ヘンリー・ブリッグスの『対数算術』と『数理精蘊』の対数部分について : 会田安明『対数表起源』との関連を含めて (数学史の研究)

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$(1628) Tables des Sinus, Tangentes et Secantes; et des Logarithmes des Sinus et des Ozanam,, $J$ Tangentes; & des Nombres depuis l unite jusques \ a 1 $10\theta 00($ ( ) 10000 ( ) 4,1670( ), 1685( 1$

1 : 1 RJMS (Henry Briggs, ) $Ar ithmetica$ logarithmica ( 1624) (Adriaan Vlacq, ) 1628 [ 2. (1628) Tables des Sinus, Tangentes et Secantes; et des Logarithmes des Sinus et des Ozanam,, $J$ Tangentes; & des Nombres depuis l unite jusques \ a 1 $10\theta 00($ ( ) ( ) 4,1670( ), 1685( 1 ( 1800 ) 2 Henry Briggs, Arithmetica logarithmica( ) (John Napier, ) Logaithmoim chilias prima Aithmetica logarithmica ( ) ( ( ), ) 1628 (Adriaan Vlacq, ) ( ) ) $)$ Arithmetica $logar\dot{\tau}thmica$ ( ) 1659 (Ferdinand Verbiest, ) 3. (1685) 14 1 IREM, Histoire de Logarithmes(2006, [7]) P.113 Bruce, Biographical Notes on Henry Briggs. (2004, [3]) p.l 2 $(1983b,$ $[6])$ p.450, (1992, [10]) pp (2002, [11]) p.225

$1 ( )10 96,97 11 5 6, 00045005013 [ $ --$ $-b_{-\text{ }\underline{=}}^{-}-----\underline{=}o- $ $\xi$i Vlacq 90 70 20 6207,8077,8642,8832,9176,9354,9706,9972,12328,12398,14763,15306,16461,17509,$

2 215 5 (Joachim Bouvet, ) (Jean Frangois Gerbillon, ) 1709 ( ) 1721 ( ) ( [ ] )5 1 lo lo 10 3 [ ( ) 3.1 ( )10 96, , [ $ --$ $-b_{-\text{ }\underline{=}}^{-}-----\underline{=}o- $ $\xi$i Vlacq ,8077,8642,8832,9176,9354,9706,9972,12328,12398,14763,15306,16461,17509, 19107,19113,19195,78700,99090 Glaisher(1872)p.258 Vlacq Vlacq Lefort Glaisher(1872)p 258 Glaisher 4 $=$ (1723, [15] $p259$) 5 (1990, [14]) pp

$$\sim$ 216 Glaisher(1872)p 258 1 32 ( ) 4 1. ( ) 2. 10 $(0 \leq n\leq 54)$ 3. 3 4. Radix Method 1 4 Radix Table [ 4 1 32.1 1 5 Mirifici logarithmorum canonis constructio, 1619 pp.$

3 $\sim$ 216 Glaisher(1872)p ( ) 4 1. ( ) $(0 \leq n\leq 54)$ Radix Method 1 4 Radix Table [ Mirifici logarithmorum canonis constructio, 1619 pp The Construction of the Wonderful Canon of Logarithms (1889; $)$ reprint 1966, [12] pp pp Some Remarks by the learned Henry Briggs on the foregoing Appendix (pp.55-63) 1 $\sim$

4 217 $\log_{10}1234= \cdots(\leftrightarrow 1234=10^{ }\ldots)$ ( ) $4-1=3$ ) 3 2. $\log_{10}$ 1234 ( $10^{3}\leq 10^{ }\ldots<10^{4}$ 1234 $3+1=4$ $N$ $-1=[\log_{10}N]$ ( $[]$ ). $\log_{10}2$ $2^{10^{14}}$ (1624) 14 ( $[\log_{10}2^{10^{14}}]=[10^{14}\log_{10}2]$ ) $2^{10^{14}}$ $-1=[10^{14}\log_{10}2]$ 1014 ( 14 ), $\log_{10}2$ ( ) ( $10^{14}\log_{10}2=$ $10^{14}$ $\log_{10}2$ ). $\log_{10}2\approx\underline{(2^{10^{14}}\text{ })-1}$ 5 1: Briggs(1624), $ $ 5 p.8 ( ) 1 2 ( 15 (16 )

$(n 218 ) $[]$ 2 3 1 ( ) 2 20,21 3 4 1 1 1 2 2 2 2 2 3 2 4 1 3 $10^{n}$ $n$ 4 2 $\cross$ ($m$ ) ) $(m+n)$ $(m+n-1)$ $(m+n)$ $(m+n-1)$ 3 3 2 121 258.$

