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[6] http://www.lygeros.org/2489-gr.htm[7] http://www.math.uoc.gr/~pamfilos/gGallery/Gallery.html[8]http://www.mfdabbs.pwp.blueyonder.co.uk/Maths_Pages/SketchPad_Files/Mechanical_Linkages/Mechanical_Linkages.htmlAbstract : The teaching of simple geometrical locus constitutes a part of one Geometric knowledge, while significances, as the conic sections or the dynamic lines in the Physics or in other sciences, constitute certain extensions or even their generalisations, that however usually are presented incohesively and fragmentarily. The work proposes their instructive reconnection in the light holism and their presentation, via a dynamic software of Geometry.

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SUMMARY:

In all the religions, the god is infinite. Its comprehension, if it is feasible, passes

through the comprehension of the infinite. The instinctive accession of the meaning of the infinite, historically has leaded into faults, misapprehensions, paradoxes and fertile contradictions. Today Mathematiques have very much penetrated in the meaning of the infinite, and because of this the philosophical and theological reflection can be enhanced. Also the Mathematician logic has attempted some of its paradoxes. Having all these equipments we can access questions as the followings: Can the God create other infinite beings, being itself an infinite being? As long as it is omnipotent, can it construct a stone that it cannot lift? The meaning of human God that is given to the Christ, is it inconsistent? Could the God create a better world than the world of the wars, the fame and the injustice, that he it has created? (Not according to Leibniz). Does the faith of a man to the God and his simultaneous default through his signs make him the acme of the human absurdity? These questions are examined in the present work that tries to give answers or to enlarge the field of their reference.

1 Ceres

Ceres 1801 Piazzi , ( Ceres). . Gauss , . Gauss . 1809 , Legendre. , (xi, yi), i=1, 2, ..., . , 4 (1, 1), (0, 2), (-1,

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414011

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. 25/4/2004

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. 25/4/2004

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. 25/4/2004

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. 25/4/2004

, ' , ( ) ' (. 191) : 1) Sir Thomas Heath

... 2001 2) - - 1999 3) . . 1971 4) - - 1966 5) , -.

6) , , , : ( 1994) 7) G.S. KIRK - J.E. RAVEN - M. SCHOFIELD, , ( 89 - 109) : ( 1990) 8) , 6 ""

. 25/4/2004

1

1-26 ( )

1-26

, . , 1-28 , , . , 1-26, , . 23 ( ) , 5 () 26 . 4 5 - , . 9 .4. .

() 1. Shme n stin, o mroj o qn. 2. Gr amm d mkoj platj. 3. Gr ammj d prata shmea. 4. E qea gramm stin, tij x sou toj f' autj shmeoij ketai. 5. 'Epifneia d stin, mkoj ka pltoj m non cei. 6.'Epi f aneaj d prata gramma. 7. 'Eppedoj pifnei stin, tij x sou taj f' autj e qeaij ketai. 8. 'Eppedoj d gwna stn n pipdJ d o grammn ptomnwn lllwn ka m p' e qeaj keimnwn pr j lllaj tn grammn klsij. 9. Ot an d a pericousai t n gwnan gramma e qeai sin, e q grammoj kaletai gwna. 10.Ot an d e qea p' e qean staqesa tj fex j gwnaj saj lllaij poi, rq katra tn swn gw- nin sti, ka festhkua e qea kqetoj kaletai, f' n fsthken. 11. 'Amblea gwna stn mezwn rqj. 12.Oxea d lsswn rqj.

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2

13.Oroj stn, tin j sti praj. 14. Scm sti t p tinoj tinwn rwn periec menon. 15. K kloj st scma ppedon p mij grammj periec menon [ kaletai perifreia], pr j n f' n j shmeou tn nt j to scmatoj keimnwn psai a prospptousai e qeai [pr j t n to k klou peri- freian] sai lllaij esn. 16. Knt r on d to k klou t shmeon kaletai. 17. Dimet r oj d to k klou stn e qe tij di to kntrou gmnh ka peratoumnh f' ktera t mrh p t j to k klou perifereaj, tij ka dca tmnei t n k klon. 18.`Hmik klion d sti t periec menon scma p te t j diamtrou ka t j polambanomnhj p' a t j peri- fereaj. knt r on d to mikuklou t a t , ka to k klou stn. 19.Scmata e q gramm sti t p e qein periec - mena, t r pl eur a mn t p trin, t et r pl eur a d t p tessrwn, pol pl eur a d t p plei nwn tess- rwn e qein periec mena. 20. Tn d triple rwn schmtwn s pleuron mn tr- gwn n sti t tj trej saj con pleurj, soskel j d t tj d o m naj saj con pleurj, skal hnn d t tj trej nsouj con pleurj. 21.Et i d tn triple rwn schmtwn rqognion mn trgwn n sti t con rq n gwnan, mbl ugnion d t con mblean gwnan, xugnion d t tj trej xeaj con gwnaj. 22. Tn d tetraple rwn schmtwn tetrgwnon mn stin, s pleur n t sti ka rqognion, ter mhkej d, rqognion mn, o k s pleuron d, mboj d, s pleuron mn, o k rqognion d, omboeidj d t tj penanton pleurj te ka gwnaj saj lllaij con, o te s pleur n stin o te rqognion t d par tata tetrpleura trapzia kalesqw. 23.Parllhlo esin e qeai, atinej n t a t pi- pdJ osai ka kball menai ej peiron f' ktera t mrh p mhdtera sumpptousin lllaij.

1.'itsqw p pant j shmeou p pn shmeon e qean gramm n gagen. 2.Ka peper asmnhn eqean kat t sunecj p' e - qeaj kbalen. 3. Ka pant kntrJ ka diastmati k klon gr-

. 25/4/2004

3

fesqai. 4.Ka psaj t j r q j gwnaj saj l l l ai j enai. 5. Ka n ej d o e qeaj e qea mpptousa tj nt j ka p t a t mrh gwnaj d o rqn lssonaj poi, kballomnaj tj d o e qeaj p' peiron sump- ptein, f' mrh esn a tn d o rqn lssonej.

1.T t a t sa ka llloij stn sa. 2.Ka n soij sa prosteq, t la stn sa. 3.Ka n p swn sa faireq, t kataleip men stin sa. 4.Ka t farm zonta p' llhla sa llloij stn. (6*).[Ka n nsoij sa prosteq, t la stn nisa. ( 7*.) Ka t to a to diplsia sa llloij stn. (8*.)Ka t to a to msh sa llloij stn.] 5.Ka t lon to mrouj mezon [stin]. ( 9*)Ka d o e qeai cwron o pericousin.

