() 800 ( p.38) 50 8 8 8 64 64 r ( r r ) ( ) 6 = r = 56 8 r r = 56 8 = 3.60438 3.6 3.6 3.6 800 8 AD BC ( ) ( 4 3 3 ) = 8 8 = 63 63 64 = 8 ( ) ( 3 = 8 )



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3 (r) (r ) 3.4 r l l r dx x 0 εριϕέρεια (William Oughtred : 573 660?) 8 (Leonhard Euler (707783) (Introductio in Analysin Innitorum : 748 ) ( 8 ) a 0x n + a x n + + a n x + a n = 0 (algebraic number) (transcendental number) k x k = 0 a x a = 0 ( p e = lim + ) n = = n n n! n=0.788 884 p e e pi + = 0 (i : i = ) (Johann Heinrich Lambert : 78777) 76 (Carl Louis Ferdinand von Lindemann : 853) 88 n 500 = 3.45 6535 873 3846 6433 837 5088 47 63 3750 580 7444 530 7864 0686 08 8680 3485 34 7067 848 0865 383 06647 0384 4605 5058 37 5354 088 48 7450 840 703 85 0555 6446 48 5430 386 4488 075 6653 3446 8475 6483 37867 8365 70 0 45648 566 34603 4860 4543 6648 333 6076 04 473 7458 70066 0635 5887 4885 00 68 540 75 36436 785 0360 033 05305 4880 4665 384 465 45 604 33057 7036 5755 53 08 673 83 67 305 8548 07446 37 674 56735 8857 574 8 738 830 4

() 800 ( p.38) 50 8 8 8 64 64 r ( r r ) ( ) 6 = r = 56 8 r r = 56 8 = 3.60438 3.6 3.6 3.6 800 8 AD BC ( ) ( 4 3 3 ) = 8 8 = 63 63 64 = 8 ( ) ( 3 = 8 ) ( ) ( ) 4 6 =.44 ( ) 4 ( ) 8 B A 8 D C 3

() BM854 ( 4 pp.505) A B,0 C,0 AB ( ) 4 4 0 6 AC 0 6,40 BC = 6 4,6 4,6 6,40 AB =,4,4 ABC AB 0, 0 = 60 3 3 ( 4 p.56) 3; 7, 30 = 3 + 7 60 + 30 3600 = 5 = 3.5 8 YBC8600 ( 4 p.56) kù² 0;5 0;5 ( ) 0;0,5 4,48 sìla sìla ( ) 4,48 0;0,,30 sìla 0;0,5 0;0,5 0;5 kù² d l h = 6 ²u-si V = h = 6 ²u-si ( 0cm) l d h = 6 ²u-si ( ) l h = 4 l h V 4,48 4 4,48 0;4,48 0; 4, 48 = 4 60 + 48 3600 = 5 4 = 5 = 5 8 3

6 ( 3 p.7) : rod nindan = cubit kù² 6 m (5.4 m) cubit kù² = 30 nger ²u-si 0.5 m (0.45 m) chain = 60 cubit kù² = 5 rod nindan cable U = 60 rod nindan league = 30 cable U : sar ( ) = rod nindan 36 m (35.836 m ) sar ( ) = sar ( ) cubit kù² 8 m 3 (7.46538 m 3 ) ubu = 50 sar iku = ubu = 00 sar eshe e²è = 6 iku bùr = 3 eshe e²è : sìla l bán = 0 sìla bariga PI = 6 bán gur = 5 bariga PI : mina ma-na = 60 shekel gín 0.5 kg (0.505 kg) talent gù = 60 mina ma-na shekel gín = 80 grain ²e YBC8600 kù² 0; 5 = 5 60 = rod nindan S A O O A B P B Q C Q C R T R 4 8 ( pp.55-56 ) OABC OAC OABC = = S = 6 = BQ = CQ = CR = x OST OB = BT = OT = BT = 4 3 3 OQT = OBT OBQ OT CQ = BT OB BQ OB x = x 3 3 x = + 3 = ( 3) OQR = QR OC = 4( 3) S = 4( 3) = 48( 3) 4 < 4 < 48( 3) + 3 < 4 < 3 4 = 5 (OB : OT = BQ : QT ) x 4

