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Erwin Schrödinger (pronounced "SHROYdihng-ur"), an only child, was born to Rudolf Schrödinger and Georginer Bauer, a daughter of Rudolf's chemistry professor. Schrödinger's father, the owner of a successful oil cloth factory; was a highly gifted man with a broad education and broad interests that he pursued throughout his life. The young Erwin was first educated at home, taking lessons twice a week from a visiting elementary school teacher. He attributed his interest in scientific, literary, and philosophical matters to his father, however, whom he would later describe as "his friend, teacher, and tireless partner in conversation." At the age of eleven, Schrödinger entered the Academische Gymnasium and he studied mathematics and physics, as well as classical languages and German poetry. After his graduation in 1910, Schrödinger stayed on as an assistant at the university's Second Physics Institute until the outbreak of World War I, when he was mobilized into the Imperial Army. He spent the three years following the war studying and teaching in Vienna, Jena, Stuttgart, and Breslau. Shortly after his arrival in Breslau in 1921, he accepted an offer from the University of Zurich to assume the professorship formerly held by Albert Einstein and Max von Laue. Schrödinger remained in Zurich for six years, enjoying the friendship of his colleagues--among whom were the mathematician Hermann Weyl and the physicist Peter J. W. Debye--and pursuing a number of problems in theoretical physics. He wrote papers dealing with specific heats of solids, problems of thermodynamics from a statistical point of view, and atomic spectra. He also studied the theory of color vision and wrote a paper on color blindness. His great discovery; Schrödinger's wave equation, was made at the end of this period, during the winter of 1925-1926. INSET: Erwin Schrödinger.

Edwin Ray Lankester, the son of a respected microscopist who cofounded the Quarterly Journal of Microscopical Science, met many scientists as a child. As a schoolboy, he attended popular lectures given by Thomas Henry Huxley, who became his mentor, and, in 1861, he published his first paper, a letter to The Geologist on a fossil animal. In the mid-1860's, Lankester published several papers on earthworms, attracting the attention of Charles Darwin. After receiving a Radcliffe Traveling Fellowship, Lankester made microscopical studies of marine species in Naples, Italy, and learned German laboratory techniques and methods of teaching. In 1872, he earned his master's degree. At this time, science education was very weak at both Oxford and Cambridge Universities. Lankester became convinced that science education in England had to be reformed on the German model. At Oxford, he petitioned to teach in the museum but was refused. At University College, London, however, he taught in the zoological laboratory in the museum, where he emphasized practical laboratory work. His equipment included aquariums for saltwater and freshwater species. Lankester's efforts at University College turned it into a major center for teaching and research into evolutionary relationships. His studies in Naples instilled in Lankester the desire to establish a laboratory to study marine species in England. Lankester's public addresses played an important role in generating popular and official interest, leading to the founding of the Marine Biological Association in 1884. In 1888, its laboratory at Plymouth opened and was the only English facility able to participate in an international study of the North Sea from 1899 to 1901.In 1882, Lankester discovered a blood parasite in frogs that scientists later recognized as a relative of the parasite that causes malaria. He also rediscovered in frogs a parasite related to those causing sleeping sicknesses. INSET: Sir Edwin Ray Lankester.

Niels Kaj Jerne (pronounced "YEHR-neh") was born on December 23, 1911, in London, England. Shortly after his birth, his family moved to Denmark and later to the Netherlands, where Jerne earned his bachelor's degree at the age of sixteen. He studied physics for two years at the University of Leiden and, twelve years later, entered the University of Copenhagen to study medicine. While working for his doctorate, Jerne also held a research position at the Danish State Serum Institute. His doctoral research dealt with the strength of interaction between antigens (foreign substances such as viruses, bacteria, and toxins) and antibodies (chemicals produced by immune cells to defend against antigens). He showed that antibody binding strength increases with every exposure to antigen. As a research fellow at the California Institute of Technology (Caltech), Jerne published a landmark paper entitled "The Natural Selection Theory of Antibody Formation," in which he proposed a revolutionary theory explaining an animal's ability to produce antibodies against a virtually limitless array of antigens. This paper provided the basis for the now universally accepted clonal selection theory of Sir Frank Macfarland Burnet. From 1956 to 1962, Jerne serve as chief medical officer of the World Health Organization (WHO) and then spent four years teaching at the University of Geneva, the University of Pittsburgh, and the J. W. Goethe University. In Pittsburgh, Jerne and Albert Nordin developed the Jerne plaque assay, a now-standard method for counting antibody-producing cells. In 1966, Jerne became director of the Paul Ehrlich Institute in Frankfurt, Germany, where he stayed until 1969. He left to become the founding director of a new institute dedicated to basic research in immunology. INSET: Niels K. Jerne.

