People From London

"Consideration of motives brings up the matter of free will. I had better say once, that my project of taking animal comparisons seriously does not involve a slick mechanistic or deterministic view of freedom. Animals are not machines; one of my main concerns is to combat this notion. Actually only machines are machines."

"But understanding and explaining motives does not compromise freedom; nor does even predicting acts necessarily do so. A person committed to a political cause may vote predictably, and intelligibly in an election. He does not vote less freely than someone that flips a coin at the last minute. So if we find comparison with animals any help in understanding motives, it will not mean that conduct is not free. And since animals are not (as Descartes supposed) automata, the issue of freedom does not make comparing man with any other species and downgrading irrelevance."

"The notion that we "have a nature" far from threatening the concept of freedom, is absolutely essential to it. If we were genuinely plastic and indeterminate at birth, there could be no reason why society should not stamp us into any shape that might suit it. The reason people view suggestions about inborn tendencies with such indiscriminate horror seems to be that they think exclusively in one particular way in which the idea of such tendencies has been misused, namely, that where conservative theorists invoke them uncritically to resist reform. But liberal theorists who combat such resistance need them just as much, and indeed, usually more. The early architects of our current notion of freedom made human nature their cornerstone. Rousseau's trumpet call "Man is born free, but everywhere he is in chains," makes sense only as description of our innate constitution as something positive, already determined, and conflicting with what society does to us. Kant and Mill took similar positions. And Marx, though he officially dropped the notion of human nature and attacked the term, relied on the idea as much as anybody else for his crucial notion of Dehumanization."

"... It is clear that if I am a mere effect of my body—a form of its activity—I shall cease when the body ceases. And it is also clear that, if I could not exist without this particular body, then the destruction of the body will be a sign that I have ceased to exist. But, besides death, there is another characteristic of nature which tends to make us doubt our immortality. Of all things around us, from a to a , science tells us that they are transitory. Each of them arose out of something else, each of them will pass away into something else. What is a man that he should be exempt from this universal law?"

"On the determinist hypothesis an omnipotent God could have prevented all sin by creating us with better natures and in more favourable surroundings. … Hence we should not be responsible for our sins to God."

"The really fundamental aspect of the dialectic is not the tendency of the finite category to negate itself but to complete itself."

"Religion has always, I think, implied a belief in some fundamental harmony, some sort of reconciliation between the claims of our own nature and the facts of the universe."

"What mathematics, therefore are expected to do for the advanced student at the university, Arithmetic, if taught demonstratively, is capable of doing for the children even of the humblest school. It furnishes training in reasoning, and particularly in deductive reasoning. It is a discipline in closeness and continuity of thought. It reveals the nature of fallacies, and refuses to avail itself of unverified assumptions. It is the one department of school-study in which the sceptical and inquisitive spirit has the most legitimate scope; in which authority goes for nothing. In other departments of instruction you have a right to ask for the scholar’s confidence, and to expect many things to be received on your testimony with the understanding that they will be explained and verified afterwards. But here you are justified in saying to your pupil “Believe nothing which you cannot understand. Take nothing for granted.” In short, the proper office of arithmetic is to serve as elementary 268 training in logic. All through your work as teachers you will bear in mind the fundamental difference between knowing and thinking; and will feel how much more important relatively to the health of the intellectual life the habit of thinking is than the power of knowing, or even facility of achieving visible results. But here this principle has special significance. It is by Arithmetic more than by any other subject in the school course that the art of thinking—consecutively, closely, logically—can be effectually taught."

"I have undertaken to say a few words to you on the "Art of Questioning". It is a subject of great importance to all of you who desire to become good teachers ; for, in truth, the success and efficiency of our teaching depend more on the skill and judgment with which we put questions than on any other single circumstance."

"It is very possible for a teacher to be fluent in speech, earnest in manner, happy in his choice of illustration, and to be a very inefficient teacher, nevertheless. We are often apt to think it enough if we deliver a good lesson, and to forget that, after all, its value depends upon the degree in which it is really received and appropriated by the children. Now, in order to secure that what we teach shall really enter their minds, and be duly fixed and comprehended there, it is above all things necessary that we should be able to use effectively the important instrument of instruction to which our attention is now to be drawn."

