The Common Sense of Political Economy

Wicksteed, Philip H.
(1844-1927)
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1910
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London: Macmillan and Co.
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1910
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CHAPTER II

ON THE DIAGRAMMATIC METHOD OF REPRESENTING AREAS OF SATISFACTION AND MARGINAL SIGNIFICANCES*13

II.2.0

Summary.—The method of representing economic phenomena by curves demands closer examination than we have yet given it, and turns out on inspection to present many problems both of interpretation and construction. The measurements on the axis of Y indicate limiting rates of marginal significance, and, while expressed in an objective rate-unit, they must ultimately rest on estimates based on psychic experience. Hence difficulties arise as to the relation between objective and psychic units, the possibility of keeping that relation stable, the meaning we are to attach to accuracy of estimate and the conditions which limit that accuracy. If we express the data of Book I. Chapter II. as to the significance of tea in the form of a tea curve we are led to examine (a) the implications of the special formula to which our data conformed, and (b) the possibility of any simple mathematical formula approximately representing the facts. An attempt accurately to interpret the curve further leads us to distinguish between a curve of total satisfaction and marginal significance on the one hand, and a curve of price-and-quantity-purchased on the other hand. We find that these curves can, at best, only coincide approximately, and that an individual curve purporting to represent both series of phenomena can theoretically only be a "temperamental" compromise.

II.2.1

In the preceding chapter I have represented satisfactions by areas bounded by curves, though with the express reservation that this procedure raised questions and required explanations upon which it was not convenient to enter at the time. We will now proceed to a more careful examination of this method. We shall frequently employ it hereafter.

II.2.2

The representation of a given satisfaction by an area of any kind, whether rectilinear or curvilinear, involves by implication the conception of a unit to which different satisfactions can be reduced, and in which they can be expressed for diagrammatic comparison with each other. And though this idea is far from familiar and presents great difficulties when first expressly suggested to the mind, we have nevertheless seen that it is directly implied in all our practical dealings and deliberations; and it underlies all the investigations upon which we have hitherto been engaged. For to say that two things are of equal value to us, and that another thing is just as valuable to us as both of them put together, is to say that the latter is worth twice as much to us as either one of the former, or that we anticipate a satisfaction twice as great from the one as from either of the others. If we say that a thing is just worth a penny, we are thereby equating the satisfaction we expect it to yield with all the other satisfactions which we believe a penny would secure at the margins of other branches of expenditure, and if we went on to say that something else was worth exactly three shillings and not a penny more, we should be saying that we expect it to yield as large a satisfaction as any thirty-six things we could get for a penny each, or a satisfaction thirty-six times as large as that which any one thing just worth a penny is expected to yield. Now it is quite true that such estimates are often vague, and almost casual, and that they are subject to every kind of fluctuation and inconsistency; but every deliberate act of choice, or of administration of resources, is an attempt to make them more precise and consistent; and even an impulsive choice is a declaration that at any rate one thing is more valued by us than another, and this involves an act of quantitative comparison. Such as they are, these choices, impulsive or deliberate, are verdicts as to comparative volumes of satisfaction, considered as magnitudes, and they often express themselves in units of pence and shillings.

II.2.3

Now all commodities, services, or opportunities that enter into the circle of exchange are ultimately estimated not as physical or objectively measurable magnitudes, but as sources of anticipated satisfaction; and we frequently estimate things that are not in the circle of exchange in terms of things that are, and constantly choose between things that are and things that are not in this circle, weighing them against each other. Thus it is clear that for each one of us, at any given moment, the ordinary conduct of life unmistakably implies and involves the conception of satisfactions as magnitudes, and therefore as expressible ideally in units, which may be represented diagrammatically by unit lines, or areas, or otherwise, as suits our convenience. And just as, in measuring and comparing lengths with a view to determining their relative magnitudes, it does not matter whether our unit is an inch, a metre, or a mile (the difference being only in the numerical expression of the results obtained, not in the results themselves), so it is of no consequence whether we take our unit of satisfaction as that represented by 1d. or that represented by £1. But in comparing different satisfactions, expressed as areas, we must always remember that to be comparable as magnitudes the satisfactions must be estimated by the same person. With these reservations we may now proceed to the diagrammatic representation of the estimates dealt with in the second chapter of Book I. and generally to the interpretation of curves of total and marginal satisfaction.

II.2.4

We may take (arbitrarily) a small square on the ruled paper of Figure 9.  Click to enlarge in new window.Fig. 9 to represent one-quarter of the satisfaction anticipated from the expenditure of a farthing. Then four squares will represent the satisfaction corresponding to a farthing, sixteen squares that corresponding to a penny, and 12 × 16 = 192 that corresponding to a shilling. Any rectangular or curvilinear area, irrespective of its shape, if equal to 192 small squares would then represent this shilling volume of satisfaction. It might, for instance, be a rectangle with a base of 1 and an altitude of 192, or one with a base of 16 and an altitude of 12.

II.2.5

Taking a side of a small square as our linear unit, let us now agree that the unit length (not area) measured along any base line shall represent a periodic (monthly or as otherwise defined) supply of one ounce of tea, and a base of 16 such units a supply of one pound. We can now represent diagrammatically any of the data as to tea which we assumed in Book I. Chapter II. For instance, the fourth pound was expected to yield a satisfaction equal to the significance of 8s. in any other application. This would be represented by an area of 8 × 192; and as we have agreed that a basis of 16 shall represent a pound, a rectangle of base 16 and altitude 8 × 12 (=96) will be the proper representation of the satisfaction anticipated from the consumption of the fourth pound per month (Fig. 9 (a)). But of this fourth pound we saw that the first half was estimated at 4s. 5¼d. and the second at 3s. 6¾d. These values would be represented respectively by rectangular areas containing 852 and 684 small squares, and since the basis of each would, by our convention, be 8 (corresponding to ½ lb.), their altitudes would be respectively 106½ and 85½ (Fig. 9 (b)). We can now interpret units of altitude. They will not signify positive quantities, as the units of the base do, but penny rates of satisfaction per pound of the commodity, or halfpenny rates of satisfaction per half-pound, and so forth.

