Mathematical Essays and Recreations

ProjectGutenberg’sMathematicalEssaysandRecreations,byHermann
Schubert

ThiseBookisfortheuseofanyoneanywhereatnocostandwith
almostnorestrictionswhatsoever.Youmaycopyit,giveitawayor
re-useitunderthetermsoftheProjectGutenbergLicenseincluded
withthiseBookoronlineatwww.gutenberg.org

Title:MathematicalEssaysandRecreations

Author:HermannSchubert

Translator:ThomasJ.McCormack

ReleaseDate:May9,2008[EBook#25387]

Language:English

Charactersetencoding:ISO-8859-1

***STARTOFTHISPROJECTGUTENBERGEBOOKMATHEMATICALESSAYS***

INTHESAMESERIES.

ONTHESTUDYANDDIFFICULTIESOFMATHEMATICS.By
Au-
gustusDeMorgan
.Entirelynewedition,withportraitoftheau-
thor,index,andannotations,bibliographiesofmodernworksonal-
gebra,thephilosophyofmathematics,pan-geometry,etc.Pp.,

.
Cloth,$

.

(

s.).

LECTURESONELEMENTARYMATHEMATICS.By
JosephLouis
Lagrange
.TranslatedfromtheFrenchby
ThomasJ.McCormack
.
WithphotogravureportraitofLagrange,notes,biography,marginal
analyses,etc.OnlyseparateeditioninFrenchorEnglish.Pages,

.
Cloth,$

.

(

s.).

HISTORYOFELEMENTARYMATHEMATICS.By
Dr.KarlFink
,
l
W
at
o
e
os
P
te
r
r
ofe
W
ss
o
o
o
r
dr
i
u
n
ff
T
B
u¨
e
b
m
in
a
g
n
ena.ndTrParnofs.la
D
te
a
d
vi
f
d
ro
E
m
uge
th
n
e
eS
G
m
er
it
m
h
.an(InbyprPerpoaf-.
ration.)

THEOPENCOURTPUBLISHINGCO.

dearbornst.,chicago.

MATHEMATICALESSAYS

NAD

RECREATIONS

YB

HERMANNSCHUBERT

PROFESSOROFMATHEMATICSINTHEJOHANNEUM,HAMBURG,GERMANY

FROMTHEGERMANBY

THOMASJ.McCORMACK

Chicago,


Transcriber’snotes

ProducedbyDavidWilson

Thise-textwascreatedfromscansofthebookpublishedat
Chicagoin

bytheOpenCourtPublishingCompany,
andatLondonbyKeganPaul,Trench,Truebner&Co.

Thetranslatorhasoccasionallychosenunusualformsof
words:thesehavebeenretained.

Somecross-referenceshavebeenslightlyrewordedtotakeaccount
ofchangesintherelativepositionoftextandfloatedfigures.
DetailsaredocumentedintheL
A
TEXsource,alongwithminor
typographicalcorrections.

TRANSLATOR’SNOTE.

he
mathematicalessaysandrecreationsinthisvolumearebyoneofthemost
T
successfulteachersandtext-bookwritersofGermany.Themonisticconstruc-
tionofarithmetic,thesystematicandorganicdevelopmentofallitsconsequences
fromafewthoroughlyestablishedprinciples,isquiteforeigntothegeneralrunof
AmericanandEnglishelementarytext-books,andthefirstthreeessaysofProfessor
Schubertwill,therefore,fromalogicalandestheticside,befullofsuggestionsfor
elementarymathematicalteachersandstudents,aswellasfornon-mathematical
readers.Fortheactualdetaileddevelopmentofthesystemofarithmetichere
sketched,wemayreferthereadertoProfessorSchubert’svolume
Arithmetikund
Algebra
,recentlypublishedintheGo¨schen-Sammlung(Go¨schen,Leipsic),—anex-
traordinarilycheapseriescontainingmanyotheruniqueandvaluabletext-booksin
mathematicsandthesciences.
Theremainingessayson“MagicSquares,”“TheFourthDimension,”and“The
HistoryoftheSquaringoftheCircle,”willbefoundtobethemostcompletegener-
allyaccessibleaccountsinEnglish,andtohave,oneandall,adistincteducational
andethicallesson.
Inalltheseessays,whichareofasimpleandpopularcharacter,anddesigned
forthegeneralpublic,ProfessorSchuberthasincorporatedmuchofhisoriginal
research.

LaSalle
,Ill.,December,1898.

ThomasJ.McCormack.

CONTENTS.

NotionandDefinitionofNumber...
MonisminArithmetic......
OntheNatureofMathematicalKnowledge
TheMagicSquare.......
TheFourthDimension......
TheSquaringoftheCircle.....

