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It may be the case that either or both of these increases does not take place. If P is a negative term, it may have no depth, and therefore adds nothing to the depth of S. If S is a particular term, it may have no breadth, and then adds nothing to the breadth of P. This latter case often occurs in metaphysics, and, on account of not-P as well as P being predicated of S, gives rise to an appearance of contradiction where there really is none; for, as a contradiction consists in giving to contradictory terms some breadth in common, it follows that, if the common subject of which they are predicated has no real breadth, there is only a verbal, and not a real contradiction. It is not really contradictory, for example, to say that a boundary is both within and without what it bounds. There is also another important case in which we may learn that “ S is P,” without thereby adding to the depth of S or the breadth of P. This is when, in the very same act by which we learn that S is P, we also learn that P was covertly contained in the previous depth of S, and that consequently S was a part of the previous breadth of P. In this case, P gains in extensive distinctness and S in comprehensive distinctness.

We are now in condition to examine Vorländer's objection to the inverse proportionality of extension and comprehension. He requires us to think away from an object all its qualities, but not, of course, by thinking it to be without those qualities, that is, by denying those qualities of it in thought. How then? Only by supposing ourselves to be ignorant whether it has qualities or not, that is, by diminishing the supposed information ; in which case, as we have seen, the depth can be diminished without increasing the breadth. In the same manner we can suppose ourselves to be ignorant whether any American but one exists, and so diminish the breadth without increasing the depth.

It is only by confusing a movement which is accompanied with a change of information with one which is not so, that people can confound generalization, induction, and abstraction. Generalization is an increase of breadth and a decrease of depth, without change of information. Induction is a certain increase of breadth without a change of depth, by an increase of believed information. Abstraction is a decrease of depth without any change of breadth, by a decrease of conceived information. Specification is commonly used (I should say unfortunately) for an increase of depth without any change of breadth, by an increase of asserted information. Supposition is used for the same process when there is only a conceived increase of information. Determination, for any increase of depth. Restriction, for any decrease of breadth ; but more particularly without change of depth, by a supposed decrease of information. Descent, for a decrease of breadth and increase of depth, without change of information. : Let us next consider the effect of the different kinds of reasoning upon the breadth, depth, and area of the two terms of the conclusion.

In the case of deductive reasoning it would be easy to show, were it necessary, that there is only an increase of the extensive distinctness of the major, and of the comprehensive distinctness of the minor, without any change in information. Of course, when the conclusion is negative or particular, even this may not be effected.

Induction requires more attention. Let us take the following example: –

S', S", S'", and ȘI have been taken at random from among the M's; S' S", S'", and S" are P: .. any M is P.

We have here, usually, an increase of information. M receives an increase of depth, P of breadth. There is, however, a difference between these two increases. A new predicate is actually added to M; one which may, it is true, have been covertly predicated of it before, but which is now actually brought to light. On the other hand, P is not yet found to apply to anything but S', S", S'', and S", but only to apply to whatever else may hereafter be found to be contained under M. The induction itself does not make known any such thing. Now take the following example of hypothesis :

M is, for instance, P', P", P"", and P" ;
S is P', P", P'', and P'v :
:: S is all that M is."

Here again there is an increase of information, if we suppose the premises to represent the state of information before the inferences. S receives an addition to its depth ; but only a potential one, since there is nothing to show that the M's have any common characters besides P', P", P", and P". M, on the other hand, receives an actual increase of breadth in S, although, perhaps, only a doubtful one. There is, therefore, this important difference between induction and hypothesis, that the former potentially increases the breadth of one term, and actually increases the depth of another, while the latter potentially increases the depth of one term, and actually increases the breadth of another.

Let us now consider reasoning from definition to definitum, and also the argument from enumeration. A defining proposition has a meaning. It is not, therefore, a merely identical proposition, but there is a difference between the definition and the definitum. According to the received doctrine, this difference consists wholly in the fact that the definition is distinct, while the definitum is confused. But I think that there is another difference. The definitum implies the character of being designated by a word, while the definition, previously to the formation of the word, does not. Thus, the definitum exceeds the definition in depth, although only verbally. In the same way, any unanalyzed notion carries with it a feeling, - a constitutional word, — which its analysis does not. If this be so, the definition is the predicate and the definitum the subject, of the defining proposition, and this last cannot be simply converted. In fact, the defining proposition affirms that whatever a certain name is applied to is supposed to have such and such characters; but it does not strictly follow from this, that whatever has such and such characters is actually called by that name, although it certainly might be so called. Hence, in reasoning from definition to definitum, there is a verbal increase of depth, and an actual increase of extensive distinctness (which is analogous to breadth). The increase of depth being merely verbal, there is no possibility of error in this procedure. Nevertheless, it seems to me proper, rather to consider this argument as a special modification of hypothesis than as a deduction, such as is reasoning from definitum to definition. A similar line of thought would show that, in the argument from enumeration, there is a verbal increase of breadth, and an actual increase of depth, or rather of comprehensive distinctness, and that therefore it is proper to consider this (as most logicians have done) as a kind of infallible induction. These species of hypothesis and induction are, in fact, merely hypotheses and inductions from the essential parts to the essential whole; this sort of reasoning from parts to whole being demonstrative. On the other hand, reasoning from the substantial parts to the substantial whole is not even a probable argument. No ultimate part of matter fills space, but it does not follow that no matter fills space.

