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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 ScienceStable URL: http://www.jstor.org/stable/1750548 .

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high levels of the.hydroxy compound (1or 2 mg per tube) show a markedgrowth-depressing effect. This growthdepression by high levels of hydroxyly-sine is eliminated by addition of largeramounts of L-lysine.Both Leuconostoc mesenteroides P-60and Streptococcusfaecalis are commonlyused for microbiological assay of lysinein hydrolyzatesof foods and tissues andeach of these organisms has been ac-cepted, on the basis of earlier work (6,7), as having a highly specific require-ment for lysine. However, in view of theresults reported here, it seems likely thatthe presence of hydroxylysine in samplehydrolyzates could interfere with thequantitave microbiological determina-tion of lysine when these organisms areused with basal media that contain no

high levels of the.hydroxy compound (1or 2 mg per tube) show a markedgrowth-depressing effect. This growthdepression by high levels of hydroxyly-sine is eliminated by addition of largeramounts of L-lysine.Both Leuconostoc mesenteroides P-60and Streptococcusfaecalis are commonlyused for microbiological assay of lysinein hydrolyzatesof foods and tissues andeach of these organisms has been ac-cepted, on the basis of earlier work (6,7), as having a highly specific require-ment for lysine. However, in view of theresults reported here, it seems likely thatthe presence of hydroxylysine in samplehydrolyzates could interfere with thequantitave microbiological determina-tion of lysine when these organisms areused with basal media that contain nohydroxylysine.ydroxylysine. C. S. PETERSEN

R. W. CARROLLC. S. PETERSENR. W. CARROLLResearch Laboratories,Quaker Oats Company, Chicago, Illinois

References and Notes1. S. Lindstedt, Acta Physiol. Scand. 27, 377(1953).2. S. Bergstrom and S. Lindstedt, Acta Chem.Scand. 5, 157 (1951).3. L. M. Henderson and E. E. Snell, J. Biol.Chem. 172, 15 (1948).4. Both were CfP grade products obtained fromCaliforniaFoundation for Biochemical Research,Los Angeles.5. P. B. Hamilton and R. A. Anderson, J. Biol.Cheni. 213, 249 (1955).6. A. D. McLaren and C. A. Knight, ibid. 204,417 (1953).7. J. L. Stokes et al., ibid. 160, 35 (1945).13 September 1955

What Are Variables and Constants?The notion variable has not in theliterature attained the degree of claritythat would justify the almost universaluse of that term without explanations.Recent investigations (1) have resolvedit into an extensive spectrum of mean-ings, some pertaining to reality as inves-tigated in science, others belonging tothe realm of symbols studied in logic-two altogether different worlds. As acorollary, these distinctions yield a clari-fication of the notion constant.Science and applied mathematics. Byquantity we mean a pair of which thesecond member (or value) is a numberwhile the first member (or object) maybe anything. We call a class of quanti-ties consistent if it does not contain two

quantities with equal objects and un-equal values. If p(y) denotes the pres-sure (in a chosen unit) of a gas sampley (2), then the pair [y, p (y)] is a quan-tity, and the class of all such pairs for anyy is consistent. This class reflects the phy-sicist's idea of gas pressure, p. Gas vol-ume, v, and temperature, t, can be de-30 MARCH 1956

Research Laboratories,Quaker Oats Company, Chicago, IllinoisReferences and Notes

1. S. Lindstedt, Acta Physiol. Scand. 27, 377(1953).2. S. Bergstrom and S. Lindstedt, Acta Chem.Scand. 5, 157 (1951).3. L. M. Henderson and E. E. Snell, J. Biol.Chem. 172, 15 (1948).4. Both were CfP grade products obtained fromCaliforniaFoundation for Biochemical Research,Los Angeles.5. P. B. Hamilton and R. A. Anderson, J. Biol.Cheni. 213, 249 (1955).6. A. D. McLaren and C. A. Knight, ibid. 204,417 (1953).7. J. L. Stokes et al., ibid. 160, 35 (1945).13 September 1955

