e. g. for thirty their points were brought together and stretched forward; for fifty the thumb was bent like the Greek г and brought against the ball of the index. The same set of signs, if executed with the thumb and index of the right hand, meant hundreds instead of tens, and the unit signs, if performed on the right hand, meant thousands.'

If we compare this gesticular system with others, articulate or written, which are familiar to us, we see that the artifices of expression resorted to are in all of them as nearly identical as the different natures of the medium of expression allow. Be the symbols articulate sounds, gestures, or written marks, they are in all three cases subjected to analogous operations which are symbolic of the conceptual process. Apart from the meaning assigned to each symbol as a separate entity, symbolic operation, i.e. the putting of the symbols into context, temporal or spatial, determines by rule or custom the meaning of that symbol in that particular context. In the naming of numbers, as compared with the naming of other conceptions, the distinguishing trait, as we have already remarked, is the fixed serial association of the names in memory. But, comparing the name-system with other symbolic systems in aid of conceiving and expressing numbers, it is seen that serial association of the symbols is not a peculiarity of the name-system; it is characteristic of them all. Obviously, neither arithmetical symbols nor the system of finger symbols above described would retain their significance apart from their being remembered as a series. This is not inconsistent with the opinion that the initial conceiving of numbers as identities in aggregation is independent of the conception of fixed serial order, either of things in space or of acts in time; it is as an effective aid in establishing these identities that the conception of fixed serial order is indispensable. A series of number-symbols is thus, one might say, a reference series of individuals, each of which has a mark which assigns to it its place in the series, only the individuals have vanished and left their place-marks behind. A number-symbol may thus be considered and intended, according to the purpose of our thought, either as the symbol of an identity in aggregation without regard to order, or as that of a place in a serial order of places, temporal or spatial.

In the exposition of mathematical thought the terms Number,

Quantity, Magnitude, and Measure, meet us at every turn. But while, in applied mathematics, writers who avoid looseness of terminology are careful to indicate, either by definition or by clear implication and example, the precise meaning which they attach to these terms, in pure mathematics it is a common if not invariable custom for writers to use these terms loosely, without any clear intimation of the shades of meaning intended, if any are intended. Thus, in textbooks of Algebra, the term quantity' is often suddenly introduced, without a word of explanation, into expressions where it is apparently intended to replace the word 'number' as a synonym. But as neither in the ordinary nor in the philosophical use of these terms are they customarily intended as synonyms, we can feel no security, in the absence of explanation, as to the writer's real intention. Is there confusion between two distinct though closely related conceptions, or is there merely an arbitrary use of the two terms as synonyms? For, at first sight, there seems to be no choice but between these alternatives.

Let us begin by taking a mathematician's own view of the distinction between number and quantity, expressed in a perfectly simple and clear manner. The distinction is drawn in the course of a careful explanation of the mathematical theory of measurement, as follows:

'NUMBER and QUANTITY. When the unit is stated the magnitude of the object is precisely determined by its measure in terms of the unit, and this measure is always a number. The "object" may be anything which we can think of as measurable in respect of any property, and the phrase "magnitude of an object" is thus co-extensive in meaning with the word "quantity". The quantity does not change when the unit chosen to measure it changes, and the quantity is not identical with the number expressing it.

A number can express a quantity only when the unit of measurement is stated or understood. When the unit is stated or implied the number expresses the quantity.'1

In this explanatory statement the terms Number, Quantity, Magnitude, and Measure are all involved, and the explanation makes the conceptual distinctions intended in the use of these terms as clear as any one could wish, though perhaps it would

1 Theoretical Mechanics, by A. E. H. Love, M.A., F.R.S. (Univ. Press, Camb.), p. 372.

have been better to have substituted for 'expressing' and express', in the first and second paragraphs respectively, the words 'measuring' and 'measure' in accordance with the definition of the measure of a quantity as being always a number. The distinction is again quite clearly marked in the two paragraphs which follow immediately upon those already quoted:

'Mathematical equations, and inequalities, are relations between numbers, expressing that a certain number which has been arrived at in one way is equal to, greater than, or less than, a certain number which has been arrived at in another way.

