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Download E-books The Commentary of al-Nayrizi on Books II-IV of Euclid's Elements of Geometry (Studies in Platonism, Neoplatonism, and the Platonic Tradition) PDF

"The remark of al-Nayrizi (circa 920) on Euclid's parts of Geometry" occupies a tremendous position either within the historical past of arithmetic and of philosophy, really Islamic philosophy. it's a compilation of unique paintings through al-Nayrizi and of translations and commentaries made by means of others, resembling Heron. it's the so much influential Arabic mathematical manuscript in life and a precept motor vehicle wherein arithmetic was once reborn within the Latin West. in addition, the statement on Euclid via the Platonic thinker Simplicius, completely reproduced by means of al-Nayrizi, and nowhere else extant, is vital to the research of the try to turn out Euclid's 5th Postulate from the previous 4. "Al-Nayrizi" was once one of many major assets from which Albertus Magnus (1193-1280), the health practitioner Universalis, realized arithmetic. This paintings offers an annotated English translation of Books II-IV and of a hitherto misplaced section of publication I.

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Extra resources for The Commentary of al-Nayrizi on Books II-IV of Euclid's Elements of Geometry (Studies in Platonism, Neoplatonism, and the Platonic Tradition)

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And so, permit line EH be a line drawn from the guts E, and allow us to believe that it intersects arc undesirable at element Θ, and so line EA is the same as line EΘ, L16and from this we're confronted with the like of what we're confronted with from line EH and line EΘL16, and so line EΘ is the same as line EH. So, because the semicircle doesn't fall, with reference to congruence, both open air or within, or with a view to intersect, yet really in order to coincide completely on either side, the 2 of them, with none doubt, are equivalent to each other. Euclid stated, “And a semicircle is a determine which a diameter and the arc that it subtends enclose. And a phase of a circle is a determine immediately line and an arc of the circle, even if larger or smaller than a semicircle, enclose. ” Simplicius acknowledged: As for the determine that's known as a semicircle, it's in truth half a circle, and this can be definitely proved via what we previously acknowledged. And as for the determine line compounded from a instantly line and a curved line encloses, he defines it after the straightforward figures. 14 bankruptcy one Euclid acknowledged, “Rectilinear figures are what instantly strains enclose. And as for trilateral figures, they're what 3 immediately traces enclose, and as for quadrilateral figures, they're what 4 instantly traces enclose, and as for multilateral figures, they're what greater than 4 instantly traces enclose. ” Simplicius stated: And as for Euclid, after he pointed out the best figures (and these are those that one basic round line encloses, that's, one with out variation), and the determine which a instantly line and a round line enclose, he turns his awareness to the rectilinear figures, and he starts off during this type with the determine that 3 strains enclose, and that's simply because one line encloses the circle and traces enclose the semicircle. And as for a rectilinear determine, lo, now not purely one line, and never in simple terms traces, enclose it. and the way is it attainable that one directly line enclose a determine, on the grounds that it truly is stretched into straightness with none bending in it in any of its elements, in order that it doesn't enclose whatever? And it is because (as regards one directly line, it truly is clear), so far as directly strains are involved, they too don't enclose a floor, and this is often one of many postulates, and we will make clear this within the position the place Euclid mentions it. And the 1st rectilinear determine is the only with 3 aspects, and the second one is the only with 4 aspects, and the 3rd is the single with extra aspects. Euclid acknowledged, “And of the trilateral figures, there's the equilateral triangle, and that's the one the place 3 facets are equivalent to each other. And as for the isosceles triangle, it's the one with of its 3 facets equivalent to each other. And as for the triangle with 3 assorted facets, it's the one whose 3 facets are unequal to each other. ” Simplicius stated: As for the triangle with unequal facets, lo, the Greeks name it scalene from σκάζει, that's, “It limps,” for it really is as though equality is the reason for stability, and on that account inequality is the reason for move, and so if a guy walking strikes, and his legs are of alternative lengths, it's a necessity that he limp.

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