"We are so accustomed to hear arithmetic spoken of as one of the three fundamental ingredients in all schemes of instruction, that it seems like inquiring too curiously to ask why this should be. Reading, Writing, and Arithmetic—these three are assumed to be of co-ordinate rank. Are they indeed co-ordinate, and if so on what grounds? In this modern “trivium” the art of reading is put first. Well, there is no doubt as to its right to the foremost place. For reading is the instrument of all our acquisition. It is indispensable. There is not an hour in our lives in which it does not make a great difference to us whether we can read or not. And the art of Writing, too; that is the instrument of all communication, and it becomes, in one form or other, useful to us every day. But Counting—doing sums,—how often in life does this accomplishment come into exercise? Beyond the simplest additions, and the power to check the items of a bill, the arithmetical knowledge required of any well-informed person in private life is very limited. For all practical purposes, whatever I may have learned at school of fractions, or proportion, or decimals, is, unless I happen to be in business, far less available to me in life than a knowledge, say, of history of my own country, or the elementary truths of physics. The truth is, that regarded as practical arts, reading, writing, and arithmetic have no right to be classed together as co-ordinate elements of education; for the last of these is considerably less useful to the average man or woman not only than the other two, but than 267 many others that might be named. But reading, writing, and such mathematical or logical exercise as may be gained in connection with the manifestation of numbers, have a right to constitute the primary elements of instruction. And I believe that arithmetic, if it deserves the high place that it conventionally holds in our educational system, deserves it mainly on the ground that it is to be treated as a logical exercise. It is the only branch of mathematics which has found its way into primary and early education; other departments of pure science being reserved for what is called higher or university instruction. But all the arguments in favor of teaching algebra and trigonometry to advanced students, apply equally to the teaching of the principles or theory of arithmetic to schoolboys. It is calculated to do for them exactly the same kind of service, to educate one side of their minds, to bring into play one set of faculties which cannot be so severely or properly exercised in any other department of learning. In short, relatively to the needs of a beginner, Arithmetic, as a science, is just as valuable—it is certainly quite as intelligible—as the higher mathematics to a university student."

"Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition."

"Suppose then I want to give myself a little training in the art of reasoning; suppose I want to get out of the region of conjecture and probability, free myself from the difficult task of weighing evidence, and putting instances together to arrive at general propositions, and simply desire to know how to deal with my general propositions when I get them, and how to deduce right inferences from them; it is clear that I shall obtain this sort of discipline best in those departments of thought in which the first principles are unquestionably true. For in all 59 our thinking, if we come to erroneous conclusions, we come to them either by accepting false premises to start with—in which case our reasoning, however good, will not save us from error; or by reasoning badly, in which case the data we start from may be perfectly sound, and yet our conclusions may be false. But in the mathematical or pure sciences,—geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves,—we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline. When Plato wrote over the portal of his school. “Let no one ignorant of geometry enter here,” he did not mean that questions relating to lines and surfaces would be discussed by his disciples. On the contrary, the topics to which he directed their attention were some of the deepest problems,—social, political, moral,—on which the mind could exercise itself. Plato and his followers tried to think out together conclusions respecting the being, the duty, and the destiny of man, and the relation in which he stood to the gods and to the unseen world. What had geometry to do with these things? Simply this: That a man whose mind has not undergone a rigorous training in systematic thinking, and in the art of drawing legitimate inferences from premises, was unfitted to enter on the discussion of these high topics; and that the sort of logical discipline which he needed was most likely to be obtained from geometry—the only mathematical science which in Plato’s time had been formulated and reduced to a system. And we in this country [England] have long acted on the same principle. Our future lawyers, clergy, and statesmen are expected at the University to learn a good deal about curves, and angles, and numbers and proportions; not because these subjects have the smallest relation to the needs of their lives, but because in the very act of learning them they are likely to acquire that habit of steadfast and accurate thinking, which is indispensable to success in all the pursuits of life."

"Marshall did something much more effective than changing the answer. He changed the question. For Ricardo the Theory of Value was a means of studying the distribution of total output between wages, rent and profit, each considered as a whole. This is a big question. Marshall turned the meaning of Value into a little question: Why does an egg cost more than a cup of tea? It may be a small question but it is a very difficult and complicated one. It takes a lot of time and algebra to work out the theory of it. So it kept all Marshall’s pupils preoccupied for fifty years. They had no time to think about the big question, or even to remember that there was a big question, because they had to keep their noses right down to the grindstone, working out the theory of the price of a cup of tea."

