2 1 17 1.1 1.1.1 1650

1 3 5 1 1 2 0 0 1 2 I II III J.

2 1 17 1.1 1.1.1 1650

1.1 3 3 6 10 3 5 1 3/5 1 2 + 1 10 ( = 6 ) 10 1/10 2000 19 17 60 2 1 1 3 10 25 33221 73 13111 0. 31 11 11 60 11/60 2 111111 3 60 + 3 332221 27 x y xy + (x y) = 183, x + y = 27

4 1 y = y + 2 xy = 210, x + y = 29 3 x = 15, y = 12 1.1.2 20 12 3 8 3 { x + y = p xy = q (x > y > 0 ) p 2 /4 q = (x + y) 2 /4 xy = (x y)/2 x+y = p, x y = 2 p 2 /4 q x = p/2+ p 2 /4 q, y = p/2 p 2 /4 q 1990

1.1 5 Thales 624 546 4 4 A.K. 1976 1 1

6 1 5 2 1.1.3 300 Euclid 365 275 5 I 1979 1 1

1.1 7 1 1 2 3. 23 1 2 3 4 5 1 2 3 4 5 300 6 5 585 525

8 1 540 480 460 370 440 430 515 440 490 430 600 500 400 300 200 BC A B A B B A A B A B

1.1 9 1960 6 5 1 Parmenides 515 440 Zenon 490 430 7 P P P 7 1 6 K. 1978 1974 http://redshift.hp.infoseek.co.jp/scilib.html 1977 1990 7 M. 2 1983

10 1 1 2 8 Platon 427 347 9 8 1 9

1.1 11 Theaitetos 415 369 10 13 Eudoxos 390 337 5 12 Aristoteles 384 322 p q q r p r 1.1.4 1 2 3 1 2 3 4 one, two, three, first, second, third, (=1st, 2nd, 3rd, ) 1, 2, 3, 2 17

12 1 1m, 2.3kg, 4 5 600 10 7 1 2 19 G. 11 7 µoνάς 5 1 = 5 6 = 6 7 = 7 m, kg 10 T.L. I 3 11 2 29

1.1 13 1 12 Archimedes 287 212 π 1.1.5 50 Pythagoras 572 494 540 40 12 A.K. 1976 I 3

14 1 13 3 3 : 4 : 5 5 : 12 : 13 x = 1 2 (m2 1), y = m, z = 1 2 (m2 + 1) m 14 2 : 1 3 : 2 4 : 3 2 ρ l T f T/ρ f = 2l 15 ρ T f l 2 1 2 8 2 3 13 3 x, y, z z x 2 + y 2 = z 2 14 p.6 1 2 15 +α 5.3

1.1 15 3 2 = 1.5 5 5 3 : 2 16 0 32 4 1 128 16 2 3 64 8 1 81 9 2 243 27 3 4 81 2 3 9 2 3 2 2 3 2 3 2 2 3 2 3 17 1Å (= 10 8 cm) 16 1 2 3 1 2 1 2 1 2 2 3 3 4 5 http://math-info.criced.tsukuba.ac.jp/forall/project/history/ 2003/monocode/monocode_index.htm 17 p.12 I 3 1 2 (a)

16 1 Å Å Å 5 : 18 18 + α 11.6

1.1 17 Hippasus 5 12 19 1 20 B C E D B A A C D 430 A E C B D E 19 : - Wikipedia 20 AB AC AB = n AC = m m, n AC AB = (m n) m, n m n AB = CD CE = A C AC AB = AC AE = CE = A C AB A C = AE D A = D E = A B ABCDE AB AC A B C D E A B A C A B C D E A B A C (> 0)

18 1 1 a b b 2 = a 2 + a 2 = 2a 2 1cm a, b b 2 = 2a 2 a, b b 2 = 2a 2 a, b 21 X 1 II 4 (a + b) 2 = a 2 + b 2 + 2ab 2 2 a b b a 21 p.16 1.8.2 http://www.h6.dion.ne.jp/ hsbook_a/kakushoupdf.html