Ejnar Hertzsprung was born in Frederiksberg, Denmark, in 1873. His father, Severin Hertzsprung, had studied astronomy at the University of Copenhagen but decided that he could not support his family well by pursuing science. Therefore, he took a position with the Danish government, rising rapidly to director of the state life insurance company. Severin retained his love for astronomy and mathematics, however, and Ejnar became interested in these sciences as well. Although he studied chemical engineering and worked as a chemist after his graduation from college, Ejnar Hertzsprung decided to make astronomy his life's work. In 1902, he began work at the observatory of the University of Copenhagen and at the Urania Observatory in Frederiksberg. In 1905 and 1907, Hertzsprung published two papers that are now considered classics. The title for both was "Zur Strahlung der Sterne," which translates as "on the radiation of the stars." These papers dealt with spectroscopy, the study of which colors (or wavelengths, in the language of physics) make up the light from glowing objects. Hertzsprung showed that stars that have a particular kind of light spectrum are more luminous than the rest. His research laid the foundation for methods of finding the distances to far-off stars and galaxies. Hertzsprung's extensive studies showed him that stars can be classified into two overall categories. He knew the absolute brightness of thousands of stars and from their spectra could determine the temperature of each. In 1911, Hertzsprung made a plot to show the relation of the brightness (or magnitude, in the language of astronomers) to the temperature. On such a plot, stars fall into two overall categories. Most stars fall on what is called the main sequence area of the graph, but some, such as blue giant stars and red dwarf stars, fall on a much different part of the plot. INSET: Ejnar Hertzsprung.

Luigi Galvani (pronounced "gahl-VAH-nee") was born in 1737 in Bologna, Papal States (Italy), to an old and prosperous family. The Galvani family had many ancient ties to the city of Bologna and to its university, and Luigi was never inclined to live elsewhere. Galvani studied medicine at the University of Bologna, obtaining degrees in medicine and philosophy there in 1759. The first years of his professional career were divided between medical practice, research on anatomy; and lecturing in medicine. In 1762 and 1767, he published papers on anatomical studies; the first was on skeletal structure and function, and the latter described the kidneys of birds. In the 1770's, Galvani's interest shifted from anatomy to what is now called physiology: He wanted to understand the function of nerves and muscles. Earlier researchers had described the effect of electrical discharges (sparks) on muscles, and, in 1772, Galvani presented a paper on this subject to the Institute delle Scienze (science institute) of Bologna. INSET: Luigi Galvani.

The early career of Louis-Jacques-Mandé Daguerre (pronounced "dah-GEHR") was as an artist at the Paris Opera in France. He was apprenticed to the chief designer in 1803 and later worked as an assistant to the panorama painter. After his successful exhibit at the Salon in 1814, he worked as an independent stage designer. His set designs were widely acclaimed. In 1822, Daguerre formed a partnership with Charles Marie Bouton, with whom he developed the Diorama. The Diorama consisted of massive pictures painted on both sides of translucent, or transparent, linen. These pictures could be lit from either the top or the rear in order to show the front or back picture. By controlling the lighting, different seasons or times of day were represented. The result was a startling visual effect that fascinated audiences. Bouton and Daguerre established theaters for the Diorama in both Paris and London. In order to create the paintings, Daguerre used a camera obscura, or dark room. The camera obscura was a pinhole camera that used light falling through a small opening to project an image on a screen or sheet of paper. Daguerre became fascinated by the idea of permanently fixing the camera obscura image on paper. In 1826, Daguerre learned of research that Joseph-Nicéphore Niépce was conducting on chemically fixing images. Niépce had already fixed outlines of objects on light-sensitive plates. Daguerre had not yet been successful in his experiments. Niépce and Daguerre became partners in 1829. Niépce had discovered that reverse (or negative) images could be produced using silver chloride to fix the image. He abandoned the process in order to work on producing a direct positive. Daguerre developed a method to reduce exposure time and produced a better lens. INSET: Jacques Daguerre.

