$XJXVWR0DULQRQL
/¶DULWPHWLFDGL/HRQDUGR
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
L’aritmetica di Leonardo
1HOODFRQILJXUD]LRQHGHO©PLWRªGL/HRQDUGRTXDOHVLqGHWHUPLQDWR DOOD
ILQHGHOORVFRUVRVHFRORODPDWHPDWLFDQRQRFFXSDLOSULPRSRVWR&LzQRQWRJOLH
FKH WUD L ORGDWRUL FKH QRQ VL FXUDQR GL GRFXPHQWDUH OH SURSULH DIIHUPD]LRQL VLD
FRQVXHWDO¶LQFOXVLRQHGHOODPDWHPDWLFDWUDOHLQILQLWHVSHFLDOL]]D]LRQLVFLHQWLILFKH
GHOO¶ ©RPR VDQ]D OHWWHUHª (VVL SHQVDQR HYLGHQWHPHQWH D FHUWH VROHQQL
DIIHUPD]LRQL GL /HRQDUGR ©1HVVXQR PL OHJJD FKH QRQ q PDWHPDWLFR QHOOL PLD
SULQFLSLª 4X ,9 Y ©1HVVXQD FHUWH]]D q GRYH QRQ VL Sz DSSOLFDUH XQD GHOOH
VFLHQ]HPDWHPDWLFKHª*Y4XHVW¶XOWLPDIUDVHVHPEUDULFDOFDUHSHUIHWWDPHQWH
TXHOOD GL 1LFROz GD &XVD ©1LKLO FHUWL KDEHPXV LQ QRVWUD VFLHQWLD QLVL QRVWUDP
PDWKHPDWLFDPª GHOOD SULPD FRQRVFLDPR VROR RJJL LO SUHFLVR VLJQLILFDWR
$FFLQJHQGRVLDVFULYHUHXQQXRYR7UDWWDWRGHOOH3LWWXUD/HRQDUGRKDLQQDQ]LDVpL
SUHFHGHQWL7UDWWDWLGL/HRQ%$OEHUWLHGL3LHURGHOOD)UDQFHVFDLTXDOLLPSRVWDQR
LOORURGLVFRUVRVXL©SULQFLSLªRGHILQL]LRQLGLSXQWROLQHDVXSHUILFLHFRPXQLDOOD
JHRPHWULDHDOODSURVSHWWLYDODJUDQGHVFRSHUWDGHOVHFRORUHVDSRVVLELOHSHUzGDL
ILORVRIL FKH QHL GXH VHFROL SUHFHGHQWL DYHYDQR WUDVIRUPDWR O¶RWWLFD GD VFLHQ]D
PHWDILVLFD D VFLHQ]D SXUDPHQWH PDWHPDWLFD 2UEHQH O¶$OEHUWL H 3LHUR GRYHQGR
SDUODUH D SLWWRUL H QRQ D ILORVRIL DGRWWDQR SLHWRVDPHQWH GHOOH GHILQL]LRQL
HPSLULFKH /HRQDUGR LQYHFH DVVHUHQGR SROHPLFDPHQWH FKH OD SLWWXUD q VFLHQ]D
HODERUDIDWLFRVDPHQWHGHOOHGHILQL]LRQLR©SULQFLSLªGLSXQWROLQHDVXSHUILFLHFKH
YRUUHEEHUR HVVHUH DVVROXWDPHQWH VFLHQWLILFKH &Lz WXWWDYLD QRQ VLJQLILFD FKH
/HRQDUGR VLD VWDWR XQ HVSHUWR QHO FDOFROR R FKH FRQRVFHVVH SURIRQGDPHQWH OH
GRWWULQH GHL PDWHPDWLFL GHO WHPSR VXR ( 7URLOR QHOOD VXD LSHUEROLFD
LQWHUSUHWD]LRQH GHO SHQVLHUR ILORVRILFR GL /HRQDUGR FRQWUDSSRQH OD PDWHPDWLFD
YLQFLDQD D TXHOOD GL 3LWDJRUD 3ODWRQH H GHL 1HRSODWRQLFL ©4XHOOD q
HVVHQ]LDOPHQWHPHWDILVLFDHSHUILQRWHRORJLFDODPDWHPDWLFDOHRQDUGLDQDLQYHFH
FRPH TXHOOD FKH IUD SRFR OD VHJXLUj JORULRVDPHQWH GL *DOLOHRq VFLHQWLILFDª ,O
FKH q YHUR FRPH q YHUR FKH LO PRGHVWR IDU GL FRQWR GHOD VFXROD HOHPHQWDUH q
VFLHQWLILFR H QRQ WHRORJLFR PD VHPEUD DGGLULWWXUD SURSRUUH VHQ]D DOFXQ VHQVR
GHOOHSURSRU]LRQLXQFRQIURQWRWUD3LWDJRUD3ODWRQHHFFGDXQDSDUWHH/HRQDUGR
GDOO¶DOWUD
1HOOD VFKLHUD GHJOL VWXGLRVL FKH VHQWRQR O¶HOHPHQWDUH GRYHUH GL
GRFXPHQWDUHOHORURGLFKLDUD]LRQLLOGLVFRUVRVXOODPDWHPDWLFDYLQFLDQDQRQqPDL
VWDWR Qp DPSLR Qp VLFXUR ,O PDJJLRUH GL HVVL SHUFKp HVSHUWR PDWHPDWLFR H
OHRQDUGLVWD q 5 0DUFRORQJR L TXDOH GRSR DYHU GHSORUDWR FKH SRFKLVVLPL
DYHVVHURVWXGLDWR©LULFFRPDWHULDOHVSDUVRQHLPDQRVFULWWLYLQFLDQLFKHULJXDUGD
8OWHULRULQRWL]LHVXTXHVWRDUJRPHQWRLQ$0$5,121,L’essere del nulla. ,Q©5LQDVFLPHQWRª
(752,/2Ricostruzione e interpretazione del pensiero filosofico GLLeonardo da 9LQFL©0HPRULHGHOO¶,VWLW9HQHWRGL
6FLHQ]H/HWWHUHHG$UWL&ODVVHGL6F0RUDOLH/HWWHUHYR,;;;,IDVF,9HQH]LDS
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
ODPDWHPDWLFDSXUDHGHVDPLQDWRVWRULFDPHQWHHFRQPRGHUQRVSLULWRFULWLFRL
FRQWULEXWR DUUHFDWR D TXHVWD VFLHQ]D H OH VWHVVH FRQRVFHQ]H PDWHPDWLFKH GL
/HRQDUGRª VWXGLz FULWLFDPHQWH PROWL SDVVL YLQFLDQL FRQQHVVL FROOD PDWHPDWLFD LOOXPLQDQGRQH DOFXQL PD VRUYRODQGR VX DOWUL H JLXQJHQGR D FRQFOXVLRQL
IHUYLGDPHQWH DPPLUDWLYH FKH QRQ SRVVLDPR VRWWRVFULYHUH VHQ]D ULVHUYH '¶DOWUD
SDUWH QRQRVWDQWH LO VXR GHVLGHULR GL VFRSULUH QXRYH VFRSHUWH OHRQDUGLDQH HJOL
GRYHWWHDPPHWWHUHFKH/HRQDUGR©QRQIXXQPDWHPDWLFRGLSURIHVVLRQHªFKHFHUWL
VXRLFDOFROLULYHODQRLQFHUWH]]HFKHSUDWLFDPHQWHLJQRUDO¶DOJHEUDJLDFFKpULVROYH
HTXD]LRQL OLQHDUL ©SHU YLD WXWW¶DOWUR FKH DOJHEULFD H SRL LQ TXDOFKH FDVR SL
FRPSOHVVRHUUDªHFKHLQILQH©QRQVLRFFXSDFKHGLJHRPHWULDª.
