By Marshall Clagett
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Extra resources for Archimedes in the Middle Ages. III. The Fate of the Medieval Archimedes. Part iv
P E a d F 86- 87 A . . E2: G Eddl -2 86-88 Et . . D om. pM arguit per el quinto di Euclide Ed3 86 sunt I 87 erit: erit proportio O 88 ergo etp / A ad B: E ad F Ed3 / ad: et b / proponebatur: proportionatur + lac. + quia A est equalis C et B equalis A, O aqua et K aqua sunt corpora eiusdem generis igitur que est proportio eorum in pondere eadem est in magnitudine per 4 petitionem et per consequens eadem est proportio in magnitudine corporum quorum unum est equale p proponebatur + lac. + quia A est equalis C et B equalis A Eddl -2 89 3 mg.
Let circle A E C be [described] about center D [see Fig. 6], while the side of the [regular] hexagon will be A E and the side of the triangle C E , by the fifteenth [proposition] of the second chapter of this [work, thc D e a r te m e n su ra n d i ]. Therefore, on each side let a semicircle be drawn. I say that the lunes drawn [on these sides] taken together are equal to A A E C , which is one-third of the [regular] hexagon inscribed in the circle. 30 (=Gr. 31). Hence, by the penultimate [proposition] of [Book] I [of the E le m e n ts ], A C 2 = A E 2 + E C 2.
1 propositio Pe, om. PD 4 DB Pd AB Pe 12 Erit Pd Est Pe 24 propositio Pe, om. Pd 1321 1322 ARCHIMEDES IN THE MIDDLE AGES Sit circulus AEC super centrum D [Fig. 6], latus vero hexagoni A E , latus trigoni erit C E per 15 2' huius. Super igitur utrumque latus 30 35 semicirculus figuretur. Dico lunulas protractas simul sumptas equales esse triangulo AEC, qui est tertia pars hexagoni circulo inscripti. Angulus enim A E C rectus est per 30 311. Ergo per penultimam primi quadratum A C valet duo quadrata A E , E C \ igitur et circulus circulos, et semicirculus semicirculos, per 2 12mi.