Three women of mathematics.

Watson, Jane

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Readers of E. T. Bell's Men of Mathematics might conclude that the only woman mathematician of any note in history was the Russian Sonya Kovalevsky. This is not so, and as the number of women in all fields of mathematics is on the increase, it is appropriate to fill in some of the gaps in our knowledge of the contributions made to mathematics by women. The first two discussions are followed by examples which are suitable for secondary schools and illustrate the opportunity for combining mathematical history and lore with the curriculum. The third portrait leads into a delightful story with a moral for teachers.

The first woman mathematician of whom we have any record was Hypatia. She flourished about 400 AD in Alexandria and was the daughter of the mathematician Theon, who was noted for his work on Euclid's Elements. Theon was associated with the museum in Alexandria and undoubtedly gave Hypatia the best education possible for that day and situation. As none of her works survive we have no record of her achievements as an original mathematician. Her talents as a teacher, however, were well-known and she was the leader of the neo-Platonist school of philosophy. Her philosophical views in fact were what led to her demise, as she was brutally murdered in 415 AD by a fanatical Christian sect which feared her influence on the Roman Prefect of Alexandria, Orestes. Caught up in the political situation of her day, Hypatia would not compromise her principles and consequently suffered a martyr's fate. (1)

By the time of Theon and Hypatia, the golden age of Greek learning was drawing to a close. Already the great library at Alexandria had been burned by Roman soldiers and the only remaining library was sacked by a Christian mob in Hypatia's day.

Mathematics suffered along with the other disciplines. The earlier emphasis on geometry had declined and interest in algebraic methods had increased, perhaps due to the influence of the Babylonians.

Diophantus (c. 250 AD) represents the apex of Greek algebra. He wrote several works, including thirteen books which made up his famous Arithmetica. Only six of these books survived and according to one theory (that of Tannery) it was Hypatia who ensured their survival. According to other ancient writers Hypatia wrote a commentary on the Arithmetica (perhaps cut short by her untimely death?). This information combined with evidence that the seven lost books disappeared very early from the mathematical scene, leads to the conjecture that it was through Hypatia's hand that the only surviving material passed. She is also credited with a commentary on the Conics of Apollonius and some other works, all of which have been lost.

To see what interested Hypatia and would have been discussed in her classroom, let us consider the Arithmetica. In Diophantus' work we see the innovations which led the way in improving algebraic notation. The most ancient algebraic writers composed their problems and solutions in complete words with no abbreviations, which naturally hampered complicated logical thinking. The intermediate step between this and our modern symbolic notation was what is termed syncopated algebra, which was used by Diophantus, Hypatia and those who followed them for over one thousand years. Syncopated algebra consisted mainly of abbreviations. To signify an unknown (our x) to her students, Hypatia would have used a symbol similar to [??] which probably developed from a contraction of [alpha][rho], the first two letters of [TEXT NOT REPRODUCIBLE IN ASCII] the Greek word for number. The powers of the unknown were again denoted by abbreviations of the words they represented; hence, [x.sup.2] would have been [[DELTA].sup.[PSI]] from [TEXT NOT REPRODUCIBLE IN ASCII] (power) and [x.sup.3] would have been [K.sup.[PSI]] from [TEXT NOT REPRODUCIBLE IN ASCII] (cube). The presence of units was denoted by [??]. Addition was accomplished by juxtaposition and subtraction by [??], which meant to subtract everything following. Coefficients were placed following the unknown or its power. Thus using the Greek numerals [alpha] = 1, [beta] = 2, [gamma] = 3, etc., we can illustrate a typical expression used by Hypatia:

[K.sup.[PSI]] [alpha] [zeta] [gamma] [??] [[DELTA].sup.[PSI]] [epsilon] M [beta]

In our modern symbolism this would represent [x.sup.3] - 5[x.sup.2] + 3x - 2. As can be seen from this notation, only one unknown was allowed at any given time. Various devious methods were used to make it possible to solve complicated problems.

