Logicians From England

"It was always my affectation even from a child, in all pieces of Learning or Knowledge, not merely to learn by rote, which is soon forgotten, but to know the grounds or reasons of what I learn; to inform my Judgement, as well as furnish my Memory; and thereby, make a better Impression on both."

"At Christmass 1631, (a season of the year when Boys use to have a vacancy from School,) I was, for about a fortnight, at home with my Mother at Ashford. I there found that a younger Brother of mine (in Order to a Trade) had, for about 3 Months, been learning (as they call'd it) to Write and Cipher, or Cast account, (and he was a good proficient for that time,) When I had been there a few days; I was inquisitive to know what it was, they so called. And (to satisfie my curiosity) my Brother did (during the Remainder of my stay there before I return'd to School) shew me what he had been Learning in those 3 Months. Which was (besides the writing a fair hand) the Practical part of Common Arithmetick in Numeration, Addition, Substraction, Multiplication, Division, The Rule of Three (Direct and Inverse) the Rule of Fellowship (with and without, Time) the Pule of False-Position, Rules of Practise and Reduction of Coins, and some other little things. Which when he had shewed me by steps, in the same method that he had learned them; and I had wrought over all the Examples which he before had done in his book; I found no difficulty to understand it, and I was very well pleased with it: and thought it ten days or a fortnight well spent. This was my first insight into Mathematicks; and all the Teaching I had."

"I made no Scruple of diverting (from the common Road of Studies then in fashion) to any part of Useful Learning. Presuming, that Knowledge is no Burthen; and, if of any part thereof I should afterwards have no occasion to make use, it would at least do me no hurt; And what of it l might or might not have occasion for, I could not then foresee."

"As to Divinity, (on which I had an eye from the first,) l had the happiness of a strict and Religious Education, all along from a Child: Whereby I was not only preserved from vicious Courses, and acquainted with Religious Exercises; but was early instructed in the Principles of Religion, and Catachetical Divinity, and the frequent Reading of Scripture, and other good Books, and diligent attendance on Sermons. (And whatever other Studies I followed, I was careful not to neglect this.) And became timely acquainted with Systematick and Polemick Theology. And had the repute of a good Proficient therein."

"Being encouraged by... success, beyond expectation; I afterwards ventured on many others and scarce missed of any, that I undertook, for many years, during our civil Wars, and afterwards. But of late years, the French Methods of Cipher are grown so intricate beyond what it was wont to be, that I have failed of many; tho' I have master'd divers of them. Of such deciphered Letters, there be copies of divers remaining in the Archives of the Bodleyan Library in Oxford; and many more in my own Custody, and with the Secretaries of State."

"About the year 1645 while, I lived in London (at a time, when, by our Civil Wars, Academical Studies were much interrupted in both our Universities:) beside the Conversation of divers eminent Divines, as to matters Theological; I had the opportunity of being acquainted with divers worthy Persons, inquisitive into Natural Philosophy, and other parts of Humane Learning; And particularly of what hath been called the New Philosophy or Experimental Philosophy. We did by agreement, divers of us, meet weekly in London on a certain day, to treat and discourse of such affairs. ...Some of which were then but New Discoveries, and others not so generally known and imbraced, as now they are, with other things appertaining to what hath been called The New Philosophy; which, from the times of Galileo at Florence, and Sr. Francis Bacon (Lord Verulam) in England, hath been much cultivated in Italy, France, Germany, and other Parts abroad, as well as with us in England. About the year 1648, 1649, some of our company being removed to Oxford (first Dr. Wilkins, then I, and soon after Dr. Goddard) our company divided. Those in London continued to meet there as before... Those meetings in London continued, and (after the King's Return in 1660) were increased with the accession of divers worthy and Honorable Persons; and were afterwards incorporated by the name of the Royal Society, &c. and so continue to this day."

"On March 4. 1644, 5. I married Susanna daughter of John and Rachel Glyde of Northjam in Sussex; born there about the end of January 1621, 2. and baptised Feb. 3 following. By whom I have (beside other children who died young) a Son and two Daughters now surviving; John born Dec. 26 1650. Anne born June 4. 1656. and Elizabeth born Sept. 23 1658. ...My Wife died at Oxford Mar. 17. 1686, 7. after we had been married more than 42 years."

"Newton opposed entangled and tedious algebraic calculations to simple, elegant geometric constructions; Wallis opposed difficult, embrangled geometric diagrams to simplified, rationally lettered diagrams and the symbolic expressions they enabled."

