Struik: A Concise History of Mathematics: Greece
by Carson Reynolds
As the Mediterranean basin transitioned from the Bronze Age to the Iron Age there were enormous economic and political changes. Iron’s introduction changed warfare but also cheapened production, causing a surplus, and allowing more people to be involved in public life. Additionally the introduction of coined money and the alphabet stimulated trade. Culture began to flourish, no longer exclusively the domain of the elite. Sea-raiders upset cultural, scientific, and mathematical progress as they destroyed Minoan civilization, and disrupted Egyptian and Babylonian development. The rise of the Greek city-state in 7th-6th centuries BCE allowed new freedoms to citizens. More leisure (begotten from slavery and wealth) stimulated the growth of rationalism, philosophy, and science.
In contrast to oriental approaches, the Greeks did not just “how?” but sought to know “why?” Thales of Milete (who had traveled to Babylon and Egypt) is considered the father of Greek math. In seeking a rational scheme to the universe his tradition added an element of rationalism to mathematics. Only small fragments remain, but through careful analysis a consistent picture of Greek math emerges.
The rise of Persia led to conflict and an eventual Greek victory, which expanded the power of Athens. Democratic ideals flourished from 450-400 BCE, paving the path for the Golden Age of Greece. Sophists had greater freedom to examine ideas more abstract than useful. The only complete fragment of this period is written by Hippocrates of Chios. The work shows perfected mathematical reasoning and deals with the impractical: the lunulae (crescents bounded by two circular arcs). The problem addressed is trying to find the area of crescents, relates to the problem of quadrature of the circle. The work shows such developments as an ordered plane geometry, logical deduction (apagoge), and also the start of axiomatics. He had written an Elements the title of all greek axiomatic treatise, including Euclid’s. There were three famous mathematical problems of antiquity:
- the trisection of the angle
- the Delic problem (finding the side of a cube whose volume is twice a given cube)
- the quadrature of the circle (finding the square of an area equal to a given figure) [struik says square, but it seems like an error]
These problems are important because they could not be solved using finite steps with straight lines and circles; the necessitated development of equations which spurred interest in algebraic numbes and group theory.
In contrast to the the democratically-inclined sophists were the aristocratically inclined Pythagoreans. Sophists emphasized the reality of change, especially the atomists (followers of Leuippus and Democritus); Pythagoreans studied the unchangeable elements. From study of the quadrivium the Pythagoreans (led by Archytas of Tarentum (400 BCE)) branched into primitive number theory. They divided numbers into classes (i.e. prime, square, etc.). While geometric patterns of triangle numbers and square numbers appear from Neolithic times, the Pythagoreans reasoned about and classified them (into arithmetical 2b=a+c, geometrical b^2=ac, and harmonical 2/b=1/a+1/c). Pythagoreans knew about the properties of some regular polygons and bodies. While the pythagorean theorem can be seen in Hummurabi’s Babylon, it’s proof is likely to be a product of the Pythagoreans. The irrational numbers were another discovery of theirs. This discovery upset the harmony between arithmetics and geometry. This along with Zeno’s paradox (Achilles, Dichotomy, Stadium, Arrow) led to a crisis of Greek math, a crisis that was mirrored by Athens’ fall.
The increased wealth of the rich and misery of the poor led to increased study of philosophy and ethics. Plato and Aristotle encouraged the study of the quadrivium. Three mathematicians were connected to Plato’s Academy: Archytas, Theatetus (d. 969 BCE) and Eudoxus (c. 408-355 BCE). Euclid’s Elements documents Theatetus theory of irrationals, and Eudoxus’ exhaustion method. Eudoxus solved the crisis in Greek math and determined the course of Greek math with rigorous formulations that discarded the distinction between incommensurable and commensurable. Struick credits him with the Axiom of Archimedes (a condition for the equality of ratios, and a property of magnitudes). The exhaustion method discarded infinitesimals in favor of formal logic and proof by contradiction. This became a common mode of rigorous proof for area and volume computation, but required that result be known in advance through some less rigorous method. The “Method” letter from Archimedes to Eratosthenes (c. 250 BCE) describes a method employing geometrical atoms to find new results in a less rigorous manner. In nearly all texts, the more formal Exhaustion method prevailed, a reflection the triumph of Plationic idealism over Democritian materialism.
Alexander the Great’s conquest of Persia (334 BCE) led to the spread of hellenistic arts, letters, and sciences, and their fusion with Oriental administration and astronomy. This contact was fertile (e.g. Egypt under the Ptolemies). During this period Babylon began to fade, although Babylonian math and astronomy were improved by the Seleucid empire. Athens, Alexandria and Syracuse became educational centers, Syracsuse being famous for producing Archimedes, the greatest Greek mathematician.