In Hypatia's classroom no zero, negative or irrational solutions were allowed. Hence, although subtraction was allowed, care was always taken to ensure that the subtrahend was smaller than the minuend. It was acknowledged that "a subtraction times a subtraction produced an addition" and that "an addition times a subtraction produced a subtraction" but this was only for use in expansions like (a-b) (c-d). The Arithmetica dealt with both determinate and indeterminate equations. The rule for solving the general quadratic equation was not to be found in the surviving books but from certain specific problems it appears that the method of solving [alpha][x.sup.2] + bx + c = 0 involved multiplying by a and completing the square much as we do today. Even when both roots were positive, only one was given (the one corresponding to the positive square root in the modern formula x = [-b [+ or -] ([square root of ([b.sup.2] - 4ac)]]/2a). The work with indeterminate equations was more extensive but the methods involved varied greatly and often a judicious guess or the assignment of a fixed value to an unknown would produce a solution. Generalisation as we would expect today, was not well developed and most problems were special cases with exact numbers supplied.

As an example of a determinate problem consider Problem 17 from Book I of the Arithmetica: Find four numbers, the sum of every arrangement three at a time being given; say, 22, 24, 27, and 20.

Today we would probably attack this problem by assigning the unknowns x, y, z, w, to the four numbers and setting up four equations in the four unknowns to solve. Without this freedom of notation Hypatia would have simply let x (or [zeta] in her notation) be the sum of the four numbers, the numbers then being x - 22, x - 24, x - 27, x- 20. Hence:

x = (x - 22) + (x - 24) + (x - 27) + (x - 20), or x = 31. The numbers were then 9, 7, 4, and 11.

Problems like the above can be easily introduced in today's classroom and they provide an excellent springboard for a discussion of some 'history of mathematics'. Some of the late nineteenth century and early twentieth authors are a good source of interesting problems (see Ball, 1893; Heath, 1921; A manual of Greek mathematics, 1931).

The following example of a quadratic problem shows how the setting of limits was used to obtain a rational solution, along with judicious selection within the limits. Problem 30 of Book V says:

   A man buys a certain number of measures of wine, some at 8 drachmae
   a measure, the rest at 5. He pays for them a square number of
   drachmae, such that if 60 be added to it, the resulting number is a
   square, the side of which is equal to the whole number of measures.

Find the number he bought at each price.

Using our modern notation but the reasoning of Hypatia and Diophantus, we let x be the number of measures of wine. The total cost is then [x.sup.2] - 60 which is also a square, say [(x - m).sup.2]. From this x = ([m.sup.2] + 60)/2m. Now [x.sup.2] - 60 (the total cost) must be divided into two parts such that 1 of one part plus 1/8 of the other part is x. Hence no (real) solution is possible unless 8x > ([x.sup.2] - 60) and 5x < ([x.sup.2] - 60). In this situation Diophantus concluded that x must be such that x is not less than 11 and not greater than 12. Within these limits we are then called to use the other condition that x = ([m.sup.2] + 60)/2m; hence if, 11 < x < 12, we have 22m < [m.sup.2] + 60 < 24m and m is constrained to be not less than 19 and not greater than 21. With no further questions the ancients put m = 20. This makes x = 11 1/2, [x.sup.2] = 132 1/4 and [x.sup.2] - 60 = 727 1/4 = (=[(17/2).sup.2]). Now we must divide 72 1/4 into two parts such that 1/5 of one part plus 1/8 of the other is 11 1/2. Letting the first part be 5y and the second 8(11 1/2 - y) we have 5y + 92 - 8y = 72 1/4 or y = 79/12. Hence, the man bought 79/12 five-drachmae measures and 59 eight-drachmae measures.

Most commentators agree that the death of Hypatia marked the end of Greek mathematics. From this point the interest of historians turns to the Arabic world for the link to the mathematical awakening which began in Europe in the late Middle Ages. Our interest must skip to the eighteenth century before we encounter another woman mathematician of note.