The father of Charles-Augustin Coulomb (pronounced "koo-LOHM") was an army officer-turned-tax collector who came from an important family in Montpellier. His mother, Catherine Bajet, came from a wealthy family. (The family name is sometimes given as "de Coulomb," but the use of "de," which implies nobility, is unwarranted. Although the Coulomb family was well connected, no firm evidence exists for a claim to nobility and Coulomb himself never used or claimed a right to such a title.) While he was young, Coulomb's family moved from his birthplace at Angoulême to Paris. There, Coulomb attended lectures at the local colleges. Upon losing his fortune through financial speculation, his father returned to Montpellier. Sometime later, when Coulomb disagreed with his mother about career plans, he also left Paris and joined his father in Montpellier. Coulomb joined the Société des Sciences de Montpellier in March, 1757, and delivered several papers on mathematics and astronomy over the next two years. Coulomb returned to Paris in 1758 for tutoring and then entered the École du Génie, an engineering school in Mézières in February, 1760. He was graduated in November, 1761, with the rank of first lieutenant in the Royal Corps of Engineers. After spending two years at his first assignment in Brest, he was transferred to Martinique in the Windward Islands of the Caribbean. In Martinique, Coulomb supervised several hundred workers as they constructed Fort Bourbon and rebuilt other forts destroyed or damaged by British troops during the Seven Years' War (or the French and Indian War, as it is called in North America). He became seriously ill in Martinique, however, and never fully recovered. In poor health, Coulomb returned to France at the age of thirty-six. In 1773, he published a paper on the strength of materials as related to architecture. INSET: Charles-Augustin Coulomb.

Gustave-Gaspard de Coriolis (pronounced "koh-ryoh-LEES") was born in Paris, France, on May 21, 1792, to a family of nobles. His father was an army officer loyal to Louis XVI. The family fled Paris during the French Revolution, and Coriolis' father became an industrialist. In 1808, the young Coriolis entered the Napoleonic École Polytechnique, which was noted as a training ground for civil servants. At the time of his graduation, he ranked second in his class. After graduation, Coriolis served in the corps of engineers of the École des Ponts et Chaussées, the department of bridges and highways. His poor health and his family's need for a substantial source of income, however, forced Coriolis to leave his post in 1816. In 1816, Coriolis became a tutor in analysis at the École Polytechnique, which would lead to further teaching positions. He continued to teach until 1838, when he became director of studies at the École Polytechnique. Coriolis was appointed to the chair of mechanics at the École Centrale des Arts et Manufactures in 1829. He also published his first book in the same year. Du calcul de l'effet des machines dealt with the definition, calculation, and application of mechanical power. In 1830, when the great mathematician Augustin-Louis Cauchy was exiled, Coriolis was offered Cauchy's chairmanship at the École Polytechnique. Coriolis declined the offer, preferring to continue his work with mechanics. Subsequently, he published a paper on accelerations and the transmission of work that appeared in the school's journal. A second paper followed in 1832. Two years after declining the chair position at the École Polytechnique, Coriolis agreed to work with Claude-Louis-Marie-Henri Navier at the École des Ponts et Chaussées. With Navier, Coriolis engaged in work on applied mechanics. INSET: Gustave-Gaspard de Coriolis.

A biography of Indian-born British photorealist painter Michael Leonard is presented. Topics covered include his birth in Bangalore in 1933, education at St. Martin's School of Art in London, England, and his preferred media of alkyd oil paint on Amazonite and pencil on paper. Also mentioned are his homoerotic figure studies of males in different states of undress, revisionist view on the traditional nude, and his response to the AIDS crisis through his art.

Samuel Stanley Wilks was the son of Chance C. and Bertha May Gammon Wilks. The family owned a farm near Little Elm, Texas. Wilks began his education in a one-room schoolhouse and later attended high school in Denton. During his final year of high school, he skipped study hall in order to attend a mathematics course at nearby North Texas State Teachers College. He continued studies at the college, earning an a degree in architecture in 1926. Wilks, however, believed that his eyesight was too poor for a career in architecture, so he began graduate study in mathematics at the University of Texas. During his first year in Austin, Wilks taught high school mathematics. The following year, he was named a part-time instructor in mathematics at the University of Texas, and he received an M.A. in mathematics in 1928. He remained as a full-time instructor for another year, until he received a fellowship to attend the University of Iowa. His dissertation, "On the Distributions of Statistics in Samples from a Normal Population of Two Variables with Matched Sampling of One Variable," was completed in 1931 and published the following year. He received a Ph.D. in mathematical statistics from Iowa and married Gena Orr of Denton. Wilks was unable to obtain permanent employment. He received fellowships from the National Research Council that allowed him to work at Columbia University, University College in London, and Cambridge University. Wilks met and worked with some of the most influential statisticians of the time, and he began to publish a number of important papers on multivariate analysis. Eventually, Wilks received an offer to teach mathematics, and later statistics, at Princeton University. INSET: Samuel Stanley Wilks.