"Wallis used Arithmetic of infinities to produce genuine understanding rather than brutal persuasion."

"[W]hereas Nature, in propriety of Speech, doth not admit more than Three (Local) Dimensions, (Length, Breadth and Thickness, in Lines, Surfaces and Solids;) it may justly seem improper to talk of a Solid (of three Dimensions) drawn into a Fourth, Fifth, Sixth, or further Dimension. A Line drawn into a Line, shall make a Plane or Surface; this drawn into a Line, shall make a Solid. But if this Solid be drawn into a Line, or this Plane into a Plane, what shall it make? A Plano-plane? This is a Monster in Nature, and less possible than a Chimera or a Centaure. For Length, Breadth and Thickness, take up the whole of Space. Nor can our Fansie imagine how there should be a Fourth Local Dimension beyond these Three."

"I came across the mathematical writings of Torricelli... which... I read in... 1651... where... he expounds the geometry of indivisibles of Cavalieri. ...His method, as taught by Torricelli... was indeed all the more welcome to me because I do not know that anything of that kind was observed in the thinking of almost any mathematician I had previously met; for what holds for most... concerning the circle... usually had by polygons with an infinite number of sides, and... the circumference by... an infinite number of infinitely short lines... could.., it seemed to me, with... changes, be... adjusted to other problems; and... by that means examine... Euclid, Appolonius and especially... Archimedes. ...I began to think ...whether this might bring ...light to the quadrature of the circle."

"You may find this work (if I judge rightly) quite new. For I see no reason why I should not proclaim it; nor do I believe that others will take it wrongly. ...it teaches all by a new method, introduced by me for the first time into geometry, and with such clarity that in these more abstruse problems no-one (as far as I know) has used..."

"Perhaps it would have been more prudent, if I were only writing to seek fame, to have presented some few particular propositions—as something admirable or stupefying—with apagogic proofs, concealing the method by which they were reached... Quite often they [the ancients] seem to have thought of doing this in order that others would marvel at them rather than understand; at least, so that these others, being compelled, produce their assent to those utterances of the mathematicians rather than understand a genuine investigation of the problem."

"It hath been my Lot to live in a time, wherein have been many and great Changes and Alterations. It hath been my endeavour all along, to act by moderate Principles, between the Extremities on either hand, in a moderate compliance with the Powers in being, in those places, where it hath been my Lot to live, without the fierce and violent animosities usual in such Cases, against all, that did not act just as I did, knowing that there were many worthy Persons engaged on either side. And willing whatever side was upmost, to promote (as I was able) any good design for the true Interest of Religion, of Learning, and the publick good; and ready so to do good Offices, as there was Opportunity; And, if things could not be just, as I could wish, to make the best of what is: And hereby, (thro' God's gracious Providence) have been able to live easy, and useful, though not Great."

"It is not unknown to those who know any Thing of publike Affairs, of how great Concernment it is, especially in civill Commotions, for those who are to manage such Transactions, to be furnished with continuall Intelligence from their Correspondents, yet so as to conceal their Councells and Resolutions from the adverse Party. And to this Purpose, in all Ages, much Care and lndustry hath been still used, how in Matters of Consequence, to convey Intelligence safely and secretly to those with whom they hold Correspondence, so as not to bee intercepted by the Enemy, or if intercepted, at least not discovered. And as this is no where of more Concernment, so no where more difficult, than in civill Wars, where the intermingling of opposite Parties makes it difficult, if not impossible, to distinguish Friends and Foes."

"Upon this Occasion many Methods have been invented of secret Writing, or Writing in Cipher, a Thing heretofore scarce known to any but the Secretaries of Princes, or others of like Condition; but of late Years, during our Commotions and civill Wars in England, grown very common and familiar, so that now there is scarce a Person of Quality, but is more or lesse acquainted with it, and doth as there is Occasion, make use of it."

"I was... informed, that Baptista Porta, and one or two more, had written somewhat of that Subject, upon this Information I was willing to see whether I might from any of them find any Directions, that might help mee, if I should afterwards have the like Occasion: But I found very little in any of them for my Purpose. Their Businesse being for the most Part, onely to shew how to write in Cipher, (which was not my Work,) and that Things so written were beyond the Skill of Men to decipher. Onely in Baptista Porta (who alone if I mistake not, hath written any Thing to Purpose about deciphering, and was it seemes famous in his Time for his Abilities that Way;) I found that there were some general Directions, such as were obvious from the Nature of the Thing, and which I had before of myself taken Notice of, and made use of so far as the Nature of an intricate Cipher would permit. But the Truth of it is, there are scarce any of his Rules, which the present Way of Cipher (which is now much improved, beyond what, it seemes, it was in his Days) doth not in a Manner wholly elude..."

