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The Project Gutenberg EBook of The Number Concept, by Levi Leonard Conant

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Title: The Number Concept

Its Origin and Development

Author: Levi Leonard Conant

Release Date: August 5, 2005 [EBook

#16449]

Language: English

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THE NUMBER CONCEPT ***

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THE MACMILLAN COMPANY

NEW YORK · BOSTON · CHICAGO · DALLAS

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MACMILLAN & CO., Limited

LONDON · BOMBAY · CALCUTTA

MELBOURNE

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OF CANADA, Limited

TORONTO

THE NUMBER CONCEPT

ITS ORIGIN AND DEVELOPMENT

byLEVI LEONARD CONANT, Ph.D

ASSOCIATE PROFESSOR OF MATHEMATICS IN

THE WORCESTER POLYTECHNIC INSTITUTE

New YorkMACMILLAN AND CO.

All rights reserved—no part of this bookmay be reproduced in any form withoutpermission in writing from the publisher.

Set up and electrotyped Published July,

1896

Norwood Press

J S Cushing Co.—Berwick & Smith Co

Norwood, Mass., U.S.A

In the selection of authorities whichhave been consulted in the preparation ofthis work, and to which reference is made

in the following pages, great care has beentaken Original sources have been drawnupon in the majority of cases, and nearlyall of these are the most recent attainable.Whenever it has not been possible to citeoriginal and recent works, the author hasquoted only such as are most standard andtrustworthy In the choice of orthography

of proper names and numeral words, theforms have, in almost all cases, been

written as they were found, with noattempt to reduce them to a systematicEnglish basis In many instances thiswould have been quite impossible; and,even if possible, it would have beenaltogether unimportant Hence the forms,whether German, French, Italian, Spanish,

or Danish in their transcription, are leftunchanged Diacritical marks are omitted,however, since the proper key couldhardly be furnished in a work of this kind

With the above exceptions, this studywill, it is hoped, be found to be quitecomplete; and as the subject hereinvestigated has never before been treated

in any thorough and comprehensivemanner, it is hoped that this book may befound helpful The collections of numeralsystems illustrating the use of the binary,

the quinary, and other number systems,are, taken together, believed to be themost extensive now existing in anylanguage Only the cardinal numerals havebeen considered The ordinals present nomarked peculiarities which would, in awork of this kind, render a separatediscussion necessary Accordingly theyhave, though with some reluctance, beenomitted entirely.

Sincere thanks are due to those whohave assisted the author in the preparation

acknowledgment should be made toHoratio Hale, Dr D G Brinton, FrankHamilton Cushing, and Dr A F.Chamberlain

Worcester, Mass., Nov 12, 1895.

CONTENTS.

CHAPTER I.

CHAPTER II.

CHAPTER III.

CHAPTER IV.

Origin of Number Words (continued)

74

CHAPTER V.

Miscellaneous Number Bases 100

CHAPTER VI.

CHAPTER VII.

THE NUMBER

CONCEPT: ITS ORIGIN AND DEVELOPMENT.

CHAPTER I.

COUNTING.

Among the speculative questions whicharise in connection with the study ofarithmetic from a historical standpoint, theorigin of number is one that has provokedmuch lively discussion, and has led to agreat amount of learned research amongthe primitive and savage languages of thehuman race A few simple considerationswill, however, show that such researchmust necessarily leave this questionentirely unsettled, and will indicateclearly that it is, from the very nature of

things, a question to which no definite andfinal answer can be given.

Among the barbarous tribes whoselanguages have been studied, even in amost cursory manner, none have ever beendiscovered which did not show somefamiliarity with the number concept Theknowledge thus indicated has often proved

to be most limited; not extending beyondthe numbers 1 and 2, or 1, 2, and 3.Examples of this poverty of numberknowledge are found among the foresttribes of Brazil, the native races ofAustralia and elsewhere, and they areconsidered in some detail in the nextchapter At first thought it seems quiteinconceivable that any human being should

be destitute of the power of countingbeyond 2 But such is the case; and in a

few instances languages have been found

to be absolutely destitute of pure numeralwords The Chiquitos of Bolivia had noreal numerals whatever,1 but expressed

their idea for “one” by the word etama,

meaning alone The Tacanas of the samecountry have no numerals except thoseborrowed from Spanish, or from Aymara

or Peno, languages with which they havelong been in contact.2 A few other SouthAmerican languages are almost equallydestitute of numeral words But even here,rudimentary as the number senseundoubtedly is, it is not wholly lacking;and some indirect expression, or someform of circumlocution, shows a

conception of the difference between one

a n d two, or at least, between one and

These facts must of necessity deter themathematician from seeking to push hisinvestigation too far back toward the veryorigin of number Philosophers haveendeavoured to establish certainpropositions concerning this subject, but,

