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What Are Variables and Constants? Author(s): Karl Menger Source: Science, New Series, Vol. 123, No. 3196 (Mar. 30, 1956), pp. 547-548 Published by: American Association for the Advancement of Science Stable URL: http://www.jstor.org/stable/1750548 . Accessed: 15/05/2013 10:25 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, rese
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  What Are Variables and Constants?Author(s): Karl MengerSource: Science, New Series, Vol. 123, No. 3196 (Mar. 30, 1956), pp. 547-548Published by: American Association for the Advancement of Science Stable URL: http://www.jstor.org/stable/1750548. Accessed: 15/05/2013 10:25 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org. .  American Association for the Advancement of Science is collaborating with JSTOR to digitize, preserve andextend access to Science. http://www.jstor.org This content downloaded from 146.155.94.33 on Wed, 15 May 2013 10:25:37 AMAll use subject toJSTOR Terms and Conditions  highlevels of the.hydroxy compound(1or 2mgpertube)showamarkedgrowth-depressingeffect.Thisgrowthdepressionbyhighlevelsofhydroxyly-sineiseliminatedbyadditionoflargeramounts ofL-lysine.Both LeuconostocmesenteroidesP-60andStreptococcusfaecalisarecommonlyused formicrobiological assayoflysineinhydrolyzatesoffoods and tissues andeach of theseorganismshas beenac-cepted,on the basis ofearlierwork(6,7),ashavingahighlyspecificrequire-ment forlysine.However,in view oftheresultsreportedhere,itseemslikelythatthepresenceofhydroxylysineinsamplehydrolyzatescouldinterferewith thequantitavemicrobiologicaldetermina-tion oflysinewhentheseorganismsareused with basal mediathat containnohighlevels of the.hydroxy compound(1or 2mgpertube)showamarkedgrowth-depressingeffect.Thisgrowthdepressionbyhighlevelsofhydroxyly-sineiseliminatedbyadditionoflargeramounts ofL-lysine.Both LeuconostocmesenteroidesP-60andStreptococcusfaecalisarecommonlyused formicrobiological assayoflysineinhydrolyzatesoffoods and tissues andeach of theseorganismshas beenac-cepted,on the basis ofearlierwork(6,7),ashavingahighlyspecificrequire-ment forlysine.However,in view oftheresultsreportedhere,itseemslikelythatthepresenceofhydroxylysineinsamplehydrolyzatescouldinterferewith thequantitavemicrobiologicaldetermina-tion oflysinewhentheseorganismsareused with basal mediathat containnohydroxylysine.ydroxylysine. C.S.PETERSENR.W.CARROLLC.S.PETERSENR.W.CARROLL ResearchLaboratories,QuakerOatsCompany,Chicago,Illinois ReferencesandNotes1. S.Lindstedt,ActaPhysiol.Scand.27,377(1953).2.S.Bergstromand S.Lindstedt,Acta Chem.Scand.5,157(1951).3. L. M.Henderson and E. E.Snell,J.Biol.Chem.172,15(1948).4. Bothwere CfPgradeproductsobtainedfromCaliforniaFoundationfor BiochemicalResearch,LosAngeles.5. P. B. Hamilton and R.A.Anderson,J.Biol.Cheni.213,249(1955).6. A. D. McLarenand C.A.Knight,ibid.204,417(1953).7.J.L.Stokes etal.,ibid.160,35(1945).13September1955 WhatAreVariablesandConstants?The notionvariablehas notin theliteratureattainedthedegreeofclaritythatwouldjustifythe almostuniversaluse ofthattermwithoutexplanations.Recentinvestigations (1)have resolvedit intoan extensivespectrumof mean-ings,somepertainingtorealityas inves-tigatedinscience,othersbelongingtotherealm