"Though a skilled mathematician, he used mathematics sparingly. He saw that excessive reliance on this instrument might lead us astray in pursuit of intellectual toys, imaginary problems not conforming to the conditions of real life: and further, might distort our sense of proportion by causing us to neglect factors that could not easily be worked up in the mathematical machine."

"The increment of the commodity which he is only just induced to acquire (whether by... direct labor or by purchase) may be called its Marginal Increment; because he is on the margin of doubt whether it is worth his while to incur the outlay required to obtain it. And the benefit-giving power, or Utility, of that increment to him may be called the Marginal Utility of the commodity to him."

"And thus the law may be worded:— The of a commodity to anyone diminishes with every increase in the amount of it he already has."

"If we compare one country of the civilized world with another, or one part of England with another, or one trade in England with another, we find that the degradation of the working-classes varies almost uniformly with the amount of rough work done by women. The most valuable of all capital is that invested in human beings; and of that capital the most precious part is the result of the care and influence of the mother, so long as she retains her tender and unselfish instincts, and has not been hardened by the strain and stress of unfeminine work."

"[E]ach several want is limited, and... with every increase in the amount of a thing which a man has, the eagerness of his desire to obtain more of it diminishes; until it yields place to the desire for some other thing, of which perhaps he hardly thought, so long as his more urgent wants were still unsatisfied. There is an endless supply of wants, but there is a limit to each separate want. This familiar and fundamental law of human nature may pass by the name of the Law of Satiable Wants or the Law of Diminishing Utility. It may be written thus:— The Total Utility of a commodity to a person (...the total benefit or satisfaction yielded ...by it) increase with every increase in his stock of it, but not as fast as his stock increases. ...[i.e.,] the additional benefit ...from an additional increment of his stock of anything, diminishes with every increase in the stock ..."

"I have not been able to lay my hands on any notes as to Mathematico-economics that would be of any use to you. I have very indistinct memories of what I used to think on the subject. I never read mathematics now: in fact I have even forgotten how to integrate a good many things. But I know I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very well unlikely to be good economics: and I went more and more on the rules—(1) Use mathematics as a shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can’t succeed in (4), burn (3). This last I do often."

"There has always been a temptation to classify economic goods in clearly defined groups, about which a number of short and sharp propositions could be made, to gratify at once the student’s desire for logical precision, and the popular liking for dogmas that have the air of being profound and are yet easily handled. But great mischief seems to have been done by yielding to this temptation, and drawing broad artificial lines of division where Nature has made none. The more simple and absolute an economic doctrine is, the greater will be the confusion which it brings into attempts to apply economic doctrines to practice, if the dividing lines to which it refers cannot be found in real life. There is not in real life a clear line of division between things that are and are not Capital, or that are and are not Necessaries, or again between labour that is and is not Productive."

"Our understanding of how markets and businesses operate was passed down to us more than a century ago by a handful of European economists — Alfred Marshall in England and a few of his contemporaries on the continent. It is an understanding based squarely upon the assumption of diminishing returns: products or companies that get ahead in a market eventually run into limitations, so that a predictable equilibrium of prices and market shares is reached. The theory was roughly valid for the bulk-processing, smokestack economy of Marshall’s day. And it still thrives in today’s economics textbooks. But steadily and continuously in this century, Western economies have undergone a transformation from bulk - material manufacturing to design and use of technology — from processing of resources to processing of information, from application of raw energy to application of ideas. As this shift has occurred, the underlying mechanisms that determine economic behavior have shifted from ones of diminishing to ones of increasing returns."

"Political Economy or Economics is a study of mankind in the ordinary business of life; it examines that part of individual and social action which is most closely connected with the attainment and with the use of the material requisites of wellbeing."

"One of the most important skills of the economist, therefore, is that of simplification of the model. Two important methods of simplification have been developed by economists. One is the method of partial equilibrium analysis (or microeconomics), generally associated with the name of Alfred Marshall and the other is the method of aggregation (or macro-economics), associated with the name of John Maynard Keynes."

"I am myself an uncompromising anti-Jingoe, a peace-at-almost-any-price man. Chamberlain is the only Unionist public man whom I have ever thoroughly distrusted. Excepting Napoleon, I believe that England's true greatness has had no such dangerous enemy since Lord North. When a radical, he delighted to dish his colleagues even more than to flout his opponents. He is now engaged in dishing his new colleagues and flouting his old friends."

