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or even suspecting other resemblances as real,—and the very circumstance of agreement, which we perceive,-at the time when we class objects together, as related, may involve, or comprehend, certain circumstances to which we then paid no attention, and which occur to us, only in that intellectual analysis of ratiocination, of which I spoke. It is as if we knew the situation and bearings of all the great cities in Europe, and could lay down, with most accurate precision, their longitude and latitude. To know this much, is to know that a certain space must intervene between them, but it is not to know what that space contains. The process of reasoning, in the discoveries which it gives, is like that topographic inquiry which slowly fills up the intervals of our map, placing here a forest, there a long extent of plains, and beyond them a still longer range of mountains, till we see, at last, innumerable objects connected with each other, in that space which before presented to us only a few points of mutual bearing. The extent of space, indeed, is still precisely the same, and Paris, Vienna, and London, are to each other what they were before. The only difference is, that we know what is contained, or a part, at least, of what is contained, in the long lines that connect them.

The reasoning which proceeds from the complex to the less complex, detecting, at each stage, some unsuspected element of our thought, may be termed strictly analytical reasoning,—the relation, involved in each separate proposition of the series, being simply, as we have seen, the relation of parts to the whole. It is exactly the same relation, however, which is felt, in reasonings that seem to proceed in an opposite way, exhibiting to us, not the whole first, and then some element of that whole, but first the elements, and then the whole which they compose. When we say, five and eight added together make thirteen, and when we say thirteen may be divided into eight and five, we express equally the comprehension of eight and five in thirteen, which is all that is felt by us in that particular proposition. Every synthesis, therefore, as much as its corresponding analysis, since one relation alone is developed at every step, implies the same elementary consideration of a whole and its parts; the difference being mere. ly in the order of the propositions, not in the nature of the feel. ing of relation, involved in any one of the separate propositions.

To this relation of comprehension, or the relation of a whole


and its parts, I have said, the other relations of coexistence, in all the propositions which express them, might, in strictness of analysis, be reduced,—even that relation of proportion which is of such importance in the reasonings of geometry and arithmetic ;-50 that every species of reasoning would be, in the strictest sense of the word, analytical, evolving only qualities essential to the very nature of the subjects of the different proportions. When, therefore, in developing one of the relations of proportion, I say, four

I are to five as sixteen to twenty, I state a relation of the number four, which may be regarded, as comprehended in my notion of that number, as any other quality is comprehended in any other subject.

It is one of the many properties of the number four, that when considered together with those other numbers, five, sixteen, twenty, it impresses us with a feeling of the relation of proportion, a feeling that its proportion to five is the same as the proportion of sixteen to twenty ; and it is a property, which, as soon as the relation is felt by us, it is impossible for us not to regard as essential to the number four,—as when we discover any new quality of a material substance, it is impossible for us not to adduce this quality, as another part, to our previous complex notion of the sum. We cannot, indeed, perceive this property of the number four, till we have considered it at the same time with the other numbers. But, as little can we know the physical qualities which form parts of our complex notion of any substance, till we have considered the substance together with other substances. For example, who could have predicted, on the mere sight of an alkaline solution, that, if mixed with oil, it would convert the oil into a soap, or, if added to a vegetable infusion, would change the colour of the infusion to green? We must have observed these mixtures, or, at least, have read or heard of the effects, before we could regard the changes as effects of the presence of the alkali,--that is to say, before we could include in our complex notion of the alkali, as a substance, the qualities of forming soap with oils, and of giving a peculiar tinge to vegetable infusions. But, having seen, or read, or heard of these effects, we feel that now, in our complex notion of the alkali, is included, as a part in its comprehending whole, the conception of these particular qualities. In like manner, the affinity of one metal to another, with which it admits of amalgamation, may be said to form a part of our complex notion of the metal; and it is the same with every other substance, the various properties of which, as soon as these properties are discovered by us, so as to admit of being stated to others, seem to us to be truly included in the notion of the substance itself, though before they could be so included, various other substances must have been considered at the same time. When, therefore, I say four are to five as sixteen to twenty, I state truly a property included in the number four,—the property, by which it affects us with a certain feeling of relation when considered together with certain other numbers,—though, for discovering the property originally, and for feeling it afterwards, it was necessary that the other numbers should be considered together with it; as, when I state that mercury admits of being amalgamated with other metals, I state a property included in my complex notion of mercury, though, for originally discovering the property, and for feeling it afterwards, I must have considered the mercury together with other metals, with which I state its readiness of entering into chemical union. When I consider the same number four togeth. er with other numbers, I discover various other relations, as when I endeavour to form new combinations of mercury, or of other chemical substances, I discover new relations, which I add to my complex notions of the substances themselves. As my original conception of mercury becomes more complex by all the new relations which I trace, so my original conception of the number four, which seemed at first a very simple one, becomes gradually more complex, by the detection of the various relations of proportion, which are truly comprehended in it as a subject of our thought,-as every new relation which I discover in a chemical substance, is comprehended in my widening conception of the substance itself, and the arithmetical or geometrical proportion, like the chemical quality, may thus strictly be reduced to the general class of the relations of comprehension.

