794 ANALYSIS as an assertion to be accepted, and lias to be shown true ; a problem is given out as an act to he done, and has to be shown possible. In the case of a theorem, Euclid accord ingly, after enunciating the proposition, proceeds generally to show, with more or less of construction on a particular diagram, and working always with fixed definitions, that the assertion follows deductively from certain truths, either assumed as evident (axioms), or formerly proved therefrom, and seen to be applicable to the present case by inspection of the figure as constructed. The grounding propositions are allowed by the reader as they are brought forward, though he may for the moment have not the least idea whither the author is tending, and at the end the con clusion is accepted, because the successive premises, being allowed, have been combined logically. In the case of a problem, after an express construction for which no reason is given, the object is to show that what has been brought to pass really supplies what was sought ; but the procedure is not different from what it was in the case of a theorem, because the object is attained by showing again that certain truths allowed, in their particular application to the figure constructed, involve as a conclusion some relation which the figure is seen to exhibit. Now if this is Euclid s procedure in general there is an exception, afterwards to be noted, where he proves his point in directly it is undeniably synthetic, in any meaning that can be ascribed to that term, the result being obtained by a massing or combining of elements or conditions. But on Euclid s part the process is one of demonstration, not of discovery. Still less is the reader s mind in the attitude of discovery : he is led on to a result which is indeed indicated, but by a way which he does not know, and, as it were, blind fold. There must, however, have been discovery before there could be such demonstration ; or how should the proposi tion admit of definite enunciation at the beginning 1 ? Thus there is, in the background, an earlier question of procedure or method, and it is this that the ancient geometricians had chiefly in view when speaking of analysis and synthesis. Now, some propositions are so simple that they must have been seen into almost as soon as conceived, and con ceived as soon as the human mind began to be directed to the consideration of forms or figures; in which case no method of discovery, to speak of, can have been necessary. There is, again, another class of propositions, more complex though still simple, which probably were established by a process of straightforward synthesis. An inquirer must have in his head some knowledge in the shape of principles more or less fixed, or he would not be an inquirer ; and either the accidental combination of such principles may lead in his mind to particular results, or the first time a particular question suggests itself to him, it may be seen at once to involve, or to foflow from, certain of the principles. Many propositions in the Elements, giving the most apparent properties of triangles, circles, &c., it can hardly be doubted, were arrived at by this way of dis covery, even when a more elaborate process of synthesis was employed for their formal demonstration ; as, for ex ample, in the case of the famous fifth proposition of Book I. But the same process of direct composition (understood always as joined with inspection) is no longer applicable, or is not effective, when the question is of less obvious properties, or of construction to be made under special conditions. To discover the fact or the feasibility in such cases is so much the real difficulty, that the question of demonstration becomes of merely secondary importance. And there is even a still prior question of discovery; for it has to be determined that some points rather than others should be made the subject of express inquiry. This, how ever, may be left aside. To any one engaged in geometri cal inquiry, in the constant inspection of figures for the understanding of their properties and mutual relations, questions must incessantly be occurring so incessantly and inevitably that it is needless, if it were not vain, to seek out a reason for the particular suggestions. As in all discovery to the last, so more especially at the first stages, there is an element of instinctive tact in the mind s action which eludes expression ; and there is also an element of what might be called chance, were it not that those only get the benefit of it who are consciously on the look-out, either generally or in some special direction. A particular question being started by whatsoever suggestion, how shall the mind arrive at certain knowledge regarding it ? Such, practically, is the form which is assumed by geometrical inquiry. Besides the thing sought there is nothing else given, or at least there is nothing else immediately given or suggested. But the mind is supposed to have some knowledge pertain ing to the matter though not extending to the particular aspect of it in question, also some knowledge of such mat ters generally. In such circumstances the aim of the inquirer must be to bring what is sought into some definite relation with what is known. Direct composition or synthesis of the known, with more or less of construction, if it led to that which is sought as a result, would determine the re lation for the inquirer, and determine it in like manner for all who allow the principles whence the conclusion is logi cally deduced, being thus at one stroke both discovery and demonstration. But synthesis, arbitrarily made, as it must be where the question is at all difficult, may fail, however often it is attempted. Without a proper start it avails nothing; and what is to determine the start] There is always one course open. Let the objective itself be made the starting-point, and let it be seen whether thence it may not be possible by some continuous route to get upon known ground. In other words, a thing sought, when itself assumed, may admit of being brought into re lation, upon some side or other, with the body of ascertained knowledge. If it can be so brought, through whatever number of steps, there is then attained as a result what before it was impossible to light upon as a beginning ; and now nothing hinders from making the start originally desired, and from reaching as a proper conclusion the assumed beginning, if the path struck out before is mea sured over again in the opposite direction. The course thus becomes once more synthetic, but only because of what was first accomplished. Till the point in question was made to yield up its own secret by a process fitly called analysis or resolution, nothing certain could be determined, At the analytic stage, however, the line taken may be twofold. The proposition, assumed at starting as something definite to work from, cither may be held as following deductively from some other, which again is dependent on still another or others, till one is worked up to that is known to be true ; or it may be taken as itself a premiss leading deductively to some other proposition, which in turn, by one or more steps, leads to a true pro position as conclusion. In either case the implication is that a proposition must itself be true, if by any line of formally correct logic it leads to a proposition known to bo true. And though the expression must be modified for questions in the form of problems, requiring something to be done -to which form of question, indeed, the analytic pro cess is peculiarly applicable the point of logical principle remains there exactly the same. But is the process, thus stated as it was understood by the ancient geometricians, logically valid 1 In the first of the two alternative forms, it is valid : the proposition assumed at starting will undoubtedly be true, if a proposi tion on which it is shown to be ultimately dependent is
true. At the same time, there is in this case no guarantee
