Archimedes, Conics, and Trig

Math 390 Lecture 7

Dr. Janssen

Archimedes

Archimedes of Syracuse (287–212 BCE)

  • Considered the greatest mathematician of ancient history
  • Inventor, engineer
  • Anticipated calculus via method of exhaustion

Statue at Oxford

Law of the Lever

  • Ancients were aware of the balancing principle at work on e.g., a see-saw
  • As far as is known, no one before Archimedes had modeled the lever
  • Idealized lever by assuming rigidity and weightlessness to make the math easier in Planes in Equilibrium

Proposition 3

Proposition 1 Suppose \(A\) and \(B\) are unequal weights with \(A > B\) which balance at a point \(C\). Let \(AC = a\), \(BC = b\). Then \(a < b\). Conversely, if the weights balance and \(a < b\), then \(A > B\).

Archimedes the Engineer

  • Plutarch: Demonstrated the law of the lever by moving a ship
  • Devised weapons of war: designed huge cranes which could lift Roman ships out of the water or dump out the crew
  • Hydrostatics: a solid heavier than a fluid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced
  • Archimedean screw

Measurement of the Circle

Proposition 2 The area \(A\) of any circle is equal to the area of a right triangle in which one of the legs is equal to the radius and the other to the circumference.

Archimedes’s The Method: Treatise on Geometry

  • Distinction from Euclid: Archimedes first presents method of discovery and motivation before presenting proof.
  • The Method: copy dating to tenth century discovered in a Greek monastery library in Constantinople in 1899
  • Written to Eratosthenes, librarian at the Library in Alexandria
  • Figures are “composed” of their indivisible cross-sections
  • Cross-sections of a given figure could be “balanced” against the cross-sections of a known figure to obtain formulas for area, volume, etc.

Conic Sections

Apollonius and The Conics

  • Generalized definition of a conic section; no longer need right circular cone
  • Goal: generalize theorems on circles proved in Elements Book III
  • Means: derive the symptom of the curve, the relation between the abscissa (\(x\)-coordinate) and ordinate (\(y\)-coordinate)
  • See the derivation of the parabola on p. 115-116
  • Could then calculate, e.g., tangents to curves (p. 120)

Astronomy and Practical Mathematics

Astronomy before Ptolemy

  • What did the ancients know about astronomy?
  • Sun and moon rise in the east and set in the west; sun rises due east around the equinoxes
  • Tracking this information led to the observation that the solar year is about 365 days
  • Structures like Stonehenge built to identify precisely those times

Passage Grave

Months

Why are our months about 30 days long?

Model of the heavens

  • Concentric spheres: the sphere of the earth, and the celestial sphere
  • Belief that the sphere is the most perfect shape helped
  • Required the study of properties of spheres
  • Eudoxus the likely inventor of the two-sphere model

Ptolemy and the Almagest

  • Claudius Ptolemy, ‘the Alexandrian’
  • Mathematiki Syntaxis, 13 books giving a complete mathematical description of Greek astronomy
  • Replaced all that came before; influential until the 1500s
  • Spherical geometry, trig, planetary predictions, etc

The beginning of trigonometry