ofsymbolsstudiedinlogic-twoaltogetherdifferent worlds.Asacorollary,thesedistinctionsyieldaclari-fication of thenotionconstant.Scienceandappliedmathematics.Byquantitywemean apairofwhich thesecondmember(orvalue)is a numberwhile the first member(orobject)maybeanything.Wecalla classofquanti-tiesconsistentif it does not containtwoquantitieswithequal objectsand un-equalvalues. Ifp(y)denotes thepres-sure(inachosenunit)ofagas sampley(2),then thepair[y, p (y)]is aquan-tity,and the class ofallsuchpairsforanyyisconsistent. This class reflectsthephy-sicist'sideaofgaspressure,p.Gas vol-ume,v,andtemperature,t,canbe de- 30 MARCH 1956 ResearchLaboratories,QuakerOatsCompany,Chicago,Illinois ReferencesandNotes1. S.Lindstedt,ActaPhysiol.Scand.27,377(1953).2.S.Bergstromand S.Lindstedt,Acta Chem.Scand.5,157(1951).3. L. M.Henderson and E. E.Snell,J.Biol.Chem.172,15(1948).4. Bothwere CfPgradeproductsobtainedfromCaliforniaFoundationfor BiochemicalResearch,LosAngeles.5. P. B. Hamilton and R.A.Anderson,J.Biol.Cheni.213,249(1955).6. A. D. McLarenand C.A.Knight,ibid.204,417(1953).7.J.L.Stokes etal.,ibid.160,35(1945).13September1955 WhatAreVariablesandConstants?The notionvariablehas notin theliteratureattainedthedegreeofclaritythatwouldjustifythe almostuniversaluse ofthattermwithoutexplanations.Recentinvestigations (1)have resolvedit intoan extensivespectrumof mean-ings,somepertainingtorealityas inves-tigatedinscience,othersbelongingtotherealm ofsymbolsstudiedinlogic-twoaltogetherdifferent worlds.Asacorollary,thesedistinctionsyieldaclari-fication of thenotionconstant.Scienceandappliedmathematics.Byquantitywemean apairofwhich thesecondmember(orvalue)is a numberwhile the first member(orobject)maybeanything.Wecalla classofquanti-tiesconsistentif it does not containtwoquantitieswithequal objectsand un-equalvalues. Ifp(y)denotes thepres-sure(inachosenunit)ofagas sampley(2),then thepair[y, p (y)]is aquan-tity,and the class ofallsuchpairsforanyyisconsistent. This class reflectsthephy-sicist'sideaofgaspressure,p.Gas vol-ume,v,andtemperature,t,canbe de- 30 MARCH 1956 finedsimilarly.Consistentclassesofquantities,suchasp,v,andt,arewhatscientists and mathematiciansinappliedfieldsmeanbyvariablequantitiesandwhat Newtoncalledfluents.Theclassofallgas sampleswillbereferredtoasthedomainofp;the classof all numbersp(y)as therangeofp.Avariablequantitywhoserangecon-sists of asinglenumberissaid to becon-stant.Anexampleis thegravitationalaccelerationgat adefinitepointontheearth,definedas theclassof allpairs[a,g(a)]foranyobjectafallingin avacuum,whereg(a)denotes the accel-eration ofa.Foranya,thisvalueofgisfoundto beequalto one andthesamenumber,g. (Throughoutthispaper,sym-bols forfluents areitalicized andsymbolsfornumbersrareprintedinromantype.)Alessimportant,because lesscompre-hensive,exampleisthe distancetraveledbyaspecificcarCwhile C isparked.