In this way, every new proportion which is traced out, in a long series of such arithmetical or geometrical propositions, may be considered as the result of a mere analysis, by which elements existing before, but unsuspected, are evolved, as in the other spe. cies of reasoning, more obviously analytic. It is evident, indeed, that the statement of any property inherent in any subject, must,

in rigid accuracy of arrangement, be analytical. But, without insisting on so subtile a process, it may be easier, at least, though it should not be more accurate, to regard our reasonings of this kind, in the same manner as we formerly regarded our feelings of the simple relation of proportion, involved in each proposition of the reasoning, as forming a class apart; the reasonings we may call, in distinction from our more obvious analytic reasonings, proportional reasonings, as we termed the simple relative suggestions, which they involve, relations of proportion.

Whatever be the species of reasoning, however, it is necessary, that the propositions which form the reasoning, should follow each other in a certain order, for without this order, though each proposition might involve some little analysis, and consequently some little accession of knowledge, the knowledge thus acquired must be very limited. There could be no deduction of remote conclusions, by which the primary subject of a distant proposition might be shown, through a long succession of analyses, to have properties, which required all these various evolutions, before they could themselves he evolved to view. In the proportional reasonings of geometry, we know well, that the omission of a single proposition, or even a change of its place, might render apparently false, and almost inconceivable by us, a conclusion, which, but for such omission or change of place of a few words of the demonstration, we should have adopted instantly, with a feeling of the absolute impossibility of resisting its evidence.

How is it then, that, when order is so essential to discovery, the propositions which we form in our own silent reasoning, arrange themselves, as they rise in succession, in this necessary order; and what are we to think of that art, which, for so many ages, was held out, not so much as an auxiliary to reason, as with the still higher praise of being an instrument that might almost supply its place, by the possession of which, the acute and accurate might argue still more acutely and accurately, and imbecility itself become a champion worthy of encountering them; and though not perhaps the victor, at least not always the vanquished.

But to these subjects I must not proceed till my next Lecture.


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GENTLEMEN, after considering and classing our feelings of relation,--as they arise in any particular case, from the simple perception or conception of two or more objects, I proceeded in my last Lecture, to consider them, as they arise in those series which are denominated reasoning-series, that correspond, of course, with the division which we have made of the species of relations involved in the separate propositions that compose them; but of which the most important, are those which I termed analytical, as involving in every stage the consideration of a whole and its parts, or those which I termed proportional, as involving some common relation of intellectual measurement. To the former of these orders, indeed, the analyticalthe others might, as I stated to you, and endeavoured to prove, admit of being reduced; but as the process which reduces them all to this one great order, might seem too subtile, and could afford no additional advantage in our

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