'Mathematical equations, and inequalities, between numbers expressing quantities are valid expressions of relations between the quantities only if they hold good for all systems of units.'

The contrast between these two paragraphs clearly exhibits the distinction between the science of number and the application of that science to theorems and problems involving the measurement of quantities. In the former the subject of reasoning is number, in the latter it is both number and quantity. From this point of view it is therefore pertinent to inquire why the term 'quantity', which is intended to express a meaning clearly distinct from that of the term 'number', should be used in the science of number without clear indication of the sense in which it is to be taken.

J. B. Stallo, the author of Concepts of Modern Physics, makes the following remarks (p. 265 of that work) on this apparent incongruity in the terminology of pure mathematics :

'The error respecting the true nature and function of arithmetic and algebraic quantities has become next to ineradicable by reason of the inveterate use of the word quantity ” for the purpose of designating indiscriminately both extended objects or forms of extension and the abstract numerical units or aggregates by means of which their metrical relations are determined. The effect of this indiscriminate use is another illustration of the well-known fact in the history of cognition that words react powerfully on the thoughts of men, and by this reaction become productive of incalculable error and confusion. It is not to be expected, of course, that mathematicians will cease, at this late day, to speak of arithmetical and algebraic symbols as quantities"; but there may be some hope for the suggestion that they might return to the old phrase "geometrical (and other) magnitudes". The mischief lies, not so much in the use of a particular word, as in the employment of the same word

for the denotation of objects differing from each other toto genere.'

This criticism has an obvious bearing on the question: why do algebraists regard a generalized symbol of number as also a symbol of abstract quantity? By implication, if not directly, Stallo condemns the practice as leading to error or confusion, for there is in this respect no difference between a word and a mathematical symbol. I may say, in passing, that I do not very well see how the confusion-if there is any-is to be avoided by the substitution which Stallo recommends; but this is probably because I cannot find any evidence that this custom really does give rise to confusion of thought. The fact is, I believe, that this practice, common to all algebraists, results from a tacit process of thought or judgement which needs only to be made explicit in order to afford a justification or explanation of the habit. A number expresses or measures a quantity only, as Professor Love says, when the unit of measurement is stated or understood; and the quantity, i. e. magnitude of the object, is not identical with the number expressing it because this number changes with any change in the unit of measurement. But this distinction becomes quite indefinite when abstraction is made of any measurable 'object', and we think merely of quantity in the abstract or quantitative relation in general as expressed by number. If it is admissible to regard numerical relation as expressive of quantitative relation in general, then the numerical unit becomes identical with the unit of abstract quantity, and generalized symbols of number can properly be considered as also symbols of abstract quantity.

CHAPTER V

SCOPE AND CHARACTER OF THE ENSUING DISCUSSION

WHAT are we to think of, how shall we characterize, a mental process which might, briefly and in general terms, be indicated thus: Explanation of the derivation, from a primary conception (say that of Quantity, or again, of Space), of another conception, followed by questions such as these: What is, or What is the nature of, this derived conception? Or-say that Abracadabra is the name given to this derived conceptionWhat is the meaning of Abracadabra? The questions imply that the nature of a derived conception is not made manifest in the account of its derivation, that the meaning of a term may be something other than that which we have agreed to assign to it. Yet if a derived conception really is present to the mind, to ask what this conception is can be only an indirect way of asking for an explanation of the process by which we have come to form it, that is, of its derivation; and, if we have given the name Abracadabra to a certain idea, to ask, What is Abracadabra? is to imply that the meaning of this term is not that idea. But select what term you please to characterize a mental process such as this-I can find none more appropriate than 'mystical'-and it will probably seem to become inappropriate when you learn further that a notion which is derived, rather than that from which it derives, is the fundamental notion of the subject of thought which involves them both.

This is not a fancy picture, or a perverse interpretation, but a genuine impression, conveyed in general terms, of an important part of Cayley's presidential address to the British Association in 1883. That Cayley was himself not altogether satisfied with the explanation which he there gave of the doctrine of Imaginaries in mathematics, he made plain by remarking in the course of it that, so far as he knew, the subject had never yet been adequately discussed, and stood in need of a philosophical foundation or justification.

We are supposed, in the calculus, and by means of its symbolism,