, 1-26 . , . ====================================== .1 'Ep t j doqeshj e qeaj peperasmnhj trgwnon s - pleuron sustsasqai. : Estw doqesa e qea peperasmnh AB.

C1

c2

. 25/4/2004

4

De d p t j AB e qeaj trgwnon s pleuron((.20) sust- sasqai. Knt r J mn t A diastmati d t AB k kloj gegr- fqw BGD, (.3) ka pl in knt r J mn t B diastmati d t BA k kloj gegrfqw AGE, ka p to G sh- meou, kaq' tmnousin lllouj( ) o k kloi, p t A, B shmea peze cqwsan e qeai a GA, GB. (.1) Ka pe t A shmeon kn- tron st to GDB k klou, (.15) sh stn AG t ABplin, pe t B shmeon kntron st to GAE k klou, (.15) sh stn BG t BA. decqh d ka GA t AB sh katra ra tn GA, GB t AB stn sh. t d t a t sa ka llloij stn sa (..1)ka GA ra t GB stn sh a trej ra a GA, AB, BG sai lllaij esn. 'Is pleuron ra st t ABG trgwnon, ka sunstatai p t j doqeshj e qeaj peperasmnhj t j AB. ['Ep t j doqeshj ra e qeaj peperasmnhj trgwnon s pleuron sunstatai] per dei poisai. ===================================== 2. Pr j t doqnti shmeJ t doqesV e qev shn e qean qsqai. : Est w t mn doqn shmeon t A, d doqesa e qea BG de d pr j t A shmeJ t doqesV e qev t BG shn e qean qsqai. 'Epeze cqw gr p to A shmeou p t B shmeon e qea AB(.1), ka sunesttw p' a t j trgwnon s pleu-

1

. 25/4/2004

5

ron t DAB,(.1) ka kbeblsqwsan p' e qeaj taj DA, DB e qeai a AE, BZ,(.2) ka knt r J mn t B diastmati d t BG k kloj gegrfqw GHQ,(.3) ka plin kntrJ t D ka diastmati t DH k kloj gegrfqw HKL.(.3) 'Epe on t B shmeon kn- tron st to GHQ k klou, (.15) sh stn BG t BH. plin, pe t D shmeon kntron st to KLH k klou, sh stn DL t DH, n DA t DB sh stn. loip ra AL loip t BH stn sh. (..3) decqh d ka BG t BH sh katra ra tn AL, BG t BH stn sh. t d t a t sa ka llloij stn sa(..1) ka AL ra t BG stn sh. Pr j ra t doqnti shmeJ t A t doqesV e qev t BG sh e qea ketai AL per dei poisai. =============================================== 3. D o doqeisn e qein nswn p t j mezonoj t lssoni shn e qean felen. : Estwsan a doqesai d o e qeai

C

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nisoi a AB, G, n mezwn stw AB de d p t j mezonoj t j AB t lssoni t G shn e qean felen. Kesqw pr j t A shmeJ t G e qev sh AD (.2)ka kntrJ mn t A diastmati d t AD k kloj gegrfqw DEZ.(.3) Ka pe t A shmeon kntron st to DEZ k klou, sh stn AE t AD(.15) ll ka G t AD stin sh. katra ra tn AE, G t AD stin sh (..1)ste ka AE t G stin sh. D o ra doqeisn e qein nswn tn AB, G p t j mezonoj t j AB t lssoni t G sh f rhtai AE per dei poisai. =============================================== .4 'En d o trgwna tj d o pleurj [taj] dus pleuraj saj cV katran katrv ka t n gwnan t gwnv shn cV t n p tn swn e qein periecomnhn, ka t n bsin t bsei shn xei, ka t trgwnon t trignJ son stai, ka a loipa gwnai taj loipaj gwnaij sai son- tai katra katrv, f' j a sai pleura potenousin. : Estw d o trgwna t ABG, DEZ tj d o pleurj tj AB, AG taj dus pleuraj taj DE, DZ saj conta kat r an kat r v t n mn AB t DE t n d AG t DZ ka gwnan t n p BAG gwnv t p EDZ shn. lgw, ti ka bsij BG bsei t EZ sh stn, ka t ABG trgwnon t DEZ ti ka bsij BG bsei t EZ sh stn, ka t ABG trgwnon t DEZ trignJ son stai, ka a loipa gwnai taj loipaj gwnaij sai son- tai katra katrv, f' j a sai pleura potenou- sin, mn p ABG t p DEZ, d p AGB t

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p DZE. 'Efarmozomnou gr to ABG trignou p t DEZ t r gwnon ka t iqemnou t o mn A shmeou p t D sh- meon t j d AB e qeaj p t n DE, farm sei ka t B shmeon p t E di t shn enai t n AB t DE far- mosshj d t j AB p t n DE farm sei ka AG eqea p t n DZ di t shn enai t n p BAG gw- nan t p EDZ ste ka t G shmeon p t Z sh- meon f ar msei di t shn pl in enai t n AG t DZ. ll m n ka t B p t E fhrm kei ste bsij BG p bsin t n EZ farm sei. e g r t o mn B p t E f ar msant oj t o d G p t Z BG bsij p t n EZ o k farm sei, d o e qeai cwron perixousin per o k farm sei, d o e qeai cwron perixousin per stn d naton. (..9*) farm sei ra BG bsij p t n EZ ka sh a t stai(..4) ste ka lon t ABG trgwnon p lon t DEZ trgwnon farm sei ka son a t stai, (..4) ka a loipa gwnai p tj loipj gwnaj farm sousi ka sai a taj sontai, mn p ABG t p DEZ d p AGB t p DZE. 'En ra d o trgwna tj d o pleurj [taj] d o pleu- raj saj cV katran katrv ka t n gwnan t gwnv shn cV t n p tn swn e qein periecomnhn, ka t n bsin t bsei shn xei, ka t trgwnon t trignJ son stai, ka a loipa gwnai taj loipaj gwnaij sai sontai katra katrv, f' j a sai pleura pote- nousin per dei dexai. ================================================ 5 Tn soskeln trignwn a pr j t bsei gwnai sai lllaij esn, ka prosekblhqeisn tn swn e qein a p t n bsin gwnai sai lllaij sontai.

