WiththesecagedcaptivesIexperimentbydaylight，whereasthefreewaspsworkintheabsolutedarknessoftheirundergroundretreat。Wherelightisabsent，colorgoesfornothing。Once，therefore，thatshehasenteredthecavern，thebumblebeeflyderivesnobenefitfromheryellowbands，whicharesupposedtobehersafeguard。Whethergarbedassheisorotherwise，itiseasyforhertoeffectherpurposeinthedark，onconditionthatsheavoidsthetumultuousinteriorofthewasps’nest。Solongasshehastheprudencenottohustlethepassersby，shecandabhereggs，withoutdanger，onthepaperwall。Noonewillknowofherpresence。Thedangerousthingistocrossthethresholdoftheburrowinbroaddaylight，beforetheeyesofthosewhogoinandout。Atthatmomentalone，protectivemimicrywouldbeconvenient。
　　NowdoestheentranceoftheVolucellaintothepresenceofafewwaspsentailsuchverygreatrisks？Thewasps’nestinmyenclosure，theonewhichwasafterwardstoperishinthesununderabellglass，gavemetheopportunityforprolongedobservations，butwithoutanyresultuponthesubjectofmyimmediateconcern。
　　Thebumblebeeflydidnotappear。Theperiodforhervisitshaddoubtlesspassed;forIfoundplentyofhergrubswhenthenestwasdugup。
　　Otherfliesrewardedmeformyassiduity。Isawsome——atarespectfuldistance，Ineedhardlysay——enteringtheburrow。Theywereinsignificantinsizeandofadarkgraycolor，notunlikethatofthehousefly。Theyhadnotapatchofyellowaboutthemandcertainlyhadnoclaimtoprotectivemimicry。Nevertheless，theywentinandoutastheypleased，calmly，asthoughtheywereathome。Aslongastherewasnottoogreatanumberatthedoor，thewaspsleftthemalone。Whentherewasanythingofacrowd，thegrayvisitorswaitednearthethresholdforalessbusymoment。Noharmcametothem。
　　Insidetheestablishment，thesamepeacefulrelationsprevail。InthisrespectIhavetheevidenceofmyexcavations。Intheundergroundcharnelhouse，sorichinFlygrubs，Ifindnocorpsesofadultflies。Ifthestrangershadbeenslaughteredinpassingthroughtheentrancehall，orlowerdown，theywouldfalltothebottomoftheburrowanyhow，withtheotherrubbish。Nowinthischarnelhouse，asIsaid，thereareneveranydeadbumblebeeflies，neveraflyofanysort。Theincomersarerespected。Havingdonetheirbusiness，theygooutunscathed。
　　Thistoleranceonthepartofthewaspsissurprising。Andasuspicioncomestoone’smind:canitbethattheVolucellaandtherestarenotwhattheacceptedtheoriesofnaturalhistorycallthem，namely，enemies，grubkillerssackingthewasps’nest？Wewilllookintothisbyexaminingthemwhentheyarehatched。
　　Nothingiseasier，inSeptemberandOctober，thantocollecttheVolucella’seggsinsuchnumbersasweplease。Theyaboundontheoutersurfaceofthewasps’nest。Moreover，aswiththelarvaeofthewasp，itissometimebeforetheyaresuffocatedbythepetroleumfumes;andsomostofthemaresuretohatch。Itakemyscissors，cutthemostdenselypopulatedbitsfromthepaperwallofthenestandfillajarwiththem。ThisisthewarehousefromwhichIshalldaily，forthebestpartofthenexttwomonths，drawmysupplyofnascentgrubs。
　　TheVolucella’seggremainswhereitis，withitswhitecoloralwaysstronglymarkedagainstthebrownofthebackground。Theshellwrinklesandcollapses;andtheforeendtearsopen。Fromitthereissuesaprettylittlewhitegrub，thininfront，swellingslightlyintherearandbristlingalloverwithfleshyprotuberances。Thecreature’spapillaearesetonitssidesliketheteethofacomb;attherear，theylengthenandspreadintoafan;ontheback，theyareshorterandarrangedinfourlongitudinalrows。Thelastsectionbutonecarriestwoshort，brightredbreathingtubes，standingaslantandjoinedtoeachother。Theforepart，nearthepointedmouth，isofadarker，brownishcolor。Thisisthebitingandmotorapparatus，seenthroughtheskinandconsistingoftwofangs。Takenallround，thegrubisaprettylittlething，withitsbristlingwhiteness，whichgivesittheappearanceofatinysnowflake。Butthiselegancedoesnotlastlong:grownbigandstrong，thebumblebeefly’sgrubbecomessoiledwithsanies，turnsarussetybrownandcrawlsaboutintheguiseofahulkingporcupine。
