to modify 
and pervade all nature. It was called the from of Amalthea, 
and reckoned the guardian and director of the universe. 

Eight, or the octad, being the first cube that occurred, was 
dedicated to Cybele, the mother of the gods, whose image, in 
the remotest times, was only a cubical block of stone. From 
its even composition, it was termed Justice, and made to 
signify the highest or inerratic sphere. 

Nine, or the ennead, was esteemed as the square of the 
triad. It denotes the number of the Muses; and, being the last 
of the series of digits, and terminating the tones of music, it 
was inscribed to Mars. Sometimes it received the appellation 
of Horizon, because, like the spreading ocean, it seemed to 
flow around the other numbers within the decad ; for the same 
reason, it was also called Terpsichore, enlivening the productive 
principles in the circle of the dance. 

Ten, or the decad, from its important office in numeration, 
was, perhaps, most celebrated. Having completed the cycle, 



84 THE PHILOSOPHY OF ARITHMETIC. 

and begun a new series of numbers, it was aptly called apo- 
catastasic, or periodic, and therefore dedicated to the double- 
faced Janus, the god of the year. It had likewise the epithet 
of Atlas, the unwearied supporter of the world. 

The cube of the triad,or the number twenty-seven, expressing 
the time of the moon's periodic revolution, was supposed to 
signify the power of the lunar circle. The quaternion of 
celestial numbers, one, three, five, and seven, joined to that of 
the terrestrial numbers, two, four, six, and eight, compose the 
number thirty-six, the square of the first perfect number, six, 
and the symbol of the universe, distinguished by wonderful 
properties. 

In pursuit of these mystical relations and analogies, every 
number became, as it were, possessed of a property; and all 
numbers possessed some relative analogy with each other to 
which a name could be given. Numbers also became the sym- 
bols of intellectual and moral qualities. Thus, perfect numbers 
compared with those which are deficient or superabundant, are 
considered as the images of the virtues, regarded as equally 
remote from excess and defect, aud constituting a mean point 
between them: thus, true courage is a mean between audacity 
and cowardice, and liberality between profusion and avarice. 
In other respects, also, this analogy is remarkable, as perfect 
numbers, like virtues, are few in number, and generated in a 
constant order; while superabundant and deficient numbers 
are like vices, infinite in number, disposable in no regular 
series, and generated according to no certain and invariable 
law. 

The tracing of these analogies, accompanied, as they usually 
were, with moral illustrations of uncommon elegance and 
beauty, may be considered as furnishing a pleasing, if not a 
useful exercise of the understanding; but such analogies were 
often taken for proofs, and assumed as the bases of the most 
absurd and inconsistent theories. Thus Pythagoras considered 
"number as the ruler of forms and ideas, and the cause of 



NUMERICAL IDEAS OF THE ANCIENTS. 85 

gods and dafemons;" and again that "to the most ancient and 
all-powerful creating Deity, number was the canon, the efficient 
reason, th6 intellect also, and the most undeviating of the 
composition and generation of all things." Philolaus declared 
"that number was the governing and self-begotten bond of 
the eternal permanency of mundane natures." Another said, 
"that number was the judicial instrument of the Maker of 
the universe, and the first paradigm of mundane fabrication." 

It appears to have been a favorite practice with the Greeks 
of the latter ages to form words in which the sum of the num- 
bers expressed by their component letters, should be equal to 
some remarkable number ; of this kind were the words aSpaaat 
and afipaaata, the letters in which express numbers, which added 
together, are equal to 365 and 366, the number of days in the 
common and bissextile years respectively; and it was also 
remarked that the word vedas possessed the same property as 
the first of these words. Words in which the sums of the 
numbers expressed by the letters were equal, were called 
m>6fiaTa h6^a; and we have an example in the Greek anthol- 
ogy, where a poet, wishing to express his dislike to a fellow of 
the name of Aa^oyopaf, says, that having heard that his name 
was equivalent in numeral value to Ao<//of, a pestilence, he pro- 
ceeded to weigh them in a balance, when the latter was found 
to be the lighter. 

Observations like these, however trifling, are not without 
their portion of curiosity ; but the same indulgence cannot be 
shown to the absurdities of those Pythagorean philosophers, 
who, among other extraordinary powers which they attributed 
to numbers, maintained that, of two combatants, the one would 
conquer, the characters of whose name expressed the larger 
sum. It was upon this principle that they explained the rela- 
tive prowess and fate of the heroes in Homer, tta-pa^if, unri, 
and A,Y"LA'c, the sums of the numbers in whose names are 871, 
1225, and 1276 respectively. 

This very singular superstition continued in force as late tu> 



86 THE PHILOSOPHY OF ARITHMETIC. 

the sixteenth century, and was transferred from the Greek to 
the Roman numeral letters, I, U or V, X, L, C, D, and M, 
which correspond to the numbers 1, 5, 10, 50, 100, 500, and 
1000; thus the numeral power of the name of Maurice (Mau- 
ritius) of Saxony, was considered as an index of his success 
against Charles V. It was the fashion, also, to select or form 
memorial sentences or verses to commemorate remarkable 
ilates. Thus the year of the Reformation (1517) was found tc 
I*} expressed by the numeral letters of this verse of the Tt 
Deum, Tibi cherubin et seraphin incessabili voce proclamant, 
in which there is one M, four C's, two L's, two IPs or V's, and 
seven I's. 

The Chinese, also, are distinguished for their arithmetical 
fancies. They regarded even numbers as terrestrial, and par- 
taking of the feminine principle Yang; while odd numbers 
were regarded as of celestial extraction, and endued with the 
masculine principle Y. Even numbers were represented by 
small black circles; odd numbers by small white ones, vari- 
ously disposed and connected by straight lines. Thirty, the 
sum of the five even numbers, 2, 4, 6, 8, and 10, was called 
the number of the Earth ; twenty-five, the sum of the odd 
numbers, 1, 3, 5, 7 t 9, and also the square of five, was called 
the number of Heaven. 

The nine digits were grouped /*\ O-O O O p ^t 
in two ways called Lo-chou and \^/ 

Ho-tou. The former expres- 
sion signifies the Book of the o \ / o 
River Lo, or what the Great Yu 1 \/ o 
saw delineated on the back of J / \ o 
the mysterious tortoise which \ Q Q 
rose out of that river. It may 
be represented as follows : Nine * / *'^ 

was the head, one the tail, ^ ^ i( ^+ 

^ ' O ^^ ^^ 

three and seven its left and \ s* *\ * 

right shoulders, four and two 



NUMERICAL IDEAS OF THE ANCIENTS. 87 

its fore feet, eight and six its hind feet. The number five, 
which represented the heart, being the square root of twenty 
five, was also the emblem of Heaven. It will be noticed that 
this group of numbers is the common magic square of nine 
digits, each row of which amounts to fifteen. 

The Ho-tou was what the Emperor Fou-hi observed on the 
body of the horse-dragon which he saw spring out of the 
river Ho. It consists of the 
iirst nine numbers arranged in 
the form of across. The central 
number was ten, which, it is 
remarked by the commentators, 
terminates all the operations on 
numbers. Other facts equally 
curious will be found in the 
literature of other nations, a 
full collection of which would 
make an interesting volume. 
For the facts here presented, 
and the manner in which they are stated, I am indebted to 
Leslie. 

This passion for discovering the mystical properties of num- 
bers descended from the ancients to the moderns, and numer- 
ous works have been written for the purpose of explaining 
them. Petrus Bungus, in 1618, wrote a work on the mysteries 
of numbers, extending to seven hundred quarto pages. He 
illustrates all the properties of numbers, whether mathemat- 
ical, metaphysical, or theological ; and not content with col- 
lecting all the observations of the Pythagoreans concerning 
them, he has referred to every passage in the Bible in which 
numbers are mentioned, incorporating, in a certain sense, the 
whole system of Christian and Pagan theology. He holds that 
the number 11, which transgresses the decad, denotes the 
wicked who transgress the Decalogue, whilst 12, the numbe' 
of the apostles, is the proper symbol of the good and the just 




88 THE PHILOSOPHY OF ARITHMETIC. 

The number, however, upon which, above all others, he haa 
dilated with peculiar industry and satisfaction, is 666, the num- 
ber of the beast in Revelation, the symbol of Antichrist ; and 
he seems particularly anxious to reduce the name of Martin 
Luther to a form which may express this formidable number. 
It may also be remarked that Luther interpreted this number 
to apply to the duration of Popery, and also that his friend and 
disciple, Stifel, the most acute and original of the early math- 
ematicians of Germany, appears to have been seduced by these 
absurd speculations. 

The numbers 3 and 7 were the subject of particular specula- 
tion with the writers of that age ; and every department of 
nature, science, literature, and art, was ransacked for the pur- 
pose of discovering ternary and septenary combinations. The 
excellent old monk, Pacioli, the author of an early printed 
treatise on arithmetic, has enlarged upon the first of these 
numbers in a manner which is rather amusing, from the quaint 
and incongruous mixture of the objects which he has selected for 
illustration. " There are three principal sins," says he, "avarice, 
luxury, and pride ; three sorts of satisfaction for sin, fasting, 
almsgiving, and prayer; three persons offended by sin, God, 
the sinner himself, and his neighbor; three witnesses in 
heaven, the Father, the Word, and the Holy Spirit; three 
degrees of penitence, contrition, confession, and satisfaction, 
which Dante has represented as the three steps of the ladder 
that leads to Purgatory, the first marble, the second black and 
rugged stone, the third red porphyry. There are three Furies 
in the infernal regions; three Fates, Atropos, Lachesis, and 
Clotho; three theological virtues, faith, hope, and charity; 
three enemies of the soul, the world, the flesh, and the devil ; 
three vows of the Minorite Friars, poverty, obedience and 
chastity ; three ways of committing sin, with the heart, the 
mouth, and the act; three principal things in Paradise, glory, 
riches, and justice; three things which are especially displeas- 
ing to God, an avaricious rich man, a proud poor man, and a 



NUMERICAL IDEAS OF THE ANCIENTS. 89 

luxurious old man ; three things which are in no esteem, the 
strength of a porter, the advice of a poor man, and the 
beauty of a beautiful woman. And all things, in short, are 
founded in three, that is, in number, in weight, and in meas- 
ure." 

In these fanciful speculations, the number seven has received 
an equal, if not a greater distinction than the number three. 
In the year 1502, there was printed at Leipsic a work in honor 
of the number seven, especially composed for the use of the 
students of the university, which consisted of seven parts, 
each part consisting of seven divisions. In 1624, William 
Ingpen, Gent., of London, published a work entitled " The 
Secrets of Numbers, according to Theological, Arithmetical, 
Geometrical, and Harmonical Computation. Drawn for the 
better part, out of those ancients, as well as Neoteriques. 
Pleasing to read, profitable to understand, opening themselves 
to the capacities of both learned and unlearned, being no other 
than a key to lead men to any doctrinal knowledge whatso- 
ever." Di Borgo seems to have been influenced by the same 
principle in determining the number of the divisions of arith- 
metic; for he says: "The ancient philosophers assign nine parts 
of algorism, but we will reduce them to seven, in reverence of 
the seven gifts of the Holy Spirit; namely, numeration, 
addition, subtraction, multiplication, division, progressions, 
and extraction of roots." 

Some of these fancies are not entirely extinct at the present 
day. In England, seven constitutes the term of apprenticeship, 
the period for academical degrees, and as in our own country, 
the product of these two magic numbers three and seven con- 
stitutes the legal age of majority ; and the frequent use of the 
number seven in the Bible has given it associations which 
have caused it to be regarded as a sacred number. 



SECTION II. 



iRITHMETICAL LANGUAGE 



I. NUMERATION. 

II. NOTATION. 

III. ORIGIN OF SYMBOLS. 

IV. BASIS OF THE SCALE. 

V. OTHER SCALES OF NUMERATION 

VI. A DUODECIMAL SCALE. 

VII. GREEK ARITHMETIC. 

VIII. ROMAN ARITHMETIC. 

IX. PALPABLE ARITHMETIC. 



CHAPTER I. 

NUMERATION, OR THE NAMING OF NUMBERS. 

T)EGINNING at the Unit, we obtain, by a process of syn- 
-L) thesis, arithmetical objects which we call Numbers. 
These objects we distinguish by names, and thus obtain the 
language of arithmetic. This language is both oral and 
written. The oral language of arithmetic is called Numera- 
tion ; the written language of arithmetic is called Notation. 
Numeration treats of the method of naming numbers; Nota- 
tion treats of the method of writing numbers. As oral 
language always precedes written language, it is seen that 
Numeration precedes Notation, and that the practice of arith- 
meticians in reversing this order is illogical. 

Numeration is the method of naming numbers. It also 
includes the reading of numbers when expressed by characters. 
The oral language of arithmetic is based upon a principle 
peculiarly simple and beautiful. Instead of giving independ- 
ent names to the different numbers, which would require more 
words even to count a million than one could acquire in a life- 
time, we name a few of the first numbers, and then form groups 
or collections, name these groups or collections, and then use 
the first simple names to number the groups. The method is 
really that of classification, which performs for arithmetic 
somewhat the same service of simplification that it does in 
natural science. This ingenious, though simple and natural 
method of breaking numbers up into classes or groups, seems 
to have been adopted by all nations. With the civilized world 
and with most uncivilized tribes, these groups generally con- 

(93) 



94 THE PHILOSOPHY OF ARITHMETIC. 

sist of ten single things, suggested, undoubtedly, by the 
practice among primitive races, of reckoning by counting the 
fingers of the two hands. 

Method of Naming. The fundamental principle of naming 
numbers, then, is that of grouping by tens. We regard ten 
single things as forming a single collection or group; ten of 
these groups forming a larger group, and so on; ten groups 
of any one value forming a new group of ten times the value, 
each group being regarded and used as a single thing. In this 
way, by giving names to the first nine numbers, and names to 
the groups, and employing the first nine to number the groups, 
we are enabled to express the largest numbers in a concise and 
convenient form. The value of this method of naming may 
be seen from the consideration that, without it, the memory 
would be overwhelmed by the multiplicity of disconnected 
words, and we should require a lifetime to learn the names of 
numbers, even up to a few hundred thousands. It also enables 
us to form a clear and distinct conception of large numbers, 
whose composition we discover in the words by which they 
are expressed, or in the symbols by which they are represented. 
It serves, also, as a basis for the ingenious and useful method 
of writing numbers, without which arithmetic would be almost 
useless to us. 

Naming numbers in this way, a single thing is called one ; 
one and one more are two ; two and one more are three ; and 
in the same manner we obtain four, five, six, seven, eight, and 
nine, and then adding one more and collecting them into a 
group, we have ten. Now, regarding the collection ten as a 
single thing, and proceeding according to the principle stated, 
we have one and ten, two and ten, three and ten, etc., up to 
ten and ten, which we call two tens. Continuing in the same 
manner, we have two tens and one, two tens and two, etc , 
up to three tens, and so on until we obtain ten of these groups 
of tens. These ten groups of tens we now bind together by a 
thread of thought, forming a new group which we call a hun- 






NUMERATION, OB THE NAMING OP NUMBERS. 95 

dred. Proceeding from the hundred in the same way, we 
unite ten of these into a larger group which we name thousand, 
etc. 

This is the actual method by which numbers were originally 
named ; but unfortunately, perhaps, for the learner and for sci- 
ence, some of these names have been so much modified and 
abbreviated by the changes incident to use, that, with several 
of the smaller numbers at least, the principle has been so far 
disguised as not to be generally perceived. If, however, the 
ordinary language of arithmetic be carefully examined, it will 
be seen that the principle has been preserved, even if disguised 
so as not always to be immediately apparent. Instead of one 
and ten we have substituted the word eleven, derived from an 
expression formerly supposed to mean one left after ten, but 
now believed to be a contraction of the Saxon endlefen, or 
Gothic ainlif (ain, one, and lif, ten); and instead of two and 
ten, we use the expression twelve, formerly supposed to have 
been derived from an expression meaning two left after ten, 
but now regarded as arising from the Saxon twelif, or 
Gothic tvalif (tva, two, and lif, ten.) 

With the numbers following twelve, the principle can be 
more readily seen, though by constant use the original expres- 
sions have been abbreviated and simplified. The stream of 
speech, "running day by day," has worn away a part of the 
primary form, and left us the words as they now exist. Thus, 
supposing the original expression to be three and ten, (orig- 
inally the Anglo-Saxon thri and tyri) if we drop the conjunction 
and, we shall have three-ten ; changing the ten to teen we 
have three-teen; then changing the three to thir, and omitting 
the hyphen, we have the present form thirteen. In a similar 
manner the expression four and ten becomes fourteen; five 
and ten, fifteen ; six and ten, sixteen, etc. By the same prin- 
ciples of abbreviation and euphonic change, we might have 
obtained twenty, thirty, etc. Supposing the original form to 
be two tens, or twain tens (in the Saxon twentig, from twegen, 



96 THE PHILOSOPHY OF ARITHMETIC. 

two, and tig, ten), then changing the twain to twen, and the 
tens to ty, we shall have the common form, twenty. In three 
tens, changing the three to thir and the tens to ty, we have 
thirty. In the same way we obtain forty, fifty, sixty, etc., 
and from these by omitting the and in the expression two tens 
and one, two tens and two, etc., we have twenty-one, twenty- 
two, thirty-three, forty-seven, etc. 

To illustrate the law of the formation of these names, we 
have used the present English forms rather than those in which 
the transformations actually occurred. It will be remembered 
that these names were derived from the Anglo-Saxon, and the 
changes which we have illustrated took place in that language 
before the names were adopted in the English tongue. The 
word thirteen was actually derived from the Anglo-Saxon 
threo-tyne, which was composed of thri, three, and tyne, ten; 
fourteen from feowertyne, composed offeower, four, and tyne, 
ten, etc. We get the word twenty from the Anglo-Saxon 
twentig, which is composed of the Anglo-Saxon twegtn, two, 
and tig, ten ; thirty from thritig, which is composed of thri, 
three, and tig, ten, etc. The law of the composition of these 
original words is no doubt the same as that illustrated by 
the use of the English words given above. 

In a similar manner we name the numbers from one hundred 
to the next group, consisting of ten hundreds, to which we 
assign a new name, calling it thousand. After reaching the 
thousand, a change occurs in the method of grouping. Previ- 
ously, ten of the old groups made one of the next higher group, 
but after the third group, or thousands, it requires a thousand 
of an old group to form a new group, which receives a 
new name. A thousand thousands forms the next group 
after thousands, which we call million from the Latin mille, 
a thousand. In the same manner, one thousand millions 
gives a new group which we call billion, one thousand billions 
a new group which we call trillion, etc. 

This change in the law by which a new group is formed from 



NUMERATION, OR THE NAMING OF NUMBERS. 97 

an old one, is not an accident; it is intentional. It is due to 
science, rather than to chance. The method of counting ten 
in a group was commenced in an age anterior to science, and 
proceeded no further than hundreds and thousands, since the 
wants of the people did not require larger numbers; but when 
arithmetic began to be cultivated as a science, it was seen to 
be a matter of convenience to increase the size of the groups 
receiving a new name, and then the law became changed. 

The reason that the law of naming numbers does not appear 
in the names of the smaller numbers, is, that they became 
changed from the original form on account of their frequent 
use. The same fact appears in grammar in the irregularity of 
the verbs expressing ordinary actions, as run, go, eat, drink, 
etc., which became thus irregular in the formation of their 
tenses from the constant and careless use of the common peo- 
ple, before the language was fixed by the rules of science or 
the art of printing. 

Utility. The utility of the method of naming numbers by 
collecting them into groups or bunches, is generally imperfectly 
appreciated. The method which naturally would be first sug- 
gested to the mind, is to give each number an independent 
name, just as we distinguish rivers, cities, states, etc. This 
would, of course, require a vocabulary of names as extensive 
as the series of natural numbers, a vocabulary which, even for 
the ordinary purposes of life, could be learned only by years 
of labor. By the method of groups, the vocabulary is so sim- 
ple that it can be acquired and employed with the greatest 
ease. It may be remarked, that this method of grouping, 
though suggested by the accidental circumstance of counting 
the fingers, is in accordance with that universal operation of 
the mind by which it binds up its knowledge into bunches or 
packages. It is, in fact, based upon the principle of generaliza- 
tion and classification. 

Origin of Names. The origin or primary moaning of the 
names applied to the first ten numbers, is not known. It hn? 



98 THE PHILOSOPHY OF ARITHMETIC. 

been supposed that the names of the simple numbers were 
originally derived from some concrete objects, and there are a 
few facts which seem to indicate the correctness of this suppo- 
sition. Thus, the Persian name for five is pendje, while 
pentcha means the expanded hand, and the corresponding terms 
in the Sanskrit are said to have a similar meaning. The term 
linia, which with slight modifications is used for five through- 
out the Indian Archipelago, means hand in the language of 
the Otaheite and other islands. Among the Jaloffs, an African 
tribe, the word for five, juorum, likewise signifies hand. 
Among the Greenlanders the term for twenty is innuk, or man; 
that is, after completing the counting of fingers and toes, they 
say innuk or man; and there are also examples of the identity 
of the term for man and twenty among some of the tribes of 
South America. 

Among the Indians of Bogota, New Grenada, the term 
quicha, meaning a foot, is used to number the second decade, 
while twenty is named gueta, which signifies a house. Nearly 
all the South American tribes use the word for hand to express 
five, and in many cases the word for man is used to express 
twenty. A tribe in Paraguay denote four by an expression 
which means the fingers of the Emu, a bird common in Par- 
aguay, possessing four claws on each foot, three before, and 
one turned back ; and their word for five is the name of a 
beautiful skin with five different colors. The same number is, 
however, more commonly expressed by hanam begem, the 
fingers of one hand; ten is expressed by the fingers of both 
hands; and for twenty they say hanam rihegem cat gracha- 
haka nnomichera hegem, the fingers of both hands and feet. 
Among the Caribbeans, the fingers are termed the children of 
the hand, and the toes children of the feet ; and the phrase for 
ten, chou oucabo raim, means all the children of the hands 

Humboldt has given from the researches of Duquesnej the 
etymological signification of some of the numerals of the 
Indians of New Grenada. Thus, ata, one, signifies water; 



NUMERATION, OR THE NAMING OF NUMBERS. 99 

bosa, two, an enclosure; mica, three, changeable; muyhica^ 
four, a cloud threatening a tempest ; hisa, five, repose ; ta, 
six, harvest; cahupqua, seven, deaf; suhuzza, eight, a tail; 
and ubchica, ten, resplendent moon. No meaning has been 
discovered for aca, the numeral for nine. It would seem im- 
possible, amidst such various meanings, to discover any prin- 
ciple which may seem to have pointed out the use of these 
terms as numerals. 

In the Mexican numeral symbols there is an intelligible con- 
nection between the sign and the thing signified, though the 
association seems to be entirely arbitrary. Thus, the symbol 
for one is a frog; for two, a nose with extended nostrils, part 
of the lunar disk, figured as a face; for three, two eyes open, 
another part of the lunar disk; for four, two eyes closed; 
for Jive, two figures united, the nuptials of the sun and 
moon, conjunction ; for six, a stake with a cord, alluding 
to the sacrifice of Guesa tied to a pillar ; for seven, two ears ; 
for eight, no meaning assigned ; for nine, two frogs coupled ; 
for ten, an ear ; for twenty, a frog extended. 

The following theory, advanced by Prof. Goldstiicker, in a 
paper read before the Philological Society in 1870, in which he 
gives good linguistic evidence in support of the origin of the 
Sanskrit numerals, and consequently of our own, is at least 
plausible, and will be interesting: One, he says, is "he," the 
third personal pronoun; two, "diversity;" three, "that which 
goes beyond ;" Jour, "and three," that is, "one and three;" 
five, "coming after;" six, "four," that is, "and four," or "two 
and four;" seven, "following;" eight, "two fours," or "twice 
four ;" nine, " that which comes after" (ch. nava, new); ten, 
"two and eight." Thus, only one and two have distinct orig- 
inal meanings. After giving these, our ancestors' powers 
needed a rest; then they made three, and added to it one for 
four; then took another rest, repeated the notion of three in 
five, and the notion of four in six ; then rested once more, 
and again repeated the notion of three and five iu seven ; took 



100 THE PHILOSOPHY OF ARITHMETIC. 

another rest, and got a new idea of two fours for eight ; but 
for nine repeated for the fourth time the " coming after" notion 
of three, five, and seven ; while for ten they repeated for the 
third time the addition-notion of four and six. The Professor 
insists strongly on this seeming poverty and helplessness of 
the early Indo-European mind. He does not put forward the 
above meanings of the numerals as new, though he believes 
that his history of most of the forms of their names is so. 
The anomalous form of the Sanskrit shash, six the hardest of 
them first set him at work on the numerals, and the Zend 
form kshvas led him to the true explanation of this, and thence 
to that of the other numerals. 

In closing this chapter, we remark that the names of the 
periods above duodecillions have not been fully settled by usage. 
Prof. Henkle, who has examined the subject with considerable 
care, finds a law which he maintains should hold in the forma- 
tion of the names of the higher periods. The terms quintillions, 
sextillions, and nonillions are formed, not from the cardinals, 
quinque, sex, and novem, but from the ordinals, quintus, sextus, 
and nonus. From this he infers that analogy plainly demands 
that the names beyond duodecillions should be formed from 
the Latin ordinal numerals. For the names thus formed, see 
appendix. 



CHAPTER II. 

NOTATION, OR THB WRITING OF NUMBERS. 

A RITHMETICAL language is the expression of arithmet- 
-lA- ical ideas. These ideas may be expressed in sound to 
the ear, or in visible form to the eye ; arithmetical language is, 
therefore, both oral and written. The oral language is called 
Numeration; the written language, Notation. Numeration is 
the method of naming numbers; Notation is the method of 
writing numbers. From this consideration it would seem that 
the written language of arithmetic must bear an intimate rela- 
tion to the oral language, which we find to be the case. The 
general method of writing numbers, now adopted by all civil- 
ized nations, is the Hindoo, usually called the Arabic method. 
This method is based upon, and arises naturally out of, the 
method of naming numbers by groups. 

The fundamental principle of the Arabic system is the 
ingenious and refined idea of place value. Recognizing the 
method of naming numbers by groups, it assumes to represent 
these groups by the simple device of place. It fixes upon a 
few characters to represent a few of the first numbers, uml 
then employs these same characters to number the groups, 
the group numbered being indicated by the place of the char- 
acter. This leads to the distinction of the intrinsic and local 
value of the numerical characters. Each character has a defi- 
nite value when it stands alone, and a relative value when used 
in connection with other characters. 

The number of the arithmetical characters is determined by 
the number of units in the group. The grouping being by 

( 101 ) 



102 THE PHILOSOPHY OF ARITHMETIC. 

tens, the number of characters needed is only nine, one less 
than the number of units in the group. These characters are 
called digits, from the Latin digitus, a finger, the name com- 
memorating the ancient custom of reckoning by counting the 
fingers. In the combination of these characters to express 
numbers, it will often be required to indicate the absence of 
some group ; hence arises the necessity of a character which 
expresses no value, a character which denotes merely the 
absence of value. This character is known as naught, or zero. 
We thus have the following ten characters : 1, 2, 3, 4, 5, 6, 7, 
8, 9, 0, with which we are able to express all possible numbers. 
Utility. The Arabic system, based upon the refined idea of 
place value, is one of the happiest results of human intelli- 
gence, and deserves our highest admiration. Remarkable as is 
its simplicity, it constitutes, regarded in its philosophical char- 
acter or its practical value, one of the greatest achievements of 
the human mind. In the hands of a skillful analyst, it be- 
comes a most powerful instrument in wresting from nature her 
hidden truths and occult laws. Without it, many of the arts 
would never have been dreamed of, and astronomy would have 
been still in its cradle. With it, man becomes armed with 
prophetic power, predicting eclipses, pointing out new planets 
which the eye of the telescope had not seen, assigning orbits 
to the erratic wanderers of space, and even estimating the ages 
that have passed since the universe thrilled with the sublime 
utterance, "Let there be light !" Familiarity with it from child- 
hood detracts from our appreciation of its philosophical beauty 
and its great practical importance. Deprived of it for a short 
time, and compelled to work with the inconvenient methods of 
other systems, we should be able to form a truer idea of the 
advantages which this invention has conferred on mankind. 

Relation to Numeration. Though the methods of notation 
and numeration are intimately related, there is also an essential 
distinction between them. Though similar, they are by no means 
identical in principle. Their similarity is seen in the fact that 



NOTATION, OR THE WRITING OP NUMBERS. 103 

the method of notation could not be applied without the method 
of numbering by groups ; their distinction is seen in the fact 
that we could have the present method of numeration without 
the Arabic system of notation. The notation seems to be an 
immediate outgrowth from the numeration, yet not a necessary 
one ; for many nations who had the same method of naming 
numbers, employed other methods of writing them. 

Their true relation also appears in considering their common 
relation to the decimal scale. The decimal principle belongs 
both to our method of naming and of writing numbers. This 
coincidence is not accidental, but essential to the harmony 
of oral and written expression. The necessity of this would 
be very apparent if we should attempt to change the base 
of the scale of notation without changing the base of the 
method of naming numbers. With our present base we say 
one and ten, two and ten, etc., or at least their equivalents; 
and our written expressions are read in the same manner. 
Should we adopt any other scale of notation, retaining our 
present base in naming numbers, the reading of numbers in 
this new scale would be so awkward and inconvenient as to be 
almost impossible. Hence it follows, that for a scale of nota- 
tion to be advantageously employed, the methods of naming 
and writing numbers should possess the same basis. Thus, if 
the scale of notation be quinary, instead of naming numbers 
five, six, seven, etc., we should say Jive, one and five, two and 
five, etc.; if the scale were senary, we should say six, one and 
six, two and six, etc. 

Relation to the Base. It will also be seen that the princi- 
ple of the methods of naming and writing numbers is entirely 
distinct from the number used as the base. The intimate asso- 
ciation of the Arabic system with the base, has sometimes led 
to the idea that the base is a part of the system itself. This 
error should be carefully avoided. The Arabic method 
assumes that we name numbers by groups, and that each 
group contains ten; but it is in principle entirely independent 



104 THE PHILOSOPHY OF ARITHMETIC. 

of the number constituting a group. The number in the group 
determines the base of the scale, and consequently the number 
of characters to be used, but does not afi'ect the principle of 
the method, which is simply that of place value. Should we 
change the base of numbering, it would change the names of 
the numbers after twelve, and the base of the Arabic scale; 
but it would in no wise affect the principle of cither the method 
of numeration or of notation. 

Number of Characters. The number of characters in the 
Arabic system of notation depends upon the number of 
units in the groups of numeration. Thus, we must have as 
many simple characters as will express the different numbers 
from one until we reach within a unit of the group. We shall 
have no character for the group, since, according to the device 
of place value, it is to be indicated by changing the place of 
the symbol which represents one, it being one of the first 
group. The number of significant characters must, therefore, 
be always one less than the number denoting the base of the 
system. In the decimal scale the number of digits is nine ; 
in an octary scale it would be seven ; in a quinary scale, four, 
etc. 

Origin. The origin of this system of notation is now uni- 
versally accredited to the Hindoos. When, by whom, and how 
it was invented, we do not know. It is not improbable that it 
began with the representation of the spoken words by marks, 
or abstract characters. They may at first have given inde- 
pendent characters to the numbers as far as represented. It 
then probably occurred to them that, since they gave independ- 
ent names to a few numbers and then numbered by groups, 
they could simplify their system of notation by making it cor- 
respond to their system of numeration. Then first dawned 
upon the mind the idea of a few characters to represent the 
first simple numbers, and the use of these same characters to 
number the groups. They now stood on the threshold of one 
of the greatest discoveries of all time. Here arose the ques- 



NUTATION, OR THE WRITING OF NUMBERS. 106 

tion How are these groups to be distinguished? How shall 
we determine when a character denotes a number of units or 
tens, or hundreds, etc.? How many methods occurred to them 
before the method of place, who can tell ? This might ha've 
been done by slightly varying the character, by attaching some 
mark to it, by annexing the initial of the group, etc.; either of 
which would .have been comparatively complicated and incon- 
vonicnt. At last, to the mind of some great thinker, occurred 
the simple idea of place value, and the problem was solved. 
"Who was the man ?" is a question answered only by its own 
echo, for his name sleeps in the silence of the past. Were it 
known, mankind would feel like rearing a monument to his 
memory, as high and enduring as the Pyramids of Egypt; but 
now it can only raise its altar to the Unknown Genius. 

Origin of Character*. The origin of the characters, like 
that of the system, is shrouded in mystery ; but little light 
upon the subject comes down the historic path. Many of the 
early writers gave some ingenious speculations concerning 
their origin. Gatterer imagined that he had discovered iu 
Egyptian manuscripts written in the enchoriac character, 
tliiit the digits were denoted by nine letters; and Wachter 
supposed them to have a natural origin in the different com- 
binations of the fingers: thus, unity is expressed by the 
outstretched finger ; two by two fingers, which may have been 
represented by two marks that, by long use, passed into the 
present form, and so on for all the other symbols. In the 
absence of facts, three theories have been presented, which are 
at least interesting on account of their ingenuity, and are 
certainly somewhat plausible. One of these theories is that 
they are formed by the combination of straight lines, as tin 1 
primary representation of numbers; another is that they are 
formed by the combination and modification of angles; and 
still ;i noilicr and more recent theory is that they are the 
initial letters of the Hindoo numerals. These three theories 
may be distinguished as the theories of lines, angles, and 
initial letters. 



106 THE PHILOSOPHY OF ARITHMETIC. 

The first theory is based on the primary use of straight 
lines to represent numbers. By this method, one straight 
line, |, would represent one,- two straight lines which may 
have been connected thus, L, two; three lines, thus, ^, or with 
a connecting curve, thus, -jj, three ; four lines arranged thus, 
Q or thus, 4 four; five lines arranged thus, Jjj, five; six lines 
arranged thus, |j, six; seven lines, thus, 9, seven ; eight lines 
thus, g, or thus Y, eight; nine lines, thus, ^, nine. The 
zero is supposed to have been originally a circle, suggested 
from counting around the fingers and thumbs held in a circular 
position. 

The second theory is based upon the use of angles to repre- 
sent numbers. The ancient mathematicians were noted for 
their astronomical observations and calculations, and being 
thus familiar with the use of angles, it is not unreasonable to 
suppose that they would employ the angle in their representa- 
tion of numbers. Thus, they might very naturally have used 
one angle, / |, for one ; two angles, ", for two ; three angles ], 
for three; four angles, A for four; five angles, g, for Jive, 
six angles, , for six; seven angles, 9, for seven; eight 
angles, 0, for eight ; nine angles, ^, for nine. These char- 
acters being frequently made, would eventually assume the 
rounded form which they now possess. By this theory, the 
character for zero is easily and naturally accounted for Jf 
angles were used to represent numbers, nothing would be rep- 
resented by a character having no angles, which is the closed 
curve. 

The latest and most plausible theory for the origin of Arabic 
characters is, that they were originally the initial letters of 
the Sanskrit numerals. This theory is presented by Prin- 
seps, a profound Sanskrit scholar, and is indorsed by Max 
Miiller. Such a use of initial letters was entirely feasible 
in the Sanskrit language, as each numeral began with a dif- 
ferent letter. The plausibility of the theory further appears 
<rom the fact that it follows the general law of representing 



NOTATION, OR THE WRITING OF NUMBERS. 107 

numbers by letters, as in the Roman, Greek, and Hebrew 
systems. 

This theory does not account for the origin of the zero, the 
luost important character of them all, in fact, the key to the 
system of modern arithmetic. No other system of notation 
except the sexagesimal system, had it. Max Miiller says : 
" It would be highly important to find out at what time the 
naught first occurs in Indian inscriptions. That inscription 
would deserve to be preserved among the most valuable monu- 
ments of antiquity, for from it would date in reality the 
beginning of true mathematical science impossible without 
the naught nay, the beginning of all the exact sciences to 
which we owe the invention of telescopes, steam engines, and 
electric telegraphs." Dr. Peacock supposes that it was derived 
from the Greek o, introduced by Ptolemy to denote the vacant 
places in the sexagesimal arithmetic; the Hindoos, he says, 
having used a dot for this purpose. 

It seems to have been difficult at first to comprehend the pre- 
cise force of the cipher, which, insignificant in itself, serves only 
to determine the rank and value of the other figures. When 
they were first introduced into Europe, it was deemed necessary 
to prefix to any work in which they were used, a short treatise 
on their nature and application. These notices are often met 
with attached to old vellum almanacs, or inserted in the blank 
leaves of missals, and frequently intermixed with famous prophe- 
cies, most direful prodigies, and infallible remedies for scalds 
and burns. A sort of mystery, which has imprinted its trace 
on our language, seemed to hang over the practice of using 
the cipher; and we still speak of deciphering and writing in 
cipher, in allusion to some dark or concealed art. Indeed, in 
the early history of arithmetic in Europe, either on account of 
its association with the infidel Mohammedans from whom it 
ivas derived, or of the popular prejudice against learning which 
prevailed at that time, the system was regarded as belonging 
to black art and the devil ; and it was, no doubt, this popular 
prejudice that delayed its general introduction into Christian 
Europe. 



CHAPTER III. 

ORIGIN OF ARITHMETICAL SYMBOLS. 

rpHE symbols of arithmetic may be divided into three general 
JL classes : Symbols of Number, Symbols of Operation, and 
Symbols of Relation. What is the origin of these symbols ; 
who invented them, or first employed them? This question, a 
very interesting one, I shall endeavor to answer in the present 
chapter. 

I. SYMBOLS OF NUMBER. The Symbols of Number em- 
ployed by different nations, are the Arabic figures and the 
letters of the alphabet. Nearly all civilized nations seem to 
have made use of the letters of the alphabet to represent num- 
bers. The Greeks divided their letters into several classes, to 
represent the different groups of the arithmetical scUe. The 
Roman system employed the seven letters, I, V, X, L, C, D, 
and M, to represent numbers. The Arabs at first used the 
Greek method, and afterward exchanged it for that of the Hin- 
doos. 

There are three theories given for the origin of the Arabic 
symbols of notation, known respectively as the theory of lines, 
of angles, and of initial letters. These three theories are 
explained in the chapter on Notation. It may also be remarked 
that some of the Arabian authors who treat of astrological 
signs, allege that the Indian or Arabic numerals were derived 
from the quartering of the circle, and Leslie says that the 
resemblance of these natural marks to the derivative ones 
appears very striking. The Roman symbols are supposed to 
have originated in the use of simple straight lines or strokes, 

(108) 



ORIGIN OF ARITHMETICAL SYMBOLS. 109 

variously combined, for which were subsequently substituted the 
letters of the alphabet. This theory is explained at length in the 
chapter on Roman Notation. 

SYMBOLS OF OPERATION. The Symbols of Operation are the 
signs of addition, subtraction, multiplication, division, involution, 
evolution, and aggregation. The origin of most of these symbols 
has been definitely determined. 

The Symbols of Addition and Subtraction were first introduced 
as symbols of operation by Michael Stifel, a German mathematic- 
ian, in a work entitled Arithmetica Integra, published at Nurem- 
burg in 1544. These signs had appeared previously in a work of 
Johann Widmann called the Mercantile Arithmetic, published at 
Leipzig in 1489. They are, however, not used by him as symbols 
of operation, but merely as marks of excess or deficiency. The 
next oldest book extant in which these signs are found is that of 
Christopher Rudolff, published in 1524, though he does not use 
them as symbols of operation. Stifel was a pupil of Rudolff, and 
it is supposed that he obtained the symbols from him, for, as he 
himself admits, he took a large part of his work from that of 
Rudolff. Stifel introduces the symbols as if he had originated 
them or their new use, for he says, " thus, we place this sign," 
etc., and " we say that the addition is thus completed," etc. To 
Stifel, therefore, belongs the credit of first using these symbols as 
signs of operation. 

Why these particular signs were adopted has been a matter of 
conjecture. Prof. Rigaud supposed that + was a corruption of 
P, the initial for plus, and Dr. Davis thought that it was a cor- 
ruption of et or Sf. Stifel, however, does not call the signs phis 
and minus, but signum additorum and signum subtractorum, which 
renders these suppositions improbable. Dr. Ritchie suggested 
that perhaps + was two marks joined together in addition, and 
that was taken to indicate subtraction, since it is what is left 
after one of the marks is removed. De Morgan thought that the 
minus sign was first used, and that + was derived from it by 
putting a small cross-bar for distinction. " The sign -f ," he 



110 THE PHILOSOPHY OF ARITHMETIC. 

says, " in the hands of Stifel's printer has the vertical bar much 
shorter than the other, and when it is introduced into the wood- 
outs of the engraver, the disproportion is greater still." The 
Hindoos, from whom our knowledge of algebra was originally 
derived, used a dot for subtraction, and the absence of the dot for 
addition, and De Morgan suggests that the Hindoo dot may have 
been elongated into a bar to signify subtraction, and that an addi- 
tional line to the symbol of subtraction gave the sign of addition. 
M. Libri attributes the invention of + and to Leonardo da 
Vinci, the celebrated Italian artist and philosopher, but it is 
probable that Da Vinci used the symbol + for the figure 4. 

The most recent explanation of these signs is that they were 
originally warehouse marks. In Widmann's arithmetic they occur 
almost exclusively in practical mercantile questions. Goods were 
sold in chests which when full were expected to hold a certain 
established weight. Any excess or deficiency was indicated by 
-f- or , and these signs may have been marked with chalk on 
the chests as they came from the warehouses. Usually the 
weight of the chest, it may be supposed, would be deficient, and 
this was marked with the sign ; when a cask or chest was 
above the standard weight the line may have been crossed with 
a vertical line giving the symbol +. It may be remarked that 
these symbols were not immediately adopted by mathematicians. 
In a work on algebra, published in 1619, the signs of addition and 
subtraction are P and M with strokes drawn through them. 

The Symbol of Multiplication (X), St. Andrew's cross, was in- 
troduced by William Oughtred, an eminent English mathematic- 
ian and divine, born at Eton in 1573. The work in which this 
symbol first appeared was entitled Clavis Mathematica, " Key of 
Mathematics," and published in 1631. Oughtred was a fine 
thinker, and was honored by the title " prince of mathematicians." 
What led to this particular form for the symbol is unknown. Two 
other signs for multiplication were proposed, the (.) by Descartes 
and the curve (^) by Leibnitz, but though having the authority 
of great names they failed of adoption. 



ORIGIN OF ARITHMETICAL SYMBOLS. Ill 

The Symbol of Division (-*-) was introduced in 1630 by Dr. 
John Pell, Professor of Philosophy and Mathematics at Breda. 
This symbol was used by some old English writers to denote the 
ratio or relation of quantities. I have also noticed it used thus in 
some old German mathematical works. The Arabs used a dash, 
writing one number under the other, in the form of a fraction. 
Dr. Pell was highly regarded as a mathematician. It was to him 
that Newton first explained his invention of fluxions. 

The System of Exponents, to represent the powers of a num- 
ber, has been generally ascribed to Descartes, 1596-1650, the 
illustrious metaphysician and inventor of Analytical Geometry. 
Exponents were, however, employed by De la Roche as early as 
1520, but Descartes' extensive use of them led lo their general 
adoption. The earliest writer on algebra denoted the powers of a 
number by an abbreviation of the name of the power. Harriot, a 
mathematician of the 17th century, repeated the quantity to indi- 
cate the power ; thus, for a* he wrote aaaa. 

The Radical Sign (p/) was introduced by Stifel, the introducer 
of -f and . This symbol is a modification of the letter r, the 
initial of radix, root. The root of a quantity was formerly de- 
noted by writing the letter r before it, and this letter was grad- 
ually changed to the form i/. 

The Vinculum or Bar, placed over quantities to connect them 
together, thus, 4X3 + 5, was first used by Vieta in 1591, the in- 
troducer, in algebra, of the system of representing known quanti- 
ties by symbols. The Parenthesis and Brackets were first used by 
Albert Girard, a Dutch writer on algebra, in 1629. 

III. SYMBOLS OF RELATION. Symbols of Relation are the 
signs of equality, ratio, equal ratios, inequality and deduction. 
The origin of a few of these has been ascertained. 

The Symbol of Equality (=) was introduced by Robert Re- 
corde, an English physician and mathematician of the sixteenth 
century. It first appeared in 1556 in his work on algebra, called 
by the odd title The Whetstone of Witte. 

This sign was also employed by Albert Girard. The French 



112 THE PHILOSOPHY OF ARITHMETIC. 

and German mathematicians used the symbol oo to denote equality, 
even long after Recorde. This symbol is said to be a modifica- 
tion of the diphthong , the initial of the Latin phrase <equale est. 
It is also stated that the symbol = was often used as an abbrevia- 
tion for cst in mediaeval manuscripts. 

The Symbol of Ratio (:) is supposed to be a modification of the 
sign of division. The sign of division was frequently employed 
by the old English and German mathematicians to indicate the 
relation of quantities. Who first omitted the dash and employed 
the present form of the symbol of ratio, I have not been able to 
ascertain. It occurs in a work by Clairaut, published in 1760. 

The Symbol of Equal Ratios (: :) may be a modification of the 
sign of equality (=) or a duplication of the symbol of ratio (:), 
but this is not certain. It seems to have been introduced by 
Oughtred, in a work published in 1631, and was brought into 
common use by "Wallis in 1686. 

The Symbols of Inequality (> and <) are evidently modifica- 
tions of the sign of equality. If parallel lines denote equality, 
oblique lines would naturally be used to denote inequality, the 
lines converging toward the less quantity. They are said to have 
been introduced by Harriot in 1651. 

I have now presented, in a connected and systematic manner, 
about all that is known concerning the origin of the ordinary 
arithmetical symbols. All of them belong to the period of mod- 
ern history and are the products of the revival of learning. One 
each of the signs of operation was furnished by France, England 
and the Netherlands, and three by Germany. Of the other sym- 
bols named, all were introduced by Englishmen, with the excep- 
tion of the vinculum, which is due to Vieta, a Frenchman, and 
the parenthesis and brackets, which were invented by the Dutch 
mathematician Girard. 



CHAPTER IV. 

THE BASIS OF THE SCALE OF NUMERATION. 

rpHE Basis of our scale of numeration and notation is dec 
JL imal. This basis is not essential, but accidental. Man- 
kind commenced reckoning by counting the fingers of the left 
hand, including the thumb, and thus at first probably reckoned 
by fives. As the art of numbering advanced, they adopted a 
group, derived from the fingers of both hands, and thus ten 
became the basis of numbering. The decimal base was con- 
sequently determined by the number of fingers on each hand. 
Had there been three fingers and a thumb, the scale would 
have been octary; had there been five fingers and a thumb, 
the scale would have been duodecimal, which would have been 
a great advantage to arithmetic, whatever it might have been 
to the hand itself. 

The universal use among civilized nations of the decimal 
scale of numeration seems to imply some peculiar excellence in 
it. It appears as if nature had pointed directly to it, on 
account of some essential fitness of the number ten, as the 
numerical basis. Indeed, this opinion has been quite general, 
and the habit acquired from the use of the system has served 
to confirm the belief. Many persons get the base of numera- 
tion and the mode of notation so mingled together that they 
see in the Arabic system nothing save the decimal basis of 
numeration, and attribute to it all those high qualities which 
belong to the mode only. It is this which has led some per- 
sons to regard the decimal basis as the perfection of simplicity 
and utility. 

8 (113) 



114 THE PHILOSOPHY OF ARITHMETIC. 

A little reflection, however, will prove that such an assump- 
tion is groundless. Although the decimal scale has been 
adopted by every civilized nation, yet, as has been shown, the 
selection was accidental, and the base entirely arbitrary. The 
selection occurred before attention was given to a general sys- 
tem, in short, without reflection, and its supposed perfection is 
a mere delusion. Any other number might have been taken 
as the root of the numerical scale ; and, were a new basis to 
be selected by mathematicians familiar with the properties of 
numbers, there are several considerations that would lead them 
to adopt some other basis than the decimal. Some of the 
objections to the decimal basis will be stated, and a few consid- 
erations presented in favor of some other number as the basis 
of the language of arithmetic. 

First, the decimal scale is unnatural. It has been super- 
ficially urged that the decimal scale is the most natural one 
that could have been selected. On the contrary, there is no- 
thing natural about it, except the fingers, and a little reflection 
would have shown that these are grouped by fours instead of 
fives. In fact, a group by tens is seldom seen, either in nature 
or in art. What things exist by tens, associate by tens, or 
separate into tenths? Nature groups in pairs, in threes, 
in fours, in fives, and in sixes; but seldom, if ever, in 
tens. Man doubles and triples and quadruples his units ; he 
divides them into halves and thirds and quarters ; but where 
does he estimate by tens or tenths ? It is thus seen that the 
grouping by tens is an unnatural method, suggested neither 
by nature nor the practical requirements of art. 

Second, the decimal scale is unscientific. The confused 
idea of the relation of the base of the scale to the mode of 
notation, has led some to suppose that the decimal scale is one 
of the triumphs of science. The truth is, as has already been 
shown, that not only was it not established upon scientific 
principles, but it is really a violation of those principles. The 
decimal scale originated by chance, by a mere accident. Men 



THE BASIS OF THE SCALE OF NUMERATION. 115 

had ten fingers, including the thumbs, and found it convenient 
to reckon by counting their fingers ; and thus acquired the 
habit of counting by tens. Had science, instead of chance, 
presided at its birth, we should have a basis that would have 
given a new beauty and a greater simplicity to our already 
admirable system of arithmetical language. 

Third, the decimal scale is also inconvenient. It has been 
held not only that the decimal basis is scientific, but that 
it is the most convenient one that could have been selected. 
It needs but little reflection to see the incorrectness of this 
assumption. One essential for the basis of a scale is the 
property of its being divisible into a number of simple parts, 
so that it may be a multiple of several of the smaller numbers. 
The number ten will admit of only two such divisions, the 
half and the fifth. The third, fourth, and sixth are not exact 
parts of the denary base, in consequence of which it is incon- 
venient to express these fractions in the scale. Were the 
basis twelve instead of ten, we could obtain the half, third, 
fourth, and sixth, and these fractions could be expressed by 
the scale in a single place ; whereas the fourth now requires 
two places (.25), and the third and sixth cannot be expressed 
exactly in a decimal scale, except as a circulate. 

Essentials of a Base. It will be interesting to notice some 
of the essentials of a base, and to observe what number com- 
plies most fully with these requirements. The first essential 
of a good base is that it will admit of being divided into the 
simple fractional parts ; the second is that the number be neither 
too large nor too small. The advantage of the capability of 
being divided into simple fractional parts is that such fractions 
may be readily expressed in the terms of the scale as we now 
express decimal fractions. In the decimal scale only one-half 
and one-fifth can be expressed in one place of decimals, since 
they are the only exact parts of ten. With a scale whose basis is 
a multiple of two, three, four and six, each correspondiug frac- 
tion could be expressed in terms of the scale in a single plucv. 



116 THE PHILOSOPHY OF ARITHMETIC. 

In respect to the size of the base, if the number is too 
small, it will require too many names and places to express 
large numbers. If the number is very large, it will group 
together too many units to be apprehended and easily used in 
numerical operations. 

Other Scales. There are several other bases which have been 
recommended as preferable to the decimal ; the most important 
of which are the Binary, the Octary, and the Duodecimal. The 
Binary scale was proposed and strongly advocated by Leibnitz. 
He maintained that it was the most natural method of counting, 
and that it presented great practical and scientific advantages. 
He even constructed an arithmetic upon this basis, called Binary 
Arithmetic. The obvious objection to this base is, that it 
would require too many names and too many places in writing 
large numbers. The Octary system has also been strongly 
advocated. A very able article in an American journal says 
that the binary base is the only proper base for gradation, and 
the octary is the true commercial base of numeration and nota- 
tion. 

It is probable, however, taking all things into consideration, 
that the duodecimal scale would be the most suitable. The 
number twelve is neither too large nor too small for conveni- 
ence. Its susceptibility of division into halves, thirds, fourths, 
and sixths, is an especial recommendation to it. So great are 
these advantages, that, if the base were to be changed, the 
duodecimal base would, without doubt, be selected. 

The advantage of the duodecimal scale is especially apparent 
in the expression of fractions in a form similar to our decimal 
fractions. In the decimal scale, and \ are the only simple 
fractions that can be expressed by the scale in a single place ; 
i cannot be expressed at all as a simple decimal ; requires 
two places, and , like , gives an interminate decimal. With 
a duodecimal scale we could express ^, , , and in a 
single place; while and would require only two places. 
Thus, in the duodecimal scale, we should have =.6; =.4; 



THE BASIS OF THE SCALE OF NUMERATION. 117 

i=-3; =.2; =.16, and =.14. This is a very great sim- 
plification; and since all combinations of 2 and 3 could be 
readily expressed, and since these constitute such a large pro- 
portion of numbers, it is evident that the simplification of the 
subject, by means of a duodecimal scale, would be very con- 
siderable. 

I will arrange the expressions of these fractions in the deci- 
mal and duodecimal scales, side by side, that the advantage of 
the latter may be more clearly seen. 

DECIMAL SCALE. DUODECIMAL SCALE. 



i=.5 = .166+ 

=.333+ |=. 142857 

{=25 =.125 

=.2 =.111+ 



1=6 =.2 



=.4 



=.16 



=.2497 



It will be seen that in the decimal scale all the simple frac- 
tions used in practice, except , give circulates or require two 
or three figures to express them; while in the duodecimal scale 
all the fractions ordinarily used in business transactions are 
expressed in a single place, and even and require only two 
places. The fractions and cannot be exactly expressed in 
the scale, but these fractions are seldom used in business. It 
will be interesting to notice that and both give perfect rep- 
etends in the duodecimal scale, and that they possess the same 
properties as perfect repetends in the decimal scale. 

There seems to have been a natural tendency towards a duo- 
decimal scale. Thus, a large number of things are reckoned 
by the dozen, and this scale is even extended to the gross and 
the great-gross; that is, to the second and the third powers of 
the base. Again, in our naming of numbers, the terms eleven 
and twelve seem to postpone the forming of a group until we 
reach a dozen. A similar fact appears in extending the multi- 
plication table to include twelve times, since, with the deci- 
mal scale, it could conveniently stop with nine or ten times 
The division of the year into twelve months, the circle into 
twelve signs, the foot into twelve inches, the pound into twelve 



118 THE PHILOSOPHY OF ARITHMETIC. 

ounces, etc., are each a further indication of the same ten- 
dency. 

Change of Base. The objections to the decimal base have 
led scientific men to advocate a change in our scale of numer- 
ation and notation. Such a change would, without doubt, be 
a great advantage, both to science and to art ; yet the practi- 
cal difficulties attending such a change are so great that it 
seems to be almost impossible. A change in the base would 
require a complete change in the oral language of arithmetic. 
The decimal scale is so interwoven with the speech of nations, 
that such a change could be effected only after years of 
labor. For a while, it would be necessary to have two methods 
of arithmetic taught and in use, as in Europe at the time of 
the transition from the Roman to the Arabic system of nota- 
tion. The learned would soon adopt the new method, but the 
common people would cling with such tenacity to the old, that 
even a century might intervene before the new method would 
become generally established. 

Will this change ever be made ? is a question which is 
sometimes asked. I do not know ; but I am strongly in favor 
of it, and believe it possible. The diffusion of popular educa- 
tion will prepare the way for it, by removing the difficulties of 
its adoption. These difficulties, though great, are not insur- 
mountable. Changes of notation have taken place in several 
different nations, and some nations have changed two or three 
times. The Greeks changed theirs, first for the alphabetic, 
and afterwards, with the rest of the civilized world, for the 
Arabic system. The Arabs themselves first adopted the Greek, 
and afterwards changed it for the Hindoo method. The peo- 
ple of Europe changed from the Roman to the Arabic system 
even as late as the fourteenth century, though it took one or 
two centuries to effect the transition. What was done thus 
early in the history of science, could, with the increased intel- 
ligence of our people, be much more readily accomplished at 
the present day. A writer in one of our American periodicals 



THE BASIS OF THE SCALE OF NUMERATION. 119 

says: "The probability is that it will be done. The question 
is one of time rather than of fact, and there is plenty of time. 
The diffusion of education will ultimately cause it to be de- 
manded." 

It is a curious fact, and one worthy of remembrance, that 
Charles XII. of Sweden, a short time before his death, while 
lying in the trenches before the Norwegian fortress of Freder- 
ickshall, seriously deliberated on a scheme of introducing the 
duodecimal system of numeration into his dominions. 



CHAPTER V. 

OTHER SCALES OF NUMERATION. 

AS we have seen, any number might have been taken as the 
basis of the scale of numeration, the number ten, the 
basis of our present scale, being selected from the circumstance 
of there being ten fingers on the two hands. Some other scales 
have actually existed, and it will be interesting to notice, in 
various languages, traces of an earlier and simpler mode of 
reckoning. In order to a clearer notion of the subject, it may 
be premised that a scale whose basis is two is called Binary ; 
three, Ternary ; four, Quaternary ; five, Quinary ; six, Senary; 
seven, Septenary ; eight, Octary ; nine, Nonary ; ten, Denary; 
twelve, Duodenary, etc. 

The earliest method of numeration was that of combining 
units in pairs. It is still familiar among sportsmen, who 
reckon by braces or couples. Some feeble traces of the Binary 
system are found in the early monuments of China. Fouhi, 
the founder and first emperor of that vast monarchy, is vener- 
ated in the East as a promoter of geometry and the inventor 
of a science, the knowledge of which has been lost. The em- 
blem of this occult science appears to consist of eight separate 
clusters of three parallel lines or trigrams, drawn one above 
the other after the Chinese manner of writing, and represented 
either as entire or broken in the middle. These varied tri- 
grams were called Koua or suspended symbols, from the cus- 
tom of hanging them up in the public places. In the formation 
of such clusters, we may perceive the application of the binary 

(120) 



OTHER SCALES OF NUMERATION. 121 

scale as far as three ranks, or the number eight. The entire 
lines are supposed to signify one, two, or four, according to 
their order, while the broken lines are valueless, and serve 
merely to indicate the rank of the others. If this be true, it 
furnishes an example of a species of arithmetic with the 
device of place, possessing an antiquity of more than 3000 
years. 

The Binary scale, though never fully adopted by any nation 
as a method of counting, has been recommended by one of the 
most celebrated modern philosophers, Leibnitz, as presenting 
many advantages, from its enabling us to perform all the 
operations in symbolic arithmetic by mere addition and sub- 
traction. Such a system would, of course, require but two 
symbols, unity and zero, by means of which all numbers could 
be expressed. Thus, two would be expressed by 10, three by 
11, four by 100, five by 101, six by 110, seven by 111, eight 
by 1000, etc. This system was studiously circulated by its 
author by means of scientific journals and his extensive cor- 
respondence; and was communicated by him to Bouvet, a 
Jesuit missionary at Pekin, at that time engaged in the study 
of Chinese ambiguities, and who imagined that he had discov- 
ered in it a key to the explanation of the Cova, or lineations 
previously referred to. 

This system was also recommended by the theological idea 
associated with it, of which it was claimed to be the represent- 
ative. As unity was considered the symbol of Deity, the forma- 
tion of all numbers out of zero and unity was considered, in 
that age of metaphysical dreaming, as an apt image of the crea- 
tion of the world, by God, from chaos. It was with reference to 
this view of the binary arithmetic, that a medal was struck, bear- 
ing on its obverse, as an inscription, the Pythagorean distich, 
Numero Deu impari gaudet, and on its reverse the appropri- 
ate verse descriptive of the system which it celebrated, Omnibus 
ex nihilo ducendis sufficit Unum. The good Jesuit, who 
seemed to have caught the spirit of Chinese belief, regarded 



122 THE PHILOSOPHY 01 ARITHMETIC. 

the Cova, which were supposed to conceal great mysteries, as 
the symbols of binary arithmetic, as a most mysterious testi- 
mony to the unity of the Deity, and as containing within them 
the germ of all the sciences. 

To count by threes was another step, and this has been pre- 
served by sportsmen under the term leash, meaning the strings 
by which three dogs, and no more, can be held at once in the 
hand. The numbering by fours has had a more extensive 
application ; it was evidently suggested by the custom of tak- 
ing, in the rapid counting of objects, a pair in each hand, and 
thus reckoning by fours. English fishermen, who generally 
count in this way, call every double pair (of herring, for 
instance), a throw or cast ; and the term warp, which origin- 
ally meant to throw, is employed to denote four, in various 
articles of trade. It is alleged that the Guaranis and Sulos, 
two of the lowest races of savages inhabiting the forests of 
South America, count only by fours ; at least they express the 
number five by four and one, six by four and two, seven by 
four and three, etc. It has been inferred, also, from a passage 
in Aristotle, that a certain tribe of Thracians were accustomed 
to use the quaternary scale of numeration. 

The Quinary system, which reckons by fives, or pentads, 
has its foundation in the practice of counting the fingers of 
one hand. It appears, from the statements of travellers, to 
have been adopted by various savage nations. Thus, certain 
tribes of South America were found to reckon by fives, which 
they called hands. In counting six, seven, and eight, they 
added to the word hand the names one, two, three, etc. Mungo 
Park found that the same system was practiced by the Yolofs 
and Foulahs of Africa, who designate ten by two hands, fifteen 
by three hands, etc. The quinary system seems also to have 
been formerly used in Persia; the word pende, which denotes 
five, having the same derivation as pentcha, which signifies 
a hand. It is even partially used in England among whole- 
sale traders. In reckoning articles delivered at the warehouse, 



OTHER SCALES OF NUMERATION. 123 

the person who takes charge of the tale, having traced a long 
horizontal line, continues to draw, alternately above and below 
it, a warp, or four vertical strokes, each set of which he crosses 
by an oblique score, and calls out tally as often as the number 
Jive is completed. This custom is a very general one in 
assemblies where votes are counted, and in similar circum- 
stances elsewhere. 

The Senary method, so far as we can learn, was never used 
by any tribe or nation; at least never arose spontaneously. 
It is said to have been adopted at one time in China by the 
order of a capricious tyrant, who, having conceived an astro- 
logical fancy for the number six, commanded its several combi- 
nations to be used in all concerns of business or learning 
throughout his vast empire. 

The Septenary scale has not, so far as we can learn, been 
used anywhere. The number seven has been regarded as a 
kind of magic number, but nothing in nature suggested the 
method of counting by sevens. The division of the year into 
periods consisting of seven days each, a custom among nearly 
all nations, has given the number seven a wide distinction, and 
its frequent use in the Bible has caused it to be regarded as 
a sacred number, the basis of a celestial system of reckoning. 
The Octary scale, also, though it would possess many advant- 
ages, and has been recommended by scientific writers, has 
never made its appearance in any language. A Nonary scale 
has also never been used, and would be the most inconvenient 
of the smaller scales except the septenary. 

The Denary scale is the system which has prevailed among 
all civilized nations, and has been incorporated into the very 
structure of their language. This universal method manifests 
the existence of some common principle of numbering, which 
was the practice, so familiar in the earlier periods of society, 
of reckoning by counting the fingers on both hands. The 
origin of the terms used in the more polished ancient languages 
is not easily traced, but in the roughness of savage dialects 



124 THE PHILOSOPHY OF ARITHMETIC. 

these names vary less from the primitive words. The Muysca 
Indians were accustomed to reckon as far as ten, which they 
called quihicha or a foot, referring, no doubt, to the number of 
toes on their bare feet; and beyond this number they used 
terms equivalent to foot one, foot two, etc., for eleven, twelve, 
etc. Another South American tribe called ten, tunca, and 
merely repeated the word to signify a hundred, or a thousand, 
thus : tunca-tunca, tunca-tunca-tunca. The Peruvian language 
was actually richer in the names of numerals than the Greek 
or Latin. The Romans went no higher than mille, a thou- 
sand, and the Greeks than fmpia, or ten thousand. But the 
Peruvians had the expressions, hue, one ; chunca, ten ; pachac, 
a hundred; huaranca, a thousand; and hunu, a million. It 
appears from an early document, that the Indian tribes of New 
England used the Denary scale, and had distinct words to ex- 
press the numbers as far as a thousand. The Laplanders join 
the cardinal to the ordinal numbers ; thus, for eleven they say 
auft nubbe lokkai, that is, one to the second ten. The origin 
of the numerals in our own dialect will be found treated at 
greater length in another place. 

The mode of reckoning by twelves or dozens, may be sup- 
posed to have had its origin in the observation of the celestial 
phenomena, there being twelve months or lunations commonly 
reckoned in a solar year. The Romans likewise adopted the 
same number to mark the subdivisions of their unit of measure 
or of weight. The scale appears also in our subdivisions of 
weights and measures, as twelve ounces to a pound, twelve 
inches to a foot ; and is still very generally employed in 
wholesale business, extending to the second and even to the 
third term of the progression. Thus, twelve dozen, or 144, 
make the long hundred of the northern nations, or the gross 
of traders; and twelve times this again, or 1728, make the 
double or great gross. 

The scale of numeration by twenties has its foundation in 
nature, like the quinary and denary. In a rude state of 
society, before the discovery of other methods of numeration, 



OTHER SCALES OF NUMERATION. 125 

men might avail themselves for this purpose, not merely of the 
fingers on the hands, but also of the toes on the naked feet ; 
and such a practice would naturally lead to the formation of a 
vicenary scale of numeration. The languages of many tribes 
indicate this method, and many savage tribes do thus actually 
reckon. It is said of the inhabitants of the peninsula of Kam- 
schatka, that "it is very amusing to see them attempt to reckon 
above ten ; for having reckoned the fingers of both hands, they 
clasp them together, which signifies ten; then they begin at their 
toes and count twenty, after which they are quite confused and 
cry matcha, where shall I take more?" Among the Caribbees 
who constituted the native population of Barbadoes and other 
islands of the Caribbean sea, the numeration beyond five was 
carried on by means of the fingers and toes, and their numer- 
ical language became generally descriptive of their practical 
method of counting. The Abipones, an equestrian people of 
Paraguay, to express five show the fingers of one hand; to 
express ten, the fingers of both hands; "for twenty, their 
expression is pleasant, being allowed to show all the fingers of 
their hands and the toes of their feet." 

Traces of reckoning by scores or twenties, are found in our 
own and other European idioms. The expression threescore 
and ten is familiar. The term score itself, which originally 
meant a notch or incision made on a tally to signify the suc- 
cessive completion of such a number, seems to indicate that 
such a mode of counting was most familiarly used by our ances- 
tors. The vicenary scale seems to have prevailed very exten- 
sively among the Scandinavian nations, as is shown by the 
vestiges of it both among them and the languages partly 
derived from them. The French language has no term for the 
numbers in the second series of the denary scale above soix- 
ante or sixty. Eighty is expressed by quatre-vingts, four 
twenties, and ninety by quatre-vingts-dix, four twenties and 
ten. The people of Biscay and Armorica are said still to 
reckon by the powers of twenty, and, according to Humboldt, 
the same mode of numeration was employed by the Mexicans. 



CHAPTER VI. 

A DUODECIMAL SCALE. 

AS already explained, any number may be made the basis 
of a system of numeration and notation. The decimal 
basis is a mere accident, and in some respects an unfortunate 
one, both for science and art. The duodecimal basis would 
have been greatly superior, giving greater simplicity to the 
science, and facilitating its various applications. In this 
chapter it will be explained how arithmetic might have been 
developed upon a duodecimal basis. 

In order to make the matter clear, I call attention to two or 
three principles of numeration and notation. First, the bases 
of numeration and notation should be the same ; that is, if we 
write numbers in a duodecimal system, we should also name 
numbers by a duodecimal system. Second, in naming num- 
bers by any system, we give independent names up to the base, 
and then reckon by groups, using the simple names to number 
the groups. Bearing these principles in mind, we are ready 
to understand Numeration, Notation, and the Fundamental 
Rules in Duodecimal Arithmetic. 

NUMERATION. In naming numbers by the duodecimal 
system, we would first name the simple numbers from one to 
eleven, and then, adding one more unit, form a group, and name 
this group twelve. We would then, as in the decimal system, use 
these first names to number the groups. Naming numbers in 
this way, we would have the simple names, one, two, three, etc., 
up to tivelve. Continuing from twelve, we would have one and 
twelve, two and twelve, three and twelve, etc., up to twelve and 
twelve, which we would call two twelves. Passing on from this 

(126) 



A DUODECIMAL SCALE. 12 / 

we would have two twelves and one, two twelves and two, etc., 
to three twelves, and so on until we reach twelve twelves, when 
we would form a new group containing twelve twelves, and 
give this new group a new name, as gross, and then employ 
the first simple names again to number the gross. In this way 
we would continue grouping by twelves, and giving a new 
name to each group, as in the decimal scale by tens, as far as 
is necessary. 

These names, in the duodecimal system, would naturally 
become abbreviated by use, as the corresponding names in the 
decimal system. Thus, as in the decimal system ten was 
changed to teen, we may suppose twelve to be changed to teel, 
and omitting the " and" as in the common system, we would 
count one-teel, two-teel, thir-teel, four-teel, Jif-teel, six-teel, etc., 
up to eleven-teel. Two-twelves might be changed into two-tel, 
or twen-tel, corresponding to two-ty or twenty, and we would 
continue to count twentel-one, twentel-two, etc. Three-twelves 
might be contracted into three-tel or thirtel, corresponding to 
three-ty or thirty of the decimal system; four-twelves to 
fourtel, five twelves to fiftel, etc., up to a gross. Proceeding 
in the same manner, a collection of twelve gross would need 
a new name, and thus on to the higher groups of the scale. 

In this manner, the names of numbers according to a duo- 
decimal system could be easily applied. Were we actually 
forming such a system, the simplest method would be to intro- 
duce only a few new names for the smaller groups, and then 
take the names of the higher groups of the decimal system, 
with perhaps a slight modification in their orthography and 
pronunciation, to name the higher groups of the new scale. 
Thus, million, billion, etc., could be used to name the new 
groups without any confusion, as they do not indicate any 
definite number of units to the mind, but merely so many col- 
lections of smaller collections. Indeed, even the word thou- 
sand, with a modification of its orthography, say thousun, 
might be used to represent a collection of twelve groups, 



128 THE PHILOSOPHY OF ARITHMETIC. 

each containing a gross, without any confusion of ideas, 
Their etymological formation would not be an objection of any 
particular force, as no one in using them thinks of their pri- 
mary signification. These terms are not suggested as the 
best, but as the simplest in making the transition from the 
old to the new system. It will also be noticed that our 
departure in the decimal scale from the principle of the sys- 
tem, by using the terms eleven and twelve, would facilitate the 
adoption of a duodecimal system. 

To illustrate the subject more fully, let us adopt the names 
suggested, and apply them to the scale. Naming numbers 
according to the method explained, we would have the names 
as indicated in the following series : 

one oneteel twentel-one one gross and one 

two twoteel twentel-two one gross and two 

three thirteel twentel-eight two gross and five 

four fourteel twentel-eleven six gross and seven 

five flfteel thlrtel-one ten gross and eight 

six sixteel fortel-two eleven gross and nine 

seven seventeel flftel-six one thousun 

eight eighteel sixtel-eight one thousun and five 

nine nineteel seventel-nine one thousun four gross 

ten tenteel tentel-ten and seven 

eleven eleventeel eleventel-eleven two thousun seven gross 

twelve twentel one gross and fortel-one 

NOTATION. The writing of numbers by the duodecimal 
system would be an immediate outgrowth of the method of 
naming numbers in this system. As in the decimal system of 
notation, it would be necessary to employ a number of char- 
acters one less than the number of units in the base, besides 
the character for nothing. Since the group contains twelve 
units, the number of significant characters would be eleven 
two more than in the decimal system. For these characters 
we should use the nine digits of the decimal system, and then 
introduce new characters for the numbers ten and eleven. To 
illustrate, we will represent ten by the character * and eleven 
by n. 

These characters, with the zero, would be combined to rep- 
resent numbers in the duodecimal scale in the same manner as 
the nine digits represent numbers in the decimal scale. Thus, 



A DUODECIMAL SCALE. 129 

twelve would be represented by 10, signifying one of the 
groups containing twelve; 11 would represent one and twelve, 
or oneteel; 12 would represent two and twelve, or twoteel; 13 
would represent thirteel; 14, fourleel; 15, fifteel, etc. Con- 
tinuing thus, 20 would represent two twelves, or twentel; 21, 
twentel-one; 23, twentel-three, etc. The notation of numbers 
up to a thousun may be indicated as follows: 

one, 1 twelve, 10 thirtel, 30 

two, 2 oneteel, 11 thirtel-two, 32 

three, 3 twoteel, 12 thirtel-five, 35 

etc., etc. twentel, 20 thirtel-ten, 3* 

nine, 9 twentel-one, 21 thirtel-eleven, 3n 

ten, $ twentel-ten, 2* one gross, 100 

eleven, n twentel-eleven, 2n one thousun, 1000 

Extending the series as explained above, we should have 
the following notation table : 

TABLE. 

g s 

* i I & I 

3 s > ^ ^ 

s S s ^ o H 





P. 
p 
j> 

'- 

H 


o 

00 

00 

o 
E 


Twelves 


00 

a 

p 


V- 


O 

O 


Twelves 


Millyuns. 


*o 

00 

ao 

S 




Twelves 


Thousuns 


00 
00 

o 
1-1 


Twelves. 


