University Of Cambridge Faculty

"In 1841 Gregory published his Examples of the Processes of the Differential and Integral Calculus, a work which produced a great change for the better in the Cambridge mathematical books. It is the first in which constant use is made of the method known by the name of the separation of the symbols of operation, and the author has enlivened its pages by occasionally introducing historical notices of the problems discussed... His other mathematical work was A Treatise on the Application of Analysis to Solid Geometry, which was left unfinished at his death, and was completed and published by Walton in 1845. This is the first treatise in which the system of solid geometry is developed by means of symmetrical equations, and is a great advance on those of Leroy and Hymers."

"Since the beginning of the century, the general aspect of mathematics has greatly changed. A different class of problems from that which chiefly engaged the attention of the great writers of the last age has arisen, and the new requirements of natural philosophy have greatly influenced the progress of pure analysis. The mathematical theories of heat, light, electricity, and magnetism, may be fairly regarded as the achievement of the last fifty years. And in this class of researches an idea is prominent, which comparatively occurs but seldom in purely dynamical enquiries. This is the idea of discontinuity. Thus, for instance, in the theory of heat, the conditions relating to the surface of the body whose variations of temperature we are considering, form an essential and peculiar element of the problem; their peculiarity arises from the discontinuity of the transition from the temperature of the body to that of the space in which it is placed. Similarly, in the undulatory theory of light, there is much difficulty in determining the conditions which belong to the bounding surfaces of any portion of ether; and although this difficulty has, in the ordinary applications of the theory, been avoided by the introduction of proximate principles, it cannot be said to have been got ‘rid of. The power, therefore, of symbolizing discontinuity, if such an expression may be permitted, is essential to the progress of the more recent applications of mathematics to natural philosophy, and it is well known that this power is intimately connected with the theory of definite integrals. Hence the principal importance of this theory, which was altogether passed over in the earlier collection of examples. Mr Gregory devoted to it a chapter of his work, and noticed particularly some of the more remarkable applications of definite integrals to the expression of the solutions of partial differential equations. It is not improbable that in another edition he would have developed this subject at somewhat greater length. He had long been an admirer of Fourier’s great work on heat, to which this part of mathematics owes so much; and once, while turning over its pages, remarked to the writer,—“ All these things seem to me to be a kind of mathematical paradise.""

"In this chapter I shall collect those Theorems in the Differential Calculus which, depending only on the laws of combination of the symbols of differentiation, and not on the functions which are operated on by these symbols, may be proved by the method of the separation of the symbols : but as the principles of this method have not as yet found a place in the elementary works on the Calculus, I shall first state? briefly the theory on which it is founded."

"There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are"

"Mr. Gregory: Late Fellow of Trinity College, Cambridge, and author of the -well-known Examples. Few in so short a life have done so much for science. The high sense which I entertain of his merits as a mathematician, is mingled with feelings of gratitude for much valuable assistance rendered to me in my earlier essays."

"Symbolical algebra is … the science which treats of the combination of operations defined not by their nature, … but by the laws of combination to which they are subject....[W]e suppose the existence of classes of unknown operations subject to the same laws."

"The chief object of the present work is, as its title indicates, to furnish to the student examples by which to illustrate the processes of the Differential and Integral Calculus. In this respect it will be seen to agree with Professor Peacock's Collection of Examples ; and indeed if a second edition of that excellent work had been published I should not have undertaken the task of making this compilation. But as Professor Peacock informed me that he had not leisure to superintend the publication of a second edition of his "Examples" which had been long out of print, I thought that I should do a service to students by preparing a work on a similar plan, but with such modifications as seemed called for by the increased cultivation of Analysis in this University."

"It has always appeared to me that we sacrifice many of the advantages and more of the pleasures of studying any science by omitting all reference to the history of its progress: I have therefore occasionally introduced historical notices of those problems which are interesting either from the nature of the questions involved, or from their bearing on the history of the Calculus. ...[T]hese digressions may serve to relieve the dryness of a mere collection of Examples."

"Stone was by no means the first economist to produce national income accounts. Simon Kuznets, for example, had already done so for the United States. Stone’s distinctive contribution was to integrate national income into a double-entry bookkeeping format. Every income item on one side of the balance sheet had to be matched by an expenditure item on the other side, thus ensuring consistency. Stone’s double-entry method has become the universally accepted way to measure national income."