　　Whatbecomesofitwhenitleavestheegg？Thismywarehousingjartellsme，partly。Unabletokeepitsbalanceonslopingsurfaces，itdropstothebottomofthereceptacle，whereIfindit，daily，ashatched，wanderingrestlessly。Thingsmusthappenlikewiseatthewasps’。Incapableofstandingontheslantofthepaperwall，thenewborngrubsslidetothebottomoftheundergroundcavity，whichcontains，especiallyattheendofthesummer，aheapedupprovenderofdeceasedwaspsanddeadlarvaeremovedfromthecellsandflungoutsidethehouse，allniceandgamy，aspropermaggot’sfoodshouldbe。
　　TheVolucella’soffspring，themselvesmaggots，notwithstandingtheirsnowyapparel，findinthischarnelhousevictualstotheirliking，incessantlyrenewed。Theirfallfromthehighwallsmightwellbenotaccidental，butratherameansofreaching，quicklyandwithoutsearching，thegoodthingsdownatthebottomofthecavern。Perhaps，also，someofthewhitegrubs，thankstotheholesthatmakethewrapperresembleaspongycover，managetoslipinsidetheWasps’nest。Still，mostoftheVolucella’sgrubs，atwhateverstageoftheirdevelopment，areinthebasementoftheburrow，amongthecarrionremains。Theothers，thosesettledinthewasps’homeitself，arecomparativelyfew。
　　Thesereturnsareenoughtoshowusthatthegrubsofthebumblebeeflydonotdeservethebadreputationthathasbeengiventhem。
　　Satisfiedwiththespoilsofthedead，theydonottouchtheliving;theydonotravagethewasps’nest:theydisinfectit。
　　Experimentconfirmswhatwehavelearntintheactualnests。Overandoveragain，IbringwaspgrubsandVolucellagrubstogetherinsmalltesttubes，whichareeasytoobserve。Thefirstarewellandstrong;Ihavejusttakenthemfromtheircells。Theothersareinvariousstages，fromthatofthesnowflakebornthesamedaytothatofthesturdyporcupine。Thereisnothingtragicabouttheencounter。Thegrubsofthebumblebeeflyroamaboutthetest—tubewithouttouchingthelivetidbit。Themostthattheydoistoputtheirmouthsforamomenttothemorsel;thentheytakeitawayagain，notcaringforthedish。
　　Theywantsomethingdifferent:awounded，adyinggrub;acorpsedissolvingintosanies。Indeed，ifIprickthewaspgrubwithaneedle，thescornfulonesatoncecomeandsupatthebleedingwound。IfIgivethemadeadgrub，brownwithputrefaction，thewormsripitopenandfeastonitshumors。Betterstill:Icanfeedthemquitesatisfactorilywithwaspsthathaveturnedputridundertheirhornyrings;IseethemgreedilysuckthejuicesofdecomposingRosechafergrubs;Icankeepthemthrivingwithchoppedupbutcher’smeat，whichtheyknowhowtoliquefybythemethodofthecommonmaggot。Andtheseunprejudicedones，whoacceptanythingthatcomestheirway，provideditbedead，refuseitwhenitisalive。Likethetruefliesthattheyare，frankbodysnatchers，theywait，beforetouchingamorsel，fordeathtodoitswork。
　　Insidethewasps’nest，robustgrubsaretheruleandweaklingstherareexception，becauseoftheassiduoussupervisionwhicheliminatesanythingthatisdiseasedandliketodie。Here，nevertheless，Volucellagrubsarefound，onthecombs，amongthebusywasps。Theyarenot，itistrue，sonumerousasinthecharnelhousebelow，butstillprettyfrequent。Nowwhatdotheydointhisabodewheretherearenocorpses？Dotheyattackthehealthy？Theircontinualvisitsfromcelltocellwouldatfirstmakeonethinkso;butweshallsoonbeundeceivedifweobservetheirmovementsclosely;andthisispossiblewithmyglassroofedcolonies。