*3 



85n46H857*365 
From the explanation given it is clearly seen that a system 
of duodecimal arithmetic might be easily developed, and read- 
ily learned and reduced to practice. Employing the names 
which I have indicated, or others similar to them, the change 
from the decimal ' to the duodecimal system would be much 
less difficult than has usually been supposed. It would be 
necessary to learn the method of naming and writing num- 
bers, which we have seen is very simple, and a new addition 
9 



ISO THE PHILOSOPHY OF ARITHMETIC. 

and multiplication table, from which we could readily derive 
the elementary differences and quotients. The rest of the sci- 
ence would be readily acquired, as all of its methods and 
principles would remain unchanged. Indeed, so readily could 
the change be made, that in view of the great advantages of 
the system, one is almost ready to believe that the time will 
come when scientific men will turn their attention seriously to 
the matter and endeavor to effect the change. 

FUNDAMENTAL OPERATIONS. In order to show how read- 
ily the transition could be made, I will present the method of 
operation in the fundamental rules. We would proceed first 
to form an addition table containing the elementary sums, 
which, as in the decimal system, we would commit to memory. 
From this we could readily derive the elementary differences 
used in subtraction. Such a table is given on page 131. 

By means of this table we can readily find the sum or dif- 
ference of numbers expressed in the duodecimal system. To 
illustrate, required the sum of 487n, 5438, OPERATION 
63n7, 4>856. The solution of this would be as 487n 

follows: Adding the column of units, 6 units 5*38 

and 7 units are 11 units, and 8 units are 19 63n7 

units, and n units are 28 units, or 2 twelves 
and 8 units ; writing the units, and carrying 
2 to the column of twelves, we have 2 twelves and 5 twelves 
are 7 twelves, and n twelves are 16 twelves, and 3 twelves 
are 19 twelves, and 7 twelves are 24 twelves, or 2 gross and 
4 twelves; writing the twelves, and carrying 2 to the third 
column, we have 2 gross and 8 gross are * gross, and 3 gross 
are 11 gross, and $ gross are In gross, and 8 gross are 27 gross, 
or 2 thousuns and 7 gross; 2 thousuns and $ thousuns are 10 
thousuns, and 6 thousuns are 16 thousuns, and 5 thousuns are 
In thousuns, and 4 thousuns are 23 thousuns; hence the 
amount is 23748. 

To illustrate subtraction let it be required to find the differ- 



A DUODECIMAL SCALE. 



131 



ADDITION AND MULTIPLICATION TABLES IN THE DUODECIMAL 8CALB. 



M to 

x|x 

. .. ^ 

e 



X X 



X X 



X X 



X X 



00 -1 OS I 01 

unit 



X X 



00, GO 00 00 

xix x!x 



g 



II ii I 

*|i- * 



Cj 
f 



f 
i i 

o 



td 



ta ic 

+ + 

CC QO 

8 f 






Tlflti? 



-I - I - I 

+ '+ 



II 9 



o 

HH 

H 
I I 

O 



w 



132 THE PHILOSOPHY OF ARITHMETIC. 

ence between 6428 and 2564. We would solve this as follows- 

Subtracting 4 units from 8 units we have 4 

units remaining; we cannot take 6 twelves OPERATION. 

from 2 twelves, so we add 10 twelves and have 

12 twelves; 6 twelves from 12 twelves leaves 



8 twelves; carrying 1 to 5 we have 6 gross; 3*84 
we cannot take 6 gross from 4 gross ; adding 10 as before we 
have 6 gross from 14 gross leaves 4> gross; adding 1 thousun 
to 2 thousuns, we have 3 thousuns from 6 thousuns leaves 3 
thousuns; hence the remainder is 3*84. 

In order to multiply and divide, we first form a multiplica- 
tion table similar to that now used in the decimal system, and 
commit it to memory. This table need not extend beyond 
" twelve times," as in our present system there is no need of 
extending beyond "ten times." From this table of elementary 
products, we can readily derive the table of elementary quo- 
tients as we do in the decimal system. Such a table will be 
found on page 131. 

It will be interesting to notice several peculiarities of this 
*,able, similar to those of the decimal system. As the column 
of "five times" ends alternately in 5 and 0, making it so 
easily learned by children, so the column of "six times" in 
the duodecimal table will end alternately in 6 and 0. In our 
present table the sum of the two terms of each product in the 
column of "nine times" equals nine, so in the duodecimal 
table, the sum of the two terms of each product in the column 
of "eleven times" equals eleven. We also notice that each 
product in the column of " twelve times" ends in 0, as does 
each product in the column of "ten times" of our present 
table. 

By means of the multiplication table we can readily find 
the product or quotient of numbers expressed in the duodeci- 
mal scale. To illustrate multiplication, let it be required to 
find the product of 54$8 by 3n7. We would solve this as 
follows: Using the first term of the multiplier, 7 times 8 are 



A DUODECIMAL SCALE. 188 

48, 7 times 4> are 5$, and 4 are 62, 7 OPERATION. 

times 4 are 24 and 6 are 2*, 7 times 5 54*8 

3n7 
are 2n and 2 are 31, making the first 

partial product 3U28; multiplying by 

n we have n times 8 are 74, n times $ 14280 

are 92 and 7 are 99, n times 4 are 38 and 



9 are 45, n times 5 are 47 and 4 are 4n ; 

3 times 8 are 20, 3 times $ are 26 and 2 are 28, 3 times 4 aw 

10 and 2 are 12, 3 times 5 are 13 and 1 are 14. Adding up 
the partial products, we have as the complete product, 1953768. 

To illustrate division, let it be required to find the quotient 
of 1953768 divided by 3n7. We would OPERATION. 
solve this as follows : We find that the 3n7)1953768(54*8 
divisor is contained in the first four 179n 

terms of the dividend 5 times, and mul- 1747 

tiplying 3n7 by 5 we have 179n ; sub- 13*4 



tracting this from the dividend we 3636 

have a remainder, 174; bringing down 337$ 

the next figure of the dividend and 2788 

proceeding as before, we have for the 
quotient 54*8. 

The method of finding the square or cube root of a number 
expressed in the duodecimal scale is similar to that used in 
the decimal scale, as may be shown by an OPERATION. 
example. Thus, find the square root of ir53'01(347 

115301. The greatest square in n is 9 ; 
subtracting and bringing down a period, 
and dividing by 2 times 3 or 6, we find the 



*"O J ftQ t 7V-)nnl 

second term of the root to be 4; complet- 8n01 

ing the divisor and multiplying 64 by 4, 
we have 214; subtracting and bringing down, we have 3n01, 
and dividing by 2 times 34, or 68, we have 7 for the last figure 
of the root; completing the divisor and multiplying it by 7, 
we have 3n01, which leaves no remainder. 
The above tables and calculations seem awkward to one 



134 THE PHILOSOPHY OF ARITHMETIC. 

who is familiar with the decimal system ; but it should be 
remembered that a beginner would learn the addition and mul- 
tiplication tables and the calculations based on them, just as 
readily as he now learns them in the decimal system. The 
practical value of such a system, in addition to what has 
already been said, may be seen in the calculation of interest, 
the rules for which would be greatly simplified on account of 
the relation of the number of months in a year (12) to the 
base, and also of the relation of the rate to the same, which 
would be some S% or 9% ; that is, 8 or 9 per gross. I hope to 
be able in a few years to publish a small work in which the 
whole science of arithmetic shall be developed on the duodeci- 
mal basis. 



CHAPTER VII. 

GREEK ARITHMETIC. 

C\ REEK Arithmetic, like that of all other nations of anti- 
vT quity, began in the representation of numbers by strokes or 
straight lines. This system, in the progress of thought and 
civilization, was finally discarded, and the letters of the alpha- 
bet taken as the symbols of numbers. After adopting the 
letters of their alphabet, the Greeks seem to have had no less 
than three distinct methods of notation. They used the 
letters in their natural order, to signify the smaller ordinal 
numbers. In this way the books of Homer's Iliad and 
Odyssey are usually marked. They employed also the first 
letters of the words for numerals as abbreviated symbols, mak- 
ing use of an ingenious device for augmenting the powers of 
these symbols; thus, a letter enclosed by a line on each side and 
another drawn over the top, as Fl, was made to signify five 
thousand times its original value. 

A more complete method consisted in the distribution of the 
twenty-four letters of their alphabet into three classes, corre- 
sponding to units, tens, and hundreds, adding another character 
to each class to complete the symbols for all of the nine digits. 
This latter method was the one in common use, and that which 
was made the basis of their arithmetic. The units from one 
to nine inclusive, were denoted by the letters a, /3, y, 6, e, r, C, 7, ; 
the tens by t, K, X, //, v, f , o, JT, h I and the hundreds by P , a, T, v t $, 
x, V*) w > ) Thousands were represented by the first series 
with the iota, or dash subscribed, thus: .#,? <? etc. With 
these characters they could readily express any number under 

(135) 



136 THE PHILOSOPHY OF ARITHMETIC. 

10,000, or a myriad. Thus, 991 was expressed by 2) '/a; 1382, 
by fap; 6420, by p; 4001, by |. 

It will be noticed that neither the order nor the number of 
characters was considered in expressing numbers. The value of 
the expression was the same in whatever order the letters were 
placed ; though as regularity tended towards simplicity, they 
generally wrote the characters according to value, from left to 
right. 

Myriads, or ten thousands, were denoted by the letter M, a 
letter representing the number of myriads indicated being 
written above it. Thus, denoted 10,000; M, 20,000; M, 
30,000, etc. Thus, also, j denoted 370,000 ; 1P 43720000 ; and 
in general, the letter M placed beneath any number had the 
same effect as our annexing four ciphers. 

This is the notation employed by Eutocius in his commenta- 
ries on Archimedes, but it is evidently inconvenient in calcula- 
tion. Diophantus and Pappus expressed the myriad more 
simply by the two letters M v placed after the number, and 
afterwards by merely writing a point after it. This enabled 
them to express 100,000,000, which was the greatest extent of 
the ordinary Greek arithmetic. 

This system had been extended by Archimedes and Apollo- 
nius, for the purpose of astronomical and other scientific 
calculations. Archimedes, in order to express the number 
of grains of sand that might be contained in a sphere that had 
for its diameter the distance of the fixed stars from the earth, 
found it necessary to represent a number which, with our nota 
tion, would require sixty-four places of figures; and in order 
to do this, he assumed the square myriad, or 100,000,000, as a 
new unit, and the numbers formed with these new units he 
called numbers of the second order ; and thus he was enabled 
to express any number which in our notation requires sixteen 
figures. Assuming again 100,000, OOO 2 as a new unit, he could 
represent any number that requires in our scale twenty-four 



GREEK ARITHMETIC. 137 

figures, and so on ; so that by means of his numbers of the 
eighth order, he could express the number in question, which 
requires sixty-four figures in our scale. 

By this system all numbers were separated into periods or 
orders of eight figures. This was afterwards considerably 
improved by Apollonius, who, instead of periods of eight 
places, which were called by Archimedes octates, reduced num- 
bers to periods of four places ; the first of which, on the left, 
were units, the second period myriads, the third double myri- 
ads or numbers of the second order, and so on indefinitely. 
In this manner Apollonius was able to write any number that 
can be expressed by our system of numeration ; as for example, 
if he had wished to represent the circumference of a circle 
whose diameter was a myriad of the ninth order, he would 
have written it thus: 

y.nvie. Bci-e. y<pir6. j^A/J. yuft?. $XP/~ %uA(3. {2)v. (fond- 

3.1415 9265 3589 7932 3846 2643 3832 7950 2824 
The learned astronomer Ptolemy modified this system in its 
descending range by applying it to the sexagesimal subdivisions 
of the lines inscribed in a circle. He likewise advanced an 
important step, by employing a small or accentuated o to supply 
the place of any number wanting in the order of progression. 
The Greek method of expressing fractions was also peculiar. 
An accent set on the right of a number, made of that number 
the denominator of a fraction whose numerator was a unit, 
thus, /=$, <i'=J, f<J'=3r, pa'=Tir> e ^ c - When the numerator is 
not unity, the denominator is placed as we set our exponents. 
Thus, t'e^ represented 15 64 , or ^|, and (, l>Ka represented 7 121 , or 
Y^y. The fraction \ had a particular character, as C, <, 
C', or K. The notation of the Greeks was not adapted to the 
descending scale, and consequently they had no decimals. 

The notation of the Greeks, though much inferior to that of 
the present day, was formed upon a regular and scientific 
basis, and could be employed with considerable convenience 
as an instrument of calculation. We will present two or three 



138 THE PHILOSOPHY OF ARITHMETIC. 

examples taken from Barlow's Theory of Numbers, from 
which some of the previous facts are gathered. 

Addition. The following example in addition is from 
Eutocius, Theorem 4, of the Measure of the Circle. 

847 3921 
60 8400 



Tff ft 908 2321 

The method, it will be seen, is similar to compound addition, 
but is simpler on account of the constant ratio of ten between 
any character and the succeeding one. 

Subtraction. The following example in subtraction is from 
Eutocius, Theorem 3, on the Measure of the Circle. 
O.yxte 93636 

/8.V 9 23409 

CT~^? 70221 

The method is simple, proceeding from right to left as in 
our subtraction, which seems so obviously advantageous and 
simple that one can hardly conceive why the Greeks should 
ever proceed in the contrary way, although there are many 
instances which make it evident that they did, both in addition 
and subtraction, work from left to right. 

Multiplication. In multiplication they most commonly pro- 
ceeded in their operations from left to right, as we do in mul- 
tiplication in algebra, and their successive products were 
placed without much apparent order; but as each of their 
characters retained always its own proper value, in whatever 
order they stood, the only inconvenience of this was, that it 
rendered the addition of them a little more troublesome. 
The following example is 

from Eutocius. As it is dim- pvy 153 

' l/'ft'S 

cult to remember the value I L _ 

of all the Greek characters, a '"^ pv ^//'l/'/g/q//^ 

we will indicate the opera- '* Q Z rf l rf 5 f $ 

tion by writing 1, 2, 3, ^ yve 2 3 ///4// go 

etc., for the series of units ' ' 



GREEK ARITHMETIC. 139 

1', 2', 3', etc., for the series of tens ; I", 2", etc., for the 
hundreds, etc., and denote the myriads by writing m as an 
exponent. 

Division. The division of the Greeks was still more intri- 
cate than their multiplication, for which reason it seems they 
generally preferred the sexagesimal division, and no example 
is left at length by any of those writers except in the latter 
form ; but these are sufficient to throw some light on the pro- 
cess they followed in the division of common numbers, and 
Delambre has accordingly supposed the following example : 
ay 332 W 3'"3"2'9 (1" / 8"2'3 



P.B 



182 


s 






1"'8"2'3 


150 
145 




8 


3 
4 


2 


9 


4 
3 


1 
6 


9 
4 


2 
6 


9 ' 




5 
5 


4 

4 


6 
G 


9 
9 



This example will be found, on a slight inspection, to resem- 
ble our compound division, or that sort of division that we 
must necessarily employ, if we were to divide feet, inches 
and parts by similar denominations, which, together with the 
number of different characters that they made use of, must 
have rendered this rule extremely laborious ; and that for the 
extraction of the square root was, of course, equally difficult, 
though the principle was the same as ours, except in the 
difference of the notation. It appears, however, that they fre- 
quently, instead of making use of the rule, found the root by 
successive trials, and then squared it in order to prove the truth 
of their assumption. 

This beautiful system was vastly superior in simplicity and 
practical utility to that transmitted to and retained by the 
Romans, and by them bequeathed to the nations of modern 
Europe. It was, at least when it had reached its highest 
development through the genius of Archimedes and Apollo 



140 THE PHILOSOPHY OF ARITHMETIC. 

nius, quite well fitted for an instrument of calculation; and 
though somewhat cumbrous in its structure, was capable of 
performing operations of very considerable difficulty and mag- 
nitude. 

It will be seen, however, that though much more refined and 
pliant than that of the Romans, the notation of the Greeks is 
very much inferior to the common or Hindoo method ; and one 
cannot help wondering that so ingenious and philosophical a 
people failed to conceive the simple idea of place value, and 
construct a system of notation upon it. This seems all the 
more astonishing when we remember that Archimedes invented 
a system of octates, or system of eights, which was subse- 
quently improved by Apollonius, by making the periods con- 
sist of only four places, and dividing all numbers into orders 
of myriads. In this form, as Barlow remarks, it seems most 
astonishing that he did not perceive the advantage of making 
the periods to consist of a less number of characters; for, hav- 
ing given a local character to his periods of four, it was only 
necessary to have done the same for the single digits, in order 
to have arrived at the system in present use. And this is 
the more singular, as the use of the cipher was not unknown 
to the Greeks, being always employed in their sexagesimal 
operations where it was necessary ; and consequently the step 
between this improved form of their notation and that of the 
present system was extremely small, although the advantages 
of the latter when compared with the former are incalculably 
great. It seems to have been the lot of the metaphysical 
mind of the Hindoos to make this "brilliant invention of the 
decimal scale," one of the greatest improvements in the whole 
circle of the sciences, and to which we are indebted for all the 
remarkable advances in modern analysis. 



CHAPTER YIII. 

ROMAN ARITHMETIC. 

1'HE arithmetic of the Romans was quite inferior to that of 
the Greeks, a necessary consequence of the inferiority of 
the method of notation adopted. The method of notation, 
though usually ascribed to the Romans, was probably invented 
by the Greeks, and communicated by them to the Romans, who 
in turn transmitted it to their successors in modern Europe. 
It no doubt originated in the use of simple strokes, variously 
combined, to represent numbers. Subsequently it was found 
convenient to represent numbers by the letters of the alphabet, 
and the numerical strokes were finally displaced by such alpha- 
betic characters as most nearly resembled them. 

The origin of the Roman characters is not certainly known; 
but the theory, as given by Leslie, and by many regarded as 
correct, is interesting and plausible. It is certain that the 
first numerical characters consisted simply of strokes or straight 
lines. This was the method primarily used by nearly every 
nation of antiquity, and was the beginning of a philosophical 
and universal system alike intelligible to all nations. Such 
characters are still preserved in the Roman notation with very 
little change, and were probably adopted before the importation 
of the alphabet itself, by the Grecian colonies that settled 
Italy and founded the Latin commonwealth. Assuming, then, 
a perpendicular line | to signify one, two such lines 1 1 to signify 
two, three lines 1 1 1 to signify three, and so on up to ten, and we 
have the first series of the numerical scale. They might then 

(141) 



142 THE PHILOSOPHY OF ARITHMETIC. 

be supposed to throw a dash across the last stroke or unit, to 
mark the completion of the series; and thus, a cross, X, 
would come to signify ten. The continued repetition of this 
mark would denote twenty, thirty, etc., until they reached a 
hundred, or ten tens, which completes the second series, and 
might be denoted by adding another dash to the mark for ten, 
or by merely connecting three strokes, thus Q. The repetition 
of this symbol would, in like manner, indicate the successive 
hundreds, the tenth of which would be marked by the addition 
of another stroke, so that four combined strokes, M> would 
express a thousand. 

Such were probably the symbols originally employed in the 
Roman notation ; in process of time it would be perceived that 
the inconvenience in writing, arising from so many repetitions 
of the same character, might be avoided by adopting symbols 
for the intermediate numbers; and it was seen that these 
might be furnished by the division of the symbols already in 
use. Thus, having parted in the middle the two strokes, X, 
either the under half, /\, or the upper half, V, was employed 
to signify Jive, or the half of ten. Next, for fifty, the half of 
a hundred, the symbol Q was divided into two equal parts, (~~ 
and |_, either of which represented fifty. Again, the symbol 
for thousand having come to assume a rounded shape, thus 
ft, or thus CD, the half of this, either CI, or ID, was taken to 
represent the half of one thousand or five hundred. The 
symbol Q, to represent a hundred, would, in process of time, 
being frequently made, have its corners rounded and attain 
the form C- Lastly, noticing that these characters closely 
resemble some of the letters of the alphabet, it was agreed to 
employ those letters as the symbols of the numbers mentioned. 

The notation of numbers by combined strokes, was evi- 
dently founded in nature, and may be regarded as the begin- 
ning of a philosophical language of arithmetic. That this 
was the foundation of the Roman system is confirmed from the 
analogous practice of other nations. It is quite clear that the 



ROMAN ARITHMETIC. 143 

Egyptians and Chinese must have followed the same method. 
The inscriptions on the ancient obelisks present a few numerals 
which are easily distinguished. The substitution of capital 
letters for the combined strokes which they chanced most to 
resemble, though it gave uniformity to the system of notation, 
prevented any farther improvements of the system. The only 
simplification which the Romans appear to have introduced, 
was to diminish the repetition of letters by reckoning in some 
cases backwards, as in IY, which was originally represented 
by four strokes, and IX, which was probably at first written 

vim. 

Their method of representing large numbers was a little 
diiferent from that now used, as may be seen by the following 
examples : 

DorD MorCD 133 CC133 1333 CCC1333 
500 1000 5000 10,000 50,000 100,000. 

In illustration, it is interesting to notice that Cicero in his 
fifth oration against Verres expresses 3600 by CI3 CI3 CI3 13C. 
The Romans often contracted or modified the forms of their 
numerals, especially in carving inscriptions upon stones, in 
which case the abbreviated letters were called lapidary char- 
acters. 

The marks for any number could also be augmented in power 
one thousand times, either by enclosing them with two hooks or 
C's, or by drawing a line over them. Thus, CXO, or X denoted 
10,000, and CLVIM given by Pliny, means 156,000,000. 
Sometimes a letter was placed over another to indicate their 
product ; thus, ^ would express 500,000. The multiplier was 
also sometimes written like an exponent, thus IIP was used 
to express three hundred. In expressing very large numbers, 
points were sometimes interposed: thus, Pliny writes XVI. 
XX.DCCCXXIX for 1,620,829. It may be remarked that if this 
practice had become more general it would probably have 
effected a material improvement of the system. 



144 THE PHILOSOPHY OF ARITHMETIC. 

In the latter ages of the Roman Empire, the small letters 
of the alphabet seem to have been used in imitation of the 
numeral system of the Greeks. The letters a, b, c, d, e, f, 
g, h, and i represented the nine digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 ; 
the next series k, 1, m, n, o, p, q, r, and s expressed 10, 20, 30, 
40, 50, 60, 70, 80, and 90 ; and the remaining letters t, u, x, 
y, and z denoted 100, 200, 300, 400, and 500. To represent 
the rest of the hundreds it was necessary to employ capitals 
or other characters, and 600, 700, 800, and 900 were repre- 
sented by I, V, hi and hu. But this mode of notation never 
obtained any degree of currency, being mostly confined to 
those foreign adventurers from Greece, Egypt or Chaldea, who, 
pretending to skill in judicial astrology, were enabled to prey 
on the credulity of the wealthy Romans. 

In modern Europe the Roman numerals were supplied by 
Saxon characters. Thus, in the accounts of the Scottish Ex- 
chequer for the year 1331, the sum of 6896 5s. 5d. stated as 
paid to the King of England is thus marked: 
o c xx 
vj. viij. iiij. xvj. Ij. v. s. v. d. 

The Roman system, as now used, employs seven characters, 
of which I represents one, V five, X ten, L fifty, C one hun- 
dred, D five hundred, M one thousand. To express other 
numbers these characters are combined according to the fol- 
lowing principles : 

1. Every time a letter is repeated its value is repeated. 

2. When a letter is placed after one of greater value, the 
sum of their values is the number expressed. 

3. When a letter is placed before one of a greater value, the 
difference of their values is the number expressed. 

4. When a letter stands between two letters of a greater 
value, it is combined with the one following it. 

5. A letter is placed before one of its own order only, or the 
unit of the next higher order. 

6. A dash over a letter increases its value a thousand fold. 



HUMAN ARITHMETIC. 1-ifi 

In accordance with the fifth principle it would be incorrect 
to write VC for ninety-Jive, or 1C for ninety-nine. It is also 
to be noticed that the letter V is never used before a letter of 
greater value, since the only case in which it could be thus 
used according to the fifth principle is before X, giving VX 
for five, which is more concisely expressed by Y itself. 

In expressing numbers by the Roman method we always 
write the different orders of units successively, beginning with 
the higher orders. Thus, in expressing four hundred and 
ninety-nine, we would not write ID, though this, by principle 
second, would be the difference of one and five hundred, but 
we first write CO CO for four hundred, then XC for ninety, 
and then IX for nine, giving CCCCXCIX. 

It may be interesting to notice, however, that though the 
Roman method was not employed in numerical calculations, it 
might have been so employed by slightly modifying the usual 
mode of notation. Thus, by not using the third principle, but 
writing IIII for IV, and YIIII for IX, or by using some 
mark to show that the letters written according to that prin- 
ciple are taken together, as XXIV, we can perform the four 
fundamental operations without much inconvenience. To illus- 
trate, we give a problem in multiplication, with its explanation. 

Explanation. VIII multiplied 
by VII equals LVI, X multiplied 

by VII equals LXX, L multiplied XXXVI 

by VII equals CCCL ; III multi- 
plied by X equals XXX, X multi- 



plied by X equals C, L multiplied DCLXXX 

by X equals DCL ; multiplying by DCLXXX 
X a second and third time, and MM?) XVI 

taking the sum of the four partial 

products, we have MMDXVI, or two thousand five hundred 
and sixteen. This result may be obtained by multiplying by 
VII and XXX; or by II, V, X, and XX, etc. The multipli- 
cand also may be variously separated in the multiplication. 
10 



146 THE PHILOSOPHY OF ARITHMETIC. 

It is clear, however, that this operation would be very com- 
plicated with large numbers, so much so, indeed, as 
to be unfitted for general use, and it is believed that it was not 
used in performing numerical calculations. These calculations 
were performed by means of counters, or other palpable em- 
blems. The instrument generally used was called the Abacus. 
Leslie says that "the system of characters among the Romans 
was so complex and unmanageable as to reduce them to the 
necessity in all cases of employing the Abacus." 

The Abacus appears to have continued in use among the 
people of Europe until quite a recent period. The counters or 
pebbles were, from a corruption of the word algorithm, called 
in England augrim, or awgrym, stones. Thus, in Chaucer's 
description of the chamber of Clerk Nicholas, he says : 

" His almageste and bokes grete and smale, 
His astrelabre, longing for his art, 
His augrim stones layen faire apart 
On shelves couched at his beddes head." 

Indeed, the modern method of arithmetic was not known in 
England until about the middle of the sixteenth century ; and 
the common people, imitating the clerks of former times, were 
still accustomed to reckon by the help of the awgrym stones. 
Thus, in Shakespeare's comedy of the Winter's Tale, written 
at the beginning of the seventeenth century, a clown, staggered 
at a very simple multiplication, exclaims that he must try it 
with counters. 

CLO. Let me see ; Every 'leven wether tods ; every tod yields 
pound and odd shilling; fifteen hundred shorn, What comes 
the wool to? . . . I cannot do't without counters. 

The Roman method is now chiefly used to denote the vol- 
umes, chapters, sections and lessons of books, the pages of pre- 
faces and introductions, to express dates, to mark the hours on 
clock and watch faces, and in other places for the sake of prom- 
inence and distinction. 



CHAPTER IX. 

PALPABLE ARITHMETIC. 

earliest methods of representing numbers in arithmetical 
JL calculation were by means of counters and other palpable 
emblems. The objects most generally used among all primitive 
nations were little stones or pebbles, from which we derive our 
word calculation. Beginning with pebbles or some such sim- 
ple objects, as they advanced in civilization these were found to 
be insufficient for their purposes, and they invented instruments 
to represent numbers, by means of which they were enabled 
to calculate with great rapidity and correctness. The Japan- 
ese and Chinese at the present day, with their arithmetical 
instruments, can add, subtract, multiply and divide as rapidly 
and correctly as we can with the Arabic system of notation. 
So extensively was this method used by the early nations before 
the method of calculating by figures was adopted, that Leslie, 
in his treatise on arithmetic, gives it a distinct and detailed 
explanation under the head of Palpable Arithmetic. The sub- 
ject is so full of interest, both for its own ingenuity and its 
relation to our present system, that I think it proper to devote 
a chapter to it, and finding a clearer statement of it in Leslie 
and Peacock than I could hope to give myself, I have tran- 
scribed their description, sometimes word for word. 

The early Egyptians performed their computations mainly 
by the help of pebbles, and so did the early Greeks and 
Romans. In the schools of ancient Greece, the boys acquired 
the elements of knowledge by working on the ABAX, asmooth 

(H7) 



148 THE PHILOSOPHY OF ARITHMETIC. 

board with narrow rim, so named evidently from the combina- 
tion of the first three letters of their alphabet, and resembling 
the tablet on which children were formerly accustomed to begin 
to learn the art of reading. Pupils were taught to calculate by 
forming progressive rows of counters, which consisted of round 
bits of bone or ivory, or even silver coins, according to the 
wealth or fancy of the individual. The same board, strewed 
with fine green sand, a color soft and agreeable to the eye, 
served equally for teaching the rudiments of writing and the 
principles of geometry. 

The ancient writers make frequent allusions to these calculat- 
ing boards. Solon, the great Athenian statesman, used to 
compare the passive ministers of kings to the counters or 
pebbles of arithmeticians which, according to the place they 
hold, are sometimes most important, and sometimes utterly 
insignificant. The Grecian orators, in speaking of balanced 
accounts, picture the settlements by saying that the pebbles 
were cleared away and none left. It thus appears that the 
ancients, in keeping their accounts, did not arrange the debits 
and credits separately, but set down pebbles for the former, and 
took up pebbles for the latter. As soon as the board became 
cleared, the opposite claims were exactly balanced. It may 
be observed that the common phrase to clear one's scores or 
accounts, meaning to settle or adjust them, still preserved in the 
popular language of Europe, was suggested by the same prac- 
tice of reckoning with counters, which prevailed, indeed, until 
a comparatively late period. 

The Romans borrowed their Abacus from the Greeks, and 
seem never to have aspired higher in the pursuit of numerical 
science. To each pebble or counter required for the board 
they gave the name of calculus, meaning a small white stone, 
and applied the verb calculare to express the operation of com- 
bining or separating such pebbles or counters. The use of 
the Abacus, called also the Mensa Pythagorica, formed an 
essential part of the education of every noble youth. A small 



PALPABLE ARITHMETIC. 149 

box or coffer, called a Loculus, having compartments for hold- 
ing the calculi, or counters, was considered as a necessary 
appendage. Instead of carrying a slate and satchel to school, 
the Roman boy was accustomed to trudge to school loaded 
with those ruder implements, his arithmetical board and his 
box of counters. 

In the progress of luxury and refinement, dice made of ivory, 
called tali, were used instead of pebbles, and small silver 
coins came to supply the place of counters. Under the Em- 
perors, every patrician living in a spacious mansion and 
indulging in all the pomp and splendor of Eastern princes, 
generally entertained, for various functions, a numerous train 
of foreign slaves or freedmen in his palace. Of these, the 
librarius, or miniculator, was employed in teaching the 
children their letters, the notarius registered expenses, the 
rationarius adjusted and settled accounts, and the tabularius 
or calculator, working with his counters and board, performed 
what computations might be required. 

To facilitate the working by counters, the construction of the 
Abacus was afterwards improved. Instead of the perpendic- 
ular lines, or bars, the board had its surface divided by sets of 
parallel grooves, by stretched wires, or even by successive 
rows of holes. It was easy to move small counters in the 
grooves, to slide perforated beads along the wires, or to stick 
large knobs or round-headed nails in the different holes. To 
diminish the number of marks required, every column was 
surmounted by a shorter one, wherein each counter had the 
same value as five of the ordinary kind. The Abacus, instead 
of wood, was often, for the sake of convenience and durabil- 
ity, made of metal, frequently brass, and sometimes silver. 
Two varieties of this instrument seem to have been used by 
the Romans. Both of them are delineated from antique 
monuments the first kind by Ursinus, and the second by 
Marcus Velserus. In the former, the numbers are represented 
by flattish perforated beads, ranged on parallel wires; and in 



150 THE PHILOSOPHY OP ARITHMETIC. 

the latter, they are signified by small round counters, moving 
in parallel grooves. These instruments contain each seven 
capital divisions, expressing in regular order units, tens, hun- 
dreds, thousands, ten thousands, hundred thousands, and 
millions, and as many shorter divisions, of five times the rela- 
tive value of the larger ones. With four beads on each of 
the long grooves or wires, and one on each corresponding short 
one, it is evident that any number could be expressed up 
to ten millions. The Roman Abacus also contained grooves 
to mark ounces, half-ounces, quarter-ounces, and thirds of an 
ounce. 

The Romans likewise applied the same word Abacus to an 
article of furniture resembling in shape the arithmetical board, 
but often highly ornamented, which was destined for the 
amusement of the opulent. It was used in a game apparently 
similar to that of chess, in which the infamous and abandoned 
Nero took particular delight, driving over the surface of the 
Abacus with a beautiful ivory quadriga or chariot. 

The Chinese have, from the remotest ages, used in all their 
computations, an instrument similar in shape and construction 
to the Roman Abacus, but more complete and uniform. It 
is admirably adapted to the decimal system of weights, meas- 
ures, and coins, which prevails throughout the empire. The 
whole range includes ten bars, and the calculator may begin 
at any one and reckon upwards or downwards with equal 
facility, treating fractions exactly like integers an advantage 
of the utmost consequence in practice. Accordingly these 
arithmetical machines, of various sizes, have been adopted by 
all ranks, from the man of letters to the humblest shopkeeper, 
and are constantly used in all the bazaars and booths of Can- 
ton and other cities, being handled, it is said, by the native 
traders with a rapidity and address quite astonishing. 

Among the various nations which regained their independ- 
ence by the fall of the Roman Empire, it was found convenient 
in all transactions where money was concerned, to follow the 



PALPABLE ARITHMETIC. 151 

procedure of the Abacus, in representing numbers by counters 
placed in parallel rows. During the Middle Ages, it became 
the usual practice over Europe for merchants, auditors of 
accounts, or judges appointed to decide in matters of revenue, 
to appear on a covered bank or bench, so called from an old 
Saxon or Franconian word signifying a seat. The term 
scaccarium, a Latinized Oriental word, from which was 
derived the French and then the English name for the 
Exchequer, anciently indicated merely a chess-board, being 
formed from scaccum, one of the pieces in that game. 

The Court of Exchequer, which takes cognizance of all 
questions of revenue, was introduced into England by the 
Norman Conquest. Fitz-Nigel, in a dialogue on the subject, 
written about the middle of the twelfth century, says that the 
scaccarium was a quadrangular table about ten feet long and 
five feet broad, with a ledge or border about four inches high, 
to prevent anything from rolling over, and was surrounded on 
all sides by seats for the judges, the tellers, and other officers. 
It was covered every year, after the term of Easter, with fresh 
black cloth, divided by perpendicular white lines or distinc- 
tures, at intervals of about a foot or a palm, and again parted by 
similar transverse lines. In reckoning accounts, they pro- 
ceeded according to the rules of arithmetic, using small coins 
for counters. The lowest bar exhibited pence, the one above 
it shillings, the next pounds, an d the higher bars denoted suc- 
cessively tens, twenties, hundreds, thousands, and ten thou- 
sands of pounds; though, in those early times of penury and 
severe economy, it very seldom happened that so large a sum 
as the last ever came to be reckoned. The teller sat about the 
middle of the table ; on his right hand, eleven pennies were 
heaped on the first bar, and nineteen shillings on the second, 
while a quantity of pounds was collected opposite to him, on 
the third bar. For the sake of expedition he might employ a 
different mark to represent half the value of any bar, a silver 
penny for ten shillings, and a gold penny for ten pounds. 



162 THE PHILOSOPHY OF ARITHMETIC. 

In early times, a checkered board, the emblem of calculation, 
was hung out, to indicate an office for changing money. It 
was afterwards adopted as the sign of an inn or hostelry, 
where victuals were sold, or strangers lodged and entertained. 
It is said that traces of this ancient practice may be found even 
at the present day. 

The use of the smaller Abacus in assisting numerical com- 
putation was not unknown during the Middle Ages. In 
England, however, it appears to have scarcely entered into 
actual practice, being mostly confined to those few individuals 
who, in such a benighted period, passed for men of science 
and learning. The calculator was styled, in correct Latin, 
abacista ; but in Italian, abbachista, or abbachiere. The 
Arabians, having adopted an improved species of numeration, 
to which they gave the barbarous name of algarismus or algo- 
rithmus, from their definite article al, and the Greek word for 
number, this compound term was adopted by the Christians 
of the West, in admiration of their superior skill, to signify 
calculation in general, long before the peculiar method of per- 
forming it had become known and practiced among them. 
The term algarism was converted in English into augrim 
or awgrym, and applied even to the pebbles or counters used 
in ordinary calculation. The same word, algorithm, is now 
applied by mathematicians to express any peculiar sort of 
notation. 

The Abacus had been adopted merely as an instrument 
for facilitating the process of computation. It became 
necessary, however, to adopt some simpler and more conveni 
ent method of expressing numbers. A very ancient practice 
consisted in employing the various articulations and disposi- 
tions of the fingers and the hands, to denote the numerical 
series. On this narrow basis, the Romans framed a system of 
considerable extent. By the inflexion of the various fingers 
of the left hand, they proceeded as far as ten, and by combin- 
ing these with some other given inflexions, as changes in the 



PALPABLE ARITHMETIC. 153 

position of the thumb, they could advance to a hundred ; and 
using the right hand in a similar manner, they proceeded as 
far as a thousand and ten thousand. This is as far as the 
system appears to have been carried by the ancients ; but the 
venerable Bede, by referring these signs to the various parts 
of the body, as the head, the throat, the side of the chest, the 
stomach, the waist, the thigh, etc., has shown how they could 
be again multiplied a hundred times, and raised to the extent 
of a million. In this numerical play, the Romans especially 
had acquired great dexterity. Many allusions to the practice 
are made by their poets and orators, and without some knowl- 
edge of the principle adopted, many passages of the classics 
would lose their whole force. 

A species of digital arithmetic seems to have existed among 
nearly all the Eastern nations. The Chinese have a system of 
indigitation by which they can express on one hand all num- 
bers less than 100,000 The thumb nail of the right hand 
touches each joint of the little finger, passing first up the 
external side, then down the middle, and afterwards up the 
other side of it, in order to express the nine digits; the tens 
are denoted in the same way on the second finger ; the hun- 
dreds on the third; the thousands on the fourth; the tens of 
thousands on the thumb. It would be only necessary to pro- 
ceed to the right hand in order to be able to extend this system 
of numeration much further than could be required for any 
ordinary purposes. The Bengalese count as far as 15 by 
touching in succession the joints of the fingers ; and merchants 
in concluding bargains, the particulars of which they wish 
to conceal from the by slanders, put their hands beneath a 
cloth and signify the prices they offer or take by the contact 
of the fingers. The same custom is prevalent also in Barbary 
and Arabia, where they conceal their hands beneath the folds 
of their cloaks, and possess methods which are probably pecu- 
liar and national, of conveying the expression of numbers to 
each other. 



154 THE PHILOSOPHY OF ARITHMETIC. 

Juvenal states it as a peculiar felicity of Nestor that he 
counted the years of his age on his right hand. The image of 
Janus was represented, according to Pliny, with the fingers 
so placed as to represent 365, the number of days in the year. 
Some authors have supposed that Solomon in the passage, 
"Length of days is in her right hand, and in her left hand 
riches and honor," referred to this practice. The common 
phrases, ad digitos redire, in digitos mittere, have the same 
meaning as computare, and distinctly refer to digital numera- 
tion ; and the phrase micare digitis, of frequent occurrence, 
alludes to a game extremely popular among the Romans, 
and which was probably the same as the morra of modern 
Italy. This noisy game is played by two persons, who stretch 
out a number of their fingers at the same moment, and instantly 
call out a number; and he is the winner who expresses the 
sum of the number of fingers thrown out. The same game 
is found amongst the Sicilians, Spaniards, Moors, and Persians, 
and under the name tsoimoi, is practiced also in China. 

These signs were merely fugitive, and it became necessary 
to adopt other marks of a permanent nature for the purpose of 
recording numbers. But of all the contrivances adopted with 
this view, the rudest undoubtedly is the method of registering 
by tallies, introduced into England along with the Court of 
Exchequer, as another badge of the Norman Conquest. These 
consist of straight, well-seasoned sticks of hazel or willow, so 
called from the French verb tattler, to cut, because they are 
squared at each end. The sum of money was marked on the 
side with notches, by the cutter of tallies, and like wise in scribed 
on both sides in Roman characters, by the writer of the tallies. 
The smallest notch signified a penny, a larger one a shilling, 
and one still larger a pound; but other notches, increasing suc- 
cessively in breadth, were made to denote ten, a hundred, and 
a thousand. The stick was then cleft through the middle by 
the deputy-chamberlains, with a knife and mallet, the one por- 
tion being called a tally, or sometimes the scachia, stipes, or 



PALPABLE ARITHMETIC. 155 

kancia, and the other portion named the counter-tally or 
folium. 

This strange custom might seem the practice of untutored 
Indians, and can be compared only to the rude simplicity of 
the ancient Romans, who kept their diary by means of lapilli 
or small pebbles, casting a white pebble into the urn on fortu- 
nate days, and dropping a black one when matters looked 
unprosperous ; and who sent, at the close of each year, the 
Praetor Maximus, with great solemnity, to drive a nail in the 
door of the right side of the temple of Jupiter, next to that of 
Minerva, the patron of learning and inventor of numbers. 

The use of counters was general throughout Europe as late 
as the end of the 15th century: about that period they were no 
longer used in Italy and Spain, where the early introduction 
of the Arabic figures and the number of treatises on the use of 
these figures had rendered them unnecessary. Recorde, in his 
Ground of Arts, prefaces his second dialogue, entitled " The 
Accounting by Counters," by observing, "Now that you have 
learned Arithmetic with the pen, you shall see the same art in 
counters, which feat doth not onely serve for them that cannot 
write and read, but also for them that can do both, but have 
not at the same time their pen or tables with them." 

We shall now proceed to give some account of the method 
of performing operations by this palpable or 
calcular arithmetic. They commenced by 
drawing seven lines with a piece of chalk, - 
on a table, board, or slate, or by a pen on 
paper, as in the margin ; the counters, which " 
were usually of brass, on the lowest line 
represented units, on the next tens, and so 
on as far as millions on the uppermost line; _ 
a counter placed between two lines repre- 
sented five counters on the line next below - 
it; thus, the number represented in the 
margin is 3629638, and the number of lines * * 
may evidently be increased so as to represent any number. 



166 



THE PHILOSOPHY OF ARITHMETIC. 





, A 


A A 


V 









A A 




m 











To add two numbers, such as 788 and 383, we divide tho 
lines as in the margin, so as to form three columns, writing 
the first number in the first 
column, numbering from the 
left, the second in the second, 
and the result in the third 
column. The sum of the 
counters on the lowest line 
in the first two columns is 6; we therefore place one on that 
line in the third column, and carry one to the space above 
which, added to the one already there, makes one on the second 
line ; adding this counter to the six already there, we have 
7, and therefore place 2 on the line and carry one to the space 
above ; adding the counters on that space, we find there are 3, 
hence we leave one in the space and carry one to the next line, 
in which the sum of the counters is six ; we leave one on the 
line and carry one to the space above, and adding to the 
counter already there we have two counters, hence we leave 
no counter there, but place one on the fourth line ; the sum 
thus obtained will be 1171. 

The principle of this operation is extremely simple, and the 
process could, with a little practice, be performed with much 
rapidity. In practice, the last column would not be used, as 
the counters on each line would be removed as the addition 
proceeded, and replaced by those which denoted their sum. 

We will illustrate the method of subtraction by taking 682 
from 1375. The two count- 
ers on the first line have 
none to correspond from 
which theycau be subtracted ; 
we therefore bring down 
the counter from the space 
above and replace it by 5 counters on the line ; we shall then 
have 3 counters left on the line and none on the space ; 
we then bring down 1 counter from the second space, leaving 



_ -- 



--- 



--- 



---- 



PALPABLE ARITHMETIC. 



157 



a remainder of 4 counters on the line ; then bring down 1 
counter from the third line to the second space, and we have 1 
counter left ; and so we proceed until the subtraction is com- 
plete, and we shall have as a remainder 693. Recorde writes 
the smaller number in the first column, and commences sub- 
tracting at the upper line. 

To illustrate the process .of multiplication, let us find the 
product of 245T by 43. We express the multiplicand in the 
first column and the 
multiplier in the 
second ; multiply 
first by 3, and 
place the product 
in the third column 
and the product by 
4 in the fourth; 
add the numbers in 
these two columns, 
and the sum is the product required. 

Division may be illustrated by dividing 12832 by 608. 
Since six hundreds is contained in 12 thousands 2 tens times, 
we place two 
counters on the 
second line of the 
quotient; multiply- 
ing 6 hundreds by 
2 tens and subtract- 
ing, we have no re- 
mainder; multiply- 
ing 8 by 2 tens, we 
have 16 tens; but since 16 tens equal 1 hundred and 6 tens, 
we take off 1 from the 3 in the third or hundreds line, leaving 
2 remaining; then take off 1 of those 2 and replace it by 2 in 
the second space, and then take 1 from the second space and 1 
from the second line; then transfer the remaining counters 




























A A 

















-- 


--- 


~* 




-A 



A 






























A-* 


ft.* A . 


* 


- * 





158 THE PHILOSOPHY OF ARITHMETIC. 

to the column of the first remainder, and we have as a re- 
mainder 672. The operation is repeated, placing the quotient 
1 on the lowest line of the quotient column ; and in this case 
we merely subtract the divisor from the first remainder, obtain- 
ing 64 for the last remainder, and 21 for the quotient. This 
process may evidently be repeated to any extent ; but in prac- 
tice it was much simplified by removing the counters of the 
dividend to form the first remainder, and so on until the opera- 
tion was complete. 

Recorde mentions two different ways of representing sums 
of money by means of counters, one of which he calls the 
merchant's and the other the auditor's 

account. In the margin, 198 19s. lid. 
is expressed by the first method, the low- 

est line being pence, the second shillings, 
the third pounds, and the fourth scores of 
pounds; the spaces represent half a unit 
of the next superior line, and the detached 

counters at the left are equivalent to five counters at the right. 
The operations of addition, subtraction, etc. would be per- 
formed in a manner similar to those already given. 

The same sum would be represented by the auditor's 
account as in the margin; the first group to the right being 

pence, the second shillings, 



the next pounds, and the 

left hand group scores 
, 

of pounds; the two lower 

lines denote units of their respective classes, while in the third 
line those on the left denote one quarter and on the right one 
half of the next superior class. 

The Chinese Computing Table or Swan-Pan, previously 
mentioned, is represented by the accompanying engraving. 
It consists of a small oblong board surrounded by a frame or 
ledge, and parted downwards near the left side by a similar 
ledge. It is then divided horizontally by ten smooth and 



PALPABLE ARITHMETIC. 



169 




slender rods of bamboo, on which are strung two small balls 
of ivory or bone in the narrow compart- 
ment, and five such balls in the wider 
compartment ; each of the latter on the 
several bars denoting one, and each of the 
former expressing five. The progressive 
bars, descending after the Chinese manner 
of writing, have their values increased ten 
fold at each step. The arrangement here 
figured denotes, reckoning downwards, 
the number 5,804,712,063. The Swan- 
Pan advances to the length of ten billions, 
or a thousand times further than the 
Roman Abacus. But the most admirable feature of the in- 
strument is, that by beginning the units at any particular bar 
the decimal subdivisions of the unit may be represented. 
The Japanese make use of a similar instrument, and the 
facility with which they perform arithmetical operations is 
truly surprising. 

Several persons of eminence, during our own times, have 
advocated the revival of the practice of calculation by means 
of counters. Prof. Leslie considers this method as better cal- 
culated than any other to give a student a philosophical knowl- 
edge of the classification of numbers, and the theory of their 
notation ; and he has given, in great detail, examples of the 
representation of numbers in different scales of notation by 
counters, and of operations by means of them. 

There are other species of Palpable Arithmetic, some of 
which have been adapted especially for the use of blind 
people: the celebrated Saunderson invented an instrument 
for this purpose with which he is said to have worked arith- 
metical questions with extraordinary rapidity. Arithmetical 
instruments of this kind possess considerable interest and im- 
portance from their use in lessening the privations consequent 
upon one of the greatest human calamities. 



160 



THE PHILOSOPHY OF ARITHMETIC. 



Among other arithmetical machines for shortening the work 
of calculation or relieving the operator from any troublesome 
or difficult exercise of the memory, are Napier's virgulae, or 
rods, which were formerly much celebrated and generally used. 
The work in which they were first described was published in 
1617, under the title of Babdologia. In the dedication to 
Chancellor Seton, he says, that the great object of his life had 
been to shorten and simplify the business of calculation ; and 
the invention of logarithms, which he had just promulgated, 
was a noble proof that he had not lived in vain. These virgu- 
lae, rods, or bones, as they were ofteu called, were thin pieces 
of brass, ivory, bone, or any other substance, about two inches 
in length and a quarter of an inch in breadth, distributed into 
ten sets, generally of five each ; at the head of each of these, 
in succession, was inscribed one of the nine digits or zero, and 
underneath them in each piece the products of the digit at the 
top with each of the nine digits in succession, in a series of 
eight squares divided by diagonals, in the upper part of which 
were put the digits in the place of tens, and in the lower the 
digits in the place of units. In order to multiply any two num- 
bers together, such as 3469 and 574, those rods are to be 
placed in contact which are headed by the digits 1, 3, 4, 6, 9, 
and the successive products of the terms of the multiplier into 
the multiplicand are found by adding successively the digit on 
the upper half of the square to the right to that in the lower half 
of the square to the left, in the line of squares which are oppo- 
site to the figure of the multiplier which is used ; thus, to mul- 
tiply 3469 by 4, we take the 13469 
line of squares opposite 4, 
represented in the margin, 
and the product is 13876, 
being found by writing 6, the sum of 4 and 3, of 6 and 2, etc.. 
carrying when necessary. In case of division, those rods are 
arranged in contact which are headed by the figures of the 
divisor, and we are thus enabled to obtain the products formed 
by the divisor and successive terms of the quotient. 




PALPABLE ARITHMETIC. 



161 




16 



8 




64 



8 



In the case containing these rods, which Napier calls mul- 
tiplicationis promptuarium, there are usually found also two 
pieces with broader faces, one consisting of three longitudinal 
divisions, and the other of four ; one of which is adapted to 
the extraction of the square, and the other of the cube root ; 
in the first, one column contains the nine digits, the second 
their doubles, and the third their squares ; in the second, the 

first column contains the digits, the 

second their squares, and the third and 
fourth their cubes, two columns being 
necessary for this purpose when 
the cube consists of three places ; 
thus, the last division but one in 
each of these rods is represented 
as in the margin, the digits occupying the right-hand column. 
In our times, when the multiplication table is so much more 
perfectly learned than formerly, the eagerness with which this 
invention was welcomed will excite some surprise, considering 
that its only object was to relieve the memory of so light and 
trivial a burden; but it is in accordance with some of the pro- 
cesses elsewhere noticed, by which early authors endeavored 
to simplify arithmetical operations. 

Pascal, in 1642, at the age of 19, invented the first arith- 
metical machine, properly so called. It is said to have cost 
him such mental efforts as to have seriously affected his health, 
and even to have shortened his days. This machine was im- 
proved afterwards by other persons, but never came into prac- 
tical use. In 1673, Leibnitz published a description of a 
machine which was much superior to that of Pascal, but more 
complicated in construction and too expensive for its work, 
since it was capable of performing only addition, subtraction 
multiplication and division. But these machines are entirely 
eclipsed by those of Babbage and Scheutz. In 1821, Mr. 
Babbage, under the patronage of the British government, began 
the construction of a machine, and in 1833 a small portion of it 
was put together, and was found to perform its work with the 
11 



162 THE PHILOSOPHY OF ARITHMETIC. 

utmost precision. In 1834 he commenced to design a still 
more powerful engine, which has not yet been constructed. 
The expense of these machines is enormous, $80,000 having 
been spent on the partial construction of the first. They are 
designed for the calculation of tables or series of numbers, 
such as tables of logarithms, sines, etc. The machine pre- 
pares a stereotype plate of the table as fast as calculated, so 
that no errors of the press can occur in publishing the result 
of its labors. Many incidental benefits have arisen from 
this invention, among which the most curious and valua- 
ble was the contrivance of a scheme of mechanical notation by 
which the connection of all parts of a machine, and the precise 
action of each part, at each instant of time, may be rendered 
visible on a diagram, thus enabling the contriver of machinery 
to devise modes of economizing space and time by a proper 
arrangement of the parts of his own invention. 

A machine invented by G. and E. Scheutz, of Stockholm, 
and finished in 1853, was purchased for the Dudley Observa- 
tory, at Albany. The Swedish government paid $20,000 as a 
gratuity towards its construction. The inventors wished to 
attain the same ends as Mr. Babbage, but by simpler means. 
It can express numbers decimally or sexagesirnally, and prints 
by the side of the table the corresponding series of numbers 
or arguments for which the table is calculated. It has already 
calculated a table of the true anomaly of Mars for each -fa of a 
day. In size, it is about equal to a boudoir piano. Other 
attempts have been made, but so far nothing has been accom- 
plished which is entirely satisfactory, though the utility of 
some such engine in the calculation of astronomical and other 
tables is so great, that it is quite probable that efforts will be 
continued until complete success is attained. 



SECTION III. 

ARITHMETICAL REASONING. 



I, TUEEE is REASONING IN ARITHMETIC. 



II. NATURE OF ARITHMETICAL REASONING. 



III. REASONING IN THE FUNDAMENTAL OPERATIONS 



IV ARITHMETICAL ANALYSIS. 



V. THE EQUATION IN ARITHMETIC. 



VI. INDUCTION IN ARITHMETIC. 



CHAPTER I. 

THERE IS REASONING IN ARITHMETIC. 

ALL reasoning is a process of comparison ; it consists in 
comparing one idea or object of thought with another. 
Comparison requires a standard, and this standard is the old, 
the axiomatic, the known. To these standards we bring the 
new, the theoretic, the unknown, and compare them that we 
may understand them. The law of correct reasoning, there- 
fore, is to compare the new with the old, the theoretic with the 
axiomatic, the unknown with the known. 

This process, simple as it seems, is the real process of all 
reasoning. We pass from idea to truth, and from lower truth 
to higher truth, in the endless chain of science, by the simple 
process of comparison. Thus the facts and phenomena of the 
material world are understood, the laws of nature interpreted, 
and the principles of science evolved. Thus we pass from the 
old to the new, from the simple to the complex, from the known 
to the unknown. Thus we discover the truths and principles 
of the world of matter and mind, and construct the various 
sciences. Comparison is the science-builder ; it is the architect 
which erects the temples of truth, vast, symmetrical, and beauti- 
ful. 

In mathematics this process is, perhaps, more clearly exhib- 
ited than in any other science. In geometry, the definitions 
and axioms are the standards of comparison ; beginning in 
these, we trace our way from the simplest primary truth to the 
profoundest theorem. In arithmetic we have the same basis, 

(166) 



166 THE PHILOSOPHY OF ARITHMETIC. 

and proceed by the same laws of logical evolution. Defini- 
tions, as a description of fundamental ideas, and axioms, as the 
statement of intuitive and necessary truths, are the foundation 
upon which we rear the superstructure of the science of num- 
bers. 

These views, though admitted in respect of geometry, have 
not always been fully recognized as true of arithmetic. The 
subject, as presented in the old text-books, was simply a col- 
lection of rules for numerical operations. The pupil learned 
the rules and followed them, without any idea of the reason 
for the operation dictated. There was no thought, no deduc- 
tion from principle; the pupil plodded on, like a beast of burden 
or an unthinking machine. There was, in fact, as the subject 
was presented, no science of arithmetic. We had a science of 
geometry, pure, exact, and beautiful, as it came from the hand 
of the great masters. Beginning with primary conceptions 
and intuitive truths, the pupil could rise step by step from the 
simplest axiom to the loftiest theorem ; but when he turned his 
attention to numbers, he found no beautiful relations, no inter- 
esting logical processes, nothing but a collection of rules for 
adding, subtracting, calculating the cost of groceries, reckoning 
interest, etc. Indeed, so universal was this darkness, that the 
metaphysicians argued that there could be no reasoning in the 
science of numbers, that it is a science of intuition ; and the 
poor pupil, not possessing the requisite intuitive power, was 
obliged to plod along in doubt, darkness, and disgust. 

Thus things continued until the light of popular education 
began to spread over the land. Men of thought and genius 
began to teach the elements of arithmetic to young pupils ; 
and the necessity of presenting the processes so that children 
could see the reason for them, began to work a change in the 
science of numbers. Then came the method of arithmetical 
analysis, in that little gem of a book by Warren Colburn. It 
touched the subject as with the wand of an enchantress, and 
it began to glow with interest and beauty. What before 



THERE IS REASONING IN ARITHMETIC. 167 

was dull routine, now became animated with the spirit of 
logic, and arithmetic was enabled to take its place beside its 
sister branch, geometry, in dignity as a science, and value as 
an educational agency. 

Before entering into an explanation of the character of arith- 
metical reasoning, it may be interesting to notice the views of 
some metaphysicians who have touched upon this subject. It 
has been maintained, as already indicated, by some eminent 
logicians, that there is no reasoning in arithmetic. Mansel 
says, " There is no demonstration in pure arithmetic," and the 
same idea is held by quite a large number of metaphysicians. 
This opinion is drawn from a very superficial view of the sub- 
ject of arithmetic, a not uncommon fault of the metaphysician 
when he attempts to write upon mathematical science. The 
course of reasoning which led to this conclusion, is probably 
as follows : 

First, addition and subtraction were considered the two fun- 
damental processes of arithmetic ; all other processes were 
regarded as the outgrowth of these, and as contained in them. 
Second, there is no reasoning in addition ; that the sum of 2 
and 3 is 5, says Whewell, is seen by intuition ; hence subtrac- 
tion, which is the reverse of addition, is pure intuition also; 
and therefore the whole science, which is contained in these 
two processes, is also intuitive, and involves no reasoning. 
This inference seems plausible, and by the metaphysicians and 
many others has been considered conclusive. 

That this conclusion is not only incorrect but absurd, may 
be seen by a reference to the more difficult processes of the 
science. Surely, no one can maintain that there is no reason- 
ing in the processes of greatest common divisor, least common 
multiple, reduction and division of fractions, ratio and pro- 
portion, etc. If these are intuitive with the logicians, it ia 
very certain that they require a great deal of thinking on the 
part of the learner. These considerations are sufficient to dis- 
prove their conclusions, but do not answer their arguments; it 



163 THE PHILOSOPHY OF ARITHMETIC. 

becomes necessary, therefore, to examine the matter a little 
more closely. 

Whether the uniting of two small numbers, as three and two, 
involves a process of reasoning, is a point upon which it is 
admitted there may be some difference of opinion. The differ- 
ence of two numbers, however, may be obtained by an infer- 
ence from the results of addition, and, as such, involves a 
process of reasoning. The elementary products of the multi- 
plication table are not intuitive truths: they are, as will be 
shown in the next article, derived, as a logical inference, from 
the elementary sums of addition. The same is also true in the 
case of the elementary quotients in division. Even admitting, 
then, that there is no reasoning in addition or subtraction, it 
can clearly be shown that the derivation of the elementary 
results in multiplication and division does require a process of 
reasoning. Passing from small numbers, which may be 
treated independently of any notation, to large numbers ex- 
pressed by the Arabic system, we see that we are required to 
reduce from one form to another, as from units to tens, etc., 
which can be done only by a comparison, and also that the 
methods are based upon, and derived from such general princi- 
ples, as " the sum of two numbers is equal to the sum of all 
their parts," etc. 

The great mistake, however, in their reasoning, is in assum- 
ing that all arithmetic is included in addition and subtraction. 
If it could be proved that addition and subtraction, and the 
processes growing immediately out of them, contain no rea- 
soning, a large portion of the science remains which does not 
find its root in these primary processes. Several divisions of 
arithmetic have their origin in and grow out of comparison, 
and not out of addition or subtraction ; and since comparison 
is reasoning, the divisions of arithmetic growing out of it, it 
is natural to suppose, involve reasoning processes. Ratio, the 
comparison of numbers ; proportion, the comparison of ratios ; 
the progressions, etc., certainly present pretty good examples 



THERE IS REASONING IN ARITHMETIC. 169 

of reasoning. These belong to the department of pure arith- 
metic. A proportion is essentially numerical, as is shown 
in another place, and belongs to arithmetic rather than to 
geometry. If, in geometry, the treatment of a proportion 
involves a reasoning process, as the logicians will surely 
admit, it must certainly do so when presented in arithmetic, 
where it really belongs. It must, therefore, be admitted that 
there is reasoning in pure arithmetic. 

Again, if there is no reasoning in arithmetic there is no 
science, for science is the product of reasoning. If we admit 
that there is a science of numbers, there must be some reason 
ing in the science. And again, arithmetic and geometry are 
regarded as the two great co-ordinate branches of mathematics. 
Now it is admitted that there is reasoning in geometry, the 
science of extension ; would it not be absurd, therefore, to sup- 
pose that there is no reasoning in arithmetic, the science of 
numbers ? 

Mansel, as already quoted, says : " Pure arithmetic contains 
no demonstrations." If by this he means and I presume he 
does that pure arithmetic contains no reasoning, he is 
answered by the previous discussion. If, however, ne meam 
that arithmetic cannot be developed in the demonstrative form 
of geometry that is, by definition, axiom, proposition, and 
demonstration he is also in error. Though arithmetic has 
never been developed in this way, it can be thus developed. 
The science of number will admit of as rigid and systematic a 
treatment as the science of extension. Some parts of the sci- 
ence are even now presented thus; the principles of ratio, 
proportion, etc., are examples. I propose, at some future time, 
to give a complete development of the subject, after the manner 
of geometry. The science, thus presented, would be a valua- 
ble addition to our academic or collegiate course, as a review 
of the principles of numbers. Assuming, then, that there is 
reasoning in arithmetic, in the next chapter I shall consider 
the nature of reasoning, as employed in the fundamental opera- 
tions of arithmetic. 



CHAPTER II. 

NATURE OF ARITHMETICAL REASONING. 

IN" order to show the nature of the reasoning of arithmetic, 
a brief statement of the general nature of reasoning will be 
presented. All forms of reasoning deal with the two kinds of 
mental products, ideas and truths. An idea is a simple notion 
which may be expressed in one or more words, not forming a 
proposition ; as, bird, triangle, four, etc. A truth is the 
comparison of two or more ideas which, expressed in language, 
give a proposition ; as, a bird is an animal, a triangle is a 
polygon, four is an even number. The comparison of two 
ideas directly with each other, is called a judgment; as, a 
bird is an animal, or five is a prime number. Herejfrue is one 
idea, and a prime number is another idea. Judgments give 
rise to propositions ; a proposition is a judgment expressed in 
words. 

Nature of Reasoning. If we compare two ideas, not 
directly, but through their relation to a third, the process is 
oalled reasoning. Thus, if we compare A and B, or B and C, 
and say A equals B or B equals C, these propositions are 
judgments. But if, knowing that A equals B, and B equals 
C, we infer that A equals C, the process is reasoning. Rea- 
soning may, therefore, be defined as the process of comparing 
two ideas through their relation to a third. Judgment is a 
process of direct or immediate comparison ; reasoning is a pro- 
cess of indirect or mediate comparison. 

(170) 



NATURE OP ARITHMETICAL REASONING. 171 

In thus comparing two ideas through their relation to a 
third, it is seen that we derive one judgment from two other 
judgments; hence we may also define reasoning as the pro- 
cess of deriving one judgment from two other judgments; 
or as the process of deriving an unknown truth from two 
known truths. The two known truths are called premises, and 
the derived truth the conclusion; and the three propositions 
together constitute a syllogism. The syllogism is the simplest 
form in which a process of reasoning can be stated. Its usual 
form is as follows : A equals B ; but B equals C ; therefore A 
equals C. Here "A equals B" and "B equals C" are the pre- 
mises, and "A equals C" is the conclusion. 

The premises in reasoning are known either by intuition, by 
immediate judgment, or by a previous course of reasoning. 
In the syllogism "All men are mortal ; Socrates is a man ; 
therefore, Socrates is mortal" the first premise is derived by 
induction, and the second by judgment. In the syllogism 
"The radii of a circle are equal; R and R' are radii of a cir- 
cle ; therefore R and R' are equal " the first premise is an intu- 
ition, and the second is a judgment. In the syllogism "A 
equals B, and B equals C ; therefore A equals C" both pre- 
mises are judgments. 

It should also be remarked that truths drawn from the first 
steps of the reasoning process, do themselves become the 
basis of other truths, and these again the basis of others, and 
so on until the science is complete. This method of reasoning 
is called Discursive (discursus) ; it passes from one truth to 
another, like a moving from place to place. We start with the 
simple truths which are so evident that we cannot help seeing 
them ; and travel from truth to truth in the pathway of science, 
until we reach the loftiest conceptions and the profoundest 
principles. 

Reasoning, as we have stated, is the comparison of two 
ideas through their relation to a third; or it may be defined as 
the derivation of one judgment from two other judgments 



172 THE PHILOSOPHY OF ARITHMETIC. 

These two judgments are not always both expressed ; indeed, 
in the usual form of thought, one is usually suppressed ; but 
both are implied, and may be supplied if desired to show 
the validity of the conclusion. Every truth derived by a pro- 
cess of reasoning, may be shown to be an inference from two 
propositions which are the premises or ground of inference, 
and this is the test of the validity of the truth derived. 

There are two kinds of reasoning, inductive and deductive. 
Inductive reasoning is the process of deriving a general truth 
from several particular ones. It is based upon the principle 
that what is true of the many is true of the whole. Thus, if 
we see that heat expands many metals, we infer, by induction, 
that it will expand all metals. Deduction is the process of 
deriving a particular truth from a general one. It is based 
upon the axiom, that what is true of the whole is true of all 
the parts. Thus, if we know that heat will expand all metals, 
we infer, by deduction, that it will expand any particular 
metal, as iron. 

Mathematics is developed by the process of deductive rea- 
soning. The science of geometry begins with the presentation 
of its ideas, as stated in its definitions, and its self-evident 
truths, as stated in its axioms. From these it passes by the 
process of deduction to other truths; and then, by means of 
these in connection with the primary truths, proceeds to still 
other truths ; and thus the science is unfolded. In arithmetic, 
no such formal presentation of definitions and axioms is made, 
and the truths are not presented in the logical form, as in 
geometry. From this it has been supposed that there is no 
reasoning in arithmetic. This inference, however, is incorrect; 
the science of numbers will admit of the same logical treat- 
ment as the science of space. There are fundamental ideas 
in arithmetic as in geometry; and there are also fundamental, 
self-evident truths, from which we may proceed by reasoning 
to other truths. In this chapter I shall endeavor to show the 
nature of the reasoning in the Fundamental Operations of 
Arithmetic. 



NATURE OF ARITHMETICAL REASONING. 173 

Arithmetical Ideas. The fundamental ideas of arithmetic, 
as given in the process of counting, are the successive 
numbers one, two, three, etc. These ideas correspond to the 
different ideas of geometry, and the definitions of them will 
correspond to the definitions of geometry. In geometry, we 
have the three dimensions of extension, giving us three distinct 
classes of ideas, lines, surfaces, and volumes; in arithmetic 
there is only one fundamental idea of succession, giving us 
but one fundamental class of notions. The primary ideas of 
arithmetic are one, two, three, four, Jive, etc., which correspond 
to the idea of line, angle, triangle, quadrilateral, pentagon, 
etc., in geometry. These ideas may be defined as in the cor- 
responding cases in geometry. Thus, two may be defined as 
one and one; three as two and one, etc.; or, in the logical form 
three is a number consisting of two units and one unit. 
There are other ideas of the science growing out of relations, 
such as factor, common divisor, common multiple, etc. 

Arithmetical Axioms. The axioms of arithmetic are the 
self-evident truths that relate to numbers. There are two 
classes of axioms in arithmetic as in geometry, those which 
relate to quantity in general, that is, to numbers and space ; and 
those which belong especially to number. Thus, " Things 
that are equal to the same thing are equal to each other," and 
" If equals be added to equals the sums will be equal," etc., 
belong to both arithmetic and geometry. In geometry we 
have some axioms which do not apply to numbers, as "All 
right angles are equal," "A straight line is the shortest dis- 
tance from one point to another," etc. There are also axioms 
which are peculiar to arithmetic, and which have no place in 
geometry. Thus, "A factor of a number is a factor of a mul- 
tiple of that number," "A multiple of a number contains all 
the factors of that number," etc. These two classes of axioms 
are the foundation of the reasoning of arithmetic, as they arc 
of the science of geometry. 

Arithmetical Reasoning. The reasoning of arithmetic is 



174 THE PHILOSOPHY OF ARITHMETIC. 

deductive. The basis of our reasoning is the definitions and 
axioms; that is, the conceptions of arithmetic, and the self- 
evident truths arising from such conceptions. The definitions 
present to us the special forms of quantity upon which we 
reason ; the axioms present the laws which guide us in the 
reasoning process. The definitions give the subject-matter of 
reasoning ; the axioms give the principles which determine the 
form of reasoning, and enable us to go forward in the discovery 
of new truths. Thus, having defined an angle, and a right 
angle, we can by comparison, prove that "the sum of the 
angles formed by one straight line meeting another, is equal to 
two right angles." Having the definition of a triangle, by 
comparison we can determine its properties, and the relation 
of its parts to each other. So in arithmetic, having defined 
any two numbers, as four and six, we can determine their 
relation and properties ; or having defined least common mul- 
tiple, we can obtain the least common multiple of two or more 
numbers, guiding our operations by the self-evident and neces- 
sary principles pertaining to the subject. 

Axioms in Reasoning. In this explanation of reasoning, 
it is stated that reasoning is a process of comparing two ideas 
through their relations to a third, and that axioms are the laws 
which. guide us in comparing. This view of the nature of 
axioms differs from the one frequently presented. Some logi- 
cians tell us that axioms are general truths which contain par- 
ticular truths, and that reasoning is the process of evolving 
these particular truths from the general ones. The axioms of 
a science are thus regarded as containing the entire science; 
if one knows the axioms of geometry, he knows the general 
truths in which are wrapped up all the particular truths of the 
science. All that is necessary for him to become a profound 
geometer is to analyze these axioms and take out what is con- 
tained in them. 

The incorrectness, or at least inadequacy of this view of 
the nature of axioms and their use in reasoning, I cannot now 



NATURE OF ARITHMETICAL REASONING. 175 

Btop to consider. Its fallacy is manifest in the extent of the 
assumption. It may be very pleasant for one to suppose that 
when he has acquired the self-evident truths of a science, he 
has potentially, if not actually, in his mind the entire science; 
such an expression may do as a figure of speech, but does not, 
it seems to me, express a scientific truth A general formula 
may be truly said to contain many special truths which may 
be derived from it; thus Lagrange's formula of Mechanics 
embraces the entire doctrine of the science ; but no axiom can 
oe, in the same sense, said to contain the science of arithmetic 
or geometry. 

But whatever may be thought of this view of the nature 
and use of axioms, it cannot be denied that the explanation of 
reasoning which I have given is correct. Reasoning is the 
comparison of two ideas through their relation to a third, the 
comparison being regulated by self-evident truths. This is the 
view of Sir William Hamilton, and it has been adopted by sev- 
eral modern writers on logic. Even if the other view is right 
that the axioms may be regarded as general truths, from 
which the particular ones are evolved by reasoning their 
practical use in reasoning coincides with the explanation of the 
nature of the reasoning powers which I have presented ; and this 
idea of the subject will be found to be much more readily under- 
stood and applied. The simpler view is that the axioms are 
laws which guide us in the comparison, or they are the laws 
of inference. Thus, if I wish to compare A and B: seeing 
that they are each equal to C, I can compare them with each 
other, and determine their equality by the law that things 
which are equal to the same thing are equal to each other. 
So, if I have two equal quantities, I may increase them equally 
without changing their relation, according to the law enun- 
ciated in the axiom that if the same quantities be added to 
equals, the results will be equal. This view of the subject of 
axioms and of their use in the process of reasoning, may be 
supported by various considerations, and will be found to 



176 THE PHILOSOPHY OF ARITHMETIC. 

throw light upon several things in logic upon which writera 
are sometimes not quite clear. In the following chapter I shall 
apply this view of reasoning to the fundamental operations of 
arithmetic. 



CHAPTER III. 

REASONING IN THE FUNDAMENTAL OPERATIONS. 

QCIENCB, as already stated, consists of ideas and truths. 
O Truths are derived either by intuition or reasoning. Intu- 
itive truths come either by the intuitions of the Sense or 
the Reason ; derivative truths by the discursive process of 
induction or deduction. The primary ideas of arithmetic are 
the individual numbers, one, two, three; its primary truths are 
the elementary sums and differences of addition and subtrac- 
tion. How these primary truths are derived, is a question 
upon which opinion is divided. On the one hand it is claimed 
that they are intuitive ; on the other, that they are derived by rea- 
soning. Thus, tivo and one are three, three and two are Jive, etc., 
are regarded by some as pure axioms, neither requiring nor 
admitting of a demonstration ; while others regard them as 
deductions from the primary process of counting. Let us ex- 
amine the subject somewhat in detail, and also consider the 
process of deriving other truths growing out of these. 

Addition. It is generally assumed that the primary sums 
of the addition tables are axioms. They are intuitive truths 
growing out of an analysis of our conceptions of a number into 
its parts, or a synthesis of these parts to form the number. 
Thus, given the conception of nine, by analysis we see that it 
consists or is composed of four and ./rue; or given four and^/Jue, 
by synthesis we immediately see that it gives a combination of 
nine units, or is equal to nine. This view is maintained by some 
eminent logicians. "Why is it," says Whewell, "that three 
12 ( 177 ) 



178 THE PHILOSOPHY OF ARITHMETIC. 

and two are equal to four and one? Because if we look at 
five things of any kind we see that it is so. The five are four 
and one ; they are also three and two. The truth of our asser- 
tion is involved in our being able to conceive the number five 
at all. We perceive this truth by intuition, for we cannot see, 
or imagine we see, five things, without perceiving also that the 
assertion above stated is true." 

