History of Mathematics Week 4 Notes

The Age of Plato and Aristotle

• “The influence of Socrates on mathematics is negligible, if not nega- tive”: Socrates-“I cannot satisfy myself that, when one is added to one, the one to which the addition is made becomes two, or that the two units added together make two by reason of addition. I cannot understand how when separated from the other, each of them was one and not two, and now, when they are brought together, the mere juxtaposition or meeting of them should be the cause of their becoming two.”

• Plato, student of Socrates, gained mathematical bent from the Pythagorean

• Platonic solids represent qualities: Tetrahedron-fire, Cube-earth, Octahedron- air, Icosahedron-water, Dodecahedron-universe

• It may have been Plato’s influence that made straightedge and compass constructions so prevalent in Greece, as he regarded other constructions as contrivances

• Plato preferred circles and lines because of their infinite symmetries (any point on a line or diameter of a circle)- tied to Platonic apotheosization. Also loved triangles, particularly right triangles, and viewed all polygons as made up of triangles.

• Plato describes physiology: “When the frame of the whole creature is young and the triangles of its constituent bodies are still as it were fresh from the workshop, their joints are firmly locked together...Accrodingly, since any triangles composing the meat and drink...are older and weaker than its own, with its newmade triangles, it gets the better of them and cuts them up, and so causes the animal to wax large.”

• Plato formalized the “analytic method” of starting with a desired state- ment and proceed through logical, reversible steps to a known statement.

• Eudoxus, student of Plato, gives the first working definition of ratio that withstands incommensurables. He says two things have a ratio if a multiple of one exceeds the other and vice-versa. This excludes 0 from ratios and does not allow comparison of lengths and areas, for example.

• Eudoxus developed the for working with curvi- linear objects. So Eudoxus could be considered the father of integral calculus.

• Manaechmus seems to have discovered the ellipse, hyperbola, and parabola and some of the directrix-focus properties, although the extent of his analytic geometry is questioned.

1 • Manaechmus knew how to duplicate the cube using conic sections.

• Dinostratus, brother of Manaechmus, used the of to square the circle.

Euclid of Alexandria

• His life remains mysterious, but due to his mathematical bent it is thought he is an academic descendant of Plato, if not a student himself.

• wrote almost a dozen books, on topics ranging from optics, as- tronomy, music, mechanics, and conics, but his Elements is his most famous work.

• Most of Euclid’s works did not survive to us. Euclid’s Division of Figures did not survive in the original Greek, but did through Arabic trans- lations.

• To Euclid is not attributed any new results, but he is known as an exceptional teacher. His Elements was considered an elementary textbook, not including basic calculations or advanced mathematics.

• Elements - 6 books on plane geometry, 3 on number theory/arithmetic, 1 on incommensurables, 3 on solid geometry. He began with a list of 23 defini- tions. “Point”-that which has no part, “Line”-breadthless length, “Surface”- that which has length and breadth only, “the extremities of a line are points”, circularity of definitions

• Euclid’s Five Postulates - (1) A line is determined by two points (2) A line segment can be extended to a line (3) A center and radius defines a circle (4) All right are equal (5) - “All else equal, the fewer the postulates the better”

• Euclid lacks postulates on continuity and uniqueness, but to his credit does give the converse of the .

• Second Book deals with geometric algebra, for distributive properties and foil ((a + b)2, a2 − b2) and solving quadratic equations, ends with (pg. 112)

• Book VII starts defining even and odd, prime and composite, plane and solid (products of two or three numbers), and a prefect number (“that which is equal to its parts”). Euclid speaks of numbers being measured by other number, not being multiples of. It also gives Euclid’s Algorithm.

• Book IX proves the infinitude of the primes, gives sum of a finite geo- metric sequence, and gives a formula for even perfect numbers

• Geometry so important for algebra because there was no concept of real number in arithmetic, so geometry was necessary for its generality

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