"The usefulness and importance of a system of national accounts can most readily be appreciated by returning to origins of the method and at the same time considering the actual necessity of this type of analysis. This new analytical technique was first introduced in Great Britain during the Second War. John Maynard Keynes was at that time an expert adviser to the Treasury on problems of war finance, and his assistants included Richard Stone. Keynes took as his starting point a balance between total current resources (including real gross national product) on the supply side and total consumption, investments and expenditure for the war effort on the demand side. Richard Stone's experiments in the systematic processing of the copious flood of statistical material in the form of national accounts moved Keynes to exclaim: "We are in a new era of joy through statistics". Stone's ideas on the design of national accounts were from the beginning aimed at full integration of national accounts for the various sub-sectors which between them represented the entire national management of resources. Every item of income and expenditure on one side of an account must recur as the opposite item - expenditure and income respectively - in another account. An integral system of accounts included, for example, household income and expenditure, the expenditure and revenues of the enterprise sector (expenditure, for example, including wage payments), national saving and investment, public sector spending and revenues and, finally, balances of payments vis-à-vis other countries. This double-entry accounting provided opportunities of cross-checking statistics for the numerous transactions. Figures from different sources had to tally."

"In 1945, the war ended and I was chosen to be the first director of the newly-established Department of Applied Economics in Cambridge. Between leaving the government service and taking up my new post, I had a break of about three months which I spent at the Institute of Advanced Study in Princeton. I intended to use my time there writing up my ideas on a social accounting system for the measurement of economic flows, a thing I had wanted to do for years but had not had time for during the war. What happened was that, in Princeton, I met , the Director of Intelligence at the League of Nations, who wanted a paper on the problems of defining and measuring the national income and related totals for consideration by the League's Committee of Statistical Experts. He asked me if I would undertake the work and naturally I accepted. I soon had a memorandum ready and it was discussed in Princeton while I was still there by a subcommittee convened by Loveday. Their report was eventually published by the United Nations in Geneva in 1947 under the title, Measurement of National Income and the Construction of Social Accounts, with my memorandum as an appendix."

"I have tried to make clear in this brief sketch of the nature of deductively formulated theories, at least as they occur in economics, that, although their origin lies in common observation and introspection, nevertheless they are capable of a purely mathematical development through the exact statement of a of a set of postulates which, however, since they must inevitably contain assumptions about human behaviour, require to be tested by reference to actual events."

"There is no hope, as far as I can see, of pushing the postulates of economic theory back to a set of irreducible axioms."

"A choice may have to be made between a biased estimator which is believed to be relatively accurate and an unbiased estimator which is demonstrably highly inefficient, and it is by no means obvious that the latter is always to be preferred."

"The usefulness of observation and measurement in testing economic theories arises because the theorems of economics are supposed to relate to the actual world... Any economic theorem rigorously deduced from given postulates may be regarded as a hypothesis about the actual world which xperience may show to be false."

"The object of this paper is fivefold. The first objective is to derive a practical system of demand equations which possess properties usually considered desirable from the standpoint of elementary economic theory. The second is to consider the statistical problems involved in applying the system of equations. The third is to analyse the pattern of demand for consumers' goods in terms of this system on the basis of annual data relating to the United Kingdom over the years 1920-38. The fourth is to compare the results of this system, and systems like it, with the actual state of demand in 1900. Finally, the fifth is to compare the post-war structure of demand with what might be expected from the inter-war relationshipp under free-market condition."

"Econometrics may be defined as the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference."

"In the first place, economic theory provides a comparatively detailed specification of the demand relationships for individual consumers' goods, but is by no means so specific about the system into which these relationships fit. It would be possible to close the system in very many ways, and the precise way selected would influence the regression estimates in a model involving simultaneous equations."

"We usually find a considerable degree of correlation between indicators and also between different sets of weights that we should ever think of using. Consequently we are likely to find that alternative index numbers using different weights show similar and often approximately proportionate movements. That this is so in most economic applications is due to the nature of the actual world and not simply to the formal characteristics of index number techniques."

