The Works of Archimedes Volume 1: The Two Books On the by Reviel Netz
By Reviel Netz
The works translated here--the books at the Sphere and Cylinder--were a resource of serious satisfaction for Archimedes, the best scientist of antiquity. Accompanying this translation is the 1st medical version of the diagrams, which contains new details from the lately came across Archimedes Palimpsest. the amount additionally comprises the 1st English translation of Eutocius's observation. Reviel Netz's statement analyzes Archimedes's paintings from modern learn views similar to medical type and the cognitive background of mathematical texts.
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Example text
P r e l i m i na ry n ot e s : a rc h i m e d e s ’ wo r k s equations. Nothing like the “elementary” nature of SC I, then. The ancillary objects, constructed for the sake of the proofs, result from complex proportion manipulations, not from any direct geometric significance. Greek geometrical tools are stretched to the limits and beyond: the mini-treatise at the end of the work relies essentially upon conic sections; the notion of the exponent is adumbrated in Proposition 8. Both treat geometrical objects in a semi-algebraic way, as objects of manipulation in calculation.
Hence the inequality (2) is impossible: it is not the case that (2) SUR>4CIR. Arguing analogously against the alternative inequality (SUR<4CIR) only the equality remains, the QED: (7) SUR=4CIR. In sum, we look for equalities holding between simple objects. We go indirectly, in three different ways: r We throw in complex objects that we compare with the simple objects, r We start with proving inequalities (direct as well as proportion inequalities) rather than equalities, r We assume that the desired equality does not hold.
It is even less clear which texts Dositheus (or any other intended reader) was expected to know. 1 So nothing can be taken for granted. The text must be read and understood in the light of what it says, how it is used, and the related material in the Archimedean corpus. This calls for a separate study, which I shall not pursue here. /1/ If a polygon is circumscribed around a circle, the perimeter of the circumscribed polygon is greater than the perimeter of the circle. For let a polygon – the one set down15 – be circumscribed around a circle.