The other view makes counting the fundamental process, 
and derives the judgments expressed in the elementary sums 
by inference. Thus, the process of finding the sum of five 
and/bur may be stated as follows: 

The sum of five and/our is that number which is four units after five; 
By counting we find that the number four units after five is nine; 
Hence, the sum of five and four is nine. 

This is a valid syllogism, and shows that the sums might be 
thus obtained, whether they are actually so obtained or not. 
It may be objected, however, that they can be obtained only 
in one way ; and if intuitive, then it is not possible to derive 
them by any process of reasoning. This does not necessarily 
follow, for we can often obtain, by a process of reasoning, a 
truth which we could also derive in some other way. If we 
discover a new metal, it can be immediately inferred that heat 
will expand it, since heat expands all metals, which is a pro- 
cess of deductive reasoning. This truth may also be obtained 
by direct experiment. Many examples may be given to show 
that a truth may be derived by reasoning, which might also 
be derived in some other way. 

These fundamental truths may be used in obtaining the rela- 
tions of different combinations of numbers, and such an 
operation will be a process of reasoning. Thus, it is not evi- 
dent to the learner, neither is it intuitive with any one, that 7 
plus 2 equals 4 plus 5 ; or, what is less readily seen, that 25 
plus 37 equals 19 plus 43. These are not axioms, since they 
cannot be seen to be true without an examination of the 
grounds of thr> relation. The process of reasoning to prove 



REASONING IN THE FUNDAMENTAL OPERATIONS. 179 

the propositions is as follows : 7 plus 2 equals 9 ; but 4 plus 5 
equals 9 ; therefore, 7 plus 2 equals 4 plus 5 ; or, as Whewell 
puts it, thus : 7 equals 4 and 3, therefore 7 and 2 equals 4 and 
H and 2 ; and because 3 and 2 are 5, 7 and 2 equals 4 and 5. 
In the former case the result depends on the axiom, " Things 
that are equal to the same thing are equal to each other ;" in 
the latter case, the reasoning process is based upon the axiom, 
" When equals are added to equals the results are equal." It 
will be noticed that Whewell's method of proof is very similar 
to the ordinary demonstration of the theorem that "When one 
straight line meets another straight line, the sum of the two 
angles equals two right angles." 

That this is a valid process of reasoning is evident from its 
similarity to the geometrical process A plus B equals C ; but 
D plus E equals C ; therefore, A plus B equals D plus E. It 
is readily seen that many such cases will arise in which the 
operations are entirely independent of the notation employed, 
from which it cannot be doubted that there is reasoning in 
addition in pure arithmetic. When we proceed to the addition 
of large numbers, expressed by the Arabic system, which may 
not be regarded as pure arithmetic, we base the operation upon 
the axiom that the sum of several numbers is equal to the sum 
of all the parts of those numbers. That the derivation of a 
result from this general axiomatic principle is a process of rea- 
soning, cannot be doubted by any one who is competent to 
understand in what reasoning consists. 

Subtraction. Subtraction, like addition, embraces two cases, 
the finding of the difference between numbers independently of 
the notation employed to express them, that is, the elementary 
differences of the subtraction table, and the finding of the dif- 
ference between large numbers expressed in the Arabic system. 
The elementary differences in subtraction may be obtained 
in two ways. First, we may find the difference between two 
numbers by counting off from the larger number as many units 
as are contained in the smaller number. Thus, if we wish to 



180 THE PHILOSOPHY OF ARITHMETIC. 

subtract four from nine, we may begin at nine and count back- 
ward four units, and find we reach five, and thus see that 
four from nine leaves five. The other method consists in 
deriving the elementary differences by inference from the ele- 
mentary sums. The former method is regarded by some as 
intuitive, although it admits of a syllogistic statement; the 
latter method, without doubt, involves a process of reasoning. 
To illustrate, suppose we wish to find the difference between 
nine and five. The ordinary process of thought is as follows: 
Since four added to five equals nine, nine diminished by five 
equals four. This process, put in the formal manner of the 
syllogism, is as follows: 

The difference between two numbers is a number which added to 
the less will equal the greater ; 

But four added to five, the less, equals nine, the greater ; 
Therefore, four is the difference between nine and five. 

This, of course, is too formal for ordinary language, but is 
all implied in the practical form, "five from nine leaves four, 
since five and four are nine." In subtracting large numbers 
expressed by the Arabic system of notation, we proceed upon 
the principle that the difference between the parts of numbers 
equals the difference between the numbers themselves, which 
shows that the process is one of deduction. 

Multiplication. Multiplication, like addition and subtrac- 
tion, embraces two cases the finding of the elementary pro- 
ducts of the multiplication table, and the use of these in 
ascertaining the product of two numbers expressed by the 
Arabic system. The elementary products are obtained by 
deduction from the elementary sums of addition. Thus, in 
obtaining the product of three times four, the logical form of 
thought is as follows: 

Three times four are the sum of three fours; 
But the sum of three fours is twelve; 
Hence, three times four are twelve. 

The first premise is an immediate inference from the defini- 



REASONING IN THE FUNDAMENTAL OPERATIONS. 181 

tion of multiplication ; the second premise we know to be true 
from addition ; the conclusion is a deductive inference from the 
two premises. In the common form of thought we omit one 
of the premises, saying, "three times four are twelve, since 
the sum of three fours is twelve." The multiplication of large 
numbers depends on these elementary products thus derived 
by deduction, and also employs the principle, that the sum of 
the products of the parts equals the whole product. 

Division. The reasoning in division is similar to that in 
multiplication. The elementary quotients of the division table 
may be obtained in two distinct ways by subtraction or 
reverse multiplication, but in either case they are an inference 
from things already known, and are thus derived by a process 
of reasoning. By the method of subtraction we say, "four is 
contained in twelve three times, since four can be subtracted 
from twelve three times ; by the method of reverse multiplica- 
tion we say, "four is contained in twelve three times, since 
three times four are twelve." Each of these may be expressed 
in the form of a syllogism, as in multiplication. The division 
of larger numbers is based on these elementary quotients, and 
also upon the principle that the sum of the partial quotients 
equals the entire quotient. 

The view here given concerning the origin of the elementary 
products and quotients may be presented in another way. 
When we begin addition we have no idea of multiplication ; 
by and by the idea of a product arises in the mind, and it is 
immediately seen that the product of the number is the sum 
arising from taking one number as many times as there are 
units in another. Suppose then we wish to know the product 
of 3 times 4, we reason as follows : 

The product of 3 times 4 equals the sum of 4 taken 3 times ; 
ut the sum of 4 taken 3 times we find is 12; 
Hence, the product of 3 times 4 equals 12. 

Primary quotients may be obtained in a similar manner, and 
both art valid forms of reasoning. But whatever view may 



182 THE PHILOSOPHY OF ARITHMETIC. 

be taken of the origin of the elementary truths of the funda- 
mental operations and the fact of a difference of opinion indi- 
cates a reason for it it certainly cannot be denied, by one who 
will examine, that there is reasoning in the processes growing 
out of these fundamental operations, and also in those which 
have their origin in comparison. These fundamental judgments 
of the tables of the four "ground rules" are committed to 
memory, and are employed in the reasoning processes by which 
we derive other truths in the science. 

Other Forms. As we leave the fundamental operations, 
however, the processes of reasoning grow more and more dis- 
tinct. As each new idea is presented, new truths arise intui- 
tively, which become the basis for the derivation of other 
truths, the same as in geometry. To illustrate, take the sub- 
ject of Greatest Common Divisor. As soon as the idea of a 
common divisor is clearly apprehended, several truths are per- 
ceived as growing immediately out of this conception. These 
truths are intuitively apprehended, and are the axioms pertain- 
ing to the subject. From these self-evident truths, we proceed 
to other truths by a process of reasoning usually called demon- 
stration. Thus, in the subject of greatest common divisor we 
have these axioms : 

1. A divisor of a number is a divisor of any number of times that num- 
ber. 

2. A common divisor of several numbers is the product of some of the 
common factors of these numbers. 

3. The greatest common dinsor of several numbers is the product of all 
the common prime factors of these numbers. 

4. The greatest common divisor of several numbers contains no factors 
but those which are common to all the numbers. 

These truths are self-evident and necessary, and are seen 
to be so as soon as a clear idea of the subject is attained. 
They may be illustrated, but cannot be demonstrated. They 
bear precisely the same relation to the arithmetical concep- 
tion of greatest common divisor that the axioms of geometry 



REASONING IN THE FUNDAMENTAL OPERATIONS. 183 

do to some of the geometrical conceptions. Thus, in geometry, 
as soon as we have the conception of a circle, it is intuitively 
seen that all the radii are equal to each other ; or that the 
radius is equal to one-half of the diameter, etc. Such truths 
are made the basis of the reasoning by which we derive the 
other truths relating to the circle. If the process of obtaining 
these derivative truths in geometry is regarded as reasoning, 
surely the similar processes in arithmetic are also reasoning. 

Having a clear conception of the idea of greatest common 
divisor, and of the self-evident truths or axioms, belonging to 
it, we are prepared to derive other truths relating to the sub- 
ject, by the process of reasoning. As an example of a truth 
derived by demonstration, take the following: The greatest 
common divisor of two quantities is a divisor of their sum 
and their difference. 

In order to demonstrate this theorem, take any two numbers, 
as 20 and 12. We see that the greatest common divisor is 4. 
We also know that 20 is 5 times 4 and 12 is 3 times 4. We 
then reason as follows : 

The sum of the two numbers equals 5 times 4 plus 3 times 4 or 8 
times 4; 

But 4, the G. C. D., is evidently a divisor of 8 times 4 ; 

Hence, 4, the G. C. D., is a divisor of the sum of the two numbers. 

In this syllogism " 8 times 4" is the middle term, the " sum 
of the two numbers" the major term, and " 4, the greatest 
common divisor," the minor term; and the syllogism is entirely 
valid. In a similar manner we may prove that the greatest 
common divisor is a divisor of the difference of the two num- 
bers. The method of reasoning with 20 and 12 is seen to be 
applicable to any two numbers having a common divisor ; 
hence the truth is general 

It should be remarked that a large portion of the reasoning 
in arithmetic consists in changing the form of a quantity, so 
that we may see a property which was concealed in a previous 
form, and then inferring that it belongs also to the quantity in 



184 THE PHILOSOPHY OP AKITHMETIC. 

its first form, since the value of the quantity is not changed by 
changing its form. 

It is thus seen that the science of arithmetic, like geometry, 
consists of ideas and truths; that some of these truths are 
self-evident, and others are derived by a process of reasoning ; 
and that the process of reasoning in the two sciences is simi- 
lar. We proceed now to consider some of these forms of rea- 
soning, and especially the subject of arithmetical analysis, 
which will be treated in the next chapter. 



CHAPTER IV. 

ARITHMETICAL ANALYSIS. 

4 RITHMETICAL Analysis is the process of developing 
-Q- the relation and properties of numbers by a comparison 
of them through their relation to the unit. All numbers con 
sist of an aggregation of units, or are so many times the single 
thing ; and hence bear a definite relation to the unit. This 
relation the mind immediately apprehends in the conception of 
a number itself. From this evident relation to the unit, all 
numbers may be readily compared with each other, and their 
properties and relations discovered. Let us examine the pro- 
cess a little more in detail. 

Unit the Basis. The basis of this analysis is the Unit. The 
Unit is the primary and fundamental idea of arithmetic. It is 
the basis of all numbers, a number being a repetition of the 
Unit, or a collection of units of the same kind. The relation of a 
number to the Unit, or of the Unit to a number, is consequently 
immediately seen from the conception of a number itself. The 
collection is intuitively conceived to be so many times the 
Unit, or the Unit such a part of the collection. The import- 
ance of the Unit, as the base of the comparison of numbers, 
is thus apparent. Integers may be readily compared with each 
other, through their relation to the fundamental elements out 
of which they are formed. 

A Unit is one of the several things considered; and, since a 
fraction is a number of equal parts of a Unit, it is seen tbax 
we have a second class of units which we may call fractional 

(185) 






186 THE PHILOSOPHY OF ARITHMETIC. 

units. These two classes of units may be distinguished as the 
Unit and the fractional unit. A number of fractional units 
gives rise to a class of numbers called fractions. The same 
principle of comparison obtains in the comparison of these as 
in the comparison of integral numbers. A fractional unit 
being one of several equal parts of the Unit, its relation to the 
latter is simple and immediately apprehended. We can thus 
compare different fractional units by their relation to the Unit, 
as we did integral numbers by their relation to it. The com- 
parison of fractions, which at first might have seemed difficult, 
thus becomes simple and easy. 

From this consideration we are enabled to see the import- 
ance of the Unit in the process of arithmetical analysis. As 
the basis of numbers, it becomes the basis of reasoning with 
numbers. We compare number with number or fraction with 
fraction by their intermediate relation to the Unit. The Unit 
thus becomes the stepping-stone of the reasoning process, the 
central point around which the circle of logic revolves. 

Comparison of Integers. Numbers are compared, as has 
already been remarked, by their relation to the Unit. In the 
comparison of numbers, the relation between them is not imme- 
diately apprehended; but knowing the relation that each sustains 
to the Unit, we can ascertain their relation to each other by 
this simple intermediate relation. To illustrate this, suppose we 
wish to compare any two numbers, as 3 and 5 ; let the problem 
be " What is the relation of 3 to 5 ?" or "3 is what part of 5?" 
We would reason thus : One is 1 fifth of 5, and if one is 1 fifth 
of 5, 3, which is three times one, is three times 1 fifth, or 3 
fifths of 5. Hence, 3 is 3 fifths of 5. In this example we cannot 
compare 3 directly with 5; we therefore make the comparison 
indirectly, by considering their intermediate relation to the 
Unit, which is readily apprehended. Again, take the problem, 
"If 3 times a number is 12, what is 5 times the number?" 
Here, it may be remarked, 3 times the number is the known 
quantity, and 5 times the number is the unknown quantity, 



ARITHMETICAL ANALYSIS. 187 

which we wish to find by comparing it with the known quan- 
tity. How shall we make this comparison, and thus pass 
from the known to the unknown? We cannot compare them 
directly, since the relation between them is not readily per- 
ceived ; we must compare them indirectly by means of their 
relation to the Unit. The process of reasoning is as follows: 
If 3 times a number is 12, once the number is of 12 or 4; 
and if once a number is 4, five times the number is 5 times 4, 
or 20. Thus we readily pass from three times the number to 
five times the number from the known to the unknown first 
passing from three to one and then from one to five. In the 
same manner all numbers may be compared with each other, 
their relation being determined by this intermediate relation to 
One, the Unit, the basis of all numbers. 

Comparison of Fractions. Fractions are also compared by 
means of their relation to the Unit. A Fraction is a number 
of fractional units. The fractional unit is one of several 
equal parts of the Unit; hence the relation between it and the 
Unit is simple and readily perceived. When we have a num- 
ber of fractional units that is, a Fraction in comparing it 
with the Unit, we must first pass from the number of fractional 
units to the fractional unit itself, and then from the fractional 
unit to the Unit. From this we can readily pass to a num- 
ber, or to any other fractional unit, and then to any number 
of such fractional units, that is, to any fraction. This will be 
more clearly seen by its application to a problem. 

Take the problem, " If | of a number is 24, what is f of the 
Dumber?" We reason thus: If two-thirds of a number is 24, 
one-third of the number is of 24, or 12; and Mree-thirds, or 
once the number, is 3 times 12, or 36. If once the number is 
36, OHe-fourth of the number is \ of 36, or 9 ; and three-fourths 
of the number is 3 times 9, or 27. In this problem we compare 
the two fractions -f and f , by passing from two-thirds down to 
one-third, then rising up to the Unit, then passing down to one- 
fourth, and then up to Mree-fourths. In other words, we pasa 



188 



THE PHILOSOPHY OF ARITHMETIC. 



from a number of fractional units to the fractional unit, then 
to the Unit, then to another fractional unit, and then to a 
number of those fractional units. We first go down, then up, 
Jien down again, and then up again to the required point. 

Another excellent example of this method of comparison is 
given in the solution of the following problem: What is the 
relation of f to f ? Here 4 is the basis of comparison with 
which it is required to compare . This relation cannot be 
immediately seen, but it can readily be determined by the 
method of analysis. The solution is as follows: One-fifth is \ 
of f , and if one-fifth is ^ of f , ^ue-fifths, or One, is 5 times \ 
or | of f . If One is f of , one-third is of or -% of , and 
two-thirds is 2 times ^, or ; hence |- is of f. In this prob- 
lem we see the same law of comparison, and this law runs 
through the entire subject. 

Having given this general idea of the process, I will state 
the several simple cases of arithmetical analysis, and illustrate 
the process of thought by means of a diagram. The central 
relation of the Unit to the thought process, and the transition 
from the Unit and to the Unit, will be readily seen. 

CASE I. To pass from the Unit to B 

any number. Take the problem : If 1 
apple costs 3 cents, what will 4 apples 
cost? If 1 apple costs 3 cents, 4 apples, 
which are 4 times 1 apple, will cost 4 times 
3 cents, or 12 cents. In this problem the 
mind starts at the Unit A, and ascends 4 
steps to B. 

CASE II. To pass from any number to 
the Unit. Take the problem : If 4 apples 
cost 12 cents,'what will 1 apple cost? The 
solution is as follows: If 4 apples cost 12 
cents, 1 apple, which is 1 fourth of 4 apples, 
will cost 1 fourth of 12 cents, or 3 cents. 
In this problem the mind starts at the num- 



Init. 



Unit. 



ARITHMETICAL ANALYSIS. 189 

her 4, four steps above the basis, and steps down to the Unit, 
or basis of numbers. 

CASE III. To pass from a number to a number. Take the 
problem: If 3 apples cost 15 cents, what will 4 apples cost? 
The solution is : If 3 apples cost 15 cents, 1 apple will cost 
^ of 15 cents, or 5 cents, and 4 apples will cost 4 times 5 cents, 
or 20 cents. In this case we are to pass from the collection 
three to the collection four. In comparing three and four, 
their relation is not readily seen ; but knowing the relation of 
three to the Unit, and of the Unit to four, we make the transi- 
tion from three to four by passing through the Unit. This 
may be illustrated as follows: Suppose one standing at A and 
wishing to pass over to C. 
Unable to step directly from 
A to C, he first steps down to 
the starting point, B, and then 
ascends to C. So in compar 
ing numbers, when we cannot 
pass directly from the one to unit. 

the other, we go down to the 

Unit, or starting-point of numbers, and then go up to the other 
number. These relations are intuitively apprehended, being 
presented in the formation of numbers. In the given problem 
we stand three steps above the Unit, and we wish to go four 
steps above the Unit. To do this we first descend the three 
steps, and then ascend the four steps. 

CASE IV. To pass from a unit to a fraction. Take the 
problem : If one ton of hay cost $8, what will f of a ton cost ? 
The solution is as follows: If one ton of hay costs $8, one-fourth 
of a ton will cost \ of $8, or $2, and three-fourths of a ton 
will cost 3 times $2, or $6. 

In this problem we pass from the Unit to the fourth, one of 
the equal divisions of the unit, and then to a collection of such 
equal divisions. In other words, we descend from the integral 



B 



190 



THE PHILOSOPHY OF ARITHMETIC. 



Unit. 



4th. 



n 



r 



Cult. 



Unit to the fractional Unit, and then ascend among the 

fractional units. It is as 

if we were standing at A, 

and wished to pass to C ; 

we first take four steps 

down to B, and then three 

steps up to C, instead of 

trying to step immediately 

over from A to C. 

CASE V. To pass from a fraction to a unit. Take the 
problem: If f of a ton of hay cost $6, what will one ton 
cost? The solution is as follows: If tfiree-fourths of a ton of 
hay cost $6, one-fourth of a ton will cost | of $6, or $2; and 
/owr-fourths of a ton, or one ton, will cost 4 times $2, or $8. 

In this problem we pass 
from a collection of frac- 
tional units to the frac- 
tional unit, and then to 
the integral Unit. It is as 
if we were standing at A, 
and wished to pass to C. 
We cannot make the transi- 
tion directly, so we step three steps down to B, and then four 
steps up to C. 

CASE VI. To pass from a fraction to a fraction. Take 
the problem If f of a number is 15, what is 4 of the 
number? The solution is as follows: If //iree-fourths of a 
number is 15, one-fourth of the number is^of 15 or 5, and 
/owr-fourths of the number, or once the number, is 4 times 5 
or 20. If the number is 20, one-fifth of the number is 1 tifth 
of 20, or 4 ; and /bur-fifths of the number is 4 times 4 or 16. 

In this problem we wish to compare the two fractions f and 
; but since we cannot perceive the relation of them directly, 
we must compare them through their relation to the Unit. To 
do this we first go from three-fourths to one-fourth, then from 



4th 



ARITHMETICAL ANALYSIS. 191 

one-fourth to the Unit, then from the Unit to one-fifth, and 
then to four-fifths. In other words, we first go down from 
the collection of fractional units to the fractional Unit and then 
up to the integral Unit ; we then descend to the other frac- 
tional unit, and then ascend to the number of fractional units 
required. It is as if we were standing at A and wished to 
pass to E ; we cannot step directly over from one point to the 

c 

Unit. ' 

"L r 



D 

other so we pass from A three steps down to B, then four 
steps up to C, then five steps down to D, and then four steps 
up to E. 

These diagrams, it is believed, present a clear illustration of 
the subject, and enable one to understand the process of thought 
in the elementary operations of arithmetical analysis. The 
Unit is thus seen to lie at the basis of the process, the mind 
running to it and from it in the comparison of numbers. It 
will be remembered, however, that these are merely illustra- 
tions, and are not designed to convey a complete idea of the 
process in all of its details. This can only be seen by a care- 
ful analysis of the process itself. 

Analysis Syllogistic. The process of arithmetical analysis 
is a process of mediate comparison, and is consequently a 
reasoning process. This will appear from the fact that it may 
be presented in the syllogistic form. Take the simplest case: 
If 4 apples cost 12 cents, what will 5 apples cost? Expressed 
in the form of a syllogism, we have the following: 

The cost of 1 apple is \ of the cost of 4 apples; 

But \ of the cost of 4 apples is \ of 12 cents, or 3 cents ; 

Hence the cost of 1 apple is 3 cents. 



192 THE PHILOSOPHY OP ARITHMETIC. 

The cost of five apples is 5 times the cost of 1 apple; 

But, 5 times the cost of 1 apple is 5 times 3 cents, or 15 cents ; 

Hence, the cost of five apples is 15 cents. 

It is thus seen that the process of analysis is purely syllo- 
gistic, and is, consequently, a reasoning process. It is not 
usually presented in the syllogistic form, since it would be too 
stiff and formal, and moreover would be more difficult for the 
young pupil to understand. 

Direct Comparison. The comparison of numbers, so far as 
explained, is indirect and mediate, that is, through their relation 
to the Unit. After becoming familiar with this process, the 
mind begins to perceive the relations between numbers them- 
selves, and is thus enabled to reason by comparing the numbers 
directly, instead of employing their intermediate relations to 
the common basis. To illustrate, take the problem : If 3 
apples cost 10 cents, what will 6 apples cost? We may reason 
thus: If 3 apples cost 10 cents, 6 apples, which are two times 
3 apples, will cost two times 10 cents, or 20 cents. Primarily 
we would have gone to the Unit, finding the cost of one apple ; 
but now we may omit this and compare the numbers directly. 

With integral numbers this direct comparison is simple and 
easy ; but with fractions it is much more complicated and diffi- 
cult. Thus, if f of a number is 20, it is difficult to see directly 
that 4 of the number is f of 20 ; that is, that the relation of 
to | is ; hence, though we should avail ourselves of the 
direct relation of integral numbers, it will be found much sim- 
pler to compare fractions by their intermediate relations to the 
Unit. 



CHAPTER V. 

THE EQUATION IN ARITHMETIC. 

THE comparison of mathematical quantities is mainly con- 
cerned with the relations of equality. The relation of 
equality gives rise to the Equation, one of the most important 
instruments of mathematical investigation. The Equation 
lies at the basis of mathematical reasoning; it is the key with 
which we unlock its most hidden principles ; the instrument 
with which we develop its profoundest truths. The equation 
is a universal form of thought, and is not restricted to any one 
branch of mathematics. In its simple form it belongs to 
arithmetic and geometry, as well as to algebra. The simplest 
process of arithmetic, one and one are two (1-4-1=2), is really 
an equation, as much as x*+ax=b. 

In the higher departments of the subject of arithmetic, the 
equational form of thought and expression becomes indispensable. 
Much of the reasoning of arithmetic, which is not formally 
thus expressed, may be put in the form of the equation. As 
an example, take the question, " If of a number is 24, what 
is the number?" The solution of this maybe expressed as 
follows: Since of the number = 24, of the number = 12, 
and 4 of the number=36. Here, "the number" is the un- 

o 

known quantity, which is ascertained by comparing it with the 
known quantity, 24 ; and then, by the analysis, passing from 
two-thirds of the number to once the number. The illustra- 
tion given is of a very simple case, but the same principle 
13 ( 193 ) 



194 THE PHILOSOPHY OF ARITHMETIC. 

holds in the most complicated processes of arithmetical analy- 
sis. If, instead of a number, we had value, cost, weight, 
labor, etc., the method of comparison and analysis would be 
the same. We can thus see the use of the equation, the great 
instrument of analysis, even in the elementary processes of 
arithmetic. Here it begins that wondrous career which ends 
in the deepest analysis and the broadest generalization. Here 
we find the germ of that power which, in its higher develop- 
ment, comprehends the whole science of Mechanics in a single 
formula, thus holding, potentially, in its mighty grasp, the 
mathematical laws of the universe. 

The equation in arithmetic assumes several different forms. 
We begin by comparing quantities the comparison of equal 
quantities giving an equation. A comparison of unequal quan- 
tities gives us ratio, and a comparison of equal ratios gives us 
another kind of equation, an equation of relations, usually called 
a proportion. The proportion 4 : 2 : : 6 : 3, is in reality an equa- 
tion, as much so as 2=2, for it really means 4-i-2=6-7-3. The 
treatment of the equation gives rise to several special forms of 
logical procedure, such as transposition, elimination, etc. 

The equation, I have said, belongs to arithmetic ; and this 
thought I desire to impress. The equation is a formal compar- 
ison of two equal quantities. This comparison is being made 
continually; all of our reasoning involves it; we cannot think 
without it ; hence, the equation must enter into the reasoning 
of arithmetic. We compare one thing with another, the 
known with the unknown, and thus attain to new truths ; and 
all such forms of comparison involve the equation, and are 
only possible by means of it. The simplest arithmetical pro- 
cess, 1 + 1 = 2, is as much an equation as Du=6u-\-du, though 
the latter may express one of the profoundest generalizations 
to which the human mind has attained. 

Substitution. A prominent element of arithmetical reason- 
ing, accompanying the equation, is substitution. By this we 
mean the using of one quantity in place of another, to which 



THE EQUATION IN ARITHMETIC. 195 

it is equal. The object of this is that if we have an expression 
consisting of a combination of several different quantities, and 
know the relation of these quantities, we may so substitute 
their values that the expression for the combination may be 
obtained in terms of one single quantity, the value of which 
may much more readily be determined ; and then the values 
of the other quantities, from their relation to this quantity, 
may also be found. 

To illustrate, suppose we have the two conditions, twice a 
number plus three times another number equals 48, and three 
times this second number equals four times the first. We 
can readily solve this by substituting for one of these numbers 
its value in terms of the other, thus obtaining a number of 
times a single quantity, equal to the known quantity 48. The 
operation maybe exhibited thus: 

2 times the first number + 3 times the second = 48 ; 
but, 3 times the second number = 4 times the first number; 

hence, 2 times the first number -f 4 times the first number = 48 ; 
or, 6 times the first number = 48, 

and, once the first number = 8, 

and from this we may easily find the second number. 

Substitution is a form of deductive reasoning, as may be 
seen by an analysis of the process. Take the simple example, 
A-fB=24, and B=3A. We usually reason as follows: If 
A+B = 24, and B 3A, then A+3A=24, or 4A = 24, eta 
That the logical character of the process may appear, we 
should reason thus: If B=3A, A-j-B will equal A-f3A, from 
the axiom, " If equals be added to equals the sums will be 
equal." And since A+B=24, and A+B A + 3A, A+3A 
must equal 24, from the axiom, " Things that are equal to the 
same thing are equal to each other." Substitution is thus 
seen to be strictly a deductive process. In practice these log- 
ical steps are omitted for brevity and conciseness, the argument 
being sufficiently clear to be readily understood. 

Substitution is almost an essential accompaniment of the 



196 THE PHILOSOPHY OF ARITHMETIC. 

equation. The comparison of two equal quantities without 
some other truth, would often be of little value in attaining 
new truth. By substituting one value for another, we can 
often so change the equation that it expresses a relation which 
will immediately lead to some new relation of the known to 
the unknown, by which we can attain to the value of the un- 
known. Substitution has been supposed to be restricted to 
algebraic reasoning; but this is not correct. It is extensively 
employed in geometrical reasoning, and is just as appropriate 
in arithmetic as in algebra. 

Transposition. In the equational form of thought, so con- 
stantly recurring in arithmetic, it sometimes occurs that we 
have a multiple of a quantity compared with another multiple 
of the same quantity, increased or diminished by some other 
quantity. In such cases it is natural to desire to unite these 
two multiples into one, which is done by so changing them as 
to bring them on the same side of the equation. This is what 
is known as transposition. It is consequently seen that trans- 
position is a process not foreign to arithmetic, but one entirely 
legitimate and natural in the comparison of arithmetical ideas. 

Other processes of thought analogous to those which occur 
in algebra are employed in arithmetical reasoning. The mind 
here takes the first step in equational thought, which, when 
generalized, leads it to the high altitudes of mathematical sci- 
ence. Here it plumes its wings to follow the master minds in 
their lofty flights in a region of thought far beyond that of 
which the mere arithmetician could even dream. The object 
of this chapter is not to give a philosophical discussion of the 
equation in general, but to show that it has a place even in 
arithmetical reasoning, which has sometimes been doubted or 
denied. 



CHAPTER VI. 

INDUCTION IN ARITHMETIC. 

"Tl MATHEMATICS is a deductive science, and all of its 
-1YJL truths, not axiomatic, may be derived by a deductive pro- 
cess of reasoning. Is it possible, however, to obtain any of 
these truths by Induction ? This is a disputed question; it 
will therefore, it is thought, be of interest to enter somewhat 
into details in its discussion. I believe it can be shown that 
there are many truths in mathematics that can be proved by 
induction ; and, furthermore, that many of its truths were 
'originally obtained by an inductive process ; and still further, 
that induction is, in many cases, a legitimate method of math- 
ematical investigation. 

Induction, as is generally known, is a process of thought 
from particular facts and truths to general ones. It is the 
logical process of inferring a general truth from particular facts 
or truths. Thus, if I observe that heat will expand the sev- 
eral metals, iron, tin, zinc, lead, etc., I may infer, since these 
are representatives of the class of metals, that heat will ex- 
pand all metals. It is thus seen to be a process of reasoning, 
based upon the principle that what is true of the individuals 
is true of the class. The basis of Induction is the general 
proposition that what is true of the many is true of the wiiole; 
or, as Esser states it, " What belongs or does not belong to 
many things of the same kind belongs or does not belong to 
all things of the same kind." 

That this method of reasoning can be employed in arithme- 

(197) 



198 THE PHILOSOPHY OF ARITHMETIC. 

tic appears evident a priori. It is certainly not unreasonable 
to suppose that we may, upon finding a truth which holds in 
several particular cases in arithmetic, infer that it will hold 
good in all similar cases. This conclusion is strengthened by 
the fact that arithmetic is somewhat special in its nature, par- 
ticularly so as compared with algebra. Its symbols represent 
special numbers, and dealing thus with special symbols, it is 
to be expected that we would discover some truths which hold 
in particular instances, before we know of their general applica- 
tion. That it is not only possible to reason inductively in 
arithmetic, but that we do reason thus, may be shown by act- 
ual examples. 

First, take the property of the divisibility of numbers by 
nine. Suppose that, not knowing this property, I divide a 
number by 9, and then divide the sum of the digits by 9, and 
thus see that both remainders are the same. Suppose I should 
try this with several different numbers, and seeing that it holds 
good in each case, infer that it is true in all cases ; should I 
not have entire faith in my conclusion, and would not this 
inference be well founded ? This is an inductive inference, 
and is as legitimate as the inference that heat expands all 
metals, because we see that it expands the several particular 
metals, iron, zinc, tin, etc. 

Second, take a number of two digits, as 37 ; invert the 
digits, and take the difference between the two numbers, and 
we have 73 37 equal to 36, in which the sum of the two 
digits, 3 and 6, equals 9. If we take several other numbers of 
two digits and do the same, we shall find the sum of the two 
digits to be also 9 ; and observing that this is true in several 
cases, we may infer that it is true in all cases, in which we 
again have a true inductive inference. 

Third, take a proportion in arithmetic, and, by actual mul- 
tiplication, we shall see that the product of the means equals 
the product of the extremes. Examining several proportions, 
we shall see that the same is true in each case, and from these 



INDUCTION IN ARITHMETIC. 199 

we can infer that it is true in all cases, in which we again 
arrive at a general truth by induction. This is not only legit- 
imate inference, but it is actually the way in which pupils 
naturally derive the truth before they understand how to 
demonstrate it. 

Now, of course each of the above principles will admit of 
rigorous demonstration by deduction; what I hold, and what 
I think is clearly shown, is, that they can also be derived by 
induction. Deduction would prove that they must be so ; in- 
duction merely shows that they are so. Many other examples 
from arithmetic might be given in illustration of the same 
thing. But the use of induction in mathematics is not con- 
fined to arithmetic; if we go to algebra we shall find that the 
same method of reasoning may be, and indeed is, employed 
there. The theorem, x n y n is divisible by x y, may be 
proved by pure induction. Try the several cases x* y 2 , 
a; 3 y 3 , x 4 y*, etc., and seeing that the division is exact in the 
several cases, it is entirely legitimate to infer that it will be 
exact in all similar cases, or that x n y n is divisible by xy. 
The same thing may be shown in many other cases, but it is 
needless to multiply examples. Even in geometry the same 
method may be applied. I knew a young person who, before 
he studied geometry, derived by trial and induction the fact 
that there may be a series of right-angled triangles, whose 
sides are in the proportion of 3, 4, and 5 ; and there is no doubt 
that the ancients knew that the square of the hypothenuse 
equaled the sum of the squares on the other two sides, long 
before Pythagoras demonstrated it. 

I have said that some of the truths of mathematics were 
discovered by induction; among these the most prominent, 
perhaps, is Newton's Binomial Theorem. Newton discovered 
this theorem by pure induction. He left no demonstration of 
it, and yet it was considered of so much importance that it 
was engraved upon his tomb. His first principles of Calculus 
were somewhat inductive in their origin, as may be seen in 
his Principia. 



THE PHILOSOPHY OF ARITHMETIC. 

The following formula is used for finding the number of 
primes up to the number x, when a; is a large number : 



, 

A log x B ' 

in which N denotes the number of primes, and A and B are 
constants to be determined by trial. This formula was 
derived by a process of induction. It is found to satisfy the 
tables of prime numbers, but no deductive demonstration of it 
has yet been given, and it must therefore be regarded as empir- 
ical. 

In the theory of numbers we have the following remarkable 
property: Every number is the sum of one, two, or three 
triangular numbers; the sum of one, two, three, or four 
square numbers ; the sum of one, two, three, four, or five 
pentagonal numbers, and so on. This law, though known to 
be entirely general, has never been demonstrated except for 
the triangular and square numbers. It was discovered by 
Fermat, who intimates, in his notes on Diophantus, that he 
was in possession of a demonstration of it; which, however, is 
doubtful, since such mathematicians as Lagrange, Legendre, 
and Gauss have failed to demonstrate it. The general law is 
at present accepted on the basis of induction. 

It is thus clearly seen that many of the truths of mathemat- 
ics can be derived by induction; that is, by inferring general 
truths from particular cases. It is not claimed, however, that 
this changes the nature of the science. I have before said 
that mathematics is a deductive science; my object has been 
merely to show the error of those who hold that it is impos- 
sible to derive any of the truths of mathematics by induction. 

I have called especial attention to this subject, on account 
of the obscure and conflicting views which seem to exist con- 
cerning it. Several authors speak of the inductive methods of 
treating arithmetic, while others as positively assert that there 
can be no inductive treatment of the science. The logicians 
lead us to infer that induction cannot be applied to mathe- 



INDUCTION IN ARITHMETIC. 201 

matics, and not a few of them distinctly assert it. Dr. 
Whewell says, in speaking of mathematics : " These sciences 
have .... no process of proof but deduction." Prof. 
Podd wrote several pamphlets to prove that there can be no 
such thing as inductive reasoning in arithmetic ; and several of 
those whom he criticised in these articles, have acknowledged 
the correctness of his views, and consequently, their own 
mistakes. 

These views, I have already shown, are only partially true. 
Arithmetic is a deductive science ; all of its truths may prob- 
ably be derived by deduction ; but it is equally true that some 
of them may also be obtained by induction, as has been shown 
above ; and also, that some of them are accepted alone on 
induction, having never been demonstrated. 

Great care should be exercised, however, in the use of induction 
in mathematics. Several supposed truths which were derived 
by induction were subsequently found to be untrue. Fermat 
asserted that the formula, 2 m -f 1 is always a prime, when 
m is taken any term in the series 1, 2, 4, 8, 16, etc., but 
Euler found that 2 : "-f 1 is a composite number. Lagrange 
tells us that Euler found by induction the following rule for 
determining the resolvability of every equation of the form 
# 2 +At/=B, when B is a prime number: the equation must be 
possible when B shall have the form, 4An-fr 2 , or 4An+r i .A. 
This proposition holds good for a large number of cases, and 
was thought by many mathematicians to be entirely general, 
but the equation, x 1 t9y 2 =101, Lagrange proves to be an 
exception to it. 

The danger of inductive inference in mathematics is also seen 
in some of the formulas which have been presented for finding 
prime numbers. Several of these hold good for many terms, 
and were supposed to be general, but were at last found to be 
only special. Thus, the formula x l -J-&+41 holds good for forty 
values of x. The formula x' l +x+ 17 gives seventeen of its first 
values prime, and 2ar-}-29 gives twenty-nine of its first values 
prime. 



202 THE PHILOSOPHY OF ARITHMETIC. 

Having shown that mathematics, though a deductive sci- 
ence, will admit, in some instances, of an inductive treatment, 
it may be remarked that such treatment is especially adapted to 
young pupils in the elementary processes of arithmetic. It is 
difficult for them to draw conclusions from the principles estab- 
lished by a deductive demonstration ; hence, in some cases, it 
may be well for them to employ the inductive method. The 
rules for working fractions may be derived by an inductive 
inference from the solution of a particular example ; and this 
method will be much more readily understood than the deriva- 
tion of them from general principles deductively established. 
The method is to solve a particular problem by analysis, and 
then derive a general method by an inductive inference from 
such analysis. Thus analysis and induction become, as it 
were, golden keys with which we unlock the complex combina- 
tions of numbers. 

It will be well, however, to lead the pupils to the deductive 
method as soon as possible. Most students will make the 
transition naturally. The better reasoners among them will 
themselves rise from this inductive method, being satisfied 
only with a deductive demonstration; and in this they should 
be encouraged. They will often see the deductive, or necessary 
idea, behind the inductive process, and thus pass spontaneously 
from the particular fact to the general truth. They will some- 
times discover a truth by trial and inference, that is, by induc- 
tion, and then learn to demonstrate it deductively ; and it will 
be a useful exercise for pupils to have some special drill in this 
manner. They will thus see the relation of the two methods 
of reasoning, and be impressed with the deductive nature of 
the science of arithmetic, and the necessary character of its 
truths. 



PART II. 

SYNTHESIS AND ANALYSIS. 



SECTION I. 
FUNDAMENTAL OPERATIONS. 



SECTION EL 
DERIVATIVE OPERATIONS. 



SECTION I. 

FUNDAMENTAL OPERATIONS OF SYNTHESIS 
AND ANALYSIS 



I. ADDITION. 



II. SUBTRACTION. 



III. MULTIPLICATION 



IV. DIVISION. 



CHAPTER I. 

ADDITION. 

rp HE fundamental synthetic process of arithmetic is Addi- 
JL tion. Beginning at the Unit as the primary numerical 
idea, numbers arise by a process of synthesis. By it we pass 
from unity to plurality; from the one to the many. This 
mental process which gives rise to numbers, we naturally 
extend to the numbers themselves, and thus synthesis becomes 
the primary operation of arithmetic. This general synthetic 
process is called Addition. 

Definition. Addition is the process of finding the sum of 
two or more numbers. The sum of two or more numbers is 
a single number which expresses as many units as the several 
numbers added. The sum is often called the amount. 

Addition may also be defined as the process of uniting sev- 
eral numbers into one number which expresses as many units 
as the several numbers united. This last definition includes 
both of the previous ones, and avoids the use of the word sum. 
The former definition is, however, preferred on account of 
its conciseness and simplicity, and is the one usually adopted 
by arithmeticians. 

Principles. The process of addition is performed in 
accordance with certain necessary laws which are called prin- 
ciples. The most important of these are the following: 

I. Only similar numbers can be added. Thus, we cannot 
find the sum of 4 apples and 5 peaches, for if we unite the 
numbers we shall have neither 9 apples nor 9 peaches. It has 

(207) 



208 THE PHILOSOPHY OF AKITHMETIC. 

been claimed, that the sum is & apples and peaches ; in proof 
of which it is said we speak properly of " 12 knives and forks." 
meaning 6 knives and 6 forks. Such a combination is, how- 
ever, popular rather than scientific ; it is not what we mean by 
a strict use of the word addition. 

It may also be observed that dissimilar numbers may be 
brought under the same name and thus become similar, when 
they can be united in one sum. Thus, 4 sticks and 5 stones 
may be regarded as so many objects or things, and their sum 
will be 9 objects or 9 things. So in writing units and tens in 
the Arabic system ; they cannot be combined directly, but by 
reducing both to tens or both to units, the addition can be 
effected. 

II. The sum is a number similar to the numbers added. 
This is evidently an axiomatic truth. The sum of 4 cows and 
5 cows is 9 cows, and cannot be horses or sheep, or anything 
besides cows. An apparent exception which will be under- 
stood by what is said above is, that the sum of 3 horses and 
5 cows is 8 animals. 

III. The sum is the same in whatever order the numbers 
are added. This is evident from the consideration that in any 
case we have the combination of the same number of units, 
and consequently the same sum. 

Cases. Addition is divided philosophically into two gen- 
eral cases. The first case consists in finding the sums of 
numbers independently of the notation used to express them. 
The second case consists in finding the sum of numbers as 
expressed in written characters, and thus grows out of the use 
of the Arabic system of notation. The former deals with small 
numbers which can be united mentally, and may be called 
mental addition ; the latter is used with large numbers as 
expressed with written characters, and may be called written 
addition. The former is a process of pure arithmetic; the 
latter is incidental to the system of notation which may be 
employed, and is not essential to number in itself considered 



ADDITION. 209 

The former method is an independent process, complete in 
itself; the latter is dependent upon the former for the elements 
with which it works. By the former case we obtain what we 
may call the primary sums of addition, or what is generally 
known as the Addition Table, which we make use of in adding 
large numbers expressed by the Arabic method of notation. 

Treatment. The primary synthetic arithmetical process is 
that of increasing by units. This process is presented in the 
genesis of numbers where, by counting, we pass from one num- 
ber to another immediately following it, by the addition of a 
unit; and it also lies at the foundation of the method by which 
we find the sum of any two or more numbers. By it we obtain 
the elementary sums of the first case, and then we use these 
sums in solving the problems of the second case. The method 
of treating both of these cases will be presented somewhat in 
detail. 

CASE I. To find the primary sums of arithmetic. The 
primary sums of arithmetic are found by the same process of 
counting by which our ideas of numbers are generated. The 
sum of two numbers is primarily determined by beginning at 
one number and counting forward from it as many units as are 
in the number to be added to it. Thus, to find the sum of any 
two numbers, asjfrue and four, we begin o-tftve and count four 
successive numbers, six, seven, eight, nine, and seeing we 
reach nine, we know that^iue and four are nine. In this way 
we obtain the sums of all small numbers, and then commit 
them to memory, that we may know them when we wish to 
use them without passing through the steps by which they 
were obtained. 

To be assured that this is the real method, we have but to 
watch young children when adding, and we shall see that they 
do actually find the sums of numbers in the manner explained. 
They may often be seen counting their fingers, or marks on the 
slate, in performing addition. The elementary sums thus 
found are the basis of addition. We fix them in the memory 
14 



210 THE PHILOSOPHY OF ARITHMETIC. 

as we do the elementary products of the multiplication table, 
and employ them in finding the sums of larger numbers. 

These primary sums may be regarded as the axioms of 
addition. They are intuitive truths, that is, truths which can- 
not be demonstrated, but are seen by intuition. "Why is it," 
says Whewell, "that three and two are equal to four and one? 
Because if we look at five things of any kind, we see that it is 
so. The Jive are four and one; they are also three and two 
The truth of our assertion is involved in our being able to 
conceive the number five at all. We perceive this truth by 
intuition, for we cannot see, or imagine we see, five things, 
without perceiving also that the assertion above stated is 
true." 

CASE II. To add numbers expressed by the Arabic system 
of notation. The principle by which we find the sum of 
larger numbers expressed by the Arabic system, is that of 
adding by parts. Having learned the sums of small numbers, 
we separate larger numbers into parts corresponding to these 
small numbers, and then find the sum of these parts which, 
united, will give the entire sum. Thus in practice we first add 
the units group, then the tens group, and thus continue until 
all the groups are added. If the sum of any group amounts 
to more than nine units of that group, we incorporate the tens 
term of the sum with the sum of the next higher group. 

Solution Thus, in adding the two numbers 368 and 579, 
are write the numbers so that similar terms 
stand in the same column, and begin at the OPERATION. 
right to add. 9 units and 8 units are 17 units, 

or 1 ten and 7 units; we write the 7 units, and 

q A * 

add the 1 ten to the sum of the next column. 
7 tens and G tens are 13 tens, and 1 ten are 14 tens, or 1 hundred 
and 4 tens; we write the 4 tens, and add the 1 hundred to the 
next column. 5 hundreds and 3 hundreds are 8 hundreds, and 
1 hundred are 9 hundreds, which we write in hundreds place 
The entire sum is therefore 947. 



ADDITION. 211 

This method of adding by parts is the result of the beautiful 
system of Arabic notation, whereby figures in different positions 
express groups of different value. It is peculiar to this method 
of expressing numbers, and illustrates its great convenience and 
utility. In adding large numbers, it would be exceedingly dif- 
ficult, if not impossible, for the mind to unite them directly into 
one sum ; but by adding the groups separately, the process is 
simple and easy. 

Rule. One of the most common errors of arithmetic is found 
in the statement of the rules of the fundamental operations. 
This error consists in confounding the meaning of the words 
figure and number. Thus, it is usual to speak of "adding the 
figures," of "carrying the left-hand figure to the next column,'' 
etc. This is a mistake involving a looseness of thought that 
ought not to be permitted to remain in the text-books. We 
cannot add figures, we can add only the numbers which they 
express. 

This error can be avoided in several ways. The method 
here suggested is the use of the word term for figure. The 
word term is already employed in a similar manner in algebra. 
It may be used in a dual sense, embracing both the figure and 
the number expressed by the figure. Numbers and figures 
have a definite signification, and one cannot be used for the 
other without a mistake ; but it will be both correct and con- 
venient to use one word for both. No ambiguity will be occa- 
sioned by it, as the particular meaning may be determined by 
the application. In this way we may avoid the error of speak- 
ing of "adding figures," and also the inconvenient expression 
sometimes employed of "adding the numbers denoted by the 
figures." 

Why do we write the numbers as suggested, and why do 
we begin at the right hand to add, are questions very fre- 
quently asked of the arithmetician. In adding numbers we 
write them one under another, so that figures of the same 
order stand in the same vertical column, for convenience ID 



212 THE PHILOSOPHY OF ARITHMETIC. 

adding. We begin at the right hand to add as a matter of 
convenience also, so that when the sum of any column exceeds 
nine units of that column, we may unite the number denoted 
by the left hand term to the next column. We can also add by 
beginning at the left, but it will be seen on trial to be much less 
convenient. We commence at the bottom of a column to add 
as a matter of custom ; in practice it is sometimes more con- 
venient to begin at the bottom and at other times at the top. 

Were the scale any other than the decimal, the principle and 
method of adding would be the same. In addition of denom- 
inate numbers, where the scales are irregular, the same general 
principle is employed. We find the sum of a lower order of 
units, reduce this to the next higher order, etc. The difference 
in practice is that, with the decimal scale, the reduction is evi- 
dent from the notation, while in the irregular scales we must 
divide to make the reduction. The general principle of 
thought in the two cases is, however, identical. 



CHAPTER II. 

SUBTRACTION. 

rpHE fundamental analytical process of arithmetic is Sub- 
JL traction. This process arises from the reversing of the 
fundamental synthetic process. The primary operation of 
arithmetic, as previously seen, is synthesis. Every synthesis 
implies a corresponding analysis; hence, the second operation 
of arithmetic, as a logical consequence, must be the oppo- 
site of the primary synthetic process. In the former case we 
united numbers to find a sum ; here we separate numbers to 
find a difference. This general analytic process has received 
the name of Subtraction. 

Definition. Subtraction is the process of finding the differ- 
ence between two numbers. The difference between two num- 
bers is a number which added to the less will give a sum equal 
to the greater. The greater number is called the Minuend,' the 
less number is called the Subtrahend. Subtraction may also 
be defined as the process of finding how much greater one 
number is than another; or, as the process of finding a num- 
ber which, added to the smaller of two numbers, will equal 
the greater. The definition first presented is, however, pre- 
ferred. 

Gases. Subtraction is philosophically divided into two 
general cases, like addition. The first case consists in finding 
the difference between two numbers, independent of the nota- 
tion used to express them. The second case consists in find- 
ing the difference between numbers as expressed in written 

(213) 



214 THE PHILOSOPHY OF ARITHMETIC. 

characters, and thus grows out of the use of the Arabic nota 
don. The first is a case of pure arithmetic, independent of any 
notation ; the latter is incidental to the notation adopted to 
express numbers. The former deals with small numbers, and 
the process being wholly in the mind maybe called. Mental 
Subtraction; the latter is employed in subtracting large num- 
bers expressed with written characters, and may be called 
Written Subtraction. The former is an independent process 
complete in itself; the latter has its origin in the Arabic system 
of notation, and is dependent upon the former for its elementary 
differences. In the ordinary text-books, the second case is 
usually divided into two separate cases, depending upon the 
size of the terms in the minuend and subtrahend; but such 
division is designed to simplify the subject in instruction, and 
is, therefore, a practical rather than a logical division df the 
-abject. 

Principles. The operations in subtraction depend upon some 
general laws called principles. The most important of the fun- 
lamental principles of subtraction are the following: 

1. Similar numbers only can be subtracted. Thus, we can- 
jot find the difference between 9 apples and 4 peaches, for if 
we take the difference between the numbers 9 and 4, which is 
5, it will be neither 5 apples nor 5 peaches. Suppose, how- 
ever, that we have 9 apples and peaches, consisting of 5 apples 
and 4 peaches; can we then subtract 4 peaches, and will not the 
remainder be 5 apples? Or suppose we have a collection of 
knives and forks consisting of half a dozen of each, which are 
sometimes spoken of as "12 knives and forks;" can we not 
take away 6 forks and leave remaining 6 knives ? In reply, we 
remark that such a " taking away" is not what we mean by 
subtraction, which is defined as the process of finding the 
difference of two numbers. 

It is also manifest, as in addition, that if we regard the dis- 
similar numbers as having the same generic name, they 
will then become similar and we can subtract them. Thus, 9 



SUBTRACTION. 216 

apples and 4 peaches may be regarded as 9 objects and 4 
objects, the difference of which is 5 objects. So in subtracting 
the different orders of units in the Arabic scale, we cannot sub- 
tract them directly as different orders, but by reducing them 
to the same denomination, the subtraction is readily performed. 

2. The difference is a number similar to the minuend and 
subtrahend. This is a necessary truth intuitively apprehended. 
Thus 4 men subtracted from d men, leaves 5 men, and not 
5 girls, or 5 women. If we have a group consisting of 9 
persons, 5 men and 4 women, and take away 4 women, there 
will remain 5 men ; hence we might infer that 4 women taken 
from 9 persons leaves 5 men ; but this is not a universal truth; 
neither, as stated above, is such a taking away, what we mean 
by subtraction. 

3. If the minuend and subtrahend be equally increased 
or diminished, the remainder will be the same. This is in- 
cluded in the axiom that the difference between two numbers 
equals the difference between them when equally increased or 
diminished. The truth of such a proposition is seen to be 
necessary as soon as the proposition is clearly apprehended by 
the mind. 

4. The minuend equals the sum of the subtrahend and 
remainder; the subtrahend equals the difference between the 
minuend and remainder. These two principles flow from the 
conception of subtraction, and the relation of the several terms 
to one another. Given a clear idea of the process of subtraction, 
and the relation of the three terms in the process, and these 
truths immediately follow. 

Method. The two cases of subtraction, as of addition, 
require distinct methods of treatment. In the former case wo 
subtract directly as wholes, finding the difference by reversing 
the process of addition. In the latter case we subtract by 
parts, using the elementary differences to find the differences 
of the corresponding parts. An explanation of both cases will 
be presented. 



216 THE PHILOSOPHY OF ARITHMETIC. 

CASE I. To find the primary differences in arithmetic. 
The elementary differences are obtained by a reversion of 
the process of finding the elementary sums. This may be 
done in two distinct ways. First, we may find the difference 
between two numbers by counting off from the larger number 
as many units as are contained in the smaller number. Thus, 
if we wish to subtract four from nine, we may begin at nine 
and count backward four units: thus, eight, seven, six, five ; 
and finding that we reach five, we know that four from nine 
leaves five. This is the reverse of the process by which we 
obtained the elementary sums in addition. In one case we 
count on for the sum ; in the other we count off for the differ- 
ence. 

The other method consists in finding the elementary differ- 
ences by deriving them by inference from the elementary 
sums. Thus, in finding the difference between five and nine, 
we may proceed as follows: since four added to five equals 
nine, nine diminished by five, equals four. This process, put 
in a formal manner, is as follows : The difference between two 
numbers is a number which, added to the less, will equal the 
greater; but, four added to five, the less, equals nine, the 
greater ; hence, four is the difference between nine and five. 
In other words, we know that five from nine leaves four, 
because four added to five equals nine. 

The difference between these two methods is radical. By 
the former method we derive the difference by direct intuition, 
as we obtained the sums in addition. We see that the differ- 
ence is five. By the second method we infer that the differ- 
ence \sfive, without directly seeing it. The latter is a process 
of reasoning, and will admit of being reduced to the form of 
a syllogism, as is shown above. The point made here is an 
important one, and will throw some light on the nature of 
the science of arithmetic, which, by the metaphysicians, has 
been somewhat imperfectly understood. 

The second method is preferred in practice to the first, as we 



SUBTRACTION. 217 

can make use of the elementary sums in finding the elementary 
differences. If the first method is used, it will be necessary to 
commit the elementary differences as well as the elementary 
sums. By making the differences depend upon the sums, this 
labor will be avoided. 

CASE II. To subtract numbers expressed by the Arabic 
scale of notation. With large numbers we cannot subtract the 
one directly from the other as with small numbers ; we there- 
fore divide the labor, subtracting by parts; that is, we find 
the difference between the corresponding groups of each term. 
By this means the labor of subtracting is greatly facilitated, so 
that with large numbers, which it would be almost, if not 
quite impossible otherwise to subtract, the operation becomes 
simple and easy. 

In the subtraction of numbers expressed in the Arabic scale 
of notation, two distinct cases arise; first, when the number 
of each group of the subtrahend does not exceed the correspond- 
ing number of the minuend ; second, when the number of a 
group in the subtrahend exceeds the corresponding number in 
the minuend. In the first case we readily subtract each group 
in the subtrahend from the corresponding group in the minu- 
end. In the second case a difficulty arises, for which we have 
two distinct methods of explanation, called respectively the 
Method by Borrowing, and the Method by Adding Ten. 

To illustrate these methods, suppose it be required to sub 
tract 526 from 874. 

First Method. Having the numbers writ- OPERATION. 
ten as in the margin, we commence at tho 
right to subtract, and reason thus: we cannot 
take 6 units from 4 units, we will therefore 
take 1 ten from the 7 tens, and add it to the four units, which 
will give 14 units. We then subtract 6 units from 14 units, 
which gives 8 units. We then subtract 2 tens from the 6 tens 
which remain after taking away the 1 ten, which leaves 4 
tens. We also subtract 5 hundreds from 8 hundreds, leaving 
3 hundreds; hence the difference is 348. 



218 



THE PHILOSOPHY OF ARITHMETIC. 



Second Method. By the second method we reason thus: 
"We cannot subtract 6 units from 4 units, hence we add 10 to 
the 4, making 14 units, and then say, G units from 14 units 
leave 8 units. Now, since \ve have added 10 to the minuend, 
that the remainder may be correct we must add one ten to the 
subtrahend ; hence we have 3 tens from 7 tens leave 4 tens, 
and also as before, 5 hundreds from 8 hundreds, 3 hundreds 
This solution is founded upon the principle that the difference 
between two numbers equals the difference between the two 
numbers equally increased. 

The first method seems preferable on account of its simpli- 
city of thought, as it merely changes the form of the minuend. 
Pupils see the reason of the process by this method more 
readily than by the method of adding ten. The second method, 
however, is preferred by some teachers for at least two reasons. 
First, it is the method generally used in practice; nearly all 
persons increasing the next lower term after " borrowing," 
instead of diminishing the upper one. Second, it is, in many 
cases which arise, much more convenient than the other 
method, as in subtracting 12345 from 20000. By the second 
method, the solution of this problem will be much simpler 
than by the first. 

Another Method. There is still another method of subtract- 
ing, which, if not of any practical value, is at least 
of sufficient interest to be worthy of mention. It 74682 

*7 Q C *\ 

consists in subtracting the terms of the subtrahend 
from 10, and adding the difference to the corrcs- 46817 
ponding terms of the minuend. Thus, in subtracting 27865 
from 74682, we say 5 from 10 leaves 5, and 2 are 7; G and 1 to 
carry arc 7, and 7 from 10 leaves 3, and 8 arc 11 ; set down the 
1; 8 from 10 leaves 2, and G arc 8; 7 and 1 to carry arc 8, and 
8 from 10 leaves 2, and 4 arc 6, etc. 

Rule. In the rule for subtraction, arithmeticians make the 
same mistake as in the rule for addition. Thus, they say, 
" Subtract each figure of the subtrahend from the figure above 




SUBTRACTION. 219 

it in the minuend," or " take each figure of the subtrahend 
from the figure above it," or, "if a figure in the lower number 
is larger than the one above it," etc. These errors are almost 
inexcusable. We cannot subtract figures, we subtract num- 
bers. If we "take one figure from another" the other figure 
will be left, not the difference of the numbers expressed by 
them. A figure is larger or smaller according to the kind of 
type in which it is printed. The figure two may be large (2) 
or small (2). One figure may be larger than another, and 
express a smaller number; as, 3 and 8. 

This error may be avoided by the use of the word term for 
the number expressed by the figure. The rule will then read, 
"Begin at the right and take each term of the subtrahend from 
the corresponding term of the minuend," etc. "If a term of 
the subtrahend is greater than the corresponding term of the 
minuend," etc. 

Remarks. We write terms of the same order in the same 
vertical column for convenience in subtracting, since only num- 
bers of the same group can be subtracted. We commence at 
the right, so that when a term of the subtrahend expresses 
more units than the corresponding term of the minuend, we 
may take it from the next higher group of the minuend ; or, if 
we use the other method of subtracting, that we may add 10 
of a group to the minuend, and 1 of the next higher group to 
the subtrahend; in other words, we commence at the right as 
a matter of convenience, as will be seen in the attempt to sub- 
tract by commencing at the left. 

The taking one from the next term of the minuend is called 
"borrowing," and the adding one to the next term of the sub- 
trahend is called "carrying." The accuracy of these words 
has been questioned. To borrow is to obtain that which we 
expect to return to the one from whom we borrow. It does 
not seem much like " borrowing" to take from one thing and 
return what we take to another. It is something like "robbing 
Peter to pay Paul." In regard to the term "carrying," it 



220 THE PHILOSOPHY OF ARITHMETIC. 

may be asked in what it is carried ; though we may answer, 
as the boy did, " we carry in the head." Notwithstanding 
these objections, the terms borrowing and carrying have been 
sanctioned by good usage ; and, since custom is the lawgiver 
in language, we may accept them as correct. Their use is a 
matter of convenience, also, as they indicate operations for 
which we have no other technical terms. It may be remarked 
that it required many years for the people of Europe to become 
familiar with the processes of borrowing and carrying. In a 
work on arithmetic by Bernard Lamy, published at Amster- 
dam in 1692, the author states that a friend sends him the 
mode of using the carriage in subtraction, he having previ- 
ously borrowed from the upper line ; and this is presented as 
a novelty. 



CHAPTER III. 

MULTIPLICATION. 

THE general process of synthesis is Addition. Having 
become familiar with this general synthetic process in ac- 
cordance with the law of thought, from the universal to the 
particular, we begin to impose certain conditions upon it. 
The numbers primarily united were of any relative value; if, 
now, we impose the condition that the numbers united shall be 
all equal, with the new idea of the times the number is used, 
we have a new process of synthesis, which we call Multiplica- 
tion. 

Multiplication is thus seen to be a special case of addition, 
in which the numbers added are all equal. The idea of mul- 
tiplication is contained in addition, and is an outgrowth of it. 
They are both synthetic processes one being a general, and 
the other a more special synthesis. Multiplication, however, 
involves the idea of " times," which does not appear in addi- 
tion. This notion of "times," originating in multiplication, is 
one of the most important in mathematics, and is itself the 
source of a large portion of the science. Thus, in involution 
there is no apparent trace of the idea of addition, and the same 
is true in respect of other processes. If, however, we follow 
these processes back far enough, we shall find they have their 
origin in the primary process of addition. Even involution 
may be performed by successive additions. 

Definition. Multiplication is the process of finding the 
product of two numbers. The Product of two numbers is 

(221) 



222 THE PHILOSOPHY OF ARITHMETIC. 

the result obtained by taking one number as many times as 
there are units in the other. The number multiplied is called 
the Multiplicand. The number by which we multiply is called 
the Multiplier. 

This definition of multiplication, introducing the word Pro- 
duct, makes it similar to the definitions of addition and sub- 
traction, in which the terms sum and difference are used. 
Defining Division in a similar manner by using the word Quo- 
tient, we shall have a harmony in the definitions of the four 
fundamental rules, which has not hitherto existed. I have 
adopted this method in my Higher Arithmetic, and shall intro- 
duce it into my other mathematical works. 

Multiplication is usually defined as the process of taking one 
number as many times as there are units in another. This 
definition is not entirely satisfactory. It says nothing about 
finding a result, which is specified in the definitions of addition 
and subtraction, and which seems to be necessary also here. 
To supply this omission, I have previously defined multiplica- 
tion as the process of finding the result of taking one number 
as many times as there are units in another. After a very 
careful consideration of the subject, however, I have concluded 
to adopt the method of defining multiplication as the process of 
finding the product, thus securing a uniformity in the defini- 
tions of the fundamental operations. 

Principles. The operations of multiplication are founded 
upon certain necessary truths called principles. The most 
important of the principles of multiplication are those which 
follow : 

1. The multiplier is always an abstract number. For, the 
multiplier shows the number of times the multiplicand is 
taken, and hence must be abstract, since we cannot take any- 
thing yards times or bushels times, etc. From this it follows 
that such problems as "Multiply 25 cts. by 25 cts.," or "2s. 6d. 
by itself" are impossible and absurd. In finding areas and 
volumes, we speak of multiplying feet by feet for square feet, 



MULTIPLICATION. 223 

square feet by feet for cubic feet, etc. It should be remem- 
bered, however, that this is merely a convenient expression, 
which does not indicate the actual process. In finding the 
area of a rectangle, we multiply the number of square feet on 
the base by the number of such rows; the multiplicand being 
square feet and the multiplier an abstract number. 

2. The product is always similar to the multiplicand. This 
is manifest from the fact that the product is merely the sum of 
the multiplicand used as many times as there are units in the 
multiplier. Thus, 3 times 4 apples are 12 apples, and cannot 
be 12 pears or peaches. 

3. The product of two numbers in the same, whichever is 
made the multiplier. This may be seen by placing # * # * 

3 rows of 4 stars each in the form of a rectangle, * . * * 
as in the margin. Now these may be regarded * # * * 
as 3 rows of 4 stars each, or 4 rows of 3 stars 
each ; hence 3 times 4 is the same as 4 times 3 ; and the same 
may be shown for any other two numbers. 

4. If the multiplicand be multiplied by all the parts of the 
multiplier, the sum of all the partial products will be the true 
product. This grows out of the general principle that the 
whole is equal to the combination of all of its parts. It is 
applied in finding the product of two numbers expressed by 
the Arabic system. 

5. The multiplicand equals the quotient of the product 
divided by the multiplier ; the multiplier equals the quotient of 
the product divided by the multiplicand. These two principles 
are manifest to the mind as soon as it attains a clear idea of 
the processes of multiplication and division, and the relation 
of the two to each other. 

Cases. Multiplication is philosophically divided into two 
general cases. The first case consists in finding the products 
of numbers independently of the method of notation used to 
express them. The second case is that which grows out of the 
use of the Arabic system of notation. The former deals with 



224 THE PHILOSOPHY OF ARITHMETIC. 

small numbers mentally, and may be called Mental Multiplier 
tion; the latter deals with large numbers, expressed by means 
of written characters, and may be called Written Multiplica- 
tion. The former is an independent process complete in itself, 
and belongs to pure number; the latter has its origin in the 
Arabic system, and is dependent upon the former for its ele- 
mentary products. 

Method. The general method is to find the product of small 
numbers by addition, and then use these in the multiplication 
of large numbers. The first case is thus made to depend upon 
addition, and the second case upon the first case. Both cases 
will be formally presented. 

CASE I. To find the elementary products of arithmetic. 
The first object in multiplication is to find the elementary pro- 
ducts. By the elementary products are meant the products of 
small numbers which, arranged together, constitute what is 
called the Multiplication Table. These elementary products 
are derived by addition. Thus, we ascertain that four times 
five are twenty, by finding, by actual addition, that the sum 
of four fives is twenty. In this manner all the elementary 
products of the table were originally obtained. This table is 
committed to memory in order to save labor and facilitate the 
process of calculation. We are thus able to tell immediately 
the product of two small numbers, which otherwise we should 
be obliged to obtain by an actual addition. 

The elementary products are not derived by intuition, and 
are therefore not axioms; they are the result of a process of 
reasoning. Thus, in order to find the product of three times 
four, we may reason as follows: Three times four is equal 
to the sum of three fours; but the sum of three fours, we 
find by addition, is twelve ; hence, three times four is twelve. 
This is as valid a syllogism as "A is equal to B ; but B is 
equal to C ; hence, A is equal to C." 

The extent of the table, for all practical purposes, is limited 
by "nine times nine." That is, with our Arabic system of 



MULTIPLICATION. 225 

notation and the decimal method of numeration, it is not neces- 
sary that the elementary products should extend beyond " nine 
times." It is not at all inconvenient, however, but quite nat- 
ural that it should include eleven and twelve times, since the 
names eleven and twelve are a seeming departure from the dec- 
imal system of numeration. 

CASE II. To multiply numbers expressed by the Arabic 
system of notation. When the numbers are small, as we have 
seen, we multiply them directly as wholes; when we extend 
beyond the elementary products, the principle is to multiply 
by parts. Thus, instead of multiplying the multiplicand as a 
single number, we multiply first one group, then the next group, 
and so on, as we united numbers in addition. Also, when the 
multiplier exceeds nine or in practice, twelve that is, when 
it is expressed in two or more places, we multiply first by the 
units term, then by the tens term, etc.; and then take the sum 
of these partial products. 

To illustrate, let it be required to multiply 65 by 37. To 
multiply by thirty-seven as a single number, would be quite a 
difficult task. We do not attempt this, however, but first mul- 
tiply by 7 units, one part of 37, and then by 3 tens, the other 
part of 37, and then take the sum of these products. It is also 
seen that the number 65 is not multiplied as a single number, 
but by using its parts, 5 units and 6 tens. The method of 
explaining the process is as follows: 

Solution. Thirty-seven times 65 equals 7 OPERATION. 
times 65 plus 3 tens times 65. Seven times 5 65 

units are 35 units, or 3 tens and 5 units ; we 37 

write the 5 units, and reserve the 3 tens to add 455 

to the product of tens. Seven times 6 tens 
are 42 tens, which, increased by 3 tens, equals 
45 tens, or 5 tens and 4 hundreds, which we write in ita 
proper place. Multiplying similarly by 3 tens, we have 5 tens 
9 hundreds and 1 thousand; and taking the sum of these two 
partial products, we have 2405. 
15 



226 THE PHILOSOPHY OF ARITHMETIC. 

This method of multiplication is founded upon, and is only 
possible with a system of notation similar to the Arabic. 
Without some such method of expressing numbers in char- 
acters, the multiplication of large numbers would be exceed- 
ingly laborious, if not altogether impossible. We are thus 
continually reminded of the advantages of the Arabic system 
of notation, and learn almost to venerate the people and 
country that conferred so great a boon upon the human race 
by its invention. 

Rule. The error of confounding the meaning of figure and 
number is repeated in the rule for multiplication. The rule, as 
usually given is, "Multiply each figure of the multiplicand by 
the multiplier," etc., or "Multiply the multiplicand by each figure 
of the multiplier," etc. This error is easily avoided by the use of 
the word term for figure. It should be remembered that we have 
two distinct things, the number and the numerical expression. 
The parts of the numerical expression are figures ; the parts of the 
entire number are numbers. The word term may be employed 
to express both of these, without any obscurity and with much 
convenience. The rule will then read, " Multiply each term of 
the multiplicand by the multiplier," etc., or, "by each term of 
the multiplier," etc. 

Remark. We write the numbers as indicated above for con- 
venience in multiplying. The placing of the multiplier under 
the multiplicand, instead of over it, and multiplying from 
below, is a mere matter of custom, corresponding with the 
method of adding and subtracting. We begin at the right 
hand to multiply so that when any product exceeds nine, we 
may incorporate the number expressed by the left hand figure 
with the following product. The convenience of this will be 
readily appreciated by performing the multiplication by begin- 
ning at the left. It was formerly the custom, however, to 
begin at the left, writing the partial products in their order and 
subsequently Collecting them. 



CHAPTER IV 

DIVISION. 

general process of analysis is Subtraction. After the 
JL mind becomes familiar with this general process, it begins 
to extend and specialize it, and thus arises a new process called 
Division. Division is, therefore, a special case of subtraction, 
in which the same number is to be successively subtracted with 
the object of finding how many times it is contained. The idea 
of Division is thus seen to be contained in that of Subtraction, 
and is the outgrowth of it. 

Division may also be regarded as arising from a reversing 
of the process of multiplication. In multiplication, we obtain 
the product of two numbers ; and since the product is a number 
of times the multiplicand, we may regard it as containing 
the multiplicand a number of times. Thus, since four times 
five are twenty, twenty may be considered as containing 
five, four times. Division is thus regarded as an analytic 
process, arising from reversing the synthetic process of multi- 
plication. 

It thus appears that Division may have originated in either 
of two different ways. In which way it did actually arise, it 
is impossible for us to decide with certainty. It has generally 
been supposed, judging from the old definition that " Division 
is a concise method of Subtraction," that it had its genesis in 
Subtraction. My own opinion, however, is that it originated 
by reversing multiplication, for which I state the following 
reasons : First, as subtraction arose from reversing the pro- 

(227) 



228 THE PHILOSOPHY OP ARITHMETIC. 

cess of addition, so is it natural to suppose that division, a 
concise subtraction, would arise from reversing multiplica- 
tion, a concise addition. Second, division involves, as essen- 
tial to it, the idea of "times," which had already appeared 
in multiplication. It seems much more natural to take the 
idea of times from multiplication, where it already existed, than 
to originate it from the process of subtraction. 

Definition. Division is the process of finding the quotient 
of two numbers. The quotient of two numbers is the number 
of times that one number contains the other. The number 
divided is the Dividend ; the number we divide by is the 
Divisor. The definition usually given is, "Division is the 
process of finding how many times one number is contained in 
another." This is regarded as correct, but is less simple and 
concise than the one above suggested. 

Defining division in this manner, we have a simple and con- 
cise definition, easily understood and logically accurate. It 
follows the method generally adopted for addition and sub- 
traction, and which I have also suggested for multiplication ; 
and presents a happy uniformity in the definitions of the four 
fundamental operations of arithmetic. The objects of these 
four fundamental processes, as thus presented, will respectively 
be to find the Sum, the Difference, the Product, and the Quo- 
tient of numbers. 