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: Est w t r gwnon soskel j t ABG shn con t n AB pleurn t AG pleur ,(.20) ka prosekbeblsqwsan p' e qeaj taj AB, AG e qeai a BD, GE (.2)lgw, t i mn p ABG gwna t p AGB sh stn, d p GBD t p BGE. Elfqw gr p t j BD tuc n sh- meon t Z, ka fVrsqw p t j me- zonoj t j AE t lssoni t AZ sh AH, (.3)ka peze cqwsan a ZG, HB e qeai. (.1) 'Epe on sh st n mn AZ t AH d AB t AG, d o d a ZA, AG dus taj HA, AB sai esn katra katrv ka gwnan koin n pericousi t n p ZAH bsij ra ZG bsei t HB sh stn, ka t AZG trgwnon t AHB trignJ son stai, ka a loipa gw- nai taj loipaj gwnaij sai sontai katra katrv, f' j a sai pleura potenousin, mn p AGZ t p ABH, d p AZG t p AHB. (.4)ka pe lh AZ lV t AH stin sh, n AB t AG stin sh, loip ra BZ loip t GH stin sh. (..3) decqh d ka ZG t HB sh d o d a BZ, ZG dus taj GH, HB sai esn katra katrv ka gwna p BZG gwnv t p GHB sh, ka bsij a tn koin BG ka t BZG ra trgwnon t GHB trignJ son stai, (.4) ka a loipa gwnai taj loipaj gwnaij sai son- tai katra katrv, f' j a sai pleura potenousin sh r a st n mn p ZBG t p HGB d p BGZ t p GBH. pe on lh p ABH gwna lV t p AGZ gwnv decqh sh, n p GBH t p BGZ sh, loip ra p ABG loip t p AGB stin sh(..3) ka esi pr j t bsei to ABG trignou. decqh d ka p ZBG t p HGB sh ka esin p t n bsin. Tn ra soskeln trignwn a pr j t bsei gwnai sai lllaij esn, ka prosekblhqeisn tn swn e qein a p t n bsin gwnai sai lllaij sontai per dei dexai. ================================================ .6'En trignou a d o gwnai sai lllaij sin, ka a p tj saj gwnaij potenousai pleura sai lllaij sontai.

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: Estw trgwnon t ABG shn con t n p ABG gw- nan t p AGB gwnv lgw, ti ka pleur AB pleur t AG stin sh. E gr nis j stin AB t AG, tra a tn mezwn stn. ( .1) stw mezwn AB, ka fVrsqw p t j mezonoj t j AB t lttoni t AG sh DB, (.3) ka peze - cqw DG. 'Epe on sh stn DB t AG koin d BG, d o d a DB, BG d o taj AG, GB sai esn katra katrv, ka gwna p DBG gwnv t p AGB stin sh bsij ra DG bsei t AB sh stn, (.4)ka t DBG trgwnon t AGB trignJ son stai, t lasson t mezoni per topon( .2) o k ra nis j stin AB t AG sh ra. ( .3) 'En ra trignou a d o gwnai sai lllaij sin, ka a p tj saj gwnaj potenousai pleura sai lllaij sontai per dei dexai. ================================================= .7 'Ep t j a t j e qeaj d o taj a taj e qeaij llai d o e qeai sai katra katrv o sustaqsontai pr j llJ ka llJ shmeJ p t a t mrh t a t prata cousai taj x rc j e qeaij.

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: E gr dunat n, p t j a tj e qeaj t j AB d o taj a taj e qeaij taj AG, GB llai d o e qeai a AD, DB sai katra katrv sunesttwsan pr j llJ ka llJ shmeJ t te G ka D p t a t mrh t a t prata cou- sai, st e shn enai t n mn GA t DA t a t praj cousan a t t A, t n d GB t DB t a t praj cousan a t t B, ka peze cqw GD. 'Epe on sh stn AG t AD, sh st ka gwna p AGD t p ADG (.5)mezwn ra p ADG t j p DGB( .4) poll ra p GDB mezwn st t j p DGB. ( .5) plin pe sh stn GB t DB, sh st ka gwna p GDB gwnv t p DGB.(.5) decqh d a - t j ka poll mezwn per stn d naton. O k ra p t j a t j e qeaj d o taj a taj e - qeaij llai d o e qeai sai katra katrv susta- qsontai pr j llJ ka llJ shmeJ p t a t mrh t a t prata cousai taj x rcj e qeaij per dei dexai. ================================= 8.'En d o trgwna tj d o pleurj [taj] d o pleuraj saj cV katran katra, cV d ka t n bsin t bsei shn, ka t n gwnan t gwnv shn xei t n p tn swn e qein periecomnhn. : Estw d o trgwna t ABG, DEZ tj d o pleurj tj AB, AG taj d o pleuraj taj DE, DZ saj con- ta katran katrv, t n mn

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AB t DE t n d AG t DZ ct w d ka bsin t n BG bsei t EZ shn lgw, ti ka gwna p BAG gwnv t p EDZ stin sh. 'Efarmozomnou gr to ABG trignou p t DEZ trgwnon ka t iqemnou t o mn B shmeou p t E shmeon t j d BG e qeaj p t n EZ farm sei ka t G shmeon p t Z di t shn enai t n BG t EZ farmosshj d t j BG p t n EZ farm sousi ka a BA, GA p tj ED, DZ. e g r bsi j mn BG p bsin t n EZ farm sei, a d BA, AG pleura p tj ED, DZ o k farm sousin ll parallxou- sin j a EH, HZ, sustaqsontai p t j a t j e qeaj d o taj a taj e qeaij llai d o e qeai sai katra katrv pr j llJ ka llJ shmeJ p t a t mrh t a t prata cousai. o sunstantai d(.7)o k ra farmozomnhj t j BG bsewj p t n EZ bsin o k farm sousi ka a BA, AG pleura p tj ED, DZ. farm sousin ra ste ka gwna p BAG p gwnan t n p EDZ farm sei ka sh a t stai. (..4) 'En ra d o trgwna tj d o pleurj [taj] d o pleu- raj saj cV katran katrv ka t n bsin t bsei shn cV, ka t n gwnan t gwnv shn xei t n p tn swn e qein periecomnhn per dei dexai. ========================================= 9. T n doqesan gwnan e q grammon dca temen. : Estw doqesa gwna e q grammoj p BAG. de