II.2.6

Now, taking ad in Fig. 9 (b) at an altitude of 96 as in Fig. 9 (a), it is obvious that the rectangle ab, which is added to the original rectangle at the left, is equal to cd subtracted from it at the right, since the total area of the two differentiated rectangles is to be exactly equal to that of the integral rectangle that represents the satisfaction yielded by the whole pound; and we may suppose that this differentiation between half-pounds, quarter-pounds (or any other fractions, for it is not necessary to proceed by bisection of a pound rather than trisection, for instance), may be carried as far as we choose. The area of any succession of differentiated rectangles will always remain equal to that of the integral areas that present them collectively as a single magnitude. In Figure 10.  Click to enlarge in new window.Fig. 10 let us carry out this process to different degrees of advancement for the different pounds; and let us draw a curve such that in the case of the small and the large rectangles alike it always adds on an area to the left equal to that which it cuts off to the right, so that for any base the area bounded above by the curve shall be exactly equal to the rectangle standing on the same base. Such a curve may be regarded as integrating any number of contiguous rectangles which we choose to take in succession. That is to say, the area intercepted by the curve above any line measured along OX will be exactly identical with the area contained in the whole series of rectangles standing upon the same base.

II.2.7

This is a curve of total satisfaction, and its meaning is now obvious. We have seen that ideally, and in the limit, the significance of any commodity is a magnitude continuously changing as we recede from the origin, so that, however small the increment we are considering, the change cannot be regarded as suspended during the progress of its consumption. The whole process, then, ideally considered, is properly represented not by a series of steps or discrete areas, however small, but by a curve-bounded space. Such a curve, could we obtain it, would give us at a single view the whole infinity of facts to be registered. If we take any portion of the weekly, monthly, or other periodic supply of a given commodity (whatever our conventional units may be), e.g. the third unit, or the quarter of a unit between 7 1/3 and 7 7/12, or generally the portion represented by the line ab on the axis of X (Figure 11.  Click to enlarge in new window.Fig. 11), then the curve is constructed so as to bound an area, ap1p2b, exactly representing the satisfaction anticipated from the consumption of the portion of the commodity represented by ab. And note that whereas in Book I. Chapter II. we directly assumed data as to pounds and binal fractions of pounds only, a curve assumes that we have all conceivable data, and can begin and end anywhere we like.

II.2.8

This continuity and entire accuracy of data is, of course, purely ideal. We may conceive approximations to it, but to imagine that any one could distinguish between the rate at which tea was ministering to his satisfaction at the beginning and at the close of his consumption, say, of the 7.9432th pound, and could express this difference in fractions of a shilling-per-pound rate, is an absurdity. Indeed the reader who has some tincture of mathematical culture will perceive that even an underlying assumption of commensurability between the satisfactions accruing from successive conventional units of the commodity and those represented by the conventional units of the currency is inconsistent with ideal accuracy. These reflections reveal at once the great convenience and the ingrained artificiality of the method of representing economic quantities by curves. The very nature of a curve is incompatible with the nature of the phenomena we are investigating; but it is of high value as an ideal simplification, and as a means of mentally arresting phenomena, which in their actual existence are unmanageably complex and fluctuating. If we professed in our diagrams to present possible or actual facts, we should have to undertake the hopeless task of determining in each case what degree of accuracy might reasonably be assumed; whereas by frankly presenting the unattainable limit in every case we declare at once the ideal nature of our hypothesis and of our representation of it.

II.2.9

This being understood, the reader will have no difficulty (if he turns back to our investigations as to "limiting rates" on pages 60 sqq.) in recognising the height ap1 of the curve above any point a as the graphic representation of the limiting rate of significance (in whatever unit measured) of increments or decrements of the commodity taken from the point a. For on considering the errors (p3fp1 and p1gp2 respectively) that would be involved in treating the areas above ab and ac as equal to each other and to the rectangle on base ab (or ac) and with altitude ap1, we shall find that they become smaller not only absolutely but proportionally to the areas themselves as we make the increments ab and ac smaller; and this without limit. For if we halve the lengths ab and ac and erect perpendiculars on them and then compare the rectangles on these bases, and with altitude ap1, with the areas above the bases bounded by the curve, we shall see that the error involved in treating them as equal is in each case less than half of the corresponding error for the wider basis. The proportion of error, therefore, decreases, without limit, as smaller bases are taken. Thus the height ap1 represents a rate of satisfaction per unit to which no increment or decrement taken from the point a ever conforms, as a whole, but which always lies between the rates proper to any given increment and to the corresponding equal decrement, and to which those rates approximate without limit as they decrease in magnitude. The units on OY, therefore, measure limiting rates of the significance of units of the commodity (per unit of time) as the increments are taken smaller. Or, in abbreviated terminology, the ordinates represent the marginal significance of the commodity for any given supply. So, too, in Fig. 10 the areas p1cd and p2ed respectively will be not only smaller, but smaller in proportion to the rectangles da and db as c or e approaches d.

II.2.10

We have now a provisional conception of what a curve of marginal significance would mean if we had it, and we may go on to the examination of the bearing upon the determination of the form of such a curve of any data we may suppose ourselves actually to command. Let us rule our paper, as in Fig. 10, so as to mark rectangles of base 16 and altitude 12. Returning to our example of tea, we may retain the significance of all our units, and for convenience may register successive pounds (each pound being 16 ounce-units) of supply along OX, and successive shilling-per-pound rates of significance (each being 12 times a penny-per-pound rate) along OY. Each large rectangle, containing 192 small squares, will indicate, as before, the area of satisfaction represented by a shilling.