　　Iseethemfussilycrawlingonthesurfaceofthecombs，curvingtheirnecksfromsidetosideandtakingstockofthecells。Thisonedoesnotsuit，northatoneeither;thebristlycreaturepasseson，stillinsearch，thrustingitspointedforepartnowhere，nowthere。Thistime，thecellappearstofulfiltherequisiteconditions。Alarva，glowingwithhealth，openswideitsmouth，believingitsnursetobeapproaching。Itfillsthehexagonalchamberwithitsbulgingsides。
　　Thegluttonousvisitorbendsandslidesitsslenderforepart，abladeofexquisitesuppleness，betweenthewallandtheinhabitant，whoseslackrotundityyieldstothepressureofthisanimatedwedge。Itplungesintothecell，leavingnopartofitselfoutsidebutitswidehindquarters，withthereddotsofthetwobreathingtubes。
　　Itremainsinthispostureforsometime，occupiedwithitsworkatthebottomofthecell。Meanwhile，thewaspspresentdonotinterfere，remainimpassive，showingthatthegrubvisitedisinnoperil。Thestranger，infact，withdrawswithasoft，glidingmotion。Thechubbybabe，asortofIndiarubberbag，resumesitsoriginalvolumewithouthavingsufferedanyharm，asitsappetiteproves。Anurseoffersitamouthful，whichitacceptswitheverysignofunimpairedvigor。AsfortheVolucellagrub，itlicksitslipsafteritsownfashion，pushingitstwofangsinandout;then，withoutfurtherlossoftime，goesandrepeatsitsprobingelsewhere。
　　Whatitwantsdownthere，atthebottomofthecells，behindthegrubs，cannotbedecidedbydirectobservation;itmustbeguessedat。Sincethevisitedlarvaremainsintact，itisnotpreythattheVolucellagrubisafter。Besides，ifmurderformedpartofitsplans，whydescendtothebottomofthecell，insteadofattackingthedefenselessreclusestraightway？Itwouldbemucheasiertosuckthepatient’sjuicesthroughtheactualorificeofthecell。
　　Insteadofthat，weseeadip，alwaysadipandneveranyothertactics。
　　Thenwhatistherebehindthewaspgrub？Letustrytoputitasdecentlyaspossible。Inspiteofitsexceedingcleanliness，thisgrubisnotexemptfromthephysiologicalillsinseparablefromthestomach。Likeallthateats，ithasintestinalwastematterwithregardtowhichitsconfinementcompelsittobehavewithextremediscretion。Likesomanyotherclose—cabinedlarvaeofWaspsandBees，itwaitsuntilthemomentofthetransformationtoriditselfofitsdigestiverefuse。Then，onceandforall，itcastsouttheuncleanaccumulationwhereofthepupa，thatdelicate，rebornorganism，mustnotretaintheleasttrace。Thisisfoundlater，inanyemptycell，intheformofadarkpurpleplug。But，withoutwaitingforthisfinalpurge，thislump，thereare，fromtimetotime，slightexcretionsoffluid，clearaswater。WehaveonlytokeepaWaspgrubinalittleglasstubetorecognizetheseoccasionaldischarges。Well，IseenothingelsetoexplaintheactionoftheVolucella’sgrubswhentheydipintothecellswithoutwoundingthelarvae。Theyarelookingforthisliquid，theyprovokeitsemission。Itrepresentstothemadaintywhichtheyenjoyoverandabovethemoresubstantialfareprovidedbythecorpses。
　　Thebumblebeefly，thatsanitaryinspectoroftheVespinecity，fulfilsadoubleoffice:shewipesthewasp’schildrenandsheridsthenestofitsdead。Forthisreason，sheispeacefullyreceived，asanauxiliary，whensheenterstheburrowtolayhereggs;forthisreason，hergrubistolerated，naymore，respected，intheveryheartofthedwelling，wherenonemightstraywithimpunity。
　　IrememberthebrutalreceptiongiventotheSaperdaandHylotomagrubswhenIplacethemonacomb。Forthwithgrabbed，bruisedandriddledwithstings，thepoorwretchesperish。ItisquiteadifferentmatterwiththeoffspringoftheVolucella。Theycomeandgoastheyplease，pokeaboutinthecells，elbowtheinhabitantsandremainunmolested。Letusgivesomeinstancesofthisclemency，whichisverystrangeintheirascibleWasp。