Vito Volterra (pronounced "vahl-TEHR-raw") was the only child of Angelica and Abramo Volterra. When his father died when Vito was two years old, his mother moved in with her brother, Alfonso Almagià, a bank employee in Florence. Alfonso treated Vito well, but the family was always poor. Vito's interest in mathematics and science became apparent early. At fourteen, he learned differential calculus from a textbook, then integral calculus without a book. His family wanted him to learn a trade, but a relative recognized his interest and ability in mathematics and science and encouraged him to continue those studies. Volterra moved quickly through the University of Pisa, publishing several important papers as a student. He was employed by the university upon graduation and became a full faculty member at twenty-three. He had already made important contributions to mathematics. Once established at Pisa, Volterra sent for his mother, who lived with him for the rest of her life. Volterra married Virginia Almagià after he moved to Rome; they had six children. The two basic operations of the calculus are differentiation and integration. Both are used to solve mathematical relationships that are often related to problems in physics, engineering, economics, biology, and other fields. In fact, the calculus is often essential for the solution of practical problems. The fundamental methods of both operations were well known before Volterra was born, but he developed techniques for solving more complex problems. He discovered methods for working out complex differential equations to solve relationships that came to be called functionals. He worked out new methods of integration. INSET: Vito Volterra.

Alan Mathison Turing was the son of Julian Mathias Turing, an official in the Indian Civil Service, and Ethel Sara Stoney. Unwilling to subject their son to the rigors of life in India, the Turings arranged for Alan to lodge with a retired military officer and his wife, although they frequently returned to England to spend time with Alan and his brother, John. Alan Turing was admitted to a private day school in 1918 and to boarding school in 1922. He entered Cambridge University in 1931 and was elected a Fellow of King's College after submitting a dissertation. Mathematical logic was in crisis at the beginning of the twentieth century. Set theory had suggested that all mathematical relationships could be expressed as membership in sets and relations between sets. This idea was threatened by the discovery of serious paradoxes inherent in the concept of membership. As British logician Bertrand Russell had pointed out, a set, such as the set of all infinite sets, could belong to itself, but the set of all sets not belonging to themselves could neither be said to belong to itself nor be said not to belong to itself. One response to the crisis was to try to establish mathematical notions on the basis of statements in symbolic form. In this form, all the true mathematical statements could be derived by manipulating symbols according to explicit rules. Proving that this was so constituted the Entscheidungsproblem, or decidability problem, posed by the great German mathematician David Hilbert. Turing published a paper on computable numbers in which he described an idealized machine, now called a Turing machine, the forerunner of the digital computer, that could execute computations. This would make it possible to equate a statement about whole numbers having particular properties to a claim that a certain machine would finish computing given a certain string of symbols as input. INSET: Alan Mathison Turing.

On August 18, 1685, Brook Taylor was born to the life of an English gentleman. Because his parents, John and Olivia Taylor, were members of the aristocracy, Brook Taylor was well educated. Even as a student, he showed exceptional talent in mathematics and physics, as well as promise in art, music, and writing. In 1701, he entered St. John's College at Cambridge University and was graduated with a bachelor of law degree in 1709. By the time that he was twenty-three, Taylor had already solved a major problem in physics about vibrating strings. On the strength of this work and other papers presented to the Royal Society of London, he was elected a Fellow to the society at the age of twenty-seven. Taylor's thirtieth year was a significant milestone, for in that year he published two groundbreaking books. The first, Methodus incrementorum directa et inversa (1715), dealt with the calculus, the preeminent field of mathematics during Taylor's era. INSET: Brook Taylor.

