matics is as follows.

When we start with a conclusion, supposed to be set before us for proof, whether this go by the name of a problem or a theorem, and reason 'backwards' from it until we come to some axiom, or other proposition which is admitted to be true, then we are said to reason analytically. When, on the other hand, we start from such axioms or other propositions and thus reach the problem or theorem which has been set before us, then we are said to reason synthetically. A typical instance of the former process is any 'problem' such as is proposed in examination papers: a similar instance of the latter is furnished in most of the examples in Euclid's Elements.

Between this usage of the terms, and that described above, there is more difference than may appear at first sight; and this difference is indicated in the accounts given respectively of their etymology. In the common interpretation, analysis is understood to be a breaking up into parts, a process of dissolution; and in accordance with this sense we generally find that the stress is laid upon the fact that the path backwards from the conclusion branches out, so as to lead us to a plurality of data or axioms. In the mathematical sense analysis is understood to be a retrograde process, that is, one from the unproved (i.e. the problem before us) to the proved (the axioms which establish the problem). Its etymology here is supposed to come, not from the breaking of a whole into pieces, but from the undoing of a knot in which we may best succeed by going back step by step along the path which any one would have taken in the act of making the knot. Thus in proving a theorem, or solving a problem which is supposed to be set before us, we take the result provisionally for granted as a starting-point, and say; If this be true, then would that, and if that be true so would some other; and so on, until we come to some already recognized truth. The fact of being led back to this point establishes the conclusion. It is obvious that it is in this way only that we could generally expect to be able to solve any proposed problem, since we are not supposed to have any hint given to us as to what premises we had best select as our starting-point, and it would be absurd to keep on trying one after another until we had hit on such as would answer our purpose.

The logical reader may be surprised here at the implied assumption that if we can derive any true proposition from the proposition proposed to us, this latter may be at once accepted as true; since this may seem to conflict with his well-known doctrine that from false premises we may yet possibly derive a true conclusion. There is however no real conflict. The difference arises mainly from a very important. characteristic of mathematics as contrasted with Logic. In the former science most of our propositions are of the nature of equations, rather than ordinary predications. Hence they are simply convertible, in the sense that if from X we obtain Y, we know that from Y we can obtain X. Take a simple example of a geometrical problem. Let it be required to find the point on a given straight line from which, if straight lines be drawn to two given points, they shall make equal angles with the given straight line.

Let P and Q be the given points, and ST the straight line. We assume that the point X is found, so that the angles SXP, TXQ, shall be equal.

What we then try to do is to deduce some consequence from this assumed result which shall be obvious, or in some way already known. We say, If QXT=PXS, then RXS-PXS (for SXRQXT); and if RXSPXS, then RSPS (where PSR is perpendicular to ST). Then comes the simple conversion; If PS=SR (a result we can readily secure by our own construction), RXSPXS; and if RXSPXS then QXR=PXS. This process of conversion is called synthesis, whilst the previous process, of which it is the conversion, is considered as one of analysis.

This regressive process is a very common one in mathematics. It is peculiarly appropriate for solving a desired problem, or proving some proposition whose truth we can

at best suspect, for thus we secure a definite starting-point. Even so we have of course to act at random to a certain extent, for we may try one path after another without finding that it leads us to any known axiom or to any obviously feasible construction. There is large opening therefore for skill and sagacity; but this procedure nevertheless involves a whole order less of vagueness than if we attempted to begin from the other end. In this latter case we should have exactly the same number of paths opening out before us whatever starting-point we selected; but in addition to this the number of such possible starting-points, in other words the number of axioms or theorems at our choice, is practically infinite. Accordingly the Analytical method, as giving us one definite starting-point, offers great advantages.

On the other hand the Synthetic, or progressive method, in which we start from axiomatic premises and derive one conclusion after another from them till at last we come to the assigned problem or theorem, is the best order for exposition. There is an obvious simplicity and naturalness in thus making every step that we take certain from the first, instead of having the whole procedure uncertain up to the last moment.

