proportional, and therefore the rectangles themselves are equal; which we could not say, unless _rectangles have their sides reciprocally proportional_, and _rectangles are equal_, were terms convertible. Now in every analysis, that which is sought is the proportion of two quantities; by which proportion, a figure being described, the quantity sought for may be exposed to sense. And this _exposition_ is the end and solution of the question, or the construction of the problem. And seeing analysis is reasoning from something supposed, till we come to principles, that is, to definitions, or to theorems formerly known; and seeing the same reasoning tends in the last place to some equation, we can therefore make no end of resolving, till we come at last to the causes themselves of equality and inequality, or to theorems formerly demonstrated from those causes; and so have a sufficient number of those theorems for the demonstration of the thing sought for. And seeing also, that the end of the analytics is either the construction of such a problem as is possible, or the detection of the impossibility thereof; whensoever the problem may be solved, the analyst must not stay, till he come to those things which contain the efficient cause of that whereof he is to make construction. But he must of necessity stay, when he comes to prime propositions; and these are definitions. These definitions therefore must contain the efficient cause of his construction; I say of his construction, not of the conclusion which he demonstrates; for the cause of the conclusion is contained in the premised propositions; that is to say, the truth of the proposition he proves is drawn from the propositions which prove the same. But the cause of his construction is in the things themselves, and consists in motion, or in the concourse of motions. Wherefore those propositions, in which analysis ends, are definitions, but such as signify in what manner the construction or generation of the thing proceeds. For otherwise, when he goes back by synthesis to the proof of his problem, he will come to no demonstration at all; there being no true demonstration but such as is scientifical; and no demonstration is scientifical, but that which proceeds from the knowledge of the causes from which the construction of the problem is drawn. To collect therefore what has been said into few words; ANALYSIS _is ratiocination from the supposed construction or generation of a thing to the efficient cause or coefficient causes of that which is constructed or generated. And_ SYNTHESIS _is ratiocination from the first causes of the construction, continued through all the middle causes till we come to the thing itself which is constructed or generated_.