Principles. The operations in division are controlled by 
certain necessary laws of thought to which we give the name 
of principles. The following are the most important of the 
principles of division: 

1. The dividend and divisor are always similar numbers. 
This is true of division scientifically considered, as may be 
seen by regarding it as originating in subtraction or multipli- 
cation. Supposing that it has its root in subtraction, and 
remembering that in subtraction the two terms must be alike, 
we see that this principle follows of necessity. Thus, if we 
inquire how many times one number is contained in another, 



DIVISION. 229 

it is evident that these numbers must be similar. We may inquire 
how many times 4 apples are contained in 8 apples, but not 
how many times 4 peaches are contained in 8 apples. Neither 
can we say "How many times is 4 contained in 8 apples?" for 8 
apples will not contain the abstract number 4 any number of 
times. The same conclusion is reached if we regard division 
as originating in multiplication. If we assume that 4 is con- 
tained in 8 apples 2 apples times, it would follow that 2 apples 
times 4 equals 8 apples, which is absurd. 

Several recent writers take the position that a concrete number 
may be divided by an abstract number, because in practice we 
hus divide a concrete number into equal parts. This is a 
(subordination of science to practice, which is neither philo- 
sophical nor necessary. The practical case which they thus 
try to include in the theory of the subject, admits of a scientific 
and simple explanation, without any modification of the funda- 
mental idea of division ; and when thus explained it becomes 
apparent that the two terms are similar numbers. 

2. The quotient is always an abstract number. This results 
from the fundamental idea of division, whether we regard it as 
originating in subtraction or multiplication. The quotient shows 
how many times one number is contained in another, and one 
number cannot be contained in another number yards times, or 
apples times, etc., from which it follows that the quotient 
must be abstract. The quotient shows how many times one 
number may be subtracted from or taken out of another before 
exhausting the latter, and must therefore be a number of times, 
and consequently abstract. Or, regarding it as arising from 
multiplication, the quotient is the number of times the divisor 
which equals the dividend ; and, as such, is a multiplier,- and 
must, consequently, be abstract. Suppose it were said that 2 
is contained in 8 apples, "4 apples times," and all authors 
agree as to the quotient denoting the number of times the 
divisor is contained in the dividend then it would follow that 
"4 apples times" 2 are 8 apples; which is, of course, absurd. 



THE PHILOSOPHY OF ARITHMETIC. 

3. The remainder is always similar to the dividend. This 
is evident, since the remainder is an undivided part of the divi- 
dend. In practice, as above intimated, some of these princi- 
ples seem to be violated, but if the analysis be given, it will be 
seen that the violation is merely seeming, and not actual. 

4. The following principles show the relation of the terms 
in division : 

1. The dividend equals the product of the divisor and quo- 
tient. 

2. The divisor equals the quotient of the dividend and 
quotient. 

3. The dividend equals the product of the divisor and quo- 
tient, plus the remainder. 

4. The divisor equals the dividend minus the remainder, 
divided by the quotient. 

5. The following principles show the result of multiplying 
or dividing the terms in division: 

1. Multiplying the dividend or dividing the divisor by any 
number multiplies the quotient by that number. 

2. Dividing the dividend or multiplying the divisor by any 
number divides the quotient by that number. 

3. Multiplying or dividing both divisor and dividend by the 
same number does not change the quotient. 

Cases. Division is philosophically divided into two general 
cases. The first case consists in finding the quotient of num- 
bers independently of the method of notation used to express 
them. The second case is that which grows out of the use of 
the Arabic system of notation. The former case deals with 
small numbers mentally, and may be called Mental Division ; 
the latter deals with large numbers, expressed by means of 
written characters, and may be called Written Division. 
The former is an independent process, belonging to pure num- 
ber, and is complete in itself; the latter operates by means of 
the Arabic characters, and is dependent upon the former for its 
elementary quotients. 



DIVISION. 231 

Method. In division we first find the elementary quotients 
corresponding to the elementary products of the multiplicatioji 
table. These may be obtained in two different ways, as will 
be explained. In the second case we operate by parts, using 
the elementary quotients as a basis of operation. The two 
cases will be formally presented. 

CASE I. To find the elementary quotients of arithmetic. 
The first object in division is to find the elementary quotients 
corresponding to the elementary products of the multiplication 
table. These quotients admit of a double origin ; that is, they 
may be derived by the method of concise subtraction, or of 
reverse multiplication. Thus, if we wish to ascertain how 
many times Jive is contained in twenty, we may find how many 
times five can be taken out of twenty by subtraction, and this 
will show how many times twenty contains five. This is the 
method of subtraction, and as thus viewed, division may be 
regarded as a method of concise subtraction. Again, since we 
know that four times five are twenty, we can immediately 
infer that twenty contains four fives, or that twenty contains 
five four times. This is the method of multiplication, and as 
thus viewed, division may be regarded as a method of reverse 
multiplication. 

Either of these two methods may be used for finding the 
elementary quotients, but the method of reverse multiplication 
is much more convenient in practice. The quotients are imme- 
diately derived from the products of the multiplication table, 
and we are thus saved the labor of forming and committing a 
table of division. If, however, the elementary quotients be 
derived by subtraction, it will be necessary to construct a 
division table, and commit the quotients, as we do the products 
in multiplication. 

These elementary quotients, whether derived by multiplica- 
tion or subtraction, are the result of a process of reasoning. 
The process of thought may be illustrated in the problem, "Fire, 
is contained how many times in twenty?' 1 and is as follows: 



232 THE PHILOSOPHY OF ARITHMETIC. 

Five is contained as many times in twenty as twenty is times 
five ; but twenty is four times five ; hence, five is contained in 
twenty, four times. In ordinary language, this is abbreviated 
thus : five is contained four times in twenty, since four times 
five are twenty. 

By the method of subtraction we reason thus : five is con- 
tained as many times in twenty as five can be successively sub- 
tracted from or taken out of twenty ; but five can be suc- 
cessively subtracted from twenty, four times; hence, five is 
contained/bur times in twenty. The ordinary form of thought 
is, five is contained four times in twenty, since it can be sub- 
tracted from twenty, four times. By "subtracted from," as here 
used, we mean subtracted successively from until twenty is 
exhausted. 

CASE II. To divide when the numbers are expressed in 
the Arabic scale of notation. When the numbers are small, 
we divide them, as we have seen, directly as wholes ; when we 
extend beyond the elementary quotients, the principle is to 
divide by parts. The dividend is not immediately divided as 
a whole, but is regarded as consisting of parts or groups; and 
these are so divided that, when remainders occur, they may be 
incorporated with inferior groups, and thus the whole number 
be divided. This method, as in multiplication, is due to the 
system of Arabic notation, and enables us to divide large num- 
bers, which would be exceedingly difficult, if not impossible, 
with a different system of notation. 

In. Written Division, or division of large numbers, two 
cases are presented. First, when the divisor is so small that 
only the elementary dividends and divisors are used ; second, 
when the divisors and dividends are larger than those employed 
in obtaining the elementary quotients. The methods of treat- 
ing these two cases are distinguished as Short Division and 
Long Division. In Short Division, the partial dividends are 
not written ; in Long Division, the partial dividends and other 
necessary work are written. 



DIVISION. 233 

Illustration. To illustrate the method of Short Division, 
divide 537 by 3. Here we cannot divide the given number as 
a whole, that is, as Jive hundred and thirty-seven, but by sep- 
arating it into parts, we can readily divide these parts, as they 
give only the elementary quotients. Thus, we first divide 
Jive hundred, reduce the remainder of the group to tens and 
incorporate with the tens group, making 23 tens, divide this as 
before, and thus continue until the whole of the number hafa 
been divided. 

When the divisor is greater than 12, the division can no 
longer be performed by using the elementary dividends and 
quotients. The process then becomes more difficult, although 
it involves the same principles as when smaller numbers are 
used. As the elementary quotients were derived from multipli- 
cation, so in Long Division we determine the quotient by mul- 
tiplying. We multiply the divisor by some number which 
we suppose to be the quotient term, and if the product does 
not exceed the partial dividend, nor the difference between the 
product and partial dividend exceed the divisor, we know that 
we have obtained the correct quotient figure. The method 
described is so common that it need not be illustrated by a 
problem. 

Rule. The mistake of using figure for number is also made 
in stating the rule for division. One author says, " Find how 
many times the divisor is contained in the fewest figures on 
the left of the dividend," etc.; another says, "Take for the first 
partial dividend the fewest figures of the given dividend," 
etc.; another says, "Take for the first partial dividend the 
least number of figures on the left that will contain the divisor," 
etc. Of course, figures will not contain the divisor; the num- 
ber expressed by the figures is what is intended, and therefore 
should be expressed. The error may be corrected by saying, 
"Divide the number expressed by the fewest figures on the 
left that will contain the divisor," or, " by the fewest terms," 
etc. 



234 THE PHILOSOPHY OP ARITHMETIC. 

Remark. We write the divisor at the left of the dividend 
and the quotient at the right as a matter of custom. Some pre- 
fer writing the divisor at the right and placing the quotient 
under the divisor. We begin at the left to divide, so that the 
remainder, when one occurs, may be united with the number 
of units of the next lower order, giving a new partial divi- 
dend. If we attempt to divide by beginning at the right, we 
will see the advantage of the ordinary method. 



SECTION II. 

DERIVATIVE OPERATIONS OF SYNTHESIS 
AND ANALYSIS. 



I. INTRODUCTION. 



II. COMPOSITION. 



III. FACTORING. 



IV. COMMON DIVISOB. 



V. COMMON MULTIPLE. 



VI. INVOLUTION. 



VII. EVOLUTION. 



CHAPTER I. 

INTRODUCTION TO DERIVATIVE OPERATIONS. 

THE four Fundamental Operations are the direct and imme- 
diate outgrowth of the general processes of synthesis and 
analysis as applied to numbers. They are called Fundamental 
Operations because all the other operations involve one or 
more of these, and may be regarded as being based upon them. 
They are the foundation or basis upon which the others are 
built up, the germ from which they are evolved, the soil out of 
which they grow. 

Several of the processes of arithmetic are so intimately 
related to the fundamental operations that they may be 
regarded as directly originating in and growing out of them. 
Such are the processes of Factoring, Common Multiple, Com- 
mon Divisor, etc. These processes have their roots in the 
general notions cf the fundamental operations, and are 
evolved from them by a modification and extension of the pri- 
mary analytic and synthetic processes. They are developed 
by the thought process of comparison, though they have not 
their basis in comparison, like the processes of Ratio, Propor- 
tion, etc. Being thus derived from the fundamental operations, 
they may be called the Derivative Operations of synthesis and 
analysis. Let us notice the origin and nature of these deriva- 
tive operations. 

If two or more numbers are multiplied together, and the 
result is considered with respect to its elements, we have the 
idea of a Composite Number. The general process of forming 
composite numbers may be called Composition. The numbers 

(237) 



238 THE PHILOSOPHY OF ARITHMETIC. 

synthetized in forming a composite number are called Factors 
of that number. If we form a composite number consisting 
of two equal factors, we have a square ; of three equal factors, 
a cube, etc., and the process is called Involution. If we find 
a composite number which is a number of times each of several 
numbers, or is so composed that each of them is one of its 
factors, it is called a common multiple of these numbers, and 
the process is known as finding Common Multiples. 

These processes are distinct from Multiplication, though 
related to it. They employ multiplication and are the out- 
growth of the general multiplicative idea, but pass beyond the 
primary idea of multiplication. In multiplication, the main 
idea is the operation of repeating one number as many times 
as there are units in another to obtain a result; here the thought 
is the result of the operation compared with the numbers 
multiplied together. In the former case, the process is purely 
synthetic ; here comparison unites with synthesis, and employs 
it for a particular object. The operation of multiplying is 
assumed as a fact, and employed for the purpose of attaining 
a result bearing some relation to the elements combined. 

Having obtained composite numbers, and the idea of their 
being composed of factors, we naturally begin to analyze them 
into their elements in order to discover these factors. This 
gives rise to an analytic process, the converse of Composition. 
The general process of analyzing a number into its factors is 
called Factoring. If we resolve a number into several equal 
factors for the purpose of seeing what factor must be repeated 
two, or three, etc., times to produce the number, we have a 
process known as Evolution. If we have given several num 
bers, and proceed to find a common factor of these numbers, 
we have the process known as Common Divisor. 

These processes, though related to Division, are clearly dis- 
tinguished from it. They are an outgrowth of the general 
idea of division, but extend beyond it. In division it is the 
operation of finding how manv times one number is Contained 



INTRODUCTION. 239 

in another that is the prominent idea; here the idea is the 
result considered in relation to the number or numbers 
operated upon. In Factoring, the process of comparison 
enters as an important element. Division is a process purely 
analytical ; Factoring is analysis, and more ; it is analysis plus 
comparison. It has its root in Analysis, and is developed by 
the thought-process of Comparison. 

There are, therefore, two general derivative processes, Com- 
position and Factoring, each of which embraces corresponding 
and opposite processes. The terms, Composition and Factor- 
ing, are in practice restricted to the general processes; the 
special processes are known by their particular names. We 
have thus three pairs of derivative processes, Composition 
and Factoring, Multiples and Divisors, and Involution and 
Evolution. These will be treated in successive chapters. 



CHAPTER II. 

COMPOSITION. 

/COMPOSITION is the process of forming composite num- 
\J bers when their factors are given. It is a general process 
which contains several subordinate and special ones. When 
fully analyzed, it will be seen to present several interesting 
cases besides the more particular ones of Involution and Mul- 
tiples. From the previous analysis it is seen that there is a 
real case of Synthesis, the converse of the analytic process 
of Factoring. 

This new generalization, and the term I have applied to it, 
will, I trust, receive the approval of mathematicians. Its 
importance as a logical necessity, is seen in its relation to 
Factoring. In the fundamental operations each synthetic pro- 
cess has its corresponding analytic process. Thus, addition is 
synthetic, subtraction is analytic ; multiplication is synthetic, 
division is analytic. It follows, therefore, that there should 
be a synthetic process corresponding to the analytic process 
of Factoring. This process I have presented under the name 
of Composition, or the process of forming composite numbers. 

Cases. There are several interesting and practical cases of 
Composition, some of the most important of which are the 
following: 

I. To form a composite number out of any factors. 

II. To form a composite number out of equal factors. 

III. To form a composite number out of factors bearing any 
definite relation to each other. 

(240) 



COMPOSITION. 241 

IV. To form composite numbers which have one or more 
given common factors. 

V. To form several or all of the composite numbers possible 
out of given factors. 

VI. To determine the number of composite numbers that 
can be formed out of given factors. 

Method of Treatment. The method of treatment is to com- 
bine these factors by multiplication in such a manner as to 
attain the result desired. I will briefly state the manner of 
treating each case. 

CASE I. To form a composite number out of any factors 
In Case I. we find the result by simply taking the product of 
the factors. Thus the composite number formed from the fac- 
tors 2, 3, and 4 equals 2x3x4, or 24. 

CASE II. To form a composite number out of equal fac- 
tors. Case II. may be solved in the same manner as Case I., or 
we may multiply a partial result by itself or by another partial 
result, to obtain the entire result. Thus, if we wish to find 
the composite number consisting of eight 2's, we may multi- 
ply 2 by 2, giving 4, then multiply 4 by 4, giving 16, and then 
multiply 16 by 16, giving 256, the number required. 

CASE III. To form a composite number out of factors 
bearing any definite relation to each other. In this case we 
may have given one factor and the relation of the other factors 
to it ; we first find the factors and then take their product. 
Thus, required the number consisting of three factors, the first 
being 4, the second twice the first, and the third three times the 
second. Here, we first find the second factor to be 8, and the 
third to be 24, and then take the product of 4, 8, and 24, which 
we find to be 768. 

CASE IV. To form composite numbers which have one or 
more given common factors. This case maybe solved by tak- 
ing the given common factor, and multiplying it by any other 
'actors we choose. If it is required that the factor given be 
the largest common factor of the numbers obtained, the mul- 
tipliers selected must be prime to each other. To illustrate, 
16 



242 THE PHILOSOPHY OF ARITHMETIC. 

find three numbers whose largest common factor shall be 12. 
If we multiply 12 by 2, 4, and 6, we will have 24, 48, and 72, 
three numbers whose common factor is 12; but since the num- 
bers used as multipliers have a common factor, 12 is not the 
largest factor common to these three numbers. To find three 
numbers having 12 as their largest common factor, we may 
multiply 12 by 2, 3, and 5, which gives us the numbers 24, 3(5, 
and 60, in which 12 is the largest common factor. 

CASE Y. To form several or all of the composite numbers 
possible out of given factors. In this case we may take the 
factors two together, three together, etc., until they arc taken 
all together; or we may multiply 1 and the first factor by 1 
and the second factor, the products thus obtained by 1 and the 
third factor, etc., until all the factors are used. To illustrate, 
form all the possible composite numbers out of 2, 3, 5, and 7. 

We first find all the possible pro- 
ducts taking them two together; OPERATION. 

o v q _ p q v c _ IK 

then all the products taking them 



> 
three together, and then the products 2x7 = 14 5x7=35 

taking them four together, as is 2x3x5=30 

shown in the margin. Another 2x3x7=42 

method, not quite so simple in a i; 7 IAK 

thought but more convenient in 2x3x5x 7==10 
practice, is as follows: 

Multiplying 1 and 2 by 1 and 3, will give 1, 2, 3, and all the 
composite numbers that can be formed out of 2 and 3; these 
multiplied 



by 1 and 5 
will give 1, 
2, 3,5, and 
all the com- 
posite num- 
bers that 
can be 
formed out 


1 
1 


2 
3 






OPERATION. 


1 

1 


2 
5 


3 


G 


1 

1 


2 

7 


3 


5 


G 10 


15 


30 




1 


2 


3 


6 


G 10 


15 


30 7 


14 21 42 35 70 105 210 



of 2, 3, and 5 ; these multiplied by 1 and 7 will give 1, 2, 3, 5, 



COMPOSITION. 243 

T, and all the composite numbers that can be formed out of 2, 
3, 5, and 7. Omitting 1, 2, 3, 5, and 7 in the last result, and 
we have all the composite numbers that can be formed out of 
2, 3, 5, and 7. 

If some of the given factors are alike, we have an interesting 
modification of this case. Thus, suppose we wish to find the 
composite numbers which 
can be composed out of 2, 2, OPERATION. 

2, 3, and 3. In this problem [ ? 8 
since 2 is used three times 



1 2 3 4 G 8 9 12 18 24 36 72 

we may make the first series 

1, 2, 2-, and 2 3 , or 1, 2, 4, and 8; and since 3 is used twice, the 
second series will be 1, 3, and 3 2 , or 1, 3, and 9 ; and the 
products of these, omitting 1, 2, and 3, will be the composite 
numbers required. 

CASE VI. To determine the number of composite num- 
bers that can be formed out of given factors. We may solve 
this case by increasing the number of times each factor is used 
by unity, take the product of the results and diminish it by 
the number of different factors used increased by one. Tho 
reason for this method may be readily shown. Suppose we 
wish to find how many composite numbers can be formed with 
three 2's and two 3's. 

Here we sec that 2 used three times as a factor gives with 1 
a scries of four terms; and 3 used twice as a factor gives 
with 1 a scries of lliree terms; hence the product will give a 
series of 4x3 or 12 terms, and omitting the unit and 2 and 3, 
we have nine terms. The inference from this solution will 
give the method stated above. 



CHAPTER IIL 

FACTORING. 

is the process of finding the factors of com 
J- posite numbers. It is the reverse of Composition. In 
Composition we have given the factors to find the number; in 
Factoring we have given the number to find the factors. Com- 
position is a synthetic process ; it proceeds from the parts by 
multiplication to the whole. Factoring is an analytic process ; 
'it proceeds from the whole by division to the parts. 

A Factor, as now generally presented in arithmetic, is 
regarded as a divisor of a number, rather than a maker or pro- 
ducer of the number. This I regard as an error. The origin 
of the word, facio, I make, indicates its original meaning to be 
a maker of a composite number. The fact of a Factor 
of a number being a divisor of it is a derivative idea, re- 
sulting from the primary conception of its entering into the 
composition of the number. This primary idea of the office of 
a Factor is the one that should be primarily presented to pupils, 
rather than the secondary or derivative idea. We should 
define according to the fundamental, rather than the derivative 
office. To do otherwise is to invert the logical relation of ideas, 
and must, as I have known it, tend to confusion. Thus taught, 
it is seen that the proposition, a factor of a number is a divi- 
sor of the number, is an immediate inference, which would have 
to be inverted if the secondary office of a factor is made the 
fundamental idea. 

(244) 



FACTORING. 245 

Cases. Factoring presents several cases analogous to those 
of Composition. Some of the principal ones are the following, 
which, it will be noticed, are the correlatives of those givei 
under Composition. 

I. To resolve a number into its prime factors. 

II. To resolve a number into equal factors. 

III. To resolve a number into factors bearing a certain rela- 
tion to each other. 

IV. To find the divisors common to two or more num- 
bers. 

V. To find all the factors or divisors of a number. 

VI. To find the number of divisors of a number. 
Method. The general method of treatment is to resolve the 

number or numbers into their prime factors, and then combine 
these factors when necessary so as to give the required result. 
The prime factors of a number are found by division, and con- 
sequently it is convenient to know before trial what numbers 
are composite and can be factored, and the conditions of their 
divisibility. Hence, the subject of Factoring gives rise to the 
investigation of the methods of determining prime and com- 
posite numbers, and the conditions of the divisibility of com- 
posite numbers. This subject will be treated under the head 
of Prime and Composite Numbers. The method of treating 
each of the above named cases of factoring will be briefly stated. 

CASE I. To resolve a number into Us prime factors. In 
Case I. we divide the number by any prime number greater 
than 1 which will exactly divide it; divide the quotient, 
if composite, in the same manner; and thus continue until the 
quotient is prime. The divisors and the last quotient will be 
the prime factors required. 

Thus, suppose we have given 105 to find its prime 
factors. Dividing 105 by the prime factor 3, and 8)105 
the quotient 35 by 5, we see that 105 is composed 6)35 
of the three factors 3, 5, and 7, and since these are 7 

prime numbers, its prime factors are 3, 5, aud 7. 



246 THE PHILOSOPHY OF ARITHMETIC. 

CASE II. To resolve a number into equal factors. In Case 
II. wq resolve the number into its prime factors and then com- 
bine by multiplication one from each set of two equal factors, 
when we wish one of the two equal factors of the number ; one 
from each set of three equal factors when we wish one of 
three equal factors, etc. 

Thus, suppose we wish to find the (2x2x2x 

three equal factors of 216, or one of its = (3x3x3 

three equal factors. We first resolve 2x3=6 

216 into its prime factors, finding 216=2x2x2x3x3x3. 
Since there are three 2's, one of the three equal factors will 
contain 2; and since there are three 3's, one of the three equal 
factors will contain 3 ; hence one of the three equal factors is 
2 x 3, or 6. 

CASE III. To resolve a number into factors bearing a cer- 
tain relation to each other In this case we may divide the 
given number by the product of the numbers representing the 
relation of the other factors to the smallest factor, then resolve 
the quotient into equal factors, and then multiply this equal 
factor by the numbers indicating the relation of the other fac- 
tors to it. 

Thus, resolve 384 into three factors, such that the second 
shall be twice the first and the third three times the first. 
Since the second factor equals 2 times the 

first and the third equals 3 times the first, 6)384 

the product of the factors-, will equal 2x3, 64=4x4x4 
or 6 times the first factor, used three times ; ~^ 

hence if we divide 384 by 6, the quotient, 
64, will be the product of the smallest factor used three times ; 
therefore, if we resolve 64 into three equal factors, one of these 
factors will be the smallest of the three factors required. One 
of the three equal factors of 64, found by the previous case, is 
4 ; hence, the smallest factor is 4, the second is 4x 2 or 8, and 
the third is 4x3 or 12. 

CASE IV. To find the divisors common to two or more 



FACTORING. 247 

numbers In this case we resolve the numbers into their 
prime factors, and the common prime factors and all the num- 
bers which we can form by combining them will be all the 
common divisors. 

Thus, find the divisors common OPERATION. 

to 108 and 144. Resolving the 108=2 2 x3 3 

numbers into their prime factors, 144=2 4 x3' 

we find the common factors to be om - foctor=2'x3' 

2'x3*; hence, 1, 2, 4, 3, 9, and 2 4 
all the possible products arising i 3 9 o A i 4 10 36 
from their combination, will be all 
the divisors of 108 and 144. 

CASE V. To find all the factors or divisors of a number 
In this case we resolve the number into its prime factors, form 
a scries consisting of 1 and the successive powers of one fac- 
tor, and under this write 1 and the successive powers of an- 
other factor, and take the products of the terms of this scries, 
etc. Thus, find all the different divisors of 108. 

The factors of 108 

are two 2's and three OPERATION. 

3's. Since 3 is a factor J ^ 2 ^2 X 3 x 3 X 3 
3 times, 1, 3, 3 2 , 3 3 , is ! 2 4 ' 

the first scries of divis- i 3 9 2 7 2 6 18 54 4 12 36 108 
ors ; and since 2 is a 

factor twice, 1, 2, 2 2 is the second scries of divisors; and the 
products of. the terms of these two scries will give the prime 
factors and all possible products of them ; and therefore, all the 
divisors of the number. 

CASE VI. To find the number of divisors of a number. 
In this case we resolve the number into its prime factors, in- 
crease the number of times each factor is used by 1, and take 
the product of the results. Thus, find the number of divisors 
of 108. 



248 THE PHILOSOPHY OF ARITHMETIC. 

Factoring, we find 108 equals OPERATION. 

2 2 x3 3 . Now it is evident that 1 108=2 2 x3 3 

with the first and second powers of (2 + l)x(3+l)=12 
2 will give a series of three divisors; and 1 with the first, 
second and third powers of 3, will give a series of four divis- 
ors; hence their products will give a series of three times 
four, or 12 divisors. 



CHAPTER IV. 

THE GKEATEST COMMON DIVISOR. 

A DIVISOR of a number is a number which will exactly 
divide it. A number is said to exactly divide another 
when it is contained in it a whole number of times without a 
remainder. A Common Divisor of two or more numbers is a 
divisor common to all of them. The Greatest Common Divi- 
sor of several numbers is the greatest divisor common to all 
of them. By using the word factor to denote an exact integral 
divisor, we may define as follows: 

A Divisor of a number is a factor of the number. A Com- 
mon Divisor of two or more numbers is a factor common to 
all of them. The Greatest Common Divisor of several num- 
bers is the greatest factor common to all of them. These defi- 
nitions employ the term factor with a derivative signification. 
A factor is primarily one of the makers of a number, entering 
into its composition multiplicatively. From this it follows, 
however, that a factor is an integral divisor of a number, and 
as such, it may be conveniently and legitimately used in defin- 
ing a common divisor. 

In the subject of greatest common divisor, the term " divisor" 
is used in a sense somewhat special. It signifies an exact 
divisor a number which is contained a whole number of 
times without a remainder. The word measure was formerly 
used instead of divisor, and is in some respects preferable to 
divisor. A common divisor of several numbers is appropri- 
ately called their common measure, since it is a common unit 

(249) 



250 THE PHILOSOPHY OF ARITHMETIC. 

of measure of those numbers. The term measure, iu this sense, 
originated in Geometry, where a line, surface, or volume which 
is contained in a given line, surface, or volume, is called the 
unit of measure of the quantity. In arithmetic, the term 
divisor is generally preferred. 

Gases. There are two general cases of greatest common 
divisor, growing out of a difference in the method of treatment 
adapted to the problems. When numbers are readily factored, 
we employ one method of operation ; when they are not readily 
factored, we are obliged to employ another method. This dual 
division of the subject into two cases is thus seen to be founded, 
not upon any distinctions in the idea of the subject, but upon 
the method of operation adapted to the numbers given. These 
two cases arc formally stated as follows: 

I. To find the greatest common divisor when the numbers 
are readily factored. 

II. To find the greatest common divisor when the numbers 
are not readily factored. 

Treatment. The general method of treatment in the first 
case is. to analyze the numbers in^o their factors, and take the 
product of the common factors. In the second case the num- 
bers arc operated upon in such a manner as to remove all the 
factors not common, and thus cause the greatest common divi- 
sor to appear. These two methods will be made clear by their 
application. 

CASE I. To find the greatest common divisor when the 
numbers are readily factored. 

This case may be solved by two distinct methods. The first 
method consists in writing the numbers one beside another, 
and finding all their common factors by division, and then tak- 
ing the product of these common factors. To illustrate, re- 
quired the greatest common divisor of 42, 84, and 126. 

1st Method. We place the numbers one beside another 
as in the margin. Dividing by 2, we see that 2 is a common 
factor of the numbers. Dividing the quotients by 3, we see 



THE GREATEST COMMON DIVISOR. 25 J 

that 3 is a common factor of the OPERATION. 

numbers. Dividing these quotients 2)42 84 126 

by 7, we see that 7 is a common 3)21 42 G3 

factor of the numbers; and since 7)7 14 21 

the final quotients 1, 2, and 3 arc 123 

prime to each other, 2, 3, and 7 are GL C. D. =2x3x7=42 
all the common factors of the given numbers. Hence 2x3x7 
or 42, is the greatest common divisor required. This method, 
so far as I can learn, was published first by the author of 
this work, in 1855. It is now in several different text-books. 

The second method consists in resolving the numbers into 
their prime factors, and taking the product of all the common 
factors. To illustrate, take the problem already solved by the 
first method. 

2d Method. Resolving the nurn- OPERATION. 

bers into their prime factors, wo 42=2x3x7 ' 

find that 2, 3, and 7, are factors 84=2x2x3x7 

1 Q/_9 vx Q xx tjy 17 

common to the three numbers; n y* _o o T_^O 
.... ._ . \x. U. U. ^XoXT 4J 
hence their product, which is 42, is 

a common divisor of the numbers; and, since these are all 
the common factors, 42 is the greatest common divisor. 

CASE II. To find the greatest common divisor ivhen the 
numbers are not readily factored. The second case may be 
solved by a process which may be entitled the method of suc- 
cessive division. It consists in dividing the greater number 
by the less, the less number by the remainder, etc., until the 
division terminates, the last divisor being the greatest common 
divisor. To illustrate, suppose it be required to find the great 
est common divisor of 32 and 5G. OPERATION 

Method. We first divide 5G by 32, then 32)50(1 
divide the divisor, 32, by the remainder, 32 

24; then divide the divisor, 24, by the 24)32(1 

remainder, 8, and find there is no remain- 
der; then is 8 the greatest common divi- 8)24(3 
sor of 32 and 56. 



252 



THE PHILOSOPHY OF ARITHMETIC. 



OPERATION. 

3215611 

24|32|1 



8,24 
|24 



o 



This method is applicable to all numbers, and may therefore 
be distinguished from the methods of the previous case by 
naming it the general method, those being 
adapted to only a special case. A more conveni- 
ent method of expressing the successive divis- 
ion, and one which I recommend for general 
adoption, is that represented in the margin. It 
is observed in this method that the quotients 
are all written in one column at the right, and that the num- 
bers in the other columns become divisors and dividends iu 
turn. 

Explanation. In the explanation of the rationale of the 
general method of successive division, there are two distinct 
conceptions of the nature of the process. These two methods 
may, for convenience in this discussion, be entitled the Old and 
the New methods of explanation. By the Old Method of 
explanation I mean the one generally given in the text-books 
on arithmetic and algebra. The New Method is the one which 
is found in my own mathematical works. I will present each, 
pointing out the difference between them. Both methods are 
based upon the following general principles of common 
divisor: 

1. A divisor of a number is a divisor of any multiple of 
that number. 

2. A common divisor of two numbers is a divisor of their 
sum, and also of their difference. 

The Old Method of explaining the process of successive 
division is briefly stated in the following propositions: 

1. Any remainder which exactly divides the previous divi- 
sor, is a common divisor of the two given quantities. 

2. The greatest common divisor will divide each remainder, 
and cannot be greater than any remainder. 

3. Therefore, any remainder which exactly divides the 
previous divisor is the greatest common divisor. 

Whatever the special form of the old method of explanation, 



THE GREATEST COMMON DIVISOR. 263 

and we find it considerably varied by different authors, it 
involves, more or less distinctly, the principles just stated 

The New Method of conceiving of the nature of the process 
and explaining it, may be presented in the following princi- 
ples: 

1. Each remainder is a NUMBER OF TIMES the greatest com- 
mon divisor. 

2. A remainder cannot exactly divide the previous divisor 
unless such remainder is ONCE the greatest common divisor. 

3. Hence, the remainder which exactly divides the previous 
divisor, is QSQ& the greatest common divisor. 

The first of these principles is evident from the considera- 
tion that a number of times the greatest common divisor, sub- 
tracted from another number of times the greatest common 
divisor, leaves ^number of times the greatest common divisor. 

The second of these principles becomes evident from the 
consideration that of any remainder and the previous divisor, 
the numbers denoting how many times the greatest common 
divisor is contained in each are prime to each other ; hence, 
one cannot divide the other unless one of these numbers is a 
unit, or the remainder becomes once the greatest common 
divisor. 

These principles may be readily seen by factoring the two 
numbers and then dividing. Thus, in the problem already 
given, knowing the greatest com- 
mon divisor to be 8, we may re- OPERATION. 

solve 32 and 56 into a number of 
times 8, and then divide. Observ- . .- 

ing the operations in this factored ' 



form, we see that each remainder Ix8)3x8t'3 

is a number of times the greatest 3x8 

common divisor, and that the fac- 

tors 7 and 4, and also 4 and 3, are respectively prime to each 
other; and also that the division terminates when we reach a 
divisor which is once the greatest common divisor, and that it 



254 THE PHILOSOPHY OF ARITHMETIC. 

/ 
cannot terminate until we come to once the greatest common 

divisor. 

In arithmetic I find it simpler to pre- OPERATION. 
sent this New Method, in a manner 32 )^( 1 
slightly varied from the above, preserving 
its spirit, but slightly changing the form 4 

to adapt it more fully to the comprchen- "Ihoifq 

sion of younger minds. To illustrate, let 9 , 

it be required to find the greatest common 
divisor of 32 and 5G. Dividing as previously explained, we 
have the work in the margin. The explanation, showing that 
this process will give the greatest common divisor, is as fol- 
lows: 

I 1st. The last remainder, 8, is a number of limes the great- 
est common divisor. For, since 32 and 5G arc each a number 
of times the G. C. D., their difference, 24, is a number of times 
the G. C. D.; and since 24 and 32 are each a number of times 
the G. C. D., their difference, 8, is also a number of times the 
G. C. D. 

2d. The last remainder, 8, is ONCE the greatest common 
divisor. For, since 8 divides 24, it will divide 24-|-8, or 32; 
and since it divides 32 and 24, it will divide 24-J-32, or 56; 
and now since 8 divides 32 and 56, and is a number of limes 
the G. C. D., and since once the G. C. D. is the greatest num- 
ber that will divide 32 and 56, therefore 8 is once the G. C. D. 

This second method of conceiving the subject is believed to 
be the true one. It is simpler than the old method, and 
reaches the root of the matter, which the other does not. It 
looks down into the process and sees the nature of the remain- 
ders, and their relation to each other. All the remainders are 
seen to be a number of times the greatest common divisor, 
each being a less and less number of times the greatest com- 
mon divisor; and consequently, if the division be continued far 
enough, we will at length arrive at once the greatest common 
divisor. The object of dividing is thus seen to be a search for 



THE GREATEST COMMON DIVISOR. 255 

a smaller number of times the greatest common divisor, know- 
ing that eventually we will arrive at once this factor, which will 
be indicated by the termination of the division. The experience 
of the class-room, especially in the sudden revelation of the 
philosophy of the division to those who thought they had a 
clear idea of the subject by the old method, has frequently 
demonstrated the superiority of the method now suggested. 
It is also readily seen, from this conception of the subject, that 
the secret of the method of finding the greatest common divi- 
sor is not in the division of the numbers, but in the subtrac- 
tion of them knowing that when we subtract one number of 
times a factor from another number of times the factor, the 
remainder is a less number of times the factor, and that the 
object is to continue the subtraction until we reach once the 
required factor. 

Abbreviation. This view of the subject leads us to discover 
a shorter process of obtaining the greatest common divisor 
than that of the ordinary method of dividing. 
Thus, suppose we wish to find the greatest OPERATION. 
common divisor of 32 and 116. If we divide in 32'llG 4 
the ordinary way, we will find that it requires five 



123 
123 



divisions and five quotients. If we take 4 times 
32 and subtract 116 from it, we get a smaller re- 
mainder than if we subtract 3 times 32 from 116, 
and hence are nearer once the greatest common divisor. If we 
then subtract 32 from 3 times 12, we obtain a smaller remainder 
than if we subtract 2 times 12 from 32, and hence arc nearer 
once the greatest common divisor, etc. This latter method 
requires but three multiplications and subtractions, and hence 
saves two-fifths of the work. In many problems nearly one- 
half the labor is saved by this method. 

The method of conceiving and explaining the greatest com- 
mon divisor here given, is perhaps most clearly exhibited by 
the use of general symbols. Thus, let A and B be any two 
numbers, of which A is the greater; let c be their greatest 



256 THE PHILOSOPHY OF ARITHMETIC. 

common divisor, and suppose Aac and B=bc] then dividing 
the greater by the less, the smaller by the remainder, and thus 
continuing, we have the operation in 
the margin, which may be explained b.c.)a. c(q 

as follows: 

in, tJdiC* t* (* 
1st. Each remainder is a number 

of times the G. C. D. This is shown r.c) b. f c(q f 
bv the division, since each remainder r< ^ ' 

J /7j r fjl\f,- = -^r n 

is a number of times c, the first being 

(abq) times c, which we indicate r/ - c ) r - c (<7" 

r'q".c 

by r times c, etc. 

1 (j r f Q 'f\ c r ff c 

2d. A remainder cannot exactly . * ' 

divide the previous divisor unless 

such remainder is ONCE the G. C. D. To prove this it must 
be shown that b and r are prime to each other; also, that r and 
r' are prime to each other, etc. Now, if b and r are not prime 
to each other, they have a common factor, and hence, r+bq or 
a contains this factor of b ; but a and b are prime to each 
other, since c is the greatest common factor of a and b; there- 
fore, b and r are prime to each other. In the same way it may 
be shown that r aud r' are prime to each other, r' and r", etc. 
Hence, since of two numbers prime to each other one cannot 
contain the other unless the latter is a unit, a remainder can- 
not exactly divide the previous divisor unless such remainder 
is once the G. C. D. 

3d. Hence, the remainder which does exactly divide the pre- 
vious divisor is ONCE the Greatest Common Divisor. 



CHAPTER V. 

THE LEAST COMMON MULTIPLE. 

A MULTIPLE of a number is one or more times the num- 
ber. A Common Multiple of two or more numbers is a 
number which is a multiple of each of them. The Least 
Common Multiple of several numbers is the least number 
which is a multiple of each of them. 

This conception of a multiple is that it is a number of time* 
some number. It regards the subject as a special case of form- 
ing composite numbers. A common multiple is a synthesis of 
all the different factors of two or more numbers, giving rise to 
a number which is one or more times each of those numbers. 
The relation of the subject to multiplication is also seen in th 
term multiple itself. The primary idea is, what number is one 
or more times each of several numbers ? 

This view of a multiple differs from that usually presented 
by our writers of text-books. The usual definition is A mul- 
tiple of a number is a number which exactly contains it. This 
puts containing as the primary idea, and makes the subject 
seem to originate in division rather than in multiplication. 
Indeed, some have gone so far in this direction as to change 
the name from multiple to dividend, calling it a common divi- 
dend instead of a common multiple. That this idea is incor- 
rect is evident both from the term multiple, and the nature of 
the subject. There can be no question of the subject having 
its origin in multiplication, and it should certainly be denned 
in accordance with this view. 

17 (257) 



258 THE PHILOSOPHY OP ARITHMETIC. 

It will be observed that the subject of Greatest Common 
Divisor is placed before that of Least Common Multiple ; that 
is, a special case of Factoring before a special case of Compo- 
sition, thus reversing the general order of synthesis before 
analysis. The reason for this is that Common Multiple is a 
synthesis of factors, and in some numbers these factors are 
most conveniently found by the method of greatest common 
divisor. This order is thus a matter of convenience in per- 
forming the operation, and not that of logical relation. 

Cases. There are two general cases of Least Common 
Multiple, as of Greatest Common Divisor. This distinction of 
cases, as in the corresponding analytic process, is not founded 
in a variation of the general idea, but rather in the practical 
ease or difficulty of finding the factors of the numbers. When 
the numbers are readily factored we employ one method of 
operation ; when they are not easily factored we employ an- 
other method. These two cases are formally stated as follows : 

I. To find the least common multiple when the numbers are 
readily factored. 

II. To find the least common multiple when the numbers 
are not readily factored. 

Treatment. The general method of treatment in the first 
case is to resolve the numbers into their different factors 
by the ordinary method of factoring, and take the product of 
all the different factors. In the second case, the different fac- 
tors are found by the process of determining the greatest com- 
mon divisor, and are then combined as before. 

CASE I. To find the least common multiple when the num- 
bers are readily factored. This case may be solved by two 
distinct methods. The first method consists in resolving the 
numbers into their prime factors, and then taking the product 
of all the different prime factors, using each factor the greatest 
number of times it appears in either number. Thus, required 
the least common multiple of 20, 30, and TO. 

We first resolve the numbers into their prime factors 



THE LEAST COMMON MULTIPLE. 259 

Since the factors of 20 are 2 X 2 x 5, the multiple must con 
tain the factors 2, 2, and 5 ; 

since the factors of 30 are ^^ TI ^\ 

20=2x 2x5 
2, 3, and 5, it must contain 30=2x3x5 

the factors 2, 3, and 5; and 70=2x5x7 

for a similar reason it must L.C. M.=2x 2x3x5x 7=420. 

contain the factors 2, 5, and 

7 ; hence, the least common multiple of 20, 30, and 70 must 

contain the factors 2, 2, 3, 5, and 7, and no others; and 

their product, which is 420, is the least common multiple 

required. 

The second method consists in writing the numbers one 
beside another and finding all the different factors by division, 
and then taking the product of these factors. To illustrate, 
find the least common multiple of 24, 30, and 70. 

Placing the numbers beside one another, and dividing by 2, 
we find that 2 is a factor of all the numbers ; it is therefore a 
factor of the least common multiple. Divid- 
ing the quotients by 3, we see that 3 is a factor OPERATION. 

2^24 30 70 
of some of the. numbers; it is therefore a factor i 

of the least common multiple. Continuing to * 
divide, we find all the different factors of the 5)_4_5_35 

417 

numbers to be 2, 3, 4, 5, and 7 ; hence, their pro- 
duct, which is 840, will be the least common multiple required. 
CASE II. To find the least common multiple when the num- 
bers are not readily factored. The second case is solved by a 
method which may be called the method of greatest common 
divisor. By it, when there are two numbers, we find the 
greatest common divisor of the two numbers and multiply one 
of them by the quotient of the other divided by their greatest 
common divisor. When there are more than two numbers, 
we find the least common multiple of two of the numbers, and 
then of this multiple and the third number, etc. To illustrate, 
required the least common multiple of 187 and 221. 



260 THE PHILOSOPHY OF ARITHMETIC. 

We first find the greatest common divisor to be 17. Now, 
the least common multiple of OPERATION. 

187 and 221 must be composed jg7 221 

of all the factors of 187, and all 170 187 



the factors of 221 not contained 17 

in 187. If we divide 221 by 

the greatest common divisor, we L Q jyj _ i^^ =2431 

shall obtain the factors of 221 not If 

belonging to 187 ; hence, the least common multiple is equal 

to 187x221-7-17, which we find is 2431. 

Another statement for this method is, divide the product of 
the two numbers by their greatest common divisor. The value 
of this method may be seen by attempting to find the least 
common multiple of 1127053 and 2264159 by each method. 

This method is very clearly OPERATION. 

exhibited by the following gen- A=axc 

eral explanation. Let A and B S=bxc . 

be any two quantities, and let L. C. M.=ax6xc= XT? 
their greatest common divisor be 

represented by c, and the other factors by a and b, respectively ; 
then we shall have the L. C. M.=ox6xc, Case L; but 6Xc= 

A A 

B, and a=-; hence, L. C. M.= -XJS. 
c ~c 



CHAPTER VI. 

INVOLUTION. 

INVOLUTION is the process of forming composite numbers 
jL by the synthesis of equal factors. It is, as has been pre- 
viously explained, a special case of Composition. If in the 
general synthesis of factors, we fix upon the condition that 
all the factors are to be equal, the process is called Involution, 
and the composite number formed is called a Power of that 
factor. 

Involution may, therefore, be defined as the process of rais- 
ing numbers to required powers. The power of a number is 
the product obtained by using the number as a factor any num- 
ber of times. The different powers of a number are called, 
respectively, the square, the cube, the fourth power, etc. The 
square of a number is the product obtained by using the num- 
ber as a factor twice. The cube of a number is the product 
obtained by using the number as a factor three times. These 
definitions, which are beginning to be adopted by authors, are 
regarded as improvements upon those framed from the usual 
conception of the subject. 

Symbol. The power of a quantity is indicated by a figure 
written at the right, and a little above the quantity. Thus, 
the third power of 5 is indicated by 5 s . The earlier writers on 
mathematics denoted the powers of numbers by an abbrevia- 
tion of the name of the power. Harriot, an eminent math- 
ematician of the 16th century, repeated the quantity to indi- 
cate the power; thus, for a fourth power he wrote aaaa. The 
present convenient system of exponents was introduced by 

(261) 



262 THE PHILOSOPHY OF ARITHMETIC. 

Descartes, an eminent philosopher and mathematician cele- 
brated for his " cogilo, ergo sum," and the invention of the 
method of Analytical Geometry. 

Cases. To raise a number to each different power is a vari- 
ation of the general idea, and might be regarded as presenting 
distinct cases; but the methods of operation in each one of these 
cases are so similar, that they may all be considered under 
one head. In raising a number to a given power, we may 
have two objects in view: first, merely to find the required 
power ; and second, to ascertain the law by which the different 
parts of a number, as expressed in the Arabic system, are 
involved. These two objects require different methods of pro- 
cedure, and upon this difference of method we may found 
two distinct cases of involution. In practice, it is convenient 
to divide the second case into the consideration of the square 
and the cube, thus making three cases. These cases, formally 
expressed, are as follows: 

I. To raise a number to any required power. 

II. To raise a number to the second power, and ascertain the 
jaw by which the power is formed. 

III. To raise a number to the third power, and ascertain the 
law by which the power is formed. 

Treatment. The general method of treatment is to involve 
the factors by multiplication. In the first case a variation 
occurs for the purpose of abbreviation, giving two methods. 
In the second and third cases the number is resolved into parts 
and involved in two different ways, giving also two distinct 
methods. The treatment of both of these cases will now 
be presented. 

CASE I. To raise a number to any required OPERATION. 
power. This case may be solved by forming a 4 

product by using the number as a factor as many 
times as there are units in the index of the power. 
Thus, to find the third power of 4, we multiply 
4 by 4 giving 16, and then multiply 16 by 4 






INVOLUTION. 263 

giving 64, which is the cube of 4, since the number is used aa 
a factor three times. 

In all powers higher than the cube, we may abbreviate the 
process by taking the product of one power by another. Thus, 
in finding the 8th power of 2, we may first find 
the square of 2, which is 4, then multiply 4, OPERATION. 
the square, by itself, obtaining 16, the 4tb 
power of 2, and then multiply 16, the 4th power, 
by itself, giving 256, the 8th power of 2. This 
method may be applied to all powers higher j^r 

than the third, and is much more convenient in 15 

practice. Thus, in finding the 5th power, we ~256 
may take the product of the 2d and 3d powers, 
or the product of the square by the square by the first power ; 
in finding the 6th power, we may cube the 2d power, or square 
the 3d power, or multiply the 4th power by the square, etc. 

CASE II. Squaring Numbers and finding the law. This 
case may be solved by two distinct methods. The first 
consists in separating the number into its elements of units, 
tens, etc., and multiplying as in algebra so as to exhibit the 
law by which the parts are involved. The second method per- 
forms the process of involution as determined by the building 
up of a figure in geometry. These two methods may be dis- 
tinguished as the algebraic and geometric, or the analytic and 
synthetic methods. The ultimate object of these methods 
is to derive a law of involution by which we may be able to 
derive methods of evolution. These two methods apply both to 
the squaring and cubing of numbers. The synthetic method 
cannot be extended beyond the cubing of numbers; the analytic 
method is general and will apply to all powers, but is of no 
practical use in arithmetic beyond the cube. We will, there- 
fore, apply these two methods only to the squaring and cubing 
of numbers. 

ANALYTIC METHOD. By the Analytic Method of squaring 
numbers, we separate the number into its units, tens, etc , and 



THE PHILOSOPHY OF ARITHMETIC. 



keep these elements distinct in the involution of the number, 
so that the law of the process becomes apparent. To illustrate, 
find the square of 25. 

Twenty-five equals 20+5, or 2 tens 
and 5 units. Writing the number as 
in the margin, and multiplying by 5 
and by 20, and taking the sum of these 
products, we have 20 2 +2 (5x20)+5 2 . 
From this we .see that the square of a 



OPEBATION. 

20+5 
20 + 5 

5X20 + 5 2 
20M-5X20 

20 2 +2(5x20)+5 s 



number consisting of tens and units, equals the tensf+S times 
tens x units+units*. 

If we involve in the same manner a number consisting of 
hundreds, tens, and units, we shall find the following law: 
The square of a number consisting of hundreds, tens, and units 
equals hundreds' +2 x hundreds x fens+fens 2 +2 X (hundreds-\- 
tens) x units+units 1 . 

SYNTHETIC METHOD. The Synthetic Method of solving the 
same problem is as follows: Let the line AB represent a 
length of 20 units, and BH, 5 units. 
Upon AB construct a square: the 
area will be 20 2 =400 square units. 
On the two sides DC and BC con- 
struct rectangles each 20 units long 
and 5 broad, the area of which will be 
5 X 20=100, and the area of both will 
be 2x100=200 square units. Now 
add the little square on CG, whose 

area is 5 2 =25 square units, and the sum of the different areas, 
400+200+25=625, is the area of a square whose side is 
25. 

When there are three figures, after completing the second 
square as above, we must make additions to it as we did to the 
first square. When there are four figures there are three addi- 
. tions, etc. 

CASE III. Cubing Numbers to find the law. This case 




B6H 



INVOLUTION. 265 

may also be solved bj two distinct methods, as in squaring 
numbers, which we distinguish as the analytic and synthetic 
methods. The former involves the number by the method of 
algebra ; the latter by the principles of geometry. 

ANALYTIC METHOD. By the Analytic Method we resolve the 
number into its elements of units, tens, etc., and keep it in 
this form as we perform the process of involution, that we 
may exhibit the law by which the elements of a number entei 
iuto its cube. To illustrate, find the third power of 25. 

Resolving the number OPERATION. 

into its units and tens 25 2 =20 2 + 2(5x20) + 5* 

and squaring as above, we 20+5 

have 20 2 +2(5x20)+5 2 . 5x 20*+2x5 2 X 20+5 3 

Multiplying the square by 20 3 +2x5x20'+5 2 x20 

5 and then by 20, and 20 3 +3x5x20'+3x5'x20+5 3 
taking the sum of the products, we have the cube of 25, as 
given in the margin. Examining the result, we see that the 
cube of a number of two digits equals tens 3 +3Xtens*Xunits+ 
3x tensXunits*+units 3 . 

Cubing a number of three digits, we obtain the following law:. 
The cube of a number of three digits equals hundreds 3 +3x 
hundreds 2 X tens + 3 X hundreds Xiens 2J rtens 3 +3X(hundreds-{- 
tensfXunits+3 X(hundreds+tens)X units 2 +units 3 . 

SYNTHETIC METHOD. By the Synthetic Method we use a 
cube to determine the process of involution. To illustrate, let 
us find the cube of 45 by this method. 

Let A, Fig. 1, represent a cube whose sides are 40 units; its 
contents will be 40 3 = 64000. We then wish to increase the 
size of this cube so that its sides will be 45 units. To in- 
crease its dimensions by 5 units, we must add first the three 
rectangular slabs, B, C, D, Fig. 2; 2d, the three corner pieces, 
E, F, G, Fig. 3 ; 3d, the little cube H, Fig. 4. The three slabs 
B, C, D, are 40 units long and wide and 5 units thick; hence, 
their contents are 40 2 X 5X3=24000; the contents of the cor- 
ner pieces, E, F, G, Fig. 3, whose length is 40 and breadth 
12 



266 



THE PHILOSOPHY OF ARITHMETIC. 



40 X 5 s X 3 =3000, and 

Fig. 1. 



and thickness 5, equal 
of the little cube 
H, Fig. 4, equal 
5 3 =125; hence the 
contents of the 
cube represented by 
Pig. 4 are 64000+ 
24000 + 3000+125= 
91125. Therefore, 
the cube of 45, etc. 
Here we see that 
40 s is the cube of 
the tens ; 40 2 x 5 X 3 
is tens 3 x units x 3 ; 
40x5 2 x3is3xtens 
X units*; and 5 s is 
units 3 ; hence we have, as before, the 
cube of a number of tens and units 
equals fens 3 +3 x tens 2 x units+B x tens 
J + units 8 . 



the contents 

Fig. 2. 




When there are three figures in the 
number, we complete the second cube 



OPEEATION. 

40 3 =64000 

40 2 X 5x3=24000 

40x5 2 x3= 3000 

5 3 = 125 



Hence, 45 3 =91125 

as above, and then make additions and complete the third in 
the same manner. If there are still some figures, and no more 
blocks to make additions, let the first cube represent the cube 
already found, and then proceed as at first. 



CHAPTER VII. 

EVOLUTION. 

"INVOLUTION is the process of finding one of the several 
J-J equal factors of a number. It is an analytic process, the 
converse of the process of Involution. Involution is a synthe- 
sis of equal factors ; Evolution is an analysis into equal factors. 
The former is a special case of composition; the latter is a 
special case of factoring. One finds its origin in multiplica- 
tion ; the other in division. Both are contained in the primary 
synthetic and analytic ideas, and are the result of pushing for- 
ward and specializing those notions. 

Any one of the several equal factors of a number is called 
a root of that number. The degree of a root depends upon the 
number of equal factors. The square root of a number is one 
of its two equal factors. The cube root of a number is one of 
its three equal factors, etc. These definitions are regarded as 
an improvement upon the old ones, that the square root of a 
number is a number which multiplied by itself will produce 
the number, and similarly for the other roots. Evolution may 
also be defined as the process of finding any required root of 
a number. 

Symbol. The Symbol of Evolution is ^/, called the radical 
sign. This sign was introduced by Stifelius, a German math- 
ematician of the 15th century. It is a modification of the 
letter r, the initial of radix, or root. Formerly, the letter r 
was written before the quantity whose root was to be extracted, 
and this gradually assumed its present form, v / - 

To indicate the degree of the root to be extracted, a figure 

(267) 



268 THE PHILOSOPHY OF ARITHMETIC. 

is prefixed to the radical sign; thus, jS, $/, J/, etc., denote 
respectively the square root, cube root, fourth root, etc. This 
figure is called the index of the root, because it indicates the 
root required. The index of the square root is usually omitted, 
perhaps because the symbol was applied to the square root for 
some time before its use was extended to the higher roots. 
The roots of numbers are also indicated by fractional expo- 
nents; as 4^, 8*, etc. 

(7ases. Each different root might be regarded as constitut- 
ing a distinct case, but it is most convenient to treat the sub- 
ject under three general cases, as in Involution. These three 
cases correspond to those of Involution, and may be formally 
expressed as follows : 

I. To extract any root of a number when it can be conven- 
iently resolved into its prime factors. 

II. To extract the square root of a number when it can not 
be conveniently factored. 

III. To extract the cube root of a number when it can not 
be conveniently factored. 

Treatment. The general method of treatment is to analyze 
the number into the parts required. In the first case, we ana- 
lyze the number into its prime factors, and then make a syn- 
thesis of some of these factors. In the second and third cases, 
we separate the number into parts by several distinct methods, 
corresponding to those of Involution. 

CASE I. To extract any root when the number can be readily 
factored. This case is solved by resolving the number into 
its prime factors, and then involving the factors so as to obtain 
the equal factor required. For the square root we take the 
product of each of the two equal factors ; for the cube root we 
take the product of each of the three equal factors, etc. 

Thus, to find the square root of 
1225, we resolve the number into its OPERATION. 

prime factors, 5, 5, 7, 7, and take the l J 86 H^S* X J > l! 

c 1 A / Sq. rt. =5x7=35 

product of one of the two 5's and one of 

the two 7's, giving us 5 x 7, or 35. 



EVOLUTION. 269 

To find the cube root of 1728 OPERATION. 

we resolve the number into its ^ 28 = 3 * 3 * 3x4x 4x4 

, L-U. rt.=oX 4=iz 
prime factors, as shown in the 

margin, and take the product of one of the three 3's, and one 
of the three 4's, giving 3X4, or 12. In a similar manner we 
find any root of any perfect power that can be resolved into 
its prime factors. 

CASE II. To extract the Square Root of a number. The 
Square Eoot of a number is one of the two equal factors of the 
number. The square root of a number may also be deflned to 
be a number which, used as a factor twice,- will produce the 
given number. The former definition is somewhat analytic ; 
the process of thought is from the number to its elements. The 
latter is rather synthetic ; the process of thought is from the 
elements to the number. 

The method of extracting the square root of a number con- 
sists in analyzing the number into two equal multiplicative 
parts. This is done by first finding the highest term of the 
root, taking its square out of the number, and using it, accord- 
ing to the laws of involution, to determine the next term of 
the root, etc. The method being found in all the works on 
arithmetic, need not be stated here. 

Explanation. There are two 'methods of deriving the rule 
for square root, or of explaining the reason for the operation. 
These methods are distinguished as the Analytic and Synthetic 
methods. The former consists in resolving the number into 
its elements by the laws obtained by the analytic method of 
involution ; the latter consists in finding the root by means of 
a geometrical diagram by reversing the process of the corres- 
ponding method of involution. The synthetic method will 
apply to both the square and cube root of numbers, but cannot 
be extended beyond the cube root. The analytic method is 
general, and can be applied to the determining of any root of 
a number. 

In order to determine how many figures there are in the root, 



270 THE PHILOSOPHY OF ARITHMETIC. 

and where to begin the extraction of the root, we employ the 
following principles: 

1. The square of a number consists of twice as many fig- 
ures as the number, or of twice as many less one. 

This principle may be demonstrated as follows: Any integral 
number between 1 and 10 consists of one figure, and any num- 
ber between their squares, 1 and 100, con- 
sists of one or two figures: hence the 1 2 =1 

10 2 =100 
square of a number of one figure is a num- inn 2 i n nnn 

ber of one or two figures. Any number io00 2 =l 000000 
between 10 and 100 consists of two figures, 
and any number between their respective squares, 100 and 
10,000, consists of three or four figures; hence, the square of 
a number of two figures is a number of three or four figures, 
etc. Therefore, etc. 

2. If a number be pointed off into periods of two figures 
each, beginning at units place, the number of full periods, 
together with the partial period at the left, if there be one, will 
equal the number of places in the square root. 

This is evident from Prin. 1, since the square of a number 
contains twice as many places as the number, or twice as many 
less one. 

ANALYTIC METHOD. By the analytic method of explaining 
the process of extracting the square root of numbers, we re- 
solve the number into its elements, and derive the method of 
operation by knowing the law of the synthesis of these elements. 
It is appropriately named the analytic method, because it ana- 
lyzes a number into its elements, and operates by reversing the 
synthetic process of involution. We will illustrate this method 
by extracting the square root of 625. 

Explanation. By the principles of involution we see that 
there will be two figures in the root, hence the number con- 
sists of the square of the tens plus the units of the root, which 
equals the square of the tens, plus twice the tens into the 
units, plus the square of the units. The greatest number of 



EVOLUTION. 271 

tens whose square is contained in 625 f-f 2fw|-w 2 =: 6*25(25 

is 2 tens ; squaring the tens and sub- < 2 = 20 J =400 

tracting we hare 225, which equals Ztu+u* =225 

twice the tens into the units, plus the 2t=^0 

i/ == 5 
square of the units. Now, since 2^-4-^=225 

twice the tens into the units is usually 

much greater than the units squared, 225 consists principally 
of twice the tens into the units; hence if we divide 225 by 
twice the tens, we can ascertain the units. Twice the tens 
equals 20x2, or 40; dividing 225 by 40, we find the units to 
be 5, etc. 

In the margin the law of the involution of the elements is 
shown by the use of the letters t and u, the initials of tens and 
units. This representation of the law of the formation of the 
number enables us to separate it into its elements. 

SYNTHETIC METHOD. By the synthetic method we use a 
geometrical figure and derive the process from the method of 
forming a square v/hose area shall equal the given number. It 
is called synthetic because we commence with a smaller square 
and add parts to it, until we find a square of the required area. 
The method of forming the square will give us a method of 
finding the square root. To illustrate, let it be required to ex- 
tract the square root of 625. 

Explanation. The greatest number of tens whose square 
is contained in 625 is 2 tens. Let A, Fig. 1, represent a square 
whose sides are 2 tens or 20 units, its area 
will be the square of 20, or 400. Subtract- 
ing 400 from 625, we have 225, hence our 
square is not large enough by 225 ; we must 
therefore increase it by 225. To do this we 
add the two rectangles B and C, each of 
which is 20 units long, and since they near- 
ly complete the square, their area must be nearly 225 units ; 
hence, if we divide by their length we can find their width. 
Their length is 20x2=40, hence their width is 225-*- 40 or 5 




272 THE PHILOSOPHY OP ARITHMETIC. 

Now complete the square by the addition of the little corner 
square whose side is 5 units, and then the entire length of all 
the additions is 40+5, or 45 units, and multiplying by the 
width we find their area to be 225 square units. Subtracting, 
nothing remains; hence, the side of a square which contains 
625 square units is 25 units. 

The same method will apply when there are more than two 
figures in the root. The methods of operation indicated by 
both the analytic and synthetic methods of explanation, are 
the same. These methods give the usual rule for the extrac- 
tion of the square root. 

CASE III. The Cube Hoot of Numbers. The Cube Hoot of 
a number is one of the three equal factors of the number. 
The cube root of a number may also be defined to be a number 
which, used as a factor three times, will produce the given 
number. Again, the cube root of a number may be defined as 
a number which, raised to the third power, will produce the 
given number. These definitions are all correct, though they 
differ in idea. The first is analytic ; the thought is from the 
number to its elements. The second and third are synthetic; 
the process of thought is from the elements to the number. 

The method of extracting the cube root of a number consists 
in analyzing it and finding one of its three equal multiplicative 
parts. This is done by first finding the highest term of the 
root and taking its cube out of the number, then finding the 
second term by means of the first term, taking their combina- 
tion out of the number, etc. There are several methods of 
doing this, the three most important of which may be distin- 
guished as the Old Method, a New Method, and Homer's 
Method. There are several other methods, which I do not 
regard of sufficient importance to consider in this work. 

Old Method. The Old Method is so called because it is the 
one which has for a long time been taught and practiced. It 
may be distinguished by the use of 300 and 30 in finding trial 
and complete divisors. By a slight modification of the method 



EVOLUTION. 273 

the ciphers of these multipliers may be omitted, and this form 
of the method is now generally preferred. The method may 
be stated as follows: 

RULE. I. Separate the number into periods of three figures 
each, beginning at units place. 

II. Find the greatest number whose cube is contained in the 
left-hand period ; place it at the right and subtract its cube 
from the period, and annex the next period to the remainder 
jor a dividend. 

III. Take 3 times the square of the first term of the root 
regarded as lens for a TRIAL DIVISOR; divide the dividend by 
it, and place the quotient as the second term of the root. 

IV. Take 3 times the last term of the root multiplied by the 
preceding part regarded as tens; write the result under the 
trial divisor, and under this write the square of the last term 
of the root ; their sum will be the COMPLETE DIVISOR. 

V. Multiply the COMPLETE DIVISOR by the last term of the 
root; subtract the product from the dividend, and to the 
remainder annex the next period for a new dividend. Take 
8 times the square of the root now found, regarded as tens, for 
a trial divisor, and find the third term of the root as before; 
and thus continue until all the periods have been used. 

Explanation. This process of extracting cube root maybe 
explained by two distinct methods, distinguished as the ana- 
lytic and synthetic methods. The analytic method consists in 
resolving the number into its elements by the laws obtained 
from the analytic method of involution. The synthetic method 
consists in ascertaining the different terms of the root by the 
building up of a geometrical cube. 

In order to determine the number of figures in the root and 
with what part of the number to begin the evolution, it is 
necessary to state and demonstrate the following principle: 

1. The cube of a number consists of three times as many 
figures as the number, or of three times as many less one ot 

two. 

18 






274 



THE PHILOSOPHY OF ARITHMETIC. 



a Q00 



This principle may be demonstrated as follows: Any inte- 
gral number between 1 and 10 consists of one figure, and any 
integral number between their cubes, 1 
and 1000, consists of one, two, or three 
figures; hence the cube of a number of 
one figure is a number of one, two, or 
three figures. Any number between 10 and 100 consists of 
two figures, and any number between their cubes, 1000 and 
1,000,000, consists of four, five, or six figures; hence the cube 
of a number of two figures consists of three times two figures, 
or three times two, less one or two figures. 

2. If a number be pointed off into periods of three figures 
each, beginning at units place, the number of full periods 
together with the partial period at the left, if there be one, will 
equal the number of figures in the root. 

This is evident from Prin. 1, since the cube of a number eon- 
tains three times as many places as the number, or three times 
as many, less one or two. 

ANALYTIC METHOD. By the analytic method of explaining 
the process of extracting the cube root of numbers, we resolve 
the number into its elements and derive the process by knowing 
the law of the synthesis of these elements in the process of 
involution. We \yill illustrate the method by the solution of 
the following problem: Required the cube root of 91125. 

Solution. Since the cube of 
a number consists of three limes 
as many places as the number 
itself, or of three times as many 
less one or two, the cube root of 
91125 will consist of two places, 
and hence consist of tens and 



91-125(40 
40 3 =64 000 _5 

40 2 X3=4800|27125l5 
40X5X3= 600 
5 2 = 25 



5425 



27125 



units, and the given number will consist of the cube of the tens, 
plus three times the square of the tens into the units, plus three 
times the tens into the square of the units, plus the cube of the 
rinitf 



EVOLUTION. 275 

The greatest number of tens whose cube is contained in the 
given number is 4 tens. Cubing the tens and subtracting, we 
have 27125, which equals three times the square of the tens 
into the units, etc. Now, since three times the square of the 
tens into the units is much greater than all the rest of the ex- 
pression, 27125 must consist principally of three times the 
square of the tens into the units ; hence if we divide by three 
times the square of the tens we can ascertain the units. Three 
times the tens squared equals 3X40 2; =4800; dividing by 
4800 we find the units to be 5. We then find three times the 
tens into the units equal to 40x5x3=600, and units squared 
equals 5 2 =25. Taking the sum and multiplying by the units, 
we have 27125, and subtracting, nothing remains. Hence the 
cube root of 91125 is 45. From this solution we readily derive 
the rule given above. 

SYNTHETIC METHOD. By the synthetic method of explana- 
tion we use a geometrical figure, a cube, and derive the process 
from the method of forming a cube whose contents shall equal 
the number of units in the given number. The number is 
regarded as expressing the number of cubic units in a cubical 
block, the number of linear units in whose side will be the 
cube root of the number. It is appropriately called synthetic, 
since we begin with a cube and add parts to it until we find a 
cube of the required contents. The method of forming the 
cube indicates the process of finding the cube root. This 
method may be illustrated by the solution of the problem 
already given: Required the cube root of 91125. 

Solution. We find the number of figures in the root aa 
before, and then proceed as follows : The greatest number of 
tens whose cube is contained in the given number is 4 tens. 

Let A, Fig. 1, represent a cube whose sides are 40, its con- 
tents will be 40 3 =64000. Subtracting from 91125 we find a 
remainder of 27125 cubic units; hence, the cube A is not large 
enough to contain 91125 cubic units by 27125 cubic units; we 
will therefore increase it by 27125 cubic units. 



276 



THE PHILOSOPHY OF ARITHMETIC. 



Fig. i. 




Fig. 3. 



Fig. 4. 




To do this we add 
the three rectangular 
slabs B, C, D, Fig. 
2, each of which is 
40 units in length 
and breadth ; and 
since they nearly 
complete the cube, 
their contents must 
be nearly 27125; 
hence, if we divide 
27125 by the sum of 
the areas of one of 
their faces as a base, 
we can ascertain 
their thickness. 

The area of a face of one slab 
is 40*=1600, and of the three, 
3X1600=4800; and dividing 
27125 by 4800 we have a quo- 
tient of 5; hence the thickness 
of the additions is 5 units. We 
now add the three corner pieces 
E, F, and G, each of which is 40 units long, 5 wide, and 5 
thick; hence the surface of a face of each is 40X5=200 square 
units, and of the three it is 200X3=600 square units. 

We now add the little corner cube H, Fig. 4, whose sides are 
6 units, and the surface of a face is 5 2 =25. We now take the 
sum of the surfaces of the additions, and multiply this by the 
Common thickness, which is 5, and we have their solid contents 
equal to (4800+600+25)X 5=27125. Subtracting, nothing 
remains; hence the cube which contains 91125 cubic units is 
40+5 or 45 units on a side. 

When there are more than two figures we increase the size 
of the new cube, Fig. 4, as we did the first, or let the first 
cube, Fig. 1, represent the new cube, and proceed as before. 



OPERATION. 

91-125(40 
40 3 =64 000 5 



40 2 X3=4800 
40X5X3= 600 

5 2 = 25 



5425 



27125 45 



27125 



EVOLUTION. 277 



Z'n'fc Ai\v>riOb CtjMiAAiiD. These two methods of explain- 
h.^ tfce Droenss ot extracting the square and cube roots of num- 
bers arc entirely distinct: thsy tiio based upon different ideas, 
though they give rise to the samt practical operation. The 
synthetic method is the one gcnerm]} given in the text-books 
on arithmetic; the analytic method was, until recently, confm-ed 
to algebra. It has been a question whk-L of these methods of 
explanation is the better, some preferring tho one and some the 
other. In my own opinion the analytic method is to be pre- 
ferred for several reasons, among which the following may be 
stated : 

First, it is in accordance with the genius of aiithmetic; we 
explain an arithmetical subject upon arithmetical principles. 
By the synthetic method we leave the subject of arithmetic, 
and bring in geometry to explain arithmetic. Should it be 
said in reply that by the analytic method we arc explaining 
arithmetic by algebra, let it be remembered that algebra has 
been called "universal arithmetic," and that all the algebra 
that is here used is purely arithmetical. In other words, 
though we may indicate the analysis of the number by letters, 
the idea is purely an arithmetical one, and is in no way depend- 
ent upon the principles of algebra as different from arithmetic. 

Second, I hold that a full, complete, and thorough insight 
into the subject can be obtained only by the analytic method. 
The geometric method indicates the process, as well as the 
analytic; but the analytic method shows the nature of the pro- 
cess, it exhibits the law of the formation of the square or cube 
as a pure process of arithmetic; and this gives a deeper in- 
sight into the subject than can be obtained by the other method. 
One who knows evolution only by the synthetic method, does 
not know it thoroughly. 

Third, the analytic method is general; it will explain the 
method of extracting all roots. The geometrical method is 
special; it enables us to extract the square and cube roots 
only. Thus, the square root is regarded as the side of a 



278 



THE PHILOSOPHY OF ARITHMETIC. 



Fig. 1. 



equare, the cube root as the side of a cube; but we have no 
geometrical conception of the fourth root, no figure correspond- 
ing to the fourth power, and therefore no idea of a fourth root; 
and so on for the higher roots. 

In respect of the comparative difficulty of the two methods, 
it may be remarked that it is generally supposed that the syn- 
thetic method is much easier than the analytic. This, however, 
I very much doubt; and this opinion is founded, not only upon 
theory, but also upon the experience of those who have tried 
both methods. I believe that a thorough knowledge of the 
subject can be gained much sooner by the analytic than by the 
synthetic method. My observation has been that pupils often 
are able to run over the geometrical explanation without really 
understanding it. It is, therefore, recommended that the ana- 
lytic method be introduced into our text-books and systems 
of instruction. 

The so-called synthetic methods of 
evolution may also be presented in 
an analytic form. Thus, instead of 
adding to the square A, page 271, 
we can begin with the large square, 
take out the square A, then obtain 
the width of the rectangles and the 
dimensions of the corner square, and 
then subtract. Indeed, this seems 
the more natural method, and is now 
being adopted by American writers. 
When thus presented, it would be 
better to call the two methods the al- 
gebraic and geometric methods. 

The same may be illustrated in the 
extraction of the cube root. Let Fir 

O 

I represent a cube which contains 

91125 cubic units. Taking out the 

cube, A (40 3 = 64000), we have a solid, Fig. 2, representing 

27125 cubic units. This solid consists principally of the three 




EVOLUXION. 



279 



Kig. 3. 




slabs, B, C, and D, each 40 units in 
length and breadth. Dividing 27125 
by the sum of the areas of a face of 
each, (3X40* = 4800), we find their 
thickness is 5 units. Removing the 
slabs, there remain three solids, Fig. 
3, each 40 units by 5 units, hence the 
surface of a face of the three is 3 X 40 
X 5 = 600 square miles. 

Removing E, F, and G, there re- 
mains the small cube H, Fig. 4, the 
surface of one of whose faces is 5 2 = 
25 square units. Multiplying the sum 
of all these surfaces by the common 
thickness, 5, we have (4800 +600+25) 
X 5 = 27125 cubic units. 