What Are Variables and Constants?The notion variable has not in theliterature attained the degree of claritythat would justify the almost universaluse of that term without explanations.Recent investigations (1) have resolvedit into an extensive spectrum of mean-ings, some pertaining to reality as inves-tigated in science, others belonging tothe realm of symbols studied in logic-two altogether different worlds. As acorollary, these distinctions yield a clari-fication of the notion constant.Science and applied mathematics. Byquantity we mean a pair of which thesecond member (or value) is a numberwhile the first member (or object) maybe anything. We call a class of quanti-ties consistent if it does not contain two

quantities with equal objects and un-equal values. If p(y) denotes the pres-sure (in a chosen unit) of a gas sampley (2), then the pair [y, p (y)] is a quan-tity, and the class of all such pairs for anyy is consistent. This class reflects the phy-sicist's idea of gas pressure, p. Gas vol-ume, v, and temperature, t, can be de-30 MARCH 1956

fined similarly. Consistent classes ofquantities, such as p, v, and t, are whatscientists and mathematicians in appliedfields mean by variable quantities andwhat Newton called fluents. The class ofall gas samples will be referred to as thedomain of p; the class of all numbersp(y) as the range of p.A variable quantity whose range con-sists of a single number is said to be con-stant. An example is the gravitationalacceleration g at a definite point on theearth, defined as the class of all pairs[a, g (a)] for any object a falling in avacuum, where g(a) denotes the accel-eration of a. For any a, this value of gis found to be equal to one and the samenumber, g. (Throughout this paper, sym-bols for fluents are italicized and symbolsfor numbersrareprinted in roman type.)A less important, because less compre-hensive, example is the distance traveledby a specific car C while C is parked.(The distance traveled by C is the classof all pairs [t, m(g)] for any act ,t ofreading the mileage gage in C, wherem(lt) is the number read as the resultof Lt.In a plane, the coordinates relative toa chosen Cartesian frame are variablequantities whose domain is the class ofall points (3). The abscissax is the classof all pairs [ja,x(t)] for any point ar.Inthe equation of the straight line x - y = 3the 3 denotes the constant fluent consist-ing of the quantities (t, 3) of value 3 foreach point at.Of paramount importance are the con-sistent classes of quantities whose do-mains consist of numbers. An exampleis the class of all pairs [x, log x] for anynumber x > 0, called the logarithmicfunction or the function log. It is capableof connecting consistent classes of quan-tities-for example, y = log x along alogarithmic curve, and w = log v for anisothermic expansion of an ideal gas, if wdenotes the work in a proper unit. Thatis to say, y((n) =log x(Jt) and w(y) =log v(y) for any point Jt on the curveand any gas sample y pertaining to theprocess. Moreover, log connects func-tions-for example, the exponential withthe identity function, and cos with logcos. Because of their connective power,functions are omnipresent in science aswell as in mathematics (4). Variablequantities such as p and v (whose do-mains do not contain numbers or systemsof numbers) lack this power and there-fore are confined to special branches ofscience such as gas theory. Denying thesignificance of this difference would bedenying the role of mathematics as a uni-versal tool in quantitative science.Logic and pure mathematics. The

fined similarly. Consistent classes ofquantities, such as p, v, and t, are whatscientists and mathematicians in appliedfields mean by variable quantities andwhat Newton called fluents. The class ofall gas samples will be referred to as thedomain of p; the class of all numbersp(y) as the range of p.A variable quantity whose range con-sists of a single number is said to be con-stant. An example is the gravitationalacceleration g at a definite point on theearth, defined as the class of all pairs[a, g (a)] for any object a falling in avacuum, where g(a) denotes the accel-eration of a. For any a, this value of gis found to be equal to one and the samenumber, g. (Throughout this paper, sym-bols for fluents are italicized and symbolsfor numbersrareprinted in roman type.)A less important, because less compre-hensive, example is the distance traveledby a specific car C while C is parked.(The distance traveled by C is the classof all pairs [t, m(g)] for any act ,t ofreading the mileage gage in C, wherem(lt) is the number read as the resultof Lt.In a plane, the coordinates relative toa chosen Cartesian frame are variablequantities whose domain is the class ofall points (3). The abscissax is the classof all pairs [ja,x(t)] for any point ar.Inthe equation of the straight line x - y = 3the 3 denotes the constant fluent consist-ing of the quantities (t, 3) of value 3 foreach point at.Of paramount importance are the con-sistent classes of quantities whose do-mains consist of numbers. An exampleis the class of all pairs [x, log x] for anynumber x > 0, called the logarithmicfunction or the function log. It is capableof connecting consistent classes of quan-tities-for example, y = log x along alogarithmic curve, and w = log v for anisothermic expansion of an ideal gas, if wdenotes the work in a proper unit. Thatis to say, y((n) =log x(Jt) and w(y) =log v(y) for any point Jt on the curveand any gas sample y pertaining to theprocess. Moreover, log connects func-tions-for example, the exponential withthe identity function, and cos with logcos. Because of their connective power,functions are omnipresent in science aswell as in mathematics (4). Variablequantities such as p and v (whose do-mains do not contain numbers or systemsof numbers) lack this power and there-fore are confined to special branches ofscience such as gas theory. Denying thesignificance of this difference would bedenying the role of mathematics as a uni-versal tool in quantitative science.Logic and pure mathematics. Theformula