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d a t n dca temen. Elfqw p t j AB tuc n shmeon t D, ka fVr- sqw p t j AG t AD sh AE,(.3) ka peze cqw DE, (.1) ka sunesttw p t j DE trgwnon s pleuron t DEZ, (.1)ka peze cqw AZ lgw, ti p BAG gwna dca ttmhtai p t j AZ e qeaj. 'Epe gr sh stn AD t AE, koin d AZ, d o d a DA, AZ dus taj EA, AZ sai esn katra katrv. ka bsij DZ bsei t EZ sh stn gwna ra p DAZ gwnv t p EAZ sh stn. (.8) `H ra doqesa gwna e q grammoj p BAG dca ttmhtai p t j AZ e qeaj per dei poisai. =========================================== 10. T n doqesan e qean peperasmnhn dca temen. : Estw doqesa e qea peperasmnh AB de d t n AB e qean peperasmnhn dca temen. Sunesttw p' a t j trgwnon s - pleuron t ABG(I.1), ka tetmsqw p AGB gwna dca t GD e qev(.9) lgw, ti AB e qea dca ttmhtai kat t D shmeon. 'Epe gr sh stn AG t GB, koin d GD, d o d a AG, GD d o taj BG, GD sai esn katra katrv ka gwna p AGD gwnv t p BGD sh stn bsij ra AD bsei t BD sh stn.(.4) `H ra doqesa e qea peperasmnh AB dca t- tmhtai kat t D per dei poisai. ============================================= 11.T doqesV e qev p to pr j a t doqntoj sh- meou pr j rqj gwnaj e qean gramm n gagen.

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: Est w mn doqesa e qea AB t d doqn shmeon p' a t j t G de d p to G shmeou t AB e - qev pr j rqj gwnaj e qean gramm n gagen. Elfqw p t j AG tuc n shmeon t D, ka kesqw t GD sh GE,(I.3) ka sunesttw p t j DE trgwnon s pleuron t ZDE,(I.1) ka peze cqw ZG(.1) lgw, ti t doqesV e qev t AB p to pr j a t doqntoj sh- meou to G pr j rqj gw- naj e qea gramm ktai ZG. 'Epe gr sh stn DG t GE, koin d GZ, d o d a DG, GZ dus taj EG, GZ sai esn katra katrv ka bsij DZ bsei t ZE sh stn gwna ra p DGZ gwnv t p EGZ sh stn(I.8) ka esin fexj. t an d e qea p' e qean staqesa tj fex j gwnaj saj lllaij poi, rq katra tn swn gwnin stin(O.10) rq ra stn katra tn p DGZ, ZGE. T ra doqesV e qev t AB p to pr j a t do- qntoj shmeou to G pr j rqj gwnaj e qea gramm ktai GZ per dei poisai. ================================== .12 'Ep t n doqesan e qean peiron p to doqntoj shmeou, m stin p' a t j, kqeton e qean gram- m n gagen.

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: Est w mn doqesa e qea peiroj AB t d do- qn shmeon, m stin p' a t j, t G de d p t n doqesan e qean peiron t n AB p to doqntoj shmeou to G, m stin p' a t j, kqeton e qean gramm n gagen. Elfqw gr p t tera mrh t j AB e qeaj tu- c n shmeon t D, ka knt r J mn t G diast mat i d t GD k kloj gegrfqw EZH, (.3)ka tetmsqw EH e qea dca kat t Q,(.10) ka peze cqwsan a GH, GQ, GE e qeai (.1)lgw, ti p t n doqesan e qean peiron t n AB p to doqntoj shmeou to G, m stin p' a t j, kqetoj ktai GQ. 'Epe gr sh stn HQ t QE, koin d QG, d o d a HQ, QG d o taj EQ, QG sai esn ka- tra katrv ka bsij GH bsei t GE stin sh gwna ra p GQH gwnv t p EQG stin sh. (.8) ka esin fexj. t an d e qea p' e qean staqesa tj fex j gwnaj saj lllaij poi, rq katra tn swn gwnin stin, ka festhkua e qea kq- etoj kaletai f' n fsthken. 'Ep t n doqesan ra e qean peiron t n AB p to doqntoj shmeou to G, m stin p' a t j, kqetoj ktai GQ per dei poisai. ========================================== 13 En e qea p' e qean staqesa gwnaj poi, toi d o rqj dusn rqaj saj poisei.

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: E qea gr tij AB p' e qean t n GD staqesa gwnaj poietw tj p GBA, ABD lgw, ti a p GBA, ABD gwnai toi d o rqa esin dusn rqaj sai. (.10) E mn on sh stn p GBA t p ABD, d o rqa esin. e d o , cqw p to B shmeou t GD [e qev] pr j rqj BE(.11) a ra p GBE, EBD d o rqa esin ka pe p GBE dus taj p GBA, ABE sh stn, koin proskesqw p EBD a ra p GBE, EBD tris taj p GBA, ABE, EBD sai esn. (..2)plin, pe p DBA dus taj p DBE, EBA sh stn, koin proskesqw p ABG a ra p DBA, ABG tris taj p DBE, EBA, ABG sai (..2 esn. decqhsan d ka a p GBE, EBD tris taj a taj sai t d t a t sa ka llloij stn sa (..1) ka a p GBE, EBD ra taj p DBA, ABG sai esn ll a p GBE, EBD d o rqa esin ka a p DBA, ABG ra dusn rqaj sai esn. 'En ra e qea p' e qean staqesa gwnaj poi, toi d o rqj dusn rqaj saj poisei per dei dexai. ========================================== 14 'En pr j tini e qev ka t pr j a t shmeJ d o e qeai m p t a t mrh kemenai tj fexj gwnaj dusn rqaj saj poisin, p' e qeaj sontai lllaij a e qeai. : Pr j gr tini e qev t AB ka t pr j a t sh-

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meJ t B d o e qeai a BG, BD m p t a t mrh kemenai tj fexj gwnaj tj p ABG, ABD d o rqaj saj poietw- san lgw, ti p' e qeaj st t GB BD. E gr m sti t BG p' e qeaj BD, stw t GB p' e qeaj BE. (.2) 'Epe on e qea AB p' e qean t n GBE fsth- ken, a ra p ABG, ABE gwnai d o rqaj sai esn (.13)es d ka a p ABG, ABD d o rqaj sai a ra p GBA, ABE taj p GBA, ABD sai (.2 .4) esn. koin fVrsqw p GBA loip ra p ABE loip t p ABD stin sh, lsswn t me- zoni ( ..3) per stn d naton. o k ra p' e qeaj stn BE t GB. mowj d dexomen, t i od llh tij pl n t j BD p' e qeaj ra stn GB t BD. 'En ra pr j tini e qev ka t pr j a t shmeJ d o e qeai m p t a t mrh kemenai tj fexj gwnaj dusn rqaj saj poisin, p' e qeaj sontai gwnaj dusn rqaj saj poisin, p' e qeaj sontai lllaij a e qeai per dei dexai. =============================================== .15. 'En d o e qeai tmnwsin lllaj, tj kat koruf n gwnaj saj lllaij poiosin. : D o gr e qeai a AB, GD temntwsan lllaj kat t E shmeon lgw, ti sh st n mn p AEG gw- na t p DEB, d p GEB t p AED. 'Epe gr e qea AE p' e qean t n GD fsthke gwnaj poiosa tj p GEA, AED, a ra p GEA, AED gwnai dusn rqaj sai esn. (.13)plin, pe e qea DE p' e qean t n AB fsthke gw-

