### Frases Augustus De Morgan

### „In every age of the world there has been an established system, which has been opposed from time to time by isolated and dissentient reformers. The established system has sometimes fallen, slowly and gradually: it has either been upset by the rising influence of some one man, or it has been sapped by gradual change of opinion in the many.“

Introductory p.1

A Budget of Paradoxes (1872)

### „The lowest steps of the ladder are as useful as the highest.“

Fuente: On the Study and Difficulties of Mathematics (1831), Ch. I.

### „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.“

"Short Supplementary Remarks on the First Six Books of Euclid's Elements" (Oct, 1848) Companion to the Almanac for 1849 as quoted by Sir Thomas Little Heath, The Thirteen Books of Euclid's Elements Vol.1 https://books.google.com/books?id=UhgPAAAAIAAJ, Introduction and Books I, II. Preface, p. v.

### „A large quantity of examples is indispensable.“

Preface, pp. viii

The Differential and Integral Calculus (1836)

### „The Object of this Treatise is—(1) To point out to the student of Mathematics, who has not the advantage of a tutor, the course of study which it is most advisable that he should follow, the extent to which he should pursue one part of the science before he commences another, and to direct him as to the sort of applications which he should make. (2) To treat fully of the various points which involve difficulties and which are apt to be misunderstood by beginners, and to describe at length the nature without going into the routine of the operations.“

Fuente: On the Study and Difficulties of Mathematics (1831), Chapter I. Introductory Remarks on the Nature and Objects of Mathematics.

### „Find a fraction which, multiplied by itself, shall give 6, or… find the square root of 6. This can be shown to be an impossible problem; for it can be shown that no fraction whatsoever multiplied by itself, can give a whole number, unless it be itself a whole number disguised in a fractional form, such as 4⁄2 or 21⁄3. To this problem, then, there is but one answer, that it is self-contradictory. But if we propose the following problem,—to find a fraction which, multiplied by itself, shall give a product lying between 6 and 6 + a; we find that this problem admits of solution in every case. It therefore admits of solution however small a may be… as small as you please.“

...there is such a thing as the square root of 6, and it is denoted by √<span style="text-decoration: overline">6</span>. But we do not say we actually find this, but that we approximate to it.

The Differential and Integral Calculus (1836)

### „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…“

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The Differential and Integral Calculus (1836)

### „The number of mathematical students, increased as it has been of late years, would be much augmented if those who hold the highest rank in science would condescend to give more effective assistance in clearing the elements of the difficulties which they present. If any one claiming that title should think my attempt obscure or enormus, he must share the blame with me, since it is through his neglect that I have been enabled to avail myself of an opportunity to perform a task which I would gladly have seen confided to more skilful hands.“

Author's Preface

On the Study and Difficulties of Mathematics (1831)

### „If much difficulty should be experienced in the elementary chapters, I know of no work which I can so confidently recommend to be used with the present one, as that of M. Duhamel.“

Duhamel, Cours d'Analyse de l'Ecole Polytechnique. Paris, Bachelier. vol i 1841 vol. ii. 1840.

The Differential and Integral Calculus (1836)

### „The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself.“

Introductory p.4

A Budget of Paradoxes (1872)

### „The moving power of mathematical invention is not reasoning, but imagination.“

Quoted in Robert Perceval Graves, The Life of Sir William Rowan Hamilton, Vol. 3 (1889), p. 219.

### „I am far from saying that this Treatise will be easy; the subject is a difficult one, as all know who have tried it.“

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The Differential and Integral Calculus (1836)