(3) (Anaxagorac (Anaxagoras) : 500? 48?) (<IppokrĹthc Qĩoc (Hippocrates of Chios) : 440 ) E C G ACB ABC F H AC BC AEC BGC AECF A BGCHB ABC A D B (Euclid (Eukleides : EÎkleÐdhc) : 300 ) (StoiqeÐwsic) ( 5 p.367) 5 3 () ( ( ) ) () ( ) (3) 3 ( 3 ) 3 3 p (?) 3 ( ) (>Arqimădhc (Archim ed es) : 87? ) (KÔklou mètrhsic (Dimensio Circuli)) ( ) : 4 ( 5 p.483) 5

6 3 3 7 0 7 3 0 7 < < 3 7 K L C M K L C Z H J J H Z B ( 5 pp.484487) E E A A CZ 6 CH CJ 4 CK 48 CL = CM 6 EC : CZ = 3 : > 65 : 53 EC : CH > 57 : 53 EC : CJ > 6 + /8 : 53 EC : CK > 334 + /4 : 53 EC : CL > 4673 + / : 53 AC EC 6 LM CL AC : ( 6 ) > 4673 + / : 4688 4688 4673 + / < 3 + 7 < 3 + 7 CB 6 CH CJ 4 CK 48 CL 6 AC : CB = 560 : 780 AC : CH < 303 + / + /4 : 780 AC : CJ < 838 + / : 40 AC : CK < 00 + /6 : 66 AC : CL < 07 + /4 : 66 CL 6 ( 6 ) : AC < 6336 : 07 + /4 6336 07 + /4 > 3 + 0 0 > 3 + 7 7 0 7 = 0.408450704 7 3.40845 < < 3.4857 = 0.485748 6

(4) 5000 500 30 070 ( ) 600 0 (834) 3 3 : 0.50 m = 0 = 0 = 6 = 300 405 m = 0 = = 0 0 = 0 : = 40 = 00 : 0.4 dl = 0 = 0 = 0 = 0 = 0 : 6 g = 6 = 5 = = 4 = 4 = 6 (6807) 30 ( ) 63 (74) ( ) ( ) 46 7

( : ) ( 6 pp.3536) 4 8 < > (l = S x S = xr l = xr = 3x r = (r) 3x l = = r x ) ( 6 p.3) 3.4 64 6 < < 3.4 65 65 3.4 ( : 4500) 7 ) 355 ( ) 3 (636 ) 3.456 < < 3.457 8 (

(5) 600 800 (Chandraguputa : 37? 3?) 37 3 (Ashoka : 68? 3?) 3 (Kanishka : 30?55?) 5 (Chandraguputa : 30?355?) 6 5 (Mahabharata) (R am ayana) 5 (±ulbas utra : ) 6 88 70 (Bakhshali manuscript) 4 5 ( Apastamba ulbas utra) ( 7 p.40) 3 ( ABCD) ( ) ( ) (M) ( ) ( ) (M ) (M AD AD G E ME ) (GE) 3 (GF) (MG ) (M MF ) ( ) ( )

A B E F M G D C a ABCD M MA 3 4 AD ME AD ME G ME GE 3 F MF ( ) ( MF = a + ) a = + a 3 6 ( + 6 ( ) a) = a 6 = + = 8(3 ) 3.0883 ( Aryabhaṭa : 476?550?) ( Aryabhaṭ ya) ( 7 p.6 p.8) 7 ( ) 0 04 8 6000 ( ) 0000 7 l d S V S = l d V = S S 3 ( = 04 8 + 6000 = 0000 ) (Bh askara : 4?85?) (L l avat ) ( 7 p.3) 37 50 ( ) ( ) 7 ( ) 4 0

(6) n l n n L n ( l n < < L n ) L n = l nl n l n + L n ( p.8) O M n N n M n N n P n P n A Q n Q n l n = l n L n O r OA P n OQ n = 360 n P n OQ n = 360 n OP n OQ n AOP n AOQ n M n N n n P n Q n n M n N n n P n Q n n l n = M n N n n L n = P n Q n n l n = M n N n n L n = P n Q n n 50 6 ( 60 ) ( ) 50 l 6 = 50 6 = 300 L 6 = 6 = 00 3 346.4065 3 L = 300 346.4065 300 + 346.4065 3.53030(43) l = 300 3.53030 30.58854(4647) L = 300 00 3 300 + 00 3 = 0000 3 00(3 + 3) = 00 3 3 + 3 = 400 00 3 = 00( 3) 3.53030(7347) l = 300 00( 3) = 360000( 3) = 600 3 = 600 = 600 4 3 3 = 300 ( 3 ) = 300( 6 ) 30.58854(305)