The eighteenth century produced several women talented in mathematics. Of these we choose the Italian, Maria Gaetana Agnesi (1718-1799). It is interesting to note that again, as in the case of Hypatia, it was the Christian Church which removed an extraordinary woman from the mathematical arena. Maria Agnesi was not a martyr like Hypatia, but a devout Christian with no worldly ambition, who gave up mathematics at a relatively early age (before the age of 44) to devote her life to Christian charity. A very precocious child, Maria spoke French at the age of five, was familiar with Greek, German and Spanish by eleven and published her Propositiones Philosophicae at twenty. She would have been popular with the feminist movement of today for at the age of nine she wrote a Latin treatise in defence of liberal studies as a proper exercise for members of her sex. Maria's father was a mathematician and a wealthy patron of culture who encouraged Maria to read up on scientific and philosophical questions and then debate them with the learned men of Bologna. Maria had a younger sister who was a composer and accomplished pianist. Foreign intelligentsia travelling in Italy regularly came to the Agnesi household to meet the talented sisters. Maria became seriously interested in mathematics at the age of nineteen. According to tradition she was a somnambulist. It is reported that on several occasions she went to her study in a somnambulistic state, made a light, and solved a problem she had left incomplete when awake. In the morning she would be surprised to fmd the solution carefully worked out on paper (Cajori, 1926).

Maria Agnesi lived in the century following Newton and Leibniz, the age when progress in mathematics was being made at a rapid pace. Infinitesimals, differences, and the Method of Fluxions were foremost in the minds of the leading mathematicians. The struggle for more precise definitions and a real understanding of the processes involved in the Calculus occupied many. Maria had studied and mastered the new ideas but she felt sympathy for the young students of mathematics who were finding it increasingly difficult to absorb so much material from so many branches of current mathematical thought. At the age of 30 in 1748, she published her most famous work, Instituzioni Analitiche ad Uso della Gioventu Italiana. In four parts, the book was an attempt to place the modern trends in mathematics in one volume. The work gained wide acclaim over Europe being translated into several languages. James Colson, the Lucasian Professor in Cambridge who translated Newton's work from Latin, was so impressed by this book that he taught himself Italian at an advanced age in order that British youth should have the good fortune of studying Maria Agnesi's book.

The first part of Instituzioni Analitiche dealt with finite quantities: the construction of loci, including conic sections, and simple problems of maxima, minima, tangents, and inflections. The second part introduced infinitely small quantities, which were defined as quantities so small that compared with the independent variable the proportion was less than any assigned quantity. Differences, which were variables tending to zero, and fluxions, which were finite rates of change, were treated as essentially the same thing, a remarkable step for that time. There were long discussions on maxima and minima, and flexure and evolutes. Part Three of the work discussed the integral calculus, then still a very new topic. She mentioned power series but there was no discussion of convergence. The last part was called The Inverse Method of Tangents and discussed some simple differential equations.

In 1748, Maria Agnesi was elected to the Bologna Academy of Science and in 1750, nominated to be the Professor of Mathematics at the University of Bologna. There is some debate among historians on this point. It appears that she did not accept the appointment, entering instead the cloistered life; although she very likely taught at the university during periods when her father was ill. Whether she ever taught or not, her clear and systematic written presentation of the mathematics of her day won praise from students and professionals alike.

The detailed study of higher order plane curves was popular in the eighteenth century. The curve which we now know as 'the witch of Agnesi' was first studied by Fermat in the seventeenth century.

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The curve may be constructed as follows. For a circle of radius a placed at the origin as shown, let OQ be any line through the origin forming the hypotenuse of a right triangle, with vertical leg AP being the line parallel to the y-axis passing through the intersection of OQ and y = 2a, and with horizontal leg QP being parallel to the x-axis. The witch is the locus of all points P. The Cartesian equation of the curve is then [x.sup.2]y = 4[a.sup.2] (2a - y). This equation can be easily derived and is a nice example of the use of parametric equations. If we let the coordinates of P be (x, y), then x = OT = 2a tan [theta].

