= = = =

2 24 1 + + 51 + 10 = 30547 60 60 2 60 3 21600. = 1.414212962 2. = 1.414213562373 2 30 25 42 + + 35 = 30547. = 42.42638888 60 60 2 720 30 2. = 42.4264068711

.. διά γωνία μέσος ποταμός Εὔϕράτης Τίγρις σύστημα Βαβυλών

M r 1 50 28 48

49 = 40 + 9 a a = a n 60 n +... + a 1 60 + a 0.

a n,..., a 1, a 0 a a = a n... a 1 a 0. 91 = 1 60+31, 3899 = 1 60 2 +4 60+59, 9 746 373 = 45 60 3 +7 60 2 +19 60+33. 95 = 1 60 + 35, 3635 = 1 60 2 + 35, 216035 = 1 60 3 + 35 60 1 60 2 60 1

1 60, 31 60, 59 2 60,... 3 a = a n 60 n +... + a 1 60 + a 0 + a 1 60 + a 2 60 2 + a 3 60 3 +... a a = a n a 1... a 0 ; a 1 a 2 a 3... a n, a 1,..., a 0 ; a 1, a 2, a 3,... 911/240, 71/480, 7801/600 911 240 = 3 + 47 60 + 45 60, 71 2 480 = 8 60 + 52 60 + 30 2 60, 7801 1 = 2 60 + 10 + 3 600 60,

355/113 π 355 113 = 3 + 8 60 + 29 60 2 + 44 60 3 + 4 60 4 + 14 60 5 +..., 355 113 = 3; 8 29 44 4 14..., π 3 = 3, 125 = 25/8 = 3; 7 30 a, b c a 2 + b 2 = c 2 (a, b, c) (a, b, c) a, b, c (a < c, b < c) a 2 + b 2 = c 2 a : b : c a, b, c a = m 2 n 2, b = 2mn, c = m 2 + n 2, m, n m > n m, n Πυθαγόρας

a 2 + b 2 = c 2 ϕίλος ἄνθρωπος c a a 2 /b 2 (a, b, c) b = c 2 a 2 a 2 /b 2 a c b n 0; 59 0 15 1; 59 2; 49 0; 48 54 1 40 1; 5 1; 37 0; 33 45 45 1; 15

(3367, 3456, 4825) a 2 /b 2 11336689/11943936 0; 56 56 58 14 50 6 17 49... a 2 /b 2 0; 56 56 58 14 50 6 15... (45, 60, 75) = 15(3, 4, 5). ἁρπεδονάπται ἁρπεδόνη ἅπτω

n a b c xy = (x + y)2 (x y) 2. 4 x + y = a, xy = b a b b = a 2 /4 ((x y)/2) 2 x + y 2 x y 2 = a 2, (a ) 2 = b. 2

x = a 2 + (a 2 ) 2 b, y = a 2 (a ) 2 b. 2 a 2 4b x, y xy = 60. N

x 1 > 0 N x n+1 = 1 (x n + N ), n = 1, 2, 3,..., 2 xn N x 1 N N/x1 = x 1 x 1 < N N/x 1 > x 1 x 1 > N N/x 1 < x 1 x 2 x 1 N/x 1 x 1 N/x 1 x 2 = 1 (x 1 + N ). 2 x1 x 2 x 3 N x 1

5 603 + 42 60 2 + 56 60 + 7 = 1 234 567 1 234 567 1 23 45 67 =. 1000 = x 1 x n N = 1 234 567 n x n 1 234 567. = 1111, 110705. = N N = a 2 + b a b b < a 2 N = a 2 (1 + b/a 2 ) = a 1 + b/a 2 a(1 + b/(2a 2 )) = a + b/(2a) = (1/2)(a + a + b/a) = (1/2)(a + N/a). x 1 + x 1 + x = 1 + x/2 x 2 /8 +...

x 1 + x x 1 + x/2 1 + x/2 1 + x x = 0 a + b/(2a) a2 + b x 2 1 = N x 1 = x 2 = x 3 =... = N x 2 1 N x 1, x 2, x 3,... N n 1 x n+1 = 1 (x n + N ) > 2 xn x n N x n = N, x n N n 2 n x n x n+1 = x n 1 (x n + N ) = 1 (x n N ) 2 xn 2 xn = x2 n N 2x n > 0. x = n x n x = (x + N/x)/2 x = N

f x 1 f x n+1 = x n f(x n) f (x n ) N f : x x 2 N x n+1 = x n x2 n N = 1 (x n + N ), 2x n 2 xn 2 x n+1 = 1 (x n + 2 ), x 1 = 3 = 1; 30. 2 xn 2 x 2 = 17 12 = 1; 25, x 3 = 577 = 1; 24 51 10 408

N N f : x (x + N/x)/2 [ N/3, ] x x ξ x x f(x ) f(x ) = f (ξ)(x x ). (1 N x 2 ) f (x) = 1 2 x [ N/3, ] 1/2 f(x ) f(x ) < 1 2 x x. f x x = 1 ( x + N ). 2 x x = N x 1 [ N/3, ]

x 1, x 2, x 3,... x n+1 = f(x n ) x = N y = f(x) y = x ( N, N) x 2 + px = q, x 2 = px + q, x 2 + q = px p, q, r ax 3 + bx 2 = c a, b, c ( 1 + 5) 1 x = 2. 3 < x < 4 x = 4 2; 22 42 55 46.

