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For the inner planets, the innermost sphere holds the planet along its equator and imputes a precession of the planet’s orbital plane. For the outer planets, Eudoxus structured the inner two spheres in a manner that creates retrograde motion. The planet sits on the equator of the innermost sphere. 14 PERFECTIONISTS By generating retrograde motion through this final construction, Eudoxus demonstrates his mastery of geometry. Below, we discuss some details of the construction. The presentation fully exploits mathematics that were not available to Eudoxus, including trigonometry, Cartesian coordinates, analytic geometry, linear algebra, and Euler angles.
To his slave attests to the generosity of the employment contract; one could afford a slave only with a high salary. The terms attracted the most able intellects throughout the Hellenic kingdoms. This was their guilded sanctuary in an otherwise harsh world. As noted in Chapter 2, there is a nostalgia associated with pre-Philippic Athenian culture promoting a sentiment that the culture of the Hellenistic age was somewhat inferior. In the case of mathematics, this sentiment is dead wrong. The mathematicians associated with Alexandria, building on the foundations of their predecessors, furthered mathematical knowledge well beyond the Athenians.
38 A HEAVY HAND 39 of Hannibal, the Carthaginian terror who ransacked the Italian peninsula at will. Hannibal crossed over the Alps 5 years earlier with an army of infantry, warrior elephants, and most importantly a Numidian cavalry that provided Hannibal with unmatched mobility on the battlefield. The Romans had engaged Hannibal three times, always with the same result. Hannibal’s cavalry would outflank the Romans, encircle, and defeat them. The defeat at Cannae was the most devastating. The Romans assembled an army of 86,000 to face Hannibal’s 50,000.