$-\mathrm{s}r\mathrm{s}$ Sarrus $?-$ (Takehisa Abe) (Seiji Fujino) Faculty of Information Sciences Faculty of Systems Engineering S

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1 Title 関 -Sarrusの公式をめぐって Sarrusは本当にこれを得たか? ( 数学史の研究 ) Author(s) 阿部剛久 ; 藤野清次 Citation 数理解析研究所講究録 (2001) Issue Date URL http//hdlhandlenet/2433/64844 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 $-\mathrm{s}r\mathrm{s}$ Sarrus $?-$ (Takehisa Abe) (Seiji Fujino) Faculty of Information Sciences Faculty of Systems Engineering Shibaura Institute of Technology Hiroshima City University $(1640?-1708)$ (1683) 3 ( $)$ ( ) Pierre Frederic Saffus ( ) ( -Sarrus ) (?) 19 Sarrus 1-3 ( ) - (?) 2 3 ( ) 3 ( ) 4

3 39 5 $\mathrm{s}$ 1 ( ) 5 ( )! $arrow$ 2 1 Sarrus $arrow$ [1] [4] $(\rceil)$ Pierre Frederic Sarrus ( ) (Aveyron) (Sant- Affrique) ( ) ( ) (Waterloo (1815) ) (Montpellier) $\mathrm{d}$ ( J Gergonne ) (Pezenas) (Perpignan) (1827) $\mathrm{r}$ (Strasbourg)

4 40 $\mathrm{d}^{\dagger}$ (1840) (legion honneur) (1858) ( ) ( ) ( ) ( ) (rue de Saffus) (2) [1] [4] Sarrus ( ) 1) ( ) L Euler (17 44) JLLagrange ( 1760) Sarrus (1842) (P M H Laurent Sarrus ) 2) (1843) $\mathrm{c}\mathrm{f}$gauss (19 ) (Ceres) JHPoincare 3 ( ) 3) ( ) - $\mathrm{r}_{2}$ Sarrus (Ecole polytechnique) 4) (1821) ( )

5 41 5) 10 ( ) $=\mathrm{s}_{\mathrm{a}\mathrm{r}\mathrm{r}}\mathrm{u}\mathrm{s}$ 6) (Euler $=\mathrm{s}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{u}\mathrm{s}$ ) $=\mathrm{s}m\mathrm{u}\mathrm{s}$ 6 ( 1) 3) Sarrus ) 2 Sarrus $\sim$ $\sim$ Sarrus Sarrus Sarrus Sarrus (1) 3 Sarrus $A=$ $[_{\partial ij}]$ 1) 3 $(i\downarrow 123;jarrow 123 )$ $ A $ ( ) $ A = a_{11}a_{21}a_{31}$ $a_{12}a_{22}a_{32}\iotaa_{13}- a_{2}a_{33}3\backslash$ $=a_{11}a_{2}2a_{3}3+a12a_{2}3a3\iota+a13a21a_{3}2$ $-a_{13}a_{22}a_{31}-a_{12}a\mathrm{z}1$ $a33-a_{\mathrm{i}1}a_{23}a_{3\mathrm{z}}$ Sarrus Sarrus 2) 1o

6 $\mathrm{z}^{9}$ $\text{ _{}\gamma\sim}^{-_{\mathrm{h}^{\mathfrak{o}}}}\mathrm{b}\searrow\overline{\overline{\mathrm{p}}}-/\backslash \mathrm{c}\backslash \backslash $ $\text{ } \rangle \overline{-}\text{ }\dot{\mathrm{k}}^{\mathrm{i}}\urcorner\ovalbox{\tt\small REJECT}^{\mathrm{J}\text{ }}$ }\S\Phi \mathrm{f}\mathrm{f}\mathrm{i}\sigma*\mathrm{f}\mathrm{f}\mathrm{l}^{\backslash }\#^{\overline{\pi}} =\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{j}\mathrm{e}\ovalbox{\tt\small $\mathrm{h}\ovalbox{\tt\small REJECT}\text{ }\mathrm{x} \vee\supset\text{ ^{}\mathit{1}}$ REJECT}a\ovalbox{\tt\small $4\backslash REJECT}_{\^{-?}k_{\backslash \backslash \mathrm{q}$ $x*\ovalbox{\tt\small }^{\ovalbox{\tt\small REJECT}}i^{\pm}\mathrm{X}\mathrm{B}\backslash i\backslash h\overline{arrow}-\mathrm{j}\mathrm{e}\text{ }\mathrm{a}\mathrm{a}\gamma$ }\dot{}_{l}^{b}\mathrm{f}\mathrm{f}\mathrm{i}-\text{ ^{ }}$ 42 ( ) $-$ ([5] [6] [9] ) positive products (2) 1) $[8 \cdot]$ $[\mathrm{q} ]$ 1 ( [1] $-[3 ]$ ) $\ovalbox{\tt\small REJECT}$ $(1 \cdot 6^{\iota}83)$ ) ) $\text{ (1690 ) ( $[\cdot 11]$ ) } } $\text{ }$ ( 1 ) WLeibniz (1693) ( $)$ [12] J G GCra-mer$(1750)$ ATVandermonde$(1772)$ i ALCauchy C GJJacobi$(1829)$ JJSylvester (18 50 )