NEW METHOD OP CUBE ROOT. I will now present a method 
of extracting cube root which is much more convenient than the 
ordinary one. The simplification consists in finding a general 
method of obtaining the trial and true divisors, so that any one 
divisor may be used in obtaining the next following divisor. In 
the operations the trial divisor is indicated by t. d., and the true 
divisor by T. D., the local value of the terms being distinguished 
by their position. The reason for the method of obtaining the 
trial and true divisors may be readily shown by the formula. 

1. Extract the cube root of 14706125. 

Solution We find as before the number of terms in the root 
and the first term of the root and cube, subtract and bring down 
the first period. 

We then find as before the trial divisor, 12, by taking 3 times 
the square of the first term, and, dividing, find the second term 
of the root to be 4. We then take 3 times the product of the first 
and second terms and the square of the second term, and add 
these to the trial divisor as a correction to obtain the true divisor, 
1456. We then multiply 1456 by 4, and subtract and bring 
down the next period. 



260 



THE PHILOSOPHY OF ARITHMETIC. 



12_ 
24 



To find the next trial divisor, 
we take the square of the last term, 
which is 16, and add it to the 
previous true divisor and the two 
corrections (which were added to 
the previous trial divisor), and we , .... 
have 1728 as the next trial divisor. jg_ 

To find the t,ue divisor, we add 1728- 
3 times the product of the last 360 
term of the root into the previous 25 



OPERATION. 

14-706-125(245 



t. d. 



T. D. 



(5706 



5824 



t. d. 



882125 



882125 



part of the root, and also the square 176425 T. D. 

of the last term, and have 176425 

for the true divisor. Multiplying by 5, we have 882125. 

The method is indicated in the following formulas, which show 
the formation of the trial and true divisors. 

1. TRUE DIVISOR = TRIAL DIVISOR + PRODUCT + SQUARE. 

2. TRIAL DIVISOR=SQUARE+TRUE DIVISOR+CORRECTIONS. 
To show the method with large numbers, extract the cube root 

OPERATION. of 145780276447. 

145-780-276-447(5263 



125 



75 



30 



4_ 



t. d. 



7804 
4 



T,D. 



20780 



15608 



8112- 



t. d. 



Solution We find the first 
term of the root, the first trial 
divisor, and the first true di- 
visor, as before. 

To find the second trial di- 
visor, we take the sum of the 
square of 2, the true divisor, 
and the previous correction, 
and we have 8112. We find the 
next true divisor by adding the 
usual corrections to the trial 
divisor, and have 820596. 

We find the third trial di- 
visor by taking the sum of the 
square of 6, the previous true 
divisor, and the corrections, and have 830028. We find the next 
true divisor as before. 



936 

36- 
820596 
36 



T.D. 



5172726 



4923576 



830028- 
4734 



9- 



t. d. 



830501 49 T. D. 



249150447 



249150447 



EVOLUTION. 



281 



OPERATION. 



We present another method involving the principle of using the 
previous work for obtaining trial and complete divisors. A part 
of this method is easily remembered by the formulas. 

COMPLETE DIVISOR = TRIAL DIVISOR + CORRECTION. 

TRIAL DIVISOR = CORRECTION + COM. DIVISOR -f SQUARE. 

The method is indicated in finding the cube root of 14706125. 

Solution. We find the number of figures in the root, and the 
first term of the root, as in the preceding methods. 

We write 2, the first term 
of the root, at the left at the 
head of Col. 1st ; 3 times its 
square with two dots an- 
nexed, at the head of Col. 
2d ; its cube under the left- 
hand period ; then subtract 
and annex the next period 
for a dividend, and divide 
it by the number in Col. 2d, 
as a trial divisor, for the second term of the root. 

We then take 2 times 2, the first term, and write the product, 
4, in Col. 1st, under the 2, and add ; then annex the second 
term of the root to the 6 in Col. 1st, making 64, and multiply 64 
by 4 for a correction, which we write under the trial divisor ; and 
adding the correction to the trial divisor, we have the complete 
divisor, 1456. We then multiply 1456 by 4, subtract the product 
from 6706, and annex the next period for a new dividend. 

We then square 4, the second figure of the root, write the 
square under the complete divisor, and add the correction, the com- 
plete divisor and the square for the next trial divisor, which we 
find to be 1728. Dividing by the trial divisor we find the next 
term of the root to be 5. 

We then take 2 times 4, the second term, write the product 8 
under the 64, add it to 64, and annex the third term of the root 
to the sum, 72, making 725, etc. 

A part of this method can be easily remembered by means 



IST COL. 
2 
4 


2DCOL 14-706-125(245 
12. . t. d. o 

056 




5824 


64 
8 


1456 c. D. 
16 






LI 20 . . t. Q. 

3625 


882125 
882125 


176425 c. D. 



282 



THE PHILOSOPHY OF ARITHMETIC. 



of the following formulas, which show the formation of the 
trial and complete divisors : 

1. Trial Divisor+Correction=Complete Divisor. 

2. Correction + Complete Di visor +Square= Trial Divisor. 
To show the application of the method we will extract the 

cube root of 41673648563. 



1ST COL. 

3 
6 


2D COL. A 

27 . . t. d. o 

r 376 n 


l-673-648-563(3467 
7 
4673 

2304 


94 

8 


3076 c. D. 

L lfi i 


1026 
12 




3468 . . t. d. 
r 6156 


2369648 
211773C 


10387 


352956 c. D. 

36 

359148 ..t.d. 
72709 


251912563 
251912563 


35987509 c. D. 



HORNER'S METHOD. Horner's Method of extracting the 
cube root was derived from a method of solving cubic equa- 
tions invented by Mr. Homer, of Bath, England. It was first 
published in the Philosophical Transactions for 1819; Under 
the title of "A New Method of Solving Numerical Equations 
of all orders by Continuous Approximations." Its inventor, 
Mr. W. G. Homer, was a teacher of mathematics at Bath ; he 
died in 1837. It is considered one of the most remarkable 
additions made to arithmetic in modern times. DeMorgan 
says that the first elementary writer who saw the value of 
florner's method was J. R. Young, who introduced it in 
an elementary treatise on algebra, published in 1826. Among 
the first to introduce it into arithmetic in this country was 
Prof. Perkins, of New York. 

This method differs from both of those already explained, 
and possesses merits which strongly recommend it for general 



EVOLUTION. 



283 



adoption. It is very concise the root of a large number can 
be extracted with one-half of the work required by the old 
method. Its conciseness arises from the fact that it proceeds 
upon a principle which enables us to make use of work already 
obtained, while the old method requires new calculations every 
time we find a trial or true divisor. In other words, it is an 
organized method by which the work is so economized that no 
operations are superfluous, but each result obtained is made 
use of in obtaining a subsequent result. 

It is entirely general in its character, applying to the extrac- 
tion of all the higher roots. This method can be explained 
both analytically and synthetically. It is presented in several 
of the higher arithmetics, and need not be stated here. It is 
more difficult to remember than either of the other methods, 
and this is perhaps the principal objection to its general adop- 
tion. The " New Methods " for cube root they do not apply to 
higher roots are, however, preferred to Horner's Method, being 
quite as concise, and much more readily acquired and remem- 
bered. 

APPROXIMATE ROOTS. The invention of rules for approxi- 
mating to the square and other roots of numbers, where those 
roots are surds, was a favorite speculation with earlier writers 
on arithmetic and algebra. These rules will be most readily 
understood and their relative values seen by stating them in 
algebraic language. 

1. The rule given by the Arabs is expressed by the formula, 

. y> 



This approximation gives the root in excess; but to increase 
its accuracy, we may repeat the process, making use of the 
root obtained. This is the rule given by Lucas di Borgo, and 
subsequently by Tartaglia, who derived it in common with the 
rest of h.is countrymen from Leonard of Pisa. 



284 THE PHILOSOPHY OF ARITHMETIC. 

2. The rule given by Juan do Ortega, 1534, is expressed by 
the followin formula: 



This approximation is in defect, but, generally speaking, more 
accurate than the former. 

3. The third method of approximation was proposed by 
Orontius Fincus, Professor of mathematics in the university 
of Paris, and who long enjoyed an uncommon reputation in 
consequence of his having introduced the knowledge of the 
mathematics of Italy among his countrymen. His method 
consisted in adding 2, 4, 6, or any even number of ciphers to the 
number whose root was required, and then reducing the num- 
ber expressed by the additional figures of the root resulting 
from these ciphers, to sexagesimal parts of an integer. Thus, 
in extracting the square root of 10, he would get 3 1 1G2, which 
reduced to sexagesimals, became 3. 9'. 43". 12'". 

This is the most remarkable approximation to the invention 
of decimals which preceded the age of Stevinus. If the 
author had stopped short at the first separation of the digits 
in the root, it would have expressed the square root of 10 to 
3 decimal places; but the influence of the use of sexagesimals 
diverted him from this very natural extension of the decimal 
notation, and retarded for more than half a century this im- 
provement in the science of numbers 

The method of Fineus excited the attention of contempora- 
neous mathematicians, who in adopting it, however, did not 
reduce the result to sexagesimals, but merely subscribed, as a 
denominator to the whole not considered as integral, 1 with half 
as many ciphers as had been added in the operation, giving 
x/10=f^^. It is under this form that it is noticed by Tar- 
taglia and.Recorde. Pclletier also, a pupil of Orontius Fincus, 
after noticing the second of the two methods of approxima- 
tion, describes this as more accurate and less tedious than any 
other. 

Methods of approximation were also quite numerous for the 






EVOLUTION. 285 

extraction of the cube root. That of Lucas di Borgo may be 
seen from the formula, 



which Tartaglia says he got from Leonard of Pisa, who had i*. 
from the Arabians; and he expresses his surprise that he 
should have committed so grievous an error, unless he had done 
so without consideration. 

The method of Oroutius Fineus is represented by the follow- 
in formula: 



which errs as much in excess as that of Di Borgo in defect. 
The method of Cardan is indicated by the formula, 



which Tartaglia criticises with great bitterness, as might nat- 
urally be expected from one who had been so treacherously 
defrauded by him of an important discovery, the general 
method of solving cubic equations. His own method is rep- 
resented by the formula, 



which, though more accurate than that of Cardan, errs in defect 
while the other erred in excess. 

In later times, methods of approximation have been proposed 
which give results much more accurate than any of the pre- 
ceding. One of the very best that we have met is the follow- 
ing, given by Alexander Evans, in the January number of The 
Analyst, 187G: 

N r 
For square root, --f- 

N 2r 
For cube root, ^ + ~y 

N nl 
For nth root, - -r 

.n-1 



286 THE PHILOSOPHY OF ARITHMETIC. 

To illustrate these formulas we will extract the square root 
of 2 and the cube root of 6. Suppose the square root of 2 is 
nearly 1.4, then r=1.4, and substituting in the formula we 
have 

N r 2 1 I* 99 



which is the correct root to four places ; and by substituting ffi 
in the formula we get the root correct to eight places. 

In extracting the cube root of 6, suppose that r=1.8, then 
substituting in the formula we have 

N 2r_ 6 2(i$)_50 6 

+ -' "- 1 ~~- + - 



which is true to three decimal places. The method cannot be 
relied upon, however, to give many correct terms in the ap- 
proximation. In applying it to the cube root of 3, regarding 
1.4 as the value of r, we obtain for the root, 1.44353, which 
is true to only two places. If we then take 1.44 as the value 
of r, we shall find the next approximation to be 1.442253, 
which is true to four places. If we take r=1.5 as the cube root 
of 4, the formula gives the first approximation 1.5925, which is 
true to only the first decimal place. If we had taken r=1.6, 
we would have obtained 1.5875, which is correct to three 
places. The best method is therefore the general one ; for a 
person who is familiar with the method which I have given 
under the name of the New Method will extract the root more 
rapidly than he can with the approximate methods, and may 
be always certain of the correctness of his result. 



PART III. 

COMPARISON. 



SECTION I. 
RATIO AND PROPORTION. 

SECTION II. 
THE PROGRESSIONS. 

SECTION III. 
PERCENTAGE. 

SECTION IV. 
THEORY OF NUMBERS. 



SECTION I. 

RATIO AND PROPORTION. 



19 



I. INTRODUCTION. 



II. NATURE OF RATIO. 



III. NATURE OF PROPORTION 



IV. APPLICATION OF PROPORTION. 



V. COMPOUND PROPORTION, ETC. 



VI. HISTORY OF PROPORTION. 



CHAPTER I. 

INTRODUCTION TO COMPARISON. 

A RITHMETIC consists fundamentally of three processeB ; 
JL\. Synthesis, Analysis, and Comparison. Synthesis and 
Analysis are mechanical processes of uniting and separating 
numbers ; Comparison is the thought process which directs the 
general processes of synthesis and analysis, and unfolds the 
various particular processes contained in them. Comparison 
also gives rise to several processes which do not grow out of 
the general operations of synthesis and analysis, but which 
have their origin in the thought process itself The principal 
processes originating in Comparison, are Eatio, Proportion, 
Progression, Percentage,. Reduction, and the Properties of 
Numbers. The particular manner in which these processes 
originate will appear from the following considerations. 

If two numbers be compared with eac,h other, we perceive 
a definite relation existing between them, and the measure of 
this relation is called Eatio. Numbers may be compared in 
two ways: first, by inquiring how much one number is greater 
or less than another ; ajid secondly, by inquiring how many times 
one number equals another. Thus, in comparing 6 with 2, we 
see that 6 is four more than 2, and also that 6 is three times 2. 
These relations, expressed numerically, give us the ratio of the 
numbers. The former is called arithmetical ratio ; the latter, 
geometrical ratio. The term ratio is generally restricted, how- 
ever, to a geometrical ratio, and it will be thus used here. 

The comparison of ratios gives rise to several distinct pro- 
cesses called Proportion. If two equal ratios be compared, 

( 291 ) 



292 THE PHILOSOPHY OF ARITHMETIC. 

the numbers producing the ratios being retained in the com- 
parison, we have what we call a Geometrical Proportion, or 
simply a Proportion. When the ratios are simple, we have a 
Simple Proportion; when one or both of the ratios are com- 
pound, we have a Compound Proportion. 

If we wish to divide a number into several equal parts, bear- 
ing a certain relation to each other, we have a process called 
Partitive Proportion. If we wish to combine numbers in 
certain definite relations, we have a process called Medial Pro- 
portion, usually known as Alligation. If we compare num- 
bers so that each consequent is of the same kind as the next an- 
tecedent, we have a process known as Conjoined Proportion. 

If we have a series of numbers differing by a common ratio, 
we may investigate such a series and ascertain its laws and 
principles; thus arises the subject of Progressions. If the 
ratio is arithmetical, the progression is called an Arithmetical 
Progression; if the ratio is geometrical, the progression is 
called a Geometrical Progression. 

Again, as was shown in the Logical Outline of arithmetic, we 
may take some number as a basis of comparison, and develop 
the relations of numbers with respect to this basis. It has 
been found convenient in business transactions to use one hun- 
dred as such a basis of comparison, which gives rise to the 
subject of Percentage. In Fractions and Denominate Numbers 
we have units of different values under the same general kind 
of quantity. By comparing these, it is seen that we may 
pass from a unit of one value to one of a greater or less value, 
and thus arises the process of Reduction. When we pass from 
a less to a greater unit the process is called Seduction Ascend- 
ing ; when we pass from a greater to a less unit, the process 
is called Reduction Descending. 

By a comparison of numbers, we may also discover certain 
properties and principles which belong to numbers per se, and 
also other properties and principles which have their origin in 
the Arabic system of notation. Such principles may be em- 



INTRODUCTION TO COMPARISON. 



293 



braced under the general head of the Properties of Numbers. 
It is thus seen that several divisions of the science of numbers 
are not contained in the original processes of synthesis and 
analysis that is, of addition and subtraction but have their 
roots in and grow out of the thought-process of comparison. 
These several subjects, evolved from the comparison of num- 
bers, will be considered in the order in which they have beec 
mentioned. 



CHAPTER H. 

NATURE OF RATIO. 

O ATIO originates in the comparison of numbers. It is the 
-Lv numerical measure of their relation. From it arise some 
of the most important parts of arithmetic, as proportion, pro- 
gressions, etc. Its importance, and the inadequate and diverse 
views held concerning it, make it necessary to give quite a care- 
ful and thorough discussion of the subject. 

Definition. Ratio is the measure of the relation of two 
similar quantities. This definition differs in one respect essen- 
tially from that usually given. Ratio is generally defined as 
"the relation of two quantities" relation and ratio being 
made equivalent. This is not accurate, or, at least, not suffi- 
ciently definite. The word ratio is a more precise term than 
relation, as will appear from the following illustration. If we 
inquire what is the relation of 8 to 2, the natural reply is " 8 
is four times 2 ;" but if we inquire what is the ratio of 8 to 2, 
the correct reply is "four." Here the ratio four is the num- 
ber which measures the relation of 8 compared with 2. It is 
thus seen that ratio is not merely the relation of two similar 
quantities, but the measure of this relation. This definition, 
presented in the author's own text-books, has already been in- 
troduced by one or two writers, and seems not unworthy of 
general adoption. 

The Terms. A ratio arises from the comparison of two 
similar quantities. These quantities are called the terms of the 
ratio. The first term is called the Antecedent ; the second term 
is called the Consequent. The antecedent is compared with 

(294) 



NATURE OF RATIO. 295 

the consequent; the consequent is the basis or standard of 
comparison. Thus, a ratio indicates the value of the first 
quantity as compared with the second as a standard. The 
ratio, therefore, expresses how many times the consequent must 
be taken to produce the antecedent, or what part the antece- 
dent is of the consequent. In other words, it answers the 
question the antecedent is how many times the consequent, or, 
the antecedent is what part of the consequent ? From this it 
also appears that the ratio equals the antecedent divided by 
the consequent. Thus, the ratio of 6 to 3 is 2, and the ratio of 
3 to 6 is . 

Method of Eatio. The question has recently been raised 
whether the correct method of determining a ratio is to divide 
the antecedent by the consequent or the consequent by the an- 
tecedent. An eminent author advocates the division of the 
consequent by the antecedent, and this method has been adopted 
by several American mathematicians. The old method some 
of them call the " English Method ;" the new method, the 
"French Method." The so-called "French Method" we be- 
lieve to be incorrect in principle and inconvenient in practice 
The correct method of finding the ratio of two numbers is to 
divide the antecedent by the consequent. Several reasons will 
be given in favor of the correctness of this method, which seem 
to us conclusive. For convenience in the discussion, let us 
distinguish the two methods as the Old and the New method. 

1. Nature of Ratio. First, I think the correctness of the 
Old Method will appear from the nature of ratio itself. If we 
inquire " What is the relation of 8 to 2 ?" the natural reply is, 
" 8 is four times 2." Here the number four is the measure of 
the relation; hence the ratio of 8 to 2 is four. If the inquiry 
is, '^What is the relation of 2 to 8?" the natural reply is "2 is 
one-fourth of 8 ;" hence in this case, the ratio is one-fourth. 
From this view of the subject it follows that the correct method 
cf determining a ratio is to divide the antecedent by the conse- 
quent, and not the consequent by the antecedent. 



296 THE PHILOSOPHY OF ARITHMETIC. 

If I ask the relation of 8 to 2, it would be illogical to reply, " & 
is one-fourth of 8," for this does not answer my question. To 
giving the reply, that number should be used first in making the 
comparison which was used first in the question, and it would 
be illogical and absurd to invert the order ; yet this is really what, 
those who advocate the other method must do. If the ratio of 
8 to 4 is one-half, then when I ask the question, "What is the 
relation of 8 to 4 ?" they must say, "4 is one-half of 8," unless 
it be supposed that they would say, " 8 is one-half of 4." 

This may be impressed by an illustration suggested by Prof. 
Dodd. Of two persons, A and B, suppose A to be the father 
and B the son. Now if the question be asked, " What is the 
relation of A to B ?" the correct reply is "A is the father of 
B," and it would be inconsistent to answer, "B is the son of 
A," for that is the reply to the question, " What is the relation 
of B to A ?" The same holds in regard to the comparison of 
numbers, and with even greater force, since it is necessary to 
be more explicit in science than in ordinary conversation. 
Hence, if the question is asked, " What is the relation of 8 to 
2 ?" the correct reply is, " 8 is four times 2 ;" from which we 
see that the ratio is four. It is clear, then, that the ratio of 
two numbers, which is the measure of the relation of the first 
to the second, is equal to the first divided by the second. 

2. Law of Comparison. The true method of determining 
a ratio may also be shown by the nature and object of the com- 
parison. The law of comparison is to compare the unknown 
with the known; thus, we logically write #=4, and not 4=a;. 
Now, in a ratio, one number is made the basis of comparison, 
the object being to comprehend or measure the other number 
by its relation to the basis. In this sense the basis may be 
regarded as the known quantity, and the other number as the 
unknown quantity. Now the unit is the basis of all numbers ; 
it is the standard by which all numbers are measured ; we un- 
derstand a number only as we know its relation to the unit. 
When any number, as 8, is presented to the mind, we compare 



NATURE OF RATIO. 297 

It with the unit, not the unit with it. The inquiry is, 8 is how 
many times one ? hence 8 is the first number named in the com- 
parison ; it is, therefore, the antecedent, and the ratio is the 
quotient of the antecedent by the consequent. The advocates 
of the new method of ratio would have us compare the 1 with 
the 8, the unit of measure with the thing to be measured, the 
known with the unknown. This is not only awkward, but it 
is directly opposed to the established principles of logical 
thought. 

3. Authority. One of the strongest arguments in favor of 
the division of the first term by the second is the usage of 
eminent mathematicians. That signification of scientific terms 
which custom has fixed should not be changed but for the 
strongest reasons. From the earliest periods of science, math- 
ematicians have divided the antecedent by the consequent. It 
was the method employed by Euclid, Pythagoras, and Archi- 
medes, the three great mathematicians of antiquity; and by 
Newton, LaPlacc, and LaG range, the three great mathemati- 
cians of modern times. The English and German, and nearly 
all the French mathematicians, employ this method, and have 
done so from the earliest periods. One or two French, and a 
few American authors have adopted the New Method; but 
with these few exceptions, the Old Method is the method of 
mathematicians at all times and in every country where the 
ratio of numbers has been employed. 

But not only is the authority of numbers upon this side of 
the question, but also the greater weight of the authority of 
eminence. The practice of all of the great mathematicians of 
every age is in favor of the Old Method. In its favor we may 
mention the illustrious names of Euclid, Pythagoras, Archi- 
medes, and to these add the not less illustrious names of Dio- 
phantus, Newton, Leibnitz, LaPlacc, LaGrange, the Bernoullis, 
Lcgendre, Arago, Bourdon, Carnot, Barrow, Ilcrschel, Bow- 
ditch, Pierce, etc.; names which shed a lustre over their country 
and age, and which are symbols of grand achievements in the 
13* 



THE PHILOSOPHY OF ARITHMETIC. 

science. All the great works, the masterpieces which stand as 
monuments of the loftiest triumphs of genius, are upon this side of 
the question. The Principia of Newton, the Mecanique Celeste 
of LaPlace, the Mecanique Analytique of LaGrange, the 
Theorie des Nombres of Legendre, the Analytical Mechanics 
of Pierce, all employ the Old Method. Such universal agree- 
ment among great mathematicians should be regarded as a final 
settlement of the matter. 

4. Inconvenience of the Change. Again, the Old Method 
cannot be changed without confusion. There are definitions 
in science which involve the idea of ratio, and a correct appre- 
hension of these definitions requires a precise idea of ratio. 
These definitions are founded upon the Old Method of ratio ; 
hence, if we change the method of determining ratio, we shall 
either have a wrong idea of the subjects defined, or else the 
definitions must be changed. The latter would be almost a 
practical impossibility, since they have become fixed forms in 
scientific language. Science has embalmed certain definitions, 
and it would seem almost like sacrilege to disturb them. 

Among these definitions may be mentioned those of specific 
gravity, differential co-efficient, index of refraction, and the 
geometrical symbol -. The specific gravity of a body is defined 
to be the ratio of its weight to the weight of an equal volume 
of some other body assumed as a standard. The index of re- 
fraction is the ratio of the sine of the angle of incidence to the 
sine of the angle of refraction. The differential co-efficient is 
the ratio of the increment of the function to that of the varia- 
ble. The geometrical symbol ?r is the ratio of the circumfer- 
ence to the diameter. These definitions have the authority of 
the great masters, and will, without doubt, remain as they are. 
One or two of them have been changed by the advocates of 
the New Method, but such changes will hardly extend beyond 
their own text-books. 

5. Origin of Symbol. It may further be remarked that the 
assumed origin of the symbol of ratio is in favor of the method 



NATURE OF RATIO. 299 

here advocated. It is said that the symbol of ratio is derived 
from that of division ; that is, that : is the symbol -=- with the 
horizontal line omitted. The symbol of division indicates that 
the quantity before it is to be divided by the one following it; 
hence if the theory of the origin is true, it indicates that prima- 
rily the ratio of two numbers was the quotient of the first 
divided by the second ; and this primary method should be 
followed, unless there are good reasons to the contrary. 

In this connection I remark that the Old Method of ratio 
gives us the simplest idea of a proportion. A proportion is 
an equality of ratios, and this idea is most clearly expressed 
thus: 6-v-3=8-=-4. With the other method of ratio, this sim- 
ple idea of a proportion cannot be presented. Whether the 
symbol : is a modification of -r-, is, I presume, not definitely 
known. It is so asserted by some authors ; but so far as I can 
learn, it is not known as a historical fact. It seems very reason- 
able, however, and in some old German works I have noticed 
that the symbol of division is used for indicating the ratio of 
numbers. 

The "French Method," inappropriately so called. These 
two methods of ratio have been distinguished by the names 
"English Method," and "French Method;" the Old Method 
being called the "English Method," and the New Method the 
"French Method." These names were first applied, I think, 
by Prof. Ray, although others had previously stated that the 
French mathematicians made use of the one and the English 
mathematicians of the other method. Both of these names are 
founded in error. The " French Method " is not used by the 
French; the general custom of the French mathematicians 
is opposed to it. Lacroix is the only mathematician of any 
eminence who, so far as I have examined, employs it. The 
" English Method " is not confined to the English, but it is used 
by French, Germans, Prussians, and Austrians, in fact, by the 
mathematicians of all countries, and is, therefore, incorrectly 
named the English Method. 



800 THE PHILOSOPHY OF ARITHMETIC. 

Nearly all the mathematicians of France, it has been said, 
employ the so-called English Method, and all of the most emi- 
nent ones do so. Among these may be mentioned LaPlace, 
LaG range, Legcndrc, Bourdon, Ycruier, Comte, Biot, Carnot, 
Arago, etc. In proof of this, I will quote from some of their 
own works. M. Bourdon, in his Arithmetic, page 222, says, 
"Par exemple, le rapport de 24 a 6 est %-, ou 4 ; et cehii de 6 
a 24 esl T 6 T , ou -. Legcndrc, in his Geometry, Book IV., Prop. 
XIV., says, "done le rapport de la circortference au diametre 
desiyne ci-dessus par TT =3.1415926." Vernier, in his Arith- 
metic, page 118, says, " com me la raison est le quotient qu' on 
obtient quand on diuise P antecedent par le consequent." Other 
authors might be quoted, but these are sufficient to show that 
the so-called French Method is not the method of the French. 
Legendrc and Bourdon are especially referred to, since some 
popular American text-books, supposed to be translations from 
these authors, employ the New Method, and have been instru- 
mental in leading quite a large number of American authors 
and teachers to adopt that method. 

In turning to Lacroix, we sec a departure from the general 
usage of the French mathematicians. In his Arithmetic, which 
is the only work of his that I have examined, he says, page 85, 
in comparing the numbers 13, 18, 130, and 180, we see '-que le 
deuxieme contient le premier aulant de fois que le quatrieme 
conlient le troisieme; et Us for men t ainsi ce qu'on appdle une 
proportion." Notice that he is here discussing the subject of 
proportion, and not the subject of ratio by itself. On the next 
page he remarks, "Je conlinuerai de prendre le consequent du 
rapport pour le numeraleur de la fraction qui exprime le 
rappori et V antecedent pour le denominaleur." 

This places Lacroix upon the opposite side of this question; 
and it is clear from the manner in which he expresses himself, 
that he is conscious of taking a position not authorized by the 
general custom of his countrymen. I think it can readily be 
seen how Lacroix was led into this error He commences the 



NATURE OF RATIO. 301 

subject with a problem in proportion, which he solves by anal- 
ysis, and then, by a mistake plausibly drawn from the process 
of analysis, seeming to think that the analysis dictates a divi- 
sion of consequent by antecedent, he defines his terms and an- 
nounces his method of ratio. The whole discussion is as illog- 
ical as the conclusion is incorrect. He begins the subject with 
proportion instead of ratio, thus inverting the whole problem 
and getting the method of ratio inverted also. The true method 
is to begin by comparing numbers, determining their relations; 
and then comparing their relations, make a proportion ; the first 
will give the true idea of Ratio, and the second of Proportion. 

Answer to Arguments in Favor of the New Method. This dis- 
cussion would be imperfect without an attempt to answer some 
of the arguments which have been presented in favor of the 
so-called "French Method." An eminent author and educator, 
who has done more for the adoption of the New Method than 
any other person in this country, gives a formal defense of it ; 
a few of his arguments I will notice. His first argument, which 
is founded upon the nature of comparison, has already been 
answered in the previous discussion. He says, in comparing 
numbers, "the standard should be the first number named;" 
hence, to comprehend 8, he would compare the basis of num- 
bers, or 1, with 8, instead of comparing the 8 with 1, that is, 
the number with the basis. The mistake he makes is in com- 
paring the standard with the thing measured ; that is, the 
known with the unknown ; the true law of comparison being just 
the reverse of this. 

This will be readily seen in continuous quantity which can 
be clearly understood only by comparing it with some definite 
part of itself assumed as a unit. Thus, suppose a period of time 
is considered ; it is clear that we can get a definite idea of it by 
comparing it with some fixed unit, as a day, or a week, or a 
year. In these cases it will be seen that we do not compare the 
unit with the given quantity, as the author quoted would main- 
tain, but the quantity to be measured with the unit of measure. 



302 THE PHILOSOPHY OF ARITHMETIC. 

His second argument is that the New Method gives a con- 
venient rule for Proportion ; the fourth term being equal to the 
third term multiplied by the ratio of the first to the second. 
The reply is that the Old Method gives just as convenient a rule, 
namely, " The fourth term equals the third divided by the ratio 
of the first to the second." His third argument is, that in a 
geometrical progression the ratio is the quotient of any term 
divided into the following term. This is the most plausible 
argument advanced, and demands special notice. If it be true 
that the ratio of any term to the following term is the quotient 
of the second divided by the first, then it is true that we here 
depart from the general method of ratio ; but still it would not 
follow that the general method of ratio should be changed to 
harmonize with this exceptional case. A more sensible conclu- 
sion would be that the method here used should be changed to 
correspond with the general method. That the general should 
control the special and not the special the general, is a fixed 
law of science. Let us see, however, if the form of writing a 
geometrical progression does present an exception to the 
general method of expressing a ratio. 

In a geometrical progression, the ratio is the measure of the 
relation that any term bears to the preceding term. In the 
series 1, 2, 4, 8, etc., we do not compare the 1 with the 2, the 2 
with the 4, etc., to determine the ratio, as will appear from the fol- 
lowing considerations. Suppose, for illustration, that we wish 
to find any term of the series, as the 5th term, would we not 
reason thus : the 5th term must bear the same relation to the 
4th, that the 4th does to the 3d ; and since the 4th is twice the 
3d, the 5th term must be twice the 4th, or 16. Here we follow 
the law of comparison, by comparing the unknown with the 
known, and reversing the apparent order, name the 8 first and 
the 4 after it. Should we write the comparison out in full, we 
would have 5th : 8 : : 8 : 4. If this is true, then, in a geometri- 
cal series, we do not compare a term with the following term, 
but rather with the term preceding it The ratio of the series, 



NATURE OF RATIO. 303 

it thus appears, is the ratio of any term to the preceding term, 
and not to the term following it. In other words, we compare 
backward, instead of forward, as in ordinary ratio ; and really 
divide the antecedent of the comparison by the consequent tc 
obtain the ratio. 

Some writers explain this apparent departure from the gen- 
eral signification of ratio, by saying that in a geometrical series 
we express the " inverse ratio of the terms." Says one, " It is 
less troublesome to express the common ratio inversely, as then 
one number will suffice." Says another, " Whenever we meet 
with the expression, the 'ratio of a geometrical series,' we are 
to understand the inverse ratio." It seems clearer to me to 
say that the order of writing the terms is in opposition to the 
order of thought. We write one way and compare another 
way. If the expression of the series were dictated by the 
idea of ratio, we would write it from the right toward the left. 

The fact is, however, that in a geometrical progression, it is 
the rate of the progression that we consider, rather than the 
ratio of the terms; that is, the rate at which the series pro- 
gresses, and this term would be preferable to ratio in this con- 
nection. A series of terms, increasing or decreasing by a common 
multiplier, although an outgrowth from the idea of ratio, pre- 
sents an idea not identical with that of ratio. 

This distinction is actually made by several French writers. 
They use the different words rapport and raison ; the former 
to express the ratio of two numbers, the latter to denote the 
rate of the geometrical series. Thus Bourdon, in his Arith- 
metic, page 279, says, " On appelle Progression par Quotient 
une suite de nombres tels que le rapport d'un terme quelconque 
a celui qui le precede est constant dans toute Velendue de la serie. 
Ge rapport constant, qui existe entre un terme el celui qui le 
precede immediatement se nomme la Eaison de la progression." 
Prof. Hcnkle, who has written several excellent articles upon 
this subject, quotes Biot to the same effect. He says of a geomet- 
rical progression, " Le Eapport de chaque terme au precedent se 



304 THE PHILOSOPHY OF ARITHMETIC. 

nomme Baison." It will thus be seen that some of the French 
writers distinguish between ratio and the constant multiplier of 
a progression, and should the word rate be adopted with us, 
we would avoid the objection of this seeming departure from 
the general signification of ratio. 

I have devoted so much space to the discussion of this sub- 
ject, because I think it one upon which there should be uni- 
formity of opinion and practice. Several of our most popular 
elementary text-books on mathematics have adopted the so- 
called " French Method," and are teaching it to the youth of 
the country. Pupils who have been taught the method can 
with difficulty relinquish it, and if they proceed to Philosophy 
and Higher Mathematics they will meet with difficulty in every 
subject containing definitions involving ratio. It is proper to 
remark that since this article was written, now some ten or 
twelve years, several authors who had adopted the new 
method, have discarded it and now use the old method. 



CHAPTER III. 

NATURE OF PROPORTION. 

P)ROPORTION arises from the comparison of ratios. Com- 
L parison begins with comparing numbers, giving rise to the 
idea of relation, the measure of which is ratio. After becoming 
familiar with the idea of the relations of numbers, we begin to 
compare these relations ; when eq\ial relations are compared, 
we attain to the idea of a Proportion. 

Proportion, it is thus seen, has its origin in comparison; it is 
a comparison of the results of two previous comparisons. Every 
proportion involves three comparisons; the two which give rise 
to the ratios, and a third, which compares or equates the ratios. 
All of these comparisons are exhibited in the expression of a 
proportion ; the symbol of ratio in the two couplets showing 
the first two, and the symbol of equality between the couplets 
showing the third. A proportion, therefore, involves four 
numbers, so arranged that it will appear that the ratio of the 
first to the second equals the ratio of the third to the fourth. 
Thus, the ratio of 6 to 3 being the same as the ratio of 8 to 4, 
if they are formally compared, as 6 : 3=8 : 4, we have a pro- 
portion. 

Notation. A proportion may be written by placing the sign 
of equality between the two ratios compared; thus 2 : 4=3 : 6. 
Instead of the sign of equality, the double colon is generally 
used to express the equality of ratios, the proportion being 
written, 2 : 4 : : 3 : 6. The symbol of equality, however, is 
frequently used by the French and German mathematicians, 
and is always to be preferred in presenting the subject to 
20 ( 305 J 



306 THE PHILOSOPHY OF ARITHMETIC. 

learners. A proportion may be read in several different ways 
Thus we may read the above proportion, "the ratio of 2 to 4 
equals the ratio of 3 to 6;" or "2 is to 4 as 3 is to 6." The 
latter is the method generally used. 

Definition.- A. Proportion is the comparison of two equal 
ratios; or, it is the expression of the equality of equal ratios. 
In this expression the numbers that are compared to obtain the 
ratio must be indicated. A proportion is thus seen to be an 
equation, and should be thus regarded. An equation, as gen- 
erally used, expresses the relation of equal numbers ; a pro- 
portion expresses the relation of equal ratios One arises from 
the comparison of quantities ; the other, from the comparison 
of the relations of quantities. The former is an equation 
between equal numbers ; the latter is an equation between equal 
ratios. 

The definition of proportion generally given is, "A propor- 
tion is an equality of ratios." This is true, but it is not suf- 
ficiently definite to constitute a perfect definition. There must 
be not only an equality of ratios, but a formal comparison of 
these ratios, to produce a proportion. This comparison must 
also exhibit the numbers which were compared to produce the 
equal ratios. Thus, the ratio .of 6 to 3 is 2, and the ratio of 8 
to 4 is 2 ; here is an equality of ratios, but not a proportion. 
Again, if we compare the ratios 2 and 2, we have the equation 
2=2, which is not a proportion, since it does not exhibit the 
numbers which produce the equal ratios. To give a proportion, 
it is essential that the ratios be compared, and that the com- 
parison of the numbers which give the ratios be exhibited. 
The mere equating of the ratios is not sufficient; the propor- 
tion must show the numbers which; compared, give rise to 
the equal ratios. A proportion, then, is not only an " equality 
of ratios," but it is a comparison of equal ratios, in which 
the comparison of the numbers compared for a ratio is ex- 
hibited. 

This idea of the exhibition of the numbers compared for the 



NATURE OF PROPORTION. 307 

ratios, though not formally stated in the definition which I 
have presented, may be directly inferred from it. For, if we 
compare as above, 22, so far as we can see, it is merely a 
comparison of numbers, and not a comparison of ratios. It is 
true that every ratio is a number, but the converse is not true; 
hence 2=2 may or may not be the comparison of two ratios. 
Such comparison would be indefinite; therefore, to express 
definitely and clearly the equality of ratios, we must retain the 
numbers compared, to show that the equation is an expression 
of equal ratios, and not a mere comparison of numbers. The 
definition is consequently regarded as sufficiently explicit to 
prevent any misapprehension. . Should we wish to incorporate 
this idea in the definition, we might define as follows: A Pro- 
portion is a comparison of equal ratios, in which the numbers 
producing the ratios are exhibited. 

Kinds of Proportion. There are several kinds of propor- 
tion, resulting from a modification or extension of the pri- 
mary ideas of ratio and proportion. A comparison of three or 
more pairs of numbers having equal ratios, is called Continued 
Proportion. An expression of the equality of compound ratios 
is called Compound Proportion. An Inverse Proportion 
is one in which two quantities are to each other inversely as 
two other quantities. An Harnionical Proportion is one in 
which the first term is to the last as the difference between the 
first and second is to the difference between the last and the 
one preceding the last. We have also Partitive and Medial 
Proportion, which will be defined subsequently. The propor- 
tion requiring special consideration is Simple Proportion, or 
the comparison of two simple ratios. 

Principles. The principles of Proportion are the truths 
which belong to it, and which exhibit the relations between the 
different members. The fundamental principle of Proportion 
is that the product of the means equals the product of the ex- 
tremes. From this we derive several other principles by which 
we can find the value of either of the four terms when the 



308 THE PHILOSOPHY OF ARITHMETIC. 

other three are given. There are many other beautiful princi- 
ples of Proportion, besides this fundamental one and its imme- 
diate derivatives, which are not usually presented in arithmetic, 
but may be found in works on algebra and geometry. They 
are, however, just as much an essential part of pure arithmetic 
as of geometry, and can all be demonstrated as easily here as 
there. Indeed, they belong to arithmetic rather than to geom- 
etry, since a ratio is essentially numerical, and hence should be 
treated in the science of numbers. These principles, it will be 
seen, are not self-evident ; they admit of demonstration. Re- 
membering this, it may be asked, what then becomes of the 
assertion of the metaphysicians, that there is no reasoning in 
pure arithmetic ? 

Demonstration. The fundamental principle of Proportion 
may be demonstrated in two ways. The method generally 
given is the following : Take the proportion 4:2:: 6 : 3. 
From this we have f=f ; clearing of fractions, we have 4x3 
=2x6; and, since 4 and 3 are the extremes, and 2 and 6 the 
means, we infer that the product of the extremes equals the 
product of the means. This is the method generally used in 
algebra and geometry. Although entirely satisfactory as a 
demonstration, the objection might be made that though it 
proves that the products are equal, it does not show why they 
are equal. 

Another method which, in arithmetic, is preferred to the above, 
is as follows : From the fundamental idea of ratio and propor- 
tion, we see that in every proportion we have 2d term x ratio 
. 2d term : : 4th term x ratio : 4th term. Now, in the product 
of the extremes, we have 2d term, ratio, and 4th term, and in 
the product of the means, we have the same factors ; hence 
the products are equal. This is a simple method, clearly seen, 
and shows not only that the products are equal, but that they 
must be so, and why they are so, which the other method does 
not. The products are seen to be equal because in the very 
nature of the subject they contain the same factors. 



NATTJBE OF PEOPORTION. 309 

The same demonstration may be put in the more concise 
language of algebra. Take the proportion a : 6 : : c : d, let r 
= the ratio, then we have a^-br, hence ab.r, and in the 
same way c=d.r ; hence the proportion becomes b.r : b :: d.r 
: d. Now, in the extremes we have 6, r, and d, and in the 
means we have the same factors ; hence the two products will 
be equal. 



CHAPTER IV. 

APPLICATION OF SIMPLE PROPORTION. 

QIMPLE PROPORTION is employed in the solution of prob- 
O lems in which three of four quantities are given, to find the 
fourth. These quantities must be so related that the required 
quantity bears the same relation to the given quantity of the 
same kind that one of the two remaining quantities does to the 
other. We can then form a proportion in which one term is 
unknown, and this unknown term can be found by the principles 
of proportion. Thus, suppose the problem to be, What cost 
3 yards of cloth, if 2 yards cost $8 ? 

Here we see that the OPERATION. 

cost of 3 yards bears the Cost of 3 yds. : $8 : : 3 yds. : 2 yds ; 
same relation to the cost Cost of 3 yds.=-^ =$12. 
of 2 yards that 3 yards 

bears to 2 yards ; nence we have the proportion given in the 
margin, from which we readily find the value of the unknown 
term. 

In all such problems three terms are given to find the fourth ; 
from which Simple Proportion has been called the Rule of 
Three. It was regarded as very important by the old school 
of arithmeticians, and was by them called " The golden rule of 
three." It is now falling into disrepute, the beautiful system 
of analysis having, to a great extent, taken its place. The 
method of analysis is simpler in thought than that of proportion, 
and in many cases is to be preferred to the solution by propor- 
tion, especially in elementary arithmetic; but still the rule of 

(310) 



APPLICATION OF SIMPLE PROPORTION. 311 

Simple Proportion should not be entirely discarded. The 
comparison of elements by proportion affords a valuable disci- 
pline and should be retained for educational reasons ; and 
moreover it is also valuable, if not indispensable, in the solu- 
tion of some problems which can hardly be reached by analysis. 
In algebra, geometry, and the higher mathematics, it is, of 
course, indispensable. 

Position of the Unknown Quantity. It is seen that, iu the 
solution of the preceding problem by proportion, I place the 
unknown quantity in the first term. This is not in accordance 
with general custom; other writers place the unknown quan- 
tity in the fourth term. I have ventured to depart from this 
custom, and to recommend the general adoption of such a depar- 
ture, for reasons which seem to me conclusive. These reasons 
are twofold: first, the method suggested is dictated by the 
laws of logic; and, second, it is more convenient in practice. 
Both of these points will be briefly considered. 

First. The law of correct reasoning is to compare the unknown 
with the known, not the known with the unknown. The ordi- 
nary method begins the proportion with the known quantities, 
thus comparing the known with the unknown, in violation of 
an established principle of logic. The method I have suggested 
commences with the unknown quantity, and thus compares the 
unknown with the known^va. conformity to the laws of thought. 
It seems therefore that the old method is not logically accu- 
rate, and that the correct method of solving a problem in Rule 
of Three is to place the unknown quantity in the first term. 

Second. The method proposed will be found to be much 
more convenient in practice. A proportion is more easily 
stated by beginning it with the unknown term. This will 
be especially appreciated by those who have taught Trigo- 
nometry. In stating a proportion so as to get the required 
quantity in the last term, I have seen pupils try two or three 
statements before obtaining the right one. It cannot be readily 
seen how the proportion should begin so that the unknown 



312 THE PHILOSOPHY OF ARITHMETIC. 

quantity shall come in the last term. If, however, the pupil 
begins the proportion with that which he wishes to find, the 
other terms will arrange themselves without any difficulty. 
Suppose, for instance, that we wish to obtain an unknown 
angle of a triangle. If we reason thus : sine of the required 
angle is to the sine of the given angle as the side opposite the 
required angle is to the side opposite the given angle; the 
pupil will write the proportion without any hesitation. If we 
reverse this order, it is necessary to go through the whole 
comparison mentally before beginning to write, so that we 
may be sure to close the proportion with the required quantity. 
It is therefore believed that the simplest method of stating a 
proportion is to place the unknown quantity in the first term. 

The utility of this change has been frequently illustrated in 
my own experience. I remember, while visiting a young 
women's college, hearing a recitation in geometry in which the 
professor was trying to lead a pupil to state a proportion from 
which a certain line could be determined. The young lady made 
several attempts and failed, when I said, " Professor, let her 
begin with the line she wishes to find." He accepted the sug- 
gestion, and she immediately stated the proportion correctly. 

Several authors suggest that the unknown quantity should 
be placed sometimes in one term and sometimes in another to 
test the pupil's knowledge of the subject. This is a valuable 
suggestion ; but any position of the unknown term except in 
the fourth term they regard not as a general, but as an excep- 
tional method. Their rule is to place the unknown term last; 
any other arrangement is the exception. What I claim is that 
the placing of the unknown quantity in the first term should 
be the rule, and any other arrangement the exception. It is 
recommended also that the teacher require the learner to place 
it in different terms, that he may acquire a clear and complete 
idea of the subject. 

Symbol for the Unknown. Some authors employ the letter 
x in arithmetic as a symbol for the unknown quantity. Thus, 



APPLICATION OF SIMPLE PROPORTION. 313 

in the problem previously presented, we may write x : $8 : : 3 : 
2. This practice is derived from the French, and is commend- 
able. It is sometimes objected, that it is introducing algebra 
into arithmetic ; but such objection, however, is not valid. Al- 
gebra and arithmetic are not two distinct sciences, but rather 
branches of the same science. The former, at least in its ele- 
ments, is but a more general kind of arithmetic ; and it is not 
at all improper to introduce its concise and general language 
into arithmetic. I think it well, with younger pupils, to ex- 
press the unknown term in an abbreviated form as is indicated 
in the previous solution ; when pupils become familiar with 
this, I would use the symbol a; as a representative of it. 

Three Terms Statement. It is seen that in the solution of 
the given problem in proportion, I use four terms in the state- 
ment. Many authors, however, use only three terms in stating 
a proportion. This was the method of the old authors, when 
rules reigned and principles were ignored, in what might be 
called "the dark ages" of arithmetic. Several recent writers 
have broken away from the old usage, and write the proportion 
with four terms instead of three. It is unnecessary to say 
that the old method was incomplete and incorrect. An ex- 
pression is not a proportion unless it has four terms. The old 
method was merely mechanical, and gave the pupil no idea, or 
at least a very imperfect idea, of the true nature of proportion. 
The sooner the new method is generally adopted the better for 
science and education. 

Method of Statement. No subject in arithmetic is so illogi- 
cally presented as Simple Proportion in its application to the 
solution of problems. In the statement of the proportion, all 
reasoning seems to be completely ignored, and the whole thing 
becomes a mere mechanical operation for the answer. The pro- 
cess is as follows: "Write that number which is like the answer 
sought as the third term ; then if the answer is to be greater 
than the third term, make the greater of the two remaining 
numbers the second term and the smaller the first term," etc. 
14 



314 THE PHILOSOPHY OP ARITHMETIC. 

Now, though this might do well enough as a rule for get- 
ting an answer, to require the pupils to explain the solution by 
it, as is done in many instances, is to rob the subject of any 
claims to a scientific process. The pupil thus taught to solve 
his problems has no more idea of proportion than if the subject 
were not presented in the book. The whole process becomes a 
piece of charlatanism, utterly devoid of all claims to science. 
A better rule would be this : Write the number like the answer ; 
if the answer is to be greater, multiply by the greater of the 
other two numbers and divide by the less, etc. This would be 
the better method, since it makes no claims to be a scientific 
process, as the other does. Both methods are absurd as a pro- 
cess of reasoning in Arithmetic ; but the latter less so, since it 
makes no pretensions to be a reasoning process. 

What then is the true method ? I answer, if a pupil cannot 
state a proportion by actual comparison of the elements of the 
problem, he is not prepared for proportion, and should solve 
the question by analysis. If he uses proportion, he should use 
it as a logical process of reasoning, and not as a blind mechan- 
ical form to get the answer. He should then be required to 
reason thus: Since the cost of 3 yds. bears the same relation to 
the cost of 2 yds. that 3 yds. bear to 2 yds., we have the pro- 
portion, cost of 3 yds. : $8 : : 3 yds. : 2 yds. 

If this is not evident and cannot be readily seen, then we 
should dispense with proportion until the pupil is old enough 
to understand it, and require the problems to be solved by analy- 
sis. If the unknown quantity be placed in the last term we 
would reason thus : Since 2 yds. bear the same relation to 3 
yds. that the cost of 2 yds. bears to the cost of 3 yds, we have 
the proportion, 2 yds. : 3 yds. : : $8 : cost of 3 yds. 

Cause and Effect. A new method of explaining proportion 
has recently been introduced into arithmetic, which may be 
called the method of Cause and Effect. All problems in pro- 
portion, it is said, may be considered as a comparison of two 
causes and two effects; and since effects are proportional to 



APPLICATION OF SIMPLE PROPORTION. 315 

causes, a problem is supposed to be readily stated in a propor- 
tion. To illustrate, take the problem, If 2 horses eat 6 tons of 
hay in a year, how much will 3 horses eat in the same time ? 
Here the horses are regarded as a cause and the tons of hay as 
an effect, and the reasoning is as follows: 2 horses as a cause 
bear the same relation to 3 horses as a cause, that 6 tons as an 
effect, bears to the required effect ; from which we have a pro- 
portion and can determine the required term. 

This method was first introduced into arithmetic by Prof. 
H. N. Robinson, and has been adopted by several authors. 
The same idea was presented by an arithmetician of Verona, 
who distinguished the quantities into agents and patients. It 
is supposed that it tends to simplify the subject, enabling 
learners more readily to state a proportion than by a simple 
comparison of the elements. This supposition, however, is not 
founded in truth. Instead of simplifying the subject, the method 
of cause and effect really increases the difficulty and tends to 
confuse the mind. It lugs into arithmetic an idea foreign to 
the subject, to explain relations which are much more evident 
than the relation of cause and effect. 

Another objection to the method is that the relation of quan- 
tities as cause and effect is often rather fancied than real. In 
many cases, indeed, there is no such relation existing at all. 
Take the problem, "If a man walks 6 miles in 2 hours, how far 
will he walk in 5 hours ?" Will the pupil readily see which 
is the cause and which the effect ? Will the advocates of the 
method, tell us whether the 6 miles or the 2 hours are to be 
regarded as the cause? Or take the problem, "If 18d. ster- 
ling equal 36 cts. TJ. S., what are 54d. sterling worth?'' 
Would not the pupils be puzzled to tell which is the cause and 
which the effect? Indeed, there is no relation of <cause and 
effect in a large number of such problems ; and any effort to 
establish such a relation will confuse that which is simple and 
easily understood. 

If anything further is needed to show the incorrectness of 



316 THE PHILOSOPHY OF ARITHMETIC. 

the method, take a problem in what is called Inverse Proportion 
Thus, "If 3 men do a piece of work in 8 days, in what time will 
6 men do it?" Here 3 men and 8 days would be regarded as 
the first cause and effect, and 6 men and the corresponding 
number of days as the second cause and effect. Now, if we 
form a proportion, we have the first cause is to second cause as 
the second effect is to the first effect; from which we see that 
in this case like causes are not to each other as like effects, a 
conclusion which completely contradicts the fundamental prin- 
ciple of the relation of cause and effect. 

Inverse Proportion. There is a class of problems which give 
rise to what is called Inverse Proportion. In this the two 
quantities of the same kind are to each other, not directly as the 
other two quantities in the order of their relation, but rather 
inversely as those quantities. Thus, in the problem, " If 3 
men build a fence in 12 days, in what time will 9 men build 
it?" Here we have the required time is to 12 days, not as 9 
men to 3 men, but as 3 men to 9 men; that is, inversely as 
the order indicated by the order of the terms of the first couplet. 
This is sometimes called Reciprocal Proportion, since the quan- 
tities are as the reciprocals of 9 and 3 ; that is as -^ to ^ or 3 to 9 

Many problems in Inverse Proportion may, however, be 
stated in a direct proportion. To illustrate, take the problem 
just solved. Now, if 3 men do a piece of work in 12 days, in 
1 day they will do fa of it, and if a number of men do a piece 
of work in 4 days, in 1 day they will do \ of it ; hence, since 
the number of men are to each other as the work done, we have 
the direct proportion, "the number of men required is to 3 men, 
as \ to -fa," from which we can readily find the term required. 
If, in this proportion, we multiply the second couplet by 48, it 
will become 12 : 4, which gives the same proportion as that 
which was obtained by the method of inverse proportion. It 
is thus seen that, in some cases at least, the method of inverse 
proportion may be avoided, and the problem be expressed by a 
direct proportion. 



APPLICATION OF SIMPLE PROPORTION. 317 

If, however, in the above problem the number of men in 
both cases had been given, and the number of days in one case 
required, the problem could not be conveniently stated in a 
direct proportion, since to do so would require the reciprocal 
of the unknown quantity. Should this quantity be represented 
by an algebraic symbol, however, we could still state the pro- 
portion directly, and readily find the unknown quantity. 

Proportion distinctly Arithmetical. The subject of propor- 
tion is purely an arithmetical process. Ratio is a number, 
hence proportion, arising from the comparison of ratios, must 
be numerical. These ratios may arise from comparing con- 
tinuous or discrete quantities, hence we may have a propor- 
tion wherein geometrical quantities are compared. Attention 
is called to the fact, however, that the principles of proportion 
are only generally true with respect of numbers. A propor- 
tion in geometry, comparing four surfaces or volumes, may be 
true, but the principles of a proportion can have no meaning in 
such a case. In taking the product of the means equal to the 
product of the extremes, we shall have one surface or one vol- 
ume multiplied by another, which can mean nothing unless 
they be regarded as numbers. In geometry we regard the 
product of two lines as giving a surface, and the product of a 
line and surface as giving a volume; but what idea can we 
attach to the product of two surfaces or two volumes ? It is 
thus seen that Proportion is essentially a process of numbers, 
and is, therefore, a branch of Pure Arithmetic. Since the 
principles of Proportion admit of demonstration, we inquire 
again what becomes of Hansel's assertion that " Pure Arith- 
metic contains no demonstration ?" 



CHAPTER Y. 

COMPOUND PROPORTION. 

A COMPOUND PROPORTION is aproportion in which one 
or both ratios are compound. It is employed in the solu- 
tion of problems in which the required term depends upon the 
comparison of more than two elements. In Simple Proportion 
the unknown quantity depends upon a comparison of two ele- 
ments forming one pair of similar quantities; in Compound 
Proportion it depends upon the comparison of several elements 
forming two or more pairs of similar quantities. 

A Compound Ratio has been defined as the product of two 
or more simple ratios. The expression of a compound ratio is 

(2 4) 

Jfi " 101 ' ^ Suc ^ a ra ^ be compared to an equal simple 

ratio, or if two such compound ratios be compared with each 

( n . r> -\ 

other, we have a compound proportion. Thus -< ~ | ~ > : : 7 : 56 

and I K . i A r : : j 7 i 1 4 r are examples of compound propor- 
tion. In these expressions we mean that the value of the first 
couplet equals the value of the second; thus, in the first pro- 
portion we have f X f or ^ equals ^g- ; in the second, | x y 5 ^ = 

fxA 

The subject of Compound Proportion has been even more 
unscientifically treated, if possible, than Simple Proportion. In 
no work upon Arithmetic, and indeed in no work upon Algebra, 
have I seen the subject presented in a really scientific manner. 
As a general thing, problems are given under the head of com- 
nound proportion, to be solved either mechanically by rale, or 
else by analysis, which, of course, is not compound proportion. 

(318) 



COMPOUND PROPORTION. 319 

The principles of a compound proportion are not developed, 
and in its application it is regarded, not as a scientific process, 
but as a machine for working out the answer. This, of course, 
is not as it should be. Compound Proportion is just as much 
a scientific process as Simple Proportion, and demands just as 
logical a treatment. I will enforce what I mean by calling 
attention to a few of the principles df such a proportion, and 
then showing its scientific application. 

Principles. In Compound Proportion we have certain defi- 
nite scientific principles, as in Simple Proportion. A few of 
these principles will now be stated. 

1. The product of all the terms in the means equals the pro- 
duct of all the terms in the extremes. To show the truth of 
this, take the proportion given 

in the margin. From the prin- OPERATION. 

( 9 4.") C R fi~) 

ciples of compound ratio we 1 * ! i n i :: JT-idi 
have | x fV=f X ^ ; and clear. ' ' 2 x ^ x ^ 

ing this of fractions we have 2x5x6x14=3x7x4x10. 
2x5x6xH = 3x 7x4x10, 

which, by examining the terms, we see proves the principle. 
From this principle we can immediately derive two others. 

2. Any term in either extreme equals the product of the 
means, divided by the product of the other terms in the ex- 
tremes. 

3. Any term in either mean equals the product of the extremes 
divided by the product of the other terms in the means. 

Other principles can also be derived, as in Simple Proportion, 
but the three given are all that are necessary in arithmetic. 

Application. In the application of Compound Proportion 
to the solution of problems, we should proceed upon the same 
principles of comparison employed in Simple Proportion. If 
we do not, the process is not Compound Proportion, and should 
not be so regarded. To illustrate the true method, we take the 
problem, "If 4 men earn $24 in 7 days, how much can 14 men 
earn in 12 days?" 



320 THE PHILOSOPHY OF ARITHMETIC. 

In the solution of this problem by Compound Proportion, we 
should reason thus : The sum earned is in proportion to the 
number of men and the time they labor; hence the sum 14 men 
can earn is to $24, the sum that 4 men 
earn, as 14 men to 4 men, and also as OPERATION. 

12 days to 7 days; giving the com- Sum : 24 : : -j ^ ! * 
pound proportion which is presented 24x14x12 

in the margin. From this we find the 4~x~7 

unknown term to be $144. Or we may 

enter a little more into detail, and say The sum 14 men can 
earn in 7 days is to the sum 4 men can earn in 7 days, as 14 men 
is to 4 men; and also the sum 14 men can earn in 12 days is 
to the sum that they can earn in 7 days, as 12 is to 7; hence 
we have the compound proportion given in the margin. 

By Analysis. The subject of Compound Proportion is some- 
what difficult, in fact too difficult, for young students in arith- 
metic. With such the method of analysis should be used 
instead of proportion. The analytical method is clear and 
simple, and will be readily understood. It should be borne in 
mind, however, that when we solve by analysis we are not 
solving by compound proportion, a fact that seems sometimes 
to be forgotten. 

In solving the preceding problems by analysis, it is necessary 
to pass from the 4 men to 14 men, and from the 7 days to 12 
days, the sum earned varying as we make the transposition : to 
do this we pass from the collection to the unit, and then from 
the unit to the collection. The solution is as follows, the work 
being as indicated in the margin. 

If 4 men earn $24 in 7 days one man will earn of $24, and 
14 men will earn 14 times or ^ of OPERATION. 

$24. If 14 men earn ^x $24 in 7 days, Sum= 
in one day they will earn ^ of ^ of $24, 
and in 12 days they will earn 12 times ^ of -^ of $24, which is 
\f- of 3 of $24, which by cancelling, we find equals $144. In- 
stead of putting it in the form of a compound fraction, we could 



PARTITIVE PROPORTION. 



321 



have made the reduction as we passed along ; but in compli- 
cated problems the method here used is preferred, as the can- 
cellation of equal factors will often greatly abridge the process. 

PARTITIVE PROPORTION 

The subject of ratio gives rise to several arithmetical 
processes which have received the name of Proportion. 
Among these we have Partitive Proportion, Conjoined Pro- 
portion, Medial Proportion, Geometrical Proportion, etc. Geo- 
metrical Proportion embraces Simple Proportion, Compound 
Proportion, Inverse Proportion, etc. The other kinds are 
distinguished by their special names. When we speak of pro- 
portion, without any qualifying word, we mean Geometrical 
Proportion. Geometrical Proportion has been treated in the 
preceding part of this chapter ; the other varieties of proportion 
will now be presented. 

The comparison of numbers gives rise to a division of them 
into parts which shall bear a given relation to each other. This 
process has received the name of Partitive Proportion. Parti- 
tive Proportion is the process of dividing numbers into parts 
bearing certain relations to each other. To illustrate, suppose 
it be required to divide 24 into two parts, one of which is twice 
the other. An equivalent problem is, " Given the sum of two 
numbers equal to 24, and one of the numbers twice the other ; 
what are the numbers ?" 

Origin. Partitive Proportion is a process of pure arithme- 
tic ; it originated, however, in the application of numbers to 
business transactions. Partnership is a case of Partitive Pro- 
portion. But, although the subject had its origin in the appli- 
cation of numbers, it is now, in accordance with the law of the 
growth of science, a purely abstract process. 

Gases. This subject embraces quite a large number of cases, 

arising from the various relations that may exist among the 

several parts into which a number is divided. It is evident, 

also, that the greater the number of the parts the more compli- 

21 



322 THE PHILOSOPHY OF ARITHMETIC. 

cated will become the process. The most important cases are 
the following: 

1. When the parts are all equal. 

2. When one part is a number more or less than the other. 

3. When one part is a number of times the other. 

4. When one part is a fractional part of the other. 

5. When the parts are to each other as given integers. 

6. When the parts are to each other as given fractions. 

7. When a number of times one part equals a number of 
times another. 

8. When a fractional part of one equals a fractional part of 
another. 

These simple cases, it is evident, may be combined with each 
other, giving rise to others more complicated than any of these. 
A little ingenuity will suggest a large number of such cases, 
some of which will be quite interesting. 

Method of Treatment. To illustrate the character of one of 
the simple cases and its treatment, let us take a problem and 
its solution. Case 8 will give us a problem like the following: 
Divide 34 into two pans such that | of the first part equals | 
of the second part. The solution of this case is as follows: 
If f of the first equals | of the second, of the first equals i of 
| or | of the second, and f of the first equals f of the second ; 
then | of the second, which is the first, plus 4 of the second, or 

O ' i O ' 

^ of the second part, equals 34, etc. The other cases are solved 
in my Mental Arithmetic, and need not be presented here. 

CONJOINED PROPORTION. 

The comparison of numbers also gives rise to an arithmetical 
process which has received the name of Conjoined Proportion. 
Conjoined Proportion is the process of comparing terms so 
related that each consequent is of the same kind as the next 
antecedent. The character of the subject is seen by the follow- 
ing concrete problem: "What cost 8 apples, if 4 apples are 
worth 2 oranges, and 3 oranges are worth 6 melons, and 4 
melons are worth 12 cents?" 



CONJOINED PROPORTION. 323 

An abstract problem, showing that it is a process of pure 
arithmetic, is as follows : " If twice a number equals 4 times 
another number, and 3 times the second number equals 6 times 
a third number, and 4 times the third number equals 2 times a 
fourth number, and 5 times the fourth number equals 40 ; what 
is the first number ?" 

Method of Treatment. Conjoined Proportion is treated by 
analysis, and presents a very interesting application of the 
analytical method of reasoning. The problems may be solved 
in two ways somewhat distinct ; that is, we may begin at the 
latter part of the problem, and work back, step by step, to the 
beginning ; or we may commence at the beginning of the prob- 
lem and pass from quantity to quantity, in regular order, until 
we find the value of the first quantity in terms of the last. To 
illustrate, the problem given may be solved thus: 

Solution 1. If 5 times the fourth number equals 40, once 
the fourth number equals of 40, or 8, and twice the 4th, which 
equals 4 times the 3d, equals 2 times 8, or 16. If 4 times the 
3d equals 16, once the 3d equals \ of 16, or 4, and 6 times the 
3d or 3 times the 2d equals 6 times 4, or 24 ; and so on until 
we reach once the 1st number. 

Solution 2. If twice a number equals 4 times another, once 
the number equals ^ of 4 times, or two times the 2d ; if 3 times 
the 2d equals 6 times the 3d, once the 2d equals of 6 times, 
or 2 times the 3d, and 2 times the 2d, or the 1st, equals twice 
2 times the 3d, or 4 times the 3d ; and so on until we find once 
the 1st in terms of the given quantity. 

Both of these methods are simple and logical. The first 
method will probably be preferred for its directness and sim- 
plicity. It may also be remarked that these problems can be 
solved by Compound Proportion, and perhaps might have been 
logically treated under that head. 

MEDIAL PROPORTION. 

The comparison of numbers and the combining of them in 
certain relations, give rise to an arithmetical process which 



824 THE PHILOSOPHY OF ARITHMETIC. 

has received the name of Medial Proportion. Medial Propor- 
tion is the process of finding in what ratio two or more quan- 
tities may be combined, that the combination may have a mean 
or average value. 

The subject, in its application, is usually called Alligation, 
from alligo, I bind or unite together, the name being suggested, 
probably, by the method of solution, which consisted of linking 
or uniting the figures with a line. It may, however, have been 
suggested by the nature of the process itself, in which the sev- 
eral quantities are combined. 

Origin. Medial Proportion also originated in the concrete, 
that is, in the application of numbers. Indeed, even now it is 
difficult to present it as an abstract process ; that is, as a process 
of pure number. It is so intimately associated with the combi- 
nation of things of different values, that it is very difficult to 
apply it to the combination of abstract numbers. Still it is 
evidently a process of pure arithmetic ; and its importance and 
distinctive character, even as an application of numbers, lead 
me to speak of it in this connection. 

Gases. The subject presents a number of cases, the most 
important of which are the following: 

1. Given, the quantity and value of each, to find the mean 
value. 

2. Given, the mean value and the value of each quantity, to 
find the proportional quantity of each. 

3. Given, the mean value, the value of each, and the relative 
amounts of two or more, to find the other quantities. 

4. Given the mean value, the value of each, and the quantity 
of one or more, to find the other quantities. 

5. Given, the mean value, the value of each, and the entire 
quantity, to find the quantity of each. 

Method of Treatment. As formerly treated, the subject was 
one of the most mechanical in arithmetic. The old "linking 
process," as presented in the text-books, was seldom understood 
either by teacher or pupil. Recently, however, Prof. Wood, 



MEDIAL PROPORTION. 825 

formerly of the New York State Normal School, has made a 
very happy application of analysis to the solution of this class 
of problems, and poured a flood of light upon the subject, so 
that it is now oue of the most interesting processes of arithmetic. 
It has extended the domain of the subject also, so that it includes 
some of the more difficult cases of Indeterminate Analysis, for 
an illustration of which see my Higher Arithmetic. 

The method of treatment is to compare one number above 
the average with one below it by their relation to the average, 
finding how much must be taken to gain or lose a unit on the 
one and balancing it with the loss or gain of a unit on the other. 
In this way the quantities are balanced around the average, 
and the proportional parts of the combination derived. For 
an illustration of the method of treatment, see my written 
arithmetics. 



CHAPTER VI. 

HISTORY OF PROPORTION. 

l^HE Rule of Three, emphatically called the Golden RUe, 
-L by both ancient and modern writers on arithmetic, is found 
in the earliest writings upon the science of numbers. In the 
Eiilawati the rule is divided, as among modern writers, into 
direct and inverse, simple and compound, with statements for 
performing the requisite operations, which are said to be quite 
clear and definite. 

The terms of the proportion in the Lilawati are written con- 
secutively, without any marks of separation between them. 
The first term is called the measure or argument ; the second 
is its fruit or produce ; the third, which is of the same species 
as the first, is the demand, requisition, desire, or question. 
When the fruit increases with the increase of the requisition, 
as in the direct rule, the second and third terms must be multi 
plied together and divided by the first ; when the fruit dimin- 
ishes with the increase of the requisition, as in the inverse 
rule, the first and second terms must be multiplied together 
and divided by the third. 

No proof of the rule is given, and no reference is made to 
the doctrine of proportion upon which it is founded. Under 
compound proportion is given the rule for five, seven, nine or 
more terms. The terms in these cases are divided into two 
sets, the first belonging to the argument, and the second to 
the requisition ; the fruit in the first set is called the produce 
of the argument ; that in the second is called the divisor of the 
set ; they are to be transposed or reciprocally brought from one 
set to the other, that is, the fruit is to be put in the second set 

and the divisor in the first. 

(326) 



HISTORY OF PROPORTION. 327 

The Rule of Three Direct may be illustrated by the follow- 
ing example : 

If two and a-half palas of saffron be obtained for three- 
sevenths of a nishca, say instantly, best of merchants, how 
much is got for nine nishcas ?* 

Statement : 

359 

T 2 1 Answer, 52 palas and 2 carshas. 

Rule of Three Inverse may be illustrated by the following 
examples: If a female slave, 16 years of age, bring 32 nishcas, 
what will one aged 20 cost? If an ox, which has been worked 
a second year, sell for 4 nishcas, what will one which has been 
worked 6 years cost ? 

1st question. 

Statement : 16 32 20. Answer, 25f nishcas. 

2d question. 

Statement : 2 4 6. Answer, 1 nishcas. 

In order to understand the solution it must be known that 
the value of living beings was supposed to be regulated by their 
age, the maximum value of female slaves being fixed at 16 
years of age, and of oxen after 2 years' work; their relative 
value in the given problem being as 3 to 1. The rule of five 
terms may be illustrated by the following example : If the in- 
terest of a hundred for a month be five, what is the interest of 
sixteen for a year ? 

Statement : 

1 12, or transposing 1 12 

100 16 the fruit, 100 16 

5 5 

the product of the larger set is 960, of the lesser 100 ; the quo- 
tient is -J^ or ^, which is the answer. 

The interest of money, judging from the examples in Brah- 

To understand their problems in rule of three it must be known that a 
j,ala=i carshas ; a car*/o=16 mashat; uiul u fnathnb gunjas, or 10 grain* of 
barley. Also, a nishc(t=\t> dramma* ; a dramma=16 panas ; apana=4 cucimt, 
and u eacini=~X) cowry shells. 



328 THE PHILOSOPHY OF ARITHMETIC. 

megupta and Lilawati, varied from 3^ to 5 per cent, a month, 
exceeding greatly the enormous interest paid in ancient Rome. 
It is also very high in modern India, where it is not uncommon 
for native merchants or tradesmen to give 30 per cent, per 
annum. 

The rule of eleven terms may be illustrated by the following 
example: Two elephants which are ten in length, and nine 
in breadth, thirty-six in girt, seven in height, consume one 
drona of grain ; how much will be the rations of STATEMENT. 
ten other elephants, which are a quarter more in 
height and other dimensions ? The fruit and ^ ^C 

denominator being transposed, the answer is gg 
%$*-. Dr. Peacock remarks that the principle of 7 -^ 
this very curious example would be rather alarm- 1 

ing, if extended to other living beings besides elephants. 

Lucas di Borgo tells us that at his time it was usual for 
students in arithmetic to commit to memory one or other of 
two long rules which he presents. Tartaglia mentions the first 
of these two rules in nearly the same terms as Di Borgo, and 
gives also a third, which, however, differs from it only in ex- 
pression. This rule formed part of the system in the practice 
of this subject, adapted to those who had not sufficient time to 
acquire, genius to comprehend, or memory to retain, the rules 
for the reduction and incorporation of fractious; a system 
reprobated by Tartaglia, and attributed by him partly to the 
ignorance of the ancient teachers of arithmetic at Venice, and 
partly to the stinginess and avarice of their pupils, who grudged 
the time and expense requisite for attaining a perfect under- 
standing of the peculiarities of fractions. 

An arithmetician of Verona, named Francesco Feliciano da 
Lazesio, objects to the memorial rules of Di Borgo as being too 
general in assuming that two of the quantities are of one species, 
and two, including the term to be found, of another species; and 
shows that in some cases they are all of the same denomina- 
tion. He wishes to distinguish the quantities into agents 



HISTORY OF PROPORTION. 329 

and patients, and these again into actual, or present, and future. 
The first term of the proportion is the present agent, and its 
corresponding patient is the second ; the third term is formed 
by the future agent, and its patient is the quantity to be deter- 
mined. This, it will be noticed, is similar to the method of cause 
and effect adopted by some recent authors, and supposed to be 
original with them. 

Di Borgo's method of stating and working a problem may be 
seen in the following example: "If a hundred pounds of fine 
sugar cost 24 ducats, what will be the cost of 975 pounds?" 

via. v a . 