1HL ©&ROORTXHV LQWHUQDWLRQDX[ª WHQXWL D 3DULJL QHO GDO ©&HQWUH
1DWLRQDO GH OD 5HFKHUFKH VFLHQWLILTXHª 5 6HUJHVFX SXU VHJXHQGR OD IDOVDULJD
GHJOLVWXGLGHO0DUFRORQJRFRQFOXGHHVVHUH©H[WUHPHPHQWGLIILFLOHHWKDVDUGHX[
GH SRUWHU XQ MXJHPHQW VXU O¶RHXYUH PDWKpPDWLTXH GH /pRQDUG FDU LO Q¶HQ D SDV
GRQQpXQHUpGDFWLRQGpILQLWLYHª0DFLzLPSOLFDXQSUHVXSSRVWRWURSSRLQILGRH
FLRqFKH/HRQDUGRSHQVDVVHDXQDUHGD]LRQHGHILQLWLYD'LFKHFRVD"8QWUDWWDWRGL
PDWHPDWLFD SXUD" ,SRWHVL JUDWXLWD GHULYDWD GD XQ YLHWR SUHJLXGL]LR FKH YXRO
VFRSULUH LQ RJQL VFULWWR YLQFLDQR OD SUHSDUD]LRQH GL FKLVVj TXDQWL ©WUDWWDWLª ,O
6HUJHVFXFRQIHUPDOHOLPLWD]LRQLJLjDPPHVVHGDO0DUFRORQJRPDFLWDVHQ]DXQD
FRQYHQLHQWH UHD]LRQH LO SDVVR LQ FXL /HRQDUGR / Y GLFKLDUD IDOVR LO ULVXOWDWR
ଶ ଷ
GHOODGLYLVLRQH ǣ DQ]LGLFHLO6HUJHVFX©LOGHPRQWUHTXHOHUpVXOWDWQHVDXUDLW
଼
ଷ ସ
rWUH ª $QFKH LO ELODQFLR VWHVR GD 8 &LVRWWL q PRGHVWR OH FRQRVFHQ]H GL
ଽ
/HRQDUGR©VLOLPLWDQRDOODJHRPHWULDGHOO¶DOJHEUDQRQVHPEUDFKH/HRQDUGRVLVLD
RFFXSDWR«PDDQFKHOHFRJQL]LRQLJHRPHWULFKHQRQVRQRPROWRDYDQ]DWHHSDUH
QRQROWUHSDVVLQROHQR]LRQLHXFOLGHHGHOWHPSRª
,Q VRVWDQ]D L GXH XOWLPL VWXGLRVL ULDVVXPRQR LO SHQVLHUR GHO 0DUFRORQJR
OLEHUDQGRORSHUzGDOO¶HQWXVLDVPRXQSR¶FLHFRFKHIXGLTXHOODJHQHUD]LRQHHFKH
RJJLVHPEUDHUHGLWDWRVRORGDLQRQVSHFLDOLVWL2UDQRLYRUUHPPRODVFLDWDGDSDUWH
OD JHRPHWULD FKH ULFKLHGHUHEEH SL OXQJR GLVFRUVR HVDPLQDUH FRQ DWWHQ]LRQH H
VHQ]D SUHJLXGL]L DOFXQL GRFXPHQWL FKH LOOXVWUDQR OH FRQRVFHQ]H GL /HRQDUGR
QHOO¶DPELWRGHOODVRODHVHPSOLFHDULWPHWLFD
7UDLQXPHURVLVFULWWLGHO0$5&2/21*2PLULIHULVFRLQSDUWLFRODUHDLe ricerche geometriche-meccaniche di Leonardo
da Vinci, LQ ©$WWL GHOOD 6RFLHWj LWDOLDQD GHOOH 6FLHQ]H 5RPD FIU SXUH LQ Leonardo da Vinci Artista-scienziato.
0LODQR LO FDS 9,,, ©,O SDUDGLVR GHOOH PDWHPDWLFKHª QRQFKp Leonardo matematico, LQ ©/¶LQJHJQHUH Q JHQQDLR
36(5*(6&8Léonard de Vinci et les mathématiques. LQ©/GH9HWO¶H[SpULHQFHVFLHQWLILTXHDXVHL]LqPHVLqFOHª
3UHVVHV8QLYHUVLWDLUHVGH)UDQFHSS
8&,6277,La matematica vinciana, LQ©/HRQDUGRGD9LQFLª1RYDUD,VWLW*HRJUDILFR'H$JRVWLQLDHGL]
SS
3HU XQ¶DQDOLVL SL PLQX]LRVD GHL WHVWL YLQFLDQL TXL ULFRUGDWL H GL PROWL DOWUL DQFRUD FIU $ 0$5,121, La teoria dei
numeri frazionari nei manoscritti vinciani. Leonardo e Luca Pacioli, LQ©5DFFROWD9LQFLDQDªIDVF;;SSHFIU
SXUHGHOORVWHVVRLe operazioni aritmetiche nei manoscritti vinciani, LQ©5DFFROWD9LQFLDQDªIDVF;,;SS
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
&RPLQFLDPR GDOOD SDJLQD ULFRUGDWD GDO 6HUJHVFX / Y 'RSR DYHU
HQXQFLDWRLOTXR]LHQWHGHOODGLYLVLRQH/HRQDUGRDJJLXQJH©TXHVW¶qIDOVRLPSHUz
FK¶HJOL q SL FKH QRQ q LO FKH QHO SDUWLUH GH¶ URWWL q LPSRVVLELOH FKH VLD
PDJJLRUH OD VRPPD GHO PROWLSOLFDWR FKH QRQ q OD FRVD FKH VL PXOWLSOLFD 2QGH
ଷYLD ଶ
଺
ଶ
GLUHPFRVu ID FLRq ª/HRQDUGRGXQTXHKDPROWLSOLFDWRLGXHWHUPLQL
ସ
ଷ
ଽ
ଷ
GHO GLYLGHQGR SHU LO VROR QXPHUDWRUH GHO GLYLVRUH H FRPH ULVXOWDWR QRQ SXz QRQ
ULRWWHQHUH LO SXUR GLYLGHQGR FRQFOXVLRQH VXSHUIOXD GL XQ ODYRUR LQXWLOH H
LQFRQFOXGHQWH/RVWUDQRqFKH/HRQDUGRWURYLSUHIHULELOHTXHVWDFRQFOXVLRQH©SDU
FKHIDFFLDPHJOLRªDOULVXOWDWRHVDWWRWDQWRLQFRQFHSLELOHJOLVHPEUDFKHLOTXRWR
FUHVFD LQ TXDQWLWj ULVSHWWR DO GLYLGHQGR 0D F¶q XQ¶DOWUD SDJLQD VIXJJLWD DO
6HUJHVFXFKHFLULSUHVHQWDOHVWHVVHLGHHLQPRGRSLDPSLRSLHVSOLFLWRHFRQ
HUURUL SL JUDYL ©6H YRL SDUWLUH SHU O¶RUGLQDULD UHJROD WL Gj FKH WX GHEEL
PXOWLSOLFDUH LQ FURFH FRO GLUH ³ YLD ID ´ H SRQL VRWWR LO LO TXDOH ILD
SDUWLWRUHSRLSHUO¶DOWUDOLQLDGLUDL³YLHID´HTXHVWRILDLOQXPHURFKHVL
GeSDUWLUH SHU FK¶q LPSRVVLELOH $GXQTXH UHVWD QRPLQDWR 0D TXL QRQ
V¶LQWHQGH GL FKH SHUFKp RJQL FRVD SDUWLWD GLPLQXLVFH GL VXD TXDQWLWj H TXHVWD q
FUHVFLXWD QHO SDUWLUH LPSHUz FK¶HOOL q PDJJLRUH QXPHUR FKH QRQ q ଶ
ଷ
$GXQTXHODUHJRODQRQqPROWRFKLDUDHSHUzSHUDOWUDYLDGL ³YLHID´
ଷ
ସ
HSRQLVRSUDXQDULJDSRLGLUDL³YLHID´HSRQLTXHVWRVRWWRHVVDULJDH
ଶ
ଷ
GLUDLHTXLWURYLFK¶HOOLqPLQRUHFKH(DQFRUDSRLGLUH ³YLH