Niccolò Fontana Tartaglia (pronounced "tawr-TAWL-yaw") was born to poor parents in Brescia, Italy; in 1500. His family name was Fontana, and his father was a postal messenger in Brescia. Niccolò was present at the taking of this city by the French in 1512. He, his father, and many others sought safety in a cathedral, but the soldiers found them and a massacre followed. His father was murdered and young Niccolò suffered a split skull and severe saber cuts about the face and throat. When his mother found him alive in the cathedral, she carried him home but was unable to afford medical treatment. Under the care of his mother, the boy recovered. The damage inflicted upon his palate, however, caused a speech impediment that resulted in his nickname "Tartaglia," which means "stammerer" or "stutterer." Tartaglia's formal education was very limited. His mother had saved a sum of money to hire a writing master to tutor her son. Unfortunately, the money ran out after about fifteen days. Refusing to end his education at this point, he stole his teacher's notebook and taught himself to read and write. When he and his mother could not afford to buy paper, Tartaglia would often go to the cemetery, where he could use the tombstones as writing slates. At the end of the Middle Ages, a renewed interest in scientific investigation took place. Universities were located in many of the larger cities. A university's prominence depended on the quality of its lecturers, which was determined by public debates and contests. Winners gained immediate recognition and were awarded academic positions at prestigious universities. One way to assure victory in one of these debates or contests was to discover something that no one else knew. No one had published a general solution for all types of cubic equations. It was the discovery of this solution that led to Tartaglia's fame. INSET: Niccolò Fontana Tartaglia.

Blaise Pascal (pronounced "pahs-KAHL") was the son of Etienne Pascal, a high-ranking civil servant skilled in mathematics. Pascal's mother died when he was three years old. Five years later, his father left his job, moved the family to Paris, and committed himself to the education of Blaise and his two sisters, as well as the pursuit of his own intellectual interests. Étienne's plan was to have Blaise study languages first and then introduce the boy to mathematics later. When the boy discovered on his own that the sum of the angles of a triangle was always equal to two right angles, however, Étienne decided that Blaise was ready for the study of geometry. Blaise Pascal accompanied his father to meetings of a group of intellectuals in Paris. When Pascal presented his paper Essai pour les coniques (1640; treatise on conic sections), they were impressed. Not long afterward, he developed a calculating machine, in part to help his father with his work in the tax office. Pascal conducted numerous experiments concerning vacuums. One result was that he disproved the traditional view that "nature abhors a vacuum." He discovered Pascal's principle: Pressure applied to a contained fluid is transmitted equally throughout the fluid irrespective of the area over which the pressure is applied. Practical developments from his investigations included the syringe, an improved barometer, and the hydraulic press. INSET: Blaise Pascal.

Abraham de Moivre (pronounced either "duhm WAWVR" or "duh moy-VEHR") was the son of a Protestant surgeon of modest means. Wishing that his son receive a strong education, Abraham's father arranged for him to attend the Catholic village school in Vitry, followed by studies at a Protestant academy in Sedan. Abraham studied the standard classical curriculum of the time but also found time for mathematics on the side. Continuing his studies at Saumur, de Moivre was introduced to probability upon reading Christiaan Huygens's Tractatus de ratiociniis in aleae ludo (1657; Of the Games of Chance, 1692). He completed his education at the University of Paris, where he studied physics and was tutored in mathematics by Jacques Ozanam. In 1685, the Revocation of the Edict of Nantes put an end to religious toleration for French Protestants, and de Moivre was jailed. Upon his release in 1688, he fled to the more hospitable climate of Protestant England, where he would spend the remainder of his life. Lacking contacts in his new homeland, de Moivre earned a precarious living by tutoring in mathematics and solving gambling and insurance problems for wealthy clients. He soon made the acquaintance of astronomer Edmond Halley, who encouraged him to read a paper to the Royal Society of London and supported his election as a Fellow. Through the society; de Moivre established relationships with many leading English scientists, including its president, Sir Isaac Newton. Because of his foreign birth, however, he could not obtain a university position and thus struggled to support himself. De Moivre mastered Newton's famous work Philosophiae Naturalis Principia Mathematica (1687; Mathematical Principles of Natural Philosophy, 1729) by studying several pages at a time while walking to his tutoring duties. INSET: Abraham de Moivre.

Richard Martin Edler von Mises (pronounced "MEE-zehs") was born in Lemberg, Austro-Hungarian Empire, on April 19, 1883, to Arthur Edler von Mises and Adele von Landau. Arthur's job with the Austrian railways involved much traveling, which made Richard accustomed to living in many places. Richard's brother Ludwig would become a successful economist. Richard von Mises's education in Germany turned him into a well-to-do intellectual young man with interests in poetry (especially that of Rainer Maria Rilke), philosophy, and sports. At the age of only eighteen, he published his first paper, in the area of pure geometry. At the age of twenty-four, he earned a doctorate in Vienna and two years later began his first professorship, in applied mathematics at the University of Strassburg. He stayed there through the end of World War I. Von Mises's interest in fluid mechanics led to his interest in aerospace. In 1913, he gave the first university course on powered flight. The next year, he began a four-year service in the Flying Corps of the Austro-Hungarian army. In 1915, he made a huge, 600-horsepower plane with an original wing design for the Austrian army and actually piloted it during World War I. After the war, von Mises returned home to Berlin only to find that he had lost most of his material possessions during the war. In addition to his applied work in aerospace, von Mises's contributions to other areas such as elasticity, plasticity, and turbulence led to deeper study of probability and statistics. Just after World War I ended, von Mises began publishing works on probability. INSET: Richard von Mises.