100 24_ 975 

~y~ x ~jf~ \ 

v a . 

o 

QH r V 

01 040 

03400 

23400 (234 ducati. 
950 10000 

23400 100 

1 

The following example of the same process, with fractions in 
every term, is given by Tartaglia : " If 3 pounds of rhubarb 
cost 2 ducats, what will be the cost of 23| pounds?" 



lire. ducati. lire. 

7*T nf 


x 1 1 1 

2 X 1 1 3 


99 


( 
4 

Is 

1 
Partitor 8' 

"9! 

1 


07 
49 
0590 
1330(15 ducati 

844 

[ . 8 

1 000 
1680(20 grossi 
844 
8 


665 
2 


dapartir 1330 
The quantities, in Di Borgo's solution, are exhibited under a 



330 THE PHILOSOPHY OF ARITHMETIC. 

fractional form, for the purpose of making the process more gen 
eral, being equally applicable to fractions and whole numbers. 
It is sufficiently curious that he should have considered it 
necessary to construct the galea for the division by 100. 

Different methods of representing the terms of the proportion 
were adopted by different authors. We will state a few of 
them as illustrating the solution of the problem, "If 2 apples 
cost 3 soldi, what will 13 cost?" Tartaglia states the propor- 
tion as follows : 

Se pomi 2 || val soldi 3 I che valera pomi 13. 
Other Italian authors write the numbers consecutively with 
mere spaces, and no distinctive marks between them ; thus, 
Pomi. Soldi. Pomi. 

2 3 13 

or thus, 

1 ma. 2 da. 3 tia. 

2 3 13 

In Recorde and older English writers, they are written as 
follows : 

Apples. Pence. 

2 3 

13^~ ---.19 Answer. 

Humfrey Baker, 1562, in speaking of the rule, says, "The 
rule of three is the chiefest, and the most profitable, and most 
excellent rule of all Arithmetike. For all other rules have neede 
of it, and it passeth all others; for the which cause, it is sayde 
the philosophers did name it the Golden Rule; but now in these 
later days, it is called by us the Rule of Three, because it re- 
quireth three numbers in the operation." He writes the terms 
thus: 

2 3 13 

The custom which generally prevailed during the 11th cen- 
tury, was to separate the numbers by a horizontal line, as fol 
lows: 



HISTORY OF PROPORTION. 



331 



Apples. 

2 - 



Pence. 
- 3 - 



Apples. 
- 13 



Oughtred, by whom the subject of proportion was very care- 
fully considered, and from whom the sign, : : , to denote the 
equality of ratios, seems to have been derived, states a propor- 
tion as follows : 

2. 3 : : 13 

In still later times the simple dot which separated the terms 
of the ratios, was replaced by two dots, as in the form which is 
now universally employed. 

Compound Proportion, as has been stated, was formerly 
included under the rule of five, six, etc., terms, there being no 
division of the subject into simple and compound proportion. 
To illustrate, take the problem, " If 9 porters drink in 8 days 
L2 casks of wine, how many casks will serve 24 porters 30 
days ?" In solving such problems Tartaglia usually puts the 
quantity mentioned once only in the last place but one, instead 
of in the second place. The statement will appear as follows : 




Divisor, 9x8. Dividend, 12 X 30 x 24 
Quotient, *f|4=120. 

The example, " Twenty braccia of Brescia are equal to 26 
braccia of Mantua, and 28 of Mantua to 30 of Rimini ; what 
number of braccia of Brescia corresponds to 39 of Rimini ?" 
given by Tartaglia, is solved as follows: 

Rimini Mantua Mantua Brescia Rimini 
30 28 26 20 39 




21840 



Answer, 28. 



332 THE PHILOSOPHY OF AKITHMETIC. 

We give another example with its solution derived from the 
same author. " Six eggs are worth 10 danari, and 12 danan 
are worth 4 thrushes, and 5 thrushes are worth 3 quails, and 8 
quails are worth 4 pigeons, and 9 pigeons are worth 2 capons, 
and 6 capons are worth a staro of wheat; how many eggs are 
worth 4 stara of wheat ?" 

960 



1_6 10 12 4 5 3 8 4 9 2 6 4 

622080 Answer, 648. 

ALLIGATION. The rule for Medial Proportion, or Alligation, 
is of eastern origin, and appears in the Lilawati, though under 
a somewhat limited form. It is there called suverna-ganita, or 
computation of gold, and is applied generally to the determin- 
ation of the fineness or touch of the mass resulting from the 
union of different masses of gold of different degrees of fine- 
ness. The questions mostly belong to what we call Alligation 
Medial. The only question given in illustration of Alligation 
Alternate is the following : " Two ingots of gold, of the touch of 
16 and 10 respectively, being mixed together, the weight be- 
came of the fineness of 12 ; tell me, friend, the weight of gold 
in both lumps." 

The rule given for the solution is, " Subtract the effected fine- 
ness from that of the gold of a higher degree of touch, and that 
of the one of the lower degree of touch from the effected fine- 
ness ; tell me, friend, the weight of gold in both lumps ? The 
differences multiplied by an arbitrarily assumed number, will be 
the weights of gold of the lower and higher degrees of purity 
respectively." 

Statement: 16, 10. Fineness resulting, 12. 

If the assumed multiplier be 1, the weights are 2 and 4 
mdshas respectively ; if 2, they are 4 and 8 ; if , they are 1 
and 2 : thus manifold answers are obtained by varying the as- 
sumption. 



HISTORY OF PROPORTION. 333 

This rule, though perfectly distinct and clear, applies to two 
quantities only, and there is no appearance that it was ever 
applied to a greater number; it involves, however, the princi- 
ple of the rule which is now used, recognizes the problem as 
unlimited, and shows in what manner an indefinite number of 
answers may be obtained. The extension of the rule is not 
entirely easy, but much more so than the invention of the orig- 
inal rule itself; the chief honor of the discovery of the rule 
belongs therefore to the mathematicians of Hindostan. The 
general rule was known to the Arabians and was denominated 
Sekis, a term meaning adulterous, inasmuch as it is not con- 
tent with a single, and, as it were, legitimate solution of the 
question. It was sometimes called Cecca by the Italians, who 
appear to have known nothing further of the word than its 
Arabic origin ; and it constitutes the alligation alternate of 
modern books of arithmetic. 

The earlier Italian writers on arithmetic, in imitation of the 
practice of their Arabian masters, have confined the applications 
of this rule almost entirely to questions connected with the mix 
ture of gold, silver, and other metals, with each other. This union 
was designated by the term consolare, which probably originated 
1 iu the dreams of astrologers and alchemists, whu thought it the 
peculiar province of the sun to produce and generate gold ; and 
as the process of the alchemist in transmuting the baser metals 
into gold was supposed to be under the influence of the sun, 
this gradual refinement, which they in common tended to pro 
duce, was designated by the common term consolare. In later 
times, it was applied to silver as well as gold, and still more 
generally to the common union of these metals with copper. 

To illustrate the method of Tartaglia, take the question, "A 
person has five kinds of wheat, worth 54, 58, G2, TO, 76 lire 
the staro respectively; what portion of each must be taken, so 
that the sum may be 100 stara, and the price of the mixture 66 
lire the staro ?" 



THE PHILOSOPHY OF ARITHMETIC. 



1st. In the proportion of the numbers 10, 4, 10, 8 and 16. 
54 58 62 70 76 

10 4 10 8 16 




2d. In the proportion of the numbers 14, 14, 14, 24, 24. 

_76, 
12 



54, 


58, 


62, 


to, 


10 


To" 


10 


12 


4 


4 


4 


8 


~U 


TI 


H 


4 








"24 



24 

Tartaglia has given three other solutions of this example aris- 
ing from a different arrangement of the ligatures. Among the 
English writers the method gradually assumed the form usually 
found in modern text-books. The method of explanation and 
the extension of the process as given in a few modern text- 
books may be ascribed to DeVolson Wood, formerly of the 
New York State Normal School. 

POSITION. Among the most celebrated rules to which Pro- 
portion was applied in the early text-books were those of Single 
and Double Position. These rules have been supplanted in 
this country by the simpler processes of arithmetical analysis, 
but they are still found in English arithmetics; and it has been 
suggested by a no less eminent scholar and mathematician than 
Dr. Hill, that they should be retained in our text-books on 
account of their disciplinary influences. Some historical facts 
concerning this old rule will be interesting to the reader. 

The rule of Single Position is the only one which is found 
in the Lilawati, where it is called Tshtacarman, or operation 
with an assumed number. We shall give a few examples from 
it, which, however, present nothing very remarkable beyond the 
peculiarities of the mode in which they are expressed. 

1. Out of a heap of pure lotus flowers, a third part, a fifth. 



HISTORY OF PROPORTION. 38o 

a sixth, were offered respectively to the gods Siva, Yishnu, and 
the Sun, and a quarter was presented to Bhavani ; the remain- 
ing six were given to the venerable preceptor. Tell me, quickly, 
the whole number of flowers. 

Statement : , -5-, , 4 ; known, 6. 

Put 1 for the assumed number ; the sum of the fractions , 
, , , subtracted from one, leaves ^ ; divide 6 by this, and 
the result is 120, the number required. 

2. Out of a swarm of bees, one-fifth part of them settled on 
the blossom of the cadamba, and one-third on the flower of a 
isilind'hri ; three times the difference of these numbers flew to 
the bloom of a cutaja. One bee, which remained, hovered and 
flew about in the air, allured at the same moment by the pleas- 
ing fragrance of a jasmin and pandanus. Tell me, charming 
woman, the number of bees. 

Statement: i, , ^: known quantity, 1; assumed 30. 

A fifth part of the assumed number is 6, a third is 10, differ- 
ence 4 ; multiplied by 3 gives 12, and the remainder is 2. Thee 
the product of the known quantity by the assumed one, being 
divided by the remainder, shows the number of bees 15. 

The following question is from the Manor an j an a: 

3. The third part of a necklace of pearls, broken in amorous 
struggle, fell to the ground; its fifth part rested on the couch, 
the sixth part was saved by the wench, and the tenth part was 
taken up by the lover ; six pearls remained strung. Say of 
how many pearls the necklace was composed. 

Statement: ^, A, , ^5 remained, 6. Answer, 30. 

Some authors have attributed the invention of the rules of 
position to Diophantus, though it is impossible to discover upon 
what grounds. When we consider the nature and difficulty of 
the problems solved by him, in those parts of his works which 
remain, we are fully justified in supposing that the Greeks had 
some method of analyzing and solving such problems, or they 
would not have proposed them in such number and variety. 

The Arabs were in possession of tho rules for both Single 



336 THE PHILOSOPHY OF ARITHMETIC. 

and Double Position, with all their applications, and in this 
instance had advanced far beyond their Indian masters ; and 
when we consider how small were the additions which they 
usually made to the sciences which passed through their hands, 
we might very naturally be inclined to suppose that their 
knowledge of these rules was derived from the Greeks. There 
is, however, a vast gap in the history of the sciences after 
the time of Theoh, and it is quite impossible to trace with 
certainty their transmission to the Arabs, or to ascertain 
through what channels some portion of Greek astronomy, at 
least, was transmitted to the Hindoos; we must therefore rest 
satisfied with the few hints to be gathered from authors between 
the 7th and 12th centuries, who had access to many writings 
which have since perished. 

The Italian writers on arithmetic derived the knowledge of 
these rules directly from the Arabians, distinguishing them by 
the Arabic name of El Cataym. The questions proposed by 
Di Borgo and Tartaglia are of immense variety, including 
every case of single and double position ; and the rules which 
are given for this purpose are such as would immediately result 
from the formula given in higher algebras. The following 
example is given and explained by Pi Borgo : 

4. A person buys a jewel for a certain number of fiorini, I 
know not how many, and sells it again for 50. Upon making 
his calculation, he finds that he gains 3^ soldi in each fiorino, 
which contains 100 soldi. I ask what is the prime cost. 

Suppose it to cost any sum you choose; assume 30 fiorini, 
the gain upon which will amount to 100 soldi, or \ fiorino: 1 
added to 30 makes 31 ; and you say that it makes 50 between 
capital and gain ; the position is therefore false, and the truth 
will be obtained by saying, if 31 in capital and gain arises from 
a mere capital of 30, from what sum will 50 arise. Multiply 
30 by 50, the product is 1500; divide it by 31, the result is 
48-^-f- ; and so much I make the prime cost of the jewel. 

Tartaglia says that such questions were frequently proposed 



HISTORY OF PROPORTION. 337 

as puzzles by way of dessert at entertainments, and has mixed 
up with his other questions a large number of such problems. 
The practice, from some circumstances, appears to be referable 
to the Greek arithmeticians of the 4th and 5th centuries, and 
perhaps to an earlier period. 

Both Di Borgo and Tartaglia sought to include every possi- 
ble case of mercantile practice under the Rule of Three, giving 
numerous examples and classifying them in various ways. The 
Italians were also the inventors of the rule of Practice, which 
they regarded as an application of the Rule of Three. Tar- 
taglia gives some interesting and practical examples, with var- 
ious ingenious methods of solution. The great convenience 
of these rules for performing the calculations which were con- 
tinually occurring in trade and commerce, made them a favor- 
ite study with practical arithmeticians, and they assumed from 
time to time a constantly increasing neatness and distinctness 
of form. Stevinus, though, speaks of them with some contempt 
as forming "a vulgar compendium of the rule of three, suffi- 
ciently commodious in countries where they reckon by livres, 
sous and deniers." John Mellis, in his addition to Recorde's 
arithmetic, presents the rules of Practice in a very simple and 
complete form, calling attention to them as " briefe rules called 
rules of practise, of rare, pleasant, and commodious effect, 
abridged into a briefer method than hath hitherto been pub- 
lished." Later works gave them still greater compactness and 
brevity, and in Cocker's Arithmetic, published in 1671, after 
his death, and in others printed towards the end of the 17th 
century, they assumed the form in which they are now found 
in English arithmetics. 

The subjects of Partnership and Barter, also treated by an 
application of Proportion, seem to have originated with the 
Italians. They grew out of their business transactions, and in 
many cases were so complicated as to require great skill and 
judgment in their solution. They are interesting as presenting 
the type of nearly all the questions of this kind found in modern 

text-books. 

22 



SECTION II. 

THE PROGRESSIONS. 



I. ARITHMETICAL PROGRESSION. 



II. GEOMETRICAL PROGRESSION. 



CHAPTER I. 

ARITHMETICAL PROGRESSION. 

IN comparing numbers we perceive that we may have a series 
of numbers which vary by a common law ; such a series is 
called a Progression. The more general name for such a suc- 
cession of terms is Series, which is used to embrace every 
arrangement of quantities that vary by a common law, how- 
ever simple or complicated, and whether expressed in numbers 
or in algebraic or transcendental terms. The term Progression 
is preferred in arithmetic, and is restricted to the arithmetical 
and geometrical series. 

The constant relation existing between two or more succes- 
sive terms of the series is called the Law of the progression. 
In the series 1, 2, 4, 8, etc., each term equals the preceding 
term multiplied by 2, and this constant relation constitutes the 
law of the series. It is evident that the law which connects 
the terms of a series may be greatly varied, and that we may 
thus have a large number of different kinds of series. The 
only two generally treated in arithmetic are the Arithmetical 
and the Geometrical series, or progressions. 

Definition, An Arithmetical Progression is a series of 
terms which vary by a constant difference ; as 2, 4, 6, 8, etc. 
The difference between any two consecutive terms is called the 
common difference. In the series given, the common difference 
is 2. The common difference is sometimes called an arithmet- 
ical ratio ; it is better, however, to restrict the use of the word 
ratio to a geometrical ratio, and call this what it really is, a 
difference. 

( 341) 



342 THE PHILOSOPHY OF ARITHMETIC. 

Special attention is called to the definition of an arithmetical 
progression here presented. The definition usually found in 
our text-books is, "An arithmetical progression is a series of 
numbers which increase or decrease by a common difference." 
In the definition proposed the word vary is used to include 
both the increase and the decrease of the terms; and this is re- 
garded as an improvement upon the old definition. It has 
already been adopted by two or three authors, and should be 
generally introduced into our text-books on arithmetic. 

Notation. The English and American authors express an 
arithmetical progression by writing the terms one after another 
with a comma between them. The French, with more pre- 
cision, employ a special notation for it. They place the sym- 
bol, -T-, before the progression and the period (.) between the 
terms. Thus Bourdon writes, 

-=-2. 7. 12. 17. 22. . . 47. 52. 57. 62. 

This method has been introduced into one or two American 
text-books, and may, in time, be generally adopted, though the 
tendency seems to be to adhere to the common form of expres- 
sion. 

Cases. There are five quantities in an Arithmetical Progres- 
sion ; the first term, the common difference, the number of 
terms, the last term, and the sum of all the terms. If any three 
of these are given, the other two can be found from them. 
This gives rise to twenty different cases, in which any three 
terms being given, the other two may be found. These cases 
cannot all be solved by arithmetic, since some of them involve 
the solution of a quadratic equation ; they are, however, very 
readily treated by the principles of algebra. The two principal 
cases in arithmetic are as follows: 

1. To find the last term, having given the first term, the 
common difference, and the number of terms. 

2. To find the sum of the terms, having given the first term, 
the last term, and the number of terms. 

Method of Treatment. The treatment of Arithmetical Pro- 



ARITHMETICAL PROGRESSION. 343 

gression in arithmetic is very simple. We derive the rule for 
finding the last term by noticing the law of the formation of a 
few terms and then generalizing this law. Thus we notice that 
the second term of an arithmetical progression equals the first 
term plus once the common difference, the third term equals 
the first term plus twice the common difference, etc. ; hence we 
infer that the last term equals the first term plus the product 
of the common difference by the number of terms less one. 

In finding the sum of the terms we take a series, then write 
under this series the same series in an inverted order, then 
adding the two series we see that twice the sum of the series 
is the same as the sum of the extremes multiplied by the num- 
ber of terms ; and generalizing this we obtain the rule for find- 
ing the sum. 

In algebra we reason in the same way, except that we employ 
general symbols, and use a general series instead of a special 
one. Expressing the two fundamental rules in general formu- 
lae, we can readily find the rest of the twenty cases by the alge- 
braic process of reasoning. These two simple cases, I think, 
should in arithmetic be expressed in the concise language of 
algebraic symbols. Pupils who have not studied algebra will 
have no difficulty in understanding them. The two rules of 
arithmetical progression are briefly expressed thus: 

1. J = o+(n l).d; a.* = (0+0.5. 



History. Of the origin of the progressions and the methods 
of treatment, but little is known. They were the object of the 
particular attention of the Pythagorean and Platonic arithme- 
ticians, who enlarged upon the most trivial properties of num- 
bers with the most tedious minuteness. Directing their spec- 
ulations, however, to the mysterious harmonies of the physical 
and intellectual world, they passed over, as unworthy of no- 
tice, the solutions of those problems which naturally arise from 
these progressions, and which appear in such numbers in Hin 
doo, Arabic, and modern European books on Arithmetic. 



344 THE PHILOSOPHY OF ARITHMETIC. 

Very little is known concerning the origin of the familiar 
problems usually found under this subject. The problem, 
" How many strokes do the clocks in Venice strike in 24 
hours?" is supposed to be of Venetian origin. The following 
familiar problem is attributed to Bede : " There is a ladder 
with 100 steps ; on the first step is seated one pigeon, on the 
second step two pigeons, on the third step three, and so on 
increasing by one each step ; tell, who can, how many pigeons 
were placed on the ladder." The celebrated problem, "If 
a hundred stones be placed in a right line, one yard apart 
and the first one yard from a basket, what length of ground 
must a person go over who gathers them up singly, returning 
with them one by one to the basket ?" though found in many 
modern text-books, is very old, but its origin is not known. 

The extraordinary magnitude of the numbers which result 
from the summation of a geometrical series is well calculated 
to excite the surprise and admiration of persons who are not 
fully aware of the principle upon which the increase of the 
terms depends; and examples are not wanting among the 
earliest writers, where the rash and ignorant are represented 
as being seduced into ruinous or impossible engagements. 
The most celebrated of these is that which tradition has 
represented as the terms of the reward demanded of an Indian 
prince by the inventor of the game of chess ; which was a 
grain of wheat for the first square on the chess board, two 
grains for the second square, four for the third, and so on, 
doubling continually to sixty-four, the whole number of 
squares. 

Lucas di Borgo solved the question, and found the result 
to be 18446744073709551615, which he reduces to higher 
denominations and finds it equal to 209022 castles of corn. 
Fie then recommends his readers to attend to this result, as 
they would then have a ready answer to many of those 
barbioni ignari de la arithmetica who have made wagers on 
such questions, and have lost their money. 



CHAPTER II. 

GEOMETRICAL PROGRESSION. 

A GEOMETRICAL PROGRESSION is a series of terms 
which vary by a common multiplier ; as, 1, 2, 4, 8, 16, etc. 
The common multiplier is called the rate or ratio of the pro- 
gression ; thus, in the progression given, the rate is 2. The 
rate of the progression equals the ratio of any term to the pre- 
ceding term. When the progression is ascending, the rate is 
greater than a unit ; when it is descending, the rate is less 
than a unit. The rate is by most authors called the ratio of 
the series; the reason for preferring the term rate will be 
stated presently. 

Notation. The method of writing a geometrical progression, 
generally employed by English and American authors, is the 
same as that for an arithmetical progression. The French 
authors, however, distinguish it from an arithmetical progression 
by a special notation. They place the symbol -H- before the 
series, and separate the terms by a colon (:) ; thus, 
H- 2 : 4 : 8 : 16 : 32 : 64 : 128. 

The Rate. The constant multipler, as before stated, is gen- 
erally called the ratio of the series. The term rate, it is 
thought, is much more appropriate and precise. The objection 
to the word ratio is that, in the comparison of numbers, the 
ratio is the quotient of the first term divided by the second, 
while the rate of a series is equal to any term divided by the 
previous term ; hence, there is a seeming contradiction of the 
correct meaning of the term ratio. This contradiction may be 
only seeming, but to avoid all difficulty in this respect, it will 
16* (345) 



346 THE PHILOSOPHY OF ARITHMETIC. 

be better to use a term which is appropriate and not liable to 
misconception. Rate seems to be an appropriate word, since 
we naturally speak of the rate of increase or decrease of any- 
thing ; and by the rate of a progression, we mean its rate of 
increase or decrease. 

The French mathematicians make this distinction between 
ratio and rate ; they use the word rapport, ratio, in proportion, 
and raison, rate, in progression. Bourdon says, "The con- 
stant ratio, which exists between any term, and that which imme- 
diately precedes it, is called the rate of the progression.* By 
rapport they seem to mean about what we do by ratio ; it is 
probably from the idea of produce, the ratio being the product 
of the division. Their word raison seems to mean the same as 
rate, taken probably from the idea of cause, the rate being the 
law or cause of the terms being what they are. 

The term ratio, as used in relation to a progression, has 
given rise to a good deal of discussion and misapprehension. 
Some writers who use the word have taken the pains to tell us 
that they mean, not a direct, but an inverse ratio. Prof. Dodd 
says, when we speak of the ratio of a geometrical progression 
being 2, we mean that "the terms progress in a twofold ratio, 
which simply means that each term has the ratio of 2 to the 
preceding term ;" and similar remarks are made by other writers. 
By using the word rate instead of ratio, all this difficulty and 
misapprehension will be avoided. It is to be hoped, therefore, 
that the term rate will be generally adopted in speaking of the 
law of variation of a geometrical series. 

Cases. There are five quantities considered, as in arithmet- 
ical progression ; the first term, the rate, the number of terms, 
the last term, and the sum of the terms. Any three of these 
being given the other two can be derived from them, which 
gives rise to twenty distinct cases. These cannot all be solved 

* Ce rapport constant, qui existe entre un terme et celui qui le prficecU 
immediatement, se nomine la RAISON de la progression. BOURDON'S Arith 
metlc, page 279. 



GEOMETRICAL PROGRESSION. 347 

by arithmetic ; the first fifteen are easily derived by common 
algebra, and the other five readily yield to the logarithmic cal- 
culus. The two cases generally given in arithmetic are the 
following : 

1. To find the last term, having given the first term, the rate, 
and the number of terms. 

2. To find the sum of the terms, having given the first term, 
the last term, and the number of terms. 

Treatment. The general method of treatment in a geomet- 
rical progression is the same as in an arithmetical progression ; 
and having been stated under arithmetical progression, need 
not be repeated here. Several cases cannot be obtained in 
arithmetic, since they require the solution of an equation. Four 
cases cannot be solved by elementary algebra, as they depend 
upon the solution of an exponential equation ; and in obtaining 
the numerical results we are obliged to make use of logarithms. 
The two fundamental cases should, we think, in arithmetic be 
expressed in the symbolic language of algebra; thus, 

1. Z=ar-i; 2. #=^=-. 

r 1 

THE INFINITE SERIES. An Infinite Series is a series in 
which the number of terms is infinite. In a descending pro- 
gression the terms are continually growing smaller; hence if 
the series be continued sufficiently far, the last term must be- 
come less than any assignable quantity ; and if continued to 
infinity, the last term must become infinitely small. 

In treating an infinite series, we regard this infinitely small 
quantity as zero, or nothing. Thus, in finding the sum of a 

ft 7** 

descending series, we use the formula S=~ ; and regarding 

1 r 

the last term as nothing, the term Ir disappears, and we have 
d 

jS=, , or the sum of the terms of an infinite series descend- 

1 r 

ing equals the first term divided by 1 minus the rate. 



348 THE PHILOSOPHY OF ARITHMETIC. 

This reduction of the last term to zero presents a difficulty 
not easily explained. The question arises, how can the last 
term become zero? At what point does a term become so 
small that, when multiplied by the rate, the product shall be 
nothing? To illustrate the difficulty, take the series 1, ^, ^, . 
etc., in which the rate is i. Now if this series be continued to 
infinity, the last term is supposed to be zero. This supposition 
seems to involve the idea that the term just before the last is 
so small that ^ of it is nothing. Who can conceive of such a 
term ? Who can trace the series down through all the differ- 
ent values, until we reach a term so small that one-half of it is 
nothing? This of course cannot be done. The mind shrinks 
from the effort ; it is unable to grasp the infinitely small. In- 
deed, neither the infinitely great nor the infinitely small can be 
positively conceived ; an infinite quantity and an infinitesimal 
are both beyond the grasp of the human mind. 

What shall we do then ? Shall we deny that the last term 
is infinitely small, or zero ? Certainly not : to assume that it 
is not infinitely small involves a greater difficulty than the sup- 
position that it is infinitely small. Fix upon any term, how- 
ever small, and we see that it can be continually divided, and 
that the division will continue as long as there is a term to be 
divided, and can only terminate when the term becomes too 
small to divide, or zero. Hence, to conceive that the infinite 
term is not zero, is to suppose that the division stopped when 
it could have proceeded, which is absurd ; consequently, it is 
absurd to suppose that the last term is not zero. The question 
then stands thus: we cannot comprehend that the last term is 
zero, and to conceive that it is not zero is absurd. We are 
thus in the dilemma that we must believe either the absurd or 
the incomprehensible. We cannot believe the absurd; we 
rather accept the incomprehensible. We are therefore forced 
to the conviction that the last term is zero, even though we 
cannot fully conceive it to be so. We believe that which we 
cannot fully understand, because not to believe it leads to an ab 



GEOMETRICAL PROGRESSION. 349 

nrdity, and the mind is so constituted that it will accept the 
jicomprehensible sooner than the absurd. We take it upon 
faith; it is the place in science "where reason falters" and 
faith accepts. 

This method of considering the subject presents an excellent 
illustration of the operation of the intuitive power in many 
questions of religious faith. I may not be able to comprehend 
a first cause ; but I know there must be one, or else I am in- 
volved in an absurdity, and the human mind cannot rest in the 
absurd. It may be remarked that the point of difficulty here 
considered, is one that frequently occurs in mathematics. The 
infinitely small is an important element in mathematical inves- 
tigations. We make use of it in geometry, and in calculus it is 
the fundamental idea upon which the science is based. 

The most satisfactory method of removing any doubt that 
one may have upon the assumption that the last term reduces 
to zero, is to take a problem which may be solved by an infinite 
series, and which can also be solved without it. If the result 
obtained by supposing the last term to be zero, agrees with the 
result otherwise obtained, the conclusion that the last term 
is zero must be accepted, whether we can conceive it or not. 
Such a problem is the following: "Abound and fox are 10 
rods apart, and the hound pursues the fox ; how far will the 
hound run to overtake the fox, if the latter runs -fa as fast as 
the hound?" 

Looking at this problem in one way, we see that when the 
hound has run the 10 rods the fox has run 1 rod, and they 
are then 1 rod apart. When the hound runs this rod, the 
fox has run y 1 ^ of a rod ; hence they are then -j 1 ^ of a rod apart. 
When the hound runs this -fa of a rod, they are -fa of -fa, or -j-J-y 
of a rod apart; hence the distance the hound will run to catch 
the fox is correctly represented by the sum of the series 10-j-l 

+T^+Tfo+TnVH-io&oo+ etc -> to an incite number of terms 
The sum of this series, obtained by the method of infinite series, 
which regards the last term as zero, equals 10-h(l 1^)= 



350 THE PHILOSOPHY Otf ABITHMBTIC. 

=11^ rods. Hence the hound runs 11| rods to catch 



the fox. 

The problem may also be solved by the following simple 
method of analysis : By the conditions, ten times the distance 
the fox runs equals the distance the hound runs ; and this di- 
minished by the distance the fox runs, is 9 times the distance 
the fox runs, which equals what the hound gains on the fox, 
or 10 rods, the distance they were apart ; then once the dis- 
tance the fox runs equals ^ of a rod, and 10 times the distance 
the fox runs, which is the distance the hound runs, equals 
10x J ^=- L ^ J -, or 11 rods. Or, we may solve it even more 
simply thus: the hound gains 9 rods in running 10, hence to 
gain 1 rod he will run -^ of a rod, and to gain 10 rods, so as 
to catch the fox, he will run 10 times *-, or 1^=11^ rods. 
This result corresponds with that obtained by the summation 
of the infinite series ; hence the supposition involved in that 
solution, that the last term of the series equals zero, must be 
correct. 

This problem is sometimes given as a puzzle, in which it is 
said that since there is always one-tenth of the previous dis- 
tance between them, the hound will never catch the fox. The 
fallacy consists in inferring that because there is an infinite 
number of successive operations, it must require an infinite 
length of time to perform them. 

A problem similar to this is the following : "A ball falls 8 feet 
to the floor and bounds back 4 feet, then falling bounds 2 feet, 
and so on; how far will it move before coming to rest?" Solv- 
ing this, we find the distance to be 24 feet. It is sometimes 
supposed in this problem, that the body will never come to 
rest ; this is a mistake, for though there will be, in theory at 
least, an infinite number of motions, they will be accomplished 
in a finite period of time. The reason of this is, that the infi- 
nitely small motions are made in infinitely small periods of 
time, the sum of which does not exceed a finite period. 

It should be remarked that some writers maintain that the 



GEOMETRICAL PROGRESSION. 351 

results in the infinite series are not absolutely correct, but are 
merely approximations ; thus, that the sum of the series a+i+i 
+etc., is not absolutely 1, but only approximately so; in other 
words, that all we can affirm concerning it is that it comes nearer 
and nearer to 1 as we increase the number of terms, though it can 
never reach 1. Unity is the limit towards which it is always 
approaching, which it never can exceed, and indeed, which it 
never can reach. 

This is the doctrine of limits, and is the one usually preferred 
by modern mathematicians. By this doctrine, in summing the 
infinite series descending, we are attempting to find the limit 
towards which the series is approaching, but which it can never 
reach. This is regarded as the most logical method of consider- 
ing the subject. Logic may admit self-evident propositions, but 
it does not admit conclusions that cannot be logically derived 
from these self-evident assumptions. Thought cannot follow an 
infinite series step by step to the zero term ; hence a conclusion 
based on the assumption of a zero term is regarded as illogical 
and inadmissible. 

This doctrine of limits as applied to the infinite series, while 
apparently logical, is not without its difficulties. It would seem 
to lead to the conclusion that in the case of the " fox and hound 
problem," given above, the hound would never catch the fox ; 
unless, as a boy once remarked, " he gets near enough to grab 
him." So in respect to the elastic ball dropped upon a pave- 
ment; if the result is only approximately true, does it not follow 
that the ball never comes to rest, but continues bounding forever ? 
Here, as in many other cases, faith in the incomprehensible seems 
more satisfactory than a timid skepticism. 

It will be interesting to notice that the two different series, + 

i~f~3T~^'8T"'~ etc ' an< ^ i+^+TV^TJ"^ 610 -' are eac ^ ec l ua l to tne 
same fraction ^. It is also an interesting truth that the pum of 
the series beginning with ^, and decreasing at the rate of ^, is 
just equal to 1. 



SECTION IIL 

PERCENTAGE. 



23 



I. NATURE OP PEBCENTAOB. 



II. NATURE OF INTEREST. 



CHAPTER I. 

NATURE OF PERCENTAGE. 

T)ERCENTAGE is a process of computation in which the 
-C basis of the comparison of numbers is a hundred. The 
same idea may also be expressed more briefly in the definition, 
Percentage is the process of computing in hundredths. 

The former definition was first presented in one of the author's 
arithmetical works. Up to this time no definition had been 
given of Percentage as a process of arithmetic. In the text- 
books, the word was merely defined as meaning so many of a 
hundred. Soon after this publication appeared, one or two 
other authors adopted a definition similar to the one given 
above, presenting the subject as a department of the science ; 
and in time, it is presumed, all will define it as a process of 
arithmetic. 

It will be readily seen that Percentage has its origin in the 
third division of the science of arithmetic; namely, Comparison. 
We may compare numbers and determine their relations with 
respect to their common unit or basis. This is the first and 
simplest case of comparison, and gives rise to Ratio and Pro- 
portion. We may also compare numbers with respect to some 
number agreed upon as a basis of comparison, and develop 
their relations with respect to this basis. When this number 
is one hundred, we have the process of Percentage. It is thus 
seen that the idea of the subject presented in the definition 
given above is correct. 

Percentage originated in the fact of the convenience of esti- 
mating by the hundred, in a decimal scale. It derives its im- 

(355) 



356 THE PHILOSOPHY OF ARITHMETIC. 

portauce and has received so full a development, partly at least, 
from the fact of our having a decimal currency. It occupies a 
more prominent place in American than in English text-books, 
where the money system is not decimal. Its principal use is in 
its application to business transactions relating to money, as 
will be seen in the various ways in which it is employed. It 
admits, however, of a purely abstract development, entirely 
independent of concrete examples ; and is, therefore, a process 
of pure arithmetic. 

Quantities. Percentage embraces four distinct kinds of quan- 
tities, the base, the rate, the percentage, and the amount or 
difference. 

The Base is the number on which the percentage is estimated. 
The Bate is the number of hundredths of the base. The Per- 
centage is the result of taking a number of hundredths of the 
base. The Amount or Difference is the sum or the difference 
of the base and percentage. 

The Amount and Difference are the same kind of quantities, 
and it would be well, in Percentage, to have some one term 
which would include them both. In several of the applications 
we have such a word ; as selling price in Profit and Loss, 
proceeds in Discount, etc. The expression Resulting Number 
has been used, but this is a little awkward and inconvenient. 
The term Proceeds, meaning that which results or comes forth, 
I have sometimes thought of adopting, and indeed have adopted 
in one of my works. Some term, in place of amount and 
difference as used in percentage, is a scientific necessity, and 
Proceeds is recommended. 

The Eate was originally expressed as a whole number, and 
the methods of operation based upon such expression. Latterly 
it is becoming the custom to represent the rate as a decimal, 
and to operate with it as such. This is much the better way, 
and will probably become universal. It gives greater simplicity 
to the rules, makes the treatment more scientific, and is quite 
as readily understood by pupils. It may be remarked that 



NATURE OF PERCENTAGE. 357 

the definition of the rate will vary according to which of these 
forms is taken. The definition above given regards the rate as a 
decimal. 

It will thus appear that there is a slight distinction between 
the term Rate and the expression the rate per cent. Per cent. 
means by the hundred ; rate per cent, means a certain number 
of or by the hundred ; while Hate means a certain number of 
hundi'edths. When money is loaned at 6 per cent, the rate 
per cent, is 6; but the Hate is .06. Thus Hate and rate by the 
hundred, are about identical in meaning. We may conse- 
quently define the Rate to be the number by which we multiply 
the base in order to obtain any required per cent, of it; and 
this is what is intended in the definition, The rate is a num- 
ber of hundredths of the base. 

Gases. It has been a question among arithmeticians under 
how many cases Percentage should be presented. There being 
four distinct classes of quantities five, if like some authors we 
regard the amount and difference as distinct any two of which 
being given, the others may be found, it will be seen that there 
are quite a large number of possible theoretical cases. What is 
the simplest and most scientific classification of these various 
cases? In other words, what are the general cases of Per- 
centage? It has been quite customary to present the subject 
under six distinct cases, and this affords a very practical view 
of the subject. Authors, however, have not been uniform in 
their treatment. I believe that the best way is to present the 
subject under three general cases, each of which will contain 
two or three special cases, as we regard the amount and differ- 
ence, as one or two classes of quantities. Uniting the amount 
and difference under one general term, as proceeds, we shall 
have throe .general cases, each including two special cases, 
making six cases in all ; regarding the amount and difference 
as two distinct quantities, we shall have three special cases 
under cjicli general case, making nine cases in all. 

These three general cases may be formally stated as follows: 



358 THE PHILOSOPHY OF ARITHMETIC. 

1. Given, the base and the rate, to find the percentage and 
the proceeds. 

2. Given, the base and either the percentage or the proceeds, 
to find the rate. 

3. Given, the rate and either the percentage or the proceeds, 
to find the base. 

Treatment. There are two distinct methods of treatment in 
Percentage, which may be distinguished as the Analytic and the 
Synthetic methods. The Analytic Method consists in reducing 
the rate to a common fraction, and taking a fractional part of 
the base for the percentage, and operating similarly in the other 
cases. It differs particularly from the other method in the 
solution of the second and third cases, as will be seen by the 
solution of a problem. It is the method for mental analysis, 
and is especially suited to the subject of Mental Arithmetic. 
To illustrate the analytic method, take the problem, "What is 
25% of 360?" We reason thus: 25% of 360 is -^ or { of 
360, which is 90. To find the base take the problem, " 90 is 
25% of what number?" The solution is, If 90 is 25%, or 
\, of some number, of the number is 4 times 90, or 360. The 
case of finding the rate per cent, is solved in a similar manner. 

The Synthetic Method consists in preserving the rate in the 
form in which it is presented, and operating accordingly. In 
the synthetic method there are two ways of operating : the 
first consists in using the rate as a whole number, and dividing 
or multiplying by a hundred ; the second operates with the 
rate in the form of a decimal, according to the principles of 
decimal multiplication and division. There has, for several 
years, been a tendency towards the latter method, and arithme- 
ticians are now generally agreed in its favor. 

This latter method is greatly to be preferred on account of 
its simplicity and scientific character. The difference may 
be shown by a rule for one of the cases. When the rate is 
used as a whole number, the rule for finding the percentage is, 
Multiply the base by the rate, and divide the product by 100 



NATURE OF PERCENTAGE. 359 

When the rate is used as a decimal, the rule is, Multiply the 
base by the rate. A similar difference will be found to exist in 
the rules for all the cases. Another consideration in favor of 
using the rate as a decimal is the ease with which the rules for 
the other cases are derived from the first. Assuming that the 
percentage equals the base multiplied by the rate ; it immedi- 
ately follows that the base equals the percentage divided by 
the rate, or the rate equals the percentage divided by the base. 

To illustrate the method preferred, suppose we have the 
problem in Case 1., "What is 25% of 360?" We would rea- 
son thus : Twenty-five per cent, of 360 equals 25 hundredths 
times 360, or 360 x. 25, which by multiplying we find to be 90. 

To illustrate Case 2, take the problem, " 90 is 25% of what 
number?" We would solve this as follows: If 90 is 25% of 
some number, then some number multiplied by .25 equals 90; 
hence this number equals 90 divided by .25, or 90-=-. 25, which 
by dividing we find is 360. 

To illustrate Case 3, take the problem, " 90 is what 
per cent, of 360 ?" The solution is as follows : If 90 is some 
per cent, of 360, then 360 multiplied by some rate equals 90 ; 
hence the rate equals 90 divided by 360, or 90-^-360, which is 
.25, or 25%. 

The solution of problems including the proceeds is quite 
similar, and need not be presented here in detail. The particu- 
lar method of explanation will be found in my Higher Arith- 
metic. 

Formulas. These synthetic methods and rules may all be 
presented in general formulas, as follows: 

CASE I. CASE II. CASE III. 

1. bxr=p 1. p-r-r=b 1. p-i-b=r 

2. &x(l+r)=4 2. A+(l + r)=b 2. A+b=\+r 

3. 6x(l r)=D 3. X>-4-(l r)=b 3. D+b=l r 
The 2d and 3d formulas of each case may be united in one ; 

thus, using P for proceeds, P=&x(lr); &=P-*-(lr); 
r=P-j-6 1, or 1 P^b. 



360 THE PHILOSOPHY OF ARITHMETIC. 

Applications. The applications of Percentage are very ex- 
tensive, owing to the great convenience of reckoning by the 
hundred in financial transactions. These applications are of 
two general classes ; those not including the element of time, and 
those which include this element. The following are the most 
important of these two classes of applications : 



IST CLASS. 

1. Profit and Loss. 

2. Stocks and Dividends. 

3. Premium and Discount. 

4. Commission. 

5. Brokerage. 

6. Insurance 

7. Taxes. 

8. Duties and Customs. 

9. Stock Investments. 



2o CLASS. 

1. Simple Interest. 

2. Partial Payments. 

3. Discounting. 

4. Banking. 

5. Exchange. 

6. Equation of Payments. 

7. Settlement of Accounts. 

8. Compound Interest 

9. Annuities. 



The different cases of the first class are solved as in pure 
percentage, and the rules are almost identical, the technical 
terms being substituted for base, percentage, etc. The solutions 
of the various cases of the second class are somewhat modified 
by the introduction of the element of time. The development 
of these various cases would occupy too much space for this 
work, and moreover does not constitute a part of the philosophy 
of arithmetic; we shall, therefore, give only a single chapter 
on the genera] nature of Interest. 



CHAPTER II. 

NATURE OF INTEREST. 

T)ERCENTAGE embraces two general classes of problems, 
JL those that involve the element of time, and those that do 
not involve this element. The most important application of 
percentage into which this element enters is Interest ; and in- 
deed all such applications may be embraced under this general 
term. 

Interest may be defined as money paid, or charged for the 
use of money. It is usually reckoned as so many units on a 
hundred, and is thus included under the general process of Per- 
centage. The sum upon which interest is reckoned is called 
the Principal, in distinction from the interest or profit, which 
is subordinate to it. The sum of the interest and principal is 
called the Amount. 

Interest is either Simple or Compound. Simple Interest is 
that which is reckoned or allowed upon the principal only, 
during the whole time of the loan. Compound Interest is 
reckoned, not only on the sum loaned, but also on the interest 
as it becomes due. Interest unpaid is regarded as a new loan 
upon which interest should be paid. 

Simple Interest. In considering the subject of simple inter- 
est, the primary object is to find the interest on a given princi- 
pal for a given time and rate. Various methods have been 
devised for the solution of this problem. The simplest in 
principle and most natural, is to find the interest for one year 
by multiplying the principal by the rate, and multiplying this 
interest by the time expressed in years. The objection to this 
16 ( 361) 



362 THE PHILOSOPHY OF ARITHMETIC. 

method in practice arises from the fact that the time is often 
given in months and days, which frequently reduce to an incon- 
venient fractional part of a year. This difficulty has led to a 
modification of the rule proposed above, which is known as 
the method of " aliquot parts." 

The importance of a method that can be readily applied in bus- 
iness, has led to the exercise of considerable ingenuity in order 
to discover the shortest and simplest rule in practice. The 
method now regarded as the simplest is that known as the 
"six per cent." method. It is based on the rate of 6%, which is 
the usual rate in this country, and may be expressed as follows : 
Gall half the number of months cents, and one-sixth of the 
number of days mills, and multiply their sum, which will be 
the interest of $1 for the rate and time, by the principal. 
Another way of stating this rule is, Regard the months as 
cents, and one-third of the days as mills, and multiply their 
sum by one-half of the principal. For short periods a modi- 
fication of the rule, which may be popularly expressed, Mul- 
tiply dollars by days and divide by 6000, is the most convenient 
in practice, and is very generally employed by business men. 
There are also many other methods of working interest which 
need not be stated here. 

The general method of finding the interest of a principal may 
be expressed in a general formula as in Percentage. The gen- 
eral formula is i=ptr, which is readily remembered by the sen- 
tence which it suggests "I equals Peter." The several cases 
which arise in interest can be readily derived from this funda- 
mental formula. These several rules may be expressed as fol- 
lows: 

1. i=ptr. 3. t=i-t-pr. 

2. p = i-i-tr. 4. r = i-i-pt. 

It is objected to the "six per cent, method," that it gives too 
great an interest, since it reckons only 360 days in a year ; and 
it has been suggested that to compute the interest on a loan by 
this method would be to take usury, and in some states would 



NATURE OF INTEREST. 363 

result in a forfeiture of the debt, or some other penalty. This 
seems like putting a very nice point on the matter, though it is 
true that the six per cent, method gives a little more interest 
than when we reckon 365 days to the year. To obtain exact 
interest, we find the interest for the years, multiply the interest 
of one year by the number of days, and divide by 365, and take 
the sum of the two results. A full presentation of the applica- 
tions of interest to business and the latest methods of treatment 
may be found in the author's Higher Arithmetic. 

Mates of Interest. It is a noteworthy fact that the propriety 
of receiving interest for the use of money, has been questioned. 
Indeed, the practice has been censured in both ancient and 
modern times as an immorality and a wrong to society. It 
may seem that so absurd a notion hardly needs a passing no- 
tice, for it is clear that a similar objection may be made to the 
charge of rents, or even to profits of any kind. A capitalist 
may invest his money in business and receive a certain return 
for it ; and if he chooses to let some one else invest it and have 
the care of such investment, it is clear that he should receive 
some remuneration for surrendering to another the profit he 
might have made himself. Again, the borrower can with cap- 
ital secure a large return of profit in business, and is not only 
entirely willing to pay for the use of such capital, but is in 
equity under obligations to do so. Interest on loans is, there- 
fore, a benefit to both the borrower and lender; and should 
therefore be both required and allowed. 

The rate of interest is determined strictly by the principle 
of competition. When the capital to be invested exceeds the 
demands of borrowers, the rate of interest is low ; when the 
demand is in excess of the capital, the rate will be high. The 
rate will vary also with the security of the loan ; thus the rate 
on landed mortgages is usually lower than on property less 
secure and certain, and consequently state loans are usually 
made at low rates. A lender assumes that he must be paid 
something for the risk of a loan, and that the greater the risk 



364 THE PHILOSOPHY OF ARITHMETIC. 

the greater the charge. It is on this principle that high inter 
est is often said to be synonymous with bad security. A high 
rate of interest may also be due to large profits on capital. In 
a community where the returns on capital are large, as in rich 
mining districts for instance, all who have capital would desire 
to invest, and consequently the difficulty of obtaining a loan 
would increase and higher rates would obtain. In such cases 
the opportunity for large gains by the capitalist and the in- 
creased demand by the borrower would both conspire to increase 
the rate of interest. 

The rates of interest have usually been regulated by govern- 
ments. This action is founded upon a variety of reasons. It 
has been argued that lenders are unproductive consumers of 
part of the profit which is produced by labor. Such a notion 
leaves out of sight, however, that production is impossible 
without capital, and that capital is accumulated and employed 
with a view to profit. It is also held that if the state does not 
regulate rates, borrowers will be open to fraud and extortion 
on the part of unprincipled lenders. This is the principal con- 
sideration in favor of state control of interest rates ; and yet 
there are valid if not unanswerable objections to it. It is, of 
course, the duty of the government to protect the citizen against 
usury and fraud ; but most of the considerations in favor of 
regulating rates of interest will apply to the regulation of the 
prices of food, land, wages, etc. It seems to be a growing 
opinion that capital should seek investment at rates determined 
by natural laws of demand and supply, as the prices of other 
property are regulated, and not be controlled by legislative en- 
actment. 

Historical. The payment of interest on money has been 
the custom from very early times. We learn from the New 
Testament that it was paid on bankers' deposits in Judea, 
though the Jews were forbidden by the laws of Moses to exact 
interest from one another. In Europe, interest was alternately 
prohibited and allowed, the church being generally hostile to 



NATURE OF INTEREST. 365 

the practice. In Italy, the trade in money was recognized, 
and the custom of borrowing and lending was common. In 
England, it was first sanctioned by the Parliament in 1546, the 
rate being fixed at 10 per cent.; but in 1552 it was again pro- 
hibited. Mary, however, borrowed at 12 per cent., which ap- 
pears to have been the usual rate at that period at Antwerp. 
In 1571, it was again made legal at 10 per cent., a rate at 
which the Scotch Parliament fixed it in 1587. The rate fell at 
the beginning of the seventeenth century, James I. having 
borrowed in Denmark at 6 per cent. In 1624, it was reduced 
to 8 per cent.; in 1651, to 6 per cent.; in 1724, to 5 per cent, 
at which legal rate it remained until all usury laws were re- 
pealed, an event which occurred only a few years ago. In 
1773, it was limited to 12 per cent, in India. In 1660, the rate 
in Scotland and Ireland was from 10 to 12 per cent.; in France 
7 per cent.; in Italy and Holland 3 per cent.; in Spain from 10 
to 12 per cent.; in Turkey 20 per cent.; but the East India 
Company, while the legal rate was 6 per cent., continued to 
borrow at 4 per cent. 

The term Usury, meaning the " use of a thing," was origi- 
nally applied to the legitimate profit arising from the use of 
money, and meant merely the taking of interest for money. 
Laws were established in various countries fixing the amount 
of interest or usury, and the evasion of these laws by charging 
excessive usury, led to the present use of the term. By the old 
Roman law of the Twelve Tables, the rate of interest allowed 
as legitimate was the usura centesima, which was strictly 1 
per cont. a month ; and has been supposed by some to have 
amounted to 12, and by others to 10 per cent, a year. The 
Roman laws against excessive usury were frequently renewed 
and constantly evaded, and the same is true of other countries. 
In England, during the reign of Henry VIII., 10 per cent, was 
allowed; by 21 James I., 8 percent.; by 12 Charles II., 8 per 
cent.; by 12 Anne, 5 per cent. Subsequently to the passage 
of the latter act, the usury laws wore relaxed by several 



866 THE PHILOSOPHY OF ARITHMETIC. 

statutes, and they were ultimately repealed in 1854. Any rate 
of interest, however high, may now be legally stipulated for, 
but 5 per cent, remains the legal interest recoverable on all 
contracts, unless otherwise specified. 

Much concern has been shown by governments in attempt- 
ing to fix rates of interest, and prevent usury. The legislation 
of Solon relieved the Athenian mortgagors ; and during many 
years of the Roman Republic, the regulation of loans, the limi- 
tation of the rate of interest, and the relief of insolvent debtors, 
formed a perpetual topic of agitation, and finally of legislation. 
In most of the European countries the administration has 
busied itself, from time to time, in fixing rates of interest, and 
in denouncing or forbidding usurious bargains. Such legisla- 
tion has, however, proved vain ; for while the most stringent 
laws were in force, high rates of interest on loans were com- 
mon, the law being incompetent to provide against evasion of 
the statute. 

The legal rate of the United States government is 6 per 
cent. Each State fixes its own rate, and attaches its special 
penalties for usury. In several of the States the usury laws 
have been repealed, and the general tendency is to allow an 
open market to the investment of capital. 

Origin of Methods. The importance of a knowledge of the 
principles of interest, discount, etc., led arithmeticians to notice 
these subjects at an early day. Interest was early divided 
into Simple and Compound. Compound Interest was properly 
called usura, and was rarely practised in the transactions of 
merchants with each other. Stevinus terms compound interest, 
interest prouffitable, or celuy qu'on ajouste au capital, whilst 
the corresponding discount is termed interest dommageable, or 
celuy qu'on soubstrait du capital. 

Problems in simple interest were by Tartaglia and his pre- 
decessors, solved by the Rule of Three. In calculating the 
interest of a sum from one day to another, the determination 
of the number of days in the interval seemed somewhat embar- 



NATURE OF INTEREST. 367 

rassing, and Tartaglia gives a rule for this purpose of which he 
seems somewhat proud. In passing from one city of Italy to 
another an additional source of embarrassment presented itself 
in the different days on which the year was supposed to com- 
mence, being reckoned at Venice from the 1st of March, at 
Florence from the Annunciation of the Virgin, and in most 
other cities of Italy from Christmas day. 

Tartaglia has noticed five methods of finding the amount of 
a sum of money at compound interest. Suppose the question 
to be to find the amount of L300 for 4 years at 10 per cent, a 
capo d'anno ; the first method is by the following four state- 
ments : 

100 : 300 : : 110 : 330 

100 : 330 : : 110 : 363 

100 : 363 : : 110 : 399^ 

100 : 399 T % : : 110 : 
The second method merely replaces 100 and 110 by 10 and 
11 in the proportion ; the third, which is his own method, mul- 
tiplies 300 four times successively by 11, and divides the last 
product by 10,000 ; the fourth consists in adding four suc- 
cessive tenths to the principal; the last in calculating the 
amount for L100, and then finding the amount of L300, or any 
other proposed sum, by a simple proportion. 

With the exception of discount at compound interest and its ap- 
plication to correct in part the conclusion respecting the values 
of annuities, there are few, if any, other questions of compound 
interest which Tartaglia and his contemporaries can be said to 
have resolved. A very natural difficulty arose in the solution 
of questions of this kind : " What is the interest of 100 for 6 
months, interest being reckoned at the rate of 20 per cent, per 
annum ?" Lucas di Borgo and others made out that this would 
be 10 ; that is, they calculated that, simple interest only being 
allowed, it was a matter of indifference into how many por- 
tions of time the whole period was divided, whether into months 
or half-years. 

Lucas di Borgo has an article on calculating tables of inter- 



368 THE PHILOSOPHY OF ARITHMETIC. 

eat in which he speaks of their great utility, thereby showing 
that such tables were in use in Italy, although no work of that 
date containing them is known to be extant. The first com- 
pound interest tables now known are those which are presented 
by Stevinus in his arithmetic, which give the present worth of 
10,000,000 from 1 to 30 years, in sixteen tables, the interest 
being reckoned successively from 1 to 16 pejr cent., and in eight 
other tables, where the interest is differently reckoned, accord- 
ing to the custom of Flanders. 

The origin of the various modern methods of calculating in- 
terest is not known. The method by " aliquot parts" is a fav- 
orite rule of the English arithmeticians, and probably originated 
with them. The " six per cent, method" has been attributed 
to a Mr. Adams, author of a work on arithmetic. The partic- 
ular form of the six per cent, method popularly stated, "multi- 
ply dollars by days and divide by 6000," was used among 
business men before it was introduced into any arithmetic, and 
is presumed to have had its origin in some counting-house, but 
it is not known where. 



SECTION IV. 

THE THEORY OF NUMBERS. 



I. NATURE OF THE SUBJECT. 



II. EVEN AND ODD NUMBERS. 



III. PRIME AND COMPOSITE NUMBERS. 



IV. PERFECT, IMPERFECT, ETC., NUMBERS. 



V. DIVISIBILITY OF NUMBERS. 



VI. DIVISIBILITY BY THE NUMBER SEVEN. 



VII. PROPERTIES OF THE NUMBER NINE. 



CHAPTER I. 

NATURE OF THE SUBJECT. 

T'HE Theory of Numbers, as generally presented, embraces 
the classification and investigation of the properties of 
numbers. This subject has engaged the attention and enlisted 
the talents of many celebrated mathematicians. The ancient 
writers, who did little for the development of arithmetic as a 
science or an art, spent much time in theorizing upon the pro- 
perties of numbers. The science of arithmetic with them was 
mainly speculative, abounding in fanciful analogies and mys- 
terious properties. 

Pythagoras attributed to numbers certain mystical properties, 
and seems to have conceived the idea of what are now termed 
Magic Squares. Aristotle, amongst other numerical specula- 
tions, noticed the practice, in almost all nations, of dividing 
numbers into groups of tens, and attempted to give a philo- 
sophical explanation of the cause. The earliest regular system 
of numbers is that given by Euclid in the 7th, 8th, 9th, and 
10th books of his " Elements," which, notwithstanding the 
embarrassing notation of the Greeks, and the inadequacy of 
geometry to the investigation of numerical properties, is still 
very interesting, and displays, like all other parts of the same 
celebrated work, that depth of thought and accuracy of demon- 
stration for which its author is so eminently distinguished. 

Archimedes, also, paid particular attention to the powers and 
properties of numbers. His tract, entitled " Arenarius," con- 
tains a method of multiplying and dividing which bears a con- 
siderable analogy to that which we now employ in multiplication 

(371) 



372 THE PHILOSOPHY OF ARITHMETIC. 

and division of powers, and which some modern writers have 
thought inculcated the principles of our present system of loga- 
rithms. Before the invention of algebra, however, but little 
progress could be made in this branch of the science ; accord- 
ingly we find that comparatively few principles had been dis- 
covered until the time of Diophantus. This eminent mathema- 
tician, who is the author of the most ancient existing work on 
the subject of algebra, presents many interesting problems in 
the properties of numbers ; but, owing to the difficulties of a 
complicated notation and a deficient analysis, little progress 
was made, compared with the advance of modern times. 

From the time of Diupliantus the subject remained unnoticed, 
or at least unimproved, until Bachet, a French analyst, under- 
took the translation of Diophantus into Latin. This work, 
which was published in 1621, contained many marginal notes 
of the translator, and may be considered as presenting the first 
germs of our present theory. These were afterward consider- 
ably extended by Format, in his posthumous edition of the 
same work, published in 1670, which contains many of the most 
elegant theorems in this branch of analysis; but they are gen- 
erally left without demonstration, which he explains in a note 
by saying that he was preparing a treatise of his own upon 
the subject. Legendre accounts for the omission by saying 
that it was in accordance with the spirit of the times for learned 
men to propose problems to each other for solution. They 
generally concealed their own method in order to obtain new 
triumphs for themselves and their nation ; and there was about 
this time an especial rivalry between the English and French 
mathematicians. Thus it has happened that most of the demon- 
strations of Fcrmat have been lost, and the few that rema'ti 
only make us regret the more those that are wanting. 

The most of these theorems remained undemonstrated until 
the subject was again renewed by Euler and Lagrange. Euler, 
in his "Elements of Algebra," and some other publications, de- 
monstrated many of the theorems of Fermat, and also added 



NATURE OF THE SUBJECT. 373 

some interesting ones of his own. Lagrange, in his additions 
to Euler's Algebra and in other writings, greatly extended the 
theory of numbers by the discovery of many new properties. 
The subject has received its largest contributions, however, from 
the hands of Gauss and Legendre. 

Legendre, in his great work, " Essai sur la Theorie des 
Nombres," was the first to reduce this branch of analysis to a 
regular system. Gauss, in his " Disquisitiones Arithmetic, " 
opened a new field of inquiry by the application of the proper- 
lies of numbers to the solution of binomial equations of the 
form, x" 1 0, 011 the solution of which depends the division of 
the circle into n equal parts. This solution he accomplished in 
several partial cases; whence the division of the circle into a 
prime number of equal parts is performed by the solution of 
equations of inferior degrees; and when the prime number is 
of the form 2"-f 1 the same may be done geometrically a prob- 
lem that was far from being supposed possible before the publi- 
cation of the work mentioned. 

The most celebrated English work on the subject is that of 
Peter Barlow, published in 1811, from the preface of which 
most of the preceding historical facts have been culled. It pre- 
sents a clear and concise statement of the principles of the sub- 
ject, and contains several original contributions, among which 
may be mentioned a demonstration of Fermat's general theorem, 
on the impossibility of the indeterminate equation x n y"=z", 
for every value of n greater than 2. This demonstration, how- 
ever, has been tacitly ignored by mathematicians; and tho 
French Institute and other learned societies have continued 
to propose the problem for solution. 

Almost every modern mathematician of eminence, however, 
has contributed more or less to the advancement of the theory. 
Ln the collected works of Euler, Gauss, Jacobi, Cauchy, 
Dirichlet, Lagrange, Eisenstein, Poinsot, and others, numerous 
memoirs on the subject will be found ; whilst the recent mathe- 
matical journals and academical transactions contain researches 



374 THE PHILOSOPHY OF ARITHMETIC. 

in the same field, by all the ablest living mathematicians. One 
of the most complete treatises on the subject is that of Prof. 
H. J. S. Smith in the article entitled, " Reports on the Theory 
of Numbers," which commenced in the Transactions of the 
British Association for 1859. It embraces a lucid, critical his- 
tory of the subject, rendered doubly valuable by copious refer- 
ences to the original sources of information. 

It will be seen from this brief statement that the subject of 
the theory of numbers is one of great magnitude and difficulty 
requiring the application of the principles of algebra for its de- 
velopment. It is, therefore, not appropriate to treat of it in 
this work, except so far as to show its logical relation to the 
general divisions of the science, and to present a few simple 
properties that may be readily understood by means of the or- 
dinary principles of arithmetic. These will be interesting to 
young arithmeticians, and perhaps the 'means of cultivating a 
taste for a more thorough study of the subject. 

The subjects to which the attention of the reader will be 
briefly directed are the following: 

1. Even and Odd Numbers. 

2. Prime and Composite Numbers. 

3. Perfect, Imperfect, etc., Numbers. 

4. Divisibility of Numbers. 

5. Divisibility by the Number Seven. 

6. Properties of the Number Nine 



CHAPTER II. 

EVEN AND ODD NUMBERS. 

"VTUMBERS have been divided into many different classes, 
-Li founded upon peculiarities discovered by investigating 
their properties. The series 1, 2, 3, 4, etc., is called the series 
of Natural Numbers. The Natural Numbers are classified 
with respect to their relation to the number two, into Odd and 
Even numbers. They are also divided into two classes with 
respect to their composition, called Prime and Composite 
numbers. Composite Numbers are divided into two classes, 
Perfect and Imperfect numbers, this classification being based 
upon the relation of the numbers to the sum of their factors. 
Imperfect Numbers are also divided into two classes with re- 
*pect to the numbers being greater or less than the sum of their 
factors. Numbers which are equal each to the sum of the di- 
visors of the other, are called Amicable Numbers. A few re- 
marks will be made on each one of these classes. 

Of the various classes of numbers, the simplest and most 
natural division is that of Ei>en and Odd numbers. This di- 
vision is founded upon the relation of numbers to the number 
2. Even numbers are those which are multiples of 2 ; Odd 
numbers are those which are not multiples of 2. In the series 
of natural numbers the increase is by a unit ; in the series of 
even numbers the scale of increase is dual. The former arise 
from counting by 1's, beginning with the unit; the latter in 
counting by 2's, beginning with the duad. The even numbers 
are divided into the oddly even numbers, 2, 6, 10, 14, etc.; and 
the evenly even numbers, 4, 8, 12, 16, etc. The odd numbers 
are divided into the evenly odd numbers 1, 5, 9, 13, etc; and 
the oddly odd numbers, 3, 7, 11, 15, etc. 

(375) 



376 THE PHILOSOPHY OF ARITHMETIC. 

The formula for the even numbers is 2n ; the formula for the 
odd numbers is 2/i-fl. In the oddly even numbers n is an odd 
number ; in the evenly even numbers n is an even number. In 
the evenly odd numbers n is even; in the oddly odd numbers 
n is odd. The evenly odd numbers are of the form 4?i-|- 1 ; the 
oddly odd numbers are of the form 4n-|-3. 

There are many interesting principles relating to even and 
odd numbers, a few of which will be stated. 

1. Every prime number except 2 is an odd number. 

2. The differences of the successive square numbers produce 
the odd numbers. 

3. The sum or difference of two even numbers or two odd 
numbers is an even number. 

4. The sum or difference of an even number and an odd num- 
ber is odd. 

5. The sum of any number of even numbers is even ; the 
sum of an even number of odd numbers is even, and the sum 
of an odd number of odd numbers is odd. 

6. The product of two even numbers is even ; of two odd nuui- 
oers is odd ; of an even number and an odd number is even. 

T. The quotient of an even by an odd number, when exact, 
is even; the quotient of an odd by an odd, when exact, is odd; 
the quotient of an even by an even, when exact, is either even 
or odd. 

8. An odd number is not exactly divisible by an even num- 
ber, and the remainder is odd. 

9. If an even number is not exactly divisible by an even 
number, its remainder is even. 

10. If an even number is not exactly divisible by an odd 
number, then when the quotient is even the remainder is even, 
and when the quotient is odd, the remainder is odd. 

11. If an odd number is not exactly divisible by an odd 
number, then when the quotient is odd the remainder is even, 
and when the quotient is even the remainder is odd. 

12. If an odd number divides an even number, it will also 



EVEN AND ODD NUMBERS. 



377 



divide one-half of it ; if an even number be divisible by an odd 
number, it will be divisible by double that number. 

13. Any power of an even number is even ; and conversely 
the root of an even number which is a complete power is even 

14. Any power of an odd number is odd ; and conversely 
the root of an odd number which is a complete power is odd. 

15. The sum or difference of any complete power and its root 
is even. 

These principles can be readily proved by the ordinary meth- 
ods of arithmetical reasoning. To illustrate, take the third 
principle, the reasoning of which is as follows : Two even 
numbers are each a number of 2's, hence their sum will be the 
sum of two different numbers of 2's, which must be a number 
of 2's, and their difference will be the difference between two 
different numbers of 2's, which is also a number of 2's. In add- 
ing two odd numbers we will have a number of 2's-f-l, added 
to another number o/2's+l, which will give us a number of 
2's + 2, or an exact number of 2's, etc. 

The simplest method is by using the general notation of al- 
gebra. Thus in the given principle, these two even number? 
will be represented by 2ra and 2n' ; their sum will be 2n+2n', 
or 2 (n-f w'), which is of the form of 2n, and is thus even ; their 
difference will be 2n 2n', or 2(n n f ), which is of the form of 
2n, and is even. The two odd numbers are of the form 2/i-fl 
and 2n'+ 1, and their sum is 2 (n-fn'-f 1), which is of the form 
of 2n, and even ; their difference is 2rc 2n', or 2 (n n'), which is 
evidently even. All the other principles may be demonstrated 
in a similar manner. 



CHAPTER III. 

PEIME AND COMPOSITE NUMBERS. 

rpHE most celebrated classification of numbers is that of Prime 
JL and Composite. This classification is with respect of their 
formation by multiplication or the possibility of their being re- 
solved into factors. The Composite number is one which can 
be produced by the multiplication of other numbers ; the Prime 
number is one which cannot be produced by the multiplication 
of other numbers. The distinction may be regarded as having 
reference to the dependence or independence of their existence. 
The composite number is regarded as deriving its existence 
from other numbers which make it; the prime number does 
not derive its being from any other numbers, but is indepen- 
dent and self-existent. 

Perhaps no subject in arithmetic has received more attention 
from mathematicians than that of Prime and Composite Numbers. 
The object has been to discover some general method of find- 
ing prime numbers, and of determining whether a given num- 
oer is prime or composite. Such a method, though laboriously 
sought for by the best mathematical minds, has not, beyond 8 
certain limit, been discovered. 

The problem of ascertaining prime numbers was discussed 
as far back as the days of Eratosthenes, a mathematician of 
Alexandria, distinguished also as having first conceived the 
plan of measuring the earth. He invented a method of obtain- 
ing primes by excluding from the series of natural numbers 
those that are not prime, and thus discovering those that are. 
This method consisted in inscribing the series of odd numbers 
upon parchment, and then cutting out the composite numbers, 

(378) 



PRIME AND COMPOSITE NUMBERS. 379 

and leaving the primes. The parchment, with its holes, resem- 
bled a sieve ; hence the method is called Eratosthenes' sieve. 
His method may be illustrated as follows: 

Suppose we write the series of odd numbers from 1 to 99 in- 
clusive. Since the series increases by 2, the third term from 
3 is 3+3 x 2, which is divisible by 3 ; hence every third term 
is divisible by 3, and is therefore composite. In a similar 
manner we see that every fifth term after 5 is divisible by 5, 
and therefore composite ; and every seventh term after 7 is di- 
visible by 7, and therefore composite. Cutting out these com- 
posite numbers, we have all the prime numbers below 100. By 
this method, assisted by some mechanical contrivance, Vega 
computed and published a table of prime numbers from 1 to 
400,000. 

This method is, however, very tedious and inconvenient, and 
mathematicians have earnestly sought for properties of prime and 
composite numbers to guide them in ascertaining primes. The 
following principles are useful in discovering or determining 
prime numbers: 

1. All prime numbers except 2 are odd, and consequently 
terminate with an odd digit. The converse of this, that all odd 
numbers are prime, is not, however, true. 

2. All prime numbers, except 2 and 5, must terminate with 
1, 3, 7, or 9 ; all other numbers are composite. This is the 
series of odd digits with the omission of 5, since any number 
terminating with 5, can be divided by 5 without a remainder. 

3. Every prime number, except 2, if increased or diminished 
by 1, is divisible by 4. In other words, every prime number, 
except 2, is of the form 4n 1. This will admit of demonstra- 
tion. 

4. Every prime number, except 2 and 3, if increased or di- 
minished by 1, is divisible by 6. In other words, every prime 
number, except 2 and 3, is of the form 6n 1. This may also 
be demonstrated. 

5. Every prime number, except 2, 3, and 5, is a measure of 



380 THE PHILOSOPHY OF ARITHMETIC. 

the number expressed, in common notation, by as many 1's as 
there-are units, less one, in the prime number. Thus, 7 is a 
measure of 111,111 ; and 13 of 111,111,111,111. 

6. Every prime number, except 2 and 5, is contained with- 
out a remainder in the number expressed in the common nota- 
tion by as many 9's as there are units, less one, in the prime 
number itself. Thus, 3 is a measure of 99 ; 7 of 999,999; and 
13 of 999,999,999,999. 

7. Three prime numbers cannot be in arithmetical progression, 
unless their common difference is divisible by 6 ; except 3 be 
the first prime number, in which case there may be three prime 
numbers in such progression, but in no case can there be more 
than three. 

8. This last principle is generally true, and may be stated 
as follows : There cannot be n prime numbers in arithmetical 
progression unless their common difference be divisible by 
2. 3. 5. 7. 11... n; except the case in which n is the 
first term of the progression, in which case there may be n such 
numbers, but not more. 

Though we have no general method for finding prime num- 
bers, there are several ways of detecting whether an assigned 
number is or is not a prime. Several remarkable formulas have 
been discovered which contain a large number of prime num- 
bers. The formula x*+x+ 41, by making successively #=0, 
1, 2, 3, 4, etc., will give a series 41, 43, 47, 53, 61, 71, etc., the 
first forty terms of which are prime numbers. This formula is 
mentioned by Euler in the Memoirs of Berlin, 1772. Of the 
two formulas # 2 +#-|-17, and 2a? 2 -f 29, the former gives seven- 
teen of its first terms primes, and the latter twenty-nine. Fer- 
mat asserted that the formula 2-f-l is always a prime when 
in is taken any term in the series 1, 2, 4, 8, 16, etc.; but Euler 
found that 2 :i2 +l=641 x 6,700,417 is not a prime. 

One of the most celebrated theorems for investigating primes 
is that discovered by Fermat and known as FermaVs Theorem. 
The theorem may be stated thus: If p be a prime, the (p l)th 



PRIME AND COMPOSITE NUMBERS. 381 

power of every number prime to p will, when diminished 
by unity, be exactly divisible by p. Expressed in algebraic 
language, we have the theorem P p ~ ' 1, is a multiple of p when 
p and P are prime to each other. Thus, 25 s 1 is exactly 
divisible by 7. 

Fermat is said to have been in possession of a proof of the 
theorem, though Euler was the first to publish its demonstra- 
tion. Euler's first demonstration was a very simple one, and 
is that usually given in the text-books. Amongst the other 
demonstrations of the theorem, those given by Lagrange are 
highly esteemed. 

It has been demonstrated by Legendre (Essai sur la Theorie 
des Nombres), that every arithmetical progression, of which the 
first term and common difference are prime to each other, con- 
tains an infinite number of prime numbers. It has been also 
shown by him that if N represents any number, then will the 
formula 

N 

h.logy 1.08366 

represent the number of prime numbers that are less than N, 
very nearly. 

Another celebrated theorem is that invented by Sir John 
Wilson, known as Wilson's Theorem. This theorem may be 
stated as follows : The continued product, increased by unity, 
of all the integers less than a given prime, is exactly divisible 
by that prime. The algebraic formula which expresses the the- 
orem, 1 + 1.2.3... (n 1), is divisible by n , n being a prime 
number. Thus 1 + 1.2.3.4.5. 6=721, is exactly divisible 
by 7. 

This theorem was first demonstrated by Lagrange ; his pro 
cess of reasoning, as might be expected, was very ingenious. 
It was afterward demonstrated by Euler, and finally byQausb, 
who extended the theorem by proving that " The product of 
all those numbers less than, and prime to, a given number, 
al, is divisible by a ;" the ambiguous sign being , when a 



382 THE PHILOSOPHY OF ARITHMETIC. 

is of the form p m , or 2p m , p being any prime number greater 
than 2 ; and, also, when a=4; but positive in all other cases. 

Wilson's Theorem furnishes us with an infallible rule, in 
theory, for ascertaining whether a given number be a prime or 
not ; for it evidently belongs exclusively to those numbers, as it 
fails in all other cases ; but it is of no use in a practical point 
of view, on account of the great magnitude of the product even 
for a few terms. 