ଷ
ସ
ID´HSRQLVRSUDXQDULJDSRLGL¶³YLH´HSRQLWDOVRWWRORHGLUDL
FKHqPHGHVLPDPHQWHPLQRUHFKHª
6L WUDWWD GL XQ SUREOHPD FKH HUD SUREDELOPHQWH WUDWWDWR QHOOH VFXROH GHJOL
DEEDFKLVWL,QIDWWL/XFD3DFLROLQHOODVXDSumma de Aritmetica VLIHUPDDVSLHJDUH
FRPHPDLGLYLGHQGRSHUXQDIUD]LRQHSURSULD©VHPSUHQHYLHQSLFKHODTXDQWLWj
SDUWLWDGLFKHSHUFHUWRSDUVWUDQLRDFLDVFXQRODSDUWHTXDOHqO¶DYHQLPHQWRHVVHU
PDJJLRUH GHO WXWWR 6u FRPPR D SDUWLUH SHU QH YHQ TXDOH VHQ]D
GXELR q SL H PDJJLRUH FKH QRQH ª 0D VXELWR VSLHJD FKH ©LQWHUURJDWLYH YRL GLUH GH TXDQWL WHU]L VH QH IDULD"ª H FRVu LPSRVWDWR LO
SUREOHPD DSSDUH HYLGHQWH FKH entra XQD YROWD H PH]]R LQ 0D QRQ
ULXVFLDPR RQHVWDPHQWH DG DWWULEXLUH D /HRQDUGR OR VWHVVR LQWHQWR H OD VWHVVD
LPSRVWD]LRQH'RYHLO3DFLROLGLFH©SDUVWUDQLRDFLDVFXQRª/HRQDUGRDIIHUPDXQ
SHUHQWRULR ©TXHVW¶q IDOVRª FKH FRQGDQQD UHFLVDPHQWH OD UHJROD ©RUGLQDULDª HG
HVDWWD $O ULJRUH GHO 3DFLROL VL RSSRQJRQR OH FRQWLQXH LQFHUWH]]H GL /HRQDUGR ©
SDU FKH IDFFLD PHJOLRª ©QRQ q PROWR FKLDUDª LQROWUH LQYHFH GL FRQGXUUH DOOD
FRQIHUPD GHOOD UHJROD HVDWWD LO GLVFRUVR YLQFLDQR VL GLVSHUGH LQ SURFHGLPHQWL
DVVXUGL FKH FRQGXFRQR D ULVXOWDWL GLYHUVL H SHUFLz FRQWUDGGLWWRULL FRO SULPR LO
3RLFKpLOSDUWLUH GH¶URWWLRYYHURODGLYLVLRQHGHOHIUD]LRQLDYYLHQH PROWLSOLFDQGRLQFURFHO¶XQRSHUO¶DOWURRVVLDLO
SULPR SHU O¶LQYHUVR GHO VHFRQGR ©OD VRPPD GHO PXOWLSOLFDWR VLJQLILFD LO TXR]LHQWH ©OD FRVD FKH VL PXOWLSOLFDª q LO
GLYLGHQGRLQDOWUHSDUROHLOTXRWRQRQSXzHVVHUHPDJJLRUHGHOGLYLGHQGR
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
TXRWRFRLQFLGHFROGLYLVRUHFROVHFRQGRFRLQFLGHFROGLYLGHQGRHSSXUHYHQJRQR
DPEHGXHGLFKLDUDWLDFFHWWDELOL,QROWUH/HRQDUGRQRQYHGHFKHPROWLSOLFDQGRLGXH
WHUPLQLGHOODIUD]LRQHSHUXQRVWHVVRQXPHURODIUD]LRQHQRQ FDPELDHFRQFOXGH
FKHRSSXUHVRQRPLQRUL GL4XHVWLHUURULSRVVRQRVHPEUDUHHFFHVVLYL
HG LQFUHGLELOL VH QRQ SHQVLDPR FKH HUDQR FRPXQL DL ©VDQ]D OHWWHUHª QRQ
ULHQWUDQGR OH IUD]LRQL QHO QRUPDOH © IDU GL FRQWRª 'REELDPR GXQTXH FROORFDUH
/HRQDUGRIUDJOLLQGRWWLSHUTXDQWRULJXDUGDTXHVWRSUREOHPD"$TXHVWDGHFLVLRQH
FLVSLQJRQRDQFKHDOWULHQRQPHQRJUDYLDUJRPHQWL
,OIUEGHOO¶$WODQWLFRFRQWLHQHXQDWDEHOOLQDGHOOHUDGLFLTXDGUDWHGHL
SULPL FLQTXH QXPHUL GL q GL q GL q GL q GL q /D
GLPRVWUD]LRQH q IDWWD PROWLSOLFDQGR SHU FKH ID RVVLD 1RQ EDVWD
1HOORVWHVVRFRGLFHIUVLYHGHODWDEHOODGHOOHUDGLFLFXELFKHDPSLDPHQWHH
LQHTXLYRFDELOPHQWHFRPPHQWDWD
©5DGLFHFXEDGLVRQSHUFKpGLFHQGRµQRQLYLHQRQLIDQRQL
FLRqXQLQWHURHYROWHQRQLIDQRQLFKHVRQLQWHUL¶DGXQTXHqYHURFKH
qODUDGLFHFXEDGL5DGLFHGLq(GLUDLYLHIDFLRq
HYLDIDFLRqª(VLQRWLQROHVFULWWXUHLPSUHFLVHLQYHFH
GLSHULQGLFDUHJOLLQWHUL$QFKHTXLLOGLVFRUVRqWURSSRHVSOLFLWRSHUGDU
OXRJRDGHTXLYRFL1HOPROWLSOLFDUHOHIUD]LRQL/HRQDUGRWUDVFXUDLOGHQRPLQDWRUH
ଵ
ଵ
DOPHQRTXDQGRHVVRqFRPXQH(JOLqDELWXDWRDFRQVLGHUDUHʹ FLRqʹ ൅ FRPH
ଶ
ଶ
ଵ
ଶ
VHIRVVH FLRqʹš FRQIRQGHGXQTXHODVRPPDFROODPROWLSOLFD]LRQHHGLIURQWH
ଶ
ଶ
ଶ
ସ
ଶ
ସ
D š RSSXUH š HJOL DJLVFH FRPH TXDQGR GHYH VRPPDUH GXH IUD]LRQL FRQ
ଶ ଶ
ଵ଺ ଵ଺
GHQRPLQDWRUHFRPXQH,QIDWWLQHOWUDVFULYHUHLOVXRFDOFRORRUDOHGLFH©HYROWH
ଷ
QRQLªH©YLDªWUDVFXUDQGRLOIDWWRFKHVLWUDWWDQRQGLRGLPDGL HGL
ସ
ଶ ଷ ସ ହ
ଽ
e SXUH HYLGHQWH FKH /HRQDUGR QRQ V¶DFFRUJH FKH ǡ ǡ ǡ UDSSUHVHQWDQR
ଶ ଷ ସ ହ
FLDVFXQRO¶XQLWDHFKHXQRVDUHEEHLQGHILQLWLYDODUDGLFHTXDGUDWDGLWXWWLLQXPHUL
LQWHUL H QRQ VL DYYHGH FKH QHO FDOFROR GHOOH UDGLFL FXELFKH ULGXFLELOL DOOD VHULH
ଵ ଵ ଵ
ǡ ǡ «SHULQXPHULVLYHULILFDXQDGLPLQX]LRQHGHOODTXDQWLWjGHOOH
ଷ ସ ହ
UDGLFLSURSRU]LRQDOHDOFUHVFHUHGHLUDGLFDQGL
,OTXDGURGHOOHODFXQHGL/HRQDUGRQHOFDOFRORGHLQXPHULIUD]LRQDULVLYD
GXQTXHDPSOLDQGRHQRQVHPEUDSRVVLELOHDWWULEXLUHDOOHVXHSDUROHXQ¶LQWHQ]LRQH
GLGDWWLFD SDUWLFRODUH XQD GLPRVWUD]LRQH SHU DVVXUGR DSSHQD LPSRVWDWD H QRQ
FRQFOXVD" R VLJQLILFDWL DPELJXL TXDQWR PLVWHULRVL 7XWWR LQYHFH VL SXz VSLHJDUH
ULFRQRVFHQGR DOO¶RPR VDQ]D OHWWHUH LO GLULWWR GL QRQ SDGURQHJJLDUH DQFRU EHQH LO
FDOFRORGHL©URWWLªDOORUDULVHUYDWRDJOLLQL]LDWL$OWULDSSXQWLYLQFLDQLFRQIHUPDQR
