The method of reductio ad absurdum is a variety of analysis. Starting from a hypothesis, namely the contradictory of what we desire to prove, we use the same process of analysis, carrying it back until we arrive at something admittedly false or absurd. Aristotle describes this method in various ways as reductio ad absurdum, proof per impossibile, or proof leading to the impossible. But here again, though the term was new, the method was not. The paradoxes of Zeno are classical instances.
Lastly, the Greeks established the form of exposition which still governs geometrical work, simply because it is dictated by strict logic. It is seen in Euclid’s propositions, with their separate formal divisions, to which specific names were afterwards assigned, (1) the enunciation (προτασις), (2) the setting-out (εκθεσις), (3) the διορισμος, being a re-statement of what we are required to do or prove, not in general terms (as in the enunciation), but with reference to the particular data contained in the setting-out, (4) the construction (κατασκευη), (5) the proof (αποδειξις), (6) the conclusion (συμπερασμα). In the case of a problem it often happens that a solution is not possible unless the particular data are such as to satisfy certain conditions; in this case there is yet another constituent part in the proposition, namely the statement of the conditions or limits of possibility, which was called by the same name διορισμος, definition or delimitation, as that applied to the third constituent part of a theorem.
We have so far endeavoured to indicate generally the finality and the abiding value of the work done by the creators of mathematical science. It remains to summarize, as briefly as possible, the history of Greek mathematics according to periods and subjects.
The Greeks of course took what they could in the shape of elementary facts in geometry and astronomy from the Egyptians and Babylonians. But some of the essential characteristics of the Greek genius assert themselves even in their borrowings from these or other sources. Here, as everywhere else, we see their directness and concentration; they always knew what they wanted, and they had an unerring instinct for taking only what was worth having and rejecting the rest. This is illustrated by the story of Pythagoras’s travels. He consorted with priests and prophets and was initiated into the religious rites practised in different places, not out of religious enthusiasm ‘as you might think’ (says our informant), but in order that he might not overlook any fragment of knowledge worth acquiring that might lie hidden in the mysteries of divine worship.
This story also illustrates an important advantage which the Greeks had over the Egyptians and Babylonians. In those countries science, such as it was, was the monopoly of the priests; and, where this is the case, the first steps in science are apt to prove the last also, because the scientific results attained tend to become involved in religious prescriptions and routine observances, and so to end in a collection of lifeless formulae. Fortunately for the Greeks, they had no organized priesthood; untrammelled by prescription, traditional dogmas or superstition, they could give their reasoning faculties free play. Thus they were able to create science as a living thing susceptible of development without limit.
Greek geometry, as also Greek astronomy, begins with Thales (about 624-547 B. C.), who travelled in Egypt and is said to have brought geometry from thence. Such geometry as there was in Egypt arose out of practical needs. Revenue was raised by the taxation of landed property, and its assessment depended on the accurate fixing of the boundaries of the various holdings. When these were removed by the periodical flooding due to the rising of the Nile, it was necessary to replace them, or to determine the taxable area independently of them, by an art of land-surveying. We conclude from the Papyrus Rhind (say 1700 B. C.) and other documents that Egyptian geometry consisted mainly of practical rules for measuring, with more or less accuracy, (1) such areas as squares, triangles, trapezia, and circles, (2) the solid content of measures of corn, &c., of different shapes. The Egyptians also constructed pyramids of a certain slope by means of arithmetical calculations based on a certain ratio, se-qeṭ, namely the ratio of half the side of the base to the height, which is in fact equivalent to the co-tangent of the angle of slope. The use of this ratio implies the notion of similarity of figures, especially triangles. The Egyptians knew, too, that a triangle with its sides in the ratio of the numbers 3, 4, 5 is right-angled, and used the fact as a means of drawing right angles. But there is no sign that they knew the general property of a right-angled triangle (= Eucl. I. 47), of which this is a particular case, or that they proved any general theorem in geometry.
No doubt Thales, when he was in Egypt, would see diagrams drawn to illustrate the rules for the measurement of circles and other plane figures, and these diagrams would suggest to him certain similarities and congruences which would set him thinking whether there were not some elementary general principles underlying the construction and relations of different figures and parts of figures. This would be in accord with the Greek instinct for generalization and their wish to be able to account for everything on rational principles.
The following theorems are attributed to Thales: (1) that a circle is bisected by any diameter (Eucl. I, Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I. 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I. 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I. 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle, which must mean that he was the first to discover that the angle in a semicircle is a right angle (cf. Eucl. III. 31).
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)....’