　　Foracoupleofhours，IfixmyattentiononaVolucellagrubestablishedinacell，sidebysidewiththeWaspgrub，themistressofthehouse。Thehindquartersemerge，displayingtheirpapillae。Sometimesalsotheforepart，thehead，shows，bendingfromsidetosidewithsudden，snake—likemotions。Thewaspshavejustfilledtheircropsatthehoneypot;theyaredispensingtherations，areverybusilyatwork;andthingsaretakingplaceinbroaddaylight，onthetablebythewindow。
　　Astheypassfromcelltocell，thenursesrepeatedlybrushagainstandstrideacrosstheVolucellagrub。Thereisnodoubtthattheyseeit。Theintruderdoesnotbudge，or，iftroddenon，curlsup，onlytoreappearthenextmoment。Someofthewaspsstop，bendtheirheadsovertheopening，seemtobemakinginquiriesandthengooff，withouttroublingfurtheraboutthestateofthings。Oneofthemdoessomethingevenmoreremarkable:shetriestogiveamouthfultothelawfuloccupantofthecell;butthelarva，whichisbeingsqueezedbyitsvisitor，hasnoappetiteandrefuses。
　　Withouttheleastsignofanxietyonbehalfofthenurslingwhichsheseesinawkwardcompany，thewaspretiresandgoestodistributeitsrationelsewhere。InvainIprolongmyexamination:
　　thereisnoflusterofanykind。TheVolucellagrubistreatedasafriend，oratleastasavisitorthatdoesnotmatter。Thereisnoattempttodislodgeit，toworryit，toputittoflight。Nordoesthegrubseemtotroublegreatlyaboutthosewhocomeandgo。
　　Itstranquillity，tellsusthatitfeelsathome。
　　Hereissomefurtherevidence:thegrubhasplunged，headdownwards，intoanemptycell，whichistoosmalltocontainthewholeofit。Itshindquartersstickout，veryvisibly。Forlonghours，itremainsmotionlessinthisposition。Ateverymoment，waspspassandrepasscloseby。Threeofthem，atonetimetogether，atanotherseparately，comeandnibbleattheedgesofthecell;theybreakoffparticleswhichtheyreducetopasteforanewpieceofwork。Thepassersby，intentupontheirbusiness，maynotperceivetheintruder;butthesethreecertainlydo。Duringtheirworkofdemolition，theytouchthegrubwiththeirlegs，theirantennae，theirpalpi;andyetnoneofthemmindsit。Thefatgrub，soeasilyrecognizedbyitsqueerfigure，isleftalone;
　　andthisinbroaddaylight，whereeverybodycanseeit。Whatmustitbewhentheprofounddarknessoftheburrowsprotectsthevisitorwithitsmysteries！
　　IhavebeenexperimentingallalongwithbigVolucellagrubs，coloredwiththedirtyredwhichcomeswithage。Whateffectwillpurewhiteproduce？Isprinkleonthesurfaceofthecombssomelarvaethathavelatelylefttheegg。Thetiny，snow—whitegrubsmakeforthenearestcells，godownintothem，comeoutagainandhuntelsewhere。Thewaspspeaceablyletthemgotheirway，asheedlessofthelittlewhiteinvadersasofthebigredones。
　　Sometimes，whenitentersanoccupiedcell，thelittlecreatureisseizedbytheowner，thewaspgrub，whichnabsitandturnsandreturnsitbetweenitsmandibles。Isthisadefensivebite？No，thewaspgrubhasmerelyblundered，takingitsvisitorforaprofferedmouthful。Thereisnogreatharmdone。Thankstoitssuppleness，thelittlegrubemergesfromthegripintactandcontinuesitsinvestigations。
　　Itmightoccurtoustoattributethistolerancetosomelackofpenetrationinthewasps’vision。Whatfollowswillundeceiveus:
　　Iplaceseparately，inemptycells，agrubofSaperdascalariaandaVolucellagrub，bothofthemwhiteandselectedsoasnottofillthecellentirely。Theirpresenceisrevealedonlybythepalenessofthehindpartwhichservesasaplugtotheopening。A