7 $\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{a}\nearrow$ la \mathrm{a}\mathrm{t}\mathrm{e}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{t}\tilde{\mathrm{a}}$ cet rang $\partial^{1\mathrm{i}}$ $b^{1\mathrm{i}}$ $C^{\mathrm{I}\mathrm{I}}$ \mathrm{b}$ droite 43 Sarrus 3 (Sarrus ) ( ) Sarrus Saffus ) 64 ( [2] [3] ) $0$ $\mathrm{p}\mathrm{j}$ EFinck [6] ; Elements $d A\iota gebre$ 2nd ed Strasbourg No 52 p 95 Sarrus $\mathrm{z}$ $\mathrm{x}\mathrm{y}$ Pour calculer dans un example donn\ e les valeurs de et MSarrus $\mathrm{d}^{\uparrow}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{d}$ $\mathrm{m}6\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{d}\mathrm{e}$ a imagin6 la practique suivante qui est fort ingenieuse $\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{n}\nearrow $\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}_{\mathrm{c}\mathrm{i}\mathrm{e}}\mathrm{n}\mathrm{t}\mathrm{s}$ on peut calculer le effect on eecrit les des inconnues ainsi a $b$ $c$ $a$ 1 $b^{l}$ $C^{l}$ On r\ ep\ ete les trois premiers $bc$ et les trois suivants $\theta b c $ $a$ $\mathrm{b}\mathrm{a}\mathrm{s}$ Actuellement partant de a on prend diagonalement du haut en en $\mathrm{d}^{\uparrow}\mathrm{a}\mathrm{u}\tan\iota $c^{11}$ $\mathrm{d}^{\dagger}\mathrm{u}\mathrm{n}$ fois et reculant ab on part de a de m\^eme et on a a $b c$ ; de 11 et on trouve $a bc $ on a $a$ $\mathrm{c}^{1}\mathrm{e}\mathrm{s}\mathrm{t}- ainsi les trois termes positifs ( \mathrm{a}$-dire \ a prendre avec leur signes) du d\ enominateur On commence ensuite par et descendant de m\^eme vers la $c$ gauche on a $c b a$ $cb \mathrm{a} $ $c b$ a ou les trois termes n\ egatifs (ou plut\^ot les termes faut changer de $\mathrm{q}\mathrm{u}^{\uparrow}\mathrm{i}1$ signe) 1893 GWeicold [7] ; Lehrbuch und deren Anwendungen p 25 $derdete\pi nin\mathrm{a}ncen$ Erster Theil Stuttgart

8 $\mathrm{s}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{u}\mathrm{s}^{\dagger}\mathrm{s}$ order 44 Sarrus 1833 T Muir ( ) 2 3 Finck 1910 $\hslash$ $\mathfrak{b}-[13]$ ; The Fukudai and Determinants in Japanese Maffiemafics $\mathrm{p}$ $\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{s}\dagger$ This is quite the same as the well known rule of expanding the $\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{d}$ $(\cdots\cdots\ovalbox{\tt\small REJECT})$ determinants of the Thus we may conclude that the Japanese mathematicians have made use of determinants since the year 1683 and have been able to expand them by a mechanical method such as for the determinants of the third order 1912 [1 4] ; $3^{\mathrm{I}}4$ pp Strasbourg Sarrus 3 Finck 2 (1846 ) 1913 YMikami( ) [15] ;The Develop- $me\dot{n}\mathrm{i}t$ of Mathematics in China and Japan Taubne $\sim$ Leibziger pp (Reprint 1974 Chelsea $NY) 3)

9 $\mathrm{r}\mathrm{g}\mathrm{h}$ TMuir [8] ; The theory of general determinants in the historical order of its development up to 1846 Proceedings RSocEdinbu- $ \mathrm{x}\mathrm{x}\mathrm{v}$ p39 Muir Weicold PFSarrus Nouvelle methode pour la resolution des equations numeriquesparis Bachelier pp31 Muir 1911TMuir [9]([8] p39 $-\text{ }$ ) ;The theory of determinants in the historical order of development Dover Publications $\mathrm{v}\mathrm{o}\mathrm{l}2$ $\mathrm{p}\cdot 39$ Inc $ $ Muir Finck [6] 1846 $\mathrm{p}\mathrm{j}$efinck [6] (1833) Sarrus (Cauchy Saffus?) Sarrus ( ) Sarrus Finck Sarrus