"In the second place, a system would involve more variables than those which may be supposed to enter into demand relationships, and so the preliminary work of estimating the necessary time series would also be increased. In the third place, in the simultaneous method of estimation the estimates of the original parameters come to depend on most of the remaining estimates, and so it is quite likely that they will have large sampling variances."

"From 1931 to 1935, I was an undergraduate at Cambridge in my father's old college, Gonville and Caius, which was particularly strong in medicine and the law. However, after two years of law I switched to economics, much to my father's disappointment. At that time the world was in the depth of the great depression and my motive for wanting to change subject was the belief, bred of youthful ignorance and optimism, that if only economics were better understood, the world would be a better place."

"The value of a yellow metal, originally chosen as money because it tickled the fancy of savages, is clearly a chancy and irrelevant thing on which to base the value of our money and the stability of our industrial system.""

"Islands of conscious power in this ocean of unconscious co-operation like lumps of butter coagulating in a pail of buttermilk."

"Professor Sir Dennis Robertson and Keynes were closely associated in the development of their ideas in monetary economics during the 1920s. Robertson had published his textbook Money in 1922 which went through a number of editions."

"Looking back the great American ‘stabilisation’ [and boom] of 1922-1929 was really a vast attempt to destabilise the value of money in terms of human effort by means of a colossal programme of investment [driven by too easy credit]... which succeeded for a surprisingly long period, but which no human ingenuity could have managed to direct indefinitely on sound and balanced lines. [and therefore it ended dramatically in the huge 1929 stock market crash followed by the Great Depression.]"

"I regard Mr. Hawtrey as my grandparent and Mr. Robertson as my parent in the paths of errancy, and I have been greatly influenced by them."

"I seek, after George Berkeley, especially, to elaborate a notion of the real itself as linguistic, and as divine language, and, after Robert Lowth and Johann Georg Hamann, to develop a theory of human being as linguistic being which participates in the divine linguistic being."

"In focusing in this section on Lowth and Berkeley, I am also deliberately trying to disinter a spirit within British tradition which runs counter to the myth of British identity to which it has nonetheless surrendered: that it is empiricist, philistine and basely pragmatic."

"Perhaps the most surprise, the most shock, should arise when what is said is really most orthodox and ancient, since the tradition is so rarely re-performed in practice today."

"In some respects, cities are bottom-up processes, but they are also top-down initiatives – and elites soon develop. Religion is older than the city, but the elites quickly build temples and turn themselves into priests. As they become more powerful the temples they build also have huge storehouses. What you see is that the man who has a surplus of material is the powerful man. This is because everyone is living hand-to-mouth, and if you have a bad harvest one year, well, you're stuffed."

"Writing certainly develops as an elite initiative at this [ancient] time. It helps them to harness the workforce – it's a way of communicating with the gods; and it's a way for the elites to list what you have, and what they have."

"If you are looking at fine buildings or the literature of the period, you have to be aware that you are only dealing with the mindset of the elite."

"Civilisation as a term suggests human agency. […] It is the way in which people have articulated how and why they wanted to live in communities. It's about how we imagine the perfect community. […] I think the most ambitious thing humans have ever done is deciding to live together with people whom they didn't know in cities. It's really difficult to build a community, to learn to celebrate difference and to live harmoniously together, and we fail all the time."

"When I was going through graduate school at Cambridge it appeared that humanities scholarship, stung by a decade of unsympathetic Thatcherite rule, had withdrawn into its own world. Academic terminology, partly fuelled by a mania for post-modernist dogma, had become evermore complex and impenetrable to anyone outside of the rarified circle of academe."

"Civilisation as a term suggests human agency. Things don't come together organically."

"But archaeology […] often acts as a parallel narrative to history: if you start digging with a textbook in your hand, you are soon going to get confused. You have to be comfortable with the idea that there are many different versions of history."

"There are winners and losers – and human will created the world we live in."

"Had Cain been Scot, God would have changed his doom; Not made him wander, but continued him home."

"Like an ambassador that beds a queen With the nice caution of a sword between."

"He that to ancient wreaths can bring no more From his own worth, dies bankrupt on the score."

"The marigold, whose courtier's face Echoes the sun, and doth unlace Her at his rise, at his full stop Packs and shuts up her gaudy shop."