Although however this mathematical sense of Analysis has the warrant of antiquity and long usage in its own appropriate province, it seems now to be acquiring a much vaguer acceptation. Owing to the wide prevalence of analysis' in this retrogressive sense, in the domain of Algebra, any algebraic treatment of a subject is coming to be termed analytic. Thus those who speak of 'Analytic Geometry' probably have little in view beyond the symbolic treatment. They are not in any way contrasting their method with the Synthetic, but rather with one which depends upon intuition and geometrical construction. They would consider the word equally appropriate even though they adopted a method as closely as possible analogous to that of Euclid in respect of systematic exposition, provided only the successive steps were expressed in symbolic or algebraic form.

CHAPTER XV.

THE SYLLOGISM IN RELATION TO INDUCTION.

THE discussion in the last chapter was intended to convey a general notion of the nature of the Inductive process as here conceived. There is however one particular aspect of the question which needs more minute investigation in a systematic work on Logic. This is the relation of Induction to the ordinary Syllogism. As Mill's doctrine on this subject is familiar to most students of philosophy, and as this doctrine seems to me,--with considerable reservations and modifications, to be tenable, we will begin with a brief statement of it and then proceed to offer some criticisms upon it.

The doctrine in question may be briefly stated as follows. All knowledge is originally derived from experience of particulars, so that every really general proposition must have been obtained by our having generalized beyond the limits of observation. Now when we examine a valid syllogism of the first figure' we see that it starts with a general proposition, and that this proposition actually contains the conclusion. (Thus, 'All M is Q, P is M, therefore P is Q'; where P is clearly nothing else than a sample of those M's of which the major premise speaks.) The question is then raised; Under these circumstances, is the process of passing from the major, by aid of the minor, to the conclusion, a begging of the question, or not? In other words, at the time that we stated

1 It deserves notice that Mill makes no explicit reference in his discussion, to any other figure than the first, so that he must be presumed to hold that this is the only fundamental form of syllogistic reasoning; the other figures requiring to be reduced to the first. If we were to apply his description to the other figures, we should find that the functions of generalization and explanation respectively attributed to the major and minor premises would not hold; and the attempt to transfer the explanation leads to very awkward results.

that 'All M is Q', did we or did we not know that P was Q? If we did, then there was no need to go through the parade of stating two premises and pretending that there was an inference: if we did not, then we had clearly no right to state the major premise thus broadly and confidently.

Such is the difficulty. The solution is found in frankly admitting that there is no inference within the limits of the syllogistic process itself, but that the inference was secured in the act of obtaining the major premise. The true original premises were the observed original facts. Directly we had generalized these into our major premise we had already performed the whole inference, in that direction, which was warranted by them. This accounts for the major premise; but of course the next question is, What is the use of the syllogistic form? To this the reply is that such a form is one of considerable practical convenience. Though all inference is essentially one from particulars to particulars, it is nevertheless a great safeguard to make two distinct operations by separating the processes of recording and of interpreting our generalizations. The major premise records the inference in its widest extent: the minor interprets it by applying it to any particular cases as these arise.

Such is the Theory, stated in the briefest terms; which, it must be remembered, has found acceptance not only amongst those who are in general accord with Mill's philosophy, but also on the part of some who are strongly opposed to his general principles. For instance, such a strenuous antagonist as Whewell admits that the doctrine "that the force of the syllogism consists in an inductive assertion with an interpretation added to it, solves very happily the difficulties which baffle the other theories of the subject" (Philosophy of Discovery, p. 289).

Instead of proceeding directly to criticize this theory, it will be more instructive to begin by enquiring what are the grounds proposed for accepting it. These seem to me to be mainly the three following; two of them being positive, in the sense that they are direct inducements to accept the new theory, the third being negative, in the sense that it is supposed to constitute a difficulty in the way of our accepting the common theory.