II.2.11

It is obvious, to begin with, that any datum we may be able to obtain will give us some information as to the course of the curve. If we know, for instance, that the fourth pound of tea yields an area of satisfaction valued at 8s., we shall know that the curve must be such that the area ap1p2b equals the area ac, and the area p1df equals the area fcp2. (We shall express compliance with this condition by saying that the curve "satisfies the datum" of the area ac.) But there is an infinite number of curves that would fulfil this condition. Some of them might bisect dc, and others might cut it at points indefinitely near to d or c, and they might intersect the verticals from a and b at any variety of points. But if we have the additional data that the first half of the Figure 12.  Click to enlarge in new window.fourth pound corresponds to the area ag, and the second to the area bh, many of these possibilities will be excluded, for the area which the curve adds to ag must equal the area it cuts off from it; and the same must hold for bh. The course of the curve, therefore, will be more closely determined by the two rectangles ag and bh than by the one rectangle ac, which is equal to their sum. In our original hypothesis we supposed the estimates of successive pounds of tea to reveal an easily detected law which enabled us at once to calculate any smaller areas we liked to choose. This formula would absolutely determine the form of the curve, and tracing it would only be a matter of calculation. But if we assume no such property, and imagine each datum to stand alone and not to involve any derivative data (assuming only the general property of continuous decline, after a certain point, which we may take as fixed by the nature of our inquiry), then it is clear that the minuter the increments for which we can obtain estimates, the more closely can we determine the course of the curve. For instance, we have set out on Fig. 10 (page 444) a series of data as to pounds, half-pounds, etc., and we see that, so far as they shew (that is to say, apart from our knowledge that our formula would enable us to split up the larger rectangles as finely as we choose), there would be room to suppose that the curve undulated with considerable violence over the portion corresponding to the increment from 4 lbs. to 7 lbs., but that our data enable us to assert a more regular course for the portion corresponding to the increment from 2 lbs. 12 oz. to 3 lbs. 4 oz. Seeing then that if we have given any two contiguous rectangles of satisfaction, akgm and mhnb, the curve must always pass between the points g and h, it follows that if we could determine the areas corresponding to indefinitely small increments we could determine the position of the curve at any part of its course within indefinitely narrow limits; for just as we determine a point absolutely if we can determine any position we choose of points, that approach each other without limit, between which it lies,*14 so we can determine a curve absolutely if we can determine, as closely as we like, two mutually approximating points between which the value of y, corresponding to any given value of x, lies.

II.2.12

But here it will be well, for our security, to establish the fact that whereas (as we have just seen) a curve may satisfy the datum of a certain area, but may fail to satisfy the data of two smaller areas into which it can be broken up, it is not possible for it to satisfy the data of two adjacent areas, severally, without also satisfying the data of the total area which is their sum. The general proof of this proposition, to which we will now proceed, applies to all the different forms of curve shewn in Fig. 13.

II.2.13

Figure 13.  Click to enlarge in new window.We start with the two rectangles ah and ks and construct a curve, enofpg, such that it adds and subtracts equally from each of the two rectangles. The equal areas we mark by oblique or horizontal lines respectively. There are, of course, an indefinite number of such curves; but if we construct an integrating rectangle, ac, by drawing a line, bc, that makes the rectangles bh and rc equal, the area which the curve enofpg cuts off from the rectangle ac will be equal to that which it adds to it—that is to say, the area ebf will equal the area gcf. Since we have emn = nho we may substitute the latter for the former, and we shall have ebf = bmhof. Again, since we have bh = rc, we can obtain by substitution bmhof = scfor. And since we have rop = psg, we can again obtain by substitution scfor = gcf. Therefore we shall have ebf = gcf. Q.E.D.*15

II.2.14

Thus, if we have any series of rectangles arranged as in Fig. 10, on bases measured continuously along OX, a curve which adds to and cuts off equally from any contiguous pair of these rectangles, severally, will have the same property with respect to the integrating rectangle that is equal to their sum. The rectangle so obtained may then be substituted for the two rectangles of which it is the sum, and we may again integrate it with another rectangle, still relying on the same result, so that the curve will always add and subtract equally from the area of the integrating rectangle that sums any number of contiguous areas with the data of which the curve complies severally.

II.2.15

It is evident, therefore, that since we can always rely on the curve's retaining its fundamental property when we add together the data on which we build it, but never when we subdivide them, the accuracy with which we can determine it will depend on the accuracy and the fineness of the data on which we can construct it.

II.2.16

To what degree of approximation, then, can we hope actually to determine such a curve? Or, rather (since the question so put hardly admits of a definite answer), what are the principles which will determine the degree of approximation to an ideal curve that may be realised in any particular case? In the first place, let us consider the question of accuracy. In the case of the tea curve, for instance, we have to ask what will determine or influence the limits within which we can reasonably suppose our housekeeper's estimates to be exact. But on the very threshold of this inquiry we are met by a grave difficulty. What do we mean by accuracy of estimate? If we are speaking of the estimate a man forms of the length of a stick, for example, or the height of a top-hat, we are speaking of something which can be tested by actual measurement. Thus if we say that a man can be trusted to judge a yard to within a quarter of an inch, we mean that if he declares such and such a thing to be exactly a yard long, or undertakes to measure with his eye a yard length from any given point, we shall find on testing it by standard measure that what he pronounces to be a yard will not be less than 35¾ inches, nor more than 36¼ inches. But what could we mean by saying, for instance, that you could rely on a housekeeper's estimates of the significance to her of such and such an amount of tea, under such and such circumstances, to a farthing? She is making an estimate, and if that is her estimate, what is the meaning of calling it accurate or inaccurate? Even if you try to bring it to the test of experience, and ask her afterwards whether her estimate is justified by the result, she can only tell you that it has or has not procured a satisfaction equal to what she now supposes she could have got by the sum she mentioned, if she had applied it otherwise; and this is itself an estimate. Though her estimates, therefore, are based on experience, and are checked and modified by it, yet no objective standard of experience can be kept for reference, or can be applied objectively as a check, like the standard yard.

II.2.17

Apparently, therefore, what we should mean, in the first instance, by saying that a housewife's estimates, under certain conditions, will be reliable to a farthing, would be something like this:—If we are dealing with estimates, as such (and not with the experiences which might or might not correspond to them if the experiment were made), we shall find that they may be made in various ways. We might ask a housekeeper to say how much another half-pound of tea would be worth to her if she already had 2½ pounds, and then some time afterwards, when she had not that question and answer in her mind, we might ask her what half-a-pound would be worth if she had 3 pounds. Then again we might divide the amount into other fractions of a pound, thirds or fifths, and begin at some other base than 2½ pounds, but include the former area in our new inquiries. And finally, we might ask how much a whole pound would be worth if she already had 2½ pounds. Now if she answered all these questions independently, giving every answer on the strength of a direct estimate, without mental reference to previous answers, and if the answers when compared never revealed inconsistencies of more than a farthing in the pound, and if similar tests produced similar results wherever applied, we could say with confidence that her estimates were not mere guesses or random selections of prices or quantities on which her mind was accustomed to rest, but were direct and genuine quantitative estimates, accurate as estimates, and therefore consistent, to within a farthing a pound. Another test would be to present the same question at different times in such different lights or connections as to suggest different answers, and see whether such suggestions or associations influenced the answer.