3- - (3 + 1) (3 -1)is a statement about specific numbersformula

3- - (3 + 1) (3 -1)is a statement about specific numbers

designated by 3 and 1. In the more gen-eral statementx' - 1 = (x + 1) - (x - 1) for anynumberx,the remark "for any number x" stipu-lates that, in the formula, x may be re-placed by the designation of any num-ber, for example, by 3 or '/5, each re-placement yielding a valid statementabout specific numbers. A symbol that,in a certain context and according to adefinite stipulation, may be replaced bythe designation of any element of a cer-tain class is what, following Weierstrass,logicians and pure mathematicians calla variable. The said class is called thescope of the variable. The scope of x inthe formula log x2 = 2 log x for anyx > 0 is the class of all positive numbers.In (? ^3)4=9, the symbol ? V3 is avariable whose scope consists of the twonumbers '3 and - '3.A variable whose scope consists of asingle number designates that numberand, in pure mathematics, is referred toas a constant. Examples include numer-als (1, 3, .. .); e, designating the baseof natural logarithms; and brief designa-tions of numbers with unwieldy symbols,such as 22'+ eee-abbreviations intro-duced for the purpose of just one dis-cussion involving repeated references tothat number. As such ad hoc constants,one customarily uses the letters a, b, c,. , which, just as x and y, serve asvariables in other contexts, for example,in the statement involving two variablesx aa= (x+a) (x -a)for anyx and any a.Because of its vicarious character,a vari-able may always be replaced, withoutany change of the meaning, by any other-wise unused letter, for instance, a by yor x by b.Variables whose scopes are classes ofnumbers are called numerical variables.The definitions of p and g make use ofvariables y and a whose scopes consistof gas samples and falling objects, re-spectively. Statements that are valid formany fluents are conveniently expressedin terms of fluent variables; for example,If z = log u, then dz/du = 1/ufor any two fluents u and z (1)In Eq. 1, u and z may be replaced byx and y: if y =log x (which holds alongthe logarithmic curve), then dy/dx =1/x; or by v and w: if w = log v (whichholds for an isothermic expansion), thendw/dv = 1/v. But u and z in Eq. 1 mustnot be replaced by numbers such as eand 1: although 1=log e is valid,dl/de = 1/e is nonsensical.Confusion in the literature. No cleardistinction has heretofore been made be-

designated by 3 and 1. In the more gen-eral statementx' - 1 = (x + 1) - (x - 1) for anynumberx,the remark "for any number x" stipu-lates that, in the formula, x may be re-placed by the designation of any num-ber, for example, by 3 or '/5, each re-placement yielding a valid statementabout specific numbers. A symbol that,in a certain context and according to adefinite stipulation, may be replaced bythe designation of any element of a cer-tain class is what, following Weierstrass,logicians and pure mathematicians calla variable. The said class is called thescope of the variable. The scope of x inthe formula log x2 = 2 log x for anyx > 0 is the class of all positive numbers.In (? ^3)4=9, the symbol ? V3 is avariable whose scope consists of the twonumbers '3 and - '3.A variable whose scope consists of asingle number designates that numberand, in pure mathematics, is referred toas a constant. Examples include numer-als (1, 3, .. .); e, designating the baseof natural logarithms; and brief designa-tions of numbers with unwieldy symbols,such as 22'+ eee-abbreviations intro-duced for the purpose of just one dis-cussion involving repeated references tothat number. As such ad hoc constants,one customarily uses the letters a, b, c,. , which, just as x and y, serve asvariables in other contexts, for example,in the statement involving two variablesx aa= (x+a) (x -a)for anyx and any a.Because of its vicarious character,a vari-able may always be replaced, withoutany change of the meaning, by any other-wise unused letter, for instance, a by yor x by b.Variables whose scopes are classes ofnumbers are called numerical variables.The definitions of p and g make use ofvariables y and a whose scopes consistof gas samples and falling objects, re-spectively. Statements that are valid formany fluents are conveniently expressedin terms of fluent variables; for example,If z = log u, then dz/du = 1/ufor any two fluents u and z (1)In Eq. 1, u and z may be replaced byx and y: if y =log x (which holds alongthe logarithmic curve), then dy/dx =1/x; or by v and w: if w = log v (whichholds for an isothermic expansion), thendw/dv = 1/v. But u and z in Eq. 1 mustnot be replaced by numbers such as eand 1: although 1=log e is valid,dl/de = 1/e is nonsensical.Confusion in the literature. No cleardistinction has heretofore been made be-tween numerical and fluent variables.Moreover, in the literature, numericalvariables and variable quantities, not-