1.1 19 I 41 2 Eudoxos 409 355 V 3, 5 3 5 a, b, c, d a : b c : d m, n. ma > nb mc > nd ma = nb mc = nd ma < nb mc < nd. / / 22 5 2 m, n ma > nb mc > nd m, n m, n ma > nb m, n mc > nd 19 22 p.9 9 4

20 1 1.2 1.2.1 Archimedes 287 212 23 π 96 3 10 71 < π < 3 1 7 23

1.2 21 Hipparchus 190 120 α R chord(α) α Ptolemaeus 100 178 R α α chord(α) α chord(α) chord(α) = 2R sin(α/2) 16 13 1 2 1 2 180 R = 60 60 chord(α) α 24 1.2.2 0 2 2 (= 2 10 3 + 10) = 10 = 10 2 = 10 3 10 10 n (n = 0, 1, 2, ) 10 10 24 p.1 p.179

22 1 p.3 9 5 11 11 1 60 (458 14236713 25 7 9 p.1 p.264 7 10 0 0 773 10 al-khwārizmī 780 850 9 25 1998

1.2 23 0 10 12 0 algebra 825 p.27 1.2.3 2 2 2 2 p.3 3 Diophantus 3 3 (2x 3)(2x 3) = 4x 2 12x + 9. ( 3) ( 3) = +9 3 2x 3 3 2x + ( 3) 2

24 1 263 8 1 26 3 4 1 10 7 6 8 4 1 1 x 1 y { 3x + 4 = 7y 6y + 8 = 4x { 3x 7y = 4 4x + 6y = 8 27 1 =, 2 =, 3 =, 6 =, 7 = 4 =, 7 = { 3x 7y = 4 4x + 6y = 8 x + y = x + y = a > b > 0 ( a) ( b) = (a b) ( a) b = (a+b) 0 ( b) = ( b) = b { 3x 7y = 4 4x + 6y = 8 { 3 4x 7 4y = 4 4 4 3x + 6 3y = 8 3 { 3 4x 7 4y = 4 4 0x + (18 28)y = (24 + 16) { 3x 7y = 4 10y = 40 { 30x 70y = 40 10y = 40 { 30x (70 10 7)y = 40 ( 40 7) 10y = 40 { 30x = 40 + 280 = 240 y = 4 { x = 8 y = 4 26 2003 p.39 27 +α 6.4

1.2 25 2 y 18 28 = 10 18 28 = 18 + ( 28) = 10 2 7 Brahmagupta 598 668 773 628 12 Bhāskara 1114 1185 a, b, c > 0 2 a + b = c 2 ( a) + ( b) = c a + ( b) = a b. 0 ( a) = a. ( a) b = c

26 1 ( a) ( b) = c. a/( b) = c ( a)/( b) = c 2 1/8 12 x x = (x/8) 2 + 12 x 2 64x = 768 x 2 64x = 768 x 2 2 32x + 32 2 = 768 + 32 2 = 256 { (x 32) 2 = (±16) 2 32 + 16 = 48 x 32 = ±16 x = 32 16 = 16 48 16 2 x x p.11 = x x = (x /8) 2 + 12

1.2 27 1.2.4 x 3 p.23 x 6 28 10 0 825 10 39 x x 2 + 10x = 39 5 5x 5 2 x 2 + 2 5x + 5 2 = 25 + 39 x (x + 5) 2 = 8 2 x = 8 5 = 3 x 5 x 2 5x 28 p.1 p.277

28 1 21 10 x 2 + 21 = 10x (x + ( 5)) 2 = 5 2 21 = 4 5 ( 5) 2 = +5 2 x 2 + 21 = 10x x 2 + 10x = 39 x 2 + c = 2b x (b, c > 0) x 2 2b x + b 2 = b 2 c (x b ) 2 (b x) 2 b > x c = x (2b x) c 2 x 2b x = b + (b x) b 2 c b 2 c 1 b x x 2 + c = 2b x b x = b 2 c x b = b 2 c b x b 2 c b c = x(b + (b x)) x x b b x 1 2 6 a, b, c > 0 ax 2 = bx, ax 2 = c, bx = c ax 2 + bx = c, ax 2 + c = bx, bx + c = ax 2. 6 (algebra) (algorithm)