II.2.18

This must be the primary meaning of accuracy and reliability of estimates as such. But behind this we may think of the correctness of the estimates as attempts to realise hypothetical experiences. We may have a clear and consistent idea of the value we should attach to such and such a supply of a commodity if we already commanded just so much of it and no more, and it may be impossible to shake that estimate by the most skilful cross-examination; but yet if the experience comes we may find that we had formed a very erroneous conception of it, and our estimates may be very different now from what they were when the experience was only hypothetical. Thus remoteness of the supposed case from experience may either affect the precision of our estimate as such, or it may make our estimate now (whether precise or vague) unreliable as a forecast of what our estimate would really be under other circumstances. These two things must always be distinguished in our minds, though it may not always be necessary to insist on the distinction in any particular context.

II.2.19

But yet again. It is impossible to banish the idea that as well as more or less imperfect estimates there are certain definite and ultimate facts to be estimated, and that faults or errors of estimate do not affect these ultimate facts. How can we get at precise conceptions in this matter? Clearly we are still dealing with subjective experiences and not with external magnitudes. But just as we know that many impressions are received by the eye but not consciously registered by the mind, so there may be many sensations and experiences that actually go to making us happy or strong or the reverse, but of which we are not conscious as causes, or which are in themselves so slight that we have not learned to pay attention to them. An ideally perfect estimate would identify every cause and register every effect, and would actually assign to all experiences the values they would have for us if we distinctly realised them. We can reach no conception more nearly approaching objectivity than this.

II.2.20

Returning now to our actual estimates as such, we may go on to examine some of the influences which make a greater or lesser degree of accuracy, in the sense of precision and consistency, possible in any given case. But it will be well at this point to develop a distinction that has already been made, though not emphasised.*16 Accuracy is not the only valuable quality in our data, for we have seen that the curve which satisfies the minuter will always satisfy the broader data, and the minuter data determine the curve more closely than the broader. Minuter data of a certain relative inaccuracy might therefore determine the course of the curve more closely than the broader data of relatively greater accuracy. In Fig. 10, for example, we might suppose that the area of satisfaction corresponding to the sixth pound was given with great accuracy, but if we had no minuter data the curve might, for anything we should know, undulate in an indefinite number of ways, within wide limits, over that portion of its course. We should have one accurate datum, but the course of the curve would be indeterminate; whereas we might suppose a considerably higher degree of proportional inaccuracy in our data at and about the end of the third pound, and yet be more certain that we had determined the course of the curve about that point within narrow limits. The relatively inaccurate data, because narrower, would exclude many possibilities which a more accurate datum, if broader, might admit. And, as we shall see, it may very well happen that the broader data are, as a matter of fact, proportionally more accurate than the narrower. In such a case the narrower data may be of service to us in determining the general course of the curve within the limits of the broader data, but owing to their relative inaccuracy in detail their summation might give results incompatible with the broader data, and in such cases we should be guided by them only in such a general way as is consistent with compliance with the less determinate but at the same time more accurate conditions implied in the broader data.

II.2.21

With this proviso we will proceed with our examination of the conditions favourable to precision and consistency of estimate. Some general remarks on precision in estimating objectively measurable magnitudes may precede our examination of the more evasive estimate of satisfactions as magnitudes.

II.2.22

We must not blink the difficulty and complication of this problem, or the fact that any general principles we can lay down will be subject to every kind of disturbance from the personal idiosyncrasies or the special experiences of the individual who makes the estimates. It will, however, be admitted that in estimating quantities of any kind, a given individual will have a range, or theoretically a point, of maximum accuracy. Take an observer whose experience, professional or other, gives him no particular guidance in the matter, and present him successively with two pieces of wire, one an inch and the other an inch and a half long; then, successively, with diagrams shewing spaces of 1/32 in. and 3/64 in. respectively, intercepted between fine lines. Then take him to a place from which he has a variety of views, and under conditions identical as to distance, angle of observation, and so forth, ask him to notice the distance between the trunk of a tree and a boulder (known by you to be 1000 yards), and subsequently the distance between the edge of a tarn and the edge of a snow patch (which is 1500 yards). In each case ask him what proportion the first length in each pair bears to the second. You will probably expect a more accurate proportional estimate in the case of the inch and the inch and a half than in either of the other cases. Perhaps there will be some other length which he will be able to estimate more accurately still, but there will be some point, between the thirty-second of an inch and 1000 yards, in the neighbourhood of which his estimate will reach the maximum of accuracy. And as he recedes from this in either direction his estimate will become less reliable. It does not follow, however, in individual cases, that this departure from accuracy will be regular and continuous. There may be certain definite magnitudes which, for one reason or another, the individual has been accustomed to measure with unusual accuracy, and these may be irregularly distributed. Thus, if we take a carpenter who is also a professional cricketer, and who, when a boy, sometimes ran along a mile of road keeping pace with a stage-coach, and if we submit to him pairs of lengths which are really the same fractions of each other in every case, and not very remote from equality (say that one is nine-tenths of the other), probably if their mean is a foot he will estimate them with greater proportional accuracy than if their mean is 9 yards. But again he will measure them with greater accuracy if their mean is the 22 yards of a cricket pitch than if their mean is 9 yards; with less accuracy if their mean is 1000 yards than if their mean is 22 yards; but with greater accuracy again if their mean is a mile than if their mean is 1000 yards. Thus, the general principle that there is a certain magnitude in the neighbourhood of which estimates reach a maximum of accuracy from which they depart in either direction, may be qualified by any vivid experience or frequent practice which may have cultivated particularly accurate observations of certain lengths. And whatever the points of maximum accuracy may be the man will attempt to reduce his problems, when possible, to terms of the lengths he can best judge. Thus if a length is unmanageable he will try to divide it into halves, thirds, or quarters, or to multiply it by two or one and a half, and see whether these fractions give him lengths that he can judge immediately with some confidence and from which he can then calculate the others. The boy who, when asked how he would estimate the distance of the sun from the earth, answered, "Guess a quarter and multiply by four," had a confused sense of a sound method in his mind, though he was not fortunate in his application of it.

II.2.23

Now in the case of our tea curve all these complications are present, and certain others as well. The ultimate quantities to be estimated and compared, here as elsewhere in the administration of resources, are not tea-leaves and pence, but quantities of satisfaction; and yet the housewife is never accustomed to think of these as quantities at all. She thinks in pounds and ounces of tea, and in shillings and pence of money, but the half-unconscious and wholly unanalysed processes which emerge into conscious deliberations under these denominations of ounces and pence really concern lots of satisfaction. Hence a divergence between the points on which her deliberations crystallise themselves in her own consciousness and those on which they actually depend.