as might have been expected, have failed

to reach any common ground of agreement

W he w e l l has maintained that “suchpropositions as that two and three makefive are necessary truths, containing inthem an element of certainty beyond thatwhich mere experience can give.” Mill,

on the other hand, argues that any suchstatement merely expresses a truth derivedfrom early and constant experience; and inthis view he is heartily supported byTylor.3 But why this question should

provoke controversy, it is difficult for themathematician to understand Either viewwould seem to be correct, according to thestandpoint from which the question isapproached We know of no language inwhich the suggestion of number does notappear, and we must admit that the wordswhich give expression to the number sensewould be among the early words to beformed in any language They expressideas which are, at first, wholly concrete,which are of the greatest possiblesimplicity, and which seem in many ways

to be clearly understood, even by thehigher orders of the brute creation Theorigin of number would in itself, then,appear to lie beyond the proper limits ofinquiry; and the primitive conception ofnumber to be fundamental with human

In connection with the assertion that theidea of number seems to be understood bythe higher orders of animals, the followingbrief quotation from a paper by Sir JohnLubbock may not be out of place:

“Leroy … mentions a case in which a manwas anxious to shoot a crow ‘To deceivethis suspicious bird, the plan was hit upon

of sending two men to the watch house,one of whom passed on, while the otherremained; but the crow counted and kepther distance The next day three went, andagain she perceived that only two retired

In fine, it was found necessary to send five

or six men to the watch house to put herout in her calculation The crow, thinkingthat this number of men had passed by,lost no time in returning.’ From this he

inferred that crows could count up to four.Lichtenberg mentions a nightingale whichwas said to count up to three Every day

he gave it three mealworms, one at a time.When it had finished one it returned foranother, but after the third it knew that thefeast was over.… There is an amusing andsuggestive remark in Mr Galton's

interesting Narrative of an Explorer in Tropical South Africa After describing

the Demara's weakness in calculations, hesays: ‘Once while I watched a Demarafloundering hopelessly in a calculation onone side of me, I observed, “Dinah,” myspaniel, equally embarrassed on the other;she was overlooking half a dozen of hernew-born puppies, which had beenremoved two or three times from her, andher anxiety was excessive, as she tried to

find out if they were all present, or if anywere still missing She kept puzzling andrunning her eyes over them backwards andforwards, but could not satisfy herself.She evidently had a vague notion ofcounting, but the figure was too large forher brain Taking the two as they stood,dog and Demara, the comparison reflected

no great honour on the man.…’ According

t o my bird-nesting recollections, which Ihave refreshed by more recent experience,

if a nest contains four eggs, one may safely

be taken; but if two are removed, the birdgenerally deserts Here, then, it wouldseem as if we had some reason forsupposing that there is sufficientintelligence to distinguish three from four

An interesting consideration arises withreference to the number of the victims

allotted to each cell by the solitary wasps.One species of Ammophila considers onelarge caterpillar of Noctua segetum

enough; one species of Eumenes suppliesits young with five victims; another 10,

15, and even up to 24 The numberappears to be constant in each species.How does the insect know when her task

is fulfilled? Not by the cell being filled,for if some be removed, she does notreplace them When she has brought hercomplement she considers her taskaccomplished, whether the victims arestill there or not How, then, does sheknow when she has made up the number24? Perhaps it will be said that eachspecies feels some mysterious and innatetendency to provide a certain number ofvictims This would, under no

circumstances, be any explanation; but it

is not in accordance with the facts In thegenus Eumenes the males are muchsmaller than the females.… If the egg ismale, she supplies five; if female, 10victims Does she count? Certainly thisseems very like a commencement ofarithmetic.”4

Many writers do not agree with theconclusions which Lubbock reaches;maintaining that there is, in all suchinstances, a perception of greater or lessquantity rather than any idea of number.But a careful consideration of theobjections offered fails entirely to weakenthe argument Example after example of anature similar to those just quoted might

be given, indicating on the part of animals

a perception of the difference between 1

and 2, or between 2 and 3 and 4; and anyreasoning which tends to show that it isquantity rather than number which theanimal perceives, will apply with equalforce to the Demara, the Chiquito, and theAustralian Hence the actual origin ofnumber may safely be excluded from thelimits of investigation, and, for thepresent, be left in the field of purespeculation.

A most inviting field for research is,however, furnished by the primitivemethods of counting and of giving visibleexpression to the idea of number Ourstarting-point must, of course, be the signlanguage, which always precedesintelligible speech; and which is soconvenient and so expressive a method ofcommunication that the human family,

even in its most highly developedbranches, never wholly lays it aside Itmay, indeed, be stated as a universal law,that some practical method of numerationhas, in the childhood of every nation ortribe, preceded the formation of numeralwords.