5 (n 218 ) $[]$ ( ) 2 20, $10^{n}$ $n$ 4 2 $\cross$ ($m$ ) ) $(m+n)$ $(m+n-1)$ $(m+n)$ $(m+n-1)$ $(3+3-1)$ $ =241$ $\cdots$ $\cdots$ $160\cross 666=106560$ $(3+3)$ $61+241=302$ 6. $10^{14}$ Briggs(1624), $\log_{10}2^{10^{14}}=10^{14}\log_{10}2$ $\sim$ 5 p ,99956, $10^{14}\log_{10}2$ 1 Briggs(1624), $10^{14}$ p ,99956,6398 $\log_{10}2$ ( ) $\log_{10}2$ 20 $\log_{10}2= $ 14 7 $(10^{}$ $\log_{10}7)$ 2 2 $2^{2}=4,4^{2}=16,16^{2}=256,$ $\cdots$ $2^{2^{S7}}$ 7. $\log_{10}2^{ }= $ $\log_{10}2=$ $ / = $ $\cdots$ $\log_{10}2^{ }= $ $\log_{l0}2= / = \cdots$ 6 $68\cross 26=1768,14\cross 68=952$ 2 7 F ( [17])

6 219 $\log$ 1 $0^{2=} \cdots$ $10^{n}$ $2^{2^{n}}$ 1 $N$ $n$ $N^{2}$ $N$ 2 31 $31^{2}=961$ 3 $(n+n-1)$ $N$ $7^{2}=100489$ 6 $(n+n)$ $N$ $6^{2}=99856$ $(n+n-1)$ $N$ $= $ $(n+n)$ $n$ $N$ $2$ 1. $N$ 4 $N^{2}$ $2n$ $2n-1$ ( ) $n$ $\cdots$ $=3\cross 10^{n-1}-1$ 2 $3\cross 10^{n-1}-1$ $n$ 2 $2n$ $2n-1$ $(3\cross 10^{n-1}-1)^{2}=9\cross 10^{2n-2}-6\cross 10^{n-1}+1<10\cdot 10^{2n-2}=10^{2n-1}$ 1 2 $n$ 2 $2n-1$ $n$ 3 $T(\in \mathbb{n})$ 2 $2n$ $10^{2n-1}\leq T^{2}<10^{2n}$ $(n=1,2,3, \cdots)$ $\sqrt{1010^{2n-2}}\leq T<\sqrt{10^{2n}}$ $10^{n-1}\sqrt{10}\leq T<10^{n}$ (1) $T$ $10\sqrt{10}\leq T<10^{2}$ (1) $\cdots\leq T<100$ $32\leq T<100$ $(\cdot.\cdot T\in \mathbb{n})$ $T$ $10^{2}\sqrt{10}\leq T<10^{3}$ $\cdots$ $\leq T<1000$ $317\leq T<1000$ $(\cdot.\cdot T\in N)$

$$\lceil$ 220 $[10^{n-1}\sqrt{10}]=[10^{n-1}\cross 3.162277660\cdots]$ $N$ 5 $N^{2}$ 1. 2 32 30 31 2. 2 31 3 317 315 316 3. 3 316 4 3163 3161 3162 4. $[10^{n-1}\sqrt{10}]=[10^{n-1}\cross 3.162277660\cdots]$ $k$ 1 $N^{2}$ 3 $\sim$4 2 1 2 C 2 2 3 2% 3 4 0.$

7 $\lceil$ 220 $[10^{n-1}\sqrt{10}]=[10^{n-1}\cross \cdots]$ $N$ 5 $N^{2}$ $[10^{n-1}\sqrt{10}]=[10^{n-1}\cross \cdots]$ $k$ 1 $N^{2}$ 3 $\sim$ C % % % 2 31 $\cross 32=992$ $310\cross 320=99200$ $315\cross 320=100800$ $\cross 32$ 3 1 2% 322 $[i$ $\sim$ $E4$ Ozanmi,, Tables des Sinus, Tangentes et Secantes; et des Logarithmes des Sinus $J$ $1\theta\theta\theta\theta$ et des Tangentes; & des Nombres depuis l unite jusques \ a ( $)$ $)$ ( ), 1685( ) 3 De la construction des Logarithmes ( ) (1992) P $8H$. verhaeren, catalogue of the Pei-T ang Libmry