II.2.24

It is not difficult to see why this is so. In order to estimate tea with reference to other commodities we must express its value in terms of money, as the common measure between all the commodities in question; and we shall estimate it in the quantities in which we are accustomed to buy it. But our direct experience of its value is based on much smaller units, for while we pay for tea by the pound we consume it by the cupful. If a man drinks two cups of tea of a certain average strength every day for breakfast, his estimate of the value of a pound of tea must be arrived at by considering it as supplying, say, sixty-four breakfasts, and the marginal value of a quarter-pound by considering the significance of substituting a cup and a half for two cups at these sixty-four breakfasts. The enjoyment of tea at one breakfast is the quantity of satisfaction he really estimates, but in order to bring it into correspondence with his problems of expenditure he must reduce it to the terms in which he actually deals in it. If we express our estimate of one sixty-fourth of a pound of tea in terms of money we fall into manifest absurdity. For money is an instrument of practical exchange, and since we cannot give effect to these minute estimates of a fraction of a farthing in any actual transaction, this method of expression loses all its value. Hence the sense of intolerable unreality in our previous working out of the tea problem (pages 44-63). As we narrowed the areas of our estimates and so brought ourselves nearer to the actual basis of realisable experience we continued to express those estimates under a denomination that was becoming more and more hopelessly inappropriate and unconvincing.

II.2.25

Thus the point at which we deliberate as to alternative expenditures of money is likely to be remote from that at which our experience gives us the most direct and vivid sense of the immediate value of a commodity. In a word, to compare one expenditure with another we have to recede indefinitely from the points at which we can best compare one experience with another. Commodities are not practically exchanged with each other, or obtainable as alternatives, in the quantities in which the experiences they provoke are most directly comparable with each other. And as we are more accustomed to deliberate consciously as to expenditure than as to satisfaction (though our whole expenditure is ultimately regulated with a view to satisfaction), a difficulty inevitably arises. The careful administrator does occasionally revert consciously to the primary and ultimate basis. She may from time to time calculate, for instance, how many rice puddings can be made out of a pound of rice, or how many breakfasts a pound of tea will provide, in order to establish a kind of bridge along which she may pass either way from the quantities in which she buys commodities to the quantities in which she experiences their services. She sometimes travels from her expenditure per pound or per annum to her satisfaction per quarter-ounce or per diem, in order to base herself upon experience, and she sometimes calculates how much a saving too minute to be estimated in coin of the realm day by day would amount to in a month or a year, in order that she may bring one set of experiences into terms under which it may be compared with another and alternative set.

II.2.26

As we are now to deal with the ultimate limits of accuracy in the construction of a curve, it is obvious that we are concerned not directly with shillings and pence per pound, but with the estimates of satisfactions per cup, and so forth, as quantities. Obviously it is with these that the housewife must ultimately wrestle. For instance, if an economy is to be effected she may have tea at fewer meals, never supplying it at certain times of day unless it is expressly asked for, or in the last resort saying that it cannot be had; or instead of this she may make it weaker, or she may practically limit the amount of the infusion at each meal while not limiting the amount of hot water that passes through the pot, or she may look for a cheaper tea, or (horresco referens) one that will not be so popular in her household. She may or may not be subject to such more or less unsympathetic pressure from her family as is implied in some of the foregoing suppositions, but in any case she is dealing with certain alternatives, and in considering them she is estimating and comparing volumes or areas of satisfaction, and it is a reference to these that underlies her estimates in money of the marginal value of an ounce of tea, and determines at what point of pressure she will buy more or less of any given quality at any given price.

II.2.27

It is therefore here that we must apply the principle of the magnitude that is estimated with greatest proportional accuracy; for there may be some one or more of the satisfactions she habitually considers which, as magnitudes, are realised with especial distinctness and vividness, and to which others are consciously or unconsciously referred as to a kind of standard. Suppose, for example, there is one member of the household whose wants, for any reason, good or bad, the housekeeper considers it specially important to satisfy, and whom she occasionally disappoints, as to quality or quantity, in the matter of tea. The significance of this occasional contretemps may well constitute the actual unit of greatest proportional accuracy of estimate, and it may be by unconscious reference to it that the housewife can determine most accurately the relative values of all the alternative refusals, indulgences, evasions, devices, and pecuniary expenditures, with which she is concerned in the matter. Here, as in the case of the carpenter, there may be other points impressed by other experiences that give an exceptional degree of firmness to estimates of certain other quantities; but, neglecting this consideration, we may follow up the special clue we have grasped.

II.2.28

Note that our housekeeper will probably never deliberately incur or inflict the specific privation we are considering, merely in order to economise the tea needed to avert it. It will occur through some inadvertency or miscalculation, and it will be the delay, or trouble, or want of courtesy to a guest, or incidental (as distinct from primary) waste, that would be involved in correcting the error that will determine her to accept the result. But when the housekeeper is asked to make a number of hypothetical estimates as to what successive increments of the supply would be worth to her, and comes to think of a contraction of supply great enough to make this specific privation normal and permanent instead of occasional and accidental, she finds she has a very clear conception of that particular "lot" of satisfaction, that she has been accustomed to translate it into a great variety of equivalents, and that she has from time to time defined it pretty closely as worth just so much of certain other things, but not even a little more. She can now translate it, by a deliberate calculation, into so much tea per month, and can estimate it with some precision at its money value. This may form a kind of standard unit of reference, and may be the magnitude she is capable of estimating with the highest degree of proportional accuracy and precision. The area thus determined will be that of the elements out of which our curve can be constructed with greatest accuracy. For in considering the value of other increments nearer to the margin or further from it, our housewife (we are supposing) will find it easiest to make accurate estimates of areas of satisfaction of this particular magnitude; and she will find, of course, that if she has to think of herself as compelled by the further contractions of her supply to cut deeper back into the satisfactions of her household than she has ever actually done, she will realise that a smaller amount of tea, at the higher significance so reached, would yield the standard unit of satisfaction, and that in like manner at a more advanced point it would require a correspondingly larger amount to secure it. Geometrically the standard area will stand on a narrower basis as we approach the origin, and on a broader one as we recede from it.

II.2.29

Thus, subject to all the qualifications hinted at or developed, we may suppose that the ultimate elements out of which data for the curve would be obtained with the greatest proportional accuracy would consist in estimated satisfactions of a magnitude about equivalent to that of the satisfaction relinquished on the occasions of disappointment that have impressed themselves most vividly on the housewife's mind. They would be represented on our diagram (when reduced to the terms of a month's supply, and expressed in shilling and penny rates per pound) by a series of rectangles of uniform area standing on progressively larger bases as we recede from the origin.