LOQRVWURJLXGL]LR,QQDQ]LWXWWRDEELDPRODSURYDFKH/HRQDUGRVWHVVRULFRQREEHOD
VXD LQVXIILFLHQWH SUHSDUD]LRQH H LQYRFz O¶DLXWR DOWUXL ,QIDWWL QHO I UG
GHOO¶$WODQWLFR HJOL VFULVVH ©,PSDUD OD PXOWLSOLFD]LRQH GHOOH UDGLFL GD PDHVWUR
/XFDªRVVLD/XFD3DFLROL1HOOHWDEHOOHFKHDEELDPRHVDPLQDWROH©UDGLFLªVRQR
UDSSUHVHQWDWHDSSXQWRGDIUD]LRQLFKHPROWLSOLFDWHXQDRGXHYROOHSHUVHVWHVVH
ULGDQQRLOUDGLFDQGR/HRQDUGRGRYHWWHDFFRUJHUVLGHOO¶HQRUPHHUURUHFRPPHVVRH
ଵ଺
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
ULFRUVHSHUDLXWRDO3DFLROL)RUVHJOLHQHSDUOzFHUWDPHQWHHJOLVSHVHVROGL©LQ
$ULWPHWLFDGLPDHVWUR/XFDª*OLHIIHWWLEHQHILFLGLTXHVWDGHFLVLRQHVLYHGRQRQHL
II DE GHOO¶$WODQWLFR GRYH VRQR WUDVFULWWH R ULDVVXQWH GDOOH FF GHOOD
Summa de Aritmetica OH UHJROH HVDWWH SHU PROWLSOLFDUH VRPPDUH VRWWUDUUH
GLYLGHUHQRQFKp©LQILO]DUHªLURWWL
$QFKH TXHVWH WUDVFUL]LRQL GDO OLEUR GHO 3DFLROL FRQIHUPDQR L QRVWUL
SUHFHGHQWL JLXGL]L ,QQDQ]LWXWWR LO FDUDWWHUH SHGLVVHTXR GHOOH WUDVFUL]LRQL OD ORUR
FRQFLVLRQH FKH WDOYROWD RVFXUD LO VHQVR TXDOFKH RPLVVLRQH R VYLVWD R HUURUH GL
FDOFROR UHQGRQR HYLGHQWH LO FDUDWWHUH DXWRGLGDWWLFR GL TXHVWL DSSXQWL 7UD HVVL
LQROWUHHLEUDQLSUHFHGHQWHPHQWHHVDPLQDWLODIUDWWXUDqQHWWDLOFDSRYROJLPHQWR
FRPSOHWR LO FKH HVFOXGH OD SRVVLELOLWj GL ULXQLUH WXWWH TXHVWH RVVHUYD]LRQL VXOOH
IUD]LRQL H VXOOH UDGLFL LQ XQ GLVHJQR R SURJHWWR XQLWDULR SHU XQ WUDWWDWHOOR GL
DULWPHWLFD/DYRORQWjGLVFRSULUHLQTXDVLWXWWLJOLVFULWWLYLQFLDQLLPDWHULDOLRJOL
DEER]]LSHUWUDWWDWLGLRJQLVSHFLHqXQSUHJLXGL]LRRUPDLLQYHFFKLDWR/HSDJLQHLQ
FXL/HRQDUGRVFULYHRWUDVFULYHGDLOLEULSHUXVRSHUVRQDOHVRQRSLQXPHURVHGL
TXDQWRXQWHPSRQRQVLFUHGHVVH1HOFDWDORJRGHLVXRLOLEULFKH/HRQDUGRUHJLVWUD
QHOPVGHOOD%LEOLRWHFD1D]LRQDOHGL0DGULGDIIYUROWUHDOO¶$ULWPHWLFD
GLPDHVWUR/XFDWURYLDPRXQ©/LEURG¶$EDFRGHO6DVVHWWRªXQ©OLEURG¶$EDFRGD
0LODQR JUDQGH LQ DVVHª XQ ©OLEUR G¶$EDFRª SUHVWDWR D *LRYDQQL GHO 6RGR XQ
©OLEUHWWRYHFFKLRGLDULWPHWLFDªHLO©OLEURGD8UELQRPDWHPDWLFRª&KHQHFHVVLWjH
FKH SRVVLELOLWj DYHYD /HRQDUGR GL VFULYHUH XQ DOWUR WUDWWDWHOOR TXDQGR VHPEUD
HYLGHQWHFK¶HJOLQRQKDQXOODGDLQVHJQDUHHPROWRGDLPSDUDUH"(JOLQRQKDQpOD
FXOWXUD PDWHPDWLFD Qp O¶HVSHULHQ]D GLGDWWLFD GHO 3DFLROL 2 YRUUHPR DQFRUD
VRVWHQHUH FKH /HRQDUGR YDJKHJJLDVVH XQD Summa DG XVR GHJOL VFRODUL GHOOH
ERWWHJKHDUWLJLDQHFRQXQSL]]LFRGLODWLQRXQSL]]LFRGLDULWPHWLFDWUDFHQWRDOWUL
LQJUHGLHQWL"
,IIDKGHOO¶$WODQWLFRVRQRULIHULELOLDOO¶DQQRFLUFDPHQWUHLOPV
/YDGDODO'LUHPRGXQTXHFKHWUDTXHVWHGXHGDWH/HRQDUGRLPSDUz
/¶LQILO]DPHQWR GHL URWWL ©'H PRGR LQILO]DQGL IUDFWRVª QRQ q XQD VHPSOLFH VRPPD GL IUD]LRQL FRPH VHPEUD GLUH LO
0DUFRORQJR/Hricerche HFFSeXQ¶RSHUD]LRQH SDUWLFRODUH FKHVL XVDQHOGLYLGHUHXQQXPHUR PLVWRFRPHDG HV
ଵ
ʹͲ ǡSHU XQ QXPHUR LQWHUR FRPH DG HV ,QYHFH GL ULGXUUH LO GLYLGHQGR D IUD]LRQH LPSURSULD LO 3DFLROL SUHIHULVFH
ସ
ଵ
HOLPLQDUQHXQDSDUWHHFRVuFDOFRODʹͲ ‫ ʹͳ ׷‬ൌ ͳXQSULPRTXR]LHQWHFKHYLHQHDFFDQWRQDWR6XFFHVVLYDPHQWHVLGLYLGH
ଵ
ସ
ଵଵ
ଵଵ
LO UHVWR ͺ ǣ ͳʹ ൌ ǡ FKH VL infilzano FRO TXR]LHQWH SUHYHQWLYDPHQWH DFFDQWRQDWR YDOH D GLUH ͳ ǡ 3URFHGXUD XQ SR¶
ସ
ଵ଺
ଵ଺
SUROLVVD PD FKDULVVLPD QHOOD VXD LPSRVWD]LRQH 7XWWDYLD HVVD Gj OXRJR D VFULWWXUH DOPHQR SHU QRL H D SULPD YLVWD
૛
VFRQFHUWDQWL FRPH TXHVWD ଵ
૜
૚
ଵ
૝
ସ
ǡ &KH RJJL VFULYHUHPPR ʹ LO FXL SUHVXSSRVWR QRQ GLFKLDUDWR GRYUHEEH HVVHUH
XQ¶RSHUD]LRQHFRPHͳͳ ‫͵ ׷‬ǡODTXDOHVFLVVDLQGXHWHPSLGDUHEEHXQSULPRTXR]LHQWHGDDFFDQWRQDUHHODVXFFHVVLYD
ଵ
ସ
ଽ
ଷ
ଷ
GLYLVLRQHGHOUHVWRʹ ǣ ͵GDUHEEHLOVHFRQGRTXR]LHQWH ‘ GDLQILO]DUHFROJLjDFFDQWRQDWRRVVLD͵ Ǥ
ସ
ଵଶ
ସ
ସ
/¶XVR GHOOH IUD]LRQL q QRUPDOH LQ /HRQDUGR TXDQGR HVHJXH OH GLYLVLRQL LO FXL TXR]LHQWH q VSHVVR XQ QXPHUR PLVWR R
DGGLULWWXUD VL ULGXFH D XQDIUD]LRQH SURSULD TXDQGR LOGLYLGHQGR q PLQRUH GHO GLYLVRUH PD O¶XQLFD RSHUD]LRQH VXQXPHUL
IUD]LRQDUL IUHTXHQWH LQ /HRQDUGR FRQVLVWH QHO ULGXUOL DL PLQLPL WHUPLQL LO SL GHOOH YROWH GLPH]]DQGR VXFFHVVLYDPHQWH
