• 1 of top 3 mathematicians (Newton, Gauss)
• Born in Syracuse, Sicily 287 BCE
• "Give me a place to stand on and I will move the earth"
• "Eureka" (I have found it)
• 10 surviving works, lost include: optics, calendar, motion of heavenly bodies
  

• written by Archimedes
• derived correct "factor" for SA of and volume of a sphere
• SA = 4pi*r*r
• V = (4/3)pi*r*r*r
• proof by Eudoxus' method of exhaustion
• book termed induction "sterile"
  

• written by Archimedes
• showed HOW he discovered such "formula" using his method of equilibrium
• very different, since most published proofs
  

• notes on how to solve "squaring the circle"
• finding the tangent line to a spiral
• effective for parabola, conoids, spheroids

3 ¹⁰/₇₁ < pi < 3 ¹/₇

proof used a regular 96-gon

• area of a circle: pi*r*r ie ^r*cir/₂
• area/volume of figures with circles for boundries
• length/area for new classes of curves
• archimedian spiral: r=a*theta

• figure of sphere inscribed in a cylinder
• rediscovered by Cicero in 76 BCE
• rediscovered in 1957/1965

• square the circle
• trisect angles

• written by Ptolemy (c.85-165 CE)
  
• became the standard astronomical treatise for much of the world until the 17th century
• contained 13 book treatise
• gives table of chords in .5 degree intervals from .5 to 180 degrees

• simplest motion is the sun wrt the stars: eastward
• elliptic motion greatest in early January
• least in degrees/day in early July
• both in northern hemisphere

• written by Sumerians, 3000 BCE and more recent
• gave position of stars and new moons and full moons

• time from vernal equinox to next
• not constant: approximately 20 minutes more than the last one
• Noted around 150 BCE by Hipparchus who also determined mean lunar month to within 1 second

• needed trig and led to the advancement of "Babylonian" practical astronomy
• Plimpton 322 (c.1900-1600 BCE) had a table of 15 secant values

• circa 1900-1600 BCE
• early use of trig in astronomy: used sin(theta) and tan(theta) for acute theta
• first proponent of heliocentric theory of solar system

• written by Menelaus about chords
• gave us knowledge about the Greek development of spherical trig
• survived in Arabic translation

• contains tables of chords
• contains Ptolemy's theorem:
• For any cyclic quadrilateral, AB*DC+AD*BC=AC*BD
• ie sum of products of opposite sides = products of the diagonals

• Aryabhata the Elder (c.500), jya: "chord" in Sanskrit
• Arabs omit vowels to get jb
• Jaib: "cove" in Arabic
• Latin translation by Gherado of Cremona (c.1150), sinus: "cove" in Latin
• this became sine

• written by Johann Müller in 1464
• published in 1533
• first exposition of trig independently of astronomy in Europe

Number Theory terms

US v ancient Greek

 US usage                  ancient Greek usage

arithmetic                        logistic

number theory               arithmetic

• ie, number theory <=> arithmetic
• Pythagoras(& CO) made first steps

• A positive integer n>1 is a perfect number iff sum of its factors equals n
• ex) 6 and 28 are perfect
• Theorem (prop 36 of book IX of Elements): If 2ⁿ -1 is prime, then 2^n-1(2ⁿ -1) is perfect
• Euclid proved all even perfect numbers of this form
• unknown if there exist any odd perfect numbers
  

• studied by Pythagoreans
• obtained by counting dots in diagrams
• [image]
• first few: 3, 6, 10...

• Hadamand and de le Valle-Poussin, 1896
• gives asymptotic form for the prime counting function π(n) which counts the number of primes less than an integer n
[image]

• most important open question in number theory
• every number >=6 is the sum of two odd prime numbers