8 Ozanam $Min\dot fici$ logarithmorum canonis con- The Construction of the Wonderful Canon of Logarithms (1889; $)$ [12] pp.50-51) 9. Cajori(1931, structio, 1619 ( reprint 1966, $\log_{10}5$ Perhaps suggested by Napier $s$ remarks in the Constructio, this method was developed by French writers, of whom Jacques Ozanam ( I7) in 1670 was perhaps the first. [4])p.155 Ozanam Ozanam(1685, 1670) $\log_{10}9$ Ozanam $\log_{10}9$ $\log 1=0$, log lo $=$ 1, $1<9<10$ $a,$ $b$ $\frac{\log a+\log b}{2}=\frac{1}{2}\log$ $ab=1og\sqrt{ab}$ (2) $\sqrt{}$ ab $\frac{0+1}{2}=0.5$ $\sqrt{}$ 1 $\cross$ $\log 1=0$ $\log 10=1$ ( 8 ) $=$ $\log\sqrt{10}=\log =0.5$ 1 10 $\sqrt{10}= $ 9 2 $\sqrt{10}= $ $\log\sqrt{10}=0.5$ $\log 10=1$ $\frac{0.5+1}{2}=0.75$ (2) $\sqrt{10\sqrt{10}}=\sqrt{10\cross }= \cdots$ 8 (2) $\log =0.75$ 9 2 Ozanam $\log 9= $ 9 Ozanam 7 8 ( ). ( [15]pp ) 10 9 Constructio(1619) ko

$222 $\log 9=0.9542425125$ $\log 9=0.$

9 222 $\log 9= $ $\log 9= \cdots$ 7 Ozanam 8 26 [ Ozanam(1670( 1685)) Ozanam 10 2: Ozanam(1685), 3 pp ( ) Napier-Ozanam ( 2001, pp ).

10 223 4 ( ) (1800) 10. Ozanam $A\searrow$ 2 10 (1795, [1]) 1795 (2008, [9])pp $\log_{2}2=1,$ $\log_{2}4=$ $2,2<3<4$ $\log_{2}3$ $\frac{1+2}{2}=1.5$ $\sqrt{2\cross 4}= $ (2) $\log$24$f\cross$ 4 $=\log$ 2 $ =1.5$ $ <3<4$ 10 $\log_{2}3= $ 6 $\log_{10}3=\frac{\log_{2}3}{\log_{2}10}$ $\log_{2}10= [8]$ ( $\log_{2}10= \cdots$, ) 8 $\log_{10}2=1/ = $ $\log_{10}3= $ $ / = \cdots$, $ / =$ Ozanam Ozanam 5 $10$ (1981, [18]) p.9 11 [2]

11 ffi $\sim$ Ozanam(1670) 3 Constructio Ozanam 5 $)$ (1628) ( (1628) Ozanam(1670) Ozanam 2 Ozanam(1670) [ 1614 $Min\dot fici$ logarithmorum canonis descrip ( )

12 225 [1] (1795) 1016( (1881) ) [2] 1426 [3] Bruce, Ian(2004), Biographical Notes on Henry Briggs., 2004 $<$ http: $//www$-groups. dcs. st-and. ac. $uk/^{\sim}history/misceilaneous/briggs/$ index. html $>$ ( ) [4] Cajori, Florian(1931), History of Mathematics. New York: The Macmillan Company $A$ [5] (1983a) : 1959 ( 1 ) ;1983 ( 2 ) [6] $\mathscr{c}$ ( 2 ) (1983b) : 1960 ( 1 ) ;1983 [7] IREM(2006), Histoire de Logarithmes. Paris: Ellipses [S] Glaisher, J. W. L.(1872), On Errors of Vlacq s (often called Briggs s or Neper s) Tables of Ten-figure Logarithms of Numbers, Monthly Notices of the Royal Astronomical Society, Vol.32, pp [9] (2008) : [10] $\rangle$ (1992), { , pp [11] (2002) ( ) : [12] Napier, John(1899), The Construction of the Wonderful Canon of Logarithms, translated by W. R. Macdonald. Edinburgh and London: William Blackwood, ISS9; reprint, London: Dawsons of Pall Mall, 1966 [13] Ozanam, $J(1685)$, Tables des Sinus, Tangentes et Secantes; et des Logamthmes des Sinus $1\theta et des Tangentes; & des Nombres depuis l unite jusques \ a 0\theta\theta$, Paris, google. $com/books?i$d NwVBAAAAcAAJStpg-PA145&dq#v onepage&q&f $=$ $=$ $=f$ alse $>$ 9 ) http: $//books$. $<$ ( [14] (1990) ( ) : [15] ff 106, 800 [ ]: [1985] ( ), pp ([ ) $ f$ [16] 106, 801 ( ), [ ]: [1985] ( ), pp [17] (1823 ) 6420 [18] F 1981

i) M C F Richter : ) km 2800) A µm) M L = log 0 A ) ii) ph ph mol/l [H + ] ph = log 0 [H + ] = log 0 [H + ] 909 Søren Pete

i) M C F Richter : ) km 2800) A µm) M L = log 0 A ) ii) ph ph mol/l [H + ] ph = log 0 [H + ] = log 0 [H + ] 909 Søren Pete Pierre Simon Laplace : 749827) 2 Christopher Columbus : 45506) Vasco da Gama : 469524) 400 650 024 2 0 ) 048576 2 20 ) 024 048576 07374824 log 2 024 048576) 2 30 0 + 20 30 2 n!! log a MN = log a M + log