Full text: Volumen secundum. Opera geometrica. Opera astronomica. Varia de optica. (2)

GEOMET. VARIA. atque A F, F B, hoc eſt x & y. Nempe ſi in æquatione
propoſita pro x ſubſtituatur ubique x + e, & pro y, ubique
y + {ey/z}, debebit æquatio hinc formata terminos omnes ha-
bere æquales nihilo; hoc eſt
x 3 + [3exx] + 3eex + e 3 , + y 3 + [{3ey 3 /z}] + {3eey 3 /zz} + {e 3 y 3 /z 3 } = 0. - axy - [aey] - [{aeyx/z}] - {aeey/z}

In hac autem æquatione conſtat neceſſario terminos prio-
ris æquationis, ex qua formata eſt, contineri debere, nem-
pe x 3 + y 3 - axy: qui cum ſint æquales nihilo ex proprie-
tate curvæ, idcirco his in æquatione deletis, neceſſe eſt etiam
reliquos nihilo æquari, in quibus ſingulis manifeſtum quoque
eſt vel unum e vel plura reperiri, ideoque omnes per e divi-
di poſſe. Qui autem poſt hanc diviſionem non amplius ha-
bebunt e, eos, neglectis reliquis, ſcio nihilo æquari debe-
re, quantitatemque lineæ z ſive F E oſtenſuros; ſi nempe
B E jam tanquam tangens conſideretur, ideoque F G, ſeu
e, infinitè parva. Nam termini in quibus adhuc e ſupereſt,
etiam quantitates infinite parvas ſive omnino evaneſcentes
continebunt. Et his quidem hactenus Fermatianæ regulæ
origo ac ratio declaratur: nunc porro oſtendemus quomodo
eadem ad tantam brevitatem perducta ſit. Video itaque ex
æquatione totâ noviſſimâ, tantum eos terminos ſeribi neceſ-
ſe eſſe quibus ineſt e ſimplex, velut hic 3exx + {3ey 3 /z} - aey -
{aeyx/z} = 0. Qui termini quomodo facili negotio ex datis æqua-
tionis terminis x 3 + y 3 - axy = 0, deſcribi poſſint, dein-
ceps explicandum. Et primò quidem apparet 3exx +
{3ey 3 /z} nihil aliud eſſe quam ſecundos terminos cuborum ab

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