II.2.30

Now seeing that every day the housekeeper deals with the whole supply for the day, and has the opportunity of experiencing or observing the actual service rendered by every increment from the initial to the final one, we might be tempted to think that she could base her whole conjectural construction of the curve from the origin to the margin upon direct experience. But this is not so. We have seen that recurrently satisfied wants never take us back to the real initial significance of the things that satisfy them.*17 If our supplies were very much contracted (even apart from any reaction upon the organism that might ultimately take place) we should gain experience of significances that had evaded us before; for the want which to-day's first increment supplies is a different want according to the point up to which our want was satisfied yesterday. And as soon as we begin to contract or increase our supply at all this process sets in, though its effect at first may be hardly perceptible, and it may only become pronounced as we recede considerably from our present margin. Thus an additional element of uncertainty enters into all estimates far behind or far in advance of the present margin, and our ideal equal areas will become correspondingly more speculative. This speculative element may reveal itself consciously in a refusal to make equally precise estimates, or unconsciously in an inability to make equally consistent ones, as we recede from the actual margin. Past experiences, vividly remembered, may establish at irregular intervals other bases of comparatively direct and immediate estimates; or critical points may so appeal to the imagination as to give a firm but illusory precision to speculative estimates; or some changed unit of maximum accuracy may assert itself in certain regions of the curve; and throughout we must distinguish between precision and consistency in the sense explained above, and approximation to the estimates which would be formed under the pressure of immediate experience should it ever be realised.

II.2.31

When formed, our curve, such as it is, will be an estimate, or a register, more or less reliable, both of the total significance to be derived from the consumption of any given quantity of the commodity, and of its marginal significance at any point.*18

II.2.32

Before leaving this branch of the subject we may note that if we asked for estimates of the significance of a series of objectively equal increments of the commodity we should have a series of rectangles, not of equal area but on equal bases, from which to construct our curve; and we may ask what conditions would influence the delicacy and accuracy of our estimates of the difference of area between them. Two considerations are relevant here. In the first place, the same magnitude is less easily perceived and estimated as part of a larger than as part of a smaller whole. The difference of an inch is more conspicuous in the length of two men's noses than in their heights. Small differences will therefore be less delicately noted when the areas are large than when they are small, and therefore a given difference between two contiguous rectangles might escape detection near the origin but might be distinctly felt farther from it. But in the second place, our whole investigation has shewn us that the significance of successive increments of the commodity changes more rapidly in some regions than in others. Between two successive rectangles on equal bases, therefore, we shall sometimes have greater differences and sometimes have keener powers of observation. The first condition is indicated by a rapidly falling curve, and the second by a higher positive altitude of the curve. In our example of the tea, and in Fig. 14, a, these two conditions tend to counteract each other; for as the differences themselves decrease, our power of perceiving them increases. But in Fig. 14, b, they reinforce each other. As the differences themselves become greater our power of observing them also becomes more acute.

Figure 14.  Click to enlarge in new window.

II.2.33

Enough has now been said to shew in the first place how extremely precarious any actual evaluation of a curve of total significance of any commodity must necessarily be, but also, in the second place, that this value, which it is so difficult to estimate, is actually a definite and a highly significant quantity.

II.2.34

The area bounded by the curve represents what the older economists called the "value in use" of the commodity, that is to say, the total satisfaction or advantage derived from its enjoyment; and the height of the curve above any point on the abscissa represents its marginal significance, which, in the case of exchangeable things, will always tend to be brought into coincidence with its "value in exchange." And note that if our expenditure is wise a decline in marginal significance due to an increased supply will always coincide with an increased volume of satisfaction. A reduction in the "exchange value" of any commodity, taken in itself, should always result in its increased "value in use" to us.*19

II.2.35

We have now sufficiently examined the general meaning of a curve of total significance or satisfaction, and we have seen the very precarious nature of the data upon which any attempt actually to evaluate the total significance of a commodity must depend. But we have still to take note of certain points, a neglect of which might lead to erroneous or inaccurate thought.

II.2.36

It will be understood that a curve proves nothing whatever as to the facts from which we start. It is merely an idealised picture of facts and their implications. It may therefore enable us to understand the full meaning of any set of supposed facts, but it cannot establish them. At most it can only shew us the relations in which certain facts, if they exist, stand to each other. But by doing this it may bring out implications involved in our data that we had not fully realised, and this may throw back light on the validity of the data themselves. For instance, a glance at the Figure 10.  Click to enlarge in new window.tea curve at once suggests that it will not decline any further after the point to which we have carried it; and as there is no reason why the law of declining significance should become invalid after seven pounds, we begin to suspect our data of being in some way self-contradictory or impossible. And this is really the case. We supposed our original data as to the values of successive pounds of tea to conform to a perfectly rigid and easily discernible algebraical law. But this is strictly impossible. In the first place, it is impossible that the estimates should be mathematically accurate at all. That is to say, it is impossible that an infinitesimal change in the quantity of the commodity could be actually and directly appreciated, and its significance registered in terms of money. But if we are dealing only with approximations it may possibly happen that the more or less loose estimates given may conform loosely to some simple algebraic formula. Since, however, an immense number of heterogeneous factors would enter into the composition of every region of the curve, some of them changing as it proceeds, we may be very sure that no simple algebraical formula would be able to represent them all even approximately, though it might approximately fit a certain portion of the curve. So if we had assumed this precise algebraical law as determining the whole curve, we should have assumed in the first place an impossible precision, and in the second place a highly improbable (and, as it turns out, impossible) simplicity and regularity. As a matter of fact it will be found that our original data themselves assumed that after the sixth pound the law of the curve would change; for the series 23s., 17s., 12s., 8s., 5s., 3s. would give as its next term 2s., and we have constructed the curve on this estimate. But this contradicts our original data, for we started with the supposition that at 2s. a pound the purchaser would take 7 lbs.; and the figure makes it very clear that if the whole seventh pound is only worth 2s., then the first half-pound is worth more than a shilling, and the second half-pound worth less. The second half-pound therefore would not, on this supposition, be bought at all. Our curve would give about 6.42 lbs. as the ideal point at which the purchase would stop. So if we are to suppose that 7 lbs. would be bought at 2s. we must suppose the character of the curve to change after 6 lbs. It might take some such course as that indicated by the dotted line.

