1739 2011 214-225 214 : 1 RJMS 2010 8 26 (Henry Briggs, 1561-16301) $Ar ithmetica$ logarithmica ( 1624) (Adriaan Vlacq, 1600-1667 ) 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 $10\theta 00($ ( ) 10000 ( ) 4,1670( ), 1685( 1 ( 1800 ) 2 Henry Briggs, Arithmetica logarithmica( ) (John Napier, 1550-1617) 1617 1 1000 Logaithmoim chilias prima Aithmetica logarithmica ( ) 1624 1 20000 90000 100000 14 ( ( ), ) 1628 (Adriaan Vlacq, 1600-1667 ) ( ) 2 1 100000 10 ) $)$ Arithmetica $logar\dot{\tau}thmica$ (1654-1722) 1659 (Ferdinand Verbiest, 1623-1688) 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.111-115 3 (2002, [11]) p.225
215 5 (Joachim Bouvet, 1656-1730) (Jean Frangois Gerbillon, 1654-1707) 1709 (1648-1722) 1721 ( ) 1723 4( [ ] )5 1 lo lo 10 3 [ ( ) 3.1 ( )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, 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.286-287
$\sim$ 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.40-53 The Construction of the Wonderful Canon of Logarithms (1889; $)$ reprint 1966, [12] pp.52-69 pp.53-54 5 Some Remarks by the learned Henry Briggs on the foregoing Appendix (pp.55-63) 1 $\sim$
217 $\log_{10}1234=3+0.09131\cdots(\leftrightarrow 1234=10^{3.09131}\ldots)$ 1. 1234 4 ( ) $4-1=3$ ) 3 2. $\log_{10}$ 1234 ( $10^{3}\leq 10^{3.09131}\ldots<10^{4}$ 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 ), $\log_{10}2$ ( ) ( $10^{14}\log_{10}2=$ 3010299956639811952 1 30102999566398 $10^{14}$ $\log_{10}2$ ). $\log_{10}2\approx\underline{(2^{10^{14}}\text{ })-1}$ 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... 3 258 2 66564 $(3+3-1)$ $121+121-1=241$ 3 4 1 61 $\cdots$ $\cdots$ 160 241 3 666 3 $160\cross 666=106560$ $(3+3)$ $61+241=302$ 6. $10^{14}$ Briggs(1624), $\log_{10}2^{10^{14}}=10^{14}\log_{10}2$ $\sim$ 5 p.8 30102,99956,6399 1 $10^{14}\log_{10}2$ 1 Briggs(1624), $10^{14}$ p.9 30102,99956,6398 $\log_{10}2$ ( ) $\log_{10}2$ 20 $\log_{10}2=0.30102999566398119521$ 14 7 $(10^{}$ $\log_{10}7)$ 2 2 $2^{2}=4,4^{2}=16,16^{2}=256,$ $\cdots$ $2^{2^{S7}}$ 7. $\log_{10}2^{137446953472}=41375655307$ $\log_{10}2=$ $41375655307/137446953472=0.30102999566$ $\cdots$ 0.301029992021 $\log_{10}2^{137438953472}=41373247567$ $\log_{l0}2=41373247567/137438953472=0.301029995658\cdots$ 6 $68\cross 26=1768,14\cross 68=952$ 2 7 F ( [17])
219 $\log$ 1 $0^{2=}0.301029995663981\cdots$ $10^{n}$ $2^{2^{n}}$ 1 $N$ $n$ $N^{2}$ $N$ 2 31 $31^{2}=961$ 3 $(n+n-1)$ $N$ 3 317 31 $7^{2}=100489$ 6 $(n+n)$ $N$ 3 316 31 $6^{2}=99856$ $(n+n-1)$ $N$ 4 3163 31632 $=10004569$ $(n+n)$ $n$ $N$ $2$ 1. $N$ 4 $N^{2}$ $2n$ 2. 1 2 $2n-1$ ( ). 3. 3 2 4 1 2 2 9 1 2 $n$ $\cdots$ 2 2999 $=3\cross 10^{n-1}-1$ 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) $T$ 2 2 4 $10\sqrt{10}\leq T<10^{2}$ (1) 31.622 $\cdots\leq T<100$ $32\leq T<100$ $(\cdot.\cdot T\in \mathbb{n})$ $T$ 3 2 6 $10^{2}\sqrt{10}\leq T<10^{3}$ 316.227 $\cdots$ $\leq T<1000$ $317\leq T<1000$ $(\cdot.\cdot T\in N)$