"I saw, there was little or no Help to bee exspected from others; but that if I should have further Occasions of that Kind, I must trust to my owne Industry, and such Observations as the present Case should afford. And indeed the Nature of the Thing is scarce capable of any other Directions; every new Cipher allmost being contrived in a new Way, which doth not admit any constant Method for the finding of it out: But hee that will do any Thing in it, must first furnish himself with Patience and Sagacity, as well as hee may, and then Consilium in arenâ capere, and make the best Conjectures hee can, till hee shall happen upon something that hee may conclude for Truth."

"Accountants eventually became comfortable with using negative numbers... but for a long time mathematicians remained wary... the negatives were known as absurd numbers—numeri absurdi...Consider this equation:\frac{-1}{\quad 1} = \frac{\quad 1}{-1}...it states that the ratio of a smaller number, -1, to a larger number, 1, is equal to the ratio of a larger number, 1, to a smaller one, -1. The paradox was much discussed... To make sense of negative numbers, many mathematicians, including Leonhard Euler, came to the bizarre conclusion that they were larger than infinity. ...One voice of clarity among the confusion belonged to... John Wallis, who devised a powerful visual interpretation for the negative numbers. In his 1685 work A Treatise of Algebra, he first described the ","... By replacing the idea of quantity with the idea of position, Wallis argued that negative numbers were neither "Unuseful [nor] Absurd,"...It took a few years for Wallis' idea to hit the mainstream, but... it is the most successful explicatory diagram of all time."

"Vieta died in 1603, Porta died in 1615, and Dr. Wallis was born in 1617. So that, in all Probability, here is a great Deduction to be made from the many hundred Years, in which we were to have understood that the Art of Decyphering had been in being before Dr. Wallis was born."

"It is customary to consider Chebyshev, Gauss, Jacobi, and Legendre as the main creators of the theory of orthogonal polynomials. However, their contributions were directly influenced by Brouncker and Wallis who, in March of 1655, made discoveries which influenced the development of analysis for the next hundred years. Namely, Wallis found an infinite product of rational numbers converging to 4/π and Brouncker gave a remarkable continued fraction for this quantity. ...The only mathematician who understood the importance of these discoveries was Euler. ...he felt that the recovery of the original Brouncker's proof could open up new perspectives for analysis. As usual, Euler was right."

"By March of 1655 John Wallis had almost completed his Arithmetica Infinitorum in which he promoted an important method of interpolation. This was a great work. ...Wallis discovered that analytic formulas can be interpolated by their values at integer numbers. ...Wallis successfully applied his interpolation to find formulas for the areas under many curves. Only one curve remained uncovered. It was the unit circle. In 1593 Viète had found the formula \frac{2}{\pi} = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdot\cdots. Since the multipliers in Viète's formula are algebraic irrationalities of increasing order, it was not the formula which could meet Wallis' requirements. Finally in March of 1655, Wallis obtained his now well-known formula \frac{2}{\pi} = \frac{1\cdot3}{2\cdot2}\cdot\frac{3\cdot5}{4\cdot4}\cdot\frac{5\cdot7}{6\cdot6}\cdot\cdots\frac{(2n-1)\cdot(2n+1)}{2n\cdot2n}\cdot\cdots."

"During the wars between Charles I and Cromwell, Wallis's sympathies were with Cromwell, and he was of great service in reading royalist dispatches written in cipher. In fact, he was one of the most famous cryptologists of his day."

"Wallis did not become interested in mathematics till the age of thirty-one, but devoted himself to the subject for the rest of his life. One of the earliest and most important books on algebra ever written in English was his treatise published in 1685. It contains a brief historical sketch of the subject which is unfortunately not entirely accurate, but his treatment of the theory and practice of arithmetic and algebra has made the book a standard work for reference ever since."

