By Marshall Clagett
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Additional resources for Archimedes in the Middle Ages. III. The Fate of the Medieval Archimedes. Part iv
With the circle divided into 6 parts by IV. 15 and with the two [halves of the hexagon] supposed instead of one, the proposition follows. 1325 QUADRATURE BY LUNES c The D e attributed to Leon Battista Alberti lu n u la r u m q u a d r a tu r a [De lunularum quadratura] 77v 5 10 15 78r 20 78v 26 30 35 79r 40 /Modo de misurare una figura biangula contenta da due linee curve, come si vedde la figura: Controntro Toppenioni de molti che dicono che le figure contente da linee curve et circulare perfettamente no[n] si da la loro quadratura, maximamente di quelle che sono portioni de circuli.
It is 1319 1320 ARCHIMEDES IN THE MIDDLE AGES certain that angle D of A A D C is a right angle. Therefore, the square of the side opposite it, namely ABC, which [square] is A E C , equals the two squares described on the two sides opposite it (ABC), namely [the sides] A D and D C , which [squares] are D G C and the one opposite it. Therefore, it (square A E C ) is double to either of those squares, as, for example, double square D G C , since the two squares are equal. 2]. Therefore, semicircle A D C described on line ABC will be double the semicircle described on either of the two sides AD and DC, and consequently it (semicircle ADC) is equal to both [together], just as the square described on the same line ABC is equal to the [sum of the] squares described on the other two sides.
Each of these is a sixth part of the hexagon, by a common axiom. But the triangular lune is greater than the hexagonal lune, as I shall show. Therefore, this triangular lune is greater than A E D C , which is that proposed. Therefore, the hexagonal lune is by the same amount less than A A E D , which I have wished to demonstrate. The supposition [in re one lune being greater than the other] is demon strated: keeping the same figure as before [Fig. 8] line DP is added up to the top of the hexagonal lune and line DO to the top of the triangular lune.