Five hundred and eighty-eighth Meeting.

December 10, 1867. — ADJOURNED STATUTE MEETING. The PRESIDENT in the chair.

The Corresponding Secretary read letters relative to exchanges ; also a letter from Professor C. H. F. Peters, in acknowledgment of his election as an Associate Fellow of the Academy; and a letter from Henry S. Maine, in acknowledgment of his election as a Foreign Honorary Member.

The President called the attention of the Academy to the recent decease of Professor Bopp, of the Foreign Honorary Members.

On the motion of Dr. E. H. Clark, a committee was appointed to urge upon Congress the adoption of the decimal systems of weights and measures. Dr. E. H. Clark, Professor F. H. Storer, and Professor Horsford were appointed on this committee.

Professor F. H. Storer, the Librarian, moved that a committee be appointed to consider and report upon the meaning of the statutes which relate to the powers and duties of the Librarian and the Library Committee. In accordance with this motion the following committee was appointed: Professor Rogers, Mr. J. I. Bowditch, and Mr. 0. Pickering.

Mr. Roland G. Hazard, of Peace Dale, Rhode Island, was elected an Associate Fellow in Class III., Section 1.

Five hundred and eighty-ninth Meeting.

January 14, 1868. — MONTHLY MEETING. The CORRESPONDING SECRETARY in the chair.

The Corresponding Secretary announced the recent decease of Mr. Charles C. Jewett, of the Resident Fellows; of Rev. Chester Dewey, of Rochester, N: Y., of the Associate Fellows; and of M. Rayer, of Paris, of the Foreign Honorary Members.

Dr. C. T. Jackson exhibited to the Academy a series of lead-encased block-tin tubes or pipes for the conduction of VOL. VII.

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water, manufactured by the Boston Lead Company, Shaw and Willard's patent.

“ The first lot was made by placing two semicylinders of pure blocktin around a hollow core, or mandril, through which a current of cold water could be made to run, so as to keep the tin as cool as possible. The semicylinders of tin were also chilled in snow or pounded ice. When placed around the core, and held closely together by tongs, melted lead, at as low a temperature as it could be cast, was poured around the tin, and allowed to cool, the compound cylinder being eight inches long and six inches in diameter. Then, by means of a powerful hydraulic press, giving a pressure of 3,150 pounds per square inch, or between four and five hundred tons upon the ram or piston, the cylinder of lead and tin was forced out below, a mandril, or former, keeping the inner portion of the tube open and of a uniform size. When the thick cylinder was nearly driven out of its mould, the press was stopped, and a new casting was made upon the remaining metal ; so that continuous and perfectly united tubes of any desired length were formed, and were wound up on a reel in a room below. By this method, it was thought, a pipe lined with pure block-tin surrounded by the more flexible lead could be produced; but it has been found that pipes made in this way charged the water which stood in them overnight with lead.

“ Dr. Jackson was therefore called upon to make a series of analyses of the linings of many of these tubes, and discovered that the lining was composed of 22 per cent of lead and 88 per cent of tin, or an alloy. He was then invited to witness the operation of making the pipes, when every endeavor to prevent the penetration of the molten lead into the substance of the tin proved that it was impossible, so long as the molten lead was cast around the tin, — for the lead requires a temperature of 612° for its fusion, and tin only 442° ; while the great mass of hot lead rendered the tin almost a paste, into which the lead readily penetrated. There being 170° difference between the melting-points of the two metals, it was suggested that, if the lead was cast first around an iron core, and, when cold enough, on withdrawing the core and casting in the tin, the difficulty might be remedied. This, the foreman of the works said, could be easily effected, and it was soon after done with entire success; for chemical analysis showed the tin lining of such pipes to be absolutely pure tin. By this improved pro

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