1.2 29 1.2.5 4 476 1 1096 12 p.23 12 13 29 Leonardo Fibonacci 1174 1250 1202 1220 1225 9 8 7 6 5 4 3 2 1 zephirum 0 29 p.16 11.3.3

30 1 300 12 6 14 1335 30 30 p.1 p.361

1.3 18 31 1.3 18 2 3 3 1 2 3 1.3.1 14

32 1 cosa c 15 +, = 16 17 <, > Girolamo Cardano 1501 1576 1545 3 x 3 ± cx ± d = 0 31 c, d François Vite 1540 1603 3 x 3 ± cx ± d = 0 c d c, d 1.2.4 p.12 p.18 x x m 32 x 2 x 2 m 2 x 3 x 3 m 3 x 2 + 2cx = d x 2 m 2 +(2c m )(x m ) = d m 2 (x m +c m ) 2 = d m 2 +c 2 m 2 2 (square) 3 (cube) Simon Stevin 1548 1620 31 2 32 m x

1.3 18 33 10 1585 2 19 1.3.2 17 Ren Descartes 1596 1650 (1637) 33 33 p.1 p.492

34 1 6 34 1 a a 2 OEA OBC EA // BC OE OE : OA = OB : OC m O a c 1 b OC = OA OB/OE = a m b m /1 m = ab m OB = OE OC/OA = 1 m c m /a m = c/a m 2 3 ax 2 + bx + c AB BC AC B AC D ABD DBC AB : BD = DB : BC A D a E A 1 B a BD = AB BC = 1 m a m = a m B C C 34 1973

1.3 18 35 a 2, a 3, x y a, b, c, x, y 35 y = ax + b OE OX x y O ax A a b Y Ax 1 E x y = ax + b y = ax + b OA OY OE OA x, y 36 37 1 X 35 x y p.37 2 36 I p.1 p.75 absciss abscissus x ordinate ordinatus y Gottfried W. Leibniz 1646 1716 (co-) (coordinate) 37

36 1 1.3.3 (1637) 50 Isaac Newton 1642 1727 (1687) 1707 p.12 3 a a 1 a = a /1 38 Leonhard Euler 1707 1783 e iθ = cos θ + i sin θ i e iπ = 1 F = ma F m a 38 2.4.1

1.3 18 37 1 2 39 1748 1 O 1 E OE P x <0 O E P 0 1 x >0 x P P O E x = OP/OE > 0 x = OP/OE < 0 P = O x = 0 P = E x = 1 x P P x x y P x x P P(x, y) P (x, y) 1 2.3 m 4.5 l 39 2005

38 1 1.4 19 1 2 3 4 1.4.1 2 x 2 1 = 0 x 2 1 = x 2 + x x 1 = x(x + 1) (x + 1) = (x 1)(x + 1) = 0 x 1 = 0 x + 1 = 0 x = ±1 1 a, b, c, a + b = b + a, ab = ba a + (b + c) = (a + b) + c, a(bc) = (ab)c a(b + c) = ab + ac a a + 0 = a, a 1 = a 0 1 40 p.27 40 a 0 (1/a) a ( a) a = b b = a a = b b = c a = c

1.4 19 39 2 x x 2 1 = x 2 + x x 1 = x(x + 1) (x + 1) = (x 1)(x + 1) (= 0) x = ±1 x x 2 + 2x 1 = 0 x = 2 1 x = 1 + 2 4 x 4 1 = 0 x 4 1 = (x 2 1)(x 2 + 1) = (x 1)(x + 1)(x i)(x + i) = 0 (i = 1 ) 4 x = ±1, ±i x 2 + 1 = (x i)(x + i) x 4 1 = 0 α (= ±1, ±i) 4 x 4 1 (x α) n P n (x) = ax n + bx n 1 + + cx + d = 0 (a 0) α P n (α) = aα n + bα n 1 + + cα + d = 0 x k α k = (x α)(x k 1 + αx k 2 + + α k 2 x + α k 1 ) P n (x) = P n (x) P n (α) = (x α)q n 1 (x) (= 0) Q n 1 (x) n 1 β Q n 1 (x) = 0 Q n 1 (x) (x β) P n (x) (x )