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naj poiosa tj p AED, DEB, a ra p AED, DEB gwnai dusn rqaj sai esn. (.13)decqhsan d ka a p GEA, AED dusn rqaj sai a ra p GEA, AED taj p AED, DEB sai esn. (.4 .1) koin fVrsqw p AED loip ra p GEA loip t p BED sh stn (..3) mowj d deicqsetai, ti ka a p GEB, DEA sai esn. 'En ra d o e qeai tmnwsin lllaj, tj kat ko- ruf n gwnaj saj lllaij poiosin per dei dexai. ================================================ [P risma 'Ek d to tou faner n ti, n d o e qeai tmnwsin lllaj, tj pr j t tom gwnaj ttrasin rqaj saj poisousin.] ================================================ .16.Pant j trignou mij tn pleurn prosekblhqeshj kt j gwna katraj tn nt j ka penanton gw- nin mezwn stn. : Estw trgwnon t ABG, ka prosekbeblsqw a to ma pleur BG p t D lgw, ti kt j gwna p AGD mezwn stn katraj tn nt j ka penanton tn p GBA, BAG gwnin. Tetmsqw AG dca kat t E, (.10)ka pizeucqesa BE kbeblsqw p' e - qeaj p t Z, ka kesqw t BE sh EZ, (.3) ka peze cqw ZG,(.1) ka dicqw AG p t H. (.2) 'Epe on sh st n mn AE t EG, d BE t EZ, d o d a AE, EB dus taj GE, EZ sai esn katra

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katrv ka gwna p AEB gwnv t p ZEG sh stn kat koruf n gr(.15) bsij ra AB bsei t ZG sh stn, ka t ABE trgwnon t ZEG trignJ stn son, ka a loipa gwnai taj loipaj gwnaij sai esn katra katrv, f' j a sai pleura po- tenousin(.4) sh ra stn p BAE t p EGZ. me- zwn d stin p EGD t j p EGZ mezwn ra p AGD t j p BAE. ( .6) mowj d t j BG tetmh- mnhj dca deicqsetai ka p BGH, toutstin p AGD, mezwn ka t j p ABG. Pant j ra trignou mij tn pleurn prosekblh- qeshj kt j gwna katraj tn nt j ka penanton gwnin mezwn stn per dei dexai. ================================================== .17. Pant j trignou a d o gwnai d o rqn lssonj esi pntV metalamban menai. : Estw trgwnon t ABG lgw, ti to ABG trig- nou a d o gwnai d o rqn ltto- nj esi pntV metalamban menai. 'Ekbeblsqw gr BG p t D. (.2) Ka pe trignou to ABG kt j sti gwna p AGD, mezwn st t j nt j ka penanton t j p ABG. (.16) koin proskesqw p AGB a ra p AGD, AGB tn p ABG, BGA mezonj esin. ll' a p AGD, AGB d o rqaj sai esn(.13) a ra p ABG, BGA d o rqn lssonj esin.(..6*). mowj d dexomen, ti ka a p BAG, AGB d o rqn lssonj esi ka ti a p GAB, ABG. Pant j ra trignou a d o gwnai d o rqn ls- sonj esi pntV metalamban menai per dei dexai. =============================================== .18. Pant j trignou mezwn pleur t n mezona gw- nan potenei.

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: Estw gr trgwnon t ABG mezona con t n AG pleurn t j AB lgw, ti ka gwna p ABG me- zwn st t j p BGA. 'Epe gr mezwn stn AG t j AB, kesqw t AB sh AD, ka peze cqw BD. (.1) Ka pe trignou to BGD kt j sti gwna p ADB, mezwn st t j nt j ka penanton t j p DGB (.16) sh d p ADB t p ABD, pe ka pleur AB t AD stin sh mezwn ra ka p ABD t j p AGB poll ra p ABG mezwn st t j p AGB. Pant j ra trignou mezwn pleur t n mezona gwnan potenei per dei dexai. ================================================ .19.Pant j trignou p t n mezona gwnan mezwn pleur potenei. : Estw trgwnon t ABG mezona con t n p ABG gwnan t j p BGA lgw, ti ka pleur AG pleurj t j AB mezwn stn. E gr m, toi sh stn AG t AB lsswn( .7) sh mn on o k stin AG t AB sh gr n n ka gwna p ABG t p AGB (.5)o k sti d o k ra sh

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stn AG t AB. od m n lsswn stn AG t j AB lsswn gr n n ka gwna p ABG t j p AGB (.18)o k sti d o k ra lsswn stn AG t j AB. decqh d, t i od sh stn. mezwn ra stn AG t j AB. ( .8) Pant j ra trignou p t n mezona gwnan me- zwn pleur potenei per dei dexai. ==================================================== .20 Pant j trignou a d o pleura t j loipj mezonj esi pntV metalamban menai. : Estw gr trgwnon t ABG lgw, ti to ABG trignou a d o pleura t j loipj mezonj esi pntV metalamban menai, a mn BA, AG t j BG, a d AB, BG t j AG, a d BG, GA t j AB. Dicqw gr BA p t D shmeon, ka kesqw t GA sh AD, ka p- eze cqw DG. 'Epe on sh stn DA t AG, sh st ka gwna p ADG t p AGD (.5) mezwn ra p BGD t j p ADG (..5)ka pe trgwn n sti t DGB mezona con t n p BGD gwnan t j p BDG, p d t n mezona gwnan me- zwn pleur potenei, (.19) DB ra t j BG sti mezwn. sh d DA t AG mezonej ra a BA, AG t j BG mowj d dexomen, t i ka a mn AB, BG t j GA mezonj esin, a d BG, GA t j AB. Pant j ra trignou a d o pleura t j loipj me- zonj esi pntV metalamban menai per dei dexai. ============================================ .21 'En trignou p mij tn pleurn p tn pertwn d o e qeai nt j sustaqsin, a sustaqesai tn loi-