Adrien-Marie Legendre (pronounced "luh-ZHAWN-druh") grew up in a wealthy family. He received an advanced education in science and mathematics at the Collège Mazarin under the well-known mathematician Abbé Joseph-François Marie. After defending his theses in mathematics and physics at the college in 1770, Legendre used his family's money to spend all of his time on research. Even though he did not need employment, Legendre accepted a position teaching mathematics at the École Militaire, which he held from 1775 to 1780. Legendre worked to establish himself in the European mathematical community during this time. He won a prize of the Berlin Academy of Sciences in 1782 for an essay on ballistics, which caught the attention of the mathematician Joseph-Louis Lagrange. Legendre also became acquainted with Pierre-Simon Laplace, who assisted Legendre in becoming a member of the French Académie des Sciences, an important step toward scientific legitimacy in France. Legendre's name has become attached to work in analysis that he did in the 1780's: the Legendre function, Legendre polynomials, and Legendre conditions. For example, Legendre polynomials first appeared in a paper that he read to the French academy in 1784. These functions can be used in a variety of applications in mathematical physics, such as to determine heat flow on a spherical surface. INSET: Adrien-Marie Legendre.

Sofya Kovalevskaya (pronounced "kuh-vul-YAYF-skuh-yuh")--known in the West as Sonya Kovalevsky (pronounced "koh-vuh-LEHF-skee")--was born Sofya Vasilyevna Korvin-Krukovskaya in Moscow on January 15, 1850. Her family had an aristocratic background. Her father was a Russian artillery general. Being the middle of three children, Sofya often felt neglected. For much of her childhood, she was raised by a strict governess. As a result, she became nervous and withdrawn. Sofya's father refused to let her pursue mathematics, so she studied secretly. Because of a wallpaper shortage, the nursery had been papered with her father's old calculus notes. Sofya studied these notes. She also studied a physics book that she had obtained from a family friend, Professor Tyrtov. At fourteen, she taught herself trigonometry so that she could understand the physics book. Tyrtov was so impressed that he persuaded her father to let her study mathematics. Sofya went to St. Petersburg to study. After her secondary schooling, Sofya wished to attend a university. The closest university open to women was in Germany. Since single women were not permitted to travel alone, she married Vladimir Kovalevsky in September, 1868. Their marriage was solely one of convenience. They moved to Heidelberg, Germany. Sofya Kovalevskaya studied at the University of Heidelberg. In 1871, she moved to the University of Berlin to study with Karl Weierstrass. Women were not allowed to attend university lectures, but he tutored her privately for four years. Without ever attending a university lecture, she was awarded a Ph.D. from the University of Göttingen, Germany, in 1874. While working on her doctorate, Kovalevskaya had done original research on various topics and had produced four papers. INSET: Sofya Kovalevskaya.

The father of Marie-Jean-Antoine-Nicholas Caritat, the marquis de Condorcet (pronounced "kohn-dawr-SEH"), was a French cavalry officer who died in battle shortly after his son was born. Condorcet's mother became anxious and isolated her son for fear of his safety, dedicating him to the Virgin Mary and clothing him in girl's white dresses until the age of eight. His uncle, a bishop, intervened and acquired a tutor for the boy's early religious education, eventually sending him to study at a Jesuit school in Reims. Later, Condorcet attended the College of Navarre at the University of Paris and studied mathematics and philosophy. Although his family pressured him to embark on a military career, Condorcet chose to become a mathematician instead. Condorcet formed close friendships with outstanding mathematicians and members of the Académie des Sciences in Paris, including Jean Le Rond d'Alembert and Joseph-Louis Lagrange. D'Alembert became a mentor and guided Condorcet's early mathematical research. A paper that Condorcet presented to the academy systematized and generalized integral calculus and established his reputation as one of the most influential mathematicians of the era. D'Alembert's patronage helped Condorcet attain membership to the prestigious scientific organization in 1769; he rose to assistant secretary in 1773 and then permanent secretary in 1776. Condorcet became the principal spokesperson for the French scientific community; and his interests broadened. His friendship with the writer Voltaire and other men of letters instilled within Condorcet the desire for social reform. The mathematician became a leader among an influential group of Enlightenment philosophers known as the philosophes. Condorcet wrote pamphlets ferociously attacking organized religion, political institutions, and social customs. INSET: Marquis de Condorcet.