40 1 n n 41 p.32 3 42 3 2 x 3 + 3px + 2q = 0 p q x = u + v u = 3 q +, v = 3 q ( = q 2 + p 3 ) (uv) 3 = ( p) 3 uv = p 3 x 3 3 7x + 2 10 = (x 1)(x 4)(x + 5) = 0 p = 7 q = 10 = 10 2 + ( 7) 3 = 243 x = u + v = 3 10 + 243 + 3 10 243 (= 1 4 5) 16 18 300 1 4 5 41 k k p.16 2.4.3 10.3.3.3 42 2.3.2 10.3.2

1.4 19 41 1.4.2 3 1545 15 Rafael Bombelli 1526 1572 1572 a, b, c, d a + b 1 = c + d 1 a = c, b = d, (a + b 1) + (c + d 1) = (a + c) + (b + d) 1, (a + b 1)(c + d 1) = (ac bd) + (ad + bc) 1. 18 18 4 Abraham De Moivre 1667 1754 (cos θ + i sin θ) n = cos nθ + i sin nθ (i = 1 ) cos θ cos nθ 1748 e iθ = cos θ + i sin θ, (a + bi) p+qi = c + di a, b, c, d, p, q 1 i imaginary 1777 (a + bi) + (c + di) = (a + c) + (b + d)i ( ) a + b ( ) c = d ( ) a + c. b + d

42 1 z = z(r, θ) = r(cos θ + i sin θ) (r > = 0) 1 i z(1, 0 ) = cos 0 + i sin 0 = 1 z(1, 90 ) = cos 90 + i sin 90 = i z(1, 180 ) = cos 180 + i sin 180 = 1. 1 i z = x + yi = z(r, θ) z z z z = x 2 + y 2 z = r x+yi (y 0) yi = y yi i O yi θ r 1 z = x+yi z 1 = r 1 (cos θ 1 + i sin θ 1 ), z 2 = r 2 (cos θ 2 + i sin θ 2 ) z 1 z 2 = r 1 r 2 (cos θ 1 + i sin θ 1 )(cos θ 2 + i sin θ 2 ) = r 1 r 2 {(cos θ 1 cos θ 2 sin θ 1 sin θ 2 ) + i(sin θ 1 cos θ 2 + cos θ 1 sin θ 2 )} z 1 z 2 = r 1 r 2 {cos(θ 1 + θ 2 ) + i sin(θ 1 + θ 2 )}. Caspar Wessel 1745 1818 1799 Jean Robert Argand 1768 1822 1806 x x

1.4 19 43 Johann Carl Friedrich Gauss 1777 1855 1799 p.40 1790 30 n 1831 a + bi a1 + bi 1 i (complex number) i i (c + di)(c di) = c 2 + d 2 a + bi c + di = ac + bd c 2 + d 2 bc ad + c 2 + d i 2 19

44 1 1.4.3 (1637) x 2 + y 2 = r 2 2 1642 1727 1646 1716 moment x(t) 2 + y(t) 2 = r 2 t x(t), y(t) x, y x, y ẋ, ẏ 43 t o x, y x(t + o), y(t + o) o 44 x(t + o), y(t + o) ẋ, ẏ x(t + o) = x + ẋo y(t + o) = y + ẏo 45 x 2 + y 2 = r 2 x(t + o) 2 + y(t + o) 2 = (x + ẋo) 2 + (y + ẏo) 2 = r 2 2xẋo + (ẋo) 2 + 2yẏo + (ẏo) 2 = 0 o 2xẋ + 2yẏ + ẋ 2 o + ẏ 2 o = 0 o 2xẋ + 2yẏ ẋ 2 o + ẏ 2 o 0 2xẋ + 2yẏ = 0 ( dx2 dt + dy2 dt = 0 ), ẏ ẋ ( dy ) x = = dx y 43 ẋ = dx dt ẏ = dy dt x, y ẋ, ẏ 44 0 0 0 0 + 1 45 x, y ẋo, ẏo