10 $\langle$ 46 Finck $=$ (3) Sarrus? 1) 2 $0$ 1 Finck Sarrus ( ) Finck Sarrus Sarrus Saffus 2 Sarrus ( Finck ) 2) 1) $0$ $\rceil \mathrm{q}$ 1 (Robin) V GRobin ( ) $\mathrm{a}\mathrm{m}$ VASteklov Liapunov $\langle$ $\mathrm{n}\mathrm{m}$gunter Gunter (1934) SBergman (1948) $([16] -[18])$ ( ) 2o $0$ 2 Sarrus (1833) $n$ $f(x)=0$ $f (x)=0$ $0$ $n=2\rightarrow$ Sylvester ([19] ) Sarrus 3 \langle $\rangle$

11 $\text{ }\mathrm{p}_{0}1\mathrm{a}\mathrm{r}=\epsilon \text{ }\prime\supset\iota\ddagger 7\Pi\S \text{ }\ovalbox{\tt\small REJECT} \mathrm{b})k\mathrm{g}_{+}\pi 4 \mathrm{a}f -=_{E}\mathrm{E}^{\mathrm{A}\mathrm{i}}\lambda \mathrm{r}_{\frac{\varpi}{7\mathrm{j}^{\sim}})\prime\grave{\eta}}\backslash k^{1\mathrm{a}}\text{ }-\check{\mathrm{h}}\epsilon_{\mathrm{o}}\text{ ^{ }}\backslash \backslash \ovalbox{\tt\small REJECT}*x^{\mathrm{F}}\#*\mathfrak{b}\ovalbox{\tt\small REJECT} \mathrm{i}\mathrm{a}\mathrm{t}_{\sim^{f_{x}}}\mathrm{g})\mathrm{j}_{2}\gamma_{\vee}\text{ }$ REJECT}_{J^{\prime\delta^{\backslash }}}\dot{\text{ }}-\vee)\text{ _{}\mp}\backslash \mapsto \text{ }$ $\iota$ 47 ( ) Muir Finck 1846 Sarrus 1 2 Sanus Sarrus 1 $\mathrm{s}_{\mathrm{a}\mathrm{r}\mathrm{r}}\mathrm{u}\mathrm{s}$ ( ) ( $-$ ) ; $\bullet$ Joseph Diaz $\mathrm{g}\mathrm{e}\mathrm{r}\mathrm{g}_{\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}}$ $\neg\backslash \backslash$ }\backslash$ $\pm \text{ ^{}\backslash $\xi fl^{\frac{\mathrm{a}\mathrm{n}\backslash }{\mp}\text{ ( Nancy Montpellier) $1771\triangleleft19$ }}(\mathrm{n}\mathrm{a}1\mathrm{s}\mathrm{d}\llcorner \text{ ^{}\mathrm{m}\mathrm{a}}\text{ _{}\ovalbox{\tt\small REJECT}}^{\mathrm{e}_{\dagger-}}\mathrm{e}\cdot\mathrm{f}\mathrm{h}\mathrm{m}\mathrm{a}\iota\backslash \text{ }\mathrm{o}\mathrm{i}\underline{\mathrm{q}}\mathrm{u}\mathrm{e}\mathrm{p}\mathrm{f}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{a}\text{ ^{}\mathrm{s}}\text{ ^{}\mathrm{u}}\sigma)\mathrm{p}\exists\ovalbox{\tt\small REJECT} \mathrm{p}\mathrm{i}7\mathrm{g}^{1}g\mathrm{i}\mathfrak{u})\xi \mathrm{e}\mathrm{i}\wedge \mathrm{j}\mp 1\mathrm{J}(\sim(^{\mathrm{q}\mathrm{S}}\text{ _{}\grave{z}}^{\mathrm{e}}- \text{ }\hslash f 3\vee\supset\sigma)\mathrm{E}\mathrm{V}\not\in^{)}1183$ $\mathrm{c}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{e}\mathit{0}([\cdot\ddot{\{}\mathrm{o}])\ovalbox{\tt\small $\bullet$ Pierre Joseph Etienne Finck( $ $ Lauterbourg Stras bourg) $($ 1829) $( -1842)$

12 Sarrus $-$ 1833 Finck $(\sim 1868)$ ) $([2 0] [22 ])$ $\bullet$ Thomas Muir( $ $ Stonebyres Scotland Rondeboseh South Africa) Tompson ( Kelvin ) (1871) ( ) Treatise on the theory of determinants (1882) 5 $\mathrm{h}\mathrm{i}_{\mathrm{s}\mathrm{t}\mathrm{o}}\mathrm{r}\mathrm{y}$ of determinants Vol 1;1840 (1890) Vol 2; (1911) Vol 3; (1920) Vol 4; ; $(1923)\mathrm{v}_{\mathrm{o}1}5$ 1920 (1929) Vo16; $($ 1931) 35 Vol 1 Vol 2 Cape (1892) (1910) Cape Town Muir Collection ([ 21 ] [221 ) $\bullet$ Victor Gustave Robin ( Paris- $1897 $?? Paris) PrOf TMuir [3] $\mathrm{j}\mathrm{h}$ (Cape Town Webb Math DigeSt )