547

tween numerical and fluent variables.Moreover, in the literature, numericalvariables and variable quantities, not-

547

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withstanding the profound differencesbetween them, are indiscriminately re-ferred to as "variables"; and the twoconcepts have actually been confused.For instance, numerous attempts (5)have been made to define gas pressureas a symbol that may be replaced byany value of pressure, thus as a numer-ical variable, p, whose scope is the rangeof what herein has been called the fluentp. But a fluent is not determined by itsrange. Could Boyle have connected pres-sure and volume on the basis of mereinformation about the ranges of p and v?It was by transcending these ranges andreferring to the domains, namely, bycomparing v(y) and p(y) for the samesample y, that he discovered v(y) =l/p(y) (in proper units) for any y of acertain temperature or, without refer-ence to a sample variable, v= 1/p.Neither formulation involves numericalvariables. Boyle's Law connects specificfluents.

Analogously, constant fluents havebeen identified with their numericalvalues even though what primarily in-terests the physicist in gravitational ac-celeration clearly is the fact that g is itsvalue for any a rather than the numberg as such.Numerical variables and variable quan-tities belong to worlds that are not onlydifferent but nonisomorphic. The formerare interchangeable, the latter are not:x2- 4y2= (x + 2y) * (x - 2y) and

y2- 4x2= (y+ 2x) . (y- 2x)for any x and y are tantamount. Butx-y=3 and y-x=3 are differentstraight lines and w = log v is incom-patible with, and not tantamount to,v =log w.

Only the consistent maintenance of allthese distinctions makes it possible toformulate mathematical analysis as wellas its applications to science as a systemof procedures following articulate rules(6). KARL MENGERIllinois Institute of TechnologyChicago, Illinois

References and Notes1. Compare K. Menger, "The ideas of variableand function," Proc. Natl. Acad. Sci. U.S. 39,

956 (1953) and "On variables in mathematicsand in natural science," Brit. J. Phil. Sci. 5,134 (1954). A paper "Random variables and thegeneral theory of variables" is in print Proc. 3rdBerkeley Symposium Math. Statistics (1955). Asystematic exposition of the new theory of vari-ables is contained in K. Menger, Calculus. AModern Approach (Ginn, Boston, 1955).2. The term gas sample means gas in a specificcontainer at a definite instant.3. Depending on whether a physical, a postula-tional, or a pure plane is under consideration,a "point" is a physical object (for example, anink dot in a paper plane), or an undefined ob-ject satisfying certain assumptions,or an ordered

withstanding the profound differencesbetween them, are indiscriminately re-ferred to as "variables"; and the twoconcepts have actually been confused.For instance, numerous attempts (5)have been made to define gas pressureas a symbol that may be replaced byany value of pressure, thus as a numer-ical variable, p, whose scope is the rangeof what herein has been called the fluentp. But a fluent is not determined by itsrange. Could Boyle have connected pres-sure and volume on the basis of mereinformation about the ranges of p and v?It was by transcending these ranges andreferring to the domains, namely, bycomparing v(y) and p(y) for the samesample y, that he discovered v(y) =l/p(y) (in proper units) for any y of acertain temperature or, without refer-ence to a sample variable, v= 1/p.Neither formulation involves numericalvariables. Boyle's Law connects specificfluents.

Analogously, constant fluents havebeen identified with their numericalvalues even though what primarily in-terests the physicist in gravitational ac-celeration clearly is the fact that g is itsvalue for any a rather than the numberg as such.Numerical variables and variable quan-tities belong to worlds that are not onlydifferent but nonisomorphic. The formerare interchangeable, the latter are not:x2- 4y2= (x + 2y) * (x - 2y) and

y2- 4x2= (y+ 2x) . (y- 2x)for any x and y are tantamount. Butx-y=3 and y-x=3 are differentstraight lines and w = log v is incom-patible with, and not tantamount to,v =log w.