"[Y]our letter concerning my paper... has told me much... concerning the ideas current in philosophical subjects in Cambridge. I was not aware that Marshall had so long entertained notions of a quantitative theory of political economy, and think it a pity that he has so long delayed publishing something on the subject."

"The study of economics does not seem to require any specialized gifts of an unusually high order. Is it not, intellectually regarded, a very easy subject compared with the higher branches of philosophy and pure science? Yet good, or even competent, economists are the rarest of birds. An easy subject, at which very few excel! The paradox finds its explanation, perhaps, in that the master-economist must possess a rare combination of gifts. He must reach a high standard in several different directions and must combine talents not often found together. He must be mathematician, historian, statesman, philosopher – in some degree. He must understand symbols and speak in words. He must contemplate the particular in terms of the general, and touch abstract and concrete in the same flight of thought. He must study the present in the light of the past for the purposes of the future. No part of man's nature or his institutions must lie entirely outside his regard. He must be purposeful and disinterested in a simultaneous mood; as aloof and incorruptible as an artist, yet sometimes as near the earth as a politician. Much, but not all, of this many-sidedness Marshall possessed. But chiefly his mixed training and divided nature furnished him with the most essential and fundamental of the economist's necessary gifts – he was conspicuously historian and mathematician, a dealer in the particular and the general, the temporal and the eternal, at the same time."

"Jevons saw the kettle boil and cried out with the delighted voice of a child; Marshall too had seen the kettle boil and sat down silently to build an engine."

"In other branches of science, where quick publication seems to be so much desired, there may possibly be some excuse for giving to the world slovenly or ill-digested work, but there is no such excuse in mathematics. The form ought to be as 102 perfect as the substance, and the demonstrations as rigorous as those of Euclid. The mathematician has to deal with the most exact facts of Nature, and he should spare no effort to render his interpretation worthy of his subject, and to give to his work its highest degree of perfection. “Pauca sed matura” was Gauss’s motto."

"Many of the greatest masters of the mathematical sciences were first attracted to mathematical inquiry by problems relating to numbers, and no one can glance at the periodicals of the present day which contain questions for solution without noticing how singular a charm such problems still continue to exert. The interest in numbers seems implanted in the human mind, and it is a pity that it should not have freer scope in this country. The methods of the theory of numbers 271 are peculiar to itself, and are not readily acquired by a student whose mind has for years been familiarized with the very different treatment which is appropriate to the theory of continuous magnitude; it is therefore extremely desirable that some portion of the theory should be included in the ordinary course of mathematical instruction at our University. From the moment that Gauss, in his wonderful treatise of 1801, laid down the true lines of the theory, it entered upon a new day, and no one is likely to be able to do useful work in any part of the subject who is unacquainted with the principles and conceptions with which he endowed it."

"Bertrand, Darboux, and Glaisher have compared Cayley to Euler, alike for his range, his analytical power, and, not least, for his prolific production of new views and fertile theories. There is hardly a subject in the whole of pure mathematics at which he has not worked."

"The mathematician requires tact and good taste at every step of his work, and he has to learn to trust to his own instinct to distinguish between what is really worthy of his efforts and what is not; he must take care not to be the slave of his symbols, but always to have before his mind the realities which they merely serve to express. For these and other reasons it seems to me of the highest importance that a mathematician should be trained in no narrow school; a wide course of reading in the first few years of his mathematical study cannot fail to influence for good the character of the whole of his subsequent work"

"But some years after, a letter, which he received from Dr. Hooke, put him on inquiring what was the real figure, in which a body let fall from any high place descends, taking the motion of the earth round its axis into consideration. Such a body, having the same motion, which by the revolution of the earth the place has whence it falls, is to be considered as projected forward and at the same time drawn down to the centre of the earth. This gave occasion to his resuming his former thoughts concerning the moon, and Picard in France having lately measured the earth, by using his measures the moon appeared to be kept in her orbit purely by the power of gravity; and consequently, that this power decreases, as you recede from the centre of the earth, in the manner our author had formerly conjectured. Upon this principle he found the line described by a falling body to be an ellipsis, the centie of the earth being one focus. And the primary planets moving in such orbits round the sun, he had the satisfaction to see, that this inquiry, which he had undertaken merely out of curiosity, could be applied to the greatest purposes. Hereupon he composed near a dozen propositions, relating to the motion of the primary planets about the sun. Several years after this, some discourse he had with Dr. Halley, who at Cambridge made him a visit, engaged Sir Isaac Newton to resume again the consideration of this subject; and gave occasion to his writing the treatise, which he published under the title of . This treatise, full of such a variety of profound inventions, was composed by him, from scarce any other materials than the few propositions before mentioned, in the space of a year and a half."