(Thedistance traveledbyCisthe classofallpairs [t, m(g)]foranyact ,t ofreadingthemileage gageinC,wherem(lt)is thenumber readas the resultof Lt. Inaplane,the coordinates relativetoa chosen Cartesian frame arevariablequantitieswhose domainis the class ofallpoints(3).The abscissaxistheclassof allpairs[ja, x(t)] forany pointar.Intheequationof thestraightline x-y=3the3 denotes theconstant fluent consist-ingof thequantities (t, 3)ofvalue3 foreachpoint at. Ofparamount importancearethe con-sistentclassesofquantitieswhosedo-mainsconsistofnumbers.Anexampleistheclassofallpairs[x,logx]foranynumber x>0,called thelogarithmicfunctionorthefunctionlog.It iscapableofconnectingconsistent classes ofquan-tities-forexample,y=logxalongalogarithmiccurve,andw=logv foranisothermicexpansionof anidealgas,ifwdenotesthework inaproperunit.Thatis tosay,y((n)=logx(Jt) andw(y)=logv(y)forany point Jt on the curveandanygassampleypertainingto theprocess.Moreover,logconnectsfunc-tions-forexample,theexponentialwiththeidentityfunction,andcoswithlogcos. Because of theirconnectivepower,functions areomnipresentin science aswell as inmathematics(4).Variablequantitiessuchaspandv(whosedo-mains donotcontain numbers orsystemsofnumbers)lack thispowerand there-foreare confined tospecialbranches ofscience such asgastheory.Denyingthesignificanceofthisdifference wouldbedenyingtherole ofmathematicsasa uni-versaltool inquantitativescience.Logicandpuremathematics. Thefinedsimilarly.Consistentclassesofquantities,suchasp,v,andt,arewhatscientists and mathematiciansinappliedfieldsmeanbyvariablequantitiesandwhat Newtoncalledfluents.Theclassofallgas sampleswillbereferredtoasthedomainofp;the classof all numbersp(y)as therangeofp.Avariablequantitywhoserangecon-sists of asinglenumberissaid to becon-stant.Anexampleis thegravitationalaccelerationgat adefinitepointontheearth,definedas theclassof allpairs[a,g(a)]foranyobjectafallingin avacuum,whereg(a)denotes the accel-eration ofa.Foranya,thisvalueofgisfoundto beequalto one andthesamenumber,g. (Throughoutthispaper,sym-bols forfluents areitalicized andsymbolsfornumbersrareprintedinromantype.)Alessimportant,because lesscompre-hensive,exampleisthe distancetraveledbyaspecificcarCwhile C isparked.(Thedistance traveledbyCisthe classofallpairs [t, m(g)]foranyact ,t ofreadingthemileage gageinC,wherem(lt)is thenumber readas the resultof Lt. Inaplane,the coordinates relativetoa chosen Cartesian frame arevariablequantitieswhose domainis the class ofallpoints(3).The abscissaxistheclassof allpairs[ja, x(t)] forany pointar.Intheequationof thestraightline x-y=3the3 denotes theconstant fluent consist-ingof thequantities (t, 3)ofvalue3 foreachpoint at. Ofparamount importancearethe con-sistentclassesofquantitieswhosedo-mainsconsistofnumbers.Anexampleistheclassofallpairs[x,logx]foranynumber x>0,called thelogarithmicfunctionorthefunctionlog.It iscapableofconnectingconsistent classes ofquan-tities-forexample,y=logxalongalogarithmiccurve,andw=logv foranisothermicexpansionof anidealgas,ifwdenotesthework inaproperunit.Thatis tosay,y((n)=logx(Jt) andw(y)=logv(y)forany point Jt on the curveandanygassampleypertainingto theprocess.Moreover,logconnectsfunc-tions-forexample,theexponentialwiththeidentityfunction,andcoswithlogcos. Because of theirconnectivepower,functions areomnipresentin science aswell as inmathematics(4).Variablequantitiessuchaspandv(whosedo-mains donotcontain numbers orsystemsofnumbers)lack thispowerand there-foreare confined tospecialbranches ofscience such asgastheory.Denyingthesignificanceofthisdifference wouldbedenyingtherole ofmathematicsasa uni-versaltool inquantitativescience.Logicandpuremathematics. Theformula3- - (3+1)(3-1)isastatementaboutspecificnumbersformula3- - (3+1)(3-1)isastatementaboutspecificnumbersdesignatedby3and1.Inthemoregen-eralstatementx'-1=(x+1)-(x-1)foranynumberx,the