　　superficialexaminationwouldleavethenatureoftherecluseundecided。Thewaspsmakenomistake:theyextirpatetheSaperdagrub，killit，flingitonthedustheap;theyleavetheVolucellagrubinpeace。
　　Thetwostrangersarequitewellrecognizedinthesecrecyofthecells:oneistheintruderthatmustbeturnedout;theotheristheregularvisitorthatmustberespected。Sighthelps，forthingstakeplaceinthedaylight，underglass;butthewaspshaveothermeansofinformationinthedimnessoftheburrow。WhenI
　　producedarknessbycoveringtheapparatuswithascreen，themurderofthetrespassersisaccomplishedjustthesame。Forsosaythepoliceregulationsofthewasps’nest:anystrangerdiscoveredmustbeslainandthrownonthemidden。
　　Tothwartthisvigilance，therealenemiesneedtobemastersoftheartofdeceptiveimmobilityandcunningdisguise。ButthereisnodeceptionabouttheVolucellagrub。Itcomesandgoes，openly，wheresoeveritwill;itlooksroundamongstthewaspsforcellstosuitit。Whathasittomakeitselfthusrespected？Strength？
　　Certainlynot。Itisaharmlesscreature，whichthewaspcouldripopenwithablowofhershears，whileatouchofthestingwouldmeanlightningdeath。Itisafamiliarguest，towhomnodenizenofawasps’nestbearsanyillwill。Why？Becauseitrendersgoodservice:sofarfromworkingmischief，itdoesthescavengingforitshosts。Wereitanenemyormerelyanintruder，itwouldbeexterminated;asadeservingassistant，itisrespected。
　　ThenwhatneedistherefortheVolucellatodisguiseherselfasawasp？Anyfly，whethercladindrabormotley，isadmittedtotheburrowdirectlyshemakesherselfusefultothecommunity。Themimicryofthebumblebeefly，whichwassaidtobeoneofthemostconclusivecases，is，afterall，amerechildishnotion。Patientobservation，continuallyfacetofacewithfacts，willhavenoneofitandleavesittothearmchairnaturalists，whoaretoopronetolookattheanimalworldthroughtheillusivemistsoftheory，CHAPTERXIIMATHEMATICALMEMORIES:NEWTON’SBINOMIALTHEOREM
　　Thespider’swebisagloriousmathematicalproblem。Ishouldenjoyworkingitoutinallitsdetails，wereInotafraidofwearyingthereader’sattention。PerhapsIhaveevengonetoofarinthelittlethatIhavesaid，inwhichcaseIowehimsomecompensation:’Wouldyoulikeme，’Iwillaskhim，’wouldyoulikemetotellyouhowIacquiredsufficientalgebratomasterthelogarithmicsystemsandhowIbecameasurveyorofSpiders’webs？
　　Wouldyou？Itwillgiveusarestfromnaturalhistory。’
　　Iseemtocatchasignofacquiescence。Thestoryofmyvillageschool，visitedbythechicksandtheporkers，hasbeenreceivedwithsomeindulgence;whyshouldnotmyharshschoolofsolitudepossessitsinterestaswell？Letustrytodescribeit。Andwhoknows？Perhaps，indoingso，Ishallrevivethecourageofsomeotherpoorderelicthungeringafterknowledge。
　　Iwasdeniedtheprivilegeoflearningwithamaster。Ishouldbewrongtocomplain。Solitarystudyhasitsadvantages:itdoesnotcastyouintheofficialmould;itleavesyouallyouroriginality。
　　Wildfruit，whenitripens，hasadifferenttastefromhothouseproduce:itleavesonadiscriminatingpalateabittersweetflavorwhosevirtueisallthegreaterforthecontrast。Yes，ifitwereinmypower，Iwouldstartafresh，facetofacewithmyonlycounselor，thebookitself，notalwaysaverylucidone;Iwouldgladlyresumemylonelywatches，mystruggleswiththedarknesswhence，atlast，aglimmerappearsasIcontinuetoexploreit;I
　　shouldretraversetheirksomestagesofyore，stimulatedbytheonedesirethathasneverfailedme，thedesireoflearningandofafterwardsbestowingmymiteofknowledgeonothers。