"To Namier it had seemed obvious that political theories act as the merest ex post facto rationalisations of political behaviour. If we are looking for explanations of political action, he maintained, we must seek them at the level of ‘the underlying emotions, the music, to which ideas are a mere libretto, often of very inferior quality’. For critics of Namier such as Sir Herbert Butterfield, the only possible retort seemed to be to go back to a famous dictum of Lord Acton’s to the effect that ideas are often the causes rather than the effects of public events. But this response duly incurred the scorn of Namier and his followers for the alleged naiveté of supposing that political actions are ever genuinely motivated by the principles used to rationalise them."

"One of the present values of the past is as a repository of values we no longer endorse, of questions we no longer ask. One corresponding role for the intellectual historian is that of acting as a kind of archaeologist, bringing buried intellectual treasure back to the surface, dusting it down and enabling us to consider what we think of it."

"It is remarkably difficult to avoid falling under the spell of our own intellectual heritage. As we analyse and reflect on our normative concepts, it is easy to become bewitched into believing that the ways of thinking about them bequeathed to us by the mainstream of our intellectual traditions must be the ways of thinking about them. … The history of philosophy, and perhaps especially of moral, social and political philosophy, is there to prevent us from becoming too readily bewitched. The intellectual historian can help us to appreciate how far the values embodied in our present way of life, and our present ways of thinking about those values, reflect a series of choices made at different times between different possible worlds. This awareness can help to liberate us from the grip of any one hegemonal account of those values and how they should be interpreted and understood. Equipped with a broader sense of possibility, we can stand back from the intellectual commitments we have inherited and ask ourselves in a new spirit of enquiry what we should think of them."

"Within a surprisingly short space of time, however, the fortunes of the neo-roman theory began to decline and fall. … One reason for this collapse was that the social assumptions underlying the theory began to appear outdated and even absurd. With the extension of the manners of the court to the bourgeoisie in the early eighteenth century, the virtues of the independent country gentleman began to look irrelevant and even inimical to a polite commercial age. The hero of the neo-roman writers came to be viewed not as plain-hearted but as rude and boorish; not as upright but as obstinate and quarrelsome; not as a man of fortitude but of mere insensibility."

"Both parties to the dispute agree that one of the primary aims of the state should be to respect and preserve the liberty of its individual citizens. One side argues that the state can hope to redeem this pledge simply by ensuring that its citizens do not suffer any unjust or unnecessary interference in the pursuit of their chosen goals. But the other side maintains that this can never be sufficient, since it will always be necessary for the state to ensure at the same time that its citizens do not fall into a condition of avoidable dependence on the goodwill of others."

"The lack of freedom suffered by those who advise the powerful may of course be due to coercion or force. But the slavish behavior typical of such counselors may equally well be due to their basic condition of dependence and their understanding of what their clientage demands of them. As soon as they begin to ‘slide into a blind dependence upon one who has wealth and power’, they begin to desire ‘only to know his will’, and eventually ‘care not what injustice they do, if they may be rewarded’."

"The figure [neo-roman writers] wish to hold out for our admiration is described again and again. He is plain and plain-hearted; he is upright and full of integrity; above all he is a man of true manliness, of dependable valour and fortitude. His virtues are repeatedly contrasted with the vices characteristic of the obnoxious lackeys and parasites who flourish at court. The courtier, instead of being plain and plain-hearted, is lewd, dissolute and debauched; instead of being upright, he is cringing, servile and base; instead of being brave, he is fawning, abject and lacking in manliness."

"Many historians make it a principal part of their business to investigate and explain the unfamiliar beliefs we encounter in past societies. But what is the relationship between our provision of such explanations and our assessment of the truth of such beliefs? The question is obviously a highly intractable one, but no practising historian can hope to evade it, as many philosophers have recently and rightly pointed out."

"Having gestured at the concept of rationality, I ought to stress that I intend nothing very grand or precise by that much abused term. When I speak of agents as having rational beliefs, I mean only that their beliefs (what they hold to be true) should be suitable beliefs for them to hold true in the circumstances in which they find themselves. A rational belief will thus be one that an agent has attained by some accredited process of reasoning."