apier John apier(550-67) 0 2 3 4 5 6 7 8 9 0 2 4 8 6 32 64 28 256 52 024 4 32 = 28 2+5=7 2 n n 2 n 2 m n + m a 0 ;a ;a 2 ;a 3 ; a = a 0 ; r = a =a 0 = a 2 =a = a 3 =a 2 = n a n a n = ar n a r 2 a m = ar m ;a n = ar n a m a n = a(ar n+m ) a 0 ;a ;a 2 ;a 3 ;, m; n m + n a a a 0 a =0 p a p a p p a p 2 a r a a r
a a apier sin 0; 000; 000 = 0 7 sin 0 0; 000; 000 a = 0 7 ;r = 0:9999999 = 0 7 a n =0 7 ( 0 7 ) n n =0; ; 2; 3; n =0; ; 2; 3; ; 00 a n+ =0 7 ( 0 7 ) n ( 0 7 )=a n 0 7 a n 00000000000000 0000000 99999990000000 09999999 9999998000000 09999998 99999970000003 99999000004950 0; ; 2; 3; ; 00 logarithms(= ratio numbers) 9999900:0004950 logarithm 00 log 9999900:0004950 = 00 x =0 7 ( 0 7 ) y log x = y 0; 000; 000 5; 000; 000 6; 900; 000 logarithms x = ar m ;y = ar n y=x = r n m logarithms 9999900:0004950 ' 9999900 = 0 5 a =0 7 ;r = 0 5 0 7 ( 0 5 ) r ; r =0; ; 2; 3; ; 50 2
0000000000000 00000000 9999900000000 99999000 999980000000 99998000 9999700003000 999500224804 log 999500:224804 ' 50 00 = 5; 000: 999500:224804 ' 9995000 = 0000000 5000 = 0000000 5000 0000000 0 7 ( =2000) p ;p = 0; ; 2; 3; ; 20 p = 20 9900473:5780 p = 2 9895523:34 log 99000473:5780 ' 20 5000 = 00000: 9900473:5789 ' 9900000 = 0000000 00, 0 7 ( =00) q ;q =0; ; 2; 3; 68; a pq =0 7 p q ; p =0; ; 2 20; q =0; ; 2 68 2000 00 p q a pq 0000000:0000 9900000:0000 5048858:8900 9995000:0000 9895050:0000 5046334:4605 9990002:5000 989002:4750 50438:2932 9900473:5780 980468:8423 4998609:4034 5; 000; 000 a pq log a pq = plog 9995000 + qlog 9900000 logarithms log 9995000 ' 500:24506; log 9900000 ' 05026:5: 3
apier log 5000000 ' 693472:2 log 5000000 = 693247:8: apier ar n n b ar n bn apier Jost Burgi b =0;a =0 8 ;r =+0 4 apier apier Burgi 0 0 7 0 0 8 0 7 ( 0 7 ) 0 0 8 ( 0 4 ) 2 0 7 ( 0 7 ) 2 0 2 0 8 ( 0 4 ) 2 3 0 7 ( 0 7 ) 3 0 3 0 8 ( 0 4 ) 3 n 0 7 ( 0 7 ) n 0 n 0 8 ( 0 4 ) n ( + 0 4 ) 23;027 ' 0 Burgi n =23; 027 a; b 0 Burgi a =;b=0 4 Bog x = n 0 4 () x =( 0 4 ) n Bog x x 0 0 4 ( + 0 4 ) 2 0 4 ( + 0 4 ) 2 n 0 4 ( + 0 4 ) n n 0 4 = m Bog x = m () x = h (+0 4 ) 04i m : 4
0 4 0 p Bog x = m () x = h ( + 0 p ) 0pi m : ( + 0 4 ) 04 =2:78: ( + =k) k =[(k +)=k] k ;k =; 2; 3; 2 3 2 4 5 6 7 ; ; ; ; ; ; 2 3 3 4 4 5 5 6 6 Euler e e = lim + k =2:782882::: k! k log x = y () x = e y log x x a log a x = y () x = a y log a x a x Leonhard Euler(707-783) Henry Briggs apier 0, 0 log 0 4 20,000 90,000 00,000 (624) Adian Vlacq 00,000 0 628 apier P; L P 0 7 P 0 O P 0 O L L 0 L 0 P t OP x L 0 7 t L 0 L y x y og x = y x =0 7 ( 0 7 ) n og 0 7 ( 0 7 ) n ' n apier dx dt = x; y =07 t: 5