QXPHUDWRUH H GHQRPLQDWRUH FRPH VL YHGH DG HV QHO FDOFROR GHOOH SLUDPLGL RWWLFKH SHU FXL FIU $0$5,121, Le
operazioni aritmetiche«SVJJ
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
GD PDHVWUR /XFD OD PXOWLSOLFD]LRQH GHOOH UDGLFL H OH DOWUH UHJROH SHU RSHUDUH FRL
QXPHUL IUD]LRQDUL PD ELVRJQD DJJLXQJHUH FKH LO IUXWWR GL WDQWD OH]LRQH QRQ SDUH
HYLGHQWH R FRSLRVR QHOOH FDUWH GD OXL ODVFLDWH e SHUz LQWHUHVVDQWH YHGHUH FRPH
/HRQDUGRSURFHGHYDQHOORVWXGLRGHOOHIUD]LRQL
6XOYHUVRGHOIDVLOHJJRQROHVHJXHQWLIUD]LRQLFRPSRVWHLQXQDILJXUD
FKHULFKLHGHTXDOFKHVSLHJD]LRQH
ͳ
ͳ
ͳ
ͳʹ
ͳ
൐൏ …‹‘° ͳʹ
͸
ʹ
ͳʹ
Ͳ
$QFKHTXHVWDILJXUDKDLOVXRPRGHOORQHO3DFLROLHVHUYHSHUDSSOLFDUHODUHJROD
ଵ ଵ
ଵ ଵଶ
GHO WUH D XQD WULDGH GL IUD]LRQL 1RL VFULYHUHPPR ǣ ൌ ǣ GRYH LO TXDUWR
ଵଶ ଺
ଶ ଵଶ
SURSRU]LRQDOHqRWWHQXWRFROQRVWURPHWRGRPHGLDQWHODPROWLSOLFD]LRQHGHLPHGLH
OD GLYLVLRQH GHO SURGRWWR SHU LO SULPR HVWUHPR ,QYHFH FRO PHWRGR LPSDUDWR GD
/HRQDUGRVLHOLPLQDRJQLGLVWLQ]LRQHWUDPHGLHGHVWUHPLVHJXHQGROH©OLQHHGHO
YLDª FRQ XQ SURFHVVR IOXLGR H VHPSOLFH 3DUWHQGR GDO QXPHUDWRUH GHO SULPR
WHUPLQHHVFHQGHQGRDLGHQRPLQDWRULGHOVHFRQGRHWHU]RVLPROWLSOLFD[[ FKH q LO GHQRPLQDWRUH GHO TXDUWR WHUPLQH ULFHUFDWR PHQWUH SDUWHQGR GDO
GHQRPLQDWRUH GHO SULPR H VDOHQGR DL QXPHUDWRUL GHO VHFRQGR H WHU]R WHUPLQH VL
PROWLSOLFD[[ LOQXPHUDWRUHFHUFDWR'XQTXHFLRqHQRQO
FRPH/HRQDUGRVFULYHDPRGRVXR0D/HRQDUGRQRQVLDFFRQWHQWDGHOODUHJROD
,OVXRVSLULWRFULWLFRHLOVXRELVRJQRGLFHUWH]]DHVLJRQRXQDYHULILFDVSHULPHQWDOH
HSHUFLzDFFDQWRDOODILJXUDDULWPHWLFDVFULYH©6HIXVVLFKHVDUHEEH"
6DUjFLRqªHSURVHJXH©eFRPHGLUVRSUDRQFHVHXQDRQFLDIXVVL
RQFH FKH VDUHERQ " 6DUHEEH XQD OLEUDª ,Q WDO PRGR O¶DVWUD]LRQH PDWHPDWLFD q
DQFRUDWDDXQIDWWRFRQFUHWRFKHQHFRVWLWXLVFHODUDSSUHVHQWD]LRQHILVLFDHLQFHUWR
PRGRODSURYDVSHULPHQWDOH8Q¶RQFLDqLQIDWWLXQGRGLFHVLPRGLOLEEUDGXHRQFH
XQVHVWRGLLEEUDVHLRQFHPH]]DOLEEUD&RQTXHVWRUDSSRUWRWUDRQFH/HRQDUGR
KDSUDWLFDPHQWHWUDVIRUPDWRLTXDWWURWHUPLQLGHOODSURSRU]LRQHLQTXDWWURIUD]LRQL
DYHQWLORVWHVVRGHQRPLQDWRUHHGDTXHVWRSXQWRHJOLHOLPLQDLOGHQRPLQDWRUH
FRPXQH H RVVHUYD OD SURSRU]LRQH IUD L VROL QXPHUDWRUL FRQVLGHUDWL FRPH QXPHUL
LQWHUL (JOL VFULYH LQIDWWL VRSUD OD ILJXUD FKH DEELDPR HVDPLQDWR TXHVW¶DOWUD
SURSRU]LRQH_GRYHVLYHGHVXELWRHVHQ]DFDOFROLFKHFRPHqLOGRSSLR
GLOFRVuqLOGRSSLRGL1DWXUDOPHQWHSHUFKpVLDEELDQHOOHGXHSURSRU]LRQL
O¶LGHQWLFR TXDUWR SURSRU]LRQDOH RFFRUUH ULHVXPDUH LO GHQRPLQDWRUH VRSSUHVVR
FLRq
2UDVLDPRFHUWLFKHODUHJRODqVLFXUDH/HRQDUGRYXRODSSOLFDUODDIUD]LRQL
SL FRPSOHVVH 3URFHGH SHU JUDGL FRQVHUYDQGR OD WULDGH SULPD FRQVLGHUDWD ,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
PD DJJLXQJH XQ¶XQLWj D RJQL WHUPLQH WUDVIRUPDQGROR LQ QXPHUR PLVWR
FKHSRLWUDVIRUPDLQXQDIUD]LRQHLPSURSULD
ʹ
ଵ
ଵ
ଵ
ଵଶ
଺
ଶ
଺଴
଻଼
ଵ଴
ଵଷ
ଷ
ଶଵ଺
/¶LQWHUSUHWD]LRQHQRQqGLIILFLOH/DOLQHDGLDJRQDOHFKHXQLVFHRJQLXQLWj
DOGHQRPLQDWRUHLQGLFDODPROWLSOLFD]LRQHGL[FKHXQLWRDOQXPHUDWRUHGjLO
QXRYRQXPHUDWRUHGL[ GL[ ,QXRYLWHUPLQLVRQRGXQTXH
3HUWURYDUHLOGHQRPLQDWRUHGHOTXDUWRSURSRU]LRQDOHVLGHYHRUD
PROWLSOLFDUH[[ 3XUWURSSR/HRQDUGRGLPHQWLFDGLPROWLSOLFDUHSHU
HVLIHUPDD[ 3HUWURYDUHLOQXPHUDWRUHVLGHYHPROWLSOLFDUH[[ $QFKHTXL
/HRQDUGR FRPPHWWH XQD VYLVWD /¶RSHUD]LRQFLQD VFULWWD GL ILDQFR [ ULYHODFKH/HRQDUGRLQYHFHGLPROWLSOLFDUH[ KDPROWLSOLFDWR[ ଶହଶ
଼
ଶଵ଺
ଵ଴
,QYHFHGHOULVXOWDWRFRUUHWWR FLRqͳ /HRQDUGRRWWLHQH FLRq ଵହ଺
ଵଷ
଻଼
ଵଷ
$OGLVRSUDGLTXHVWDILJXUDQHWURYLDPRXQ¶DOWUDGLVHPSOLFLQXPHULLQWHUL
ହ
GRYHqIDFLOHULFRQRVFHUHFROORVWHVVRSURFHGLPHQWRJLjXVDWRSHU
ଵଷ
ଵଷ ଻ ଷ
ଵଷ ଵସ ଵ଼
OHRQFHLQXPHUDWRULGHOODWULDGH WUDVIRUPDWDLQ (DQFKHTXL
ଵଶ ଺ ଶ
ଵଶ ଵଶ ଵଶ
SHUFKpLOTXDUWRSURSRU]LRQDOHFRLQFLGDRFFRUUHUHEEHULSULVWLQDUHLOGHQRPLQDWRUH
ଵଽ
ఱ
଼
SHUFXL భయ ൌ ͳ FLzFKH/HRQDUGRQRQID
ଵଷ
ଵଶ
/¶LQJHJQRVD FXULRVLWj GL /HRQDUGR q FRQIHUPDWD GDOOD VXFFHVVLYD
ଶ
ହ
SURSRU]LRQH FKHqULFDYDWDGDOODSUHFHGHQWH ଷ
ଵଷ
HOLPLQDQGRODFLIUDGHOOHGHFLQHQHLSULPLWUHWHUPLQLFRQXQSURFHVVRLQYHUVRD
