121 $($ 3 exact scienoe \S ( evolution model (\S \infty \infty \infty $\infty$ \S : (\alpha Platon Euclid ( 2 (\beta 3 ( \S $(\beta$ ( 2 ( Era

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1 \copyright Copyright by Hisaaki YOSHIZAWA ( ( ( $=$ ( $=$ ( Kurt von Fritz [l] (

2 121 $($ 3 exact scienoe \S ( evolution model (\S \infty \infty \infty $\infty$ \S : (\alpha Platon Euclid ( 2 (\beta 3 ( \S $(\beta$ ( 2 ( Eratosthenes Eutokios ( (A (A [ ] Creta Minos Glauchos 2 geometer ( $=$ (B[ ] Hippo\alpha ates (BC440 : ( 2 ( idea (C [ ] Apolon Platon

3 122 Platon (D[ ] Platon Akademeia Archytas Eudoxos ( (! $(*1$ (E[ ] Hippo\mbox{\boldmath $\sigma$}ates 2 Plaion 650 ( 1 Platon [2] Hippoaates 800 ( 3 $-$ Platon Roma Hieron Archimedes (? $(*2$ ( BC39 Dionusiosl + ( ( neuron $(*1$ Platon ( $\text{ _{ } _{}}^{}\text{ }$ [Plutarchos ] Platon $(^{*}2$ Plut\mbox{\boldmath $\pi$}chos Archimedes ( [3]

4 123 ( ( ( [4] $(*1$ 4 Archimedes Roma ( Egypt Ptolemaios ( Archimed\mbox{\boldmath $\omega$} Alexandria : D= ( dactylos\neq 19$8\mathrm{m}\mathrm{m}$ M= ( mina\neq 437F $\mathrm{d}=\iota1\cross\sqrt[3]{\mathrm{m}\cross_{10}\mathrm{o}}$ $(*1$ [4] [5]

5 $\mathrm{h}\mathrm{i}\mathrm{p}\mathrm{p}\mathrm{o}\alpha \mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s}$ $\uparrow \mathrm{a}_{\gamma}\mathrm{c}\mathrm{h}\bm{\mathrm{m}}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{s}^{1\dagger}$ 124 (BC440 Archytas Eudoxos (BC4 Platon Eratosthenes (BC3 Archimedes (BC3 5 epilogue Archimedes ( Archimedes ( (Archimedes ( $\S 2$ ( ArcheoashonomyI : 1 Muzeion Alexandria Egypt Alexandros ( Aristoteles Euclid Caesar AD1 Plutarchos [6] ( -

6 125 - ( ( UNESCO Alexandria ( 2 ( 19 [7] BC lmilenium ( 300\alpha ( (19 20 FLindemaxm 2 ( $[_{8}]$ Platon (a Etruria BC lmilenium ( 19 Monte Loffa ( (b (Elba Alps

7 126 Piemonte ( ( 12 (a (c Alps Elba - Etmri Ehuria CeItic Galia ( Etruria Celtic ( (d Pythagoras BC50n ( $=$ (e Pythagoras Platon (Pythagoras So\alpha ates \xi = Plutarchos (ltsymposia\uparrow \dagger -- Platon rplaton \acute \supset Putarchos : + $\cdot$ - - ( (Plutarchos (f <BC200 Euclid Alexandria

8 ] Petrie 127 Dodecahedron Black steatite $\mathrm{i}^{\mathrm{c}^{\tau}}$ $\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{s}$ From Egypt Collection of London \S 3 Newton Newton lprincipia \uparrow \S 1 17 Newton 17 AWeil (Bourbaki Fermat Napier Cavalieri Wallis < Newton ( O (\uparrow lpnnncipial\dagger \epsilon -\mbox{\boldmath $\delta$} ( O

9 $\backslash \cdot -$ 128 $\mathrm{o}$ O $\mathrm{o}$fourier ORadon $\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{t}\mathrm{o}\mathrm{n}\text{ _{ } _{ } }\backslash \dot{\text{ }}\cdot \text{ }$ $\mathrm{e}$ Weil 17 (Hegel [ Zeitgeistl\dagger $\text{ }$ tautology 17 Newton Newton 1990 Newton mi\alpha 0mm ( 3 [9] lo ( 50 Newton (Gauss Newton ( ( We Newton 2 analysis\dagger l llsynffiesisll Newton ( (Euclid Newton $\uparrow\underline{ }\mathrm{p}\dot{\mathrm{n}}$ncipia \dagger ( analyti synthetic 18 analytic Newton ( chaos (

10 129 analysis/synthesis 1989 Gelfand $\mathrm{s}\mathrm{l}(2\mathrm{c}$ $ \mathrm{i}\mathrm{n}l\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l} \mathrm{t}$ infmitesimal Naimark : 1 $-\cdots$ VArnold Moscow ( Newton Copemicus Kepler 90 ( $=$ Weil : Gauss Eule handicap ( EU $\text{ }$?

11 130 ( ( \uparrow \S $\infty$ ( (1 exactness exadnes (exactn\mbox{\boldmath $\omega$}s joumal \sim larchive\sim exactness (2 evolution ( ( \S 0 biology evolution ( I evolutionll evolution context ( biologist evolution (3 nalve ( J][10]

12 131 Treppenfunktion \uparrow ( Kolmogorov 4 : ( 3 ( 1 (Platon/Euclid ( 2 ( (Newton ( 3 (Gauss (4 ( 3 ( ( 3 ( Gauss Newton (\S 3 2 : (i (ii ( \S 1 (i(ii ( (biology evolution O?? $\mathrm{o}$ /? $\mathrm{o}$? O?

13 \mathrm{a}$ History 132 ( [1] Kurt von Fritz : The discovery of incommeusurability by Hippasos of Metapentum m Math 48 [2] $\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{h}:^{1} of Greek $\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{s}^{\mathrm{t}}\uparrow$ voll Oxford(1921 at the Clarendon Press (pp244\sim 270 [3] 4 ( (pp 158\sim 166 [4] $\mathrm{e}\mathrm{w}$marsden: $ \uparrow \mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{k}$ and Roman $\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}$] $1\mathrm{e}\mathrm{r}\mathrm{y}^{1} $ Oxford at the Clarendon Pr\ es (1969\sim 71 [5] W V : ( (pp112\sim 122 [6] 9 ( (pp 159 [7] JFCH\ ese1(1830esfedorov$( $ ASchoenf1ies$( $ ( [8] Cml Louis Ferdinand von Lindemann(1852\sim 1939 : Zur Geschichte der Polyeder und der Zahlzeichen Sitzungsberichte der mathematisch-naturwissenshaftlichen Abteilung der Bayerischen Akademie der Wissenschaften zu MUnchen (1897 (pp625\sim 756 Zur Geschichte der Polyeder (1934(PP $265\sim 275$ [9] Sir Isaac Newton Manuscripts and Papers 43 reels Chadwyck-Healey $\mathrm{u}\mathrm{d}(1991$ [10] ( 1 (1934

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