Mathematical Essays and Recreations

Hermann Cäsar Hannibal Schubert

Description

Project Gutenberg’s Mathematical Essays and Recreations, by Hermann Schubert This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Mathematical Essays and Recreations Author: Hermann Schubert Translator: Thomas J. McCormack Release Date: May 9, 2008 [EBook #25387] Language: English Character set encoding: ISO 8859 1 START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL ESSAYS*** *** IN THE SAME SERIES. ON THE STUDY AND DIFFICULTIES OF MATHEMATICS. By Au- gustus De Morgan. Entirely new edition, with portrait of the au- thor, index, and annotations, bibliographies of modern works on al- gebra, the philosophy of mathematics, pan-geometry, etc. Pp., . Cloth, $. (s.). LECTURES ON ELEMENTARY MATHEMATICS. By Joseph Louis Lagrange. Translated from the French by Thomas J. McCormack. With photogravure portrait of Lagrange, notes, biography, marginal analyses, etc. Only separate edition in French or English. Pages, . Cloth, $. (s.). HISTORY OF ELEMENTARY MATHEMATICS. By Dr. Karl Fink, late Professor in Tu¨bingen. Translated from the German by Prof. Wooster Woodruff Beman and Prof. David Eugene Smith. (In prepa- ration.) THE OPEN COURT PUBLISHING CO.  dearborn st., chicago.

Informations

• Publié par: oproi
• Publié le: 08 décembre 2010
• Nombre de lectures: 20
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

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.

.)