remark foranynumberx stipu-latesthat,in theformula,xmaybere-placedbythedesignationofanynum-ber,forexample, by3 or'/5,eachre-placementyieldingavalidstatementaboutspecificnumbers.Asymbolthat,inacertaincontextandaccordingto adefinitestipulation,maybereplacedbythedesignationofanyelementofacer-tainclassiswhat,followingWeierstrass,logiciansandpuremathematicianscalla variable. The saidclassis calledthescopeof the variable.Thescopeofx intheformulalogx2 = 2logxforanyx>0 isthe class of allpositivenumbers.In(? ^3)4=9,thesymbol? V3 isavariablewhosescopeconsistsof the two numbers '3 and-'3. A variable whosescopeconsistsofasinglenumberdesignatesthat numberand,inpuremathematics,isreferredtoas a constant.Examplesinclude numer- als(1,3,...);e,designatingthebase of naturallogarithms;andbriefdesigna-tions of numberswithunwieldysymbols,suchas 22'+ eee-abbreviationsintro-ducedfor thepurposeofjustone dis-cussioninvolvingrepeatedreferencestothatnumber.Assuchadhocconstants,onecustomarilyuses the lettersa,b,c,., which,justas x andy,serveasvariables inothercontexts,forexample,in thestatementinvolvingtwo variables xaa=(x+a) (x -a) foranyxandanya.Becauseofitsvicariouscharacter,avari-ablemayalwaysbereplaced,withoutany changeofthemeaning,byanyother-wiseunusedletter,forinstance,abyyor xbyb.Variables whosescopesareclasses ofnumbersare callednumericalvariables.The definitionsofpandgmake useofvariablesyandawhosescopesconsistofgassamplesandfallingobjects,re-spectively.Statementsthatare validformanyfluentsareconvenientlyexpressedintermsoffluentvariables;forexample,Ifz=logu,thendz/du=1/uforanytwofluents u andz(1)InEq.1,uandzmaybereplacedbyxandy:ify =logx(whichholdsalongthelogarithmiccurve),thendy/dx=1/x;orbyv and w:ifw=logv(whichholds foran isothermicexpansion),thendw/dv=1/v.Butu andz inEq.1mustnot bereplaced bynumberssuchaseand 1:although1=logeisvalid,dl/de=1/eis nonsensical.Confusioninthe literature.Nocleardistinction has heretoforebeen made be-designatedby3and1.Inthemoregen-eralstatementx'-1=(x+1)-(x-1)foranynumberx,the remark foranynumberx stipu-latesthat,in theformula,xmaybere-placedbythedesignationofanynum-ber,forexample, by3 or'/5,eachre-placementyieldingavalidstatementaboutspecificnumbers.Asymbolthat,inacertaincontextandaccordingto adefinitestipulation,maybereplacedbythedesignationofanyelementofacer-tainclassiswhat,followingWeierstrass,logiciansandpuremathematicianscalla variable. The saidclassis calledthescopeof the variable.Thescopeofx intheformulalogx2 = 2logxforanyx>0 isthe class of allpositivenumbers.In(? ^3)4=9,thesymbol? V3 isavariablewhosescopeconsistsof the two numbers '3 and-'3. A variable whosescopeconsistsofasinglenumberdesignatesthat numberand,inpuremathematics,isreferredtoas a constant.Examplesinclude numer- als(1,3,...);e,designatingthebase of naturallogarithms;andbriefdesigna-tions of numberswithunwieldysymbols,suchas 22'+ eee-abbreviationsintro-ducedfor thepurposeofjustone dis-cussioninvolvingrepeatedreferencestothatnumber.Assuchadhocconstants,onecustomarilyuses the lettersa,b,c,., which,justas x andy,serveasvariables inothercontexts,forexample,in thestatementinvolvingtwo variables xaa=(x+a) (x -a) foranyxandanya.Becauseofitsvicariouscharacter,avari-ablemayalwaysbereplaced,withoutany