Professional science flourished in Alexandria, with its Museum and Library. One of the first Alexandrian scholars was Euclid (c. 306-283 BCE) known best for his 13 Elements and Data, the first full math texts we have from Greek antiquity. Elements is important because it was widely reproduced and studied, our school geometry is taken from it, and much of scientific thinking was influenced by it. Euclid’s treatment of plane geometry, Eudoxus’ theory of proportions, Pythangorean Theorem, golden sections, quadratic roots, solid geometry, and number theory was based strictly on logical deduction of theorems from definitions, postulates, and axioms. Interesting highlights include Euclids algorithm to find the greatest common divisor, and Euclid’s theorem (there are an infinite number of primes), Euclidean geometry. His treatment of algebra is in entirely geometrical form, and hs treatment of arithmetic confines itself to integers and their ratios. His purpose was to bring together 3 discoveries: Eudoxus’s theory of proportions, Theatetus’s theory of irrationals, and Plato’s theory of five regular bodies.
Archimedes (278-212 BCE) was the greatest mathematician of the Hellenistic period. He lived in Syracuse and died there wehn the city fell to the Romans. His development of integral calculus (theorems on areas and volumes). His Measurement of the Circle finds pi = 3 1/7. His On the Sphere and Cylinder provide expressions for the area of the sphere and the volume of the sphere. Other works include Quadrature of the Parabola, On Spirals, On Conoids and Spheroids, On Floating Bodies. His work is both rigorous, making consistent use of the exhaustion method. His work was influenced by Oriental sources, for instance his cattle problem leading to Pell equations, and requiring solutions of very large numbers.
Apollonius of Perga (c. 247-205 BCE) was the third great Hellenistic mathematician, following the geometrical tradition. His Conics develops conic sections, even providing with our current names (parabola y^2=px, ellipse, y^2=px-(p/d)x^2, hyperbola y^2=px+(p/d)x^2). He used no algebraic notation or coordinates (probabbly following Exodius). He also posed the tangency problem. He required geometrical constructions be confined to compass and ruler.
Mathematics is intimately tied to astronomy, which was the science first importance in Greece and the Orient. The motion of the moon was one of the most challenging problems to mathematicians. Eudoxus provided the oldest known theory of planetary motion (assuming the superposition of 4 rotating concentric spheres). This was a stark contrast to simple chronicling of celestial phenomena . . . it’s development led into modern dynamics and Fourier series. Aristarchus of Samos (c. 280 BCE) was credited with the first heliocentric model (which was unaccepted, even though many believed the earth rotated). Hipparchus of Nicea (active 161-126 BCE) was used as an authoritative source for much of Ptolemy’s Almagest. He made use of eccentric circles and epicycles. Aristarchus discovered the procession of the equinoxes, and a method to determine latitude and longitude. His work is closely associated with Babylonian astronomy, and may be seen as fruit of the Greek-Oriental cross-pollination.
The Roman period saw a decline of science toward mediocrity in the West. However, in the east during the pax Romana and the Han Dynasty Hellenistic and Oriental elements were fruitfully recombined. There was wide diffusion of scientific and mathematical knowledge (e.g. Hindu-Arabic numerals introduction to Europe). Alexandria remained a center of original work, and increasingly compilation. The decline of Greek math may be linked to its rejection of algebra. The work of Alexandria was not purely Greek though.
Nicomachus of Gerasa (c. 100 CE) was one of the earliest Alexandrian mathematicians of the Roman era. His Arithmetic Introduction is the most complete extant piece on Pythagorean artithmetic, but unlike Euclid’s Elements it uses arithmetic notation, instead of pure geometry. His work influenced Boetius and medieval arithmetic. Ptolemy’s Great Collection AKA Almagest (c. 150 CE) was an impressive work on astronomy and trigonometry. He held pi = 377/120. It also contains formula for sine and cosine. He also relates Ptolemey’s theorem (for quadrilateral inscribed in a circle). His Planisphareium and Geographia developed stereographic projection and speherical coordinates, which were necessary for the development of the astrolabe and later octant and sextant. Menelasus (c. 100 BCE) predated Ptolemy with Sphaerica which contained a spherical geometry as well as Menelaus’ theorem for the triangle in its extension to the sphere. While Ptolemy used sexagesimal fractions, Menelaus’ was purely geometrical. Heron was a contemporary who in 62 BCE described a lunar eclipse. In Metrica he derived the Heronic formula for the area of a triangle. He made use of Egyptian unit fractions.
Diophantus (c. 250 CE) was more strongly influenced by oriental mathematicians. He made use of indeterminate equations that may have been of Indian or Babylonian origin perhaps like himself. He used Diophantine analysis to find answers to sets of equations. He was only interested in positive rational solutions. He did work on Pell equations and provided some theorems in number theory. Most notably we find the first systematic use of algebraic symbols, and their applications to problems of great complexity.
Pappus (c. 400 BCE) produced the last great Alexandrian math treatise, Collection (Synagoge). This was a handbook to accompany other works commenting in chapters on topics like isoperimetric figures. D’Arcy references some of his findings about honeycombs. The Alexandrian school gradually died out branded as “pagan” and later taken by Arabs.
Greeks made a distinction between arithmetica (more akin to number theory) and logistics (computation). Interestingly, one was not considered a number. Line segments did not necessarily have an actual length associated with them. This stands in contrast to logistics were systems of numeration were used. Greek letters were assigned to numbers to form a decimal non-position system. This system stayed in use until the year 1455. Some argue that the number system retarded the growth of Greek algebra.