　　WhenIleftthenormalschool，mystockofmathematicswasofthescantiest。Howtoextractasquareroot，howtocalculateandprovethesurfaceofasphere:theserepresentedtometheculminatingpointsofthesubject。Thoseterriblelogarithms，whenIhappenedtoopenatableofthem，mademyheadswim，withtheircolumnsoffigures;actualfright，notunmixedwithrespect，overwhelmedmeontheverythresholdofthatarithmeticalcave。OfalgebraIhadnoknowledgewhatever。Ihadheardthename;andthesyllablesrepresentedtomypoorbrainthewholewhirlinglegionoftheabstruse。
　　Besides，Ifeltnoinclinationtodecipherthealarminghieroglyphics。Theymadeoneofthoseindigestibledisheswhichweconfidentlyextolwithouttouchingthem。IgreatlypreferredafinelineofVirgil，whomIwasnowbeginningtounderstand;andI
　　shouldhavebeensurprisedindeedhadanyonetoldmethat，forlongyearstocome，Ishouldbeanenthusiasticstudentoftheformidablescience。Goodfortuneprocuredmemyfirstlessoninalgebra，alessongivenandnotreceived，ofcourse。
　　Ayoungmanofaboutmyownagecametomeandaskedmetoteachhimalgebra。Hewaspreparingforhisexaminationasacivilengineer;andhecametomebecause，ingenuousyouththathewas，hetookmeforawelloflearning。Theguilelessapplicantwasveryfaroutinhisreckoning。
　　Hisrequestgavemeashockofsurprise，whichwasforthwithrepressedonreflection:’Igivealgebralessons？’saidItomyself。’Itwouldbemadness:Idon’tknowanythingaboutthesubject！’
　　AndIleftitatthatforamomentortwo，thinkinghard，drawnnowthisway，nowthatwithindecision:’ShallIaccept？ShallI
　　refuse？’continuedtheinnervoice。
　　Pooh，let’saccept！Anheroicmethodoflearningtoswimistoleapboldlyintothesea。Letushurlourselvesheadfirstintothealgebraicalgulf;andperhapstheimminentdangerofdrowningwillcallfortheffortscapableofbringingmetoland。Iknownothingofwhathewants。Itmakesnodifference:let’sgoaheadandplungeintothemystery。Ishalllearnbyteaching。
　　ItwasafinecouragethatdrovemefulltiltintoaprovincewhichIhadnotyetthoughtofentering。Mytwenty—year—oldconfidencewasanincomparablelever。
　　’Verywell，’Ireplied。’Comethedayaftertomorrow，atfive，andwe’llbegin。’
　　Thistwenty—fourhours’delayconcealedaplan。Itsecuredmetherespiteofaday，theblessedThursday，whichwouldgivemetimetocollectmyforces。
　　Thursdaycomes。Theskyisgrayandcold。Inthishorridweather，agratewellfilledwithcokehasitscharms。Let’swarmourselvesandthink。
　　Well，myboy，you’velandedyourselfinanicepredicament！Howwillyoumanagetomorrow？Withabook，ploddingallthroughthenight，ifnecessary，youmightscrapeupsomethingresemblingalesson，justenoughtofillthedreadhourmoreorless。Thenyoucouldseeaboutthenext:sufficientforthedayistheevilthereof。Butyouhaven’tthebook。Andit’snouserunningouttothebookshop。Algebraicaltreatisesarenotcurrentwares。You’llhavetosendforone，whichwilltakeafortnightatleast。AndI’vepromisedfortomorrow，fortomorrowcertain！Anotherargumentandonethatadmitsofnoreply:fundsarelow;mylastpecuniaryresourceslieinthecornerofadrawer。Icountthemoney:itamountstotwelvesous，whichisnotenough。
　　MustIcryoff？Rathernot！Oneresourcesuggestsitself:ahighlyimproperone，Iadmit，notfarremovedindeedfromlarceny。O
　　quietpathsofalgebra，youaremyexcuseforthisvenialsin！Letmeconfessthetemporaryembezzlement。
　　Lifeatmycollegeismoreorlesscloistered。Inreturnforamodestpayment，mostofusmastersarelodgedinthebuilding;andwetakeourmealsattheprincipal’stable。Thesciencemaster，whoisthebiggunofthestaffandlivesinthetown，hasnevertheless，likeourselves，hisowntwocells，inadditiontoabalcony，orleads，wherethechemicalpreparationsgiveforththeirsuffocatinggasesintheopenair。Forthisreason，hefindsitmoreconvenienttoholdhisclasshereduringthegreaterpartoftheyear。Theboyscometotheseroomsinwinter，infrontofagratestuffedfullofcoke，likemine，andtherefindablackboard，apneumatictrough，amantelpiececoveredwithglassreceivers，panopliesofbenttubesonthewalls，and，lastly，acertaincupboardinwhichIrememberseeingarowofbooks，theoraclesconsultedbythemasterinthecourseofhislessons。
