SECTION III.

Of the Import of the Words Analysis and Synthesis, in the Language of Modern Philosophy.

As the words Analysis and Synthesis are now become of constant and necessary use in all the different departments of knowledge; and as there is reason to suspect, that they are often employed without due attention to the various modifications of their import, which must be the consequence of this variety in their applications,-it may be proper, before proceeding farther, to illustrate, by a few examples, their true logical meaning in those branches of science, to which I have the most frequent occasions to refer in the course of these inquiries. I begin with some remarks on their primary signification in that science, from which they have been transferred by the moderns to Physics, to Chemistry, and to the Philosophy of the Human Mind.

I.

Preliminary Observations on the Analysis and Synthesis of the Greek Geometricians.

It appears from a very interesting relic of an ancient writer,* that, among the Greek geometricians, two different

* Preface to the seventh book of the Mathematical Collections of Pappus Alexandrinus. An extract from the Latin version of it by Dr. Halley may be found in Note (P.)

sorts of analysis were employed as aids or guides to the inventive powers; the one adapted to the solution of problems; the other to the demonstration of theorems. Of the former of these, many beautiful exemplifications have been long in the hands of mathematical students; and of the latter, (which has drawn much less attention in modern times) a satisfactory idea may be formed from a series of propositions published at Edinburgh about fifty years ago. I do not, however, know, that any person has yet turned his thoughts to an examination of the deep and subtle logic displayed in these analytical investigations; although it is a subject well worth the study of those who delight in tracing the steps by which the mind proceeds in pursuit of scientific discoveries. This desideratum it is not my present purpose to make any attempt to supply; but only to convey such general notions as may prevent my readers from falling into the common error of confounding the analysis and synthesis of the Greek Geometry, with the analysis and synthesis of the Inductive Philosophy.

In the arrangement of the following hints, I shall consider, in the first place, the nature and use of analysis in investigating the demonstration of theorems. For such an application of it, various occasions must be constantly presenting themselves to every geometer;-when engaged, for example, in the search of more elegant modes of demonstrating propositions previously brought to light; or in ascertaining the truth of dubious theorems, which

* Propositiones Geometrica More Veterum Demonstratæ. Auctore Matthæo Stewart, S. T. P. Matheseos in Academia Edinensi Profes sore, 1763.

from analogy, or other accidental circumstances, possess a degree of versimilitude sufficient to rouse the curiosity.

In order to make myself intelligible to those who are acquainted only with that form of reasoning which is used by Euclid, it is necessary to remind them, that the enunciation of every mathematical proposition consists of two parts. In the first place, certain suppositions are made, and secondly, a certain consequence is affirmed to follow from these suppositions. In all the demonstrations which are to be found in Euclid's Elements (with the exception of the small number of indirect demonstrations,) the particulars involved in the hypothetical part of the enunciation are assumed as the principles of our reasoning; and from these principles a series or chain of consequences is, link by link, deduced, till we at last arrive at the conclusion which the enunciation of the proposition asserted as a truth. A demonstration of this kind is called a Synthetical demonstration.

Suppose now, that I arrange the steps of my reasoning in the reverse order; that I assume hypothetically the truth of the proposition which I wish to demonstrate, and proceed to deduce from this assumption, as a principle, the different consequences to which it leads. If, in this deduction, I arrive at a consequence which I already know to be true, I conclude with confidence, that the principle from which it was deduced is likewise true. But if, on the other hand, I arrive at a consequence which I know to be false, I conclude, that the principle or assumption on which my reasoning has proceeded is false also. Such a demonstration of the truth or falsity of a proposition is called an Analytical demonstration.

According to these definitions of Analysis and Synthesis, those demonstrations in Euclid which prove a proposition to be true, by showing, that the contrary supposition leads to some absurd inference, are, properly speaking, analytical processes of reasoning. In every case, the conclusiveness of an analytical proof rests on this general maxim, That truth is always consistent with itself; that a supposition which leads, by a concatenation of mathematical deductions, to a consequence which is true, must itself be true; and that what necessarily involves a consequence which is absurd or impossible, must itself be false.

It is evident, that, when we are demonstrating a proposition with a view to convince another of its truth, the synthetic form of reasoning is the more natural and pleasing of the two; as it leads the understanding directly from known truths to such as are unknown. When a proposition, however, is doubtful, and we wish to satisfy our own minds with respect to it; or when we wish to discover a new method of demonstrating a theorem previously ascertained to be true; it will be found (as I already hinted) far more convenient to conduct the investigation analytically. The justness of this remark is universally acknowledged by all who have ever exercised their ingenuity in mathematical inquiries; and must be obvious to every one who has the curiosity to make the experiment. It is not, however, so easy to point out the principle on which this remarkable difference between these two opposite intellectual processes depends. The suggestions which I am now to offer appear to myself to touch upon the most essential circumstance; but I am perfectly aware that they by no meas amount to a complete solution of the difficulty.

Let it be supposed, then, either that a new demonstration is required of an old theorem; or, that a new and doubtful theorem is proposed as a subject of examination. In what manner shall I set to work, in order to discover the necessary media of proof? From the hypothetical part of the enunciation, it is probable, that a great variety of different consequences may be immediately deducible; from each of which consequences a series of other consequences will follow: At the same time, it is possible that only one or two of these trains of reasoning may lead the way to the truth which I wish to demonstrate. By what rule am I to be guided in selecting the line of deduction which I am here to pursue? The only expedient which seems to present itself, is merely tentative or experimental; to assume successively all the different proximate consequences as the first link of the chain, and to follow out the deduction from each of them, till I, at last, find myself conducted to the truth which I am anxious to reach. According to this supposition, I merely grope my way in the dark, without rule or method: the object I am in quest of, may, after all my labour, elude my search; and even, if I should be so fortunate as to attain it, my success affords me no lights whatever to guide me in future on a similar occasion.

Suppose now that I reverse this order, and prosecute the investigation analytically; assuming (agreeably to the explanation already given) the proposition to be true, and attempting, from this supposition, to deduce some acknowledged truth as a necessary consequence. I have here one fixed point from which I am to set out; or, in other words, one specific principle or datum from which all my conseU u