Unknown quantity

“ But as a measure of quantity, such as the natural foot, fathom, or handful, which is continually varying in its own quantity, can never be an accurate measure of the quantity of other things; so a commodity which is itself continually varying in its own value, can never be an accurate measure of the value of other commodities. ”

“ Mathematics contain properly the doctrine of measure; and the object of this science is commonly said to be quantity; therefore quantity ought to be defined, what may be measured. Those who have defined quantity to be whatever is capable of more or less, have given too wide a notion of it, which, I apprehend, has led some persons to apply mathematical reasoning to subjects that do not admit of it. ”

“ It must be allowed, then, that whenever two elements are necessary to the supply, and enter into the composition of commodities, their value cannot depend exclusively upon one of them, except by accident, or when the other can be considered as a given or common quantity. ”

“ II.14 The quantity theory has been one of the most bitterly contested theories in economics, largely because the recognition of its truth or falsity affected powerful interests in commerce and politics. ”

“ We can not, therefore, reason respecting causation, without introducing considerations of quantity and extension at every step; and if the nature of the phenomena admits of our obtaining numerical data of sufficient accuracy, the laws of quantity become the grand instrument for calculating forward to an effect, or backward to a cause. That in all other sciences, as well as in geometry, questions of quality are scarcely ever independent of questions of quantity, may be seen from the most familiar phenomena. ”

“ If the quantity theory is not valid here, if the price-level is not absolutely fixed, helpless to change, with these assumptions, then the quantity theory, even as a minor tendency, must be surrendered, and the quantity theorist must admit that the whole line of thought has been fallacious. ”

“ We say that a man is tall, or heavy, or that the room is hot—qualitative statements; or we may say exactly how tall, or how heavy, or how hot—quantitative statements. The distinction between qualitative analysis and quantitative analysis in chemistry implies the same idea. Thus we may speak of a piece of wealth as having a definite quantity of value, or say that the value of the piece of wealth is a definite quantity. ”

“ Every day I am more satisfied that the former enquiry is vain and delusive, and the latter only the true object of the science. You say that my proposition, 'that with few exceptions the quantity of [Pg 176] labour employed on commodities determines the rate at which they will exchange for each other, is not well founded.' I acknowledge that it is not rigidly true, but I say that it is the nearest approximation to truth, as a rule for measuring relative value, of any I have ever heard. ”

“ The process of measure, instead of being only the wrong infinite of an endless progression, in the shape of an ever-recurrent recoil from quality to quantity, and from quantity to quality, is also the true infinity of coincidence with self in another. In measure, quality and quantity originally confront each other, like some and other. But quality is implicitly quantity, and conversely quantity is implicitly quality. ”

“ And now, as to the more universal meaning of this serial structure of our world of facts, we may note in passing still one further consideration. It is a commonplace that the exact sciences of Nature have owed their exactness, up to the present time, and in the main, to their quantitative treatment of facts. The logic of the Conception of Quantity has its own very complex problems; but thus much is clear: Any system of quantities, such as distances, times, masses, temperatures, pressures, is a serial order-system of facts, or is a complex of such serial order-systems. ”

“ I will add one consideration which will not have escaped those geographers who have studied the calculation of probabilities. In a series of observations made under the same circumstances, and especially by the same observers, the greater the number, the more probable is it that their amount will approach to the total quantity required. When there is no reason that the errors committed should be on one side rather than another, they mutually destroy each other, and the more so, the more the observations are multiplied. ”

“ Every expression of a Quantity consists of two factors or components. One of these is the name of a certain known quantity of the same kind as the quantity to be expressed, which is taken as a standard of reference. The other component is the number of times the standard is to be taken in order to make up the required quantity. The standard quantity is technically called the Unit, and the number is called the Numerical Value of the quantity. ”

“ In particular we observe that the probability that the energy of an unspecified system of a canonical ensemble lies between the limits and will be represented by the integral [2]
and that the average value in the ensemble of any quantity which only varies with the energy is given by the equation [3] where we may regard the constant as determined by the equation [4] In regard to the lower limit in these integrals, it will be observed that is equivalent to the condition that the value of is the least possible. ”

“ Though proportion be measurable in its own nature, and therefore hath proper quantity, yet as things cannot have proportion which have not quantity of some other kind, it follows that whatever has quantity must have it in one or other of these three kinds, extension, duration, or number. ”

“ Weights and measures.—Another important growth in the machinery of trade is found in standards of quantity,—weights and measures of every kind. It is scarcely possible to realize the uncertainty of exchange without exact weights and measures. ”

“ Size, on the other hand, and its alteration regarded merely as size, is for the qualitative determination, when it is unable to assert itself in terms of measure, an indifferent determination. That is to say, measure is quantity, precisely in so far as it can give to itself an aspect which qualifies it, and thereby a qualitative determination is united to the purely quantitative one. ”

“ It is not easy to say how many kinds of improper quantity may in time be introduced into mathematics, or to what new subjects measures may be applied; but this, I think, we may conclude, that there is no foundation in nature for, nor can any valuable end be served by, applying measure to anything but what has these two properties. ”

“ Their actual knowledge, however, and their scientific experience were necessarily limited, because in their days the records of human thought were only beginning to accumulate. It is probable that the first exact notions of quantity were founded on the consideration of number. It is by the help of numbers that concrete quantities are practically measured and calculated. ”