changeofthemeaning,byanyother-wiseunusedletter,forinstance,abyyor xbyb.Variables whosescopesareclasses ofnumbersare callednumericalvariables.The definitionsofpandgmake useofvariablesyandawhosescopesconsistofgassamplesandfallingobjects,re-spectively.Statementsthatare validformanyfluentsareconvenientlyexpressedintermsoffluentvariables;forexample,Ifz=logu,thendz/du=1/uforanytwofluents u andz(1)InEq.1,uandzmaybereplacedbyxandy:ify =logx(whichholdsalongthelogarithmiccurve),thendy/dx=1/x;orbyv and w:ifw=logv(whichholds foran isothermicexpansion),thendw/dv=1/v.Butu andz inEq.1mustnot bereplaced bynumberssuchaseand 1:although1=logeisvalid,dl/de=1/eis nonsensical.Confusioninthe literature.Nocleardistinction has heretoforebeen made be-tweennumerical andfluent variables.Moreover,intheliterature,numericalvariablesand variablequantities,not- 547 tweennumerical andfluent variables.Moreover,intheliterature,numericalvariablesand variablequantities,not- 547 This content downloaded from 146.155.94.33 on Wed, 15 May 2013 10:25:37 AMAll use subject toJSTOR Terms and Conditions  withstandingtheprofounddifferencesbetweenthem,areindiscriminatelyre-ferredto as variables ;and thetwoconceptshaveactuallybeen confused.Forinstance,numerousattempts(5)havebeen made to definegaspressureasasymbolthatmaybereplaced byanyvalueofpressure,thus as a numer-icalvariable,p,whosescopeis therangeofwhatherein has been called the fluentp.Buta fluentis not determinedbyitsrange.CouldBoylehaveconnectedpres-sureand volumeon the basisof mereinformation about therangesofpandv?It wasby transcendingtheserangesandreferringto thedomains,namely,bycomparingv(y)andp(y)forthe samesampley,that he discoveredv(y)=l/p(y)(inproperunits)foranyyofacertaintemperatureor,withoutrefer-enceto asamplevariable,v=1/p.Neither formulation involvesnumericalvariables.Boyle'sLaw connectsspecificfluents.Analogously,constantfluentshavebeenidentified with their numericalvalueseventhoughwhatprimarilyin-terests thephysicistingravitationalac-celerationclearlyis the fact thatgisitsvalueforanyaratherthan thenumbergas such.Numerical variablesandvariablequan-titiesbelongto worlds that arenotonlydifferentbutnonisomorphic.Theformerareinterchangeable,thelatter arenot:x2-4y2=(x+2y)*(x-2y)andy2-4x2=(y+2x) . (y- 2x)foranyxandyare tantamount.Butx-y=3andy-x=3are differentstraightlines and w = logvis incom-patiblewith,andnot tantamountto,v=logw.Onlytheconsistent maintenanceof allthese distinctionsmakes itpossibletoformulatemathematicalanalysisas wellas itsapplicationsto scienceasasystemofproceduresfollowingarticulaterules(6). KARL MENGER IllinoisInstituteofTechnologyChicago,Illinois References andNotes1.CompareK.Menger, Theideasof variableandfunction, Proc.Natl.Acad.Sci. U.S.39,956(1953)and Onvariablesinmathematicsand in naturalscience, Brit.J.Phil.Sci.5,134(1954).Apaper Randomvariables andthegeneral theoryofvariables is inprintProc. 