Image: Diophantus-VI-24-20-Fermat

Description: Work by Diophantus (died in about 280 B.C.), with additions by Pierre de Fermat (died in 1665). This edition of the book was published in 1670. p. 339 contains Diophantus' problem VI.XXIV, with the note added by Fermat which became known as Fermat's last theorem for case n=4.    Area trianguli rectanguli in numeris non potest esse quadratus, hujus theorematis a nobis inventi demonstrationem, quam et ipsi tandem non sine operosa laboriosa meditatione deteximus, subiungemus. Hoc nempe demonstrandi genus miros in arithmeticis suppeditabit progressus, si area trianguli esset quadratus darentur duo quadratoquadrati quorum differentia esset quadratus: Unde sequitur dari duo quadratos quorum & summa, & differentia esset quadratus. Datur itaque numerus compositus ex quadrato & duplo quadrati æqualis quadrato, ea conditione ut quadrati eum componentes faciant quadratum. Sed si numerus quadratus componitur ex Quadrato & duplo alterius quadrati eius latus similiter componitur ex quadrato & duplo quadrati ut facillime possumus demonstrare.    Unde concludetur latus illud esse summam laterum circa rectum trianguli rectanguli & unum ex quadratis illud componentibus efficere basem & duplum quadratum æquari perpendiculo.    Illud itaque triangulum rectangulum conficietur a duobus quadratis quorum summa & differentia erunt quadrati. At isti duo quadrati minores probabuntur primis quadratis primo suppositis quorum tam summa quam differentia faciunt quadratu. Ergo si dentur duo quadrata quorum summa & differentia faciant quadratum, dabitur in integris summa duorum quadratorum eiusdem naturæ priore minor. Eodem ratiocinio dabitur & minor ista inuenta per utam prioris & semper in infinitum minores inuenientur numeri in integris idem præstantes: Quod impossibile est, quia dato numero quouis integro non possunt dari infiniti in integris illo minores. Demonstrationem integram & fusius explicatam inserere margini vetat ipsius exiguitas.    Hac ratione deprehendimus & demonstratione confirmatus nullum numerum triangulum præter vnitatem æquari quadratoquadrato. OBSERVATIO D. P. F.
Title: Diophantus-VI-24-20-Fermat
Credit: Facsimile editions from: André Weil. Number Theory: An Approach Through History From Hammurapi to Legendre (p. 78). Birkhäuser (Springer), 1984.
Author: Improved by Leonid 2
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