Noting that: OQ = 2a cos [theta] y = MQ = OQ cos = 2a cos2 [theta]

Then using the trigonometric identity 1 + [tan.sup.2][theta] = [sec.sup.2][theta], we can obtain the Cartesian equation. The area between the witch and the x-axis is 4[pa.sup.2] and the volume of revolution about the x-axis is 4[p.sup.2][a.sup.3].

It is thought that confusion in the copying of a name accounts for the title, "witch', being given to this curve. Another Italian who studied the curve before Maria Agnesi named it "a vesoria', a Latin word meaning a rope that 'guides a sail'. Perhaps the curve was suggestive of such a rope. Apparently this was what Maria meant to use when she wrote "versiera' in her book. In Italian this word means 'the devil's grandmother', a female fiend or goblin who frightens little children. This is the word which John Colson translated as witch in his English edition of the Instituzioni Analitiche. Thus we came to know of 'the witch of Agnesi'.

In our attempt to select representative women mathematicians we must omit several important figures of the eighteenth and nineteenth centuries to consider Emmy Noether (1882-1935), to many, the greatest woman mathematician of all time. Like her two predecessors, Emmy was influenced by her mathematician father. Max Noether was a professor at the southern German university of Erlangen. Unlike Maria Agnesi who was apparently appointed to a professorship by the Pope, Emmy Noether had difficulty obtaining any position in a German university of the early twentieth century. She could almost stand as a martyr to the women's movement of today. When she attended Erlangen in 1900, women were only allowed to take courses with the consent of the professor; this was often withheld. It was possible however to sit for the university certificate whether or not courses had been completed, so a woman could receive her degree. This Emmy did and in 1908 she also received her doctorate from Erlangen. Unable to obtain a position at Erlangen, Emmy moved to Gottingen to join the famous group of Blumenthal, Hilbert and Klein. For six years she taught with no salary and for the first three years with no official position. Hilbert tried very hard to obtain a position for her and it is told of Hilbert that he once declared to opponents during a University Senate meeting, "I do not see that the sex of the candidate is an argument against her admission as Privatdozent. After all, we are a university not a bathing establishment." Finally she achieved the title of "nicht-beamteter ausserordentlicher Professor" (unofficial extraordinary professor) and later a small salary for lecturing in algebra. During her time at Gottingen, Emmy Noether collected about her a school of eager students to whom she was a great inspiration. A sincere, kind and considerate person, Emmy was short in stature and rather rotund. She had short-cropped hair and wore thick glasses. She loved walking but would often become so involved with conversation that she had to be protected by her companions from traffic. Of Jewish ancestry, Emmy suffered somewhat in the same vein as Hypatia and was forced to leave Germany in 1933. She emigrated to the United States where she resided at Bryn Mawr College, giving lectures there and at Princeton until her death in 1935.

Emmy Noether made many contributions to twentieth century mathematics. Some would credit her with moulding the whole style of thinking in the field of algebra. It is beyond the scope of this article to explore her research topics, her main interests lying in the general theory of ideals and the study of non-commutative algebras (Weyl, 1935). Then, as now, there was the conflict between the general and the specific, the abstract and the concrete, the axiomatic and the constructive. She protested vigorously against those who prophesied that the axiomatic method was coming on hard times because the material which it used was becoming exhausted. In her hands this was certainly not the case. She forged ahead in areas which are still being followed up today. It is evident from her work that she saw both sides of this argument but when pushed into a corner would defend her love of the abstract to the last.

A story is told by G. Polya which illustrates this (Polya, 1973). He recalls an argument they had many years ago on generalisation and specialisation:

   Emmy was, of course, all for generalization and I defended the
   relatively concrete special cases. Then once I interrupted Emmy:
   'Now, look here, a mathematician who can only generalize is like a
   monkey who can only climb up a tree'. And then Emmy broke off the
   discussion--she was visibly hurt.