TXHOOR FRQ FXL SUHPHWWHQGR OD FLIUD DOOD SURSRU]LRQH EDVH VL HUD FRVWUXLWD OD
ଵ
ଵ
ଵ
SURSRU]LRQH ͳ ͳ ͳ &KH FRVD VSHUDYD GL FRQFOXGHUH /HRQDUGR" 'L
ଵଶ
଺
ଶ
ULWURYDUVL DO SXQWR GL SDUWHQ]D" &Lz QRQ HUD SRVVLELOH SHUFKp O¶HOLPLQD]LRQH GHO
ଵସ
GHQRPLQDWRUHDOWHUDLOYDORUHGHLWHUPLQLDOWURqDOWUR 0DQRQGREELDPR
ଵଶ
IDUH LO SURFHVVR DOOH LQWHQ]LRQL VH TXHVWH QRQ DSSDLRQR FKLDUDPHQWH GHILQLWH
GLUHPRVRORFKH/HRQDUGRVWDSURYDQGRHULSURYDQGRFRQXQDFXULRVLWjULYHODWULFH
GHOOD VXD VFDUVD FRQRVFHQ]D GL TXHVWD SDUWH GHOO¶DULWPHWLFD PD DQFKH GHOO¶DFXWD
FXULRVLWjGHOVXRVSLULWRFKHLQDOWULFDPSLGjIUXWWLSLSUH]LRVL
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
7UDODVFLDPR QXPHURVL DOWUL HVHPSL GL DSSOLFD]LRQH GHOOD UHJRODGHO WUH DL
QXPHULIUD]LRQDULHFRQVLGHULDPRXQ¶DOWUDFXULRVDILJXUDFKHOHIUD]LRQLDVVXPRQR
LQ /HRQDUGR $QFKH VWDYROWD HJOL QRQ DSSOLFD O¶RUGLQDULD UHJROD H VHJXH XQ
SURFHGLPHQWRDVVDLSUROLVVRFKHSXUFRQGXFHDXQULVXOWDWRHVDWWR)UDLQRQSRFKL
HVHPSLHVDPLQLDPRTXHOORGHOFRG$WOYGFKHFROOHVXHLQFHUWH]]HHLVXRL
ODSVXVPHJOLRULYHODORVIRU]RGHOO¶DSSUHQGLVWDLQHVSHUWR©GLFKHQXPHURILDH
"7DQWRqDGLUH'DPPLHGLTXDQWRqDGLUHGLFKHQXPHURILDH
"(WXWWRVLIDSHUXQDUHJROD
ଷ
ସ
ସ
ହ
ଷ
଺
E͵ ସ
଼
ଷ
D
ସ
,OGLb, FKHGLVFHQGHLQa, qSRVWRLQWDOVLWRVROSHUGHQRPLQDWRUHGHOOH
ଷ
ILJXUHGLVRSUDODULJDFKHVLGHEERQRLQWHQGHUHHVVHUH͵‡ GLTXDUWLFKHVLGq
ସ
SR¶ IDUH GL WXWWR TXHVWR VHGLFHVLPL FKH VRQ TXDUWL GL TXDUWL H ULWURYHUDL VWDU WDO
UHJRODQHOVXRYHURHVVHUH«'L¶FRVuYLDID3DUWLORSHOFKHVWDVRSUD
ଷ
LOQHYHUUj͵‡ (TXHVWRV¶LQWHQGHQRQXQLWjPDGLTXH¶TXDUWLSULPLH
ସ
SH¶VHJQRGLFRPHHOFKHVWDVRWWRLOPHWWHUD¶ORVRWWRLOHHGLUDLFKHQH
య
ଷୣ
ଷ
ర
YHQJD
G¶XQG¶HVVLTXDUWLRQGHULGXFHUDLHVVR͵‡ DPLQRUURWWRHGLUD¶
ସ
ସ
³ YLH ID ´ $JJLXJQLOL LO FKH VWD VRSUD OD ULJD H GL¶ ³ GL TXDUWL´ FLRq
2UDULGXFLWXWWDODWXDSURSRVWDDHVVLVHGLFHVLPLHDUDLLOWXRTXLVLWRHVVHUH
FKHYHGLFKHqYHUDPHQWHHGL(ULWRUQDQGRDTXDUWLHVDUj
HHVVHUHWUHTXDUWLLQWHULHGLTXDUWR(VWDEHQHª
ଷ
ସ
6L WUDWWD VHPSOLFHPHQWH GL GLYLGHUH ‫ ׷‬/¶RUGLQDULD UHJROD GLUHEEH GL
ସ
ହ
PROWLSOLFDUHLQFURFHHFRQGXUUHEEHLPPHGLDWDPHQWHDOULVXOWDWR/HRQDUGR
LQYHFH SURFHGH OHQWDPHQWH H IDWLFRVDPHQWH SHU XQD YLD SL OXQJD LQGLFDWD GDOOH
OLQHH GHO YLD 0ROWLSOLFD LO QXPHUDWRUH GHO GLYLGHQGR SHU LO GHQRPLQDWRUH GHO
GLYLVRUH [ FROORFD LO SURGRWWRVRWWRLOGLYLVRUHHVRWWRGLHVVRWUDVSRUWDLO
ଷ
QXPHUDWRUHGHOGLYLVRUH,OULVXOWDWRGLTXHVWDGLYLVLRQHqVFULWWRGLILDQFR͵ ସ
0D Y¶q RUD GD ULSUHQGHUH LO GHQRPLQDWRUH GHO GLYLGHQGR b FKH YLHQH FRQGRWWR
ଷ
VRWWRLOTXR]LHQWHRWWHQXWR͵ aSHUFKp LQ HIIHWWL VL WUDWWD GL TXDUWL GL TXDUWL
ସ
ଷ
ଵହ
ଷ
ସ
FLRq VHGLFHVLPL 4XL ELVRJQD WUDVIRUPDUH LO ͵ LQ FKH GLYLVR SHU TXDWWUR
ଵହ
GLYLHQHILQDOPHQWH 0D/HRQDUGRQRQqDQFRUDVRGGLVIDWWRGLFRVuOXQJDIDWLFD
ଵ଺
(JOLYXROHRUDDVVLFXUDUVLFKHWDOUHJRODVWLD©QHOVXRYHURHVVHUHªFKHVLDHVDWWD
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
HGHIILFLHQWH5LFRUUHDOORUDDOO¶HVSHGLHQWHJLjRVVHUYDWRQHOSUHFHGHQWHSDUDJUDIR
OD WUDVIRUPD]LRQH GHO GLYLGHQGR XQD IUD]LRQH DYHQWH LO GHQRPLQDWRUH FRPXQH D
TXHOOR GHO TXRWR FRVu GD SRWHUOR LJQRUDUH H FRQVWDWDUH QHO UDSSRUWR GHL VROL
QXPHUDWRUL FKH OD UHJROD ©VWD EHQHª q ©QHO VXR YHUR HVVHUHª 1HO FDVR QRVWUR
ଷ ସ
ଵହ
FRQFOXVR FKH ǣ ൌ /HRQDUGR WUDVIRUPD LO ©TXHVLWRª R ©SURSRVWDª RVVLD OD
ସ ହ
ଷ
ଵ଺
ଵଶ
IUD]LRQHGLYLGHQGR ©LQVHGLFHVLPLªRVVLDQHOODIUD]LRQHHTXLYDOHQWH TXLSHU
ସ
ଵଶ
ଵ଺
ଵଶ
ଵହ
ଵହ
ଵ଺
ସ
ଵ଺
XQ ODSVXV VFULYH ©H DUDL LO WXR TXLVLWR HVVHUH ª SHU SRWHU RSSRUUH D H
WUDVFXUDWRLOGHQRPLQDWRUHFRPXQHFRQVWDWDUH©FKHqYHUDPHQWHH GLª,O
ହ
GLVFRUVRVDUHEEHTXLQGLILQLWRPD /HRQDUGRYXROULWRUQDUHDOODIUD]LRQHLQL]LDOH
ଵଶ
ଵଶ
©(ULWRUQDQGRDTXDUWLH ªHULSHWHO¶HUURUHLQOXRJRGL PDO¶RSHUD]LRQHH
ଵହ
ଵ଺
ଵଶ
VFULWWDLQFRORQQDGLILDQFRDOODILJXUDGLPH]]DQGRGXHYROWH ILQRDJLXQJHUHD
ଷ
ଷ
ଵହ
ସ
ସ
ଵ଺
ଵ଺
ଷ
VDUj H HVVHUHWUHTXDUWLLQWHULH GLTXDUWRª4XHVW¶XOWLPDHVSUHVVLRQHVL
ଵହ
ସ
ଵଶ
ଷ
ଷ
ଷ
FKLDULVFHVFULYHQGRVHPSOLFHPHQWH ൌ ൅ ൌ ൅ ଵ଺
ଵ଺
ଵ଺
ସ
ଵ଺
/R VWHVVR SURFHGLPHQWR FROOD VWHVVD ILJXUD ULWRUQD LQ DOWUH SDJLQH YLQFLDQH 1HO