II.2.37

In very many cases a curve that approximated to a similar formula during a part of its course might reasonably be expected to change its character as it approached the origin; for we have seen that at first a commodity may have increasing significance, and may only enter upon the period of declining significance "after a certain point."*20 In the case of tea, however, there is nothing palpably absurd in supposing our curve to follow approximately the formula we have assumed, at any rate up to a very close proximity to the origin. It is easy to imagine that as tea (or coffee) became dearer and dearer a careful housekeeper, whose family still retained a taste that they were less and less able to indulge, might limit the purchases more and more till at last it was only on occasions of special festivity that the precious infusion was consumed. When the price of £1:6:4d. a pound was reached, a quarter of a pound, or two ounces, might be bought for Christmas Day, and none at all at any other time. This consumption (four or two ounces a year) would be at the rate of one-third or one-sixth of an ounce per month, and would be represented on our figure by a point only one-third or one-sixth of the side of a small square from the origin. And if we had lowered the whole curve by, say, two of the large units on Y so that it intercepted the axis of X at a little under 6 lbs. 7 oz., the whole series of marginal values from the initial increment to the one that completed the full satisfaction of the desire might, without palpable absurdity, have been supposed to be represented by this particular curve. As it is, it is clear that our original data involve the supposition that the law indicated by the successive steps in declining value from 1 lb. to 2 lbs., etc., up to 6 lbs., would not continue to hold for the decline from 6 lbs. to 7 lbs.

II.2.38

Even if we do not assume an algebraical formula for a curve, we can seldom use this diagrammatic method without expressing more and expressing it more precisely than we desire, and this constitutes a grave disadvantage in the use of curves for popular demonstrations. If, for example, we say that successive increments of a commodity will decline in significance after a certain point the statement remains general. But if we illustrate it by a curve, the "point after which" will be determined and the rate of decline at every point will be determined, and a general conception of the modes of variation will be suggested. And so the incautious student may be misled by the characteristics of the individual curve selected, and may fail to distinguish between them and those characteristics really involved in the data. The utmost caution is needed to prevent a curve from surreptitiously insinuating into our minds suppositions which are not included or involved in our data, but which we nevertheless receive into our conclusions. Nor is it beginners only that have fallen into this trap.*21 But this by the way.

II.2.39

We might now suppose that in such a diagram as Figure 15.  Click to enlarge in new window.Fig. 15, if properly constructed, we should have an ideal presentation of the amount of the commodity Ox that would be purchased by a certain individual at any given market price Oy; of the total satisfaction Oy0px that its consumption would afford; of the volume of other satisfactions Op sacrificed in the total sum paid for it; and of the surplus of satisfaction yy0p which is secured over and above what is sacrificed. If this were so, then this last-named area would represent the advantage which the consumer derived from the existence of this particular market, and the volume of satisfaction of which he would be deprived if it closed or became inaccessible to him, all other things remaining equal.

II.2.40

These conclusions, however, are still subject to sundry modifications and qualifications which we must now examine.

II.2.41

In constructing our curve, we have used denominations of shillings and pence simply as measures of certain definite satisfactions, and we have tried to shew how, ideally, the area of total satisfaction corresponding to any given supply Ox of the commodity could be actually evaluated in these denominations. But on closer inspection we become aware of a disturbing instability and ambiguity in our unit when regarded as a psychological magnitude. We have often noted that 1s. has a different psychological significance to two different men, and also to the same man if his income rises or falls. Theoretically, then, the marginal significance of a shilling will be affected by the sum the man has already paid to secure a certain satisfaction. We supposed, in our example of the tea, that the housekeeper gave us the outside value of the first pound of tea to her, and then supposing herself to have paid that sum for it went on to give us the outside value of a second pound, and so forth. If our Fig. 15 has been constructed on this system, then x1p1 will represent the marginal value of a commodity to a man, on the supposition that he has actually paid the money represented by the area Oy0p1x1 for the quantity Ox1. But will Ox1 represent the amount he would actually buy if the market price were Oy1? Not unless the sum of money represented by the whole area Oy0p1x1 is so insignificant a part of the man's total expenditure that it makes no perceptible difference to the marginal significance of a penny whether the area Oy0p1x1 or only the area Op1 has been spent upon tea. If this is not so, then the fact that he can actually get Ox1 for the expenditure of Op1 will leave him better off than on our first supposition by the area y1y0p1; and this being an appreciable sum it will enable him to get a little more of everything or anything (including the commodity under direct consideration) than he would have been able to do had he spent y1y0p1 (as well as Op1) on the supply Ox1. A little more than Ox1 may therefore be purchased. And again, since all the man's wants will be satisfied down to a lower point of urgency, the significance of what a penny will buy at this advanced margin is lowered. Thus the psychological significance of our unit will be smaller if the whole supply is purchased at the lower price than if the full sum represented by the mixtilinear area had been given for it. As we imagine Ox to advance or recede, the changing values of the total or the rectangular areas will react upon the psychological significance of the unit, and the difference between them will prevent the abscissa from accurately representing the amount that would be consumed at the price represented by the ordinate.

II.2.42

This is not a mere fanciful speculation. If a careful housekeeper were giving any such estimates as we have supposed, when she came to think of herself as paying 50s. or 60s. a month for tea instead of something like 14s., she might be perfectly conscious of the constraint she would feel in all branches of her household expenditure, and might realise that she was estimating the increments of tea in a unit of higher significance than that by which her actual expenditure is regulated.

II.2.43

The curve as constructed, therefore, does not represent the relation of price to quantity purchased with any theoretical accuracy at all, and it represents the psychological value of the satisfaction secured in a fluctuating unit.

