Next there is the method of mathematical _analysis_. This method is said to have been 'communicated' or 'explained' by Plato to Leodamas of Thasos; but, like reduction (to which it is closely akin), analysis in the mathematical sense must have been in use much earlier. _Analysis_ and its correlative _synthesis_ are defined by Pappus: 'in analysis we assume that which is sought as if it were already done, and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of principles. But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis, and, by arranging in their natural order as consequences what were before antecedents and successively connecting them one with another, we arrive finally at the construction of that which was sought.' 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 {protasis}), (2) the _setting-out_ (εκθεσις {ekthesis}), (3) the διορισμος {diorismos}, 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 {kataskeuê}), (5) the _proof_ (αποδειξις {apodeixis}), (6) the _conclusion_ (συμπερασμα {symperasma}). 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 {diorismos}, 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.