3rdBerkeley SymposiumMath.Statistics(1955).Asystematic expositionofthe newtheoryofvari-ablesiscontainedinK. Menger,Calculus.AModernApproach(Ginn,Boston, 1955).2.Thetermgas samplemeansgasin aspecificcontainer at a definite instant.3.Dependingonwhetheraphysical,apostula-tional,or apure planeis underconsideration,a point isaphysicalobject (forexample,aninkdotinapaper plane),or anundefinedob-jectsatisfyingcertainassumptions,or anordered withstandingtheprofounddifferencesbetweenthem,areindiscriminatelyre-ferredto as variables ;and thetwoconceptshaveactuallybeen confused.Forinstance,numerousattempts(5)havebeen made to definegaspressureasasymbolthatmaybereplaced byanyvalueofpressure,thus as a numer-icalvariable,p,whosescopeis therangeofwhatherein has been called the fluentp.Buta fluentis not determinedbyitsrange.CouldBoylehaveconnectedpres-sureand volumeon the basisof mereinformation about therangesofpandv?It wasby transcendingtheserangesandreferringto thedomains,namely,bycomparingv(y)andp(y)forthe samesampley,that he discoveredv(y)=l/p(y)(inproperunits)foranyyofacertaintemperatureor,withoutrefer-enceto asamplevariable,v=1/p.Neither formulation involvesnumericalvariables.Boyle'sLaw connectsspecificfluents.Analogously,constantfluentshavebeenidentified with their numericalvalueseventhoughwhatprimarilyin-terests thephysicistingravitationalac-celerationclearlyis the fact thatgisitsvalueforanyaratherthan thenumbergas such.Numerical variablesandvariablequan-titiesbelongto worlds that arenotonlydifferentbutnonisomorphic.Theformerareinterchangeable,thelatter arenot:x2-4y2=(x+2y)*(x-2y)andy2-4x2=(y+2x) . (y- 2x)foranyxandyare tantamount.Butx-y=3andy-x=3are differentstraightlines and w = logvis incom-patiblewith,andnot tantamountto,v=logw.Onlytheconsistent maintenanceof allthese distinctionsmakes itpossibletoformulatemathematicalanalysisas wellas itsapplicationsto scienceasasystemofproceduresfollowingarticulaterules(6). KARL MENGER IllinoisInstituteofTechnologyChicago,Illinois References andNotes1.CompareK.Menger, Theideasof variableandfunction, Proc.Natl.Acad.Sci. U.S.39,956(1953)and Onvariablesinmathematicsand in naturalscience, Brit.J.Phil.Sci.5,134(1954).Apaper Randomvariables andthegeneral theoryofvariables is inprintProc. 3rdBerkeley SymposiumMath.Statistics(1955).Asystematic expositionofthe newtheoryofvari-ablesiscontainedinK. Menger,Calculus.AModernApproach(Ginn,Boston, 1955).2.Thetermgas samplemeansgasin aspecificcontainer at a definite instant.3.Dependingonwhetheraphysical,apostula-tional,or apure planeis underconsideration,a point isaphysicalobject (forexample,aninkdotinapaper plane),or anundefinedob-jectsatisfyingcertainassumptions,or anordered pairof numbers.airof numbers.4.Evenif,followingthesuggestionofsomemath-ematicians,one called all consistentclassesofquantities functions, oneobviouslywouldneedaspecialtermfor functionallyconnect-ingfunctions suchaslog.5.Compare,forexample,R.Courant,DifferentialandIntegralCalculus(Interscience,NewYork,1954),vol.1, p.14.6.CompareK.Menger,Calculus. A ModernAp-proach (Ginn, Boston, 1955).9September1955 PrenatalDiagnosisofSexUsingCells from the AmnioticFluid In mostmammals,includinghuman beings,malesnormallyhavethesexchromosome constitutionXY,andfe-males the sexchromosomeconstitutionXX(1).Ithasbeenshown in avarietyof tissuesinhumanbeingsandsomeotherspecies(2)thatthere is a sex dif-ferenceinthepercentageofcellswithchromocenters,especiallythoseat thenuclearmembrane;thispresumablyisdueto thisdifferenceinsexchromosomeconstitutionin males and females.Ade-terminationof thepercentageof cellswithchromocenterscan thereforegive,insexuallynormalindividuals,adiag-nosisofsex.Thepresentstudywas