ଶ ଷ
ଶయ
ସ
ଷ ସ
ଷ
FRG$UXQGHOUODILJXUDqULSUHVHQWDWDWUHYROWHSHUOHGLYLVLRQLͷǣ ǣ ൌ
ଷ ସ
ଵହ
ସ ହ
ଵ଺
ǣ ൌ
మ
ଷ
/DSULPDGLYLVLRQHqFRVuSUHVHQWDWD
ହ
ଵ
ଷ
ସ
ଶ
͸ ଷ
5LVROYLLOLQHOURWWRUHVXOWDWRHFRVuLOHWURYHUUDLHVVHUHHOHVVHUH
FKH FRPH YHGL VRQR H GL RQGH OD UHJROD q EXRQDª 3HU OD
ଷ
VHFRQGDGLYLVLRQHGLFHFKH©GLUjSRLªHGqFKLDURFKH͸ ൌ †‹ͺ
ସ
6L FRQIHUPD GXQTXH FKH LO SURFHGLPHQWR PLUD D FRQWUDSSRUUH IUD ORUR L
QXPHUDWRULGHOOHIUD]LRQLGLYLGHQGRHTXRWRSHUDFFHUWDUHFRQLPPHGLDWDHYLGHQ]D
FKHQRQWDQWRLOVLQJRORULVXOWDWRPDLOSULQFLSLRWHRULFRODUHJRODqEXRQD,OFKH
VLJQLILFD FKH /HRQDUGR KD GLPHVWLFKH]]D FRL QXPHUL LQWHUL H QRQ DYHQGRQH XQD
DOWUHWWDOHSHLQXPHULIUD]LRQDULqFRQGRWWRDWUDVIHULUHTXHVW¶XOWLPLQHOO¶DPELWRGHL
SULPLSHUYHULILFDUHODYDOLGLWjGHOSURFHGLPHQWRHO¶LQHTXLYRFDELOLWjGHLULVXOWDWL
8Q¶HQQHVLPDSURYDGHOODVHULHWjGL/HRQDUGRFRHUHQWHFRQLVXRLSULQFLSLDQFKH
VHQRQHVSHUWRPDWHPDWLFR
6LLQWHQGD͸ ଶଷ
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
1HOIYGHOORVWHVVRFRGLFHWURYLDPRRUGLQDWDPHQWHGLVSRVWHHLQFRORQQDWHOH
଺ ଶ ଶ ଷ ଷ ସ ସ ହ ହ ଺ ଷ ସ
GLYLVLRQL ǣ Ǣ ǣ Ǣ ǣ Ǣ ǣ Ǣ ǣ Ǣ ǣ H DFFRPSDJQDWH FLDVFXQD GD XQ
ଶ ଷ ଷ ସ ସ ହ ହ ଺ ଺ ଻ ହ ଽ
EUHYHFRPPHQWR4XHVWDYROWDO¶RSHUD]LRQHQRQqLQTXDGUDWDQHOODSUROLVVDILJXUD
VRSUDHVDPLQDWDLQ$WOYGH$UUPDqULVROWDVYHOWDPHQWHFROO¶RUGLQDULD
UHJRODGHOPXOWLSOLFDLQFURFH7XWWDYLDODYHULILFDqVHPSUHHVHJXLWDFROPHWRGR
GHOODWUDVIRUPD]LRQHGHOGLYLGHQGRLQIUD]LRQHDGHQRPLQDWRUHFRPXQHFROTXRWR
7DOH GHQRPLQDWRUH q GHILQLWR GD /HRQDUGR FRPH lo intero $G HVHPSLR OD
ଷ ସ
ଵହ
ଷ
ଵଶ
GLYLVLRQH ǣ ൌ /HRQDUGRFRPPHQWD©4XLORµQWHURVLqHVXD ILD ସ ହ
ଵଶ
ଵ଺
ସ
ଵହ
ହ
ଵ଺
ସ
ଵ଺
LOTXDO ILDH GH ଵ଺
0D ELVRJQD LQWHQGHUH SHU LQWHUR QRQ PD
ଵଶ
ସ
ଵହ
ଵ଺
ହ
ଵ଺
VHGLFHVLPL q FKH q SDUL D ൈ
ସ ହ
ଶସ
ହ ଺
ଶହ
ǣ ൌ
ଵ଺
ଷ
,O GLYLGHQGR WUDVIRUPDWR LQ
ଵ଺
ସ
ସ
FRPH ͳʹ ൌ ൈ ͳͷ 3HU OD GLYLVLRQH
ହ
ଶହ
ସ
ଶହ
ହ
LOFRPPHQWRGLFH©4XLORµQWHURqªRVVLD ©H GLVLqLO
ହ
ଶସ
଺
ଶହ
TXDOH q †‹ ª ,O IDWWR FKH LO GHQRPLQDWRUH RUD FRPSDLD H VFRPSDLD
ସ
VLJQLILFD FKH HVVR YLHQH PHQWDOPHQWH WUDVFXUDWR 1RL VFULYHUHPPR
ଶ଴
ଶହ
ହ
ଶସ
଺
ଶହ
ൌ ൈ
PD SHU JOL RPLQL VDQ]D OHWWHUH OD VFULWWXUD
ହ
ଶ଴
ଶହ
ହ
ଶସ
଺
ଶହ
ൌ ൈ
ହ
ൌ
ଶ଴
H
ଶହ
QRQ KD
XQ¶HYLGHQ]D FRVu WDQJLELOH FRPH ʹͲ ൌ †‹ʹͶ e EHQVu YHUR FKH /HRQDUGR KD
VFULWWRSRFRSULPD©FRPHYHGL
ଵହ
ଷ
଺
ଷ
ଶ଴
ସ
ଷ
•‘‘‡ †‹ ªPDDQFKHFRVuVFULYHQGR
ଷ
HJOLREOLWHUDPHQWDOPHQWHLOGHQRPLQDWRUHHYHGHͳͷ ൌ ൈ ʹͲVHPSOLILFD]LRQH
ଵହ
ସ
ଷ
ଶ଴
ସ
ଷ
DVVDL SL UDSLGD GHOOH DOWUH SRVVLELOL FRPH VDUHEEHUR RYYHUR ൌ ൈ
ଶ଴
ଵହ
ଷ
ൌ ͷRSSXUH ൌ ൈ
ଷ
ସ
/HRQDUGR
ସ
ଶ଴
ଷ
ൌ
଺଴
ଵଶ
‡
଺଴ǣସ
ଵଶǣସ
ൌ
ଵହ
ଷ
ଷ
ൌ
FKHVRQRSURFHGLPHQWLLQVXHWLSHU
/D FRQFOXVLRQH q FKH /HRQDUGR SHU EXRQD SDUWH GHOOD VXD YLWD VL VHUYu
VFDUVDPHQWH GHL QXPHUL IUD]LRQDUL OLPLWDQGRVL DOOD ORUR ULGX]LRQH DL PLQLPL
WHUPLQLPDHYLWDQGRGLVRPPDUOLVRWWUDUOLPROWLSOLFDUOLHGLYLGHUOL3HUXQFHUWR
WHPSR HJOL FRQGLYLVH LO GXEELR FRPXQH DL VDQ]D OHWWHUH FKH O¶RUGLQDULD UHJROD
GHOOD GLYLVLRQH GHOOH IUD]LRQL IRVVH ©SRFR FKLDUD ª PD ULXVFu D FKLDULUOD YHUVR L
FLQTXDQW¶DQQL VLD LPSDUDQGR OD OH]LRQH GHO 3DFLROL VLD DFFHUWDQGROD FRQ PHWRGL
SHUVRQDOL FRQVLVWHQWL VRSUDWWXWWR QHOOD ULGX]LRQH GHOOH IUD]LRQL FRQ GHQRPLQDWRUL
GLYHUVLLQIUD]LRQLDGHQRPLQDWRUHFRPXQHSHUPHJOLRYHULILFDUQHLOUDSSRUWRFRO
VHPSOLFHUDSSRUWRWUDLQXPHUDWRULRUPDLFRQVLGHUDWLFRPHQXPHULLQWHUL4XHVWR
LQJHJQRVRSURFHGLPHQWRQRQWDQWRFLIDVRUULGHUHTXDQWRFLIDDSSUH]]DUHVXXQ
SLDQR PRUDOH SULPD FKH VFLHQWLILFR O¶DVVROXWR ELVRJQR GL FHUWH]]D GL SDOPDUH
HYLGHQ]DFKHVHPSUHJXLGzLOSHQVLHURGL/HRQDUGR(JOLQRQVLDIILGzDOO¶DXWRULWj
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
GL QHVVXQR QHSSXUH D TXHOOD GHO 3DFLROL H YROOH GLPRVWUDUH D VH VWHVVR FRPH H
SHUFKpODUHJROD©qEXRQDª
6H q YHUR FKH /HRQDUGR VROR YHUVR L FLQTXDQW¶DQQL LPSDUz GD PDHVWUR /XFD OD
PXOWLSOLFD]LRQH GHOOH UDGLFL H D RSHUDUH FRUUHWWDPHQWH VXL QXPHUL IUD]LRQDUL FL
FKLHGLDPR TXDOH XVR HJOL IHFH LQ VHJXLWR GL TXHJOL LQVHJQDPHQWL 4XL VLDPR
FRVWUHWWL DG HVDPLQDUH FULWLFDPHQWH OH DIIHUPD]LRQL HQWXVLDVWLFKH IDWWH GDO