“ But if it comes to be discovered that variations of quality in any class of phenomena, correspond regularly to variations of quantity either in those same or in some other phenomena; [Pg 249] every formula of mathematics applicable to quantities which vary in that particular manner, becomes a mark of a corresponding general truth respecting the variations in quality which accompany them: and the science of quantity being (as far as any science can be) altogether deductive, the theory of that particular kind of qualities becomes, to this extent, deductive likewise. ”

“ There is a notion that political economy is entirely mathematical; and its negative quantity is strongly recommended for study: it contains "the whole of the Funds, Credit, 32 parts out of 33 of the value of Land...." The mathematics are described as consisting of—first, number, or Arithmetic; secondly, the theory of dependent quantities, subdivided into dependence by cause and effect, and dependence by simultaneous variations; thirdly, "independent quantities or unconnected events, which is the theory of probabilities." ”

“ Consequently, the value of the variable quantity of produce which, under different circumstances, forms the wages of a given number of men, being composed of the values of the two elements, labour and profits, varying as above described, must be constant, and may therefore, with propriety, be proposed as a standard measure. ”

“ From the nature of quantity we may see what it is that gives mathematics such advantage over other sciences in clearness and certainty; namely, that quantity admits of a much greater variety of relations than any other subject of human reasoning; and, at the same time, every relation or proportion of quantities may, by the help of lines and numbers, be so distinctly defined as to be easily distinguished from all others, without any danger of mistake. ”

“ Changes in the price-level are not one result, but rather, are the mathematician's average of many changes, due to a host of causes, in many individual prices. The quantity theory is an effort to simplify phenomena highly complex. Of course, the simplification of complex phenomena in thought is a laudable scientific goal, but when the simplification goes so far as to group things only superficially related, and to leave out the really vital elements, it is worthless. ”

“ Yet we must here exercise great caution. One function may approach indefinitely near to another function, while some quantity determined by the first does not approach the corresponding quantity determined by the second. ”

“ The theory of discriminant functions gives a very simple expression for this probability, which is obtained by integrating the product of the differential of the quantity of which the result deduced from a great number of observations varies from the truth by a constant less than unity, dependent upon the nature of the problem, and raised to a power whose exponent is the ratio of the square of this variation to the number of observations. ”

“ The reason we should consider it uncommonly cold up there is that fewer molecules would strike us. Quantity, therefore, in our estimation replaces quality,—a possible substitution which also accounts for some reputations, literary or otherwise. ”

“ X. In general it is clear that, if the value of A increases in a duplicate ratio, yet if the value of B increases in a triplicate ratio, so far from commanding a greater quantity of B, A shall command a smaller quantity; and if A continually goes on squaring its former value, yet if B continually goes on cubing its former value, then, though A will continually augment in value, yet the quantity which it will command of B shall be continually less, until at length it shall become practically equal to nothing. ”

“ He is thus led to recognize a classification of quantities on a new principle, according to which the physical nature of the quantity is subordinated to its mathematical form. ”

“ At first, no doubt, looking at the nature of quantity as an indifferent and external character of Being, we are disposed to answer these questions in the negative. And yet, as we must admit, this indifferent increase and diminution has its limit: a point is finally reached, where a single additional grain makes a heap of wheat; and the bald-tail is produced, if we continue plucking out single hairs. ”

“ Strictly speaking, only the things which I have mentioned belong to the category of quantity: everything else that is called quantitative is a quantity in a secondary sense. It is because we have in mind some one of these quantities, properly so called, that we apply quantitative terms to other things. We speak of what is white as large, because the surface over which the white extends is large; we speak of an action or a process as lengthy, because the time covered is long ”

“ Despite the number of changing factors affecting the methods of exchange and the amount of business, the quantity theory is a rule unable at any moment. These various factors change slowly, and the quantity theory answers the question: What general change occurs in prices as a result of the increase or decrease of the money in a given community at a given moment? ”

“ Thus, the total numbers receiving poor-law relief, or the numbers apprehended in the kingdom for drunkenness, are numerical results affected by the oscillations of trade, by the character of the seasons, the value of gold, etc., etc., as well as by the acts of the Legislature. To make a valid experiment we must have a certain thing subject to certain constant conditions, and we must introduce a single definite change of condition, which will then be probably the cause of whatever phenomenon follows. ”

“ We offer to produce cases in which from double value it shall not be lawful to infer double quantity. We offer to produce cases in which from double quantity it shall not be lawful to infer double value. And thence we argue, that until the value is discovered in some other way, it will be impossible to discover whether it be high or low from any consideration of the quantity commanded; and again, with respect to the quantity commanded—that, until known in some other way, it shall never be known from any consideration of the value commanding. ”

“ If all mathematicians had considered this as carefully as Sir Isaac Newton appears to have done, some labour had been saved both to themselves and to their readers. That great man, whose clear and comprehensive understanding appears even in his definitions, having frequent occasion to treat of such improper quantities, never fails to define them, so as to give a measure of them, either in proper quantities, or in such as had a known measure. ”

“ When we consider farther, that value itself is relative, and that two things are necessary to constitute it, independently of the third thing which is to measure it; we may define a Measure of Value to be something, by comparing with which any two other things, we may infer their value in relation to one another.
In this sense, any commodity will serve as a measure of value at a given time and place; since we can always infer the proportion in which things exchange for one another, when we know the proportion in which each exchanges for any third thing. ”