13 49 2 Sarrus Robin 1 ( ) 2 $\text{ }(\text{ }\text{ })$ ( ) $-$ 3 (Sarrus Robin $\neg$ 3 ) 4 (Sarrus Fink (1 ) Robin Picard (1 ) ) 5 ( ) ( ) (Sarrus Robin ) (MBerthlot $-$ ) (Bertholot ) $-$ Sarrus Robin 1 2? ( (KGustafson ) 2-20 Robin 600 )? [1] TAbe SFujino Seki and Sarrus and Again Sarrus Relating to the Theory of Elimination Proceedings of the 4th International Symposium on the History of Math and Mathematical Edu (ISHME) Maebashi Institute of Technology August 1999 $(^{\text{ }})$ [2] Sarrus 2000-CH pp [3] Sarrus? ( ) [4] P $ $ Tome 46 No pp15-18 $\mathrm{w}\mathrm{w}$ [5] JGullberg Mathematics from the birth of numbers &Company 1997 p 647 Norton

14 $\mathrm{i}$ 50 [6] $\mathrm{p}\mathrm{j}$efinck Elements d Algebra 2nd ed Strasbourg No p 95 [7] GWeichold Lehrbuch der Determinanten und deren Anwendungen Erster Theil Stuttgart 1893 p 380 [8] TMuir The Theory of general determinants in the historical orger of its development up to 1846 Proceedings of RSoc Edinburgh XXV 1903 pp 39 [9] TMuir The Theory of detefminants in the historical order of development Dover Publications Inc Vol p 39 [1 0] FKlein Vorlesungen uber die Mathematik $Entwick\mathit{1}un\mathrm{g};der$ $im\mathit{1}\mathit{9}$ $J\mathrm{a}hThUIlg_{GT}t$ Springer 1926 ( ) $\text{ }$ [1 1] Vol pp 3-18 [1 2] EKnoblock Determinants in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (IGrattan-Guiness ed) Routledge 1994 pp [13] $-$ $ \mathrm{f}\mathrm{u}\mathrm{k}\mathrm{u}\mathrm{d}\mathrm{a}\mathrm{i}^{1} (\text{ })$ The and Determinants in Japanese Mathematics PP [14] $y$ pp [1 5] YMikami( ) The Development of in China and $M\partial them\partial rics$ - Japan Taubne Leibziger 1913 pp ( Reprint 1974 Chelsea $\mathrm{n}\mathrm{y})$ [16] $\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{i}\mathrm{n}^{\mathfrak{j}}\mathrm{s}$ KGustafson TAbeThe Third Boundary Condition-Was it? $- \mathit{7}1$ - Math Intelligencer Vol20 No pp 63 [1 7] KGustafson TAbe (Victor) Gustave Robin Math Intelligencer Vol 20 No pp 47 $- 53$ [18] [19] $\mathrm{j}\mathrm{j}$sylvester Tome 44 No pp AMethod of Determining by Mere Inspection the Derivatives from Two Equations of Any Degree Philo Magazine XVI pp [20] JShallit Origin of the Analysis of the Euclidean Algorithm Historia Mathematica pp 401 $- 419$ [2 1] HWTurnbull Thomas Muir JLnodon Math Soc Vol pp $\mathrm{h}\mathrm{t}\mathrm{t}\mathrm{p}//\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{n}\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{t}-\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{c}\mathrm{u}\mathrm{k}/-\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{t}_{\mathrm{o}}\mathrm{r}\mathrm{y}/\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}\mathrm{h}\iota \mathrm{m}1$ [22]

$\sim 22$ *) 1 $(2R)_{\text{}}$ $(2r)_{\text{}}$ 1 1 $(a)$ $(S)_{\text{}}$ $(L)$ 1 ( ) ( 2:1712 ) 3 ( ) 1) 2 18 ( 13 :

$$\sim 22$ *) 1 $(2R)_{\text{}}$ $(2r)_{\text{}}$ 1 1 $(a)$ $(S)_{\text{}}$ $(L)$ 1 ( ) ( 2:1712 ) 3 ( ) 1) 2 18 ( 13 :$ Title 角術への三角法の応用について ( 数学史の研究 ) Author(s) 小林, 龍彦 Citation 数理解析研究所講究録 (2001), 1195: 165-175 Issue Date 2001-04 URL http://hdl.handle.net/2433/64832 Right Type Departmental Bulletin Paper Textversion publisher