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pn t o t r ignou do pl eur n l t t onej mn sontai, mezona d gwnan perixousin. : Trignou gr to ABG p mij tn pleurn t j BG p tn pertwn tn B, G d o e qeai nt j sun- esttwsan a BD, DG lgw, ti a BD, DG tn loi- pn to trignou d o pleurn tn BA, AG lssonej mn esin, mezona d gwnan pericousi t n p BDG t j p BAG. Dicqw gr BD p t E. ka pe pant j trignou a d o pleura t j loipj mezonj esin, (.20) to ABE ra trignou a d o pleura a AB, AE t j BE mezonj esin koin proskesqw EG a ra BA, AG tn BE, EG mezonj esin. plin, pe to GED trignou a d o pleura a GE, ED t j GD mezonj esin, koin pros- kesqw DB a GE, EB ra tn GD, DB mezonj esin. ll tn BE, EG mezonej decqhsan a BA, AG poll ra a BA, AG tn BD, DG mezonj esin. Plin, pe pant j trignou kt j gwna t j nt j ka penanton mezwn stn, (.16)to GDE ra trignou kt j gwna p BDG mezwn st t j p GED. di ta t tonun ka to ABE trignou kt j gwna p GEB mezwn st t j p BAG. ll t j p GEB mezwn decqh p BDG poll ra p BDG mezwn st t j p BAG. 'En ra trignou p mij tn pleurn p tn pe- rtwn d o e qeai nt j sustaqsin, a sustaqesai tn loipn to trignou d o pleurn lttonej mn esin, mezona d gwnan pericousin per dei dexai. =============================================== .22 'Ek trin e qein, a esin sai tris taj doqesaij

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[e qeaij], trgwnon sustsasqai de d tj d o t j l oipj mezonaj enai pntV metalambanomnaj [di t ka pant j trignou tj d o pleurj t j loipj mezo- naj enai pntV metalambanomnaj]. (.20) : Estwsan a doqesai trej e qeai a A, B, G, n a d o t j loipj mezonej stwsan pntV metalamban - menai, a mn A, B t j G, a d A, G t j B, ka ti a B, G t j A de d k tn swn taj A, B, G trgwnon sustsasqai. 'Ekkesqw t i j eqea DE peper asmnh mn kat t D pei r oj d kat t E, ka kesqw t mn A sh DZ, t d B sh ZH, t d G sh HQ (.3)ka kntrJ mn t Z, diast mat i d t ZD k kloj gegrfqw DKL pl in knt r J mn t H, diast mat i d t HQ k kloj gegrfqw KLQ, ka peze cqwsan a KZ, KH lgw, ti k trin e qein tn swn taj A, B, G trgwnon sunstatai t KZH. 'Epe gr t Z shmeon kntron(.16) st to DKL k - klou, sh stn ZD t ZK ll ZD t A stin sh. (..1) ka KZ ra t A stin sh. (..1) plin, pe t H shmeon kntron(.16) st to LKQ k klou, sh stn HQ t HK ll HQ t G stin sh ka KH ra t G stin sh. st d ka ZH t B sh a trej ra e qeai a KZ, ZH, HK tris taj A, B, G sai esn. 'Ek trin ra e qein tn KZ, ZH, HK, a esin sai tris taj doqesaij e qeaij taj A, B, G, trgw- non sunstatai t KZH per dei poisai. ================================================== .23 Pr j t doqesV e qev ka t pr j a t shmeJ t doqesV gwnv e qugrmmJ shn gwnan e q grammon sustsasqai.

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: Est w mn doqesa e qea AB, t d pr j a t shmeon t A, d doqesa gwna e q grammoj p DGE de d pr j t doqesV e qev t AB ka t pr j a t shmeJ t A t doqesV gwnv e qugrmmJ t p DGE shn gwnan e q grammon sustsasqai. Elfqw f' katraj tn GD, GE tuc nta shmea t D, E, ka peze - cqw DE ka k trin e qein, a esin sai tris taj GD, DE, GE, trgwnon sun- esttw t AZH, st e shn enai t n mn GD t AZ, t n d GE t AH, ka ti t n DE t ZH.(.22) 'Epe on d o a DG, GE d o taj ZA, AH sai esn katra katrv, ka bsij DE bsei t ZH sh, gwna ra p DGE gwnv t p ZAH stin sh. (.8) Pr j ra t doqesV e qev t AB ka t pr j a t shmeJ t A t doqesV gwnv e qugrmmJ t p DGE sh gwna e q grammoj sunstatai p ZAH per dei poisai. ================================================== .24 'En d o trgwna tj d o pleurj [taj] d o pleuraj saj cV katran katrv, t n d gwnan t j gwnaj mezona cV t n p tn swn e qein periecomnhn, ka t n bsin t j bsewj mezona xei.

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: Estw d o trgwna t ABG, DEZ tj d o pleurj tj AB, AG taj d o pleuraj taj DE, DZ saj conta katran katrv, t n mn AB t DE t n d AG t DZ, d pr j t A gwna t j pr j t D gwnaj me- zwn stw lgw, ti ka bsij BG bsewj t j EZ mezwn stn. 'Epe gr mezwn p BAG gwna t j p EDZ gwnaj, sunesttw pr j t DE e qev ka t pr j a t shmeJ t D t p BAG gwnv sh p EDH,(.23) ka kesqw potrv tn AG, DZ sh DH, ka peze - cqwsan a EH, ZH. 'Epe on sh st n mn AB t DE, d AG t DH, d o d a BA, AG dus taj ED, DH sai esn ka- tra katrv ka gwna p BAG gwnv t p EDH sh bsij ra BG bsei t EH stin sh. (.4)plin, pe sh stn DZ t DH, sh st ka p DHZ gwna t p DZH(.5) mezwn ra p DZH t j p EHZ poll ra mezwn stn p EZH t j p EHZ. ka pe trgwn n sti t EZH mezona con t n p EZH gwnan t j p EHZ, p d t n mezona gwnan mezwn pleur potenei, mezwn ra ka pleur EH t j EZ.(.19) sh d EH t BG mezwn ra ka BG t j EZ. 'En ra d o trgwna tj d o pleurj dus pleuraj saj cV katran katrv, t n d gwnan t j gwnaj mezona cV t n p tn swn e qein periecomnhn, ka t n bsin t j bsewj mezona xei per dei dexai. ================================================ .25 'En d o trgwna tj d o pleurj dus pleuraj saj cV katran katrv, t n d bsin t j bsewj mezona cV, ka t n gwnan t j gwnaj mezona xei t n p tn swn e qein periecomnhn.

