For a theorem, the method of analysis consists in reasoning as follows: "I can prove this proposition if I can prove this thing; I can prove this thing if I can prove that; I can prove that if I can prove a third thing," and so the reasoning runs until the pupil comes to the point where he is able to add, "but I _can_ prove that." This does not prove the proposition, but it enables him to reverse the process, beginning with the thing he can prove and going back, step by step, to the thing that he is to prove. Analysis is, therefore, his method of discovery of the way in which he may arrange his synthetic proof. Pupils often wonder how any one ever came to know how to arrange the proofs of geometry, and this answers the question. Some one guessed that a statement was true; he applied analysis and found that he _could_ prove it; he then applied synthesis and _did_ prove it. For a problem, the method of analysis is much the same as in the case of a theorem. Two things are involved, however, instead of one, for here we must make the construction and then prove that this construction is correct. The pupil, therefore, first supposes the problem solved, and sees what results follow. He then reverses the process and sees if he can attain these results and thus effect the required construction. If so, he states the process and gives the resulting proof. For example: In a triangle _ABC_, to draw _PQ_ parallel to the base _AB_, cutting the sides in _P_ and _Q_, so that _PQ_ shall equal _AP_ + _BQ_. [Illustration] =Analysis.= Assume the problem solved. Then _AP_ must equal some part of _PQ_ as _PX_, and _BQ_ must equal _QX_. But if _AP_ = _PX_, what must [L]_PXA_ equal? [because] _PQ_ is || _AB_, what does [L]_PXA_ equal? Then why must [L]_BAX_ = [L]_XAP_? Similarly, what about [L]_QBX_ and [L]_XBA_? =Construction.= Now reverse the process. What may we do to [Ls] _A_ and _B_ in order to fix _X_? Then how shall _PQ_ be drawn? Now give the proof. [Illustration] [Illustration]