Alexis-Claude Clairaut (pronounced "klehr-OH") was born in Paris on May 7, 1713. His father, Jean-Baptiste Clairaut, was a corresponding member of the Berlin Academy of Sciences who designed a circular slide rule. He taught mathematics in Paris and educated his twenty children at home. Alexis-Claude and a younger brother, called "le cadet Clairaut," were both prodigies in mathematics. This brother died of smallpox at the age of sixteen; many of Clairaut's other siblings also died young. Clairaut learned to read his letters from Euclid's Stoicheia (third century B.C.E.; Elements) and could read and write by the age of four. When he was nine, his father had him study a good introduction to analytical geometry and calculus by N. Guisnée. Clairaut's solutions to the problems were simple and elegant. He already showed the inventiveness and clear thinking that characterized his work. When he was ten, he studied Guillaume Francois Antoine de L'Hôpital's book on conics and his calculus text, the first to be written. At twelve, Clairaut read his own paper before the Académie Royale des Sciences. It was on the four cubic curves he had discovered that could be used to find any number of means proportional between two given lines. When he was thirteen, Clairaut started research on a book extending the ideas of René Descartes to three dimensions. About that time, he and some friends founded the Société des Arts, in which they trained to be members of the Académie des Sciences. Clairaut's first book was published in 1731, when he was eighteen. On the basis of it, he was elected to the academy even though he was below the legal age. It was an important honor. INSET: Alexis-Claude Clairaut.

Arthur Cayley was born during a visit by his parents to England from St. Petersburg, Russia, where his father was a merchant. Cayley lived in Russia until he was eight, when his father retired to the outskirts of London and sent Arthur to a private school. At age fourteen, Cayley entered King's College School, and his natural mathematical abilities became even more apparent. Cayley then started to study at Trinity College in Cambridge University at the relatively early age of seventeen. He excelled throughout his studies and was graduated as senior wrangler and first Smith's Prizeman, which meant that he had achieved the highest scores on both of the university's prestigious mathematics examinations. Although he was appointed a Fellow of Trinity College later that year and began to tutor students, Cayley turned to legal studies in 1845 in order to support himself before he was able to secure a permanent position in academia. By the time that Cayley became a lawyer in 1849, he was already contributing to mathematics in quality and quantity. For example, he posed the two motivating problems of invariant theory before 1846. Invariants are functions that are unaltered by transformation of their variables. Cayley worked on invariants with James Joseph Sylvester, another mathematician who studied for the bar in the 1840's. The pair enjoyed a highly profitable mathematical friendship, and they dominated invariant theory for most of their lifetimes. Cayley's fame continued to grow through the three hundred mathematical papers written during his fourteen-year law career, despite their characteristically concise style. Another of Cayley's favorite topics was what developed into linear algebra. He experimented with determinants and, in 1858, developed matrix algebra. INSET: Arthur Cayley.

Georg Ferdinand Ludwig Philipp Cantor (pronounced "KAWN-tor") was born in St. Petersburg, Russia, in 1845. His father, Georg Woldemar, was of Danish-Jewish origin; his mother, Maria Anna Boehm, came from a musically gifted family that included several classical violinists. Cantor was educated in a strongly religious spirit that stressed the need to succeed professionally. In 1863, he began to study engineering at the University of Berlin, but Karl Weierstrass's brilliant lectures on analysis turned him to pure mathematics. In 1867, he received his doctorate in mathematics for a paper about indeterminate equations from number theory. In 1869, Cantor was appointed a lecturer at the University of Halle. His first publication of 1872 dealt with trigonometric series. This work already involved infinite sets and lead to point set topology. Cantor introduced the notions of accumulation points and the derived set, which were immediately recognized as a profitable tool in analysis. His beautiful treatment of the irrational numbers lead to a lifelong exchange of ideas with Richard Dedekind. Shortly before Christmas, 1873, Cantor found his first proof, that real numbers are uncountable; he published this revolutionary discovery appeared in 1874. His monograph of 1883 opened transfinite set theory as a new field of research. Both transfinite cardinals and ordinals were introduced as two different autonomous and systematic extensions of the natural numbers. INSET: Georg Cantor.

