"Life is an unfoldment, and the further we travel the more truth we can comprehend. To understand the things that are at our door is the best preparation for understanding those that lie beyond."
"All formal dogmatic religions are fallacious and must never be accepted by self-respecting persons as final."
"Reserve your right to think, for even to think wrongly is better than not to think at all."
"Fables should be taught as fables, myths as myths, and miracles as poetic fancies. To teach superstitions as truths is a most terrible thing. The child mind accepts and believes them, and only through great pain and perhaps tragedy can he be in after years relieved of them. In fact, men will fight for a superstition quite as quickly as for a living truth --- often more so, since a superstition is so intangible you cannot get at it to refute it, but truth is a point of view, and so is changeable."
Hypatia of Alexandria is considered to have been the first woman to write on the subject of mathematics. Hypatia lived and worked in the Egyptian city of Alexandria which was situated near one mouth of the Nile River. Alexander the Great founded the city in about 330 B.C. and after his death in 323 B.C., Ptolemy became the ruler of Egypt. Ptolemy chose Alexandria to be the capital of his kingdom and founded an institution of higher learning there, which was known as the Alexandrian Museum. Scholars from all over the world were invited to come and teach at the Museum. It is believed that Euclid, the famous Greek mathematician, was the first professor of mathematics at the Museum. When Hypatia was born in 370 A.D., Alexandria was a cosmopolitan center where scholars from all the civilized countries gathered to exchange ideas. Hypatia's father, Theon, was a distinguished mathematician and astronomer at the Museum and so Hypatia grew up in an atmosphere of learning and exploration. Theon was an unusually liberated person in a time when men dominated the intellectual world. He encouraged his gifted daughter to develop her mind and, thus, helped her acheive academically what no woman previously had. In fact, Theon closely supervised Hypatia's education and, acting as her tutor, he passed on to Hypatia his own love of math's beauty and logic. In addition to her mathematics training, Hypatia also received a thorough formal education in the arts, literature, science, and philosophy. According to legend, Theon was determined that Hypatia develop into a "perfect human being," which is remarkable since during this age women were often considered to be less than human. As part of his plan, Theon established a regimen of physical training for Hypatia to ensure that her body would be as healthy as her well-trained mind. Hypatia was also instructed by her father not to let any rigid system of religion take possession of her life and exclude the discovery of new scientific truths. Theon told her to "reserve your right to think, for even to think wrongly is better than not to think at all" (Hubbard p.82). As a result of her father's influence, Hypatia became an outspoken supporter of Greek scientific rational thought. Her loyalty to this school of thought, however, would eventually lead to her death. As a part of her extensive education, Hypatia traveled to Italy and Athens where she was a student at the school of Plutarch the Younger. When she returned to Alexandria, Hypatia became a teacher of mathematics and philosophy. Students from around the world converged on Alexandria to attend her lectures. Also, Hypatia's home became an intellectual center where scholars gathered to discuss scientific and philosophical questions. It is belived that Hypatia never married, choosing instead to devote her life to intellectual pursuits. There is evidence that she was regarded as physically beautiful and dressed in the tattered cloak favored by the academics of antiquity. She gave many public lectures on the writings of Plato and Aristotle and carried on mathematical discussions in the center of the city. Hypatia was reportedly not hesitant to speak her mind, even in the presence of males, because she had great confidence in her own intelligence and abilities. Although Hypatia never became the "perfect human being" her father had envisioned, she was a versatile and charismatic teacher who was beloved by her students and respected by the intellectual community. Though Hypatia gained respect and popularity through her intellectual accomplishments, she was still vulnerable to the religious war that was raging in Alexandria. Her adherence to the doctrine of Greek scientific rationalism, which was favored by the academic world, was the main factor that resulted in her murder. The Roman empire had gained control of