Only the consistent maintenance of allthese distinctions makes it possible toformulate mathematical analysis as wellas its applications to science as a systemof procedures following articulate rules(6). KARL MENGERIllinois Institute of TechnologyChicago, Illinois

References and Notes1. Compare K. Menger, "The ideas of variableand function," Proc. Natl. Acad. Sci. U.S. 39,

956 (1953) and "On variables in mathematicsand in natural science," Brit. J. Phil. Sci. 5,134 (1954). A paper "Random variables and thegeneral theory of variables" is in print Proc. 3rdBerkeley Symposium Math. Statistics (1955). Asystematic exposition of the new theory of vari-ables is contained in K. Menger, Calculus. AModern Approach (Ginn, Boston, 1955).2. The term gas sample means gas in a specificcontainer at a definite instant.3. Depending on whether a physical, a postula-tional, or a pure plane is under consideration,a "point" is a physical object (for example, anink dot in a paper plane), or an undefined ob-ject satisfying certain assumptions,or an orderedpair of numbers.air of numbers.

4. Even if, following the suggestion of some math-ematicians, one called all consistent classes ofquantities "functions," one obviously wouldneed a special term for "functionally connect-ing functions" such as log.5. Compare, for example, R. Courant, Differentialand Integral Calculus (Interscience, New York,1954), vol. 1, p. 14.6. Compare K. Menger, Calculus. A Modern Ap-proach (Ginn, Boston, 1955).9 September 1955

Prenatal Diagnosis of Sex UsingCells from the Amniotic FluidIn most mammals, including human

beings, males normally have the sexchromosome constitution XY, and fe-males the sex chromosome constitutionXX (1). It has been shown in a varietyof tissues in human beings and someother species (2) that there is a sex dif-ference in the percentage of cells withchromocenters, especially those at thenuclear membrane; this presumably isdue to this difference in sex chromosomeconstitution in males and females. A de-termination of the percentage of cellswith chromocenters can therefore give,in sexually normal individuals, a diag-nosis of sex.

The present study was undertaken inorder to show whether, in human beings,such a diagnosis can be made beforebirth, not only for aborted fetuses fromwhich pieces of tissue can be removedfor examination, but also for viablefetuses by an examination of cells fromthe amniotic fluid. In order to establishwhether amniotic fluid contains cellssuitable for diagnosis, fluid was takenbefore delivery by puncture of the mem-branes from women in the ninth monthof pregnancy. The fluid was centrifuged,and the cells were smeared on slides,fixed in alcohol-ether, and stained withFeulgen and fast green (Fig. 1).Our analysis has shown that cells suit-able for the diagnosis are present, andan examination of 35 cases in the ninthmonth, which include those reportedpreviously (3), has given 35 correct diag-noses of the sex of the fetus.

It therefore seems that this method isparticularly reliable, especially sincethere appears to be no theoretical objec-tion to it. The only apparent exceptionthat occurs to us at present is the rarecase of an intersex in which the sexualphenotype does not correspond to thesex chromosome constitution. When oneis collecting the amniotic fluid it is, ofcourse, essential to avoid contaminationwith cells from the mother.

Amniotic fluid can be obtained fromviable human fetuses from 12 weeks toterm (4). We have found, from an ex-

4. Even if, following the suggestion of some math-ematicians, one called all consistent classes ofquantities "functions," one obviously wouldneed a special term for "functionally connect-ing functions" such as log.5. Compare, for example, R. Courant, Differentialand Integral Calculus (Interscience, New York,1954), vol. 1, p. 14.6. Compare K. Menger, Calculus. A Modern Ap-proach (Ginn, Boston, 1955).9 September 1955

Prenatal Diagnosis of Sex UsingCells from the Amniotic FluidIn most mammals, including human

beings, males normally have the sexchromosome constitution XY, and fe-males the sex chromosome constitutionXX (1). It has been shown in a varietyof tissues in human beings and someother species (2) that there is a sex dif-ference in the percentage of cells withchromocenters, especially those at thenuclear membrane; this presumably isdue to this difference in sex chromosomeconstitution in males and females. A de-termination of the percentage of cellswith chromocenters can therefore give,in sexually normal individuals, a diag-nosis of sex.

