482 G. VAILATI : That is, it manifests itself as an assertion of the heuristic efficacy of that process of inquiry, which, taking as a start- ing point, idealistic and simplifying concepts and hypotheses, not having any exact counterpart in what is called the reality of things, arrives, precisely by means of deductions from these, and by means of what have been recently called (Mach) " experiments in thought " (Gedankenexperimente) at analysing, comprehending, dominating this reality and discovering in it and under it, independently of recourse to direct experiment, regularity, laws, standards, which direct and passive observation would never have been able to reveal. So understood the theory of ideas appears more intimately connected than would be generally admitted with the other great innovation in method attributed to Plato, viz. : the employment of deductive reasoning in the choice and the rejection of the various hypothetical alternatives which pre- sent themselves as possible with regard to a given subject. Plato found an example of the efficacy of both these pro- cesses together in Astronomy, understanding that term, as the Greeks always understood it, to mean the science that aimed at explaining and reducing to order (a-v^Koa^elv, to use the word employed by Aristotle (De Ccelo, ii., 13), almost in jest, against the Pythagoreans) the irregularities and anom- alies of the apparent motions of the stars on the celestial sphere, making them result as consequences of certain hy- potheses with regard to their real motions in space. It is in these first applications of mathematical doctrines to the explanation and prevision of the phenomena of the physical world, that Plato found the most convincing proof and confirmation of the power that the human mind is capable of acquiring by means of the logical discipline by which it is enabled, to use the phrase of Timseus reported by Proclus (in Eucl. i.), to recognise as connected and akin things apparently most diverse and opposite (<i'Xa TO. /j-a^6/jLva. teal (TVfMTraOr) teal Trpoa-rjyopa ra BiecrTMTa, Prologus, i., 22) i.e., to trace, in the chaos of the facts which present themselves to observation and experiment, the invariable laws to which these facts conform. Of the position in which Plato maintained that the philosopher and the scientist stood in this respect, he gives us a symbolical representation in the famous image of the cave and the bound prisoners in it who were obliged to look only at the shadows projected on a wall by objects passing behind their backs. It is to this very situation that the sentences refer, in which