　　’Amongthosebooks，’saidItomyself，’thereissuretobeoneonalgebra。Toasktheownerfortheloanofitdoesnotappealtome。Myamiablecolleaguewouldreceivemesuperciliouslyandlaughatmyambitiousaims。Iamsurehewouldrefusemyrequest。’
　　Thefuturewastoshowthatmydistrustwasjustified。Narrowmindednessandpettyjealousyprevaileverywherealike。
　　Idecidetohelpmyselftothisbook，whichIshouldnevergetbyasking。Thisisthehalf—holiday。Thesciencemasterwillnotputinanappearancetoday;andthekeyofmyroomispracticallythesameashis。Igo，witheyesandearsonthealert。Mykeydoesnotquitefit;itsticksalittle，thengoesin;andanextraeffortmakesitturninthelock。Thedooropens。Iinspectthecupboardandfindthatitdoescontainanalgebrabook，oneofthebig，fatbookswhichmenusedtowriteinthosedays，abooknearlyhalfafootthick。Mylegsgivewaybeneathme。Youpoorspecimenofahousebreaker，supposeyouwerecaughtatit！However，allgoeswell。Quick，let’slockthedooragainandgobacktoourownquarterswiththepilferedvolume。
　　Andnowwearetogether，Omysterioustome，whoseArabnamebreathesastrangemustinessofoccultloreandclaimskindredwiththesciencesofalmagestandalchemy。Whatwillyoushowme？Letusturntheleavesatrandom。Beforefixingone’seyesonadefinitepointinthelandscape，itiswelltotakeasummaryviewofthewhole。Pagefollowsswiftlyuponpage，tellingmenothing。
　　Achaptercatchesmyattentioninthemiddleofthevolume;itisheaded，Newton’sBinomialTheorem。
　　Thetitlealluresme。Whatcanabinomialtheorembe，especiallyonewhoseauthorisNewton，thegreatEnglishmathematicianwhoweighedtheworlds？Whathasthemechanismoftheskytodowiththis？Letusreadandseekforenlightenment。Withmyelbowsonthetableandmythumbsbehindmyears，Iconcentrateallmyattention。
　　Iamseizedwithastonishment，forIunderstand！Thereareacertainnumberofletters，generalsymbolswhicharegroupedinallmannerofways，takingtheirplaceshere，thereandelsewherebyturns;thereare，asthetexttellsme，arrangements，permutationsandcombinations。Peninhand，Iarrange，permuteandcombine。Itisaverydivertingexercise，uponmyword，agameinwhichthetestofthewrittenresultconfirmstheanticipationsoflogicandsupplementstheshortcomingsofone’sthinkingapparatus。
　　’Itwillbeplainsailing，’saidItomyself，’ifalgebraisnomoredifficultthanthis。’
　　Iwastorecoverfromtheillusionlater，whenthebinomialtheorem，thatlight，crispbiscuit，wasfollowedbyheavierandlessdigestiblefare。But，forthemoment，Ihadnoforetasteofthefuturedifficulties，ofthepitfallinwhichonebecomesmoreandmoreentangled，thelongeronepersistsinstruggling。Whatadelightfulafternoonthatwas，beforemygrate，amidmypermutationsandcombinations！Bytheevening，Ihadnearlymasteredmysubject。Whenthebellrang，atseven，tosummonustothecommonmealattheprincipal’stable，Iwentdownstairspuffedupwiththejoysofthenewlyinitiatedneophyte。Iwasescortedonmywaybya，bandc，intertwinedincunninggarlands。
　　Nextday，mypupilisthere。Blackboardandchalk，everythingisready。Notquitesoreadyisthemaster。Ibravelybroachmybinomialtheorem。Myhearerbecomesinterestedinthecombinationsofletters。NotforamomentdoeshesuspectthatIamputtingthecartbeforethehorseandbeginningwhereweoughttohavefinished。Irelievethedrynessofmyexplanationswithafewlittleproblems，somanyhaltsatwhichthemindtakesbreathawhileandgathersstrengthforfreshflights。
