This equation is satisfied (apparently) by making μ = 0, A =

The case is, however, of too doubtful a character to warrant us in adopting the conclusion. One thing alone I infer from it, that if any power of the distance (not of r) be assumed as the factor, it must be the inverse square. It would require that we should retrace our steps, and investigate the different formulæ corresponding to this hypothesis, before we could speak positively on the subject. I have only to add to this discussion on the probable coefficient of vibration, that an approximation has been made use of in the value of the distance between the disturbing and disturbed points, as it appears within the circular function. The approximation amounts in fact to supposing the wave elliptical, instead of circular. In the second problem I find that the square of this distance, being substituted within the circular function for the distance itself, leads to precisely the conclusions we have obtained. It is possible, therefore, that the omission of our factor, and the approximation made use of within the circular function, exactly counterbalance each other.

I cannot conclude without repeating my conviction of the importance of results such as those which Professor Forbes has just announced. It appears that the effect of scratching a piece of rock salt, &c. is to alter its power of transmitting heat in such a manner, that heat of a low temperature, or dark heat, is transmitted in greater proportions than before. If

then the two kinds of heat correspond, the one to vibrations, or transmission due to vibrations; the other to transmission due to excess of elasticity, our analysis teaches us to expect that the quantity of the former kind stopped by the wires or scratches should be in exact proportion to the space covered by them, whilst we should hardly expect to find any considerable stoppage effected on the latter. Thus I am led to hope that the Theory which I proposed in the Transactions of the Cambridge Philosophical Society, Vol. vi., pp. 274, and seq. and subsequently developed in my little work on the subject, will be strengthened in some points, although I am far from expecting that it will be confirmed in all. Perhaps subsequent results may render it necessary to modify our hypotheses, but at present I do not know that experiment is very far in advance of theory. I cannot conclude without expressing my conviction that the masterly researches of Professor Forbes will have the effect of setting right several errors even in the Theory of Light, which have crept in from the difficulty of subjecting that branch of philosophy to strict measurement.

EDINBURGH,

Jan. 23, 1840.

P. KELLAND.

X. On the Foundation of Algebra. By AUGUSTUS DE MORGAN, F.R.A.S. F.C.P.S.; of Trinity College; Professor of Mathematics in University College, London.

[Read Dec. 9, 1839.]

THE extent to which explanation of the meaning of the symbolical results of Algebra has been carried within the last half century; the complete interpretation of all which formerly appeared incongruous; the separation, as it was called, of the symbols of operation and quantity, which amounts to the use of an algebra in which the symbols represent something more than simple magnitude ;-will for some time to come suggest inquiry into the logic of this many-handled instrument of reasoning, which seems to be capable of presenting, under fixed laws of operation, all the results which arise from very distinct primary conceptions as to the things operated upon.

When several different hypotheses lead to results which admit of a common mode of expression, we are naturally led to look for something which the hypotheses have in common, and upon which the sameness of the method of expression depends. A comparison of the properties of the ellipse and hyperbola would bewilder the imagination, under any of the distinct definitions which might be given of the two curves; nor would the mind rest satisfied until it had discovered the reason of the similarity which exists between these properties.

Algebra now consists of two parts, the technical, and the logical. Technical algebra is the art of using symbols under regulations which, when this part of the subject is considered independently of the other, are prescribed as the definitions of the symbols. Logical algebra is the science which investigates the method of giving meaning to the primary symbols, and of interpreting all subsequent symbolic results. It is desirable that the word definition should not enter in two distinct senses, and I should propose to retain

it as used in the art of algebra, applying the terms explanation and interpretation to denote the preparatory and terminal processes of the science. Thus a symbol is defined when such rules are laid down for its use as will enable us to accept or reject any proposed transformation of it, or by means of it. A simple symbol is explained when such a meaning is given to it as will enable us to accept or reject the application of its definition, as a consequence of that meaning: and a compound symbol is interpreted, when, having occurred as a result of explained elements, used under prescribed definitions, a necessary meaning can be given to it; the necessity arising from the tacit supposition that the compound symbol, considered as a new simple one, must still be subject to the prescribed definitions, when it subsequently comes in contact with other symbols. The last words may need the remark, that though we sometimes appear to interpret a symbol merely for the purpose of explaining a result, ye we know that such interpretation would be subsequently rejected, if the use of the symbol, under the prescribed definitions, were not found to be logically admissible.

A symbol is not the representation of an external object absolutely, but of a state of the mind in regard to that object; of a conception formed, for the formation of which the mind knows that it is or was indebted to the presence, bodily or ideal, of the object. Those who do not remember this, the real use of a symbol, are apt to dogmatize*, declaring one or another explanation of a symbol, that is, the signification by it of one or another impression produced on their own minds, to be real, true, natural, or necessary it being neither one nor the other, except with reference to the particular mind in question. To take a very simple case, and one which bears upon our subject, let us imagine that we form successively a conception of the absence of all definite magnitude, followed by one of the existence of a certain magnitude, say a line of given length. The mind of one person may pass from the one to the other by imagining the given length to be instantaneously generated, no one portion of it coming into the thoughts before or after another; that of a second may make the transition by imagining a point to move from one extremity to the other: while that of a third may dwell rather on the relative position of the two extremities, and may think

* Of course, I use this word in its primitive sense, without any censure implied: the very sentence in which the word occurs is, and is meant to be, dogmatical.