Unearned increment

“ If we take the average amount of all incomes to represent the sum each worker might justly receive, 69we find that a number of people have far more than this sum. The surplus represents then an “unearned increment” obtained by force of circumstance. A still larger number of people, on the other hand, are wholly unable to win, by any effort they may make, the above average amount, even if they work hard and well all their lives. ”

“ The simple law observed is only the translation of the general analytical rule by which the infinitely small increment of a function is proportional to the increment of the variable. As in reality our increments are not infinitely small, but only very small, the law of proportionality is only approximate, and simplicity is only apparent. ”

“ Forming, then, a series of fractions whose denominators are the products of the number of persons indicated in the table of mortality as living at the age of that one who settles the annuity, by the successive powers of unity augmented by the rate of interest, and whose numerators are the products of the annuity by the number of persons living at the same age augmented successively by one year, by two years, etc., the sum of these fractions will be the amount required for the life annuity at that age. ”

“ With respect to these different rates of increase we may remark, that the first is the most stable of all, and that though its exact quantity is difficult to be ascertained, yet, whatever it is, it is nearly invariable, and, if it can be rightly assumed to give a rate of duplication in ten years at any particular time and place, the same assumption will be equally applicable to all times and places. ”

“ The third rate, or the actual progression, is of course the most variable of all, being influenced by the greatest variety of causes. It is observable, that, while the checks, which produce the difference between the first rate and the second, have the property of retarding, and of taking away a part of the original rate of progression, still they are not connected with any limitation of its range, and their intensity is not necessarily increased in consequence of any actual increase of number. ”

“ The Commission is not under a duty to write off the whole or any part of the balance in 100.4, if the difference between original and present cost is a true increment of value. On the contrary, only such amount will be written off as appears, upon an application for appropriate directions, to be a fictitious or paper increment. This is made clear, if it might otherwise be doubtful, by administrative construction. ”

“ Other things remaining the same, the additional products which come into being through increasing subdivision, gradually diminish in value as increment after increment is added, according to the well-known law of diminishing returns; and on the contrary the detriment to society as a whole increases as individuals are forced into narrower and narrower rounds of duty. ”

“ There can be little doubt that many of us were taught to count very badly, and that we were hampered in our arithmetic throughout life by this defect. Counting should be taught be means of small cubes, which the child can arrange and rearrange in groups. It should have at least over a hundred of these cubes—if possible a thousand; they will be useful as toy bricks, and for innumerable purposes. Our civilization is now wedded to a decimal system of counting, and, to begin with, it will be well to teach the child to count up to ten and to stop there for a time. ”

“ If you once commit the addition table thoroughly, you have it fixed for life; whereas if you do not, you have to make the calculation over again every time, and thus lose a vast amount of labor. I have no doubt that there are some in this class who are in the habit of counting, who have ascertained that seven and eight for instance, make fifteen; by counting up from seven to fifteen, hundreds of times. ”

“ Among the subjects which should be curtailed or omitted are compound proportion, cube root, abstract mensuration and the greater part of commercial arithmetic. Percentage should be reduced, and the needs of practical life—profit and loss, bank discount, compound interest, with such complications as result from fractional periods of time—are useless and undesirable. The metric system should be taught in application to actual measurements, and the weights and measures handled. ”

“ But Turner counted his pines, and did all that could be done for them, and rested content with that.
So in all the affairs of life, the arithmetical part of the business is the dominant one. How many and how much have we? How many and how much do we want? How constantly does noble Arithmetic of the finite lose itself in base Avarice of the Infinite, and in blind imagination of it! In counting of minutes, is our arithmetic ever solicitous enough? In counting our days, is she ever severe enough? How we shrink from putting, in their decades, the diminished store of them! ”

“ What I have said will suffice, I hope, as an answer to Balbus, who holds that (a) and (c) are the only possible varieties of the problem, and that to say "We cannot use addition, therefore we must be intended to use multiplication," is "no more illogical than, from knowledge that one was not born in the night, to infer that he was born in the daytime"; and also to Fifee, who says "I think a little more consideration will show you that our 'error of adding the proportional numbers together for each candidate instead of multiplying' is no error at all." ”

“ II. But if it be thought that, either now, or at any future time, the citizens of the United States may be induced to undertake a thorough reformation of their whole system of measures, weights and coins, reducing every branch to the same decimal ratio already established in their coins, and thus bringing the calculation of the principal affairs of life within the arithmetic of every man who can multiply and divide plain numbers, greater changes will be necessary. ”

“ The whole food of a country divided by the sum of its population, constitutes the share of each person. Here, the food is the numerator, and the population the denominator of a fraction. In order that this fraction shall diminish, it is not necessary that the numerator shall continue stationary while the denominator increases: it is sufficient that it shall not increase as fast; and this is the case with food, which, we know, cannot increase as rapidly as an unchecked population.
I have also stated that the channel of employment can receive as many labourers as can possibly be maintained. ”

“ Every one sees that there would be an astonishing simplification in this system of reckoning by tens—that the study of arithmetic would be immensely facilitated, and the business of the counting-house divested of puzzling calculations. Let us see whereabouts we are in the way towards its attainment.
About ten years ago, a parliamentary commission on the subject of weights and measures, advised the adoption of a decimal scale, but recommended as a preliminary step, the decimation of the Coinage. ”