Alexandria in 30 B.C. and during Hypatia's lifetime, the Romans were converting to Chritianity. Hypatia, however, refused to convert and remained loyal to her Greek religious beliefs. The Christians were hostile to these beliefs, which they considered to be pagan ideas, and even regarded them as the cause of the gradual weakening of Roman character. When Cyril became the patriarch of Alexandria in 412 A.D., he began a program of oppression against anyone he believed challenged the Christian authority in Alexandria. Hypatia, as a supporter of Greek scientific rational thought, could not be tolerated according to Cyril. In March of 415 A.D., a group of Christian zealots led by Peter the Reader seized Hypatia from her chariot as she was returning home from the Museum. They dragged her into the cathedral of Alexandria, stripped her and proceeded to dismember her with sharp shells and burn the pieces of her corpse. Hypatia's violent death has come to mark the end of the age of great Greek mathematics. The dominance of the Alexandrian Museum ended in 641 A.D. when the Arabs invaded and destroyed Alexandria. Subsequently, Western mathematics entered a dormant period that was to last about a thousand years. Hypatia is frequently the only woman mentioned in histories of mathematics. Although her writings have all been lost, numerous references to them exist and some information on her accomplishments comes from the surviving letters of her pupil, Synesius of Cyrene. There is no evidence of research mathematics on the part of Hypatia and it is believed that most of her writings originated as textbooks for her students. In her writings, Hypatia tried to enhance her students' understanding of mathematical classics while maintaining their interest. Hypatia authored a treatise called ON THE CONICS OF APOLLONIUS that was a popularization of Apollonius' own work on conic sections. Apollonius was a third century B.C. Alexandrian geometer who originated epicycles and deferents to explain the irregular orbits of the planets. Like her Greek ancestors, Hypatia was fascinated by conic sections which are the geometric figures formed when a plane is passed through a cone. After Hypatia's death, conics were neglected by mathematicians until the beginning of the seventeenth century. The letters of her pupil, Synesius, contain Hypatia's designs for several scientific instruments including a plane astrolabe. The plane astrolabe was used for measuring the positions of the stars, planets, and the sun. These letters suggest that Hypatia probably lectured on simple mechanics and astronomy as well as mathematics and philosophy. Hypatia also reportedly developed an apparatus for distilling water, an instrument for measuring the level of water, and a graduated brass hydrometer for determining the specific density of a liquid. Hypatia's most significant work, however, was in algebra. She wrote a commentary on the first six books of Diophantus' thirteen book ARITHMATICA. A commentary in Hypatia's era consisted of a rewriting of the entire text including any original problems, however, additional problems as well as alternative solutions to problems were frequently added by the commentator. Since the commentator usually didn't distinguish his or her additions from the author's work, when the commentaries were later recopied by scribes, the commentator's additions were incorporated into the new manuscript. Unfortuneately, Diophantus' original manuscript of ARITHMATICA and Hypatia's commentary have both been lost. The SUDA LEXICON, which is a tenth century encyclopedia that drew on other sources, mentions Hypatia's commentary on ARITHMATICA. This is the first mention in any source of so ancient an edition of Diophantus' work. The existing Greek manuscript of the ARITHMATICA possesses all the characteristics of a commentary made during the decline of Greek mathematics. Thus, due to the citation in the SUDA LEXICON and the evidence in the Greek text, it is theorized that all existing manuscripts of ARITHMATICA derived from the common source of Hypatia's commentary. If this is true, then some of Hypatia's work may appear in the existing manuscripts. The theory that Hypatia's commentary served as the source for existing ARITHMATICA texts is also supported by the fact that there are no existing manuscripts of Books 7 through 13 of ARITHMATICA. Since Hypatia's commentary didn't include these books, then those manuscripts that are based on her commentary would also not include them. Thus, due to Hypatia's work, much of the work of Diophantus has survived for the benefit of the mathematics world. Diophantus lived and worked in Alexandria in the third century A.D. and has been called the "father of algebra". He