0DUFRORQJR VHFRQGR LO TXDOH EDVWHUHEEH LO I Y GHO FRGLFH $UXQGHO SHU
GLPRVWUDUHO¶DELOLWjFRQFXL/HRQDUGRFDOFRODIUD]LRQLHUDGLFL
6L WUDWWD GL XQ WHRUHPD FRQ FXL /HRQDUGR WURYD FRQ HOHJDQWH SURFHGLPHQWR
JHRPHWULFR LO EDULFHQWUR GL XQD PHQVROD DYHQWH IRUPD GL WUDSH]LR LVRVFHOH
7UDFFLDWD VXOOD EDVH GHO WUDSH]LR OD SHUSHQGLFRODUH FKH OR GLYLGH LQ GXH HVDWWH
PHWj /HRQDUGR DIIHUPD FKH LQ HVVD SHUSHQGLFRODUH GHYH FDGHUH LO EDULFHQWUR
ULFHUFDWR 3HU GHWHUPLQDUOR FRQJLXQJH GXH YHUWLFL RSSRVWL GHO WUDSH]LR FRQ XQD
OLQHD FKH OR GLYLGH LQ GXH WULDQJROL FKH FRVWLWXLVFRQR ULVSHWWLYDPHQWH XQR H GXH
WHU]L GHOO¶LQWHUR 4XLQGL FRO¶LQFURFLR GHOOH PHGLDQH GHL GXH WULDQJROL QH WURYD L
VLQJROLEDULFHQWULh HL. ,OVHJPHQWRhL FRQJLXQJHQWHLGXHEDULFHQWULGHWHUPLQDLO
EDULFHQWUR GHOOD PHQVROD QHO SXQWR . LQ FXL GHWWR VHJPHQWR WDJOLD OD
SHUSHQGLFRODUHRPHGLDQDGHOWUDSH]LR,OVHJPHQWRhL qGLYLVRGD.LQPRGRFKH
hK VWD D KL QHOOD SURSRU]LRQH LQYHUVD D TXHOOD GHL GXH WULDQJROL LQ FXL YHQJRQR
ULVSHWWLYDPHQWHDJLDFHUHFLRqGXHHXQWHU]RGLKL. $TXHVWRSXQWRWHUPLQDWRLO
GLVFRUVR JHRPHWULFR /HRQDUGR LQWURGXFH HVHPSOLILFDQGR GHL YDORUL QXPHULFL
©6H OD PHQVROD SHVD LO WULDQJROR abd SHVD H LO WULDQJROR acd SHVD H OD
OLQLDhL qUDGLFHGLHHGLYLVDQHOODSURSRU]LRQHGHOOLWULDQJROLLQSXQWR.
FKH ILD LK UDGLFH GL H H hK UDGLFH GL H ª 4XHVWL YDORUL ʊ
FRPPHQWDLO0DUFRORQJRʊ©SURYDQRFKHHJOLIDVSHGLWDPHQWHHFRUUHWWDPHQWH
ULGX]LRQH GL IUD]LRQL FDOFROL GL UDGLFDOL H SURYDQR OD LQIRQGDWH]]D GL FHUWL
JLXGL]L DVVROXWL IRQGDWL VX HVDPH GL VFULWWL DSSDUWHQHQWL DL SULPL DQQL GHOOD
JLRYHQWHDQFKHDLSULPLVWXGLGL/HRQDUGRFRPHSHUHVHPSLRLOVHJXHQWHGHOOR
6FKXVWHUdies hängt mit seiner geringen mathematischen Schulung zusammenª
0D q SURSULR TXHVWR JLXGL]LR GHO 0DUFRORQJR FKH FL VHPEUD WURSSR DVVROXWR H
DIIUHWWDWR,QQDQ]LWXWWRLO0DUFRORQJRLJQRUDRWUDVFXUDLGDWLFURQRORJLFL3HUFKq
UHWURGDWDUHOH FRVLGGHWWH ©LQFHUWH]]H GL /HRQDUGRª TXDQGR VL WUDWWD GL JUDYLVVLPH
ODFXQH HG HUURUL ULVFRQWUDELOL QHOOD VXD SLHQD PDWXULWj" &RPH VL SXz GLUH FKH OH
ULJKHILQDOLGHOWHRUHPDGHOWUDSH]LRSURYDQRFKHHJOLID©VSHGLWDPHQWHªLOFDOFROR
GHLUDGLFDOLTXDQGROHVXGGHWWHULJKHFLSUHVHQWDQRVHPSOLFHPHQWHLULVXOWDWLGLXQ
FDOFRORFKHQRQVDSSLDPRGDFKLVLDVWDWRHVHJXLWRQpFRPHQpLQTXDQWRWHPSR"
1RL VDSSLDPR FKH OH QRQ SRFKH SDJLQH YLQFLDQH LQ FXL VL HVHJXRQR FDOFROL
DULWPHWLFL ULYHODQR TXDVL VHPSUH LQFHUWH]]H HG HUURUL 3HU LOI GHOO¶$UXQGHO L
FDOFROL PDQFDQR 6H HVVL IXURQR HVHJXLWL YHUDPHQWH GD /HRQDUGR GRYUHPPR
ULFRQRVFHUH FKH QHO VHL DQQL GRSR OD OH]LRQH GHO 3DFLROL /HRQDUGR DYHYD
FRPSLXWR GHL QRWHYROL SURJUHVVL QHO FDOFROR GHOOH IUD]LRQL H GHL UDGLFDOL 0D QHO
1RQVDSSLDPRSHUzVLEDGLVHVLWUDWWDGLXQRVFULWWRRULJLQDOHGL/HRQDUGRRSSXUHFRPHWDQWL
DOWULFRSLDWRRULDVVXQWRGDWHVWLDOWUXL
50DUFRORQJRLe ricerche«S
,Q³3HULRGLFRGL0DWHPDWLFKH´%RORJQDVHULH,9YRO;/9,QGLFHPEUHSS
JUXSSR GL SDJLQH GHOO¶$UXQGHO VFULWWH QHO H D FXL DSSDUWLHQH LO I QRQ
PDQFDQREUDQLFKH/HRQDUGRKDFRSLDWRRULDVVXQWRGDDOWULOLEUL1HOODSumma GHO
3DFLROL WURYLDPR OD IRQWH GHO I U ©0RGR GL WURYDUH WXWWH OH UDGLFH Gq QXPHUL
JHRPHWULFDPHQWHª H WURYLDPR SXUH PROWL WHRUHPL FRQGRWWL FROOR VWHVVR
SURFHGLPHQWRJHRPHWULFRHFRQFOXVLFRQXQ¶HVHPSOLILFD]LRQHQXPHULFDDEDVHGL
IUD]LRQLHGLUDGLFLFRPHVLYHGHQHOWHRUHPDGHOWUDSH]LR'¶DOWUDSDUWHQHOIU
VHPSUH FRHYR GHO I U QRL SRVVLDPR ULWURYDUH LPSUHFLVLRQL GL OLQJXDJJLR FKH
QRQ SRWUHEEHUR FRQIHUPDUH L SUHVXQWL SURJUHVVL GL /HRQDUGR QHO FDOFROR
PDWHPDWLFR'LIURQWHDLQXPHURVLIRJOLFKHGLPRVWUDQRLQ/HRQDUGRXQDPRGHVWD
HVSHULHQ]D QHO FDOFROR DULWPHWLFR LO I GHO FRGLFH $UXQGHO QRQ EDVWD D
GLPRVWUDUH FRPH YROHYD LO 0DUFRORQJR XQ VLFXUR SURJUHVVR FRPSLXWR GD
/HRQDUGRVHPPDLDFLQTXDQWDVHLDQQL,OJLXGL]LRGHOOR6FKXVWHUVXOODOLPLWDWDR
VFDGHQWH LVWUX]LRQH PDWHPDWLFD ULFHYXWD GD /HRQDUGR FRQVHUYD GXQTXH OD VXD
SLHQDYDOLGLWj
$0$5,121,
Scarica

L`aritmetica di Leonardo