References
- Translated by Crew Henry De Salvio Alfonso Dover Publications Inc. New York Henceforth, in all foot-notes, to be referred to as: T.N.S.
- A geometrical proof, in Galileo's view, was one carried out ‘in a rigid manner from fundamental principles’. This method was claimed to be superior to any other, including that by logic, and was identified by him with ‘demonstrative reasoning’ (see T.N.S. xx xx 2–3, 6, 137). As we shall see, however, it was Galileo's Geometry that was to lead him astray.
- It is precisely here, where Galileo quits sense perception to reach his conclusion by reason that all his troubles begin, for this false step, as we shall see, concealed from him the key to the solution of the problem (see below: p. 321 and pp. 334–335). The further he now goes, the further his geometrical methods lead him into error, with the result that he not only fails to come within sight of the true solution of the problem, but he undermines all the so-called ‘other new and remarkable facts’ he claims to derive from his original ‘proposition’ T.N.S. 20 20
- See Aristotle's Mechanics Oxford vi Chap. 1, 848a 4 ‘Another peculiarity of the circle is that it moves in two contrary directions at the same time; for it moves simultaneously to a forward and a backward position. Such, too, is the nature of the radius which describes the circle. For its extremity comes back again to the same position from which it starts.’ A diagram will make Aristotle's point clear (fig. 8).
- In this case, this cannot be according to the methods employed in present-day mathematics (see below, p. 334 Review of Metaphysics U.S.A. xvi 478 478 In brief, though I shall be dealing with the subject of refutation on another occasion: Every argument is a mental journey by which the arguer claims to reach his conclusion (say, B) from his starting-off-point (say, A) along his chosen route (A–B). As a consequence, the refutation must follow the track of the argument, and the solution must follow from the rectification of the fallacies uncovered in the tracking of the argument. Unless this is so, the refuter lays himself open to the charge of ignoratio elenchi. The refutation is sophistical rather than genuine. It appears to reufte, bnt actually does not, as happened when Samuel Johnson kicked the heavy stone outside Harwich church in 1763, in an attempt to refute Bishop Berkeley's ingenious sophistry to prove the non-existence of matter, and that everything in the universe is merely ideal.
- This is precisely what was found at the end of the first movement in our solution of the Achilles paradox. See the Review of Metaphysics U.S.A. xvi 478 478 and 481, No. 3, March 1963; and also Mind (Oxford, Great Britain), vol. lxxvii, No. 306, April 1968 for our solution of Zeno's First Paradox where a parallel distinction is equally necessary (pp. 217 and 218). This is the clue to the problems here
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