Nicolas Bourbaki (pronounced "bewr-bah-KEE") is like a complex number, having both real and imaginary parts. He is one of the most influential mathematicians of the twentieth century. He is well known within the world of mathematics and has published thousands of pages of books, articles, and other papers. Yet, he is an imaginary person. He only exists in the minds of a group of young twentieth century mathematicians as a société anonyme, the French phrase for corporation. This corporate Bourbaki originated as an outgrowth of a project started in 1934-1935 by two of the founding members, Claude Chevalley and André Weil. They decided to rewrite a classic French calculus text by Édouard-Jean-Baptiste Goursat that was considered to be in adequate and deficient. Their intent was to reintroduce rigor into the teaching of calculus. Chevalley and Weil enlisted the help of Henri Cartan, Jean Delsarte, Jean Dieudonné, Szolem Mandelbrojt, and René de Possel. The newly created group intended to finish the revision within two years. By the late 1990's, this effort had yet to be completed. The decision of this group to call themselves Nicolas Bourbaki was apparently a continuation of a joke started in the 1930's at the École Normale Supérieure in Paris, France. The joke involved a lecture for entering mathematics students given by a distinguished visitor named Nicolas Bourbaki. The lecturer was actually an amateur actor (or perhaps an upperclassman) in a fake beard, and the lecture was a collection of plausible but mathematical double talk. The point of taking a collective pseudonym, however, was to eliminate title pages with long and changing lists of names and provide a concise, simple way to refer to their project. INSET: Nicolas Bourbaki.

Félix-Édouard-Justin-Émile Borel (pronounced "baw-REHL"), the youngest of three children, was born in Saint-Affrique, France, on January 7, 1871. His father, Honoré, was a Protestant pastor. His mother, Emilie Teissie-Solier, was the daughter of a moderately wealthy merchant. Borel was educated at home by his father until age eleven, when his prodigious mathematical talents opened the door to the lycée (high school) in Montauban, the nearest town. He was such a remarkable student that he was quickly sent on to Paris to study for admission to the university. Soon after arriving, he was welcomed into the extended family circle of Gaston Darboux, an eminent mathematician whose son was Borel's friend. By 1889, Borel had been awarded first place in entrance examinations for the École Polytechnique, the leading engineering school, and the École Normale Supérieure, the leading science and mathematics school. Despite strong lobbying efforts by the École Polytechnique, Borel elected to attend the École Normale. Within his first year there, Borel published three brief mathematics papers. Borel was graduated first in his class and was offered a full-time teaching post by the University of Lille. (Obtaining a university post before a doctorate remains virtually impossible in France.) Upon completion of his doctorate of science, Borel was called back to a teaching post at the École Normale. Soon, his research and publications began to reinvent the approach to university mathematics education in France. In 1901, Borel married Marguerite Appell, the daughter of a well-known French mathematician. They had no children of their own but adopted a nephew; he would be killed in World War I. INSET: Émile Borel.

Jean-Baptiste Le Rond d'Alembert (pronounced "dah-lawn-BEHR") was the illegitimate son of Madame de Tencin, a former nun and the hostess of a well-known Paris salon where artists, authors, and philosophers gathered. One of her lovers, cavalry officer Chevalier Destouches-Canon, was Jean's father. Tencin, fearing forced return to a convent, had her infant deposited on the steps of Saint Jean-le-Rond, a baptistery of Notre Dame Cathedral from which he obtained his name. Jean's father traced his whereabouts and provided him with a foster home. Although Destouches-Canon did not publicly acknowledge his son, he provided an annuity for his support and education. Jean was enrolled in the Collège des Quatre Nations under the name "Jean-Baptiste Daremberg," which was later changed to d'Alembert. His education at the college was focused on the classics and rhetoric but also included much mathematics. Uncertain about his direction, d'Alembert was pushed toward theology but developed a marked dislike for religion. He studied law and medicine but turned to mathematics as a lifelong, largely self-taught pursuit. D'Alembert presented his first mathematical paper to the Académie des Sciences in 1739 and an additional five notes over the next two years. His work was quickly recognized, and he became associated with the group known as the philosophes and with the salons where they exchanged their views, opinions, and discoveries. He was elected to the Académie des Sciences in 1741. In 1743, he published Traité de dynamique, in which he extended the views of Sir Isaac Newton and presented his own versions of the laws of motion. His focus turned from rigid bodies to motion in liquids and air. In Réflexions sur la cause générale des vents (1747; reflections on the general cause of wind), d'Alembert presented his development of partial differential equations. INSET: Jean Le Rond d'Alembert.