Plane. Geometry, proofs of some of the easier theorems and construc- tions are left as exercises for the student, or are given in an incomplete form. In the Solid.
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PREFACEThisbookistheoutgrowthofanexperienceofmanyyearsintheteachingofmathematicsinsecondaryschools.Thetexthasbeenusedbymanydifferentteachers,inclassesofallstagesofdevelopment,andundervaryingconditionsofsec-ondaryschoolteaching.Theproofshavehadthebenefitofthecriticismsofhundredsofexperiencedteachersofmathe-maticsthroughoutthecountry.Thebookinitspresentformisthereforethecombinedproductofexperience,classroomtest,andseverecriticism.Thefollowingaresomeoftheleadingfeaturesofthebook:Tliestudentisrapidlyinitiatedintothesubject.Definitionsaregivenonlyasneeded.Theselectionandarrangementoftheoremsissuchastomeetthegeneraldemandofteachers,asexpressedthroughtheMathe-maticalAssociationsofthecountry.Mostoftheproofshavebeengiveninfull.InthePlaneGeometry,proofsofsomeoftheeasiertheoremsandconstruc-tionsareleftasexercisesforthestudent,oraregiveninanincompleteform.IntheSolidGeometry,moreproofsandpartsofproofsarethuslefttothestudent;butineverycaseinwhichtheproofisnotcomplete,theincompletenessisspecificallystated.Theindirectmethodofproofisconsistentlyapplied.Theusualmethodofprovingsuchpropositions,forexample,asArts.189and415,isconfusingtothestudent.Themethodusedhereisconvincingandclear.Theexercisesarecarefullyselected.Inchoosingexercises,eachofthefollowinggroupshasbeengivendueimportance:(a)Concreteexercises,includingnumericalproblemsandproblemsofconstruction.(6)So-calledpracticalproblems,suchasindirectmeasure-mentsofheightsanddistancesbymeansofequalandsimilartriangles,drawingtoscaleasanapplicationofsimilarfigures,..problemsfromphysics,fromdesign,etc,304026
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ivPREFACE(c)Traditionalexercisesofamoreorlessabstractnature.ThearrangementoftheexercisesispedagogicalCompara-tivelyeasyexercisesareplacedimmediatelyafterthetheoremsofwhichtheyareapplications,insteadofbeinggroupedto-getherwithoutregardtotheprinciplesinvolvedinthem.Forthebenefitofthebrighterpupils,however,andforreviewclasses,longlistsofmoreorlessdifficultexercisesaregroupedattheendofeachbook.Thedefinitionsofclosedfiguresareunique.Thestudent’snaturalconceptionofaplaneclosedfigure,forexample,isnottheboundarylineonly,northeplaneonly,butthewholefigurecomposedoftheboundarylineandtheplanebounded.Alldefinitionsofclosedfiguresinvolvethisidea,whichisentirelyconsistentwiththehighermathematics.Thenumericaltreatmentofmagyiitudesisexplicit,thefunda-mentalprinciplesbeingdefinitelyassumed(Art.336,proofinAppendix,Art.595).Thisnovelprocedurefurnishesalogical,aswellasateachable,methodofdealingwithincommensurables.Theareaofarectangleisintroducedbyactuallymeasuringit,therebyobtainingitsmeasure-number.Thismethodpermitsthesameorderoftheoremsandcorollariesasisusedintheparallelogramandthetriangle.Thecorrelationwitharithmeticinthisconnectionisvaluable.Asimilarmethodisemployedforintroducingthevolumeofaparallelopiped.Proofsofthesuperpositiontheoremsandtheconcurrentlinetheoremswillbefoundexceptionallyaccurateandcomplete.Themanyhistoricalnotesivilladdlifeandinteresttothework.Thecarefullyarrangedsummariesthroughoutthebook,thecollectionofformulasofPlaneGeometry,andthecollectionofformulasofSolidGeometry,itishoped,willbefoundhelpfultoteacherandstudentalike.Argumentandreasonsarearrangedinparallelform.Thisarrangementgivesadefinitemodelforprovingexercises,ren-dersthecarelessomissionofthereasonsinademonstrationimpossible,leadstoaccuratethinking,andgreatlylightensthelaborofreadingpapers.
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PREFACEvEveryconstructionfigurecontainsallnecessaryconstructionlines.Thismethodkeepsconstantlybeforethestudentamodelforhisconstructionwork,anddistinguishesbetweenafigureforaconstructionandafigureforatheorem.TJiemechanicalarrangementissuchastogivethestudenteverypossibleaidincomprehendingthesubjectmatter.ThefollowingaresomeofthespecialfeaturesoftheSolidGeometry:ThevitalrelationoftheSolidGeometrytothePlaneGeometryisemphasizedateverypoint.(SeeArts.703,786,794,813,853,924,951,955,961,etc.)Thestudentisgiveneverypossibleaidinforminghisearlyspaceconcepts.IntheearlyworkinSolidGeometry,theaveragestudentexperiencesdifficultyinfullycomprehendingspacerelations,thatis,inseeinggeometricfiguresinspace.Thestudentisaidedinovercomingthisdifficultybytheintro-ductionofmanyeasyandpracticalquestionsandexercises,aswellasbybeingencouragedtomakehisfigures.(See”605.)Asafurtheraidinthisdirection,reproductionsofmodelsmadebystudentsthemselvesareshowninagroup(p.302)andatvariouspointsthroughoutBookVI.Tliestudent’sknowledgeofthethingsabouthimisconstantlydraivnupon.Especiallyisthistrueoftheworkonthesphere,wherethestudent’sknowledgeofmathematicalgeographyhasbeenappealedtoinmakingclearthetermsandtherelationsoffiguresconnectedwiththesphere.ThesamelogicalrigorthatcharacterizesthedemonstrationsinthePlaneGeometryisusedthroughouttheSolid.Thetreatmentofthepolyhedralangle(p.336),oftheprism(p.345),andofthepyramid(p.350)issimilartothatofthecylinderandofthecone.ThisisinaccordancewiththerecommendationsoftheleadingMathematicalAssociationsthroughoutthecountry.Thegratefulacknowledgmentoftheauthorsisduetomanyfriendsforhelpfulsuggestions;especiallytoMissGraceA.Bruce,oftheWadleighHighSchool,NewYork;toMr.EdwardB.Parsons,oftheBoys’HighSchool,Brooklyn;andtoProfessorMcMahon,ofCornellUniversity.
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