Elementary as these things are, they represent a new departure of a momentous kind, being the first steps towards a _theory_ of geometry. On this point we cannot do better than quote some remarks from Kant's preface to the second edition of his _Kritik der reinen Vernunft_. 'Mathematics has, from the earliest times to which the history of human reason goes back, (that is to say) with that wonderful people the Greeks, travelled the safe road of a _science_. But it must not be supposed that it was as easy for mathematics as it was for logic, where reason is concerned with itself alone, to find, or rather to build for itself, that royal road. I believe on the contrary that with mathematics it remained for long a case of groping about--the Egyptians in particular were still at that stage--and that this transformation must be ascribed to a _revolution_ brought about by the happy inspiration of one man in trying an experiment, from which point onward the road that must be taken could no longer be missed, and the safe way of a science was struck and traced out for all time and to distances illimitable.... A light broke on the first man who demonstrated the property of the isosceles triangle (whether his name was Thales or what you will)....' Thales also solved two problems of a practical kind: (1) he showed how to measure the distance of a ship at sea, and (2) he found the heights of pyramids by means of the shadows thrown on the ground by the pyramid and by a stick of known length at the same moment; one account says that he chose the time when the lengths of the stick and of its shadow were equal, but in either case he argued by similarity of triangles. In astronomy Thales predicted a solar eclipse which was probably that of the 28th May 585 B. C. Now the Babylonians, as the result of observations continued through centuries, had discovered the period of 223 lunations after which eclipses recur. It is most likely therefore that Thales had heard of this period, and that his prediction was based upon it. He is further said to have used the Little Bear for finding the pole, to have discovered the inequality of the four astronomical seasons, and to have written works _On the Equinox_ and _On the Solstice_. After Thales come the Pythagoreans. Of the Pythagoreans Aristotle says that they applied themselves to the study of mathematics and were the first to advance that science, going so far as to find in the principles of mathematics the principles of all existing things. Of Pythagoras himself we are told that he attached supreme importance to the study of arithmetic, advancing it and taking it out of the region of practical utility, and again that he transformed the study of geometry into a liberal education, examining the principles of the science from the beginning. The very word μαθηματα {mathêmata}, which originally meant 'subjects of instruction' generally, is said to have been first appropriated to mathematics by the Pythagoreans.