developed indeterminate or so-called Diophantine equations which are equations with multiple integer solutions. An example of a Diophantine equation is 5n + 10d + 25q= 100 which is the equation used to determine the various ways to make change for a dollar using nickels, dimes, and quarters. In the equation, n=number of nickels, d=number of dimes, q=number of quarters and the only possible solutions are when n, d, q are non-negative integers. Diophantus was also the first writer to make an effort toward developing a symbolism for the powers of algebraic equations. He developed a system of symbols to represent the various powers of the unknown. Diophantus wrote his algebraic equations similar to the way we do today except that he used his symbols for the variable and the Greek numeral system in which letters represented numbers. (Refer to the handout given in class for an example of what an equation written by Diophantus would have looked like.) The following is a problem from a fourteenth century manuscript of ARITHMATICA: Find two numbers such that their sum is equal to 20 and the difference of their squares is 80. An algebraic solution to this problem according to Diophantus that is included in the manuscript is: Let there be x+10 10-x Squares, x^2 + 20x + 100 x^2 + 100 - 20x Difference of squares, 40x = 80 Division, x = 2 whence x+10 = 12 10-x = 8 In this method, Diophantus only used one unknown and then represented the two variables by equations involving this unknown. Since the sum of the two variables is 20, he assumed that one variable was greater than half the sum (x+10), and the other variable was less than half the sum (10-x). By choosing to represent the variables by these equations, when the difference of their squares was computed, all terms cancelled except the term involving the first power of the unknown. Then by setting this to the given value of the difference of the variables' squares, the value of the one unknown (x) was found. Lastly, by substituting the value of the unknown into the equations representing the variables, the variables themselves were found. Since the original manuscript of Diophantus' ARITHMATICA was lost and all existing manuscripts are theorized to originate from Hypatia's commentary, it is possible that the above method that is attributed to Diophantus could actually be Hypatia's method of solution. It is known that Hypatia's commentary included many new problems and subsequently these problems were probably incorporated into the existing ARITHMATICA manuscripts. One of the most likely problems to have come from Hypatia is the following "student exercise" at the beginning of Book 2. Solve the pair of simultaneous equations where a and b are known: x - y = a, x^2 - y^2 = (x - y) + b Hypatia probably meant for this problem to be solved using the same method that was attributed to Diophantus in the preceeding problem. Thus, the solution is: Let there be x= z+(1/2)a y= z-(1/2)a Squares, z^2 + az + (1/4)a^2 z^2 - az + (1/4)a^2 Difference of squares, 2az = (x - y) + b = a + b Division, z = (a + b)/(2a) whence x= (a+b)/(2a) +(1/2)a y= (a+b)/(2a) -(1/2)a The solution of this problem involved first representing the two variables (x, y) in terms of only one unknown. This was done by assuming that one variable (x) was greater than half the difference of the two variables and the other variable (y) was less than half the difference of the two variables. Thus, x could be represented by z+(1/2)a and y could be represented as z-(1/2)a. Next, when the difference of the variables' squares was calculated, by virtue of the equations used to represent them, all terms cancelled except the term involving the first power of the unknown, z [(z^2+az+(1/4)a^2) - (z^2-az+(1/4)a^2) = 2az]. Then setting this equal to the given value of the difference of the variables' squares [2az = a+b], the value of z was found [z= (a+b)/(2a)]. Finally, the values of the two variables was determined by plugging the value of the unknown, z, into the equations used to represent the variables. In conclusion, Hypatia was a renowned teacher who had a vast knowledge on a wide range of topics including mathematics, philosophy, and astronomy. Although her writings were mostly commentaries on the work of others, the math community owes Hypatia a great debt. It is through her commentaries that the work of great mathematicians such as Diophantus has survived. Regardless of the content of Hypatia's writings, the fact that they existed is sufficient to establish her as the first woman known to have written on mathematical subjects. Exercise: Solve the following pair of simultaneous equations using the algebraic method attributed to Diophantus: x - y = 16, x^2 - y^2 = (x - y) + 112