"As for Newton himself, all that he had done never seems to have inspired him with any sentiment except that of a deeper sense of the narrow and insignificant range of his discoveries as compared with the whole mighty realm of nature. A little before his death, Dr Pemberton tells us, he observed : "I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.""

"Dr. Pemberton tells us a that the first thoughts, which gave rise to Newton's Principia, occurred to him when he had retired from Cambridge into Lincolnshire, in 1666, on account of the plague. Voltaire had his information from Mrs. Catharine Barton, Newton's favourite niece, who married Conduitt, a member of the Royal Society, and one of his intimate friends: from having spent a great portion of her life in his society, she was good authority for such an anecdote, and she related that some fruit, falling from a tree, was the accidental cause of this direction to Newton's speculations."

"If so, her motion must be influenced by it; perhaps she is retained in her orbit thereby. However, though the power of gravity is not sensibly weakened in the little change of distance, at which we can place ourselves from the centre of the earth, yet it is very possible that, so high as the moon, this power may differ much in strength from what it is here. To make an estimate what might be the degree of this diminution, he considered with himself that, if the moon be retained in her orbit by the force of gravity, no doubt the primary planets are carried round the sun by the like power. And, by comparing the periods of the several planets with their distances from the sun, he found that if any power like gravity held them in their courses, its strength must decrease in the duplicate proportion of the increase of distance. This he concluded by supposing them to move in perfect circles concentrical to the sun, from which the orbits of the greatest part of them do not much differ. Supposing therefore the power of gravity, when extended to the moon, to decrease in the same manner, he computed whether that force would be sufficient to keep the moon in her orbit. In this computation, being absent from books, he took the common estimate, in use among geographers and our seamen before Norwood had measured the earth, that 60 English miles were contained in one degree of latitude on the surface of the earth. But as this is a very faulty supposition, each degree containing about 691/2 of our miles, his computation did not answer expectation; whence he concluded, that some other cause must at least join with the action of the power of gravity on the moon. On this account he laid aside, for that time, any farther thoughts upon this matter."

"When I had the honour of his conversation, I endeavoured to learn his thoughts upon mathematical subjects, and something historical concerning his inventions, that I had not been before acquainted with. I found, he had read fewer of the modern mathematicians, than one could have expected; but his own prodigious invention readily supplied him with what he might have an occasion for in the pursuit of any subject he undertook. I have often heard him censure the handling geometrical subjects by algebraic calculations; and his book of Algebra he called by the name of Universal Arithmetic, in opposition to the injudicious title of Geometry, which Des Cartes had given to the treatise, wherein he shews, how the geometer may assist his invention by such kind of computations. He frequently praised , Barrow and Huygens for not being influenced by the false taste, which then began to prevail. He used to commend the laudable attempt of Hugo de Omerique to restore the ancient analysis, and very much esteemed Apollonius's book De sectione rationis for giving us a clearer notion of that analysis than we had before."

"The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier’s Descriptio."

"The first thoughts, which gave rise to his Principia, he had, when he retired from Cambridge in 1666 on account of the plague. As he sat alone in a garden, he fell into a speculation on the power of gravity; that as this power is not found sensibly diminished at the remotest distance from the centre of the earth to which we can rise, neither at the tops of the loftiest buildings, nor even on the summits of the highest mountains, it appeared to him reasonable to conclude that this power must extend much further than was usually thought: why not as high as the moon? said he to himself."