1.4 19 45 t o dt x, y ẋo, ẏo dx = dx dy dt dy = dt dt dt 1707 1783 46 1 Augustin Louis Cauchy, 1789 1857 (limit) lim f (x) = α f (x) α (x a) x a x a x f (x) α x(t) 2 + y(t) 2 = r 2 o t t ( 0) x, y x = x(t + t) x(t), y = y(t + t) y(t) x(t + t) = x + x y(t + t) = y + y x(t + t) 2 + y(t + t) 2 = r 2 (x + x) 2 + (y + y) 2 = r 2 x 2 + y 2 = r 2 2x x + 2y y + x 2 + y 2 = 0 t x 2x x t + 2y y t + x y x + y = 0, t t y 2x + 2y x + x + y x y = 0 t 0 x 0, y 0 1, 2 x t y t lim x t 0 t = lim t 0 x(t + t) x(t) t = dx dt, lim y t 0 t = lim t 0 y(t + t) y(t) t = dy dt. 46 1960 1

46 1 3, 4 0 2x dx dt + 2y dy dt = 0, dy dx = x y. f (x) a lim x a f (x) = f (a) lim lim( f (x) + g(x)) = lim f (x) + limg(x) x a x a x a ε error δ distance ε δ 2.4 Karl T. W. Weierstraß 1815 1897 1860 lim x a f (x) = α ε δ f (x) α ε x a δ ε δ 0 < x a < δ x f (x) α < ε x a f (x) α ε δ 0 < x a < δ f (x) α < ε ε δ ε δ 19

1.4 19 47 Julius W.R. Dedekind 1831 1916 p.37 10 1 10 2 n n 10 0.101101110111101111101 1 0.101101110111101111101 α α n a n a 3 = 0.101 lim n a n = α {a n } n a n α a n α α 1 : 1 0.999999 = 1

48 1 47 48 Q 2 Q 2 A, A A A { }} { { }} { α A, A < A, A > α α =< A, A > i A A = Q, A A =, A, A ii r A, s A r < s A A iii A iv A = {r Q r < α} 49, A = {s Q s > = α} α =< A, A > A 4 3 A 4 3 iv < A, A >= 4 3 A 4 3 A 2 A 2 2 A 2 2 47 1961 48, { } r A r A r A A A A A A A A A 1 A B A B A B A B A B A = B A B 49 α =< A, A >, β =< B, B > α = β A = B α < β A B A B

1.4 19 49 < A, A > 2 2 I A a < A, A > a a II A < A, A > Q (, ) α (, α] (α, ) (, α) [α, ) 50 2 3 = 6 α(βγ) = (αβ)γ α(β + γ) = αβ + αγ 50 I

50 1 1.4.4 2300 p.6 p.7 5 1 1 1 51 19 Nikolai I. Lobachevsky 1792 1856 János Bolyai 1802 1860 2 5 19 Georg Riemann 1826 1866 n 1 2 51 5 2 + 2 = 4 1 1 1

1.4 19 51 0 20 52 2 20 David Hilbert 1862 1943 20 1899 5 I (I 1 8 ) II (II 1 4 ) III (III 1 5 ) IV (IV 1 ) V (V 1 ) 8 I 1 I 8 I I 1 2 A, B l I 2 2 A, B 1 I 3 1 2 1 3 52 A i A i j A i jk l n n (n = 0, 1, 2, ) A i jk l A i j k lm n

52 1 I 4 1 3 A, B, C α 1 I 5 1 3 A, B, C 1 I 6 l 2 A, B α l l α I 7 A 2 α, β A 1 I 8 1 4 53 2 53 D. II II 1 B A, C A, B, C 1 3 B C, A II 2 l 2 A, C l B C A, B II 3 B A, C A B, C 2 A, B AB BA A, B A, B AB A B AB AB II 4 A, B, C 1 3 ABC A, B, C l l AB 1 l BC CA 1 l α 1 A, B l α 2 A=B AB l A, B α l l α l A l α