In the later works on the Theory of Numbers it is demon- 
strated that, No algebraical formula can represent prime 
numbers only. It is also shown that, The number of prime 
numbers is infinite. The latter proposition is evident a 
priori; the former was pretty nearly evident from induction 
before it received a rigid demonstration. 

The distribution of prime numbers does not follow any known 
law; but for a given interval it is found that the number of 
primes is generally less the higher the beginning of the interval 
is taken. The whole number of primes below 10,000 is 1,230; 
between 10,000 and 20,000 it is 1,033; between 20,000 and 
30,000 it is 983 ; between 90,000 and 100,000 it is 879. The 
largest prime which had been verified when Barlow wrote, is 
2 31 1 = 2,147,483,647, which was found by Euler. 

The term prime is also applied to a species of numbers called 
complex numbers, first suggested by Gauss in 1825. Accord- 
ing to this theory, a complex integer is of the form a + b^/^j t 
in which a and b denote ordinary (real) integers. The product 
a 2 + 6 2 , of a complex number a+frv/IT^ and its conjugate, 
a fe^/ITf, is called its norm, and is denoted by the symbols 
N(a + b^ 1)> -N( a &\/ l)- The four associative numbers, 
a + fcv/^T. a\/~\ b, a b^/^}, and a<Si -f b, as well 
as their respective conjugates, have all the same norm. A com- 
plex number is said to be prime when it admits of no divisor 
except itself, its associatives, and the four units, 1, 1, \/i, 
and v/ 13 !- Many of the higher theorems, such as that of 
Fermat, may be extended to the system of complex numbers. 



CHAPTER IV. 

PERFECT, IMPERFECT, ETC., NUMBERS. 

HAVING separated numbers into their factors, the human 
mind, ever active in the attempt to discover the new, be- 
gan to compare the sum of the factors or divisors of numbers 
with the numbers themselves, and thus discovered certain re- 
lations which gave rise to three new classes of numbers. In 
some cases it was seen that a number was just equal to the 
sum of all of its divisors, not including itself, and such num- 
bers were called Perfect Numbers. Numbers not possessing 
this property were called Imperfect Numbers ; and were divided 
into two classes, Defective and Abundant, according as they 
were greater or less than the sum of their divisors. 

Pushing the comparison still further, it was also discovered 
that some numbers were reciprocally equal to their divisors; 
and this relation was so intimate that such numbers were re- 
garded as friendly or Amicable Numbers. These several classes 
will be formally defined in this chapter. Perfect and Imperfect 
numbers were known by the ancient Greek mathematicians, but 
their properties have been developed by the mathematicians of 
modern times. Amicable Numbers were first investigated by 
the Dutch mathematician Van Schooten, who lived from 1581 to 
1640. 

A Perfect Number is one which is equal to the sum of all its 
divisors, except itself; thus, 6=1+2+3; 28=1+2+4+7+14 
An J in perfect Number is one which is not equal to the sum 
of all its divisors. Imperfect Numbers are Abundant or De- 
fective. An Abundant Number is one the sum of whose di- 
visors exceeds the number itself; as, l+2+3+6+9>18. A 

(383) 



384 THE PHILOSOPHY OF ARITHMETIC. 

Defective Number is one the sum of whose divisors is less 
than the number itself; as, 1+2+4+8 < 16. 

Every number of the form (2"" 1 ) (2 n 1), the latter factor 
being a prime number, is a perfect number. The only values 
of n yet found, which make 2" 1 a prime are 2, 3, 5, 7, 13, 17, 
19, and 31 ; there are, therefore, only ten perfect numbers 
known. Substituting 2 for n in the formula, we have 2(2 2 1) 
=6, the first perfect number; the second is 2 2 (2 3 1)=28. 
The first eight perfect numbers are, 6, 28, 496, 8128, 3S55033F,, 
8589869056, 137438691328, 2305843008139952128. Each 
number, as is seen, ends in 6 or 28. 

The difficulty in finding perfect numbers consists in finding 
primes of the form of 2" 1. The greatest prime number, ac- 
cording to Barlow, yet ascertained, is 2 31 1 = 2147483647, dis- 
covered by Euler ; and the last of the above perfect numbers, 
which depends upon this, is the greatest perfect number known 
at present, and Barlow remarks that it is probably the greatest 
that will ever be discovered ; for, as they are merely curious 
without being useful, it is not likely that any person will 
attempt to find one beyond it. An author of an arithmetic gives 
two other numbers which are said to be perfect, 2417851639228- 
158837784576, 9903520314282971830448816128, but I do not 
know his authority. 

Two numbers are called Amicable when each is equal to the 
sum of the divisors of the other ; thus, 284 and 220. The for- 
mulas for finding amicable numbers are A2 n + } d and B= 
Z n+l bc, in which n is an integer, and b, c, and d are prime 
numbers satisfying the following conditions: 1st, 6=3 x 2" 1 ; 
2d, c=6x2 n -1; 3d, d=18x2 2(l 1. If we make n=l, we 
find 6=5, c=ll, and d=71; substituting these in the above 
formulas, we have .4=4x71=284, and 5=4x5x11=220, the 
first pair of amicable numbers. The next two pairs are 
17296, 18416, and 936358, 9437056. 

The first pair, 220 and 284, were found by E. Van Schooten, 
with whom the name amicable appears to have originated, though 



PERFECT, IMPERFECT, ETC., NUMBERS. 385 

Rudolphus and Descartes were previously acquainted with 
this property of certain numbers. A formula for amicable 
numbers was, in fact, given by Descartes, and afterwards gen- 
eralized by Euler and others. 

Figurate Numbers. Figurate Numbers are numbers formed 
from an arithmetical progression whose first term is unity, and 
common difference integral, by taking successively the sum of 
the first two, the first three, the first four, etc., terms of the 
series; and then operating on the new series in the same man- 
ner as in the original progression in order to obtain a second 
series, and so on. 

For example, take the series of natural numbers in which 
the common difference is 1, as repre- 

ented by A in the margin; then the A, 1-2-3-4-5-6-7 
series B, derived as stated above, will ^' |"J JQ~JJj"ge " ?1 " gjj 
be figurate numbers ; series C, derived p' 1.5.15.35.70-126-210 
as above from series B and series D, 

derived from series C, will be figurate numbers. Other seriea 
could be obtained by beginning with any other arithmetical 
series whose first term is 1, and common difference an integer. 
Thus, the series derived from the progression 1, 3, 5, 7, 9, etc., 
is 1, 4, 9, 16, 25, etc. 

A more general method of conceiving figurate numbers is to 
regard them as a series of numbers, the general term of each 
series being expressed by the formula, 

n(n+l)(n+2)(n+3) .... (n+m) 



1.2.3.4 . . . 
in which m represents the order of the series, and n represents 
the place of the required term. 

Series of figurate numbers are divided into orders; when m 
= 0, the series is of the 1st order; when m = I, the seriea is 
of the 2d order ; when m 2, it is of the 3d order, etc. 

By regarding m equal to in this formula, and substituting 
successive numbers 1, 2, 3, etc., for n, it will be seen that the 
general term is n, and we find that the figurate series of the 
first order is the series of natural numbers, 1, 2, 3, 4, etc., n. 



S86 THE PHILOSOPHY OF ARITHMETIC. 

By regarding m equal to 1, the general term of the series 

becomes -^ - > and substituting the successive values of n, 
1 . 2 

1, 2, 3, etc., we find the terms to be 1, 3, 6, 10, 15, 21, 28, etc., 
which is the series of figurate numbers of the second order. 

Ir a similar manner we find the general term of the figurate 
series of the 3d and 4th orders to be respectively, 

n(n + l)(n + 2) n(n + l)(n + 2) (n + 3) 
and 



1.2.3 1.2.3.4 

from which we can readily derive those series. These several 
series of figurate numbers are the same as those represented in 
the margin above. 

One of the most remarkable properties of the series of figu- 
rate numbers is that, if the nth term of a series of any order be 
added to the (n -f- l)th term of the series of the preceding 
order, the sum will be equal to the (n-fl)th term of the series 
of the given order. Thus, in the series marked C, if we add 
the second term, 4, to the third term, 6, in series B, we shall 
have the third term, 10, of series C ; the third term of series C 
plus the fourth term of series B equals the fourth term of series 
C, etc. 

If we begin with a series of 1's, all of the series of figurate 
numbers may be deduced in succession by the application of 
th : s principle. 

ORDERS OF FIGURATE NUMBERS. 

Series of Us 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 

1st order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 

2d order 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 

3d order 1, 4, 10, 20, 35, 56, 84, 120, 165, 220 

4th order 1, 5, 15, 35, TO, 126, 210, 330, 495, U5 

5th order 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002 

6th order 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005 

1th order 1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440 

By inspecting these series, it will be seen that the values 



PERFECT, IMPERFECT, ETC., NUMBERS. 387 

ead diagonally upward are the numerical coefficients of the 
terras in the development of (a -j- b) with an exponent corre- 
sponding to the order of the series. It is said that it was this 
principle which gave rise to a complete investigation of the 
subject of figurate numbers. 

In speaking of denning figurate numbers by giving the form 
of each of the orders, Barlow remarks that it is more simple to 
deduce the generation of figurate numbers from their form than 
to deduce their form from their generation. The principle 
given above, showing the relation of the terms of two succes- 
sive orders of figurate numbers, is ascribed to Fermat, and is 
considered by him as one of his most interesting propositions. 

Polygonal Numbers are figurate numbers which represent 
the sides of polygons. The second series of figurate numbers, 
I, 3, 6, 10, etc., are called triangular . 

numbers, because the number of units 

that they express can be arranged in * * * 

the form of a triangle. If we take 
the series 1, 3, 5, 7, 9, etc., in which 
the common difference is 2, we obtain .... 

the figurate series, 1, 4, 9, 16, 25, etc., which are called square 
numbers, because they can be arranged in a square. The 
series 1, 4, 7, 10, etc., in which the common difference is 3, 
gives the series 1, 5, 12, 22, etc., which are called pentagonal 
numbers, because they can be arranged in the form of a penta- 
gon. In a similar manner we obtain hexagonal, heptagonal, 
octagonal, etc., numbers. It will be noticed that the number 
of the sides of the polygon which they represent is always two 
greater than the common difference of the series from which 

they were derived. Common difference=l ; 1, 2, 3, 4, 5, 

When the common Triangular numbers 1, 3, 6, 10, 15, 21 

A -ff f ,1 Common difference=2; 1, 3, 5, 7, 9. 11 

difference of the Square numbers 4 | 9> 16 | ^ 36 

series in arithmeti- Common difference=3 ; 1, 4, 7,10,13,16 

cal progression is Pentagonal numbers 

I, the sums of the terms give the triangular numbers; when 



888 THE PHILOSOPHY OF ARITHMETIC. 

the common difference is 2, the sums of the terms are the squart. 
numbers ; when the difference is 3, the sums are the pentagonal 
numbers, and so on. 

These numbers are called polygonal from possessing the pro- 
perty that the same number of points may be arranged in the 
form of that polygonal figure to which it belongs. Thus the 
pentagonal numbers 5, 12, 22, 35, 51, etc., may be severally 
arranged in the form of a pentagon. Thus, 5 points will form 
one pentagon; 12 points will form a second pentagon enclos- 
ing the former; 22, a third pentagon enclosing both of the 
former, etc. 

The following property of polygonal numbers was discovered 
by Fermat : Every number is either a triangular number or 
the sum of two or three triangular numbers; every number is 
either a square number, or the sum of two, three, or four 
square numbers ; every number is either a pentagonal number 
or the sum of two, three, four, or Jive pentagonal numbers ; 
etc. This property is generally true, although it has been 
demonstrated for only triangular and square numbers. All 
the other cases still remain without demonstration, notwith- 
standing the researches of many of the ablest mathematicians. 
Permat himself, however, as appears from one of his notes on 
Diophantus, was in possession of the demonstration, although 
it was never published, which circumstance renders the theorem 
still more interesting to mathematicians, and the demonstration 
of it more desirable. 

Pyramidal Numbers are those which represent the number 
of bodies that can be arranged in pyramids. They are formed 
by the successive sums of polygonal numbers in the same man- 
ner as the polygonal numbers are formed from arithmetical 
progressions. The Triangular Pyramidal numbers are the 
series of figurate numbers derived from the series of triangular 
numbers. Thus, from the triangular numbers 1, 3, 6, 10, 15, 
etc., we have the triangular pyramidal numbers 1, 4, 10, 20, 
etc. The Square Pyramidal numbers are derived from the 
square numbers. 



CHAPTER V. 

DIVISIBILITY OF NUMBERS. 

IN factoring a composite number, we divide successively by 
exact divisors of the number till we obtain a quotient which 
is a prime number. In order to know by what numbers to 
divide, it is convenient to have some tests of divisibility, other- 
wise it would be necessary to try several numbers until we hit 
upon one which is exactly contained. There are certain laws 
which indicate, without the test of actual division, whether a 
number is divisible by a given factor, some of which are simple 
and may be readily applied The investigation of these laws 
of the relations of the factors of numbers to the numbers them- 
selves, gives rise to a subject known as" the Divisibility of Num- 
bers. 

The laws for the divisibility of numbers, as usually presented, 
embrace the conditions of divisibility by the numbers 2, 3, 4, 
etc., up to 12. These laws may be stated as follows: 

1. A number is divisible by 2 when the right-hand term is 
zero or an even digit. For, the number is evidently an even 
number, and all even numbers are divisible by 2. 

2. A number is divisible by 3 when the sum of the numbers 
denoted by its digits is divisible by 3. It will be shown here- 
after that every number is a multiple of 9, plus the sum of its 
digits; hence, since 3 is a factor of 9, the number is divisible 
by 3 when the sum of the digits is divisible by 3. 

3. A number is divisible by 4, when the two right-hand terms 
are ciphers, or when they express a number which is divisible 
by 4. If the two right-hand terms are ciphers, the number 

(389) 



390 THE PHILOSOPHY OF ARITHMETIC. 

equals a number of hundreds, aiid since 100 Is divisible by 4, 
any number of hundreds is divisible by 4. If the number ex- 
pressed by the two right-hand digits is divisible by 4, the num- 
ber will consist of a number of hundreds, plus the number ex- 
pressed by the two right-hand digits ; and since both of these 
are divisible by 4, their sum, which is the number itself, is 
divisible by 4. 

4. A number is divisible by 5, when its right-hand term is 
or 5. If the right-hand term is 0, the number is a number 
of times 10 ; and since 10 is divisible by 5, the number itself 
is divisible by 5. If the right-hand term is 5, the entire num- 
ber will consist of a number of tens, plus 5 ; and since both 
of these are divisible by 5, their sum, which is the number 
itself, is divisible by 5. 

5. A number is divisible by 6, when it is even and the sum 
of the digits is divisible by 3. Since the number is even, it is 
divisible by 2, and since the sum of the digits is divisible by 3, 
the number is divisible by 3, and since it contains both 2 and 3 
it will contain their product, 3x2, or 6. 

6. A number is divisible by 7, when the sum of the odd nu- 
merical periods, minus the sum of the even numerical periods, 
is divisible by 7. The law for the divisibility by 7 is perhaps 
of not so much practical importance as the others, being not 
quite so readily applied, but it is of too much scientific interest 
to be omitted from the series. Its demonstration will be given 
in the following chapter. 

7. A number is divisible by 8, when the three right-hand terms 
are ciphers, or when the number expressed by them is divisible 
by 8, If the three right-hand terms are ciphers, the number 
equals a number of thousands; and since 1000 is divisible by 8, 
any number of thousands is divisible by 8. If the number ex- 
pressed by the three right-hand digits is divisible by 8, the 
entire number will consist of a number of thousands, plus the 
number expressed by the three right-hand digits (thus 17368 
= 17,000 + 368) ; and since both of these parts are divisible by 
8, their sum, which is the number itself, is divisible by 8. 



DIVISIBILITY OF NUMBERS. 391 

8. A number is divisible by 9, when the sum of the digits is 
divisible by 9. This law is derived from showing that a num- 
ber may be resolved into two parts, one part being a multiple 
of 9 and the other the sum of the digits. A complete demon- 
stration is presented on a subsequent page, to which the reader 
is referred. 

9. A number is divisible by 10, when the unit term is 0. For, 
such a number equals a number of tens, and any number of tens 
is divisible by 10; hence the number is divisible by 10. 

10. A number is divisible by 11, when the difference between 
the sums of the digits in the odd places and in the even places 
is divisible by 11, or when the difference is 0. This law 
is derived by showing that a number may be resolved into two 
parts, one part being a multiple of 11, and the other part con- 
sisting of the sum of the digits in the odd places, minus the 
sum of the digits in the even places. A complete demonstra- 
tion will be presented on a subsequent page. 

11. A number is divisible by 12, when the sum of the digits 
is divisible by 3 and the number expressed by the two right- 
hand digits is divisible by 4. For, since the sum of the digits 
is divisible by 3, the number is divisible by 3, and since the 
number expressed by the two right-hand digits is divisible by 
4, the number is divisible by 4; hence, since the number is 
divisible by both 3 and 4, it is divisible by their product, or 12. 

These laws are simple, and, with the exception of those re- 
lating to the numbers 7, 9, and 1 1, readily applied. The laws of 
dividing by 9 and 11 present some interesting points, which will 
be formally discussed. It will be noticed, upon examining text- 
books on arithmetic, and also works on the theory of numbers, 
that the law of divisibility by 7 is omitted. Apparently efforts 
\vere made to discover such a law, for several writers give 
some special rules for dividing by 7 ; but it would seem that 
no general law was known to them. In the principle as above 
presented, this hiatus is filled up by a law not quite so simp].- 
as that for the other numbers, but still of scientific interest, if 



392 THE PHILOSOPHY OF ARITHMETIC. 

not of much practical value. Besides the law given, there are 
several other laws, interesting as showing the development of 
the subject, and which we therefore present. The methods of 
demonstration are similar to those used in proving the divisi- 
bility of numbers by 9 and 11; indeed, one of the laws from 
which the others were derived was discovered by the applica- 
tion of that method to the number 7. I shall therefore first 
present the demonstration of divisibility by 9 and 11, and then 
state and demonstrate the laws relating to the number 7. 

Divisibility by Nine. The law of divisibility by nine has 
been known for a long time. By whom it was discovered has 
not been ascertained. Its application to testing the correctness 
of the work in the fundamental rules, called proof by " casting 
out nines," has been attributed to the Arabs. The law, as pre- 
viously stated, is that a number is divisible by nine when the 
sum of the digits is divisible by nine. This principle depends 
on a more general law which will be first stated, and then the 
law of exact division, as well as some other interesting princi- 
ples, will be drawn from it. 

1. A number divided by 9 leaves the same remainder as the 
sum of the digits divided by 9. 

This theorem can be demonstrated both arithmetically and 
algebraically. We will first present the arithmetical demonstra- 
tion. If we take any number, as 6854, and analyze it, as in 
the margin, r 4_ 4 

we will see ft aej J 50=5x10 =5x (9+l)=5x9 +5 
thatitcon- J 800=8x100 =8x (99+l)=8x99 +8 

sists of two 1 6000=6 x 1000=6 x (999+l)=6 X 999+6 

parts: the Multiple of 9 Sumof^igits 

.-. 6854 = 5x9+8x99+6x999 + 4+5+8+6 
first p&Tu CL 

multiple of 9, and the second part the sum of the digits. 

The first part is evidently divisible by 9, hence the only re- 
mainder that can arise from dividing a number by 9 will be 
equal to the remainder arising from dividing the sum of the 
digits by 9. When the sum of the digits is exactly divisible 



DIVISIBILITY OF NUMBERS. 393 

by 9, it is evident that the number itself is exactly divisible 
by 9, which proves the theorem. From this theorem the fol- 
lowing principles may be readily inferred : 

2. A number is exactly divisible by 9 when the sum of it* 
digits is divisible by 9. 

3. The difference between any number and the sum of its 
digits is divisible by 9. 

4. A number divided by 9 gives the same remainder as any 
one formed by changing the order of the figures. 

5. The difference between two numbers, the sums of whose 
digits are equal, is exactly divisible by 9. 

The fundamental theorem may also be demonstrated algebra- 
ically as follows: Let a, b, c, d, etc., represent the digits of 
any number, aod r the radix of the scale, that is, the number 
of units in a group ; then every number may be represented 
by formula (1) below. If we now subtract b, c, d, etc., from 
one part of this expression, and add them to another part, it 
will not change the value, and we shall have formula (2) ; and 
factoring, we obtain formula (3). 

(1). ^T=a+6r+cr 2 +dr s +er < +etc. 

(2). N=br 6-Hcr 2 c+dr 3 d+er f e, etc.+a + b + c+d+e 
+etc. 

(3). N=b (rl) + c (r 2 1) + d (r s -l) + e (r 4 -!) -fete. +a 



Now, r 1, r 1 1, r 3 1, etc., etc., are all divisible by r 1 ; 
hence the only remainder which can arise from dividing the 
number by j 1, will occur from dividing a+b+c+d+etc., by 
r 1; that is, any number divided by r 1 leaves the same 
remainder as the sum of the digits divided by r 1. In our 
decimal scale r=10, hence r 1=9; and hence any number 
divided by 9 leaves the same remainder as the sum of the digits 
divided by 9. This law is the basis of some very interesting 
properties, and also of the proof of the fundamental rules called 
"casting out nines." 

Divisibility by Eleven. The law of the divisibility of num- 
17* 



394: THE PHILOSOPHY OF ARITHMETIC. 

bers by 1 1 is quite similar to that of 9. This might have been 
anticipated, as they each differ from the basis of the scale by 
unity, the former being a unit below and the latter a unit above 
the base. The law, as previously stated, is that a number it 
divisible by 11 when the difference between the sum of the 
digits in the odd places and the even places is divisible by 11. 
This principle depends upon a more general one, which will first 
be stated, and then this, as well as some other interesting prin- 
ciples, will be derived from it. 

1. Every number is a multiple of 11, plus the sum of the 
digits in the odd places, minus the sum of the digits in the 
even places. This principle may be demonstrated both arith- 
metically and algebraically. We will first give the arithmetical 
proof. If we take any number, as 65478, and analyze it as in- 

+8 
70= 7x10= 7X(H 1)= 7x117 



65478= 



400= 4x100= 4x(99-fl)= 4x99+4 
5000= 5 X 1000=5 X (1001 I)=5xl001 5 
L 60000=6xlOOOO=6x(9999+l)=6x 9999+6 

Sum of Sum of 

Multiples of 11. odd digits. even digits. 



/. 65478=7x11+4x99+5x1001+6x9999 + 8+4+6 5+7 
dicated, we shall see that it consists of two parts; the first 
being a multiple of 11, and the second consisting of the sum 
of the digits in the odd places, minus the sum of the digits in 
the even places. The first part is evidently divisible by 11 ; 
hence the only remainder that can arise from dividing a 
number by 11 will be equal to the remainder arising from 
dividing the difference between the sums of the digits in the 
odd places and the even places by 11. When this difference is 
exactly divisible by 11, it follows that the number itself is 
divisible by 11. When the sum of the digits in the even places 
is greater than the sum in the odd places, we take the difference, 
divide by 11, and subtract the remainder from 11 to find the 
true remainder. The reason for this will appear from the 
above demonstration. From this theorem the following prin 
ciples can be readily inferred : 

2. A number is exactly divisibliTby 11, when the sum of the 



DIVISIBILITY OF NUMBERS. 395 

digits in the odd places is equal to the sum of the digits in the 
even places. 

3 A number is exactly divisible by II, when the difference 
between the sums of the digits in the odd places and the even 
places is a multiple of 11. 

4. A number increased by the sum of the digits in the even 
places and diminished by the sum of the digits in the odd 
places, is exactly divisible by 11. 

5. The excess of ll's in any number is not changed by add- 
ing any multiple of 11 to the sum of the digits of either order. 

The algebraic demonstration of this property is as follows: 
Taking the same formula as for the number 9, we add b and 
then subtract b, we subtract c and "hen add c, etc., the formula 
becoming (2) below, being the same in value as the first, 
but changed in form. Then, factoring, we have (3). 

(1). N " T hr+ / dr +er*+etc. 

(2). N=br+b+cr 2 c+dr^d+er* e+etc.+a 6+c d+e, 
etc. 

(3).A^6(r+l)+c(r 2 1) + d (r s + 1 ) + e (r 4 l)+etc.+(a+ 
c fe-f-etc.) (6+d+ctc.) 

Now r-H, r 2 1, r s +l, etc., are each divisible by r+1; 
hence the only remainder that can arise from dividing this 
number by r+1 must arise from dividing (o+c+e+etc.) 
(6+d+etc.) by r+ 1 ; that is, by dividing the difference of the 
sum of the digits in the even places subtracted from the sum 
of the digits in the odd places by r+ ! . In the decimal scale, 
r=10, and r+l=ll; hence we sue that any number divided 
by 1 1 leaves the same remainder as the difference of the sum 
of the digits in the even places, subtracted from the sum of the 
digits in the odd places does when divided by 11. When this 
difference is exactly divisible by 11, the number itself is divisi- 
1)1.', which proves the principle of the divisibility by 11. This 
principle may also be used for the proof of the fundamental 
rules, but not quite so conveniently as that of the number 'J. 



CHAPTER VI. 

THE DIVISIBILITY BY SEVEN. 

THE Divisibility of Numbers, as presented by different 
authors, embraces the conditions of divisibility by the 
numbers 2, 3, etc., up to 12, with the omission of the num- 
ber 7. This omission leads us to inquire whether there is 
any general law for the divisibility of numbers by 7. A few 
of our text-books present some special truths in regard to this 
subject, among which are the following : 

1. A number is divisible by 7 when the unit term is one-half 
or one-ninth of the part on the left. Thus 21, 42, 63, 126, and 
91, 182, 273, etc. 

2. A number is divisible by 7 when the number expressed 
by the two right-hand terms is five times the part on the left, 
or one-third of it. Thus 525, 840, 1995, and 602, 903, 3612, 
etc. 

3. A number consisting of not more than two numerical 
periods is divisible by 7 when these periods are alike. Thus 
45045, 235235, 506506, etc., are divisible by 7. 

There are, however, some general laws for the divisibility 
by 7, which seem to have been overlooked by most writers on 
the theory of numbers, and which, though of not much practical 
importance, are interesting in a scientific point of view. The 
first and least simple of these laws is as follows : 

1. A number is divisible by 7, when the sum of once the 
first, or units digit, 3 times the second, 2 times the third, 6 
times the fourth, 4 limes the fifth, 5 times the sixth, once the 
seventh, 3 times the eighth, etc., is divisible by 7. It will be 

(396) 



THE DIVISIBILITY BY SEVEN. 397 

seen that the series of multipliers is 1, 3, 2, 6, 4, 5. To illus- 
trate the law, take the number 7935942, and we have for the 
sum of the multiples of the digits, 1 x 2+3 x 4+2 x 9+6 x 5+4 x 
3+5 x 9+1 x 7= 126, which is exactly divisible by 7 ; and if we 
divide the number itself by 7, we find there is no remainder. 
Assuming this principle it will be demonstrated on page 
398 we can derive several other principles of divisibility 
from it. 

In this law we see that the second half of the series of mul- 
tipliers, 6, 4, 5, equals respectively 7 minus the first half, 1, 3, 2; 
hence, instead of adding the multiples of the second series, 6, 4, 
5, we may subtract the respective multiples of the terms of the 
second period by the first series of multipliers, 1, 3, 2, which 
will give rise to the following principle : 

2. A number is divisible by 7, when the number arising 
from the sum of once the first digit, 3 times the second, 
2 times the third, minus the sum of the same multiples of the 
next three digits, plus the sum of the same multiples of the 
next three digits, etc., is divisible by 7. 

It will be seen that the series of multipliers is 1, 3, 2, the 
first products additive, the second products subtractive, etc. ; 
the odd numerical periods being additive and the even periods 
subtractive. If we take the number 5439728, we have 1x8+ 
8 x 2+2 x 71 X 98 X 32 X 4+1 x 5=7, which isdivisible by 
7. Upon trial we find the original number is also exactly di- 
visible by 7. 

This second principle may also be stated thus: A number is 
divisible by 7 when the sum of the multiples expressed by the 
numbers, 1, 3, 2, of the terms of the odd numerical periods, 
minus the sum of the same multiples of the terms of the evert 
numerical periods, is divisible by 7. 

Now, if we add exact multiples of 7 to the multiples of the 
terms which are united in the test of divisibility, it will not 
change the remainder. Thus, taking the number 5439728, if 
we add 7 X 2 to 3 x 2, we have 10 x 2, or 20 ; and adding 98 X 1 



398 THE PHILOSOPHY OF ARITHMETIC. 

X) 2x7 we have 100x7, or 700; hence we may use in place 
)f 1x8+3x2+2x7, 8+20+700, or 728, the first numerical 
period ; and in the same way it may be shown that we may 
use the second period subtractively in the test, etc. Hence 
from Principle 2 we may derive the following principle: 

3. A number is divisible by 7, when the sum of the odd nu- 
merical periods, minus the sum of the even numerical periods, 
is divisible by 7. 

To illustrate, take the number 5,643,378,762; we have for 
the sum of the odd numerical periods 762+643=1405; for the 
sum of the even periods, 378+5=383; the difference is 1022, 
which is exactly divisible by 7 ; and if we divide the number 
itself by 7, we find that there is also no remainder. 

If we apply the same reasoning to Principle 1, by which we 
derived Principle 3 from Principle 2, we shall derive from it the 
following principle : 

4. A number is divisible by 7, when the sum of the numbers 
denoted by the double numerical periods is divisible by 7. 
Thus, in the number 5,643,378,762, we have 5,643+378,762= 
384,405, which is divisible by 7, and the number is also divisi- 
ble by 7. 

The first principle, from which I have derived the other 
three, may be demonstrated arithmetically and algebraically. 
Let us take any number as 98765432 and analyze it thus : 

2= 1X2 

30= 3X10= 3x(7+3)= 3x7+3x3 

400= 4x100= 4 X (98+2)= 4x98+2x4 

5000= 5X1000= 5x (994+6)= 5x994+6x5 

60000= 6x10000= 6 X (9996+4)= 6x9996+4x6 

700000= 7x100000= 7x(99995+5)= 7x99995+5x7 

8000000= 8x1000000= 8x (999999+1)= 8x999999+1x8 

90000000=9 X 10000000=9 X (9999997+ 3) =9 X 9999997+3 X 9 

Here 98765432=a multiple of 7 plus once the 1st term, plus 
three times the second term, plus two times the third term, plus 
six times the fourth term, plus four times the fifth term, plus 
five times the sixth term, plus once the seventh term, plus three 
times the eighth term. Hence the only remainder that can occur 
must arise from dividing the sum of the multiples of the terms 



THE DIVISIBILITY BY SEVEN. 399 

by 7 ; hence when the sura of these multiples is divisible by 7, 
the number is divisible by 7, which proves the principle. 

The second principle, which is readily derived from the first, 
may be demonstrated independently, as follows: 

2= 1x2 

30= 3x10= 3x(7+3)= 3x7+3x3 

400= 4x100= 4x(98+2)= 4x98+2x4 

5000= 5X1000= 5 X (1001 1)= 5x10011x5 

60000= 6x10000= 6x(10003 3)= 6x100033x6 

700000= 7x100000= 7x (100002 2)= 7x1000022x7 

8000000= 8x1000000= 8x (999999+1)= 8x999999+1x8 

90000000=9xlOOOOOOO=9x(9999997+3)=9x 9999997+3x9 

Here 98765432=a multiple of 7, plus once, the first digit, 
plus three times the second, plus twice the third, minus once the 
fourth, minus three times the fifth, minus twice the sixth, plus 
once the seventh, plus three times the eighth. Hence the only 
remainder that can occur must arise from dividing the difference 
between the additive and subtractive multiples of the digits by 
7 ; therefore, when this difference is divisible by 7, the number 
is divisible by 7, which proves the principle. When the sum 
of the subtractive multiples of the digits is greater than the 
sum of the additive, we take the difference, divide by 7, and 
subtract the remainder from 7 to find the true remainder. 

To demonstrate the third principle, take any number, as 7,946,- 
321,675 and analyze it, and it will be seen to consist of parts 
which are multiples of 7, plus the periods in the odd places, 
minus the periods in the even places. 

675= 675 



7946321675= - 



321000= 321 X (1001 1)= 821x1001321 



946000000= 946X (999999+1) =946 X 999x 1001+940 
. 7000000000=7x (1000000001 I)=7x999001x 1001- 7 

Multiples of 7. Odd periods. Even periods. 

321 X 1001+946x999999+7x1000000001 +"675+946 - 321+7 
Now 1001 is a multiple of 7, 999999 is 999 times 1001, and 
1000000001 is also a multiple of 1001, and if we continue the 
number to still higher periods, we shall find a constant series 
of multiples of 1001, alternately 1 more and 1 less than the 
number represented by one unit of the period. Hence 
7,946,321,675 is composed of the sum of three multiples of 7, 
plus (675 + 946) (321 + 7), or the difference between t.hp nms 



400 THE PHILOSOPHY OF ARITHMETIC. 

of the even and odd periods. The first part is evidently divisi- 
ble by 7, therefore the divisibility of the number depends on 
the divisibility of the difference of the sums of the odd and 
even periods ; and when this difference is divisible by 7, the 
number itself must be divisible by 7, which proves the prin- 
ciple. 

From this demonstration, we can immediately derive the fol- 
lowing principle, more general than the one stated and from 
which that may be derived: 

3. Any number divided by 7 gives the same remainder as is 
obtained when the sum of the odd numerical periods, minus the 
sum of the even numerical periods, is divided by 7. If the sum 
of the even periods is the greater, we find the difference, divide 
by 7, and subtract the remainder from 7 for the true remainder. 

This investigation leads to a still more general principle of 
divisibility, derived from the fact that 1001, which maybe con- 
sidered as the basis of the above demonstration, is the product 
of 7, 11, and 13; hence what we have just proved for 7, is also 
true of 11 and 13. The most general form of the principle then 
is as follows: 

6. Any number divided by 7, II, or 13 gives the same re- 
mainder as is obtained when the sum of the odd numerical 
periods, minus the sum of the even numerical periods, is divided 
by 7, 11, or 13 respectively. 

A special truth growing out of this general principle, had 
been previously given in the rule that any number of not more 
than two periods, when those two periods are alike, is divisible 
by 7, 11, or 13. All such numbers, on examination, will be found 
to be multiples of 1001, and, of course, divisible by its factors. 
It may seem surprising that those who were familiar with 
this special truth, and were thus on the very brink of a dis- 
covery, did not extend it and reach the general law above pre- 
sented. 

The fourth Principle, which was derived from the first, maj 
also be demonstrated independently by a method similar to 
that used in proving the third Principle. The algebraic demon 



THE DIVISIBILITY BY SEVEN. 401 

siration of Principle 1, which is the foundation of the other 
principles, is as follows: Take the same general formula as used 
in demonstrating the divisibility by 9 and 11, add and subtract 
36, 3 z e, B 3 d, etc., and the formula is readily reduced to the form 
of (5). 

(1). N=a+br+cr*+dr 3 +er'+fr 6 +gr*+hr''+etc. 

(2). N=br 36+cr 2 3'c+dr s 3 s d+er* tfe+fr 6 3 5 /, etc. 
+a+36+9c+27d+81e+243/+ete. 

(3). #=6(r-3)+c(r 2 -3 2 )+d(r 3 3 3 )+e(V 4 -3 4 )+/(r 5 3 5 ) 
+g(r* 3 6 ) etc.+a+36+9c+27d+81e+243/+7290, etc. 

(4). N=b(r 3)+c(r 2 3'0+d(r 3 3 3 )+e (r 4 3 4 ) +/(r 3 5 ) 



(5). N= \ b (V_3) + c (V 3') + d (r 3 3 s ) + e (r 4 3 4 ) +/ 
(^S^+grCr 6 S^+etc. +7c+2W+TTe+238/+ 7280+ etc. j -fa 
+36+ 2c+6d + 4e+5/+ 1 gr+etc. 

Now the first part of this expression is exactly divisible by 
r 3, or 7 ; hence the only remainder that can arise must occur 
from dividing a+36+2c+6d, etc., by r 3, or 7 ; that is, by 
dividing by 7 the sum of once the first digit, three times the 
second, two times the third, six times the fourth, four times 
the fifth, five times the sixth, and so on in the same order; and 
when this sum is exactly divisible by 7, the number is divisi- 
ble by 7. By a slight change in the terms of the formula, the 
theorem as stated in the second form may also be derived. 

Several years after the discovery of the law expressed in 
Principle 2, I learned that Prof. Elliott had employed the same 
property as early as 1846. Whether it was known to any 
mathematicians previous to this date, I am not able to ascertain. 

Laws for Other Numbers. In a similar manner we may find 
a law for the divisibility of numbers by 13, 17, etc. The law 
26 



402 THE PHILOSOPHY OP ARITHMETIC. 

for 13 may be stated as follows: A number is divisible by 13 
when ONCE the first term, MINUS the sum of 3 times the second 
4 times the third and 1 time the fourth, PLUS the sum of the 
same multiples of the next three terms, MINUS the sum of the 
same multiples of the next three terms, etc., is divisible by 13. 

It will be noticed that after the first term, the series of num- 
bers by which we multiply is 3, 4, 1, which is easily remem- 
bered and readily applied. To illustrate, take the number 
8765432; we have 2 (3x3+4 x 4+1 x5)+(3x 6+4x7+1 x 8) 
=26, which is divisible by 13; and on trial we find the num- 
ber itself is also divisible. 

This law is derived from the more general principle that any 
number divided by 13 will give the same remainder as that ob- 
tained by dividing the result arising from the above multiples 
by 13. This principle may be demonstrated by taking any 
number, as 4987654, and analyzing it as in the previous case. 

4= +1x4 

50= 5x10= 5x(13 3)= 5x13-3x5 
600= 6X100= 6X (104-4)= 6x104-4x6 
4987654=-! 7000= 7x1000= 7x (1001-1)= 7x1001-1x7 
80000= 8x10000= 8 X (9997+ 3)= 8x9997+3x8 
900000= 9X100000= 9x (99996+4)= 9x99996+4x^ 
L 4000000 =4x 1000000 =4x(999999+l)=4x 999999+1x4 

Laws for the divisibility of numbers by 17, 19, 23, etc., may 
be obtained in a similar manner. We present a few of then) 
below, including 7, 11, and 13, already given. 

( 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, etc. 
' (orl, 3, 2, 6, 4, 5. 1, 3, 2, 6, 4, 5, etc. 

1, -1, 1, -1, 1, -1, 1, -1, 1, -I, 1, -I, etc. 
orl, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, etc. 

( 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, etc. 
' (orl, 10, 9, 12, 3, 4, 1, 10, 9, 12, 3, 4, etc. 

., (1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, etc. 
(orl, 10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, etc 

f 1, 10, 18, 16, -4, 1, 10, 18, 16, -4, etc. 
' ' (orl, 10, 18, 16, 37, 1, 10, 18, 16, 37, etc. 



( 



. THE DIVISIBILITY BY SEVEN. 403 

( 1, 10, 27, -22, -1, -10, -27, 22, 1, 10, 27, etc. 
"jorl, 10, 27, 51, 72, 63, 46, 22, I, 10, 27, etc. 
99. . . 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, etc. 

, ( 1, 10, -1, -10, 1, 10, -1, -10, 1, 10, -1, -10, etc. 
"(or 1, 10, 100, 91,1,10,100, 91,1,10,100, 91, etc. 

The laws for 99 and 101, it is seen, are very simple and 
readily applied. 



CHAPTER VII. 

PKOPERTIES OF THE NUMBER NINE. 

rpHE number Nine possesses the most remarkable .pro- 
JL perties of any of the natural numbers. Many of these 
properties have been known for centuries and have excited much 
interest among both mathematicians and ordinary scholars. 
So striking and peculiar are some of these properties that the 
number nine has been called " the most romantic " of all the 
numbers. On account of its relation to the numerical scale, if 
we get the factor 9 into a number it will cling to the expression 
and turn up in a variety of ways, now in one place and now 
in another, in a manner truly surprising. It reminds one of a 
mountain streamlet which ripples along its pathway, now buried 
beneath the ground and for awhile hidden from our sight, but 
presently gurgling to the surface at the most unexpected 
moment. It is no wonder that the property has been regarded 
as magical, and the number been called the "magical number." 
A few of these interesting properties will be here presented. 

1. The first property of this number which attracts our at- 
tention is, that all through the column of "nine times" in the 
multiplication table, the sum of the terms is nine or a multiple 
of nine. Begin with twice nine, 18; add the digits together, 
and 1 and 8 are 9. Three times 9 are 27 ; 2 and 7 are 9. So 
it goes on up to eleven times nine, which gives 99. Add the 
digits; 9 and 9 are 18; 8 and 1 are nine. Go on in the 
same manner to any extent, and it is impossible to get rid of 
the figure 9. Multiply 326 by 9, and we have 2934, the sum 
of whose digits is 18, the sum of whose digits is 9. Let the 

( 404) 



PROPERTIES OF THE NUMBER NINE. 405 

number nine once enter any calculation involving multiplica- 
tion, and whatever you do, "like the body of Eugene Aram's 
victim," it is sure to turn up again. This curious property is 
explained by the principle of divisibility of numbers presented 
in the previous chapter. All these numbers being divisible by 
9, the sums of their digits must be 9, or a multiple of 9. 

2. Another curious property of the number nine is that if 
you take any row of figures and change their order as you 
please, the numbers thus obtained, when divided by 9, leave 
the same remainder. Thus, 42378, 24783, 82734, etc., when 
divided by 9 all give the same remainder, 6. The reason of 
this is, that the sum of the digits is the same, in whatever order 
they stand ; and, as previously shown, the remainder from 
dividing a number by 9, is the same as the remainder from 
dividing the sum of its digits by 9. 

3. An interesting principle is presented in the following 
puzzle, which, to the uninitiated, seems very singular. Take 
a number consisting of two places, invert the figures, and take 
the difference between the resulting number and the first 
number, and tell me one figure of the remainder and I will 
name the other. The secret is that the sum of the two digits 
of the remainder will always equal 9. Thus take 74, invert 
the terms, and wo have 47; take the difference of the two num- 
bers and we have 27, in which we see that the sum of 7 and 2 
equals 9. In this case, suppose I had not known what number 
was taken ; if the person had named one digit, say 2, I could 
have immediately named the other digit 7, since I know that 
the sum of the two digits is always 9. 

The reason for this is that both numbers, having the same 
digits, are multiples of 9 with the same remainder; hence 
their difference is an exact multiple of 9, and consequently the 
sum of the two digits will equal 9. When the digits of the 
number are equal, the difference will be 0; and when they 
differ by unity, the difference will be 9. 

4. There is another interesting puzzle, based upon theso 



406 THE PHILOSOPHY OP AKITHMETIC. 

principles, which is very curious to one who does not see the 
philosophy of it, and interesting to one who does. You tell a 
person to write a number of three or more figures ; divide by 
&, and name the remainder; erase one figure of the number; 
divide by 9, and tell you the remainder ; and you will tell what 
figure was erased. 

This is readily done when the principle is understood. If 
the second remainder is less than the first, the figure erased is 
the difference between the remainders; but if the second 
remainder is greater than the first, the figure erased equals 
the difference of the remainders subtracted from 9. The 
reason for this is that the remainder, after dividing a number 
by 9, is the same as the remainder after dividing the sum of 
the digits by 9, and hence the sum of the digits being diminished 
by the number erased, the remainder will also be diminished 
by it. If there is no remainder either time, then the term 
erased must be either or 9. 

To illustrate, suppose the number selected were 45T; divid- 
ing by 9 the remainder is T; erasing the second term and 
dividing, the remainder is 2 ; hence the term erased is 7 less 2 
or 5. If the number were 461, dividing by 9, the remainder 
is 2; erasing the second term and dividing, the remainder is 
5; hence the term erased must be the difference between 5 
and 2, or 3, subtracted from 9, which is 6. 

5. The following puzzle also arises from the principle of the 
divisibility by 9. Take any number, divide it by 9, and name 
the remainder; multiply the number taken by some number 
which I name, and divide the product by 9, and I will name 
the remainder. To tell the remainder, I multiply the first 
remainder by the number which I named as a multiplier, and 
divide this product by 9. The remainder thus arising will 
evidently be the same as the remainder which the person 
obtained. 

6. If we take any number consisting of three consecutive 
digits and, by changing the place of the digits, make two other 



PROPERTIES OF THE NUMBER NINE. - 407 

numbers, the sum of these three numbers will be divisible by 
9. This depends on the principle that the sum of any three 
consecutive digits is divisible by 3; and consequently each 
number, if not an exact multiple of 9, is a multiple of 9 plus 3, or 
of 9 plus a multiple of 3 ; and therefore the sum of three numbers 
is a multiple of 9 plus three 3's, and thus an exact multiple of 
9. If we permutate the digits, making five other numbers, the 
sum of the six numbers will be divisible by twice 9 ; which 
may also be readily explained. 

7. From the law of the divisibility by nine, several other 
properties, especially interesting to the young arithmetician, 
may be derived. Among these may be mentioned the follow- 
ing: 1. If we subtract the sum of the digits from any number 
the difference will be exactly divisible by 9. 2. If we take 
two numbers in which the sums of the digits are the same, the 
difference of the two numbers will be divisible by 9. 3. Ar- 
range the terms of any number in whatever order we choose, 
and divide by 9, and the remainder in each case is the same. 
Such properties as these must have seemed exceedingly curioua 
to the early arithmeticians, and fully entitle the number nine 
to be regarded as a magical number. All of these properties, 
it is proper to remark, would have belonged to the numbw 
eleven, if our scale had been duodecimal instead of decimal. 



PART IV. 
FRACTIONS. 



18 



SECTION I. 



SECTION II. 
DECIMAL FRACTIONS. 



SECTION I. 

COMMON FRACTIONS 



I. NATURE OF FRACTIONS. 



II. CLASSES OF COMMON FRACTIONS 



III. TREATMENT OF COMMON FRACTIONS 



IV. CONTINUED FRACTIONS. 



CHAPTER I. 

NATURE OF FRACTIONS. 

fPHE Unit is the fundamental idea of arithmetic. Prom it 
JL arise two great classes of numbers Integers and Frac- 
tions. Integers have their origin in the multiplication of the 
Unit; Fractions arise from the division of the Unit. One is 
the result of an immediate synthesis; the other, of a primary 
analysis. Fractions have their origin in the analysis of the 
Unit, as integers arise from the synthesis of units. 

When the Unit is divided into equal parts, each part is seen 
to bear a certain relation to the Unit, and these parts may be 
collected together and numbered. This complex process of di 
vision, relation, and collection, gives us a fraction. The con 
ception of a fraction, therefore, involves three things: 1st, a 
division of the unit; 2d, a comparison of the part with the 
unit ; 3d, a collection of the equal parts considered. When a 
unit is divided into a number of equal parts, the comparison 
of the part with the unit gives the fractional idea, and the col- 
lection of the parts gives the fraction itself. Herein is clearly 
seen the distinction between an integer and a fraction. The 
former is an immediate synthesis; the latter involves a process 
of division, an idea of relation, and a synthesis of the parts. 
A fraction is, therefore, a triune product a result of analysis, 
comparison, and synthesis. 

Fractions, as has been stated, have their origin in a division 
of the Unit ; they may also be derived from the comparison of 
numbers. Thus the comparison of one with two, or of two with 
four, may give the idea of one-half ; and in a similar manner 

(418) 



414 THE PHILOSOPHY OF ARITHMETIC. 

other fractions may be obtained. This, however, is a possible 
rather than the actual origin ; fractions really originated in the 
division of the Unit. 

When the Unit is divided into equal parts, these parts are 
collected and numbered as individual things ; they may, there- 
fore, be regarded as a special kind of units. To distinguish 
them from the Unit already considered, we call them fraction al 
units. This gives us two classes of units, integral units and 
fractional units. The integral unit is known as the Unit ; 
when fractional units are meant we use the distinguishing 
term fractional. The definite conception of an integer requires 
a clear idea of the Unit ; the definite conception of a fraction 
requires a clear idea both of the integral and the fractional unit. 
The character of the thing divided, and the nature of the divis- 
ion, must be kept clearly before the mind, in order to obtain a 
distinct conception of the fraction. From this brief statement 
of the nature of the fraction we are prepared to define it. 

Definition. A fraction is a number of the equal parts of a 
Unit. This definition is an immediate inference from the con- 
ception of a fraction above presented. We divide the Unit 
into equal parts, and then take a number of these equal parts, 
and this is the fraction. A definition quite similar to this is, 
a fraction is one or more of the equal parts of a unit. This 
is not incorrect, though it is preferred to use the word "num- 
ber " for " one or more." It is believed that the idea is thus 
expressed in the most concise and elegant form, and that it will 
meet the approval of mathematicians. 

Several other definitions of a fraction have been presented 
by different authors, some of which are correct, while others are 
liable to serious objections. One writer says, "A fraction is a 
part of a unit." This is only part of the truth, for a fraction 
may be not only one part but several parts of a unit. Another 
writer says, "A fraction is an expression for one or more of 
the equal parts of a unit." In this definition the expression, 
the written or printed symbols, is made the fraction, which is 






NATURE OF FRACTIONS. 415 

evidently incorrect, as we have fractions previous to and inde- 
pendent of the expression of them. The expressions are not 
subjects of mathematical calculation, and hence they cannot be 
fractions. The same distinction holds between a fraction and 
its expression, as between a number and its expression. Thus 
we have the number four and the figure 4 ; so we have the 
fraction three-fourths, and the expression , as two distinct 
things. 

Another definition of a fraction is that it is an " unexecuted 
division." Says one writer, "A fraction is nothing more nor 
less than an unexecuted division." Says another, "A fraction 
may be regarded as an expression of an unexecuted division." 
This conception of a fraction is incorrect, as the idea of a frac- 
tion, and the idea of the division of one number by another, 
are entirely distinct. The fraction i (4 fifths), means four of 
the equal parts which are obtained by dividing a unit into five 
equal parts. The division of 4 by 5 will give the expression 
|, but the idea of 4 divided by 5 is entirely distinct from the 
fractional idea; and hence the assertion, that a fraction is nothing 
more nor less than an unexecuted division, is absurd. 

A fraction has also been defined to be the relation of a part 
of anything to the whole. This was the idea of Sir Isaac 
Newton, and is correct, though it is rather too abstract for a 
popular definition. Another form of stating the same idea is 
that " a fraction is that definite part which a portion is of the 
whole." Thus, if we divide an apple into two equal por- 
tions, one of these is one-half of the whole, and this definite 
part, one-half, is the fraction. This form of statement is not 
incorrect, though, like Newton's, it is too abstract for a popular 
definition. 

Notation. A fraction being a number of equal parts of a 
unit, it is natural that, in the notation of a fraction, we should 
indicate the number of parts used, by a figure. It would also, 
at first thought, seem natural to represent the name of the frac- 
tional unit by the words, half, third, etc., as 2 thirds, 3 fourths, 



416 THE PHILOSOPHY OF ARITHMETIC. 

etc.; or by their abbreviations, as 2-3ds, 3-4ths, etc. The let 
ters would be finally omitted altogether, and the expressions 
become 2-3, 3-4, etc. This probably was the primary form, as 
is indicated by the expressions, 2-3 for 2 thirds, 3-4 for 3 
fourths, which we meet in some of the older books. 

It has been found more convenient, however, not to express 
directly the name of the part, but rather to represent the num- 
ber of parts into which the unit is divided, from which the 
name of the part is inferred. This might have been done by 
writing one figure after another, 2-3, the 3 denoting the number 
of equal parts of the unit, and the 2 the number of parts con- 
sidered. In practice it has been agreed, however, to write the 
figure denoting the number of parts into which the unit is 
divided, under the other, separating them by a line, as in divi- 
sion. The number expressed by the figure below the line is 
called the denominator of the fraction, the number expressed 
by the figure above the line is called the numerator of the 
fraction. The primary object of the figure below the line is not 
to name the fractional unit, but to denote the number of equal 
parts of a unit ; from this the name of the fractional unit is in- 
ferred. Primarily, then, in our present notation, the denomi- 
nator of the fraction is not the denomination of the fraction, 
though from the denominator the denomination is inferred. 
The denominator thus serves the double object of showing 
directly the number of equal parts into which the unit is divided, 
and, indirectly, the name or denomination of the fraction. This 
distinction should be carefully noted. 

In integers we have one word to indicate the thing itself, 
and another to indicate the expression of it. Thus, number 
means the how many, or thing itself; and figure, the expres- 
sion of it; the thing and its symbol being distinguished by in- 
dependent names. In fractions there are no such terms to distin- 
guish the expression of a fraction from the fraction itself. We 
are therefore obliged to use the same word fraction to designate 
both. This we are authorized to do by a figure of rhetoric 



NATURE OF FRACTIONS. 417 

called Metonymy, in which the name of an object is sometimes 
given to the symbol, or expression of the object. It is conse- 
quently allowable to use the word fraction when we mean the 
expression of a fraction, though this frequently occasions con- 
fusion and calls for particular care on the part of the teacher 
to prevent it. We are sometimes obliged to make the 
same dual use of the terms numerator and denominator, but 
should always do so with extreme caution to avoid confusioa. 

The expression of a fraction in its relation to the fraction 
itself, is seen, when analyzed, to be a more complicated thing 
than at first appears. To illustrate; first, we have the fraction 
itself, as so many parts of a unit ; then we have the two figures 
to represent the fraction ; and then we have the numbers, which 
these two figures denote ; all of which should be carefully dis- 
tinguished, if we would have a clear idea of the relation of a 
fraction to its notation. If we begin with the unit and com- 
pare it with the fraction as expressed, the matter becomes still 
more complicated. Thus, first we have the Unit; then the 
equal parts into which the Unit is divided; then the relation 
of these parts to the Unit; then the expression for a number 
of these parts, consisting of two figures ; and then the numbers 
which these figures denote. It is therefore not entirely sur- 
prising that writers should have been careless and confused in 
their use of the terms relating to fractions. 

History. Before proceeding to the classification and treat- 
ment of Fractions, attention is called to a few points concerning 
their origin and history. The treatment of fractions by Ahmes 
is shown in the chapter en the origin of our system of arith- 
metic. In the Lilawati, fractions are denoted by writing tin- 
numerator above the denominator, without any line between 
them. The introduction of the line of separation is due to the 
Arabs; and it is found in their earliest manuscripts on arith- 
metic. To denote a fraction of a fraction, as $ of |, the two 
fractions are written consecutively, without any symbol between 
them. To represent u number increased by u fraction, the 
27 



418 THE PHILOSOPHY OF ARITHMETIC. 

fraction is written beneath the number ; and when the fraction 
is to be subtracted from the number a dot is prefixed to it; thus, 

2 3 

2 is denoted by A and 3 by -JL 

4 4 

In other cases, their notation is not intelligible without ver- 
bal explanation, and the same is true of the Arabs and earlier 
European writers, who were singularly deficient in artifices of 
notation. In the solution of a problem 

in the Lilawati, in which " the fourth i i STATEMENT. ^ ^ 
of a sixteenth of the fifth of three 12345164 
quarters of two-thirds of a moiety" is 

required, the work is written as indicated in the margin ; 
which gives r ^ f , or y^rr- In solving the problem, " Tell me, 
dear woman, quickly, how much a fifth, 

a quarter, a third, a half and a sixth STATEMENT. 

11111 29 
make when added together," the work 54326 20 

appears in the Lilawati as indicated in 

the margin. In solving the problem, " Tell me what is the 
residue of three, subtracting these frac- 
tions ;" they expressed the work as in- STATEMENT. 
dicated, which it is apparent could not 1^1326 20 
be understood without an explanation. 

The Lilawati contains four rules for the reduction and as- 
similation of fractions, as well as the application of their eight 
fundamental rules of arithmetic to them. These rules are clear 
and simple, and differ very little from those used in modern 
practice. That the author regarded fractions as somewhat 
difficult, is apparent from the following problem : " Tell me the 
result of dividing five by two and a third, and a sixth by a 
third, if thy understanding, sharpened into confidence, be com- 
petent to the division of fractions." 

The notation of compound fractions varied with different 
authors; thus with Lucas di Borgo of , or f xf, v a 

was represented as in the margin, where v a denotes I ? 

via, or times. Stifel denoted three-fourths of two- 






NATURE OF FRACTIONS. 419 

thirds of one-seventh by writing the fractions nearly 
under one another as in the margin ; and the same 
operation was indicated by Gemma Frisius thus : ^ 

i im 

a notation simple and convenient. 

In the writings of Lucas di Borgo, when two fractions are 
to be added together or subtracted one from another, the 
operations to be performed are indicated as follows : 
8 9 

fXlAH^-iV 

12 

where those quantities are to be multiplied together which are 
connected by the lines. There seems to be very little difference 
between the operations in fractions in ancient and modern text- 
books. In the works of Di Borgo and Tartaglia, the number of 
cases and their subdivisions are unnecessarily multiplied, and 
the reader is frequently more perplexed than instructed by the 
minuteness of their explanations. It may be remarked that the 
early writers seem to have been extremely embarrassed by the 
usage and meaning of the term multiplication in the case of 
fractions, where the product is less than the multiplicand; and 
some of their methods of explaining the seeming inconsistency 
are curious and ingenious. 



CHAPTER II. 

CLASSES OF COMMON FRACTIONS. 

TT^R ACTIONS are divided into two general classes Com 
J- mon and Decimal. A Common Fraction is a number of 
equal parts of a unit, without any restriction as to the size of 
those parts. A Decimal Fraction is a number of the decimal 
divisions of a unit ; that is, a number of tenths, hundredths, 
etc. 

This distinction of fractions originated in a difference in the 
notation, rather than in any essential difference in the fractions 
themselves. It was seen that the decimal scale of notation, 
when extended to the right of the units place, was capable of 
expressing tenths, hundredths, etc., and that there would be a 
great advantage in such an expression of them ; and thus the 
decimal fraction came to be regarded and treated as a distinct 
class. A brief discussion of each will be given. 

Common Fractions are variously classified, according to dif- 
ferent considerations. The primary division is that based upon 
their relative value compared with the Unit. Classifying them 
in reference to this relation, we have Proper Fractions and 
Improper Fractions. A Proper Fraction is one whose value 
is less than a unit ; that is, one which is properly a fraction ac- 
cording to the primary conception of a fraction. An Improper 
Fraction is one which is equal to or greater than a unit ; that 
is, one which is not properly a fraction in the primary meaning 
of the term. 

Another division of common fractions arises from the idea 
of dividing a fraction into equal parts. A fraction originated 

(420) 



CLASSES OP COMMON FRACTIONS. 421 

in the division of the Unit into equal parts; now, if we ex- 
tend this idea to obtaining a number of equal parts of a fraction, 
we get what is called a Compound Fraction. The Compound 
Fraction, it is thus seen, originated in the extension of the 
primary idea of division, which gave rise to the simple fraction 
This idea of a compound fraction leads to the division of frac- 
tions into two classes Simple Fractions and Compound Frac- 
tions. A Compound Fraction is technically denned as a fraction 
of a fraction. 

If we extend the fractional idea a little further, and suppose 
the numerator, or denominator, or both, to become fractional, we 
have what arithmeticians call a Complex Fraction. The 
Complex Fraction may be defined as a fraction whose numera- 
tor, or denominator, or both, are fractional. Whether the com- 
plex fraction agrees with the definition of a fraction, or with 
the functions ascribed to the numerator or the denominator of 
a fraction, is a point which will be considered a little further 
on ; but its origin was a natural outgrowth of the principle of 
pushing a notation to its limits. It should be noticed that the 
complex fraction may also have originated in the expression 
of the division of one fraction by another by writing the divisor 
under the dividend with a line between them ; but the proba- 
bilities are that it originated as first indicated, by an extension 
of the fractional idea. 

Fractions, therefore, are divided with regard to their value, as 
compared with the Unit, into Proper and Improper Fractions; 
with regard to their form, into Simple, Compound, and Com- 
plex. There is also another form of expressing fractional rela- 
tions, so closely connected with the common fraction that it 
may be embraced under the same general head. I refer to the 
Continued Fraction, which will be treated with the general 
subject of common fractions. 

Improper Fractions. According to the primary idea, a 
fraction is regarded as a part of a unit, and hence as less than 
a unit. But since we can speak of any number of fractional 



4:22 THE PHILOSOPHY OF ARITHMETIC. 

units as we do of integral units, there arises a fractional 
expression whose value is greater than a unit. Thus we may 
speak of 5 fourths, T fourths, etc., although in a unit there are 
only 4 fourths. These we call improper fractions; that is 
they are improperly fractions from the primary idea of a 
fraction. The improper fraction presents several points of 
difficulty and interest, which will be briefly considered. 

Take the expression $|; is this strictly a fraction? That it 
is properly a fraction, appears from the definition of a fraction 
and from the discussion just given. How, then, shall it be 
read? If we read it "f of a dollar," some one will object, 
that there are only four fourths in a dollar, and hence you 
cannot speak of five fourths of a dollar. If it be read " dollars," 
we will object, since there are not enough to make dollars, the 
plural meaning two or more. But, says some one, the gram- 
mars tell us that "the plural means more than one," and since 
$| is more than one, we may use the plural form and say "| 
dollars." This, we reply, is a mere quibble, as the grammar- 
ians contemplate only integers when they say "more than 
one," and really mean "two or more." The reading "% dol- 
lars" is, therefore, not strictly correct. 

How, then, should it be read ? I think the correct reading 
is "^ of a dollar." We mean by it five of such parts as are 
obtained by dividing a dollar into four equal parts. It is true 
there are not five fourths in one dollar, and the reading does 
not assume that there are. No one will object to saying of 
100 cents equals 125 cents, which is equivalent to saying |- of 
a dollar equals a dollar and a quarter. The fractional units, 
are fourths of a dollar, and the number of fractional units is 
five; hence the fraction is "five-fourths of a dollar." It is an 
improper fraction improperly a fraction from the primary 
idea of a fraction and in the name "improper fraction" we 
apparently enter a tittle protest against the absolute correctness 
of the reading in view of the primary idea of the fraction. If 
we have $| or $-^, we can then say |- dollars or ^ dollars, 



CLASSES OF COMMON FRACTIONS. 423 

since we then have ''two or more." This discussion seems to 
have been called for from the fact that the question is often 
raised and debated as to what is the correct reading of the 
improper fraction. 

Complex Fraction. According to the strictest definition of 
a fraction, the complex fraction is an impossibility. This is 
rendered evident from a consideration of the functions ascribed 
to the denominator by the definition. The denominator shows 

the number of equal parts into which the unit is divided ; 


hence, in the complex fraction ~, the denominator, , denotes 

that the unit is divided into f equal parts. This is an impos- 
sibility, as may be seen at least in two ways. First, we can 
divide a unit into three or two equal parts, but not into one 
part, since there will be no -division; and if we cannot divide 
it into one equal part, it is evident that we cannot divide it 
into less than one equal part. Secondly, if any one doubts 
the conclusion from this reasoning, let him take an apple and 
endeavor to divide it into f equal parts. The effort I have 
sometimes known to be in a high degree amusing, and always 
conclusive of the correctness of the position assumed above. 

A somewhat plausible argument in favor of the correctness 
of the complex fraction is the following: In the algebraic frac- 
tion r , the numerator and denominator are general expressions, 
o 

and hence may represent fractions as well as integers. If then 
6=| we shall have a complex fraction. This method of reason- 
ing is too general for arithmetic ; even in algebra it would prove 

fl *7* f* 

that clearing the equation, -=-, of fractions, does not clear it 

CL 

of fractions, since in adx=bc, each term may be a fraction. 
The expression =- means a divided by 6, and is a fraction only 

so far as it coincides with our arithmetical idea of a fraction 
We conclude, therefore, that strictly speaking, the complex 
fraction is an impossibility. It is merely a convenient expres- 
sion that one fraction is to be divided by another. 



424 THE PHILOSOPHY OP ARITHMETIC. 

Should the idea and expression of a complex fraction, there- 
fore, be discarded from arithmetic? This does not follow, and 
is not recommended. It is a convenient form of expressing the 
division of one fraction by another, and may thus be retained. 
Those who use it, however, should understand that it is not 
strictly a fraction, according to the primary idea of a fraction, 
but a representation of the division of a fraction by a fraction, 
or of a whole number and a fraction when only one term is 
fractional. 

Is a Fraction a Number ? It has been stated by some writers, 
and seems frequently to be the idea of pupils, that a fraction is 
not a number. This, however, is a mistake, as will appear 
from a slight consideration of the matter. Newton's definition 
of number provides for the fractional number when the object 
measured is a definite part of the measure ; it consequently ap- 
pears that the fraction is a number, if we accept his definition 
as correct. The definition, "A Fraction is a number of equal 
parts of unity," also makes it clear that a fraction is a num- 
ber. Again, if it is not a number, what kind of a quantity is 
it ; and why should it be treated in arithmetic, the science of 
numbers ? Five inches is certainly a number ; hence its equiv- 
alent, five-twelfths of a foot, is also a number. Numbers are 
of two classes, -integers and fractions ; and fractions are num- 
bers, as much so as integers. The fractional number, it will be 
noticed, involves two ideas first, the integral unit; and second, 
the fractional unit. In an integer we have the idea of a num- 
ber of units ; in the fraction we have, not only an idea of a 
number of units, but also the relation of the fractional unit to 
the integral unit. 

Is a Fraction a Denominate Number ? It has been affirmed 
by some authors that "fractions are a species of denominate 
numbers." This, however, is true only in a very limited or 
partial sense. Three quarts is not precisely the same as three- 
fourths of a gallon, though they are equal in value. In the 
latter case, there is a direct and necessary relation of a part to 



CLASSES OF COMMON FRACTIONS. 425 

a unit ; in the former case, no such relation is implied. To un- 
derstand the fraction, three-fourths of a gallon, the idea of the 
unit, gallon, must be in the mind ; in three quarts no such con- 
dition is necessary. In one case there are two units considered, 
the gallon and the fourth; in the other case but one unit, the 
quart, not considering the unit of the pure numbers, three 
and four themselves. Fourths have reference to the integral 
unit, and always imply this relation ; quarts have no reference 
to gallons, and do not imply gallons. 

Again, the fraction three-fourths may be used entirely dis- 
tinct from any denominate unit, and in this case it must be an 
abstract, not a denominate number. Two is one-fourth of eight ; 
here the measure of this relation, one-fourth, cannot but be ab- 
stract. It is evident, therefore, that a fraction is not a denom- 
inate number. There are abstract and denominate fractions, aa 
there are abstract and denominate integers. 



CHAPTER III. 

TREATMENT OF COMMON FRACTIONS. 

A FRACTION has been defined as a number of the equal 
parts of a unit. The parts into which the unit is divided 
arc called fractional units. A fraction may, therefore, also be 
defined as a number of fractional units. Fractions are divided, 
as previously stated, into Common and Decimal Fractions. 

A Common Fraction is a number of fractional units expressed 
with a numerator and a denominator ; as two-thirds, written f . 
The denominator of a fraction denotes the number of equal 
parts into which the unit is divided. The numerator of a frac- 
tion denotes the number of fractional units in the fraction. A 
common fraction is usually expressed by writing the numerator 
above the denominator with a line between them. Care should 
be taken not to define the denominator as the "figure below the 
line," and the numerator as the "figure above the line ;" and 
then speak of multiplying the numerator and denominator. 
This will lead one to suppose that figures may be multiplied, 
rather than the numbers which they represent. It is surpris- 
ing that so many writers upon arithmetic should have fallen 
into this error. 

Gases. Fractions admit of th same general treatment as 
integers; we therefore have the same fundamental cases in 
fractions as in whole numbers. These cases are all embraced 
under the general processes of Synthesis, Analysis, and Com- 
parison. The cases of synthesis and analysis are the same as 
in whole numbers. To perform the synthetic and analytic 
processes, we need to change fractions from one form to another ; 

(426) 



TREATMENT OF COMMON FRACTIONS. 



427 



hence Reduction enters largely into the treatment of fractions. 
The comparison of fractions gives rise to several cases called 
the Relation of Fractions, which do not appear in whole num- 
bers. The various cases of fractions then are ; Reduction, Ad- 
dition, Subtraction, Multiplication, Division, Relation, Com- 
position, Factoring, Common Divisor, Common Multiple, In- 
volution, and Evolution. 

A complete view of the fundamental processes is presented 
in the following logical outline. Composition, Factoring, Invo- 
lution, and Evolution, presenting no points different from those 
of whole numbers, are omitted in the treatment. The other 
cases arising out of Comparison apply equally to integers and 
fractions, and do not require a distinct treatment. 

1. Number to a Fraction. 

2. Fraction to a Number. 

3. To Higher Terms. 

4. To Lower Terms. 

5. Compound to Simple. 

6. Dissimilar to Similar. 

(1. The denominators alike. 
(2. The denominators unlike. 
(1. The denominators alike. 
(2. The denominators unlike. 
1. Fraction by a Number. 



Outline 
of the 
Cases 

of 
Fractions. 



1. REDUCTION. 



2. ADDITION. 



3. SUBTRACTION. 



4. MULTIPLICATION. 



2. Number by a Fraction. 

3. Fraction by a Fraction. 

1. Fraction by a Number. 

2. Number by a Fraction. 

3. Fraction by a Fraction. 

1. Number to a Number. 

2. Fraction to a Number. 

3. Number to a Fraction. 

4. Fraction to a Fraction. 
The " Relation of Fractions" is a new division of the subject 

of fractions: it was first published in the Normal Written 
Arithmetic, in 1863, arid has since been introduced into several 
other works on written arithmetic, and will probably be gen 
erally adopted. 



5. DIVISION. 



6. RELATION. 



4:28 THE PHILOSOPHY OF ARITHMETIC. 

Methods of Treatment. There are two methods of develop- 
ing the subject of common fractions, which may be distinguished 
as the Inductive and Deductive methods. These two methods 
are entirely distinct in principle and form; and the distinction, 
being new, seems worthy of special attention. 

By the Inductive Method, we solve each case by analysis, 
and derive the rules, or methods of operation, from these anal- 
yses, by inference or induction. The method is called induc- 
tive, because it proceeds from the analysis of particular problems 
to a general method which applies to all problems of a given 
class. The solutions, it will be noticed, are independent of any 
previously established principles of fractions, each case being 
treated by the method of arithmetical analysis which reasons 
to and from the Unit. 

To illustrate the method we will take the problem, "In f how 
many twentieths?" We analyze this as follows: One equals 
f$, and equals of 20 twentieths, or 5 twentieths; and f 
equals 3 times 5 twentieths, or 15 twentieths; hence f equals 
|-. Now, by examining this solution, we see that we multiply 
the numerator of f by the number which denotes how many 
times four, the given denominator, equals the required denom- 
inator, twenty, which is the same as multiplying both terms of | 
by the same number, five ; hence we derive the rule, " to reduce 
a fraction to higher terms, multiply both terms by the same 
number." 

For another illustration, take the converse of this problem, 
"In |f how many fourths?" The solution is as follows : One 
equals -f$, and ^ equals of f$, which is -fa; hence - of the 
number of 20ths equals the number of 4ths ; of 15 is 3, hence 
|-jj equals f. This is the analysis of the problem; we then 
proceed to derive a rule by which all such problems may be 
solved. By examining this analysis, we see that we take the 
same part of the numerator for the numerator of the required 
fraction that the denominator of the required fraction is of the 
denominator of the given fraction ; hence we derive the rule, 



TREATMENT OF COMMON FRACTIONS. 429 

"to reduce a fraction to lower terms divide both numerator and 
denominator by the same number." This rule is thus obtained 
by analyzing the analysis ; it may also be obtained by compar- 
ing the two fractions. Thus, comparing f and |-f, we see that 
3 equals 15 divided by 5, and 4 equals 20 divided by 5 that 
is, both divided by the same number and seeing that thif 
principle holds good in several cases, we infer the rule. 

By the Deductive Method we first establish a few general 
principles by demonstration, and then derive the rules, or 
methods of operation, from these principles. The method is 
called deductive because it proceeds from the general principle 
to the particular problem. To illustrate this method, let us 
solve the same problem, " Reduce f to twentieths." By a gen- 
eral proposition which we assume has been demonstrated, we 
have the principle, " Multiplying both terms of a fraction by any 
number does not change its value;" hence we may reduce | to 
twentieths by multiplying both terms by 5, which will give 
the required denominator, and we have f equal to ^-|. 

For another illustration, we will solve the converse problem, 
" Reduce | to fourths." By a general proposition, which we 
assume has been demonstrated, we have the principle, " Divid- 
ing both terms of a fraction by the same number does not change 
its value;" hence we may reduce -^ to fourths by dividing 
both numerator and denominator by any number which will 
give the required denominator. This number, we see, is 5; 
hence, dividing both numerator and denominator by 5, we 
have ^ equal to . 

We will illustrate the difference of these two methods still 
further by a problem in compound fractions. Take the ques- 
tion, "What is f of $?" The analysis is as follows: ^ of | is 
one of the three equal parts into which may be divided ; if each 
5th is divided into 3 equal parts, $ or the Unit will be divided 
into 5 times 3, or 15 equal parts, and each part will be-j^; hence 
^ of -^ is 3^, and of is 4 times -j^, or fa and of $ is 2 times 
&' or & Examining this analysis, we see that we have mul- 



430 THE PHILOSOPHY OF ARITHMETIC. 

tiplied the two denominators together and the two numerators 
together, from which we derive the rule for the reduction of 
compound fractions. By the deductive method we would 
reason as follows : By a principle previously demonstrated, ^ 
of f, which is the same as dividing 4 by 3, is T 4 -; and f of f 
by another principle, is T 8 ^. It will be noticed that the deduc- 
tive method is much shorter than the inductive method, because 
while the former explains every point involved, the latter makes 
use of principles previously demonstrated. If in the deductive 
solution, we should stop and demonstrate the principles we are to 
use, it would make the solution much longer. The difference 
of the two methods can also be clearly illustrated in the divi- 
sion and relation of fractions. In my higher arithmetic the two 
methods are presented in each case, where a full comparison 
may be made of them. 

