Raphael's famous Scuola di Atene covers one wall in the Apostolic Palace at the Vatican. The painting is 7.7 meters long and 5 meters high. Many scholars of the ancient world are depicted in this fresco, some of whom were mathematicians.
The lady who is dressed in white, standing toward the front-left, and gazing out of the picture toward us is thought to represent Hypatia. Hypatia was a prominent fourth-fifth century Alexandrian philosopher who wrote extensively in mathematics and astronomy. Her life and works are described elsewhere in this newsletter in The Feminine Math-tyque. The fifth century church historian Socrates Scholasticus praises her in his Ecclesiastical History:
"There was a woman at Alexandria named Hypatia, daughter of the philosopher Theon, who made such attainments in literature and science, as to far surpass all the philosophers of her own time. Having succeeded to the school of Plato and Plotinus, she explained the principles of philosophy to her auditors, many of whom came from a distance to receive her instructions. On account of the self-possession and ease of manner, which she had acquired in consequence of the cultivation of her mind, she not infrequently appeared in public in presence of the magistrates. Neither did she feel abashed in coming to an assembly of men. For all men on account of her extraordinary dignity and virtue admired her the more."
The person toward the front-left writing in a book with a couple of people peering over his shoulders probably represents Pythagoras. The diagram on the tablet being held in front of him relates numerical ratios to musical intervals, a relationship that he is said to have discovered. Little can be said about Pythagoras with certainty, but his greatness in the minds of his followers and his impact on both mathematics and the civilization in which he lived cannot reasonably be doubted. He certainly regarded numbers as the key to understanding reality, an idea that has proved especially fruitful in modern times, but his doctrine that things themselves and even virtues themselves are ultimately numbers seems more than a bit strange to the modern mind. Pythagoras gave geometry its rigorous character. He may have been the first to use definitions in mathematical work, and he may have been the first to arrange leading propositions in a logical order. He was among the earliest to use logical deduction as a means to establish the truth of mathematical propositions, although Thales was probably the first. The Pythagorean Theorem was known to the Babylonians well before Pythagoras, but their discovery may have been an achievement based on measurement. To the Pythagoreans, who attribute its discovery to Pythagoras, it was an abstract mathematical theorem. It is said that Pythagoras sacrificed 100 oxen to the gods as a token of gratitude for its discovery.
The greatest mathematician of the ancient world, Archimedes, appears to be absent, unless he is the person toward the front-right bending over to write on a tablet. But the drawing on the tablet is a geometric figure and there appears to be a number of students present. This suggests that the person depicted is Euclid, the author of the most successful text in mathematics or science in the history of the world. Euclid's Elements, after 23 centuries and more than a thousand editions, is still in use, and most of our school geometry is merely a rendering of portions of it into modern mathematical language. One modern historian has said of Euclid that "he is the only man to whom there ever came or ever can come again the glory of having successfully incorporated in his own writings all the essential parts of the accumulated mathematical knowledge of his time." The Elements contains the earliest extant evidence of a systematic arrangement of postulates, definitions, and propositions. Its simple yet logical exposition accounts for its perennial success.
No depiction of the school of Athens could fail to include Socrates, Plato, and Aristotle. Plato and Aristotle are in the center walking toward us. Plato is pointing upward, and Aristotle is on his left, gesturing horizontally. The fourth person to Plato's right speaking to a small audience is thought to be Socrates. Xenophon, a student of Socrates who, apart from Plato, is the best source of information we have about Socrates, may be among those with whom Socrates is conversing. Among the others thought to be portrayed are Plotinus (back-right in isolation), Averroes (standing behind Pythagoras), Boethius (kneeling behind Pythagoras), Epicurus (behind Averroes with the book), Heraclitus (front-center at the desk), Parmenides (behind Heraclitus), Diogenes of Sinope (lying on the steps), Strabo (front-right with beard holding celestial sphere), and Ptolemy (front-right with his back toward us holding terrestrial sphere).
Raphael himself appears in the fresco. He is the person on the lower level, visible only from the neck up, standing second to the far right, gazing out of the picture toward us. He probably represents Apelles, a famous artist from ancient Greece. Other double portrayals probably include Leonardo da Vinci as Plato and Michelangelo as Heraclitus. This brings us back to Hypatia. The representation of Hypatia is thought to be a feminine portrayal of Francesco Maria della Rovere I, the duke of Urbino and nephew of the pope, but it has also been suggested that the likeness is that of Raphael's mistress.
History of Mathematics Vol. I, D.E. Smith, Dover Publications Inc., New York
A Concise History of Mathematics, Dirk J. Struik, Dover Publications Inc., New York
A Short Account of the History of Mathematics, W.W.R. Ball, Sterling Publishing Company Inc., New York
Princeton Companion to Mathematics, Princeton University Press, Princeton, NJ
Ecclesiastical History, Socrates Scholasticus, Nicene and Post-Nicene Fathers, second series, Volume 2, Hendrickson, Peabody, MA
School of Athens, Who is Who?, Michael Lahanas, https://www.mlahanas.de/Greeks/SchoolAthens.htm