2 3 4 5 6 7 2 2222 22 22 222222 222 222 442 44 2 777 777 777 666 555 666 555 666 555 444 444 444 333 333 333 222 222 222

222 222 222 222 222 222 222 222 222 22 333 333 333 22 222 22 222 222 222 222 222 222

2 3 4 5 6 7

22 2 r 1 3 = r 1 4 = r 5 12 = 1 4 + 1 6 = r r M 1/2 r s 2/3 r 333 r222 222 1/360

5/7 1/2 + 1/7 + 1/14 1/2 + 1/6 + 1/21 1 = 0, 5 = 0, 49, 2 0, 49 = 4 10 + 9 100 + 9 1000 +... = 4 10 + 9 100 1 1 1 10 = 4 10 + 9 90 = 5 10 = 0, 5. 2/7 = 1/4 + 1/28 r r 22 2/n n ἐϕημερίδος ἐϕημερίς ἐπί ἡμέρα

λεπτά Διόϕαντος ὁ Ἀλεξανδρεύς Θαλῆς ὁ Μιλήσιος Υψικλῆς Ιππαρχος 7 30 180 7 30 30 365 = τξεʹ ʹ σνα χίλιοι μύριοι

αʹ βʹ γʹ δʹ εʹ ϛʹ ζʹ ηʹ θʹ ιʹ κʹ λʹ μʹ νʹ ξʹ οʹ πʹ ϙʹ ρʹ σʹ τʹ υʹ ϕʹ χʹ ψʹ ωʹ ϡʹ α β γ δ ε ϛ ζ η θ ι κ λ μ ν ξ ο π ϙ ϙ θϡϙθʹ 315 467 = 31 10 000 + 5 467 ευξζʹ λα ἑξηκοστὰ τμήμα, λεπτά πρῶτα δεύτερα ἑξηκοστά Κλαύδιος Πτολεμαῖος Μαθηματικὴ σύνταξις Η μεγάλη σύνταξις Η μεγίστη σύνταξις μεγίστη

τμήματα α AB 120 α/2 α AB AB = α = 120 α/2 (180 α) = 120 α/2 2 α+ 2 α = 1 2 α + 2 (180 α) = 120 2. χορδή 1/2 180 72, 36, 54, 18, 9 30, 60, 45, 15, 75 3, 1 30, 45 1 30 1

3 1 α α α/ α πδ π μα γ μϛ κε πε πα δ ιε μϛ ιδ πε πα κζ κβ μϛ γ οὐδέν

M α α = 1/2 ) α α/ α = ((α + α) α)/ α περιϕερειῶν εὐθειῶν ἑξηκοστῶν περιϕερεία εὐθεῖα ἑξηκοστά α α α/ α α α β ν α β ν ξ ξ νδ κα ϙ πδ να ι μδ κ ρκ ργ γε κγ λα ιη λϛ λζ δ νε νθ μγ 60 μοιρῶν ξ 60 = 120 30 = 60 ξ SBA 1 1. = 1; 2 50 = 1 + 2 60 + 50 3600 = 377 360. 360 377/360 = 377 π =

3, 141592653... π. = 377 120 = 3; 8 30 = 3 + 8 60 + 30 3600 = 3.1416. 90 μοιρῶν ϙ 84; 51 10 = 84 + 51 60 + 10 3600 = 30547 360. 90 = 120 45 = 60 2. 30547 24 2 = = 1; 24 51 10 = 1 + 21600 60 + 51 3600 + 10 = 1, 4142129629. 216000 2 =. 1, 414213562373 120 μοιρῶν ρκ 103; 55 23 = 103 + 55 60 + 23 3600 = 374123 3600. 120 = 120 60 = 60 3. 374123 43 3 = = 1; 43 55 23 = 1 + 216000 60 + 55 3600 + 23 = 1, 7320509259. 216000 3. = 1, 732050807568877 36 36 d a τ = (1 + 5)/2 ABD d = 60 a = d/τ 36 = a = 60/τ = 30( 5 1). = 37.08203932 = 37; 4 55 20. 36 λϛ 37; 4 55

Εὐκλείδης Στοιχεῖα ἄκρος καὶ μέσος λόγος α EC

s A B s DE S α/2 DC s ABCD AB DC + AD BC = AC BD. AB = 120 DC = (180 α) AC = BD = (180 α/2)) AD = BC = α/2 120 (180 α) + 2 α/2 = 2 (180 α/2). DEB DEC 2 α/2 + 2 (180 α/2) = 120 2, 2 α + 2 (180 α) = 120 2, 120 (180 α) + 2 α/2 = 120 2 2 α/2. 2 α/2 = 60(120 120 2 2 α). 18 ( 2 18 = 60(120 120 2 2 36 ) = 60 120 120 2 30 2 ( 5 1) 2 ). 18 = 60 120 120 30 10 + 5. 18. = 18; 46 19 41 = ιη μϛ ιθ μα.

2π 2π. = 6; 16 59 28 1 34 51 46 14 50. 2π. = 6, 28318530717958648. π 2π 2π. = 6; 16 59 28 1 34 51 46 14 49 55 12 35 26 8 58 14 20 7, 2π. = 6, 283185307179586476925286766559005768394. π