　　Wetrytogether。Discreetly，soastoleavehimthemeritofthediscovery，Ishedalittlelightonthepath。Thesolutionisfound。Mypupiltriumphs;sodoI，butsilently，inmyinnerconsciousness，whichsays:
　　’Youunderstand，becauseyousucceedinmakinganotherunderstand。’
　　Thehourpassedquicklyandverypleasantlyforbothofus。Myyoungmanwascontentedwhenheleftme;andInolessso，forI
　　perceivedanewandoriginalwayoflearningthings。
　　Theingeniousandeasyarrangementofthebinomialgavemetimetotacklemyalgebrabookfromthepropercommencement。Inthreeorfourdays，Ihadrubbedupmyweapons。Therewasnothingtobesaidaboutadditionandsubtraction:theyweresosimpleastoforcethemselvesupononeatfirstsight。Multiplicationspoiltthings。Therewasacertainruleofsignswhichdeclaredthatminusmultipliedbyminusmadeplus。HowItoiledoverthatwretchedparadox！Itwouldseemthatthebookdidnotexplainthissubjectclearly，orratheremployedtooabstractamethod。Iread，rereadandmeditatedinvain:theobscuretextretainedallitsobscurity。Thatisthedrawbackofbooksingeneral:theytellyouwhatisprintedinthemandnothingmore。Ifyoufailtounderstand，theyneveradviseyou，neversuggestanattemptalonganotherroadwhichmightleadyoutothelight。Themerestwordwouldsometimesbeenoughtoputyouontherighttrack;andthatwordthebooks，hideboundinaregulationphraseology，nevergiveyou。
　　Howgreatlypreferableistheorallesson！Itgoesforward，goesback，startsafresh，walksaroundtheobstacleandvariesthemethodsofattackuntil，atlonglast，lightissheduponthedarkness。Thisincomparablebeaconofthemaster’swordwaswhatI
　　lacked;andIwentunder，withouthopeofsuccor，inthattreacherouspooloftheruleofsigns。
　　Mypupilwasboundtosuffertheeffects。AfteranattemptatanexplanationinwhichImadethemostofthefewgleamsthatreachedmeIaskedhim:
　　’Doyouunderstand？’
　　Itwasafutilequestion，butusefulforgainingtime。Myselfnotunderstanding，Iwasconvincedbeforehandthathedidnotunderstandeither。
　　’No，’hereplied，accusinghimself，perhaps，inhissimplemind，ofpossessingabrainincapableoftakinginthosetranscendentalverities。
　　’Letustryanothermethod。’
　　AndIstartagainthiswayandthatwayandyetanotherway。Mypupil’seyesserveasmythermometerandtellmeoftheprogressofmyefforts。Ablinkofsatisfactionannouncesmysuccess。Ihavestruckhome，Ihavefoundthejointinthearmor。Theproductofminusmultipliedbyminusdeliversitsmysteriestous。
　　Andthuswecontinuedourstudies:he，thepassivereceiver，takingintheideasacquiredwithouteffort;I，thefiercepioneer，blastingmyrock，thebook，withtheaidofmuchsittingupatnight，toextractthediamond，truth。Anotherandnolessarduoustaskfelltomyshare:Ihadtocutandpolishthereconditegem，tostripitofitsruggednessandpresentittomycompanion’sintelligenceunderalessforbiddingaspect。Thisdiamondcutter’swork，whichadmittedalittlelightintothepreciousstone，wasthefavoriteoccupationofmyleisure;andIoweagreatdealtoit。
　　Theultimateresultwasthatmypupilpassedhisexamination。Asforthebookborrowedbystealth，Irestoredittotheshelvesandreplaceditbyanother，which，thistime，belongedtome。
　　Atmynormalschool，Ihadlearntalittleelementarygeometryunderamaster。Fromthefirstfewlessonsonwards，Iratherenjoyedthesubject。Idivinedinitaguideforone’sreasoningfacultiesthroughthethicketsoftheimagination;Icaughtaglimpseofasearchaftertruththatdidnotinvolvetoomuchstumblingontheway，becauseeachstepforwardrestssolidlyuponthestepalreadytaken;Isuspectedgeometrytobewhatitpreeminentlyis:aschoolofintellectualfencing。
　　Thetruthdemonstratedanditsapplicationmatterlittletome;