y = og x =0 7 log 07 x : apier og 0 7 ( 0 7 ) n = n ψ+ 0 7 + 0 4 2 3! + ' :00000005 n: Gregory St Vincent 647 x [a; b] y = =x S(a; b) t>0 S(ta; tb) =S(a; b): [a; b] n a = x 0 <x <x 2 < <x n = b; x i x i = b a ; i =; 2; n n [x i ;x i ] =x i S n (a; b) ta = tx 0 <tx <tx 2 < <tx n = tb [ta; tb] n S n (ta; tb) nx tb ta nx b a S n (ta; tb) = = = S n (a; b): tx i n x i n i= n S n (a; b) S(a; b) S n (ta; tb) S(ta; tb) i= S(ta; tb) = lim n! S n(ta; tb) = lim n! S n(a; b) =S(a; b): Vincent AA de Sarasa ( S(;x) x L(x) = S(x; ) 0 <x< <x<y L(xy) =L(x)+L(y) L(xy) =S(; xy)=s(; x)+s(x; xy) =S(; x)+s(; y)=l(x)+l(y): 6
x 3 L(x) =logx Euler ewton, Mercator ewton (642-727) 667 y = x + (x > ) [0;x] A( + x) S(; +x) x + x 2 +x ) +x x x x 2 x 2 x 2 + x 3 ewton A( + x) = = y = +x = x + x2 x 3 + : Z x 0 Z x 0 +x dx dx Z x 0 3 x dx + Z x = x x2 2 + x 3 x 4 4 + : 0 x 2 dx Z x A(( + x)( + y)) = A( + x)+a( + y) ψ! +x A = A( + x) A( + y): +y 0 x 3 dx + x = ±0:; ±0:2; A(0:8);A(0:9);A(:);A(:2) 57 2=(:2 :2)=(0:8 0:9) A(2) = 2A(:2) A(0:8) A(0:9) 7
A(2) A(3);A(5);A();A(0);A(00); ewton icolas Mercator (620-687) 668 log( + x) =x x2 2 + x 3 3 x 4 4 + : Wallis(66-703) 668 Mercator, [0;x] n h = x=n y ==( + x) [0;x], =( + x) A( + x) ' h + h A( + x) n X j= h +jh +jh = jh +(jh)2 (jh) 3 + : ' nh h 2 [ + 2 + +(n )] + h 3 [ 2 +2 2 + +(n ) 2 ] = x x2 3 x [ + 2 + +(n )] + 2 n n 3 [2 +2 2 + +(n ) 2 ] : Wallis 656 n! lim k +2 k + + n k = n k+ k + n! Mercator Wallis k» 0 Wallis log( + x) x< y ==x [x ;x 2 ] log(x 2 =x ) Vincent, Sarasa 660 Mercator 668 Euler Leonhard Euler(707-783) John(=Jean=Johann) Bernoulli (667-748) (727-74,766-783) (74-766) 8
Euler Euler, Euler a x log a x = y a y = x y Euler a 0 =, ffl a ffl =+kffl k ffl x = x=ffl a x = a ffl =(a ffl ) =(+kffl) = a x = lim n! ψ! + kx n n ψ! + kx : ψ! + kx ψ! ψ! 2 ψ! 3 kx ( ) kx ( )( 2) kx =+ + + + 2! 3! =+kx + 2! k 2 x 2 + 3! 2 k 3 x 3 + : 0= = 2 = ψ a x = n! lim + kx n! n =+ kx! + k 2! 2 x 2 + k3 x 3 3! + : a k x = ψ! a = n! lim + k n =+ k n! + k 2 2! + k 3 3! + : Euler e k = a e = n! lim + n =+ n! + 2! + 3! + 9
23 e ' 2:782882845904523536028: e x = n! lim + x n x =+ n! + x 2 2! + x 3 3! + : Euler log e ( + x) =y, +x = e y : +x = e y = + y : ( + x) = =+(y=) y = ewton ( + x) = =+ x + h ( + x) = i : 2! x 2 + = =0 2 y = x + 2! x2 + ( )( 2) x 3 + : 3! y =log e ( + x) =x x2 2 + x 3 3 : 3! x 3 + e log e ( + x) y ==( + x) [0;x] A( + x) Euler R sin A 2A cos A Euler x sin x; cos x sin 2 x +cos 2 x = sin(x ± y) = sin x cos y ± cos x sin y; cos(x ± y) = cos x cos y sin x sin y De Moivre (cos z + i sin z) n = cos nz + i sin nz i = p 0
x x ffl = x= x = ffl: De Moivre z = ffl; n = cos x + i sin x =cosffl + i sin ffl = (cos ffl + i sin ffl) : ffl cos ffl =; sin ffl = ffl (cos ffl + i sin ffl) =(+iffl) = + ix = e ix : Euler cos x + i sin x = e ix e ix Euler x x z = x + iy e z = e x+iy = e x e iy = e x (cos y + i sin y): e z e x y w = e z log w = z w C H Edwards, Jr, "The Historical Development of the Calculus", Springer- Verlag, 979