Practical methods of numeration aremany in number and diverse in kind Butthe one primitive method of countingwhich seems to have been almostuniversal throughout all time is the fingermethod It is a matter of commonexperience and observation that everychild, when he begins to count, turnsinstinctively to his fingers; and, with theseconvenient aids as counters, tallies off thelittle number he has in mind This method

is at once so natural and obvious that there

can be no doubt that it has always beenemployed by savage tribes, since the firstappearance of the human race in remoteantiquity All research among uncivilizedpeoples has tended to confirm this view,were confirmation needed of anything sopatent Occasionally some exception tothis rule is found; or some variation, such

as is presented by the forest tribes ofBrazil, who, instead of counting on thefingers themselves, count on the joints oftheir fingers.5 As the entire number system

of these tribes appears to be limited to

three, this variation is no cause for

surprise

The variety in practical methods ofnumeration observed among savage races,and among civilized peoples as well, is sogreat that any detailed account of them

would be almost impossible In one region

we find sticks or splints used; in another,pebbles or shells; in another, simplescratches, or notches cut in a stick,Robinson Crusoe fashion; in another,kernels or little heaps of grain; in another,knots on a string; and so on, in diversity ofmethod almost endless Such are thedevices which have been, and still are, to

be found in the daily habit of greatnumbers of Indian, negro, Mongolian, andMalay tribes; while, to pass at a singlestep to the other extremity of intellectualdevelopment, the German student keepshis beer score by chalk marks on the table

or on the wall But back of all thesedevices, and forming a common origin towhich all may be referred, is the universalfinger method; the method with which all

begin, and which all find too convenientever to relinquish entirely, even thoughtheir civilization be of the highest type.Any such mode of counting, whetherinvolving the use of the fingers or not, is

to be regarded simply as an extraneous aid

in the expression or comprehension of anidea which the mind cannot grasp, orcannot retain, without assistance TheGerman student scores his reckoning withchalk marks because he might otherwiseforget; while the Andaman Islander counts

on his fingers because he has no othermethod of counting,—or, in other words,

of grasping the idea of number A singleillustration may be given which typifiesall practical methods of numeration Morethan a century ago travellers inMadagascar observed a curious but

simple mode of ascertaining the number ofsoldiers in an army.6 Each soldier wasmade to go through a passage in thepresence of the principal chiefs; and as hewent through, a pebble was dropped onthe ground This continued until a heap of

10 was obtained, when one was set asideand a new heap begun Upon thecompletion of 10 heaps, a pebble was setaside to indicate 100; and so on until theentire army had been numbered Anotherillustration, taken from the very antipodes

of Madagascar, recently found its way intoprint in an incidental manner,7 and is sogood that it deserves a place beside deFlacourt's time-honoured example MomCely, a Southern negro of unknown age,finds herself in debt to the storekeeper;

and, unwilling to believe that the amount

is as great as he represents, she proceeds

to investigate the matter in her ownpeculiar way She had “kept a tally ofthese purchases by means of a string, inwhich she tied commemorative knots.”When her creditor “undertook to make thematter clear to Cely's comprehension, hehad to proceed upon a system of her owndevising A small notch was cut in asmooth white stick for every dime sheowed, and a large notch when the dimesamounted to a dollar; for every fivedollars a string was tied in the fifth bignotch, Cely keeping tally by the knots inher bit of twine; thus, when two stringswere tied about the stick, the ten dollarswere seen to be an indisputable fact.”This interesting method of computing the

amount of her debt, whether an invention

of her own or a survival of the Africanlife of her parents, served the old negrowoman's purpose perfectly; and itillustrates, as well as a score of examplescould, the methods of numeration to whichthe children of barbarism resort when anynumber is to be expressed which exceedsthe number of counters with which naturehas provided them The fingers are,however, often employed in countingnumbers far above the first decade Aftergiving the Il-Oigob numerals up to 60,Müller adds:8 “Above 60 all numbers,indicated by the proper figure pantomime,

are expressed by means of the word ipi.”

We know, moreover, that many of theAmerican Indian tribes count one ten afteranother on their fingers; so that, whatever

number they are endeavouring to indicate,

we need feel no surprise if the savagecontinues to use his fingers throughout theentire extent of his counts In rareinstances we find tribes which, like theMairassis of the interior of New Guinea,appear to use nothing but fingerpantomime.9 This tribe, though by nomeans destitute of the number sense, issaid to have no numerals whatever, but to

use the single word awari with each show

of fingers, no matter how few or howmany are displayed

In the methods of finger countingemployed by savages a considerabledegree of uniformity has been observed.Not only does he use his fingers to assisthim in his tally, but he almost alwaysbegins with the little finger of his left