ANALYSIS 795 that the most effective line for establishing it has been taken, in view of the well-known logical principle that the same conclusion may follow from different premisses, In the other form of the process, where the proposition assumed is itself used as a premiss, the case as to validity is otherwise. As Aristotle first clearly apprehended and showed, it is quite possible to reach a (materially) true conclusion by strict logical deduction from premisses either one or both false ; and thus the mere fact that the proposition assumed is found, in combination with others, to lead to a conclusion known to be true, does nothing to establish its own char acter. Yet although the process of analysis thus carried out by way of deduction, as formulated by Euclid and (in one of his expressions) by Pappus, is theoretically faulty, through neglect or ignorance of Aristotle s observation, the practice of Euclid is not therefore invalidated. It was his habit, as Pappus also enjoins, to follow up the analysis by a synthesis consisting in a reversal of it, and this would effectively get rid of error ; since the result of the analysis, if it did not follow from the assumed premiss by true im plication, but only accidentally, could not itself, when in turn used as a premiss for the synthesis, be made to yield the original proposition as a legitimate conclusion. In order, however, to validate this form of analysis it is not necessary to resort to the laborious expedient of retracing the whole path synthetically. As Duhamel, in his treatise jDes Methodes dans les Sciences de Raisonnement (pt. i. c. 5), has pointed out, it is enough if, at the different stages of the deduction, the inquirer assures himself, as he easily may do where it is the fact, that there is perfect " reci procity " among the propositions successively obtained from the one first assumed ; meaning that, in the circumstances of the deduction, each may as well follow from the one coming after as it is fitted to yield that. And the same simple expedient suffices equally to obviate the less grave defect above noted in analysis carried out by regression from consequents to conditions, or conclusions to premises ; reciprocity, if it can be made out here at the different stages, will guarantee the exclusive validity of the line of reasoning taken. So may analysis become perfectly inde pendent as a method of discovery, and give as much in sight as synthesis, where this is directly applicable, does ; while it is what synthesis is not directly applicable to every kind of question, however complex. It is unnecessary, for the purposes of the present article, to enter further into details respecting the methods anciently practised in geometry. Let it suffice to mention only the method of indirect proof known as rediictio ad absurdum, employed sometimes by Euclid in the Elements. This con forms to the type of analysis in that it starts from the question to be determined, though it is peculiar in follow ing out, not the assumption itself, but what is thereby sug gested as excluded, with the final result that the point in question is established upon the ruin of every other sup position. It is a method of discovery as well as a method of demonstration ; while the previous argument has shown that analysis, directly practised, may be made a method of demonstration by itself, besides being the most potent and unfailing instrument of discovery. Also it was seen before that synthesis may be a method of discovery, though it is more frequently employed as a method of demonstra tion in sequence upon discovery by analysis. To insist thus upon the double character alike of analysis and syn thesis, as practised in geometry, is of vital importance, be cause of the change in application which the terms have undergone among mathematicians. In modern times analysis has come to mean the employment of the alge braical and higher calculus, and synthesis any direct treat ment of the properties of geometrical figures, in the manner of the ancients, without the use of algebraical notation and transformations. The excuse for the change lies in the fact that, while the Greeks had only extremely undeveloped means of analysis, they gave the highest possible finish and exactness to their synthetic demonstrations of geometrical propositions, seldom being content to let their discoveries rest upon the ground of that analysis by which they were made. But though it has this excuse or motive, the change involves a misunderstanding, as all mathematicians allow who have turned their minds seriously to consider the rationale of their practice. It is, in the first place, clear that only by the process described above, rightly called analysis, can anything be determined about the more com plex properties and relations of geometrical figures ; hap hazard synthesis is of no avail. The ancients therefore, in their geometry, had an analysis. It is next to be re marked that the algebraical solution of problems is not so exclusively analytic in character that it may not in simple cases assume the form of direct (algebraical) synthesis ; and in all cases, for verification, it admits of being followed up by an exposition that is truly synthetic. The moderns, therefore, in their calculus, are not without their synthesis. Furthermore, the ancients, however little progress they made, comparatively speaking, in the general science of calculation, and however their special methods for the resolution of geometrical questions, even as involving direct figured construction, still more as applying calculation, fell short of the variety and pliability of modern devices, yet had their own analytical weapons, though they cannot be specified here. For our present purpose it is equally un necessary to enter into details as regards the modern devices, whether belonging to the lower or higher ana lysis, or as regards the principle for applying them de veloped by Descartes and his successors ; but to arrogate for these exclusively the name of analysis, it cannot be too pointedly declared, is to lose sight of the end in the means. II. Chemical Analysis and /Synthesis. After mathe matics, chemistry is the science in which application has most expressly been made of processes termed analysis and synthesis. In physics, regarded as the science of motion, whether abstractly taken or as manifested actually in natural bodies, the application is universal ; the resolu tion and composition of velocities, motions, and forces being fundamental processes pervading the whole science under all variety of circumstances. There is nothing, however, in such an employment of analysis and synthesis that is not easily intelligible in the light of the processes as practised either in the more general science of mathe matics, dealing with relations of quantity in number and form, or in the more special science of chemistry, which deals with those characteristic qualities of actual bodies for which no definite expression in terms of motion can be found. The concrete substances in nature are found to be siich that some by no means in our power can be brought to anything simpler, while others can be broken up into con stituents differing in character from the original substances and also among themselves. Hence a division is made of bodies into elements and compounds; elements being all such bodies, not farther reducible, as are either actually found in nature, or, though not so found, have emerged in the manipulation of actual bodies; compounds, all such as, being actually found, are reducible to two or more different elements, or have by artificial combination been constituted. The process of reduction to elements is called analysis; the process of re-combination or free combination is called synthesis. When the analysis is carried out simply with the view of detecting what elements are present in a sub stance, it is called qualitative; and quantitative, if with
the further view of determining the definite proportions
