Archimedes in the Middle Ages, Vol. 4: A supplement on the by Marshall Clagett

By Marshall Clagett

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Additional resources for Archimedes in the Middle Ages, Vol. 4: A supplement on the Medieval Latin traditions of conic sections (1150-1566). Part i

Sample text

And two points, h and t, are posited upon line ez, and at these points are erected perpendiculars to triangle abg, namely, [perpendiculars] hk and tl. I say, that dh-he / hk2 = dt-te / tl2, which is proved thus. Let us draw from points t and h two lines parallel to line b g , and let these be the lines mhn and sto. Hence the plane in which hk and mn lie is paral­ lel to the base of the cone. Therefore [by Prop. 2] this plane forms in the cone a circle whose diameter is line mhn. Hence let the circle be mkn.

Est autem et que XK dupla ipsius KP, propterea quod et que XZ ipsius XH et quia equedistans est que PH ipsi KZ. Equalis ergo que EK (/ O K ) ipsi KP. Que ergo OKP contingens yperbolam et intermedia existens asymptotarum in duo equa secatur. " 41vB -E (see Fig. 4): “ Et circa axem ZB per Z describatur parabola ita ut protracte possint penes AH: transibit ergo per T, quoniam quod sub ZAH est equale ei quod ab AT. Sit igitur descripta et sit ut que ZTK, et per B protrahatur equedistanter ipsi AT que BK et secet parabolam penes K, et per H circa asymptotas ZB, BK sit descripta yperbola.

Therefore, line EFG will be one line. And so, since that section is one line, it may be called a conic or pyramidal section. And so, if the axis of the cone, which is AD, is equal to the radius of the base, which [radius] is DB, it is evident that the cone ABC is right-angled, because the angle BAC of triangle ABC is right. e. 29 [of the Elements of Euclid], since angle BAC is right, that angle BFH will also be right, and similarly angle HFA. In such a case then, the section EFG will be called a right-angled section or parabola, and it is that [section] which the Arabs call mukefi.

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