The distinction between these two methods is broad and 
emphatic. By the Inductive Method the problem is solved 
without any reference to any previously established principle ; 
by the Deductive Method, the solution is derived from a gen- 
eral principle supposed to have been previously demonstrated. 
Both of these methods may be used in the development of frac- 
tions, and it is a question worthy of consideration which is to 
be preferred. 

The Inductive Method is believed to be simpler and more 
easily understood by young pupils. It is especially adapted to 
beginners, since it proceeds according to the simple steps of 
analysis, or the comparison of the collection with the unit. It 
also follows the law of the development of the young mind 
"from the particular to the general." It is especially suited 
to the subject of Mental Arithmetic, on account of its simplicity 
and the mental discipline it is calculated to afford. 

The Deductive Method is more difficult in thought than the 
Inductive Method. Young pupils always find a difficulty in 
founding a process of reasoning upon previously established 
principles. It is not natural for the youthful mind to reason from 



TREATMENT OF COMMON FRACTIONS. 481 

generals to particulars. Besides, the demonstrations of these 
general principles are not readily understood by young pupils. 
With much experience as a teacher, I state that it is a rare 
thing to find a pupil who can give a good logical demonstration 
of these principles, and text-books and teachers often do no 
better. The so-called demonstrations in many of our text-books 
are mere explanations or illustrations, and not logical proofs of 
the propositions. To say that " multiplying the denominator 
of a fraction increases the number of parts of the fraction, and 
diminishes their size in the same proportion," is a loose sort of 
statement that comes very far short of scientific demonstration. 
We will consider these principles and their demonstration. 

Fundamental Principles. In the Deductive Method, we 
have stated, we first establish several general principles, and 
then derive the rules or methods of operation from them. 
These principles relate to the multiplication of the numerator 
and denominator of a fraction. They may be demonstrated in 
two distinct ways. One of these is founded upon the princi- 
ples of division ; the other upon the nature of the fraction and 
the functions of the numerator and denominator. All the 
various methods in our text-books on arithmetic may be em- 
braced under these two general methods. 

The Method of Division is employed by a large majority of 
our writers on arithmetic. This method consists in regarding 
the fraction as an expression of an unexecuted division, the 
numerator representing the dividend, and the denominator the 
divisor, and the value of the fraction being the quotient. Then, 
by principles of division presumed to have beeu previously 
established, since dividing the dividend divides the quotient, 
dividing the numerator divides the fraction; and since multi 
plying the divisor divides the quotient, multiplying the denom 
inator divides the fraction, etc. 

The Fractional Method of demonstrating these fundamental 
principles is based upon the nature of the fraction itself. It 
regards the fraction as a number of equal parts of a unit, an* 



432 THE PHILOSOPHY OF ARITHMETIC. 

determines the result of these operations by comparing the 
fractional unit with the Unit. Thus, if we multiply the de- 
nominator of a fraction by any number, as three, the Unit will 
be divided into three times as many equal parts, hence each 
part will be one-third as large as before; and the same number 
of parts being taken, the value of the fraction will be one-third 
as large as before. In a similar manner all the principles may 
be demonstrated. 

The Fractional Method is undoubtedly the correct one. The 
Method of Division is liable to several objections, and should 
be discarded in teaching and in writing text-books, as appears 
from several considerations. 

First, it is illogical to leave the conception of a fraction and 
pass to that of division, to establish a principle of a fraction. 
A fraction and an expression of division are two distinct 
things, and should not be confounded. The fraction f is 
three-fourths, and does not mean 3 divided by 4. It is true 
that the expression | does also mean 3 divided by 4; but when 
we regard it as a fraction we have and should have no idea of 
the division of three by four. It is, therefore, illogical, I say, to 
convert a fraction into a division of one number by another to 
attain to a principle of the fraction. 

Secondly, it is not only illogical to treat the subject in this 
manner, but it does not give the learner the true idea of it. 
He may see that multiplying the denominator does divide the 
value of the fraction, but he will not see down into the core of 
the matter, why it does so. The method, to say the least, 
gives but a superficial view of the subject, and is therefore 
objectionable. If the fraction will admit of a simple treatment 
as a fraction, it is absurd to transform it into something else 

o 

to prove its principles. 

It may be said in favor of the method of division, that it is 
simpler and more easily understood by . Earner; but this 
both theory and experience in instruction will disprove. I 
believe that the pupil can quite as readily see thut dividing 



TREATMENT OF COMMON FRACTIONS. 433 

the numerator of a fraction divides the value of the fraction, as 
he can see that dividing the dividend divides the quotient ; and 
the same holds for the other principles. This method may 
sometimes seem a little easier to the learner, because it 
depends upon an assumed principle ; but require the pupil to 
prove that principle, and he will find it quite as difficult as to 
prove the fractional principle itself. For the method of demon- 
strating these theorems which the author prefers in arithmetic, 
the reader is referred to his arithmetical worka 



28 



CHAPTER IV. 

CONTINUED FRACTIONS. 

17^ VERY new idea, when once fixed, becomes a starting point 
-I-J from which we pass to other new ideas. The mind never 
rests satisfied with the old; it is always reaching out beyond 
the known into the unknown. " Still sighs the world for some- 
thing new," is as true in science as in society. Given a new 
conception, and the tendency is to push it forward until it leads 
us to other ideas and truths not anticipated in the original con- 
ception. Thus, from the original idea of a simple fraction 
originated the compound and complex fractions ; and thus also 
by extending the original conception, arose the Continued Frac- 
tion. 

Definition. A Continued Fraction is a fraction whose nu- 
merator is 1, and denominator an integer plus a fraction whose 
numerator is also 1 and denominator a similar fraction, and so 
on. Thus, 



Several recent authors, for convenience, write a continued 
fraction with the sign of addition between the denominators; 
1111 



Origin. Continued Fractions were first suggested to the 
world in a work by Cataldi, published in 1613, at Bologna. 
Cataldi reduces the square roots of even numbers to continued 
fractions, and then uses these fractions in approximation, though 
without the modern rule by which each approximation is educed 
from the preceding two. Daniel Schwenter, according to Fink, was 
the first to make any material contribution (1625) towards deter- 
ininingthe convergents of continued fractions. Continued fractions 
were also proposed about the year 1670, by Lord Brouncker, 
President of the Royal Society. It is known that in order to ex- 
press the ratio of the circumscribed square to the circumference 

(434) 



CONTINUED FRACTIONS. 435 

of the circle, he derived the following con- j_f.i 

tinued fraction given in the margin ; but ^^\-2f. 

by what means he was led to it, has not 
been ascertained. He was the first to investigate and make 
any use of their properties. Dr. Wallis subsequently added to 
and improved the subject, giving a general method of reducing 
all kinds of continued fraetions to common fractions. 

The complete development of these fractions, with their ap- 
plication to the solution of numerical equations and problems 
in indeterminate analysis, is due to the Continental mathemati- 
cians. Huygens is said to have explained the manner of form- 
ing the fractions by continual divisions, and to have demon- 
strated the principal properties of the converging fractions 
which result from them. John Bernoulli made a happy and 
useful application of the continued fraction to a new species of 
calculation which he devised for facilitating the construction of 
tables of proportional parts. The most complete development 
of continued fractions was given by Euler, who introduced the 
term f radio continua. 

Treatment. The subject of continued fractions is most con- 
veniently treated by the algebraic method, and may be fouud 
quite fully presented in some of the works on higher algebra. 
In this place we shall briefly consider: 1. Reducing common 
fractions to continued fractions; 2. Reducing continued frac- 
tions to common fractions; 3. Their application; 4. Their prin- 
ciples. 

We shall first show how a common fraction may be reduced 
to a continued fraction. Take the common fraction ^ 7 . Dividing 
both numerator and denominator by 68, 
we have the first expression in. the mar- J_^2i *_j_ i 
gin; dividing the numerator and denom- "" 

inator of the second fraction by 21, we 9, 

have the second expression in the margin ; 
dividing again by 5, we have the third 
expression in the margin ; which finishes the division, a* 



4:36 THE PHILOSOPHY OF ARITHMETIC. 

the numerator of the last fraction is unity. The terms -, ^. , 
etc., are called the first, second, third, etc., partial fractious. 

The same result may be obtained by dividing as in finding 
the greatest common divisor, and taking the several quotients 
for the successive denominators. Taking j^ T , and dividing as 
if to find the greatest common divisor of its terms, 
we see that the resulting quotients are the same as 68 157 



the denominators of the partial fractions. Hence we _ ." 



derive the followin rule for reducin common 







fractions to continued fractions: Find the greatest 
common divisor of the terms of the given fraction ; 
the reciprocals of the successive quotients will be the partial 
fractions which constitute the continued fraction required. 

Let us now see how a continued fraction may be reduced to 
a common fraction. This reduction may be effected in two 
ways ; by beginning at the last fraction and working up, 
or by beginning at the first fraction and working down. 

If we take the continued fraction given in the margin 
and reduce the complex fraction formed 
by the last two terms to a simple frac- 
tion, we shall have ^ 5 T . Taking this result 
and the preceding partial fraction together, 

15 21 

we have - , which reduced equals . Joining this to the 
2 + 21 47 

preceding term, we have - , which equals . Finally, 

1+47 68 

=3 , the value of the fraction. 
0+68 251 

By beginning at the first fraction, approximate values of the 
continued fraction may be obtained by respectively reducing 
two, three, or more of the partial fractions to simple fractions. 
Thus, in the fraction given above, the first approximate value 

111 111 

is ; the second - , or ; the third is - - -^-, or 

q I q /o 

-- ; the fourth - : the fifth -7^7- 
11 48 251 



CONTINUED FRACTIONS. 437 

By exhibiting this process in an analytic form, a law may be 
discovered which presents a simpler and easier method of find- 
ing approximate values than either of the others. 
Let us take the fraction in the margin and find 2 . i 
its successive approximate values, and notice the ~^ 

la^ A the derivation of one approximation from 
the previous ones. The work may be written as follows: 
2 =i, 1st approx. val. 

=f, 2d 






3x2+1 
1 

3x5+1 3x5+l 



_ _ 

3 X 5+1 (3 x 2+l)x5+2 -7x5+2 -. 

(3X5+l)X4+3 



3x(5+i)+l ' {(3 x 2+l) x 5+2}x 4+3x2+1 
16x4+3 
37x4+7 T * 5 

. We take ^, the first term of the continued fraction, for the first 
approximate value. Reducing the complex fraction formed by 
the first two terms of the continued fraction, we have ^ for the 
second approximate value. Continuing the reduction, we 
obtain ^f and -fJ^ for the remaining values. Examining the 
last two reductions, we find that the third approximate value 
is obtained by multiplying the terms of the second approximate 
fraction by the denominator of the third partial fraction, and 
adding to these products the corresponding terms of the first 
approximate fraction. We see also that the fourth approximate 
value is equal to the product of the terms of the third approxi- 
mate value by the denominator of the fourth partial fraction, 
plus the corresponding terms of the second approximation. 
Hence we derive the following rule : 

For the first approximate value take the first partial 
fraction; for the second value, reduce the complex fraction 
formed by the first two terms of the continued fraction ; for 
each succeeding approximate value, multiply both terms of 




438 THE PHILOSOPHY OF ARITHMETIC. 

the approximation last obtained by the next denominator of 
the continued fraction, and add to the products the corre- 
sponding terms of the preceding approximation. 

We will now show the application of continued fractions by 
the solution of several practical questions. 

1. Let it be required to express approximately, in the fraction 
of a day, the difference between a solar year and 365 days. 

By the old reckoning, the excess of the solar year over 365 
days was 5 hours, 48 minutes, 48 seconds. Reducing, we find this 
excess equals 20,928 seconds, and 24 hours equals 86,400 seconds. 
Therefore, the true value of the fraction =-||m=-^|f. Now, 
converting |4^ into a continued fraction, we have the expres- 
sion given in the margin, from which, 
by the last rule, we obtain the approx- iinr" 
imate values J, ^ &, & *. Hih 

The fraction agrees with the 
correction introduced into the calendar by Julius Caesar, by 
means of bissextile or leap year. The fraction -^ is the cor- 
rection used by the Persian astronomers, who add 8 days in 
every 33 years, by having f regular leap-years, and then de- 
ferring the eighth for 5 years. 

2. Required the approximate ratio of the English foot to 
the French metre containing 39.371 inches. 

The true ratio is ff^-. Reducing to a continued fraction, 
we find some of the first approximate values to be , T 3 7 , T \, 
fa, |f, yVg-- Hence the foot is to the metre as 3 to 10, nearly; 
a more correct ratio is 32 to 105. 

3. To find some of the approximate values of the ratio of 
*,he circumference of a circle to the diameter. 

Taking the value of the circumference of the circle whose 
diameter is 1, to 10 places of decimals, the ratio of the diameter 
to the circumference will be expressed by the common fraction 
Hif IHHHHHnr- Reducing to a continued fraction, some of the 
first approximate values are, \, -fa, i||, li-f . Inverting these 
tractions, we have the ratio of the circumference to the diame- 






CONTINUED FRACTIONS. 439 

ter, which is the ratio commonly used. The second gives ty 
which is the ratio said to have been found by Archimedes ; and 
the fourth gives f-f-f , which is the same as that determined by 
Metius, which is more exact than 3.141592, from which it is de- 
rived. 

Continued fractions have been employed for obtaining elegant 
approximations to the roots of surds. Thus, let it be required 
to find the square root of , or the ratio of the side of a square 
to its diagonal. 

The square root of i?, or ^/\, equals . Dividing both 



terms by the numerator we have o~ = ~i o T* Multi- 

>/ 2 l~r\/2 1 

1 
^2 i 

plying both terms of the fraction by x/2-fl, it will be 

come r~7~ = S~I To i"- Substituting, we have 

v/ 2 + 1 2 + \/ 2 1 

1 

1 1 

""TFi 



24V2 1* 

n~ 

/9 1 1 

Again, the fraction S = - becomes, as before, equalto / 2 1 

1 

and by thus continuing the process, we find to equal the 
following continued fraction : 



Some of the first approximate values of this fraction are |, f , 

7 TT lr T9 2ffT e f 

Continued fractions are also applied to the solution of inde- 
terminate problems, as may be seen in Barlow's Thec/ry of 
Numbers, or Legendre's Thforie des Nombre*. 



440 THE PHILOSOPHY OF ARITHMETIC. 

There are several beautiful principles belonging to the ap- 
proximate values of continued fractions, a few of which we 
present in this place. The values just obtained for the ratio 
of the side of a square to its diagonal are used as illustrations. 

1. The approximate fractions are alternately too small and 
too large. Thus, , ff-, $$, are too small, while \, %, f^, and 
| are too large. 

2. Any one of these fractions differs from the true value 
of the continued fraction by a quantity which is less than the 
reciprocal of the square of its denominator. Thus, j-|, which 
is the ratio much used by carpenters in cutting braces, differs 
from the true ratio by a quantity less than ( T V) 2== ^i^- 

3. Any two consecutive approximate fractions, when re 
duced to a common denominator, will differ by a unit in their 
numerators. Thus f and \%, when reduced to a common de- 
nominator, become -ffy and T 8 T \. 

4. All approximate fractions are in their lowest terms. If 
they were not, the difference of the numerators of two consec- 
utive approximate fractions, when reduced to a common denom- 
inator, would differ by more than unity. For each numerator 
is multiplied by the denominator of the other fraction, hence 
one derived numerator contains the original numerator, and 
the other the original denominator of either fraction. If then 
there were a common factor, it must be a factor of the difference 
of the numerators; and this difference would be greater than 
unity, which is contrary to the previous principle. 

The successive approximate values are called the convergents 
of the fraction. The numerator or denominator of the convergent 
is called, by Sylvester, a cumulant. A non-terminating contin- 
ued fraction whose quotients recur, is called a periodical or 
recurring continued fraction. Its value can be shown to be 
equal to one of the roots of a quadratic equation. It can also 
be shown that every quadratic surd gives rise to an equivalent 
periodic continued fraction. 



SECTION II. 

DECIMAL FRACTIONS. 



I. ORIGIN OP DECIMALS. 



II. TREATMENT OP DECIMALS. 



III. NATURE OP CIRCULATES. 



IV. TREATMENT OP CIRCULATES. 



V. PRINCIPLES OF CIRCULATES. 



VI. COMPLEMENTARY REPETENDS. 



VII. A NEW CIRCULATE FORM. 



CHAPTER I 

ORIGIN OP DECIMALS. 

rpHE invention of the Decimal Fraction, like the invention of 
JL the Arabic scale, was one of the happy strokes of genius. 
The common fraction was expressed by a notation quite dis- 
tinct from that of integers, and required not only a different 
treatment, but one much more complicated and difficult. The 
expression of the decimal divisions of the unit in the same scale 
with integers, and the possibility of reducing common fractions 
to the decimal form, wrought quite a revolution in the science 
of arithmetic, and has greatly simplified it. This new method 
of expressing fractions gave rise to a much simpler method of 
treating them ; and has elevated the decimal fraction into dis- 
tinction, and gained for it an independent consideration. 

Origin. The Decimal Fraction had its origin within the last 
three centuries. Theoretically it may have originated in either 
of two ways. There may have been a transition from the com- 
mon fraction to the decimal, by noticing that a number of tenths, 
hundredths, etc., might be expressed by the decimal scale. 
This is the manner in which the subject is usually presented in 
the text-books of the present day. Thus, after the pupil is made 
familiar with the fractions fa, y^, etc., it is stated that -fa may 
be expressed thus, .1 ; -^ thus, .01, etc. The decimal fraction 
could also have arisen directly from the decimal scale. Thus, 
since the law of the scale is, that terms diminish in value from 
left to right in a ten-fold ratio, the idea of carrying the scale on 
to the right of the unit would naturally present itself, and such 
a continuation would give rise to the decimal. As the unit 

(443) 



444 THE PHILOSOPHY OF ARITHA1ETIC. 

was one-tenth of the tens, the first place to the right of the 
unit would be one-tenth of the unit ; the second place, one-tenth 
of one-tenth, or one-hundredth, etc. These two methods of 
conceiving the origin of decimals are entirely distinct ; indeed, 
they are the converse of each other. In one case we pass from 
the common fraction to its expression in the decimal scale; in 
the other we pass from the expression in the decimal scale 
to the fraction. This distinction, it may be remarked, has a 
practical bearing upon the method of teaching the subject. In 
which way it did actually originate is not definitely known, 
though De Morgan holds that the table of compound interest 
suggested decimal fractions to Stevinus. 

History. The introduction of decimal fractions was formerly 
ascribed to Regiomontanus, but subsequent investigations have 
shown this to be incorrect. The mistake seems to have arisen 
from the confused manner in which Wallis stated that Regio- 
montanus introduced the decimal radius into trigonometry in 
place of the sexagesimal. Decimal fractions were introduced so 
gradually that it is difficult, if not impossible, to assign their 
origin to any one person. The earliest indications of the deci- 
mal idea are found in a work published in 1525 by a French 
mathematician named Orontius Fineus. In extracting the 
square root of 10, he extracts the approximate root of 10000000 
and obtains 3162. He then separates 162, which with him is 
not a fraction, but only a means of procuring fractions, and 
converts it, after the scientific custom of the times, into sexa- 
gesimal fractions (having as base 60), so that the square root 
of 10 would be expressed 3 9' 43" 12'", or a-t-^+yffor+iiiSSoo- 
He concludes that chapter of his book by stating that in this 
162, 1 is a tenth, 6 is six hundredths, etc., so that it would seem 
that he had quite a clear notion of decimals. 

Tartaglia, in 1556, gives a full account of the metnod of 
Orontius, but prefers the common fractional form 3 T WV In 
Recorde's Whetstone of Witte, 1557, the same rule is copied; 
but after obtaining three decimal places of the square root, the 



ORIGIN OF DECIMALS. 445 

remainder is written as a common fraction. Peter Raums, in 
an arithmetic published in Paris in 1584 or 1592, also quotes 
the rule of Orontius. 

In 1585, Stevinus wrote a special treatise in French, called 
" The DISME, by the which we can operate with whole numbers 
without fractions." It was first published in Dutch about the 
year 1590, and describes in very express and simple terms the 
advantages to be derived from this new arithmetic. Decimals 
are called nombres de disme: those iii the first place whose sign 
is (1) are called primes, those in the second place whose sign is 
(2) are called seconds, and so on ; whilst all integers are char- 
acterized by the sign (0), which is put over the last digit. The 
following are some of his arithmetical operations by means of 
decimals, representing multiplication and division. 

(0) (1) (2) (0) (1) (2) (3) (4) (5) (1) (2) 

3257 344352 (9 6 

8946 




1 8 6 
5114 
7637 

29137122 3 4 4 3 5 2(3 5 8 7 

96666 
999 

It will be seen that he employs the " scratch method " of 
division. The following is an example of indefinite division 
found in his work : 

(0) (1) (2) (3) 

f=l 333 

In this treatise Stevinus proposed to supersede fractions by 
c^.s/m-.s, or decimals. He enumerates the advantages which 
would result from the decimal subdivision of the units of length, 
urea, capacity, value, and lastly of a degree of the quadrant, 
in the uniformity of notation, and the increased facility of per- 
forming all arithmetical operations in which fractions of such 
units were involved. It is remarkable, however, that though 
while he confines himself to the matter of his computation he 



446 TUB PHILOSOPHY OF ARITHMETIC. 

admits his dismes, when he passes to their form he converts 
them into integers. Still, he must be regarded as the real in- 
ventor and introducer of the system of decimals. De Morgan 
says "The Disme is the first announcement of the use of deci- 
mal fractions ;" and Dr. Peacock also remarks that " the first 
notice of decimals, properly so called, is to be found in La 






This work of Steviuus was translated into English in 1608, 
by Richard Norton, under the title, "Diame, the arte of tenths, 
or decimal Arithnietike, teaching how to perform all computa- 
tions whatsoever by whole numbers without fractions, by the 
four principles of common Arithmetike : namely, addition, sub- 
traction, multiplication, and division, invented by the excellent 
mathematician, Simon Stevin." In this work the notation is 
changed to 

(1) (2) (3) (4) 

3, 7, 5, 9. 

The introduction of decimals into works on arithmetic was 
slow, even after their use had been shown by Stevinus. One 
of the earliest English works in which decimal fractions are 
really used, is that of Richard Witt, 1613, containing tables 
of half-yearly and compound interest. These tables are con- 
structed for ten million pounds ; seven figures are cut off, and 
the reduction to shillings and pence, with a temporary decimal 
separation, is introduced when wanted. Thus, when the quar- 
terly table of amounts of interest at ten per cent, is used for 
three years, the principal being 100Z., in the table stands 1372- 
66420, which multiplied by 100 and seven places cut off, gives 
tne first line of the following citation : 
" The Worke 



(1 1372 
Facit < sh 13 



66429 

2858 

4296." 



(d 3 

Giving 1372Z. 13s. 3d. for the answer. The tables are expressly 
stated to consist of numerators, with 100... for a denominator 
Napier's work, published in 1617, contains a treatise on deci- 



ORIGIN OF DECIMALS. 447 

mals, though he does not use the decimal point, except in one 
or two instances, but rather indicates the place of the decimal 
figures by primes, seconds, etc., according to the method of 
Stevinus. The author expressly attributes the origin of dec- 
imals to Stevinus. 

In 1619 we find the contents of Norton's treatise embodied 
in an English work entitled, "The Art of Tens, or Decimall 
Arithmetike, wherein the art of Arithmetike is taught in a 
more exact and perfect method, avoyding the intricacies of 
fractions. Exercised by Henry Lyte, Gentleman, and by him 
set forth for his countries good. London, 1619." It is 
dedicated to Charles, Prince of Wales, and he tells us that he 
has been requested for ten years to publish his exercises in 
decimall Arithmetike. After enlarging upon the advantages 
which attend the knowledge of this arithmetic to landlords 
and tenants, merchants and tradesmen, surveyors, gaugers, 
farmers, etc., and all men's affairs, whether by sea or land, he 
adds, "if God spare my life, I will spend some time in most 
cities of this land for my countries good to teach this art." 
This author was one of the earliest users of decimal fractions 

In the year 1619 there appeared, at Frankfort, a work on 
decimal arithmetic by Johann Hartman Beyern, in which the 
author states that he first thought upon the subject in the year 
1597, but that he was prevented from pursuing it for many 
years by the little leisure afforded him from his professional 
pursuits. He makes no mention of Stevinus, but assumes 
throughout the invention as his own. The decimal places 
ure indicated by the superscription of the Roman numerals, 
though the exponent corresponding to every digit in the 
decimal places is not always put down. Thus, 34.1426 is 
written 34.1 I 4 II 2 III 6 IV , or 34.14 II 26 IV , or 34.1426 IV . 
The author must have been acquainted with the liabdologia 
of Napier, as one chapter of his work is devoted to the 
explanation of the construction and use of these rods, which 
enjoyed a most extraordinary popularity at that period; and 



448 THE PHILOSOPHY OF ARITHMETIC. 

he could not, consequently, have been ignorant of Napier's 
notation or of the work of Stevinus ; and we may therefore 
doubt the truth of his pretensions to being the originator of 
the system of decimals. 

Albert Girard published an edition of the works of Stevinus 
in 1625, and in the solution of the equation x 3 3x 1 by a 
table of sines, of which method he was the author, 
we find the three roots as in the margin. On 1,532} 

o itr [ 

another occasion, he denotes the separation of the *; ' f 

integers and decimals by a vertical line. He 
does not always adhere to this simple notation, as we after- 
wards find the square root of 4^ expressed by 20816(4) ; and 
on another occasion we find similar vestiges of the original 
notation of Stevinus. 

Oughtred is said to have contributed much to the propaga- 
tion and general introduction of decimal arithmetic. In the 
first chapter of his Clavis, published in 1631, we find an 
explanation of decimal notation. The integers he separates 
from the decimal, or parts, by a mark, L, which be calls the 
separatrix, as in the examples, 0^56, 48^, for .56 and 48.5; 
and in giving examples of the common operations of arithmetic 
he unites them under common rules. His view of the theory 
of decimals was generally adopted, and in some cases hi* 
notation also, by English writers on arithmetic for more than 
thirty years after this period. 

In " Webster's tables for simple interest," etc., 1634, decimals 
seem to be treated as a thing generally known, though no 
decimal point is used. During the same year, 1634, Peter 
Herigone, of Paris, published a work in which he introduces 
the decimal fraction of Stevinus, having a chapter " des nombres 
de ia dixme." The mark of the decimal is made by marking 
the place where the last figure comes. Thus when 137 livres 
16 sous is to be taken 23 years 7 months, the product of 1378' 
and 23583'" is found to be 32497374"", or 3249 liv. 14 sous, 
8 deniers. In 1633, John Johnson (Survaighor) published a 



ORIGIN OF DECIMALS. 449 

work, the second part of which is called "Decimall Arithmatick 
wherby all fractionall operations are wrought, in whole num- 
bers," etc. In his decimal fractions Johnson has the rudest 
form of notation ; for he generally writes the places of decimals 

1.2.3.4.5. 

over the figures; thus, 146.03817 would be 146103817. In 
1640, the "Arithmetica Practica" of Adrian Metius contains 
sexagesimal fractions, but not decimal ones ; and a work by 
Job. Henr. Alsted, in 1641, containing a slight treatise on 
arithmetic and algebra, says nothing about decimal fractions. 

About this time the subject of decimals must have been 
pretty generally understood; for in "Moore's Arithmetick," 
1650, the subject of decimals is quite thoroughly presented 
and the contracted methods of multiplication and division are 
given. Noah Bridges, in his "Arithmetick Natural and Deci- 
mal, "has an appendix on decimals, though the author expresses 
his disapproval of the use which some would make of decimals, 
averring that the rule of practice is more convenient in many 
eases. John Wallis, 1657, uses the old decimal notation 12 345, 
but he afterwards adopts the usual point in his algebra; and 
subsequently decimals seem to have been no longer regarded 
as a novelty, but took their place along with the other accepted 
subjects and methods of arithmetic. 

It may be supposed that the publication of the tables of log- 
arithms was necessarily connected with the knowledge and use 
of decimal arithmetic ; but this, Dr. Peacock thinks, is not so. 
Tho theory of absolute indices, in its general form at least, was 
at that time unknown ; and logarithms were not considered 
as the indices of the base, but as a measure of ratios merely. 
Under this view of their theory, it was a matter of indifference 
whether we assumed the measure of the rntio of 10 to 1 to be 
one, ten, a hundred, ten millions, or ten billions, the number 
assumed by Briggs in his system of logarithms. Thus, whether 
tin- logarithms are expressed by decimals or integers, they will 
have the same characteristics, and their use in calculation is 
29 



450 THE PHILOSOPHY OF ARITHMETIC. 

exactly the same. It is under the integral forms that the loga- 
rithms are given in the earlier tables, such as those of Napier, 
Briggs, Kepler, etc. 

This statement will sufficiently explain the reason why no 
notice is taken of decimals in the elaborate explanations which 
are given of the theory and construction of logarithms by Na- 
pier, Briggs, and Kepler ; and indeed we find no mention of them 
in any English author between 1619 and 1631. In that year 
the Logarithmical Arithmetike was published by Gellibrand, 
a friend of Briggs who died the year before, with a much more 
detailed and popular explanation of the doctrine of logarithms 
than was to be found in Briggs's Arithmetica Logarithmica. It 
is there stated that the logarithms of 19695, 1969 -fa, 19 T Vinr are 
respectively 4,29435 etc., 3,29435 etc., 1,29435 etc., differing 
merely in their characteristic; and ^, r < WiJ) are called decimal 
fractions. Rules are also given for the reduction of vulgar 
fractions to decimals, by a simple proportion; and, lastly, a 
table for the reduction of shillings, pence, and farthings to deci- 
mals of a pound sterling. 

The Decimal Point. The final and greatest improvement 
in the system of decimal arithmetic, by which the notations of 
decimals and integers are assimilated, was the introduction of 
the decimal point, and much labor has been spent to ascertain 
its author. According to Dr. Peacock, the decimal point was 
introduced by Napier, the illustrious inventor of logarithms. 
In writing decimals Napier seems to have generally employed 
the method of Stevinus, which was to indicate the decimal 
places by primes, seconds, etc. ; but there are at least two in- 
stances in which he used a character as a decimal separatrix. 
The first is an example of division in which he writes 1993,273, 
using a comma, and then presents his answer in the form 1993 
2/ >j// 3/// The other instance occurs in a problem in multi- 
plication, in which he draws a line down through the places of 
the partial products that would be occupied by the decimal 
point; but in the sum he uses the exponents of Stevinus, 



ORIGIN OF DECIMALS. 451 

which thus combines both methods, and stands 1994 | 9' 1" 
6'" a"". 

The problems in which these occur are found in the Rabdol- 
ogia, published in 161*7, in which 

he mentions the invention of Stevi- , f 

uus in terms of highest praise, and \^ 

explains his notation without notic- 402 

ing his own simplification of it. 429 

The use of the comma, above re- 861094,000(1993,273 

ferred to, is presented in the ac- 3888 

companying solution, in which it 3888 

is required to divide 861094 by 1296 

432. I present but a part of the etc. 

process of division. 

...... The quotient is 1993,273, 

The use of the vertical line Qr jgg 3 %, >,,} ^//J 

is found in an example of ab- 
breviated multiplication which occurs in the solution of the 
following problem: "If 31416 be the approximate value of the 
circumference of a circle whose diameter is 10,000, what is the 
numerical value of the circumference of a circle whose diame- 
ter is 635?" This solution is said to be the first example 
found of this abbreviated multiplication ; the use of it, how- 
ever, became very popular in a short time afterward, being ee- 
pecially useful in the multiplication of the large numbers which 
were made use of in the construction of the tables of sines, etc. 
This seems like a very near approach to the decimal point, 
if it is not indeed the introduction of it ; but De Morgan main- 
tains that Napier only used his comma or line as a rest in the 
process, and not as "a final and permanent indication, as well 
as a way of pointing out where the integers end and the frac- 
tions begin." It must be admitted that the use of the separatrix 
was merely incidental, and not the practice of Napier ; but he 
seems to be the first to use a mark for this purpose, even in- 
cidentally, and there can be no doubt that even this incidental 
use had very great influence in leading to the general adoption 
of a decimal point. 



452 THE PHILOSOPHY OF ARITHMETIC. 

De Morgan thinks that Richard Witt, who published a work 
four years before Napier, " made a nearer approach to the dec- 
imal point" than Napier; yet he says, "I can hardly admit 
him to have arrived at the notation of the decimal point " Witt, 
in a work published in 1613, presents some tables of compound 
interest, in which decimal fractions are used. The tables are 
constructed for ten millions of pounds, seven figures are cut 
off, and the reduction to shillings and pence with a temporary 
decimal separatrix, in the form of a vertical line, is introduced 
when wanted, as may be seen on page 446. 

But though his tables are distinctly stated to contain only 
numerators, the denominator of which is always unity followed 
by ciphers, and though he had arrived at a complete and 
permanent command of the decimal separator, and though he 
always multiplies or divides by a power of 10 by changing the 
place of the decimal separator, which is a vertical line, yet 
De Morgan thinks he gave no "meaning to the quantity 
with its separator inserted." He thinks that if Witt had been 
"asked what his 123 | 456 was, he would have answered: It 

gives 123^. not Jt 12 3 T 4 7nrV" 

Briggs, the author of the common system of logarithms, was 
a disciple of Napier, and might have been expected to adopt 
Napier's method of writing decimals. We find, however, 
that in 1624, instead of using a decimal point he draws a line 
under the decimal terms, omitting the denominator; thus, 
5 9321. A work by Albert Girard, published in 1629 at 
Amsterdam, is remarkable as using the decimal point on a 
single occasion. Oughtred, in his Clavis, published in 1631, 
uses both the vertical and sub-horizontal separatrix, thus 
shutting up the numerator in a semi-rectangular outline, as 
23 456 for 23.456. William Webster's work, published in 



1 634, treats of decimals as a thing generally known ; but does 
not make use of the decimal point, using the partition line to 
separate integers and decimals. In 1657 John Wallis pub 
lished a work in which the old notation, 12 345, was used ; 



ORIGIN OF DECIMALS. 
but he subsequently adopted the decimal point in his algebra. 

12345 

la 1643, the notation used in Johnson's arithmetic is 3 2291 9, 
and 312500, and 34,625, and sometimes 358149411 fifths. Kav- 
anagh says that the present notation was, for the first time, 
clearly set forth in some editions of Wingate's arithmetic, 1650. 
On the Continent the notation used was 12 345 or 12[345, even 
in works of the highest repute, up to the beginning of the 18th 
century. 

The following summary presents some of the different 
methods of writing decimals which are found among the early 
writers on arithmetic, both in England and on the Continent: 
34. 1'. 4". 2'". 6"" 34 1426 

(1) (2) (3) (4) 

34. 1 . 4 . 2 . 6 34 1426 



34. 1 . 4 . 2 . 6 34'1426 

34.1426"" 34,1426 

34.1426W 

It is believed that Gunter, who was born in 1581, did more 
for the introduction of the decimal point than any one of his 
cotemporaries. He first adopted the notation of Briggs, but 
gradually dropped it and substituted the decimal point. In 
one of his works, De Morgan tells us, Briggs's notation appears 
without explanation, and 116 04 is given as the third proportional 
to 100 and 108. On a subsequent page a dot is added to 
Briggs's notation in one instance; thus 100J. in 20 years at 8 
per cent, becomes 466.095Z. At the bottom of the same page, 
Briggs's notation disappears thus: "It appeareth before, that 
100/. due at the yeares end is worth but 92 592 in ready money . 
If it be due at the end of two yeares, the present worth is 
85Z.733; then adding these two together, wee have 178/.32G for 
the present worth of 100 pound annuity for 2 yeares, and so 
forward." After this change, thus made without warning in 
the middle of a sentence, Briggs's notation does not again occur 
in the part of the work which relates to numbers. In a pre 
vious work on the sector, etc., the simple point is always used; 



454 THE PHILOSOPHY OF ARITHMETIC. 

but in explanation the fraction is not thus written, but described 
as parts./ Thus, 32.81 feet used in the operation is, in the de- 
scription of the answer, 32 feet 81 parts. 

Fink says that decimal fractions were known by Rudolff, who 
in the division of integers by powers of 10 cut off the required 
number of places with a comma. He also attributes the intro- 
duction of the decimal point to Kepler, while Cantor says it is 
found in the trigonometric tables of Pitiscus, published in 1612. 

It was some time after this, however, before the decimal point 
was fully recognized in all its uses, even in England. As long 
as Oughtred was widely used, which was until the end of the 
seventeenth century, there must have been a large school of those 
who were trained to the notation 23 I 456. The complete and 
final victory of the decimal point must be referred to the first 
quarter of the eighteenth century. 

It may seem surprising that the decimal fraction should have 
been introduced so late in the history of the science ; this delay, 
however, admits of explanation. The decimal division of the 
unit would be of no value until after the Arabic system of notation 
was adopted. Even then the introduction was necessarily slow. 
Simple as they now appear, the development of decimal fractions 
was too great an effort for one mind, or even one age. The idea 
of their use dawned gradually upon the mind, and one mathe- 
matician taking up what another had timidly begun, added an 
idea or two, until the subject was at length fully conceived and 
developed. 

The advantages of the decimal notation of fractions are so ob- 
vious that they hardly need to be specified. Many of the opera- 
tions upon fractions are thereby greatly simplified, and others are 
entirely avoided. The fundamental operations of addition, sub- 
traction, multiplication and division, are the same as in integers, 
and the cases of reduction to lower terms, common denominator, 
etc., do not occur at all. The advantages would have have been 
still greater if the basis of the numeral scale had been twelve in- 
stead of ten, as appears from a previous discussion. 



CHAPTER II. 

THE TREATMENT OF DECIMALS. 

A DECIMAL FRACTION is a number of the decimal 
divisions of a unit; or it is a number of tenths, hundredths, 
etc. Some authors define it as a fraction whose denominator 
is ten or some power of ten ; and others as a fraction whoso 
denominator is one followed by one or more ciphers. Both 
of these definitions are correct, but seem less satisfactory than 
the one first presented. They are objectionable on account of 
not expressing the kind of fractional unit, but rather indicating 
its nature by describing the denominator of the fraction. 

A Decimal Fraction may be expressed in two ways in the 
form of a common fraction, or by means of the decimal scale. 
When expressed by the scale it is distinguished from the 
general meaning of the term decimal fraction by calling it a 
Decimal. A Decimal may thus be defined as a decimal 
fraction expressed by the decimal method of notation. Thus 
&> iVff' e ^ c -' are decimal fractions, but not decimals; while 
.5, .45, etc., are both decimal fractions and decimals. This 
distinction is convenient in practice, and is believed to be 
strictly logical. It has not been generally adopted, but then; 
seems to be a growing tendency towards such a distinction. 
In popular language, however, we use the term "decimal 
fraction" as equivalent to a decimal. 

Notation. The decimal fraction, as expressed by the decimal 
scale, has no denominator written, the denominator being 
indicated by a point before the numerator. This notation, as 
already seen, arises from that of integers, and is merely un 



456 THE PHILOSOPHY OF ARITHMETIC. 

extension of it. Beginning at units' place, by a beautiful 
generalization, numbers are regarded as increasing toward the 
left and decreasing toward the right, in a ten-fold ratio, the 
result of which is a decimal division of the unit, corresponding 
to each decimal multiple of it. 

In order to distinguish between the integral and fractional 
expression and locate each term properly, a point or separatrix 
is used. Various marks have been employed for this purpose, 
at different times, but the period is now generally adopted. 
The origin of this use of the decimal separatrix is discussed in 
the previous chapter. Sir Isaac Newton held that the point 
should be placed near the top of the figures, thus, 3'56, to 
prevent it from being confounded with the period used as a 
mark of punctuation. 

Cases. The cases in decimals, it is evident, must be nearly 
the same as in whole numbers. The relation of common 
fractions to decimals would, it is natural to suppose, give rise 
to one or more new processes. A new method of notation 
having been agreed upon for a special class of common 
fractions, the inquiry naturally arises, Can other common 
fractions be expressed as decimals, and how? We thus begin 
to pass from common fractions to decimals; and, reversing 
tbis process, pass back from decimals to common fractions. 
This gives rise to a process known as the Reduction of Fractions, 
embracing the two cases of reducing common fractions to deci- 
mals, and its converse, decimals to common fractions. The 
reduction of common fractions to decimals gives rise to a par- 
ticular kind of decimals called circulates, which require an 
independent treatment. The other cases of decimals are the 
same as in whole numbers. 

Method of Treatment. The method of treating decimals is 
quite similar to that of whole numbers. Indeed, they so closely 
resemble integers that many authors have been of the opinion 
that they should be presented with them. It is claimed that 
there is but one principle in the expression of integers and 



TREATMENT OF DECIMALS. 457 

decimals, and that the processes and reasoning are the same, 
whether the scale is ascending or descending. It is therefore 
concluded that the notation of decimals should be presented 
with that of integers, and that the fundamental processes of 
addition, subtraction, etc., should be applied to them both in 
the same connection. 

There are, however, valid objections to this seemingly plausi- 
'ole inference It will be admitted that the mechanical opera- 
tions are the same ; but the reasoning processes, in at least two 
of the fundamental operations, are not identical. The fixing 
of the decimal point in multiplication and division, would be 
entirely too difficult to be presented along with the fundamental 
operations of integers. Besides, it would be illogical to separate 
one class of fractions from the general subject of fractions ; and 
moreover, one process, namely the reduction of decimals, could 
not be considered until after common fractions had been dis- 
cussed. These considerations have been sufficient to prevent 
authors of arithmetic from uniting the treatment of decimals 
with that of integers, and will, I doubt not, continue to sepa- 
rate them. 

Numeration. In the treatment of decimals, the first thing 
to be considered is the method of reading and writing them, or 
their Numeration and Notation. These processes present sev- 
eral points worthy of notice, points which seem to have escaped 
the attention of the writers on arithmetic. Having introduced 
the subject of decimals by explaining that the first place to the 
rig-lit of units is tenths, the second place hundredths, etc., it im- 
mediately follows that .45 is read "4 tenths and 5 hundredths," 
but it does not immediately follow, as many arithmeticians are 
in the habit of assuming, that it is read "45 hundredths." If, 
however, it is first explained that -^ is written .4, and y*^, .45. 
then it does not immediately follow that .45 is read "4 tenths 
and 5 huudredths." The usual method of presenting decimals 
is to explain that the first place to the right of the decimal point 
is tenths, tho second place hundredths, etc.; it should thon b 
20 



458 THE PHILOSOPHY OF ARITHMETIC. 

shown that the decimal can be otherwise read. Thus, suppose 
we have the decimal .45: this expresses primarily 4 tenths and 
5 hundredtlis ; and since 4 tenths equals 40 hundredths, and 40 
hundredths and 5 hundredths are 45 hundredths, the expression 
45 may also be read 45 hundredths. This must be explained 
if we desire to preserve the chain of logical thought in our 
treatment 

From this it is seen that in practice there are two methods 
of reading decimals, which may be expressed as follows : 

1. Begin at the decimal point and read in succession the 
value of each term belonging to the decimal, or 

2. Bead the decimal as a whole number, and annex the name 
of the right-hand decimal place. 

It will be noticed that in reading a large decimal we should 
numerate from the decimal point to derive the denominator, 
and toward the decimal point to determine the numerator. 

Notation. The writing of decimals, when conceived or read 
to us, presents several points of interest. When the decimal is 
conceived analytically, that is, as so many tenths, hundredths, 
etc., we write it by the following rule : 

1. Fix the decimal point and write each term, in its proper 
decimal place. 

If the decimal is conceived synthetically, that is, as a number 
of ten-thousandths, or a number of millionths, etc., we write 
it by the following rule : 

2. Write the numerator as an integer, and then place the 
decimal point so that the right-hand term shall express the de- 
nomination of the decimal. 

In writing a decimal in which the numerator does not occupy 
the required number of decimal places, it is not readily seen 
where to place the decimal point, and how many ciphers to pre- 
fix. The best practical rule in this case is the following. 

3 Write the numerator as an integer, and then begin at. 
the right and numerate backward, filling vacant places with 
ciphers, until we reach the required denomination, and to the 
expression thus obtained, prefix the decimal point. 



TREATMENT OF DECIMALS. 459 

Thus, to write 475 millionths, we first write 475 ; then be- 
ginning at the 5, we numerate toward the left, saying tenths, 
hundredth*, thousandths, ten-thousandths (writing a cipher), 
hundred-thousandths (writing a cipher), millionths (writing a 
cipher), and then place the decimal point. 

Several other methods have been suggested for writing 
decimals, among which is the following, by Prof. Henkle. It 
is seen that the tens of any number of tenths, the hundreds of 
any number of hundredths, the thousands of any number of 
thousandths, etc., each fall in the order of units when the 
decimal is expressed. Thus 56 tenths, is 5.6, the 5 tens falling 
in units' 1 place ; 2345 hundredths is 23.45, the 3 hundreds falling 
in units' place, etc. Hence the rule, 

1. Begin at the left and write the term corresponding to the 
denominator of the decimal in the place of units. 

Reduction. The methods of treating the two cases of reduc- 
tion are very simple. In reducing a common fraction to a 
decimal fraction, we reduce the different terms of the numerator 
to tenths, hundredths, etc., and divide by the denominator. In 
reducing a decimal to a common fraction, we express the deci- 
mal in the form of a common fraction, and then reduce it to its 
lowest terms. 

Fundamental Operations. Addition and subtraction are 
treated exactly as in integers, the same rules applying to 
both. The mechanical processes of multiplication and division 
are the same as in whole numbers ; the only difference being 
the placing of the point in the product and quotient. There 
are two .methods of explaining the location of the decimal point 
in multiplication and division, based upon the different concep- 
tions of the origin of the decimal. One locates the point by 
the principles of common fractions; the other derives the 
method from the pure decimal conception. The latter is the 
simpler and more practical method. These two methods are 
explained in my works on written arithmetic, and need not IK> 
presented here. 



CHAPTER III. 

NATURE OF CIRCULATES. 

adoption of the method of expressing fractions by the 
decimal scale opened up a new avenue of thought in the 
science of numbers. Decimals were treated without writing the 
denominator, and common fractions were frequently thrown 
into the decimal form and operated upon by means of the rules 
for whole numbers. The process of changing common fractions 
into the decimal scale led to the discovery of an interesting 
class of decimals called Circulating Decimals. These new 
forms soon attracted the attention and called forth the ingenuity 
of mathematicians; and, when investigated, were found to 
possess some remarkable and interesting properties. 

Origin. Circulating Decimals have their origin in the 
reduction of common fractions to decimals. In making this 
reduction, we annex ciphers to the numerator, and divide by 
the denominator. This division sometimes terminates with an 
exact quotient, and sometimes would continue on without 
ending. When it does terminate, the common fraction can be 
exactly expressed in a decimal ; when it does not terminate, if 
the division be carried sufficiently far, a figure or set of figures 
will begin to repeat in the same order. Such a decimal is 
called a circulating decimal, or simply a Circulate. 

It is thus seen that Circulates have their origin, not in the 
nature of number itself, but in the method of notation adopted 
to express numbers. They are an outgrowth of the Arabic 
system of notation and the decimal scale upon which it is 
based. If the scale of this system were duodecimal instead of 

(460) 



NATURE OF CIRCULATES. 461 

decimal, the subject of Circulates would be greatly modified. 
Thus %, , ^, etc., which now give circulates, would then give 
finite decimals; while i, |, J^, etc., would give circulating 
decimals. 

Notation. The part of the circulate which repeats is called 
.1 Repetend A Repetend is indicated by placing one or two 
periods or dots over it. A repetend of one figure is expressed 
oy placing a point over the figure which repeats ; thus .3 
expresses .333, etc. A repetend of more than one figure is 
expressed by placing a period over the first and the last figure; 
thus, 6.345 expresses 6.345345, etc. Sometimes the first part 
of a decimal does not repeat, while the latter part does repeat. 
Such a decimal is called a mixed circulate. The part which 
repeats is called the repeating part ; the part which does not 
repeat is called the non-repeating or finite part of the circulate. 
Thus 4.536 is a mixed circulate in which 5 is the finite, or 
non-repeating part, and 36 the repeating part. 

In an expression consisting of a whole number and a 
circulate, if the whole number contains terms similar to those 
of the repetend, the repetend may be indicated by placing one 
of the dots over a term in the whole number. Thus, suppose 
we have the circulate 54.234234, etc. ; this is usually expressed 
thus, 54.234; but, since the term just before the decimal point 
is the same as the last term of the repetend, it may also be 
expressed thus, 54.23. This indicates that 423 repeats; and. 
expanding the expression, we have 54.23423 etc., which, 
expressed in the ordinary way, becomes 54.234. In the same 
way, G.04 denotes 6.046 ; 20.12 denotes 20.1220. 

The reading of a repetend is a matter which often puzzles 
voung teachers. Thus, in the case of the repetend .3, since 
the denominator is 9, we cannot say "the decimal 3 tenths;" 
neither will it answer to say "the decimal 3 ninths;" how, 
then, shall it be read? The true reading is "the circulate 
3 tenths." Calling it a circulate distinguishes it from tho 
decimal fraction 3 tenths, and also indicates that it is equal to 
3 ninths. 



4:62 THE PHILOSOPHY OF ARITHMETIC. 

Again, how shall we read 436 ? It is not sufficiently explicit 
to say "the mixed circulate 436 thousandths," or "the mixed 
circulate 4 tenths and 36 thousandths," since neither of these 
expresses the idea exactly. The correct reading is, "the mixed 
circulate 436 thousandths, whose non-repeating part is 3 tenths 
and repeating part 36 thousandths." There may be other read- 
ings equally correct ; the one suggested is given to lead teachers 
to avoid the adoption of those which are erroneous. 

Definitions. A Circulate is a decimal in which one or more 
figures repeat in the same order. A Repetend is the term or 
series of terms which repeat. This distinction between a cir- 
culate and a repetend should be carefully noted, as it is not 
always clearly understood. Circulates are Pure and Mixed; 
Repetends are Perfect, and Imperfect, Similar and Dissimi- 
lar, and Complementary. A Perfect Repetend is one which 
contains as many decimal places, less one, as there are units in 
the denominator of the equivalent common fraction. Thus, -^= 
.142857, and ^=.0588235294117647 are each perfect repe- 
tends. 

Similar Repetends are those which begin and end respec- 
tively at the same decimal places; as .427 and .536. Dissimi- 
lar Repetends are those which begin or end at different decimal 
places. Especial attention is called to this definition of simi- 
lar repetends, as it is a departure from the view usually taken 
Repetends which begin at the same place have usually been re- 
garded as similar; and those which end at the same place, 
conterminous. It is thought, however, to be much more pre- 
cise to regard repetends beginning and ending respectively at 
the same places as similar. Repetends are surely not quite 
similar if they end at different places ; to be similar they should 
both begin and end at the same place. This view makes it 
necessary to employ some other term to indicate a similarity 
of beginning. There being no word thus used, the term 
cooriginous, expressing a coorigin, is suggested. Its appro- 
priateness may be seen by comparing it with conterminous, de- 



NATURE OF CIRCULATES. 46<J 

noting a Determination, which has already been adopted to 
denote a similarity of endings. 

Cases. Since circulates have their origin in the reduction 
of common fractions to decimals, it follows that the first case in 
the treatment of circulates is Reduction. The Reduction of 
Circulates embraces three distinct cases: 1. The reduction of 
common fractions to circulates; 2. The reduction of circulate^ 
to common fractions; 3. The reduction of dissimilar repeteurt.- 
to similar repetends. We have also Addition, Subtraction, 
Multiplication, and Division of Circulates. I have also recently 
introduced in my Higher Arithmetic the Greatest Common 
Divisor and Least Common Multiple of Circulates, subjects 
not heretofore treated in any arithmetical work. The comparison 
of circulates with common fractions gives rise to a number of 
interesting truths, which will be presented under the head of 
Principles of Circulates. 

Method of Treatment. The method of reducing common 
fractions to circulates is the same as that of reducing them to 
ordinary decimals. An abbreviation, based upon a principle of 
repetends, is sometimes employed. The method of reducing 
circulates to common fractions differs considerably from the 
method of reducing decimals to common fractions. In the 
finite decimal, the denominator understood is 1 with as many 
ciphers annexed as there are places in the decimal ; in the 
circulate the denominator of the repetend is as many 9's as 
there are places in the repetend. There are three methods of 
explaining this reduction, as will be shown in the treatment. 

Circulates can be added, subtracted, multiplied, and divided, 
by first reducing them to common fractions ; or they may be 
expanded sufficiently far so that the repeating figures may 
appear in the result. Both of these methods are objectionable 
on account of their length, and are therefore not usually 
employed. In the addition and subtraction of circulates, it i 
better to reduce them to similar repetends and then perform 
the operation. In the multiplication and division of circulates, 
a slight modification of this method is employed. 



CHAPTER IT. 

TREATMENT OF CIRCULATES. 

rpHE Treatment of Circulates embraces the operations of 
JL Reduction, Addition, Subtraction, Multiplication, Division, 
Greatest Common Divisor, Least Common Multiple, etc., and 
the Principles of Circulates. Attention will be called to the 
treatment of several of these subjects, and a distinct chapter 
will be devoted to the Principles of Circulates. 

Reduction of Circulates. The Reduction of Circulates is 
conveniently treated under four cases : 

1. To reduce common fractions to circulates. 

2. To reduce a pure circulate to a common fraction. 

3. To reduce a mixed circulate to a common fraction. 

4. To reduce dissimilar repetends to similar ones. 

1. To reduce common fractions to circulates. The gen- 
eral method of reducing common fractions to circulates is to 
annex ciphers to the numerator of the common fraction, and 
divide by the denominator, continuing the division until the 
figures of the circulate begin to repeat. Thus, to reduce -^ to 
a circulate, we annex ciphers to the numerator 5, divide by the 
denominator 12, indicate the repeating figure by placing a period 
over it ; and we have the circulate .416. 

When the circulate consists of many figures, the process of 
reduction may be abbreviated by employing some of the prin- 
ciples of repetends. Thus, suppose it be required to reduce -^ 
to a repetend. By actual division to five places, we find 

^=0.03448^. 

Now - is 8 times $, hence multiplying this by 8 we have 
^=0.27586^-. Substituting this value of -^ in the expression 
for the value of -^y, and we have 

^=0.0344827586^. 
(464) 



TREATMENT OF CIRCULATES. 465 

This, multiplied by 6, gives ^=0.20689655 17^; which, sub- 
stituted in the second expression for , gives 

3^=0.03448275862068965517^. 

Multiplying by 7, we get ^=0.24137931034482758620f ; 
which, substituted in the third expression for -fa, gives 
3V=0.0344827586206896551724137931034482758620f&. 

As the terms have begun to repeat, it is unnecessary to 
continue the process any further. It will be seen, on examina- 
tion, that the repetend consists of 28 figures, or one less than 
the denominator of -fa, and therefore is a perfect repetend. 

2. To reduce a pure circulate to a common fraction. 
There are three distinct methods of explaining this case, as has 
already been stated. In order to illustrate these methods, we 
will solve the problem, Reduce .45 to a common fraction. 

In the first method, having proved by actual division that 
1=1, .01=^, .001=^-^, etc., we derive the denominator of 
any circulate from its relation to these given circulates. To 
illustrate, reduce the circulate .45 to a common fraction. The 
method is as follows: Since .61=^, as shown by OPERATION. 
actual division, .45, which is 45 times .61, equals ,oi=A 
45 times fa, or ff, which, reduced to its lowest .45 |j? & 
terms, equals -fa. 

By the second method, we multiply the circulate by 1 with. 
as many ciphers annexed as there are places in the repetend, 
which makes a whole number of the repeating part of the 
circulate. We then subtract the two circulates, and have a 
certain number of times the given circulate equal to the differ- 
ence, from which the given circulate is readily found. We will 
illustrate by the solution of the same problem. 

Let C represent the common fraction 
which equals the circulate ; we will then ^ ] 

have C=4545 etc. ; multiplying by 100 iooC=45i4545 etc'. 
to make a whole number of the repeating qon 45 
part, we have 100 times the common C=M=-A 

fraction equal to 45.4545 etc. ; subtract- 
ing once the common fraction from 100 times the common 
PO 



4:66 THE PHILOSOPHY OF ARITHMETIC. 

fraction, we have 99 times the common fraction equal to 
45.4545 etc., minus .4545 etc., which equals 45; hence once 
the common fraction equals H, or T 5 T . 

By the third method, the repetend is regarded as an infinite 
series, the ratio being a fraction whose numerator is 1, and 
denominator 1 with as many ciphers annexed as there aro 
places in the repetend. The solution 
is as follows: The repetend .45 may .. OPERATION. 
be regarded as an infinite series, y 4 ^ 
+nnnnr^ etc - The f rmu la f r the 

Bum of an infinite series is S=- 

1 r 

Substituting the value of a~-fis> aa( l r ~Ttt> we ^ ave S=-nnj 
-f- y^g-, which equals |-f , or y^-. 

3. To reduce a mixed circulate to a common fraction. 
There are three distinct methods of reducing mixed circulates 
to common fractions, as in the preceding case. To illustrate 
these methods we will solve the problem, 
Reduce .3i8 to a common fraction. By _ OPERATION. 
the first method, we reason thus: The -318=yV of 3.18 
mixed circulate.3 18 equals y 1 ^ of 3.1 8, which = -__U =._LL 

by the preceding case equals y 1 ^ of 3^-f, or _ 35 _ 7 

j- 1 ^ of 3 T 2 r , which equals T *y%, or -fa. lllf -'*' 

By the second method, we reason as follows: Let C repre- 
sent the common fraction, then we 

u 11 u n 01010 !* i OPERATION. 

shall have C .31818 etc.; multiply- Q_ 31818 etc 

ing by 10 to make a whole number 



etc. 

of the non-repeating part, we have 10000=318.1818 etc. 
10 times the fraction equals 3.1818 
etc.; multiplying this by 100 to make 
a whole number of the repeating part, 
wt have 1000 times the fraction equals 318.1818 etc.; subtract- 
ing 10 times the fraction from 1000 times the fraction, we have 
990 times the fraction equals 315, from which we find the 
fraction equals |^|, or -fa. 



TREATMENT OF CIRCULATES. 467 

In the previous method we see that we subtract 



OPERATION. 



the finite part from the entire circulate, and divide 
by as many 9's as there are figures in the repe- 
tend, with as many ciphers annexed as there are 
decimal places before the repetend; hence, by 



r j 816 7 

generalizing this into a rule, we may perform the 

operation as in the margin. This is the method preferred in 

practice. 

This case may also be solved by regarding the repetend as 
an infinite scries, and finding its OPERATION. 

sum by geometrical progression, ,3i 
and then adding it to the 
finite part. The solution is 
presented in the margin, in 
which it is seen that we regard y^j- as the first term of the 
series, and T -^ as the rate. 

4. To reduce dissimilar repetends to similar ones. To solve 
this case it is necessary to understand the following principles: 

1. Any terminate decimal may be considered interminate, 
its repetend being ciphers; thus, .45 .450, or .45000, etc. 

2. A simple repetend may be made compound by repeating 
the repeating figure; thus, .3=. 33=. 3333, etc. 

3. A compound repetend may be enlarged by moving the 
right-hand dot towards the right over an exact number of 
periods ; thus, .245=.24545, etc. 

4. Both dots of a repetend may be moved the same number 
of places to the right; thus, .5378=.53783 or .537837, etc., 
for each expression developed will give the same result. 

5. Dissimilar rcpetends may be made cobriginous by moving 
both dots of the repetend to the right until they all begin at 
the same place. 

0. Dissimilar repetends may be made conterminous by mov- 
ing the right-hand clots of each repetend over an exact number 
of periods of each repetend until they end at the same place. 

The method of treating this case may be illustrated by the 



468 



THE PHILOSOPHY OF ARITHMETIC. 



OPERATION. 

.45 =.45454545454545 
.4362 =.43628623623623 

.813694=.81369436943694 



following example: Make .45, .4362, and .813694 similar. The 
solution is as follows: To make 
these repetends similar, they must 
be made to begin and end at the 
same place. To do this, we first 
move the left-hand dots so that they 
begin at the same place, and then move the right-hand dots 
over an exact number of periods, so that they will end at the 
same place. Now the number of places in the periods are re- 
spectively 2, 3, and 4 ; hence the number of places in the new 
periods must be a common multiple of 2, 3, and 4, which is 12 ; 
we therefore move the right-hand dot so that each repetend 
shall contain 12 places. 

Divisor and Multiple. The Greatest Common Divisor of 
two or more decimals is the greatest decimal that will exactly 
divide them. Such a divisor can be found by reducing tho 
decimals to common fractions, and applying the method for 
common fractions ; but it can also be found by keeping them in 
the decimal form ; and the latter method is generally less 
tedious and more direct. To illustrate the method, let us find 
the greatest common divisor of .375 and .423. We make tho 
two circulates similar, and sub- 
tract the finite part, which re- 
duces them to fractions having 
a common denominator. We 
then find the greatest common 
divisor of their numerators, 
1638, which is the numerator 
of the greatest common divisor, 
the denominator being of tho 
same denomination as the simi- 
lar decimals ; hence the greatest 
common divisor is 5 aWA ff> or 
,0001638. 



OPERATION. 

.3751575 .4234234 



3757572 
3813264 


4234230 
3757572 


1 

8 
9 

2 
4 

4 
. C 


476658 
501228 


55692 
49140 


24570 
26208 


6552 
6552 

snnnp=-00 


1638 
01638, G 



D 



TREATMENT OF CIRCULATES. 



469 



The method, it is seen, consists in reducing the decimals to 
a common denominator, finding the greatest common divisor of 
their numerators, writing this over the common denominator, 
and reducing the resulting fraction to a decimal. 

The Least Common Multiple of two or more decimals is the 
least number that will exactly contain each of them. Such a 
multiple can be found by reducing the decimals to common 
fractions and applying the method for common fractions; but 
it can also be found by keeping them in their decimal form ; 
and the latter method is preferred, as being generally more 
direct and less laborious. 

To illustrate the method, let us find the least common mul- 
tiple of .327, i.Oll and .075. We reduce the circulates to frac- 
tions having a common 
denominator, as in the 
previous case. The 
least common multiple 
of these numerators is 
275699700, which is 
the numerator of the 
least common multiple, 
the denominator being 
the common denomina- 



OPERATION. 



3 
4 
25 
101 


.32727 
3 


1.01110 
10 


.07575 



32724 


101100 


07575 


10908 


33700 


2525 


2727 


8425 


2525 


2727 


337 


101 


27 


337 


1 



tor of the 
Reducin 



fractions. 



3 x 4 x 25 X 101 x 27 X 337=275699700 
iL5^||^.fiJL=2757.2727, L. C. M. 

=2757.2 

the least common mul- 

tiple, to whole numbers and decimals, we have 2757.2, the 

least common multiple. 

It will be seen that the method consists in reducing the dec- 
imals to a common denominator, finding the least common 
multiple of their numerators, writing this over the common 
denominator, and reducing the resulting fraction to a decimal. 



T 



CHAPTER V. 

PRINCIPLES OF CIRCULATES. 

investigation of the relation of circulate forms to com- 
mon fractions has led to the discovery of some very inter- 
esting and remarkable properties. These will be considered 
under the head of Principles of Circulates, and Complemen- 
tary Itepetends. The subject being rather briefly treated in 
the text-books, will be presented here somewhat in detail. A 
brief and simple explanation will be given in connection with 
each principle. 

1. A common fraction whose denominator contains no other 
prime factors than 2 or 5, can be reduced to a simple decimal. 
For, since 2 and 5 are factors of 10, if we annex as many ciphers 
to the numerator as there are 2's or 5's in the denominator, the 
numerator will then be exactly divisible by the denominator. 

2. The number of places in the simple decimal to which a 
common fraction may be reduced, is equal to the greatest num- 
ber of 2's or 5's in the denominator. For, to make the numer- 
ator contain the denominator, we must annex a cipher for every 
2 or 5 in the denominator, and the number of places in the 
quotient, which is the decimal, will equal the number of ciphers 
annexed. 

3. Every common fraction, in its lowest terms, whose denom- 
inator contains other prime factors than 2 or 5, will give an 
inter minate decimal. For, since 2 and 5 are the only factors 
of 10, if the denominator contains other prime factors, the nu- 
merator with ciphers annexed will not exactly contain the 
denominator; hence the division will not terminate, and the 
result will be an interminate decimal. 

(470) 



PRINCIPLES OF CIRCULATES. 471 

i. Every common fraction which does not give a simplf 
decimal, gives a circulate. For, in reducing a common frac- 
tion to a decimal, there cannot be more different remainders 
than there are units in the denominator; hence, if the division 
be continued, a remainder must occur which has already been 
used, and we shall thus have a series of remainders and divi- 
dends like those already used, therefore the terms of the quo- 
tient will be repeated 

5. The number of figure* in, a repetend cannot exceed the 
number of units in the denominator of the common fraction 
which produces it, less I. For, in reducing a common fraction 
to a decimal, when the number of decimal places equals the 
number of units in the denominator, less 1, all the possible 
different remainders will have been used, and hence the divi- 
dends, and therefore the quotients which constitute the circu- 
late, will begin to repeat. In many cases the remainders 
begin to repeat before we have as many as the denominator 
less 1. 

6. The number of places in a repetend, when the denominator 
of the common fraction producing it is a prime, is always equal 
to the number of units in the denominator, less 1, or to some 
factor of this number. For, the repetend must end when it 
reaches the point where it has as many places less 1 as there 
are units in the denominator of the producing fraction; hence, 
if it ends before this, the number of places must be an exact 
part of the number of places in the denominator less 1, that it 
may terminate when it has as many places* as the denominator 
less 1. This is not generally true when the denominator is 
composite, as in J T , -fa, ^, ^ etc - 

1. A common fraction whose denominator contains 2'x or 
Vs iinlh other prime factors, will give a mixed circulate, and 
the number of places in the non-repeating part will equal the 
rireatest number of 2's or 5's in the denominator. Dividing 
first by the 2's and 5's, we shall have a decimal numerator 
containing as many places as the greatest number of 2's or 5'a 



472 THE PHILOSOPHY OF ARITHMETIC. 

(Prin. 2). If we now divide by the other factors, the dividends 
consisting of the terms of the decimal numerator will not give 
the same series of remainders as when we have a series of 
dividends with ciphers annexed ; hence the circulate will begin 
directly after the last place of these decimal terms. To illustrate, 
take -^Q, and factor the denominator, and we have 



dividing by the 2 and the 5's we have -^, in which it is evident 
the circulate must begin in the third decimal place, just as the 
circulate from -f- begins in the first decimal place. 

8. When the reciprocal of a prime number gives a perfect 
repetend, the remainder which occurs at the close of the period 
is 1. For, since the reduction of the fraction to a circulate 
commenced with a dividend of 1 with one or more ciphers 
annexed, that the quotients may repeat we must begin with 
the same dividend, and therefore the remainder at the close of 
the period must be 1. 

9. When the reciprocal of any prime number is reduced to 
a repetend, the remainder which occurs when the number of 
decimal places is one less than, the prime, is 1. For, since the 
number of decimal places in the period equals the denominator 
less 1, or is a factor of the denominator less 1, at the close of a 
period consisting of as many places as the denominator less 1, 
there will be an exact number of repeating periods, and .therefore 
che remainder will be 1. 

10. A number consisting of as many 9's as there are units 
tn any prime less 1, is divisible by that prime. For, if we 
divide 1 with ciphers annexed by a prime, after a number of 
places 1 less than the prime, the remainder is 1; hence 1 wilh 
the same number of ciphers annexed, minus 1, would be exactly 
divisible by the prime; but this remainder will be a series of 
9's, therefore such a series of 9's is divisible by the prime. 
Thus 999999 is divisible by 7. 

11. A number consisting of as many 1's as there are units 



PRINCIPLES OF CIRCULATES. 473 

in any prime (except 3), less 1, is divisible by that prime 
For the prime is a divisor of a series of 9's (Prin. 10), which 
is equal to 9 times a series of 1's; and since 9 and the prime 
are relatively prime, and the prime is a divisor of 9 times a 
series of 1's, it must be a divisor also of a series of 1's. Thus 
111111 is divisible by 7 ; also 1111111111 is divisible by 11. 

12. A number consisting of any digit used as many times 
as there are units in a prime (except 3), less 1, is divisible by 
that prime. For, since such a series of 1's is divisible by the 
prime, any number of times such a series of 1's will be divisible 
by the prime. Hence 222222, 333333, 444444, etc., are each 
divisible by 7. 

13. The same perfect repetend will express the value of all 
proper fractions having the same prime denominator, by 
starting at different places. Thus, |=.14285714285 etc. But 
tf-=.lf, hence the part that follows 1 in the repetend of ^ is the 
repetend off; that is, ^=.428571. Again, ^--.14f; hence the 
part that follows .14 in the repetend of \ is the repetend of ; 
that is, -f=. 285714. In a similar manner we find ^=.857142, 
$=.571428 ; and the same thing is generally true. 

14. In reducing the reciprocal of a prime to a decimal, if 
we obtain a remainder 1 less than the prime, we have one- 
half of the repetend, and the remaining half can be found by 
subtracting the terms of the first half respectively from 9. Tn k<> 
}, and let us suppose in decimating we have reached a remain- 
der of 6; now what follows will be the repetend of , and the 
repetend of $ added to the repetend of ^ must equal 1, since 
-|-|=l; hence the sum of these two repetends must equal 
.999999 etc., since .999999 etc. equals 1. Now in adding the 
terms of these two repetends together, that the sum may bo a 
series of 9's, there must be just as many places before the point 
where 6 occurred as a remainder, as after; hence G occurred as 
a remainder when we were half through the scries. 

Again, since the sum of the terms of the latter and the for- 
mer half of the repetend equals a series of 9's. each term of 



474 THE PHILOSOPHY OF ARITHMETIC. 

the Grst half of the repetend, subtracted from 9, will give the 
corresponding term of the latter half of the series. 

All perfect repetends possess this property, and a large num- 
ber of those which are not perfect. Repetends possessing this 
property are called complementary repetends. The last two 
properties are of great practical value in reducing common 
fractions to repetends. 

15. Any prime is an exact divisor of 10 raised to a power 
denoted by the number of terms in the repetend of the prime, 
less 1 ; or of 1 raised loa power denoted by any multiple of the 
number of terms, less 1. For, by Prin. 6, the number of places 
in the repetend must equal the number of units in the prime, or 
some factor of that number ; hence the dividend used in ob- 
taining a period must be 10 raised to a power equal to the 
number of terms in the period; and since the remainder at the 
end of the period is 1, the prime will exactly divide 10 raised 
to a power equal to the number of terms in the period, less 1. 

Both this and principle 6 depend on Fermat\~> Theorem, that 
pp-i i i s divisible by p when p and P are prime to each 
other." For 10, the base of the decimal system, is prime to 
any prime number except 2 and 5; hence 10 P ~ J 1 is always 
exactly divisible by p, when p is any prime except 2 and 5. 
It thus follows that in the division of 1 with ciphers annexed, 
the remainder is always 1 when the number of places in the 
quotient is equal to the number of units in the prime. From 
this we can readily derive the second part of principle 6, and 
also principle 15. 

16. Any prime is an exact divisor of a number when it will 
divide the sum of the numbers formed by taking groups of 
the number consisting of as many terms as there are figures in 
the repetend of the reciprocal of that prime. We will show 
this for a prime whose reciprocal gives a repetend of three 
places. The number 47,685,672,856, may be put in the form 
856 + 672xl0 3 + 685xl0 6 + 47xl0 9 , or 672x(10 3 1) + 685 
X (10 l)-f47x(10 9 1) + 856+ 672 + 685 + 47; but these 



PRINCIPLES OF CIRCULATES. 475 

different powers of 10, diminished by 1, are all divisible by any 
number whose reciprocal gives a number of three places, as 37; 
hence if the sum of the groups, 47 -f 685+672 + 856, is divisible 
by 37, the entire number is also divisible by 37. The same 
may be illustrated with any other number, and the principle is 
therefore general. The principle admits, also, of a general 
demonstration. 

From this general proposition we derive the following special 
principles embraced under it: 

1. Since the reciprocals of 3 and 9 give a repetend of one 
place, they will divide a number when they divide the sum of 
the digits. 

2. Since the reciprocals of 11, 33, and 99, give a repetend of 
two places, they will divide a number when they divide the 
sum of the numbers found by taking groups of two places. 

3. Since the reciprocals of 27, 37, and 111, give repetends of 
three places, they will divide a number when they divide the 
sum of the numbers formed by taking groups of three places. 

4. Since the reciprocal of 101 gives a repetend of four places, 
it will divide a number when it divides the sum of the numbers 
formed by taking groups of four places. 

5. Sinco the reciprocals of 41 and 271 give repetends of five 
places, they will divide a number when they divide the sum 
of the numbers formed by taking groups of five places. 

6. Since the reciprocals of 7, 13, 21, and 39 give repetends 
of six places, they will divide a number when they divide the 
sum of the numbers formed by taking groups of six places. 



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