"It would seem at first sight as if the rapid expansion of the region of mathematics must be a source of danger to its future progress. Not only does the area widen but the subjects of study increase rapidly in number, and the work of the mathematician tends to become more and more specialized. It is, of course, merely a brilliant exaggeration to say that no mathematician is able to understand the work of any other mathematician, but it is certainly true that it is daily becoming more and more difficult for a mathematician to keep himself acquainted, even in a general way, with the progress of any of the branches of mathematics except those which form the field of his own labours. I believe, however, that the increasing extent of the territory of mathematics will always be counteracted by increased facilities in the means of communication. Additional knowledge opens to us new principles and methods which may conduct us with the greatest ease to results which previously were most difficult of access; and improvements in notation may exercise the most powerful effects both in the simplification and accessibility of a subject. It rests with the worker in mathematics not only to explore new truths, but to devise the language by which they may be discovered and expressed; and the genius of a great mathematician displays itself no less in the notation he invents for deciphering his subject than in the results attained.... I have great faith in the power of well-chosen notation to simplify complicated theories and to bring remote ones near and I think it is safe to predict that the increased knowledge of principles and the resulting improvements in the symbolic language of mathematics will always enable us to grapple satisfactorily with the difficulties arising from the mere extent of the subject"

"Quite distinct from the theoretical question of the manner in which mathematics will rescue itself from the perils to which it is exposed by its own prolific nature is the practical problem of finding means of rendering available for the student the results which have been already accumulated, and making it possible for the learner to obtain some idea of the present state of the various departments of mathematics.... The great mass of mathematical literature will be always contained in Journals and Transactions, but there is no reason why it should not be rendered far more useful and accessible than at present by means of treatises or higher text-books. The whole science suffers from want of avenues of approach, and many beautiful branches of mathematics are regarded as difficult and technical merely because they are not easily accessible.... I feel very strongly that any introduction to a new subject written by a competent person confers a real benefit on the whole science. The number of excellent text-books of an elementary kind that are published in this country makes it all the more to be regretted that we have so few that are intended for the advanced student. As an example of the higher kind of text-book, the want of which is so badly felt in many subjects, I may mention the second part of Prof. Chrystal’s “Algebra” published last year, which in a small compass gives a great mass of valuable and fundamental knowledge that has hitherto been beyond the reach of an ordinary student, though in reality lying so close at hand. I may add that in any treatise or higher text-book it is always desirable that references to the original memoirs should be given, and, if possible, short historic notices also. I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history."

"The history of mathematics cannot with certainty be traced back to any school or period before that of the…Greeks…. Though all early races …knew something of numeration yet the rules…were neither deduced from nor did they form part of any science."

"It was, however, from Spain, and not from Arabia, that a knowledge of eastern mathematics first came into western Europe. The Moors had established their rules in Spain in 747, and by the tenth or eleven century had attained a high degree of civilisation."

"Babbage was one of the founders of the Cambridge Analytical Society whose purpose he stated was to advocate "the principles of pure d-ism as opposed to the dot-age of the university."

"The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible."

"An English agronomist, Arthur Young wrote to George Washington concerning the economics of the American experience. He wanted to know what enticed or motivated the Americans to use so much land so quickly. The letters of George Washington (1801) are a fascinating response to the questions of Mr Young. Not only did Washington respond with accounts of his estate at Mount Vernon but he asked several of his colleagues, Thomas Jefferson, Alexander Hamilton, among others, about the economics of farming... Young was interested to see if slavery, the cost of labor, had a role in the methods used, but Washington was convinced that the abundance of new and very cheap land was at fault."

"A country is signally blessed above others, which can grow ."

"A vast change had taken place in English social life within two generations... View the navigation, the roads, the harbours and all other public works. Take notice of the spirit with which manufactures are carried on. Move your eye which side you will, you behold nothing but great riches and yet greater resources."

"The Englishman, in eleven years, gets three bushels more of wheat than the Frenchman. He gets three crops of barley, tares, or beans, which produce nearly twice as many bushels per acre, as what the three French crops of spring corn produce. And he farther gets, at the same time, three crops of turnips and two of clover, the turnips worth 40s. the acre, and the clover 60s. that is 12l. for both. What an enormous superiority! More wheat; almost double of the spring corn; and above 20s. per acre per annum in turnips and clover. But farther; the Englishman's land, by means of the manure arising from the consumption of the turnips and clover is in a constant state of improvement, while the Frenchman's farm is stationary."

"I have before proved the nation to be in the possession of a vast income, highly sufficient for all demands, to possess a vigorous agriculture, flourishing manufactures, and an extended commerce, in a word, to be a great industrious country. Now I conceive that it is impossible to prove such points, without proportionably proving the kingdom to be a populous one."