1.4 19 53 1 2 1 III l 1 O O 2 A, B A=B O AB A, B O A, B O l O A l O O A III 1 A, B l 2 A l 1 l A 1 B AB A B AB A B AB A B III 2 AB A B AB A B A B A B III 3 A, B, C l 3 B A, C A, B, C l 3 B A, C AB A B BC B C AC A C l h, m h 1 O l, m O α 2 α (l h, m h ) (m h, l h ) l h, m h A, B (m h, l h ) AOB l h, m h O α m l h l m h (l h, m h ) α (l h, m h ) III 4 (l h, m h ) α l α O l 1 O l 1 l h l α 1 α h α h m h (l h, m h ) (l h, m h ) m h 1 III 5 A, B, C A, B, C 1 3 AB A B AC A C BAC B A C ABC A B C IV IV 1 l A l A A l 1 V 1 A<B A B A>B A B II A<B C A, B A<C<B B A<B<C A<B (<C) B<A (<C) V 1 1 2 1 2 1 2

54 1 I T C H T C H 1 2 C 1 T 2 T 2 C l, m I 2 2 T C 1 C l C m 1 2 2 H α, β T α, β C I 7 T A 2 H α, β H A 1 H T B I 2 2 T A, B C l 1 I 6 C l 2 T A, B H α C l α 1 I 6 C l 2 T A, B H β C l β C l 2 H α β C 2 I 7, I 2, I 6, I 6 2 T C H A, B, l, m, α, β,

1.4 19 55 p.48 P, Q P, Q P Q P Q P Q P Q P Q P Q P Q P Q P(n) n P(3) P(4) m, n P(m, n) m < n np(5, n) n(5 < n) 5 n m( n(m < n)) m m < n n 54 54 1931 Kurt Gödel 1906 1978

56 1 a a R R I i) x, y R R x, y x + y R 1 x + y = y + x, (x + y) + z = x + (y + z). 0 R x R x + 0 = x. 0 x R x R 1 x + ( x) = 0. x + ( y) x y ii) x, y R xy R 1 xy = yx, (xy)z = x(yz), (x + y)z = xz + yz. 1 R x R 1x = x. 1 x 0 (x R) 1/x R 1 x(1/x) = 1. x(1/y) x/y R 0

1.4 19 57 II. i) x, y R 1 x < y, x = y, x > y. x < y y > x x < y x = y x < = y x < = y y < = z x < = z. x = z x = y y = z x = z x < = y y < z x < z x < y y < = z x < z ii) x < = y x + z < = y + z, x < = y, 0 < = z xz < = yz. x < y, 0 < z xz < yz x > 0 x x < 0 x x III. i) x { x x > x = = 0 x x < 0. A, B R A, B A B = R, A B =, a A, b B a < b. A, B (A B) R (A B) x R x A a A(a < = x), b B(x < b) x B a A(a < x), b B(x < = b) A = (, x], B = (x, ) A = (, x), B = [x, ) R 1

58 1 2 3 x(0x = 0) x + 0 = x (x + 0)x = xx = x 2 ( ) x 2 + 0x = x 2 (x 2 + 0x) x 2 = x 2 x 2 0x = 0 1 > 0 1x = x 1 2 = 1 x 2 > 0 (x 0) x > 0 x 2 > 0x = 0 x 2 > 0 x < 0 x x < 0 x 0 < x x > 0 x < 0 x ( x) < 0 ( x) = 0 x + ( x) = 0 x 2 + x ( x) = 0 x ( x) = x 2 x 2 = x ( x) < 0 x 2 < 0 x 2 > 0 (x < 0) x 0 x 2 > 0 1 = 1 2 > 0, 1 > 0 1 0 x (1x = x) 1 = 0 1x = 0x = 0 x(0 = x) 0 < a < b (a, b R) an > b n R a (> 0) b a b 55 S R S x = sup S R S B s S b B(s < = b) B R A A B = R A B = A, B A a A b B(a < b) (A B) A B x S m s S (s < = m) m B x S 55 R S s S (s < = b) S b S x 1 s S (s < = x) x 2 a < x s S (a < s < = x) x x S x = sup S

1.4 19 59 s S s < s s S S S B = (S B) R A B =, A B = R S A A x s S (s < = x) x S A A B x S S B S n N(an > b) N { an } b 0 < an < a(n + 1) { an } { an } x n N(an < = x) a > 0 x a < x x a { an } n N(x a < an) n N(x < a(n + 1) (n + 1) N n N(x < an) x { an } n N(an > b) n N(n > b/a) 2 3