undertakeninorder to showwhether,inhumanbeings,suchadiagnosiscan be madebeforebirth,notonlyforabortedfetusesfromwhichpiecesoftissuecan beremovedforexamination,butalsoforviablefetusesbyan examinationof cellsfromtheamnioticfluid.Inordertoestablishwhether amnioticfluidcontainscellssuitablefordiagnosis,fluidwastakenbeforedeliverybypunctureof the mem-branesfromwomenintheninthmonthofpregnancy.Thefluid wascentrifuged,and the cellswere smearedonslides,fixed inalcohol-ether,and stainedwithFeulgenand fastgreen(Fig.1).Ouranalysishasshown that cells suit-ablefor thediagnosisarepresent,andan examinationof35 cases inthe ninthmonth,which includethosereportedpreviously(3),hasgiven35 correctdiag-nosesof the sexof the fetus.It thereforeseems thatthismethodisparticularlyreliable,especiallysincethereappearsto beno theoreticalobjec-tiontoit.Theonly apparentexceptionthatoccurs tous atpresentis therarecaseofanintersexin which thesexualphenotypedoesnotcorrespondto thesex chromosomeconstitution.Whenoneiscollectingthe amnioticfluid itis,ofcourse,essentialtoavoidcontaminationwithcellsfromthe mother.Amniotic fluid canbeobtainedfromviable humanfetuses from 12weekstoterm(4).We havefound,from an ex- 4.Evenif,followingthesuggestionofsomemath-ematicians,one called all consistentclassesofquantities functions, oneobviouslywouldneedaspecialtermfor functionallyconnect-ingfunctions suchaslog.5.Compare,forexample,R.Courant,DifferentialandIntegralCalculus(Interscience,NewYork,1954),vol.1, p.14.6.CompareK.Menger,Calculus. A ModernAp-proach (Ginn, Boston, 1955).9September1955 PrenatalDiagnosisofSexUsingCells from the AmnioticFluid In mostmammals,includinghuman beings,malesnormallyhavethesexchromosome constitutionXY,andfe-males the sexchromosomeconstitutionXX(1).Ithasbeenshown in avarietyof tissuesinhumanbeingsandsomeotherspecies(2)thatthere is a sex dif-ferenceinthepercentageofcellswithchromocenters,especiallythoseat thenuclearmembrane;thispresumablyisdueto thisdifferenceinsexchromosomeconstitutionin males and females.Ade-terminationof thepercentageof cellswithchromocenterscan thereforegive,insexuallynormalindividuals,adiag-nosisofsex.Thepresentstudywas undertakeninorder to showwhether,inhumanbeings,suchadiagnosiscan be madebeforebirth,notonlyforabortedfetusesfromwhichpiecesoftissuecan beremovedforexamination,butalsoforviablefetusesbyan examinationof cellsfromtheamnioticfluid.Inordertoestablishwhether amnioticfluidcontainscellssuitablefordiagnosis,fluidwastakenbeforedeliverybypunctureof the mem-branesfromwomenintheninthmonthofpregnancy.Thefluid wascentrifuged,and the cellswere smearedonslides,fixed inalcohol-ether,and stainedwithFeulgenand fastgreen(Fig.1).Ouranalysishasshown that cells suit-ablefor thediagnosisarepresent,andan examinationof35 cases inthe ninthmonth,which includethosereportedpreviously(3),hasgiven35 correctdiag-nosesof the sexof the fetus.It thereforeseems thatthismethodisparticularlyreliable,especiallysincethereappearsto beno theoreticalobjec-tiontoit.Theonly apparentexceptionthatoccurs tous atpresentis therarecaseofanintersexin which thesexualphenotypedoesnotcorrespondto thesex chromosomeconstitution.Whenoneiscollectingthe amnioticfluid itis,ofcourse,essentialtoavoidcontaminationwithcellsfromthe mother.Amniotic fluid canbeobtainedfromviable humanfetuses from 12weekstoterm(4).We havefound,from an ex-aminationofthefluid obtainedfromminationofthefluid obtainedfromFig.1.Photomicrographsofcells fromtheamnioticfluid(x1500).