“ To suppose a country’s rate of increase permanent is hardly less fallacious than to suppose an invariable order of births and deaths over the world generally. Even if we are beyond the time when we need to make any allowance for increasing accuracy and fulness, and if we may assume that no given census has any 185units to sweep into its net that through their fear or an official’s carelessness escaped its predecessor, still we cannot take the rate of increase from one census to another as a sure indication of the future. ”

“ It is no paltry question of an outlet for the surplus products of a mere nation of seventy-five millions that confronts you. Your mathematical professors will tell you that, at the ratio of increase established in this Nation by the census returns for the century just closing, its population would amount during the next century to the bewildering and incomprehensible figure of twelve hundred millions. ”

“ But when we speak of the units which go to form number, we no longer think of these units as sums, but as pure, simple, irreducible units, intended to yield the natural series of numbers by an indefinitely continued process of accumulation. It seems, then, that there are two kinds of units, the one ultimate, out of which a number is formed by a process of addition, and the other provisional, the number so formed, which is multiple in itself, and owes its unity to the simplicity of the act by which the mind perceives it. ”

“ Is it in contemplation with the House of Representatives to extend a like improvement to our measures and weights, and to arrange them also in a decimal ratio? The facility which this would introduce into the vulgar arithmetic would, unquestionably, be soon and sensibly felt by the whole mass of the people, who would thereby be enabled to compute for themselves whatever they should have occasion to buy, to sell, or to measure, which the present complicated and difficult ratios place beyond their computation for the most part. ”

“ The numerator, in life, is What We Have. The denominator is What We Think We Ought to Have. Mankind may be divided into two classes: Fools and Wise. The fools are eternally trying to get happiness by multiplying the numerator, the wise divide the denominator. They both come to the same—only one you can do and the other is impossible.
If you have only one thousand dollars and think you ought to have two thousand dollars, the answer is one thousand divided by two thousand, which is one half. ”

“ The division of the fingers into two sets of five each is still in his mind, but it is no longer the leading idea. As the count proceeds further, the quinary base may be retained, or it may be supplanted by a decimal or a vigesimal base. How readily the one or the other may predominate is seen by a glance at the following numerals:
By the time 20 is reached the savage has probably allowed his conception of any aggregate to be so far modified that this number does not present itself to his mind as 4 fives. ”

“ The comparison between the numerical results obtained at one time and place and another, and between those presented in different countries, is becoming a political method increasing in prevalence and repute. In many quarters, indeed, the value of purely numerical estimates has been much exaggerated, and its peculiar liability to error, when made a basis of political reasoning, has been too much ignored. ”

“ Conversely, in any computation of the ratio of spindles to population at different dates, a deduction must be made in 1860 and 1870 for those who were at these respective dates clothed in hand-made fabrics. Perhaps it may be said, and perhaps it may rightly be said, that if an allowance must be made according to each man's judgment for all these variable elements of the problem, what dependence can be put upon the final figures? Each one may answer his own question or doubt in his own way. ”

“ Arithmetic, without further delay, produced specimens of his four simplest kinds of sum grates, like those from which Dick had been supplied. Lubin and Nelly soon chose Addition as their first purchase from Arithmetic—a grate so small and so light that even the little girl supported the burden with tolerable ease.
"You must come back to-morrow for something a little heavier," said Mr. Arithmetic. "Addition is simple enough; but Division needs a little greater effort of strength." ”

“ I know that table like a living thing, and yet I never saw it until to-day.”
“I suppose,” said Hugh, “that you base your system on the law of average?”
The professor laughed tolerantly. “No, I don’t. Law [225] of average,—that is child’s talk to me. Of course every table has its average, every wheel has its average, every croupier, you, I; every coup that is played, is an average with something that has gone before. Ah! these foolish system-players with their talk of average and probability and phenomenon. Why, every coup is a phenomenon as regards something that has gone before. ”

“ It is not too much to say that the present development of modern science would be impossible without our number system, yet how slow the world was to accept the reform. Is not the same story being repeated in the United States and England with the decimal system of weights and measures? But the optimistic soul regards chiefly the final acceptance with the comfortable assurance that the forward movement is as sure as it is slow. ”

“ Both Price and Malthus lay down a proposition which can easily be verified by the multiplication-table. If, as Malthus said, population doubles in twenty-five years, the number in two centuries would be to the present number as 256 to 1, and in three as 4096 to 1. If, meanwhile, the quantity of subsistence increased in 'arithmetical progression,' the multipliers for it would be only 9 and 13. It follows that, in the year 2003, two hundred and fifty-six persons will have to live upon what now supports nine. ”

“ Waiving the question whether annual increases or the range of measurements relative to the age norms would be satisfactory indications of the change in the rate of growth, it seems to be fairly clear that neither of these criteria would be adequate unless we first knew that the units in which they were measured were equivalent at different portions of the scale. ”

“ He thought, also, that the ratio of representation proposed could not be safely fixed, since in a century and a half our computed increase of population would carry the number of representatives to an enormous excess; that the number of inhabitants was not the proper index of ability and wealth; that property was the primary object of society; and that, in fixing a ratio, this ought not to be excluded from the estimate. With regard to new states, he observed, that there was something peculiar in the business, which had not been noticed. ”