Polya continues on to excuse himself by concluding that her reply should have been, "And a mathematician who can only specialize is like a monkey who can only climb down a tree". The point is that a real mathematician, like a real monkey, must be able to do both things to survive. We can look to Emmy Noether as one of the forces which kept the up monkey viable for this century. As Polya concludes:

   ...[t]here is, I think, a moral for the teacher. A teacher of
   traditional mathematics is in danger of becoming a down monkey, and
   a teacher of modern maths an up monkey. The down teacher dishing
   out one routine problem after the other may never get off the
   ground, never attain any general idea. And the up teacher dishing
   out one definition after the other may never climb down from his
   verbiage, may never get to solid ground, to something of tangible
   interest for his pupils.

Surely Emmy Noether would agree completely with this statement. In conclusion one might choose to discuss Hanna Neumann, as a mid-twentieth century exemplification of the qualities inherent in the three women chosen here. This has been done in M.F. Newman's excellent biography in The Australian Mathematics Teacher (Newman, 1973). Further, some of Hanna Neumann's ideas on education of first year university students and what equipment she expected them to have are found in the same number (Neuman, 1973). Both of these articles make most interesting reading.

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Published in Vol. 30, No. 5, 1974, John Veness (Ed.)

References and further reading

Ball, W. W. R. (1893). A short account of the history of mathematics (2nd ed.). London: Macmillan and Co.

Bell, E. T. (1945). The development of mathematics, (2nd ed.). New York: McGraw-Hill Book Company Inc.

Bell, E. T. (1937). Men of mathematics. New York: Simon and Schuster.

Cajori, F. (1926). A history of mathematics (2nd ed.). New York: The Macmillan Company.

Coolidge, J. L. (1951). Six female mathematicians. Scripta Mathematica ,17, 20-31.

Heath, T. L. (1921). A history of Greek mathematics, II. Oxford: Clarendon Press.

Heath, T. L. (1931). A manual of greek mathematics. Oxford: Clarendon Press.

Kimberling C. H. (1972). Emmy Noether. The American Mathematical Monthly, 79, 136-149.

Kimberling C. H. (1972). Addendum to 'Emmy Noether'. The American Mathematical Monthly, 79, 755.

Kramer, E. E. (1955). The main stream of mathematics. New York: Oxford University Press.

Kramer, E. E. (1970). The nature and growth of modern mathematics. New York: Hawthorn Books.

Kramer, E. E. (1957). Six more female mathematicians. Scripta Mathematica, 23, 83-95.

Larsen, H. D. (1968). The witch of Agnesi. The Journal of Recreational Mathematics, 1, 49-53.

Neumann, B. H. (1973). Byron's daughter. The Mathematical Gazette, 57, 94-97.

Neumann, H. (1973). Teaching first year undergraduates: Fads and fancies. The Australian Mathematics Teacher, 29, 23-28.

Newman, M. F. (1973). Hanna Neumann, a biographical notice. The Australian Mathematics Teacher, 29, 1-22.

Patterson, E. C. (1969). Mary Somerville. The British Journal for the History of Science, 4, 311-339.

Polya, G. (1973). A story with a moral. The Mathematical Gazette, 57, 86-87.

Reid, C. (1970). Hilbert. Berlin: Springer-Verlag.

Thomas a Kempis, Sister M. (1939). The walking polyglot. Scripta Mathematica, 6, 211-217.

Van der Waerden, B. L. (1935). Nachruf auf Emmy Noether. Mathematische Annalen, 111, 469-476.

Weyl, H. (1935). Emmy Noether. Scripta Mathematica, 3, 201-220.

(1) Some insight into the times in which Hypatia lived may be obtained from Charles Kingsley's historical novel Hypatia (1853).