(Toprow)Nu-clei withachromocenteratthenuclearmembrane;from femalehumanfetusesintheninthmonth.(Bottomrow)Nucleiwithoutachromocenter;frommale hu-man fetusesinthe ninthmonth.viablehumanfetusesinthesixth andseventhmonths,that aprenataldiagnosisof sexcanbemadeat thesestages.Wehave alsofound suitablecellsinthefluidof an aborted8-week-oldhumanembryo.Itmaybepossibletoapplythismethodfor theprenataldiagnosisofsextodomesticanimals.LEO SACHS DepartmentofExperimentalBiology,WeizmannInstituteofScience,Rehovoth,Israel DAVIDM.SERR DepartmentofObstetricsandGynaecology,Rothschild-HadassahUniversityHospital,Jerusalem,Israel MATHILDE DANON DepartmentofExperimentalBiology,WeizmannInstituteofScience,Rehovoth,Israel References 1. L.Sachs,Ann.Eugen.18,255(1954);Genetica27,309(1955).2.K.L.Moore,M.A.Graham,M. L.Barr,Surg.Gynecol.Obstet.96,641(1953);K. L.Moore andM. L.Barr,ActaAnat.21,197(1954); J.L.Emeryand M.McMillan,J.Pathol.Bacteriol.68,17(1954);M. A.Graham,Anat.Record119,469(1954);K.L. MooreandM. L.Barr,Lancet269,57(1955);E. Mar-bergerandW.O.Nelson,Bruns'Beitr.klin.Chir.190,103(1955);E.Marberger,R. A.Boc-cabella,W.O.Nelson,Proc.Soc.Exptl.Biol.Med.89,488(1955);Lancet269,654(1955);L. Sachs and M.Danon,Genetica,inpress.3. D. M.Serr,L.Sachs,M.Danon,Bull.Re-searchCouncilIsrael5B,137(1955). 4. W.J.DieckmannandM.E.Davies,Am.J. Obstet.Gynecol. 25,623(1933);H. Alvarezand R.Caldeyro,Surg.Gynecol.Obstet.91,1(1950).30December1955 Fig.1.Photomicrographsofcells fromtheamnioticfluid(x1500).(Toprow)Nu-clei withachromocenteratthenuclearmembrane;from femalehumanfetusesintheninthmonth.(Bottomrow)Nucleiwithoutachromocenter;frommale hu-man fetusesinthe ninthmonth.viablehumanfetusesinthesixth andseventhmonths,that aprenataldiagnosisof sexcanbemadeat thesestages.Wehave alsofound suitablecellsinthefluidof an aborted8-week-oldhumanembryo.Itmaybepossibletoapplythismethodfor theprenataldiagnosisofsextodomesticanimals.LEO SACHS DepartmentofExperimentalBiology,WeizmannInstituteofScience,Rehovoth,Israel DAVIDM.SERR DepartmentofObstetricsandGynaecology,Rothschild-HadassahUniversityHospital,Jerusalem,Israel MATHILDE DANON DepartmentofExperimentalBiology,WeizmannInstituteofScience,Rehovoth,Israel References 1. L.Sachs,Ann.Eugen.18,255(1954);Genetica27,309(1955).2.K.L.Moore,M.A.Graham,M. L.Barr,Surg.Gynecol.Obstet.96,641(1953);K. L.Moore andM. L.Barr,ActaAnat.21,197(1954); J.L.Emeryand M.McMillan,J.Pathol.Bacteriol.68,17(1954);M. A.Graham,Anat.Record119,469(1954);K.L. MooreandM. L.Barr,Lancet269,57(1955);E. Mar-bergerandW.O.Nelson,Bruns'Beitr.klin.Chir.190,103(1955);E.Marberger,R. A.Boc-cabella,W.O.Nelson,Proc.Soc.Exptl.Biol.Med.89,488(1955);Lancet269,654(1955);L. Sachs and M.Danon,Genetica,inpress.3. D. M.Serr,L.Sachs,M.Danon,Bull.Re-searchCouncilIsrael5B,137(1955). 4. W.J.DieckmannandM.E.Davies,Am.J. Obstet.Gynecol. 25,623(1933);H. Alvarezand R.Caldeyro,Surg.Gynecol.Obstet.91,1(1950).30December1955 SCIENCE,VOL. 123CIENCE,VOL. 123 4848 This content downloaded from 146.155.94.33 on Wed, 15 May 2013 10:25:37 AMAll use subject toJSTOR Terms and Conditions
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