II.2.44

We will begin with the latter difficulty. How can we maintain the stability of our psychological unit throughout a series of estimates? What we really want is to fix in our own minds or the mind of our informant the actual psychological magnitude represented by the objective unit at the margin of our current expenditure; and then to estimate in that unit the significance of small increments of the commodity at various margins. We should then have, for any given quantity consumed, what we set out to obtain, viz. an evaluation in a stable unit of the total estimated satisfaction enjoyed, as distinct from the sum paid. These estimates, however, are such as we could only imagine experts trained in a psychological laboratory attempting to make. The naïve, however careful and acute, answers we could expect from practical administrators would never be based on so subtle a conception as that of the psychological unit. We should have to assist our informant by putting our questions in some such form as this: "If when you had bought your tea for the month and paid for it at market prices, you lost half, three-quarters, nine-tenths, or all of your stock, what in each case would you pay for a first small increment, sooner than go without it?" The smallness of the increment estimated would reduce to a vanishing point the reaction of the sum to be paid upon the psychologic value of the money unit, and the fact that in every case the full amount that is actually paid for the commodity, and no more, is already written off, would keep that psychological value uniform. The ingenious reader may still think of disturbing influences, the shock of the loss, the changed significance of other enjoyments caused by the reduction in the supply of tea and so forth; and he may imagine any system of discounts that pleases him. It is clear that in any case absolute fixity of the psychological unit is only an ideal conception, and that actual estimates in money will never be more than approximately consistent in their psychological significance. The essential point is that the total psychic value of the satisfaction derived from the consumption of a given amount of a commodity is a finite quantity, capable of ideal evaluation in a fixed unit, and that over a vast field of our current expenditure it exceeds, in our own estimate, the value of the alternatives we relinquish for it.*22 This total area of satisfaction may, in theory, be represented accurately by a figure which sets forth the marginal significance of every successive increment of the commodity; but if we have taken as our psychic unit the satisfaction which the money unit commands at the actual margin of our expenditure under existing conditions, then any hypothesis which sensibly changes those conditions (as by increasing or diminishing the amount actually spent on our commodity) will change the significance of the unit; and therefore, if we measure penny or shilling rates on the axis of Y, it follows that the same figure cannot represent, with theoretical accuracy, the meaning of a number of different hypotheses, regarded as co-existing. Given any price and the actual administration of resources that corresponds to it, we can ideally construct a curve of total satisfaction, the unit of which corresponds to the marginal satisfaction now secured by a penny or a shilling; but if the price changes we cannot preserve the same figure and get an accurate result by simply changing the point on OX at which we erect a perpendicular to cut the curve; for under the changed conditions the satisfaction secured by a penny or a shilling will have changed.

II.2.45

I have been careful to speak of the Figure as giving, ideally, a representation of the total satisfaction derived from the consumption of Ox, in the mixtilinear area above it. I have not said that the surplus of satisfaction over payment would be accurately represented by the area yy0p. For this again would only be an approximation. In evaluating the price actually paid at Op our Figure implies that if the market for the commodity in question were closed, or if the commodity ceased to exist, the purchaser, while losing the total area above Ox, would gain the released area of the rectangle Op. This means that the whole of the money now spent on this commodity could be expended on other commodities at a marginal significance represented by xp or Oy. But theoretically this is not true, for if the supplies of other commodities were increased, it would of course be at a declining significance, and consequently, when the whole sum Op had been distributed amongst them, their marginal values would have declined to some extent, however small, from the height xp. Some portion of them, therefore, would have less value than if their marginal significance had remained at xp; and in the sum they will not equal Op. And here again, as we recede from the actual point of departure towards the origin, there will be another source of disturbance in the psychic significance of the money unit, independent of the advancing margin, viz. the change in the whole significance of remaining sources of satisfaction as the one to which the Figure refers dries up. Here again, therefore, all attempts to guard against and discount sources of disturbance in the psychic value of our objective unit must be at once subtle and clumsy. The only ideal method is to conceive of a mind trained to hold a psychic magnitude firmly and apply it consistently as a unit. That magnitude would be the satisfaction represented by the money unit under existing conditions, but it would be applied to hypothetically changed conditions directly, and not through the convenient but treacherous intervention of a money unit which might be perpetually changing its significance.

II.2.46

If we traced our original curve with a stable psychic unit, based on the satisfaction secured by a penny or a shilling at present margins, and if we then allowed for the decreasing values of other commodities as the margins advanced, represented by a decline in the height of the ordinates as we pass from xp to Oy, we should have a consistent representation of total satisfaction, and of surplus of satisfaction over the sacrifice represented by the price, corresponding to the actual state of things. It would shew how much satisfaction I get and how much I pay for it, measured in a stable unit. But it would not give us accurate information as to any other than the actual state of things.

II.2.47

If, on the other hand, we were to ask, not "how much would you give for an ounce of tea under such and such circumstances?" but "how much tea would you buy if it were such and such a price?" we should get a curve with just the opposite characteristics. It would give us information about a number of different hypothetical conditions, but its different parts would have no consistent significance. Thus, by asking "how high would the price of tea have to rise before you would stop buying it altogether?" we might find a point on the axis of Y, and then, by asking how much would be bought at the several prices descending from that to zero, we might obtain points on a curve which would accurately represent the relation between price and quantity purchased for every hypothesis at once. But on each hypothesis the psychological significance of the unit would be different, and as it would always make a (theoretical) difference whether the whole sum represented by the mixtilinear area above any abscissa, or only that represented by the rectangle, were paid, the area would never represent accurately either the total sum that the consumer would pay for the amount Ox, or its psychological evaluation in any fixed unit.

II.2.48

A curve, therefore, which professes to give, for every price, (1) the quantity that would be purchased at that price, (2) either the pecuniary or the psychic evaluation of the total satisfaction it would yield, can only be a compromise, for it endeavours to comply with two incompatible sets of conditions. Its construction would illustrate the principle of "temperament" by which a note on the piano which is neither D sharp nor E flat, but a compromise between them, is made to do duty for both alike. This is only possible if the interval between them is small. In our case the errors involved in confounding the two curves become negligible in proportion as the total expenditure on the commodity in question is a negligible part of the man's whole income.

II.2.49

The psychological curve always remains the ultimate and basal fact, and though we can never rely on its precise evaluations it is essential that we should form a precise conception of its nature and should realise that it has a definite value. The price-and-quantity-purchased curve is the most accessible and is the one with which we shall usually work; but unless the contrary is expressly stated we shall assume that our curves have a "temperament" which allows us to read them either way.*23


Notes for this chapter


13.
Chapters II. and III., though important from the theoretical point of view, are of an abstract and somewhat academic character, and some readers may prefer to go on at once to Chapter IV.
14.
See page 60.
15.
This analytical proof is, strictly speaking, unnecessary; for since we have ah = aeok and ks = kogd, we have also aegd = ah + ks = ac; and this involves the equality of bef and fgc. But the proof by substitutions may probably be found the more enlightening.
16.
At the bottom of page 451.
17.
Cf. page 426.
18.
But see below, pages 467 sqq.
19.
Cf. pages 45-47.
20.
Cf. page 435.
21.
Cf. pages 552 and 568 sqq.
22.
But compare the following chapter.
23.
Cf. Appendix to Chapters II. and III. pages 490-492.

End of Notes


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