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: Estw d o trgwna t ABG, DEZ tj d o pleurj tj AB, AG taj d o pleuraj taj DE, DZ saj conta katran katrv, t n mn AB t DE, t n d AG t DZ bsi j d BG bsewj t j EZ mezwn stw lgw, ti ka gwna p BAG gwnaj t j p EDZ mezwn stn E gr m, toi sh stn a t lsswn sh mn on o k stin p BAG t p EDZ sh gr n n ka bsij BG bsei t EZ(.4) o k sti d. o k ra sh st gwna p BAG t p EDZ od m n lsswn stn p BAG t j p EDZ lsswn gr n n ka bsij BG bsewj t j EZ(.24) o k sti d o k ra lsswn stn p BAG gwna t j p EDZ. decqh d ti o d sh mezwn ra stn p BAG t j p EDZ. 'En ra d o trgwna tj d o pleurj dus pleuraj saj cV katran kterv, t n d bsin t j bsewj mezona cV, ka t n gwnan t j gwnaj mezona xei t n p tn swn e qein periecomnhn per dei dexai. =================================================== 26 'En d o trgwna tj d o gwnaj dus gwnaij saj cV katran katrv ka man pleurn mi pleur shn toi t n pr j taj saij gwnaij t n potenousan p man tn swn gwnin, ka tj loipj pleurj taj loipaj pleuraj saj xei [katran katrv] ka t n loip n gwnan t loip gwnv. : Estw d o trgwna t ABG, DEZ tj d o gwnaj tj p ABG, BGA dus taj p DEZ, EZD saj conta katran katrv, t n mn p

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ABG t p DEZ, t n d p BGA t p EZD ct w d ka man pleurn mi pleur shn, pr teron t n pr j taj saij gwnaij t n BG t EZ lgw, ti ka tj loipj pleurj taj loipaj pleuraj saj xei katran katrv, t n mn AB t DE t n d AG t DZ, ka t n loip n gwnan t loip gwnv, t n p BAG t p EDZ. E gr nis j stin AB t DE, ma a tn mezwn stn. stw mezwn AB, ka kesqw t DE sh BH, ka peze cqw HG. 'Epe on sh st n mn BH t DE, d BG t EZ, d o d a BH, BG dus taj DE, EZ sai esn katra katrv ka gwna p HBG gwnv t p DEZ sh stn bsij ra HG bsei t DZ sh stn, ka t HBG trgwnon t DEZ trignJ son stn, ka a loi- pa gwnai taj loipaj gwnaij sai sontai, f' j a sai pleura potenousin (.4)sh ra p HGB gwna t p DZE. ll p DZE t p BGA p keitai sh ka p BGH ra t p BGA sh stn, lsswn t mezoni per d naton. o k ra nis j stin AB t DE. sh ra. st i d ka BG t EZ sh d o d a AB, BG dus taj DE, EZ sai esn katra katrv ka gwna p ABG gwnv t p DEZ stin sh bsij ra AG bsei t DZ sh stn, ka loip gwna p BAG t loip gwnv t p EDZ sh stn. (.4) 'All d plin stwsan a p tj saj gwnaj pleu- ra potenousai sai, j AB t DE lgw plin, ti ka a loipa pleura taj loipaj pleuraj sai sontai, mn AG t DZ, d BG t EZ ka ti loip gwna p BAG t loip gwnv t p EDZ sh stn. E gr nis j stin BG t EZ, ma a tn mezwn stn. stw mezwn, e dunat n, BG, ka kesqw t EZ sh BQ, ka peze cqw AQ. ka pe sh st n mn BQ t EZ d AB t DE, d o d a AB, BQ dus taj DE, EZ sai esn katra katrv ka gwnaj saj pericousin bsij ra AQ bsei t DZ sh stn, ka t ABQ trgwnon t DEZ trignJ son stn, ka a loipa gwnai taj loipaj gwnaij sai sontai, f' j a sai pleura potenousin (.4)sh ra stn p BQA gwna t p EZD. ll p EZD t p BGA stin sh trignou d to AQG kt j gwna p BQA sh st t nt j ka penanton t p BGA per d naton. (.16)o k ra nis j stin BG t EZ sh ra. st d ka AB t DE sh. d o d a AB, BG d o taj

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DE, EZ sai esn katra katrv ka gwnaj saj pericousi bsij ra AG bsei t DZ sh stn, ka t ABG trgwnon t DEZ trignJ son ka loip gwna p BAG t loip gwnv t p EDZ sh. (.4) 'En ra d o trgwna tj d o gwnaj dus gwnaij saj cV katran katrv ka man pleurn mi pleur shn toi t n pr j taj saij gwnaij, t n potenousan p man tn swn gwnin, ka tj loipj pleurj taj loipaj pleuraj saj xei ka t n loip n gwnan t loip gwnv per dei dexai. .1-.26

(/) .

*

1 - 3,1 15,20 15 1

2

1 1,2,3,3 15 3,1

3 2 3 15 1 4 (2) 4,4, 9*

5 3,4 2,1 20 3,3 6 3,4 1,2,3

7 5,5 4,5

8 7 4

9 1,3,8 1

10 1,4,9

11 1,3,8 1 10

12 8,10 3,1

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13 11 2,2 1,3

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16 3,4,10,15 1,2, 6

17 13,16 2 6*

18 16 1

19 5,18 7, 8 20 5,19 5

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. . , . , , , .

. 25/4/2004

29

, .

. .( The Greek Concept of Proof 290 : Topics in the history of Mathematics, )

, , , . .6 , .7 , .16 , .17 , .19. , . 1. , .( [ = ], [] )(.6) 2. (.6)

. 25/4/2004

30

3. . ( , )[ ( ) , =](.6) 4. , .(

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. 19 Pash (1882) . ( ) Peano (1889) Pieri (1889) 20 Veblen (1904) Pash Forder (1924) , Robinson( 1940) Levi (1960).. , Hilbert- (1899) Birkhoff (1932) Pash ( ) ,,, , () () , , .

. 25/4/2004

32

: () , . Dedekint ( ) , ,, , . Sir Thomas Heath 3 , , , . , . (ii) - () .4 .8 , () . , , .. .2 .3 . , , . , , , ( Thomas Haeth) 4 , . Heath : 4, . 5, .. 4 , , ., . , , . Hilbert .4 Hilbert , .