"The true "principle of number," for Wallis as for Stevin, is the "nought". It is the sole numerical analogue of the geometric point (just as the instant is the temporary analogue... Wallis expressly rejects the accusation that he is relinquishing the unanimous opinion of the ancients and the moderns, who all saw the unit as the element of number. ...the traditional opinion can be brought into accord with his own if the following distinction is taken account of: Something can be a "principle" of something (1) which is the "first which is such" (primum quod sic) as to be of the same nature as the thing itself and (2) which is the last which is not" (ultimum quod non) such as to be of the same nature of the thing itself. In the first sense the unit may indeed be called the "principle of number," while the nought is a "principle" in the second sense. ...The ancients... overlooked the fact that the analogy which exists is not between the "point" and the "unit," but between the point and the "nought." For this reason they were able to develop their algebra only for "geometric magnitudes"..."

"Paralleling what happened in France, an English group centered about John Wallis began in 1645 to hold meetings in Gresham College, London, These men emphasized mathematics and astronomy. The group was given a formal charter by Charles II in 1662 and adopted the name of the Royal Society of London for the Promotion of Natural Knowledge."

"[E]arly analytic geometers—Descartes in particular—did not accept that geometry could be based on numbers or algebra. Perhaps the first to take the idea of arithmetizing geometry seriously was Wallis... [(1657) Mathesis universalis. Opera 1, 11-228.] Chs. XXIII and XXV, gave the first arithmetic treatment of Euclid's Books II and V, and he had earlier given purely algebraic treatment of s [(1655) De sectionibus conicus. Opera 1, 291-364.]. He initially derived equations from classical definitions by sections of the cone but then proceeded to derive their properties from the equations, "without the embranglings of the cone," as he put it."

"The greatest of modern have been so far from adding any thing of importance to the discoveries of ancient mathematicians, that some of their most splendid inventions are either wholly erroneous or remarkable instances of the possibility of deducing true conclusions from unscientific and false principles. Strange, however as this assertion may seem, the following elementary treatise demonstrates it to be true; by showing that all the leading propositions of the Arithmetic of Infinites of Dr. Wallis are false, and that the Doctrine of Fluxions is a baseless fabric, and in the language of the ingenious Bishop Berkley, "must be considered only as a presumption, as a knack, an art, or rather an artifice, but not a scientific demonstration."

"‘Yet some few of such investigations we have in the five first propositions of Euclid’s thirteenth book … seems to be the work of Theo, […] rather than of Euclid himself.’"

"As he was unacquainted with the he could not effect the quadrature of the circle, whose equation is y = (1 - x^2)^\frac{1}{2}, since he was unable to expand this in powers of x. He laid down however the principle of interpolation. He argued that as the ordinate of the circle is the geometrical mean between the ordinates of the curves y = (1 - x^2)^0 and y = (1 - x^2)^1, so as an approximation its area might be taken as the geometrical mean between 1 and \frac{2}{3}. This is equivalent to taking 4\sqrt{\frac{2}{3}} or 3.26... as the value of \pi. But, he continued, we have in fact a series 1, \frac{2}{3}, \frac{8}{15}, \frac{16}{35},... and thus the term interpolated between 1 and \frac{2}{3} ought to be so chosen as to obey the law of this series. This by an elaborate method leads to a value for the interpolated term which is equivalent to making\pi = 2\frac{2\cdot2\cdot4\cdot4\cdot6\cdot6\cdot8\cdot8...}{1\cdot3\cdot3\cdot5\cdot5\cdot7\cdot7\cdot9...}The subsequent mathematicians of the seventeenth century constantly used interpolation to obtain results which we should attempt to obtain by direct algebraic analysis."

"In 1659 Wallis published a tract on s in which incidentally he explained how the principles laid down in his Arithmetica infinitorum could be applied to the rectification of s: and in the following year one of his pupils, by name William Neil, applied the rule to rectify the x^3 = ay^2. This was the first case in which the length of a curved line was determined by mathematics, and as all attempts to rectify the ellipse and hyperbola had (necessarily) been ineffectual, it had previously been generally supposed that no curves could be rectified."

"In 1665 Wallis published the first systematic treatise on Analytical conic sections. Analytical geometry was invented by Descartes and the first exposition of it was given in 1637: that exposition was both difficult and obscure, and to most of his contemporaries, to whom the method was new, it must have been incomprehensible. Wallis made the method intelligible to all mathematicians. This is the first book in which these curves are considered and defined as curves of the second degree and not as sections of a cone."