$\lceil$ 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 0.21% 4 5 0.021% 2 31 $\cross 32=992$ 3 0 9 10 3 100 $310\cross 320=99200$ $315\cross 320=100800$ 1 1 31 2 $\cross 32$ 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 ( $)$ $)$ 1 10000 ( ), 1685( ) 3 De la construction des Logarithmes ( ) (1992) P111 8. $8H$. verhaeren, catalogue of the Pei-T ang Libmry
10 221 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 (1640-17I7) in 1670 was perhaps the first. [4])p.155 Ozanam Ozanam(1685, 1670) $\log_{10}9$ Ozanam $\log_{10}9$ 10 9 2 2 $\log 1=0$, log lo $=$ 1, $1<9<10$ 2 1 10 2 $a,$ $b$ $\frac{\log a+\log b}{2}=\frac{1}{2}\log$ $ab=1og\sqrt{ab}$ (2) $\sqrt{}$ ab $\frac{0+1}{2}=0.5$ $\sqrt{}$ 1 $\cross$ $\log 1=0$ $\log 10=1$ 31622777( 8 ) $=$ $\log\sqrt{10}=\log 3.1622777=0.5$ 1 10 $\sqrt{10}=3.1622777$ 9 2 $\sqrt{10}=3.1622777$ 10 2 9 $\log\sqrt{10}=0.5$ $\log 10=1$ $\frac{0.5+1}{2}=0.75$ (2) $\sqrt{10\sqrt{10}}=\sqrt{10\cross 31622777}=5.62341328\cdots$ 8 (2) $\log 5.62341328=0.75$ 9 2 Ozanam 90000000 26 $\log 9=0.95424251$ 9 Ozanam 7 8 ( ). ( [15]pp.263-267) 10 9 Constructio(1619) ko
222 $\log 9=0.9542425125$ $\log 9=0.9542425094\cdots$ 7 Ozanam 8 26 [ 10 33 Ozanam(1670( 1685)) Ozanam 10 2: Ozanam(1685), 3 pp.37-39 ( ) Napier-Ozanam ( 2001, pp.87-89 ).
223 4 (1747-1817) (1800) 10. Ozanam $A\searrow$ 2 10 (1795, [1]) 1795 (2008, [9])pp.295-303 $\log_{2}2=1,$ $\log_{2}4=$ $2,2<3<4$ $\log_{2}3$ $\frac{1+2}{2}=1.5$ $\sqrt{2\cross 4}=2.8284271247$ (2) $\log$24$f\cross$ 4 $=\log$ 2 $2.8284271247=1.5$ $2.8284271247<3<4$ 10 $\log_{2}3=1.584961$ 6 $\log_{10}3=\frac{\log_{2}3}{\log_{2}10}$ $\log_{2}10=3.32192[8]$ ( $\log_{2}10=3.3219280948\cdots$, ) 8 $\log_{10}2=1/3.321928=0.3010300$ $\log_{10}3=0.4771213$ $1584961/3.321928=0.4771208\cdots$, 04771219 11. $1584961/3.32192=$ 2 10 7 30 2 3 Ozanam 2 3 1 10 3 2 4 24 Ozanam 5 $10$ (1981, [18]) p.9 11 [2]
224 2 10 ffi $\sim$ Ozanam(1670) 3 Constructio Ozanam 5 $)$ (1628) ( (1628) Ozanam(1670) Ozanam 2 Ozanam(1670) [ 1614 $Min\dot fici$ logarithmorum canonis descrip 2014 400 (1780-1830 )
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 $>$ (2008 12 21 ) [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.255-262 [9] (2008) : [10] $\rangle$ (1992), { 11 2 1992, pp.109-119 [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: $//books$. $<$ (2010 8 [14] (1990) ( ) : [15] ff 106, 800 [ ]: [1985] ( ), pp.258-302 ([ ) $ f$ [16] 106, 801 ( ), [ ]: [1985] ( ), pp.1-516 [17] (1823 ) 6420 [18] F 1981