00 l L 30.58854 00 < < 3.53030 00 3.058854 < < 3.53030 3.058854 + 3.53030 = 3.606045 4 30.58854 3.53030 L 4 = 30.58854 + 3.53030 L 4 = 300( 6 ) 00( 3) 300( 6 ) + 00( 3) 35.654 l 4 = 30.58854 35.654 33.6863 l 4 = 300( 6 ) 400( 3)( 6 ) 6 4 3 + 8 = 00( 3 ) 3 6 4 3 + 8 33.6863 = 400( 3)( 6 ) 6 4 3 + 8 35.654 l 4 + L 4 00 3.36863 + 3.5654 = 3.4644775 5 ( ) p n l n L n 6 300 346.4065 3.30508075 30.58854 3.53030 3.606045 4 33.6863 35.654 3.4644775 48 6 384

(7) (François Viète : 540603) = + + + (John Wallis : 66703) = (k)(k) (k )(k ) = 4 4 6 6 8 8 3 3 5 5 7 7 k= (James Gregory : 638675) (Gottfried Wilhelm Freiherrvon Leibniz : 64676) 4 = 3 + 5 7 + + arctan x = x 3 x3 + 5 x5 7 x7 + x 67 673 3 (Isaac Newton : 6477) 6 = arcsin = + 3 3 + 3 4 5 5 + 3 5 4 6 7 4 (Carl Friedrich Gauss : 777855) 4 = arctan 8 + 8 arctan 57 5 arctan 3 5 4 = arctan + arctan 3 6 (John Machin : 68075) 4 = 4 arctan 5 arctan 3 7 + 7 (Charles Hutton : 73783) 4 = 3 arctan 4 + arctan 5 3

arctan x = x 3 x3 + 5 x5 ( ) k 7 x7 + = k + xk+ ( x ) arctan 5 5 ( ) 3 + ( ) 5 ( ) 7 + ( 3 5 5 5 7 5 5 k=0 = 5 3 5 3 + 5 5 5 7 5 7 + 5 0.73556650736507 4 arctan 5 0.785846603746037 arctan 3 3 3 4 ( ) 3 + 3 5 ( ) 5 3 7 ) ( ) 7 + 3 = 3 3 3 3 + 5 3 5 7 3 7 + 3 0.0048407600074738645 0.785846603746037 0.0048407600074738645 = 0.7853870600873 ( ) 3 3.4568404357 7 arctan 5 5 ( ) 3 ( 3 5 + ) 5 5 5 7 ( 5 ) 7 + ( 5 ) ( 5 ) + 3 = 5 3 5 3 + 5 5 5 7 5 7 + 5 5 + 3 5 3 0.735558508535 4 arctan 5 arctan 3 3 3 + 3 ( ) 3 3 + 5 ( 3 ) 3 ( ) 5 3 7 ( ) 7 3 + ( ) 3 = 3 3 3 3 + 5 3 5 7 3 7 + 3 3 + 3 3 3 0.00484076000747386 4 ( 5 ) 3 ( ) 3 4

(8) (Georges Louis Leclerc Comte de Buon : 707788) (Monte-Carlo method) 45 (John von Neumann : 0357) a l l < a l a l < a O x θ 0 x a 0 θ p x θ a x O l θ x a x = l sin θ θ x l sin θ p = = pa p 0 l sin θdθ = l pa ( ) 855 (Augustus de Morgan : 80687) a = 5 l = 3 304 3 ( p.6) 3 5 = 8.5 304 = 6 5 304 8.5 3.55354446 = 6 5 304 4 = 67 85 3.4764706 5

= 6 5 304 3 = 4 6065 3.6665 6 (Johann Rudolf Wolf : 8683) 5000 53 a = 45 mm l = 36 mm ( p.) 0 0 3 4 5 6 7 8 500 45 5 53 5 53 50 48 36 53 5 000 3 6 03 0 3 7 4 5 0 06 500 7 000? 80 ( 55) 77 ( 5) 3 M. ( ) ( ) 6 ( 37) 4 000 ( ) 5 ( ) ( ) 7 ( 47) 6 ( ) ( ) 80 ( 55) 7 ( ) ( ) 80 ( 55) 8 ( 55) 3 ( 5) 77 ( 5) 0 G. ( ) ( ) 6 ( 8) ( ) ( A3) 78 ( 53) 3 Victor J. Katz(ed.) The Mathematics of Egypt, Mesopotamia, China, India, and Islam. A Sourcebook Princeton U. P. 007 6

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