"A particle moving with a uniform velocity would be denoted by Wallis by the formula s = vt, ...while previous writers would have denoted the same relation by stating what is equivalent to the proposition s1 : s2 = v1t1 : v2t2 (see e.g. Newton's Principia, bk. I. sect. I., lemma 10 or 11)."

"Wallis rejected as absurd and inconceivable the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity."

"The foundations of the new analysis were laid in the second half of the seventeenth century when Newton... and Leibnitz... founded the Differential and Integral Calculus, the ground having been to some extent prepared by the labours of Huyghens, Fermat, Wallis, and others. By this great invention of Newton and Leibnitz, and with the help of the brothers James Bernoulli... and John Bernoulli... the ideas and methods of the Mathematicians underwent a radical transformation which naturally had a profound effect upon our problem. The first effect of the new analysis was to replace the old geometrical or semi-geometrical methods of calculating \pi by others in which analytical expressions formed according to definite laws were used, and which could be employed for the calculation of \pi to any assigned degree of approximation."

"The first result of this kind was due to John Wallis... undergraduate at Emmanuel College, Fellow of Queen's College, and afterwards Savilian Professor of Geometry at Oxford. He was the first to formulate the modern arithmetic theory of limits, the fundamental importance of which, however, has only during the last half century received its due recognition; it is now regarded as lying at the very foundation of analysis. Wallis gave in his Arithmetica Infinitorum the expression\frac{\pi}{2} = \frac {2}{1}\cdot\frac {2}{3}\cdot\frac {4}{3}\cdot\frac {4}{5}\cdot\frac {6}{5}\cdot\frac {6}{7}\cdot\frac {8}{7}\cdot\frac {8}{9}\cdotsfor \pi as an infinite product, and he shewed that the approximation obtained at stopping at any fraction in the expression on the right is in defect or in excess of the value \frac{\pi}{2} according as the fraction is proper or improper. This expression was obtained by an ingenious method depending on the expression for \frac{\pi}{8} the area of a semi-circle of diameter 1 as the definite integral \int\limits_{0}^{1}\sqrt{x-x^2}dx. The expression has the advantage over that of Vieta that the operations required are all rational ones."

"However, it is not unlikely that the Arabs, who received from the Indians the numeral figures (which the Greeks knew not), did from them also receive the use of them, and many profound speculations concerning them, which neither Latins nor Greeks know, till that now of late we have learned them from thence. From the Indians also they might learn their algebra, rather than from Diophantus."

"Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates."

"...the most interesting fact about the Tarot pack [is] that it is the subject of the most successful propaganda campaign ever launched: not by a very long way the most important, but the most completely successful. An entire false history, and false interpretations, of the Tarot pack was concocted by the occultists; and it is all but universally believed."

"Such were the lucidity of exposition and his mastery of the topic that it seems possible that, had he ever published it, the political theory of Britain would have been significantly different."

"Philosophy can take us no further than enabling us to command a clear view of the concepts by means of which we think about the world, and, by so doing, to attain a firmer grasp of the way we represent the world in our thought. It is for this reason and in this sense that philosophy is about the world."

"I need hardly stay to demonstrate that facts are valueless unless connected and explained by a correct theory; that analogies are very dangerous grounds of inference, unless carefully founded on similar conditions; and that experience misleads if it be misinterpreted."

"Pray accept my best thanks for your kindness in sending me a copy of your Memoir, and for the very courteous letter in which you draw my attention to it. When your letter came I had, indeed, already noticed in the Journal des Economistes your very remarkable theory. I felt the greater interest in the subject because my own speculations have led me in the same direction, now for the last twelve years or more. It is satisfactory to me to find that my theory of exchange, which, when published in England, was either neglected or criticised, is practically confirmed by your researches. I do not know whether you are acquainted with my writings on the subject. All the chief points of my mathematical theory were clear to my own mind by the year 1862, when I drew up a brief account of it, which was read at the meeting of the British Association at Cambridge... A very brief abstract was then alone inserted in the report, but the original paper was printed in the journal of the London Statistical Society in 1866... Finally, in 1871, I caused to be published... the Theory of Political Economy, in which is given a full explanation of the theory, with the aid of mathematical symbols. [...] You will find, I think, that your theory substantially coincides with and confirms mine, although the symbols are differently chosen, and there are incidental variations. You will see that the whole theory rests on the notion... that the utility of a commodity is not proportional to its quantity; what you call the rarity of a commodity appears to be exactly what I called the coefficient of utility at first, and afterwards the degree of utility, which... was really the differential coefficient of the utility considered as the function of the quantity of commodity. The theory of exchange is given... and may be considered to be contained in one sentence. An equation may thus be established on either side between the utility gained and sacrificed at the ratio of exchange of the whole commodities, upon the last increments exchanged. Now in my book of 1871, I show fully how this theory may be expressed in symbols. [...] Indeed, when the meaning of the terms is explained, your proposition 'Les prix courants ou prix d'equilibre sont égaux aux rapports des raretés' is seen to coincide precisely with my theory, except that you do not point out how many equations are requisite, or how many unknown quantities there are. The publication of your paper... tends to confirm my belief in the correctness of the theory, but it might lead to misapprehensions as to the originality and priority... I shall therefore take it as a favour if you will kindly inform me whether you are sufficiently acquainted with my writings, or whether you would desire me to forward a copy of my Theory of Political Economy."

