?? ??? ??? ?? ??? ?? ?? ??(500-1400AD): ???????? ? ? ?? ?? ??? ?? ??

1
???? 2 (?????)

• ?? ??? ??? ?? ??? ?? ?? ??(500-1400AD) ????????
  ? ? ?? ?? ??? ?? ??
• ??? ??, ??????
• ???? ??, ?????, ???
• ???? 1500AD
• ???? (???,?? ????? ??)

2
???? ???

• Pierro della Francesca (1412-1492)
• ??? (Desargues 1591-1661)

3
3,4????? ?

• Cardano (Fior), Tartalia(Fontana)? ?? 3, 4?
  ???? ? solution of x3 mx n.
• Notice that (a - b)3 3ab(a - b) a3 - b3 so if
  a and b satisfy 3ab m and a3 - b3 n then a -
  b is a solution of x3 mx n. But now b m/3a
  so a3 - m3/27a3 n, i.e. a6 - na3 - m3/27 0.
  Then x a - b is the solution to the cubic.
• ??? ??

4
???? 1596 in La Haye (now Descartes),Touraine,
France-1650 Sweden

• ??? ???
• ???? ??
• ???? ?? ? ??.
• ??? ???? ??? ???? ???? ???? ???? ????.
• ???? ?? ???? ???
• ???? ??? ??? ???? ? ???? ?? ????? ?? ? ??.

5
????1564 in Pisa -1642 in Arcetri (near Florence)

• ??? ??? ????? ??? ????? ???? (????? ??)
• Pendulum
• Theory of motions
• ??????
• Ptolemy, Aristotle? ??

6
?? 1643 Lincolnshire, EnglandDied 31 March 1727
in London, England

• Barrow
• ????? (???? ????)
• ????? ??
• ?? ?? (xo)nxn noxn-1 (nn-n)/2 oox n-2 .
  . .
• ????? ?? ??? ?? ??
• Principia Mathematica ?? ?? ?? (?? 300?????)

7
?????

• ?????, ???
• ????? (1646-1716) ???, ????, ???
• ???(1707-1783) ?????, ??, ???, ????(????),
  calculus of variation
• ????(1736-1813) ????
• ????, Legendre, ??, ???, ???, Hamilton, ??

8
Königsberg bridges
9
Fourier series
10
5? ???

• ?? (4??????? ??)
• ??? (?17??? ??)
• Galois (?? ??? ?? ??)
• Abel (?? ???? ??)

11
??? ???

• ????? 5?? ??
• ?? (critiques of pure reason)
• Saccheri (1667-1733)
• ???, Lambert, Lobachevsky, ????
• ??
• Beltrami, Klein

12
Euclidean Axioms

• To draw a straight line from any point to any
  other.
• To produce a finite straight line continuously in
  a straight line.
• To describe a circle with any centre and
  distance.
• That all right angles are equal to each other.
• That, if a straight line falling on two straight
  lines make the interior angles on the same side
  less than two right angles, if produced
  indefinitely, meet on that side on which are the
  angles less than the two right angles.

13
??? ? (???)

• ??? ??, ???
• ?? ?? ???
• ???
• Boole
• Hamilton (quaternion), Clifford (octanion,
  biquaternion, Clifford algebra)
• Cayley, Sylvester (??, matrix algebra)
• Group, ring, module, field, algebra, Lie algebra,
  Lie group

14
???, quaternion
15
?? ??

• Cantor (ordinal, cardinal)
• Dedekind (Real number system)
• Russel, Whitehead ??? ,,
  ,,, .

16
????

• ??????? ??? ????? ?? ?? ??? ?????
• 16?? ?? ????? ?? ??? ?? ???? ??? ????
• ????? ??? ???? ????
• ????? ?? ??? ??? ????
• ?? ?????? ????? ??? ?? ?????. ??? ??? ?????
• ????? ??? ?????
• ??? ??? ??? ??? ????