ProjectGutenbergLicensing

.

.

......

.

......

.

......

.

......

.

......

.

......

.

......

.

......

.

......

.

......

.

egap





NOTIONANDDEFINITIONOFNUMBER.

any
essayshavebeenwrittenonthedefinitionofnumber.But
M
mostofthemcontaintoomanytechnicalexpressions,bothphilo-
sophicalandmathematical,tosuitthenon-mathematician.Theclear-
estideaofwhatcountingandnumbersmeanmaybegainedfromthe
observationofchildrenandofnationsinthechildhoodofcivilisation.
Whenchildrencountoradd,theyuseeithertheirfingers,orsmallsticks
ofwood,orpebbles,orsimilarthings,whichtheyadjoinsinglytothe
thingstobecountedorotherwiseordinallyassociatewiththem.As
weknowfromhistory,theRomansandGreeksemployedtheirfingers
whentheycountedoradded.Andevento-daywefrequentlymeetwith
peopletowhomtheuseofthefingersisabsolutelyindispensablefor
computation.
Stillbetterproofthattheaccurateassociationofsuch“other”
thingswiththethingstobecountedistheessentialelementofnu-
merationarethetalesoftravellersinAfrica,tellingushowAfrican
tribessometimesinformfriendlynationsofthenumberoftheenemies
whohaveinvadedtheirdomain.Theconveyanceoftheinformation
iseffectednotbymessengers,butsimplybyplacingatspotsselected
forthepurposeanumberofstonesexactlyequaltothenumberof
theinvaders.Noonewilldenythatthenumberofthetribe’sfoesis
thuscommunicated,eventhoughnonameexistsforthisnumberinthe
languagesofthetribes.Thereasonwhythefingersaresouniversally
employedasameansofnumerationis,thateveryonepossessesadef-
initenumberoffingers,sufficientlylargeforpurposesofcomputation
andthattheyarealwaysathand.
Besidesthisfirstandchiefelementofnumerationwhich,aswehave
seen,istheexact,individualconjunctionorassociationofotherthings
withthethingstobecounted,istobementionedasecondimportant




NOTIONANDDEFINITIONOFNUMBER.

element,whichinsomerespectsperhapsisnotsoabsolutelyessential;
namely,thatthethingstobecountedshallberegardedasofthesame
kind.Thus,anyonewhosubjectsapplesandnutscollectivelytoa
processofnumerationwillregardthemforthetimebeingasobjectsof
thesamekind,perhapsbysubsumingthemunderthecommonnotion
offruit.Wemaythereforelaydownprovisionallythefollowingasa
definitionofcounting:tocountagroupofthingsistoregardthethings
asthesameinkindandtoassociateordinally,accurately,andsingly
withthemotherthings.Inwriting,weassociatewiththethingstobe
countedsimplesigns,likepoints,strokes,orcircles.Theformofthe
symbolsweuseisindifferent.Neitherneedtheybeuniform.Itisalso
indifferentwhatthespatialrelationsordispositionsofthesesymbols
are.Although,ofcourse,itismuchmoreconvenientandsimplerto
fashionsymbolsgrowingoutofoperationsofcountingonprinciplesof
uniformityandtoplacethemspatiallyneareachother.Inthismanner
areproducedwhatIhavecalled
*
naturalnumber-pictures;forexample,
••••••••••••••••
••••••••••••••••••••
etc.
Now-a-dayssuchnaturalnumber-picturesarerarelyemployed,andare
tobeseenonlyondominoes,dice,andsometimes,also,onplaying-
cards.
Itcanbeshownbyarchæologicalevidencethatoriginallynumeral
writingwasmadeupwhollyofnaturalnumber-pictures.Forexam-
ple,theRomansinearlytimesrepresentedallnumbers,whichwere
writtenatall,byassemblagesofstrokes.Wehaveremnantsofthis
writinginthefirstthreenumeralsofthemodernRomansystem.Ifwe
neededadditionalevidencethattheRomansoriginallyemployednat-
uralnumber-signs,wemightcitethepassageinLivy,VII.

,wherewe
aretold,that,inaccordancewithaveryancientlaw,anailwasannually
drivenintoacertainspotinthesanctuaryofMinerva,the“inventrix”
ofcounting,forthepurposeofshowingthenumberofyearswhichhad
elapsedsincethebuildingoftheedifice.Welearnfromthesamesource
thatalsointhetempleatVolsiniinailswereshownwhichtheEtruscans
hadplacedthereasmarksforthenumberofyears.
AlsorecentresearchesinthecivilisationofancientMexicoshow
thatnaturalnumber-pictureswerethefirststageofnumeralnotation.

*
SystemderArithmetik
.(Potsdam:Aug.Stein.

.)