"Coal in truth stands not beside, but entirely above, all other commodities. It is the material source of the energy of the country—the universal aid—the factor in everything we do. With coal almost any feat is possible or easy; without it we are thrown back into the laborious poverty of earlier times."

"It was during the year 1851, while living almost unhappily among thoughtless, if not bad companions, in Gower Street a gloomy house on which I now look with dread it was then, and when I had got a quiet hour in my small bedroom at the top of the house, that I began to think that I could and ought to do more than others. A vague desire and determination grew upon me. I was then in the habit of saying my prayers like any good church person, and it was when so engaged that I thought most eagerly of the future, and hoped for the unknown. My reserve was so perfect that I suppose no one had the slightest comprehension of my motives or ends. My father probably knew me but little. I never had any confidential conversation with him. At school and college the success in the classes was the only indication of my powers. All else that I intended or did was within or carefully hidden. The reserved character, as I have often thought, is not pleasant nor lovely. But is it not necessary to one such as I? Would it have been sensible or even possible for a boy of fifteen or sixteen to say what he was going to do before he was fifty? For my own part I felt it to be almost presumptuous to pronounce to myself the hopes I held and the schemes I formed. Time alone could reveal whether they were empty or real; only when proved real could they be known to others."

"I used to think I should like to be a bookbinder or bookseller it seemed to me a most delightful trade and I wished or thought of nothing better. More lately I thought I should be a minister, it seemed so serious and useful a profession, and I entered but little into the merits of religion and the duties of a minister. Every one dissuaded me from the notion, and before I arrived at any age to require a real decision, science had claimed me."

"I cannot consent with the Radical party to obliterate a glorious past, nor can I consent with the Conservatives to prolong abuses into the present. I wish with all my heart to aid in securing all that is good for the masses, yet to give them all they wish and are striving for is to endanger much that is good beyond their comprehension. I cannot pretend to underestimate the good that the English monarchy and aristocracy, with all the liberal policy actuating it, does for the human race, and yet I cannot but fear the pretensions of democracy against it are strong, and in some respects properly strong. This antithesis and struggle, perhaps, after all, is no more than has always more or less existed, but is now becoming more marked. Compromise, perhaps, is the only resource. Those who rightly possess the power in virtue of their superior knowledge must yield up some, that they may carry with them the honest but uncertain wills of those less educated but more numerous and physically powerful."

"It is very commonly urged, that the failing supply of coal will be met by new modes of using it efficiently and economically. ...It is wholly a confusion of ideas to suppose that the economical use of fuel is equivalent to a diminished consumption. The very contrary is the truth."

"You will perceive that economy, scientifically speaking, is a very contracted science; it is in fact a sort of vague mathematics which calculates the causes and effects of man's industry, and shows how it may be best applied. There are a multitude of allied branches of knowledge connected with mans condition; the relation of these to political economy is analogous to the connexion of mechanics, astronomy, optics, sound, heat, and every other branch more or less of physical science, with pure mathematics."

"One of the most important axioms is, that as the quantity of any commodity, for instance, plain food, which a man has to consume, increases, so the utility or benefit derived from the last portion used decreases in degree. The decrease in enjoyment between the beginning and the end of a meal may be taken as an example. And I assume that on an average, the ratio of utility is some continuous mathematical function of the quantity of commodity. This law of utility has, in fact, always been assumed by political economists under the more complex form and name of the Law of Supply and Demand. But once fairly stated in its simple form, it opens up the whole of the subject."