The present study was undertaken inorder to show whether, in human beings,such a diagnosis can be made beforebirth, not only for aborted fetuses fromwhich pieces of tissue can be removedfor examination, but also for viablefetuses by an examination of cells fromthe amniotic fluid. In order to establishwhether amniotic fluid contains cellssuitable for diagnosis, fluid was takenbefore delivery by puncture of the mem-branes from women in the ninth monthof pregnancy. The fluid was centrifuged,and the cells were smeared on slides,fixed in alcohol-ether, and stained withFeulgen and fast green (Fig. 1).Our analysis has shown that cells suit-able for the diagnosis are present, andan examination of 35 cases in the ninthmonth, which include those reportedpreviously (3), has given 35 correct diag-noses of the sex of the fetus.

It therefore seems that this method isparticularly reliable, especially sincethere appears to be no theoretical objec-tion to it. The only apparent exceptionthat occurs to us at present is the rarecase of an intersex in which the sexualphenotype does not correspond to thesex chromosome constitution. When oneis collecting the amniotic fluid it is, ofcourse, essential to avoid contaminationwith cells from the mother.

Amniotic fluid can be obtained fromviable human fetuses from 12 weeks toterm (4). We have found, from an ex-amination of the fluid obtained frommination of the fluid obtained from

Fig. 1. Photomicrographs of cells from theamniotic fluid (x 1500). (Top row) Nu-clei with a chromocenter at the nuclearmembrane; from female human fetuses inthe ninth month. (Bottom row) Nucleiwithout a chromocenter; from male hu-man fetuses in the ninth month.viable human fetuses in the sixth andseventh months, that a prenatal diagnosisof sex can be made at these stages. Wehave also found suitable cells in thefluid of an aborted 8-week-old humanembryo.It may be possible to apply thismethod for the prenatal diagnosis of sexto domestic animals. LEO SACHSDepartment of Experimental Biology,Weizmann Institute of Science,Rehovoth, Israel DAVIDM. SERRDepartment of Obstetrics andGynaecology, Rothschild-HadassahUniversity Hospital, Jerusalem, IsraelMATHILDE DANONDepartment of Experimental Biology,WeizmannInstitute of Science,Rehovoth, Israel

References1. 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 and M. L. Barr, Acta Anat. 21, 197(1954); J. L. Emery and M. McMillan, J.Pathol. Bacteriol. 68, 17 (1954); M. A. Graham,Anat. Record 119, 469 (1954); K. L. Moore andM. L. Barr, Lancet 269, 57 (1955); E. Mar-berger and W. 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); Lancet 269, 654 (1955);L. Sachs and M. Danon, Genetica, in press.3. D. M. Serr, L. Sachs, M. Danon, Bull. Re-search Council Israel 5B, 137 (1955).4. W. J. Dieckmann and M. E. Davies, Am. J.Obstet. Gynecol. 25, 623 (1933); H. Alvarezand R. Caldeyro, Surg. Gynecol. Obstet. 91, 1(1950).30 December 1955

Fig. 1. Photomicrographs of cells from theamniotic fluid (x 1500). (Top row) Nu-clei with a chromocenter at the nuclearmembrane; from female human fetuses inthe ninth month. (Bottom row) Nucleiwithout a chromocenter; from male hu-man fetuses in the ninth month.viable human fetuses in the sixth andseventh months, that a prenatal diagnosisof sex can be made at these stages. Wehave also found suitable cells in thefluid of an aborted 8-week-old humanembryo.It may be possible to apply thismethod for the prenatal diagnosis of sexto domestic animals. LEO SACHSDepartment of Experimental Biology,Weizmann Institute of Science,Rehovoth, Israel DAVIDM. SERRDepartment of Obstetrics andGynaecology, Rothschild-HadassahUniversity Hospital, Jerusalem, IsraelMATHILDE DANONDepartment of Experimental Biology,WeizmannInstitute of Science,Rehovoth, Israel

References1. 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 and M. L. Barr, Acta Anat. 21, 197(1954); J. L. Emery and M. McMillan, J.Pathol. Bacteriol. 68, 17 (1954); M. A. Graham,Anat. Record 119, 469 (1954); K. L. Moore andM. L. Barr, Lancet 269, 57 (1955); E. Mar-berger and W. 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); Lancet 269, 654 (1955);L. Sachs and M. Danon, Genetica, in press.3. D. M. Serr, L. Sachs, M. Danon, Bull. Re-search Council Israel 5B, 137 (1955).4. W. J. Dieckmann and M. E. Davies, Am. J.Obstet. Gynecol. 25, 623 (1933); H. Alvarezand R. Caldeyro, Surg. Gynecol. Obstet. 91, 1(1950).30 December 1955

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