Augustus De Morgan Quotes (1806-1871)

“I have throughout introduced the Integral Calculus in connexion with the Differential Calculus. ...Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction? The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold...”

“We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it can see better with one eye than with two.”

“I cannot see why it is necessary that every deduction from algebra should be bound to certain conventions incident to an earlier stage of mathematical learning, even supposing them to have been consistently used up to the point in question. I should not care if any one thought this treatise unalgebraical, but should only ask whether the premises were admissible and the conclusions logical.”

“The following Treatise... has been endeavoured to make the theory of limits, or ultimate ratios... the sole foundation of the science, without any aid whatsoever from the theory of series, or algebraical expansions. I am not aware that any work exists in which this has been avowedly attempted, and I have been the more encouraged to make the trial from observing that the objections to the theory of limits have usually been founded either upon the difficulty of the notion itself, or its unalgebraical character, and seldom or never upon anything not to be defined or not to be received in the conception of a limit...”

“Although there is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear. This necessarily follows in a science which consists of reasoning from the very commencement, for it is evident that every student will feel a claim to have his objections answered, not by authority, but by argument, and that the intelligent student will perceive more readily than another the force of an objection and the obscurity arising from an unexplained difficulty, as the greater is the ordinary light the more will occasional darkness be felt. To remove some of these difficulties is the principal object of this Treatise.”

“The late Professor Leslie... [i]n his Philosophy of Arithmetic... entered... into much of its history. ...[O]ne principal, thing to be cautious of is, his almost monomaniac antipathy to every thing Hindoo—a most unfortunate turn... Leslie... generalises... fearfully every now and then. He informs us that it was the practice throughout Europe to reduce the rules of arithmetic to memorial verses, and that [William] Buckley's Arithmetica Memorativa appears at one period to have gained possession of the schools and colleges of England. Now the truth... the verses attributed to Sacrobosco had never... been printed when Leslie wrote; and Buckley... was printed only once... and two or three times as an appendix to a work on logic. Dr. Peacock expresses the truth in saying... before the invention of printing, the practice of writing memorial verses was common, as appears by manuscript libraries. ...[H]ad the practice of using them been common, the presses of the fifteenth and sixteenth centuries would have given them forth in great numbers. But I cannot learn that any metrical work was printed in the fifteenth century, except the Compotus of [Magister] Anianus, and that only once.”

“One day at least in every week, The sects of every kind Their doctrines here are sure to seek, And just as sure to find.”

“Common integration is only the memory of differentiation... The artifices by which it [integration] is effected, are changes, not from the unknown to the known, but from the forms in which memory will not serve to those in which it will”

“Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum. And the great fleas themselves, in turn, have greater fleas to go on; While these again have greater still, and greater still, and so on.”

“Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum, And the great fleas themselves, in turn, have greater fleas to go on, While these again have greater still, and greater still, and so on.”

“Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum. And the great fleas themselves, in turn, have greater fleas to go on; While these again have greater still, and greater still, and so on.”

“There never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensions or developments) from the plan laid down by Euclid.”

“Gregory St. Vincent is the greatest of circle-squarers, and his investigations led him into many truths: he found the property of the arc of the hyperbola which led to Napier's logarithms being called hyperbolic. Montucla says of him, with sly truth, that no one ever squared the circle with so much genius, or, excepting his principal object, with so much success.”

“Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty [of the parallel axiom]. He went so far as to write a paper, which he took with him to the Institute, and began to read it. But in the first paragraph something struck him which he had not observed : he muttered II faut que fy songe encore, and put the paper in his pocket.”

“Montucla says, speaking of France, that he finds three notions prevalent among cyclometers: 1. That there is a large reward offered for success; 2. That the longitude problem depends on that success; 3. That the solution is the great end and object of geometry. The same three notions are equally prevalent among the same class in England. No reward has ever been offered by the government of either country. The longitude problem hi no way depends upon perfect solution; existing approximations are sufficient to a point of accuracy far beyond what can be wanted. And geometry, content with what exists, has long passed on to other matters. Sometimes a cyclometer persuades a skipper who has made land in the wrong place that the astronomers are at fault, for using a wrong measure of the circle; and the skipper thinks it a very comfortable solution! And this is the utmost that the problem has to do with longitude.”

“A very interesting detailed account of the peculiarities of the circle squarer, and of the futility of the attempts on the part of the Mathematicians to convince him of his errors, will be found in Augustus De Morgan's Budget of Paradoxes.”

“The fact is known that having very thoroughly worked at the generalisations of Mathematics in theory and practice, Mr. De Morgan was enabled to establish with perfect precision the most highly generalised conception of Logic, perhaps, which it is possible to entertain. It is no new doctrine that Logic deals with the necessary laws of action of thought, and that Mathematics apply these laws to necessary matter of thought; but by showing that these laws can and must be applied with equal precision and equal necessity to all kinds of relations, and not only to those which the Aristotelian theory takes account of, he so enlarged the scope and intensified the power of Logic as an instrument, that we may hope for coming generations, as he must have hoped... another instalment of the kind... Mathematics are, meanwhile, and perhaps will always remain, the completest and most accurate example of the generalised Logic. At any rate, in the mind of the author, Logic and Mathematics as 'the two great branches of exact science, the study of the necessary laws of thought, the study of the necessary matter of thought, were always viewed in connection and antithesis.”

“Dr. George Boole, author of The Laws of Thought had introduced himself in the year 1842 to Mr. De Morgan by a letter on the Differential and Integral Calculus then recently published. His character and pursuits were in many points like those of the author who found great pleasure in his correspondence and friendship. ...In 1847, his attention having been drawn to the subject by the publication of Mr. De Morgan's Formal Logic, he published the Mathematical Analysis of Logic and in the following year communicated... a paper on the Calculus of Logic. His great work, An Investigation into the Laws of Thought... was a development of the principle laid down in the Calculus...”