. 25/4/2004

33

(iii) 1 2 1 , , . , 2 , , . , ( ) . ( Carle B.Boyer-Uta C. Merzbach 1977) , , , . , ( ) , . Sir Thomas Haeath 1 2 , , , , . Heath , , () .4 ( ) . Heath , 2, .11 Simson , . , .

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. 25/4/2004

35

6 1847 Oliver Byrne. , . .1 , .2 , .3 http://www.math.ubc.ca/people/faculty/cass/Euclid/byrne.html : 1) .. . . . -

1952 2) Sir Thomas Heath

... 2001 3) - ... 2001 4) Euclid ,The thirteen Books of the Elements Vol1. Sir Thomas Heath Dover

Rublications Inc. --New York 5) Courla B. Boyer Uta.C. Merzbach .

1977 6) 14 - 1997 (

) 1) http://aleph0.clarku.edu/~djoyce/java/elements/elements.html 2) http://www.perseus.tufts.edu/cgi-

bin/ptext?doc=Perseus%3Atext%3A1999.01.0086&layout=&query=toc&loc=9.1

3) http://www.math.ubc.ca/people/faculty/cass/Euclid/byrne.html

.

1

, , ,

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Thomas Heath ,

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. ( [] 2 : 'Epe d ka o Puqag reioi t shmeon for - zontai monda proslabosan qsin, skepton t pote noontej lgousin 3 : t mn pntV ka qeton lgetai monj, t d pntV ka qs in con s t igm, t d monac gramm, t d dic p pedon, t d pntV ka tr ic diai retn kat t posn 4 20-24 : . ti a stigma k t noj nuprxous in; totJ mn on t gnei ka diemceto Pltwn j nti gewmetrik d gmati, ll' klei rc n grammjtoto d pollki j t qeitj t mouj grammj. ka toi ngkh totwn ena ti praj st' x o l gou gramm sti, ka stigm stin. 5 (= , , ) : es d tn toiotwn rismn te tj stigmj ka tj grammj ka to pipdou pntej gr di tn strwn t pr tera dhlosin t mn gr grammj, t d' pipdou, t d stereo f as i praj enai.

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Nayrizi ( Curtze 3)

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13 : mnon on t shmeon merj kat tn gewme- trik n lhn ka m naj kat t n riqmhtikn 14 O mn on E kle dhj di tj pof sewj t n meristn smhnen mn t n rc n pshj tj po- keimnhj a t fsewj ej qewr an. ka gr o po- fatiko l goi proskousi taj rcaj, j Parme- n dhj m j nadidskei tn te prwt sthn at an ka t n scthn di m nwn tn pofsewn paradoj. p sa gr rc tn p' a tj kaq' tran o s an fsthken ka a totwn pofseij t n ke nhj mn dhlosin di thta. t gr at ion mn totwn, odn d totwn prcon, n ati n sti, gnrimon pj g netai di to tr pou totou tj didaskal aj. Iswj d' n tij porseien, pj pnta morfwti- kj ka meristj tj fantas aj decomnhj merj ti shmeon gewmtrhj n a t qewre. m gr ti toj n diano v l gouj, ll ka tj tn noern ka qe wn edn mfseij fantas a kat t n okean dcetai fsin, t n mn mrf wn morf j t n d schmat stwn scmata prote nousa. pr j d tathn t n por an lgomen, ti tj fantastikj kinsewj

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Pasch , Veronese, Enriques , Hilbert .

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( 1020 a11)20

)

(De anima I.4 409a4)21

(100, 5-19)22

19 To : t mn pntV ka qeton lgetai monj, t d pntV ka qsin con stigm, t d monac gramm, t d dic p pedon, t d pntV ka tric diairet n kat t pos n sma ka ntistryanti d t mn dic diairet n p pe-don, t d monac gramm, t d mhdam diairet n kat t pos n stigm ka monj, mn qetoj monj d qet j stigm. 20 : Pos n lgetai t diairet n ej nuprconta n kteron kas ton n t i ka tde t i pf uken enai. plqoj mn on pos n ti n r iqmhtn , mgeqoj d n metrht n . l getai d plqoj mn t diairet n dunmei ej m sun-ec, mgeqoj d t ej sunec megqouj d t mn f' n sunecj mkoj t d' p do pltoj t d' p tr a bqoj. totwn d plqoj mn t peperasmnon r iqmj mkoj d gramm pl toj d pifneia bqoj d sma 21 To : pe fasi kinhqesan gramm n p pedon poien, s t igmn d grammn, ka a tn mondwn kinseij gramma sontai 22 : cein lgetai. Tosata per tj grammj ersqw kat tj qewrikwtraj pibolj. podexmeqa d ka toj per 'Apollnion lgontaj, ti grammj nnoian mn comen, tan t mkh m non tn dn tn to cwn nametren kelewmen o gr prospoiomeqa t te t pltoj, ll t n f' n distasin nalogiz meqa, kaqper d ka tan cwr a metrmen, t n pifneian rmen, tan d frata, t stere n. psaj gr mo tj diastseij sullab ntej pofai-n meqa tos nde enai t disthma to fratoj kat te mkoj ka pltoj ka bqoj. asqhsin d a tj lboimen n pid ntej ej toj diorismoj tn pefw-tismnwn t pwn p tn skiasmnwn ka p tj selnhj ka p tj gj. toto gr t mson kat mn pltoj distat n sti, mkoj d cei t sum- parektein menon t fwt ka t ski .

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( 98625)23 .

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:

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a = b

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Eth.Nic. VIII. 12, 1161 8 ,

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, Eth.Nic.x.8,1178 & 25

, IX, 15, 174, 32

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Simon : die

Gerade liegt in gleicher Weise wie ihre Punkte.

Simon

(Abstand) .

:die Gerade ist liegt fur

(durch) ihre Punkte gleichmassig gegeben worden,

, die Gerade ist

durch ihre Punkte gleichmassig gegeben worden,

.

( )

, Bolzano (Berachtungen

uber einige Gegenstande der Elementargeometrie, 1804,

Schotten, Inhalt und Methode des planimetrischen

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Elementi di geometria Veronese ( ., 1904, . 10):

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Leibniz Saccheri, Krafft Gauss,

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Van Swinden (Elemente der Geometrie 1834, .4)

/ (Legendre, Elements de

Geometrie I., 3,8).

Unger (Die Geometrie des Euklid, 1833):

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