These, with other observations concerning this method, and its application, led me on gradually to compose a Treatise of a much greater extent than I intended, or would have engaged in, if I had been aware of it when I began this Work, because my attendance in the University could allow one to bestow but a small part of my time in carrying it on.

In a letter to Ampère dated 3 September 1822, Faraday lamented, "I am unfortunate in a want of mathematical knowledge and the power of entering with facility into abstract reasoning, I am obliged to feel my wasy by facts closely placed together." ...
Faraday's "facts" were his experiments, both published and unpublished. During a periof of 23 years, 1831–54, he compiled the results of those experiments into three volumes, called Experimental Researches in Electricity ... A most remarkbalbe thing is that there was not a single formula in this monumental compilation, which showed that Faraday was feeling his way, guided only by geometric intuition without andy precise algebraic formulation.

The Author of the piece, is entitled to much credit for the goodness of his Pen: and I could wish he had as much credit for the rectitude of his Heart — for, as Men see thro’ different Optics, and are induced by the reflecting faculties of the Mind, to use different means to attain the same end; the Author of the Address, should have had more charity, than to mark for Suspicion, the Man who should recommend Moderation and longer forbearance — or, in other words, who should not think as he thinks, and act as he advises. But he had another plan in view, in which candor and liberality of Sentiment, regard to justice, and love of Country, have no part; and he was right, to insinuate the darkest suspicion, to effect the blackest designs. That the Address is drawn with great art, and is designed to answer the most insidious purposes. That it is calculated to impress the Mind, with an idea of premeditated injustice in the Sovereign power of the United States, and rouse all those resentments which must unavoidably flow from such a belief. That the secret Mover of this Scheme (whoever he may be) intended to take advantage of the passions, while they were warmed by the recollection of past distresses, without giving time for cool, deliberative thinking, & that composure of Mind which is so necessary to give dignity & stability to measures, is rendered too obvious, by the mode of conducting the business, to need other proof than a reference to the proceeding.

After I saw that so much had been written upon it to no good purpose, I was rather induced to delay the publication of this Treatise, til I could finish my design.

I devoted several months in privacy to the composition of a treatise on the mysteries of Three Dimensions. Only, with the view of evading the Law, if possible, I spoke not of a physical Dimension, but of a Thoughtland whence, in theory, a Figure could look down upon Flatland and see simultaneously the insides of all things, and where it was possible that there might be supposed to exist a Figure environed, as it were, with six Squares, and containing eight terminal Points. But in writing this book I found myself sadly hampered by the impossibility of drawing such diagrams as were necessary for my purpose... my life was under a cloud. All pleasures palled upon me; all sights tantalized and tempted me to outspoken treason, because I could not but compare what I saw in Two Dimensions with what it really was if seen in Three, and could hardly refrain from making my comparisons aloud.' I neglected my clients and my own business to give myself to the contemplation of the mysteries which I had once beheld, yet which I could impart to no one, and found daily more difficult to reproduce even before my own mental vision.

Coxeter also wrote a long mathematical explanation in his letter, which was beyond Escher's comprehension, as he remarked in a letter to George and Corrie.

Baarn, 28 May 1960: 'I had an enthusiastic letter from Coexter about my coloured fish, which I sent him. Three pages of explanation of what I actually did... It's a pity that I understand nothing, absolutely nothing of it...

The book deals with the problems of philosophy and shows, as I believe, that the method of formulating these problems rests on the misunderstanding of the logic of our language.

But though much may be said to Excuse the Chymists when they write Darkly, and Ænigmatically, about the Preparation of their Elixir, and Some few other grand Arcana, the divulging of which they may upon Grounds Plausible enough esteem unfit; yet when they pretend to teach the General Principles of Natural Philosophers, this Equivocall Way of Writing is not to be endur'd. For in such Speculative Enquiries, where the naked Knowledge of the Truth is the thing Principally aim'd at, what does he teach me worth thanks that does not, if he can, make his Notion intelligible to me, but by Mystical Termes, and Ambiguous Phrases darkens what he should clear up; and makes me add the Trouble of guessing at the sence of what he Equivocally expresses, to that of examining the Truth of what he seems to deliver.

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...

The numberless treatises of antiquities, philosophy, mathematics, natural and other history ... written originally in, or translated to our tongue are sufficient to lead us a great way into any science our curiosity shall prompt us to. The greatest difficulty we struggled with, was the want of a good art of reasoning, which we had not, that I know of, till that defect was supplied by Locke, whose Essay on Human Understanding makes large amends for the want of all others in that kind.

In 1673 he wrote his great work De Algebra Tractatus; Historicus & Practicus, of which an English edition appeared in 1685. In this there is seen the first serious attempt in England to write on the history of mathematics, and the result shows a wide range of reading of classical literature of the science. This work is also noteworthy because it contains the first of an effort to represent the imaginary number graphically by the method now used. The effort stopped short of success but was an ingenious beginning.

In October 1838, that is, fifteen months after I had begun my systematic enquiry, I happened to read for amusement 'Malthus on Population,' and being well prepared to appreciate the struggle for existence which everywhere goes on from long-continued observation of the habits of animals and plants, it at once struck me that under these circumstances favourable variations would tend to be preserved, and unfavourable ones to be destroyed. The result of this would be the formation of new species. Here then I had at last got a theory by which to work; but I was so anxious to avoid prejudice, that I determined not for some time to write even the briefest sketch of it. In June 1842 I first allowed myself the satisfaction of writing a very brief abstract of my theory in pencil in 35 pages; and this was enlarged during the summer of 1844 into one of 230 pages...

This Treatise, he says, was read before the Royal Academy of Sweden the third day of April 1742. ...This was eleven Months after mine was laid before the Royal Society.

"I was at first almost frightened when I saw such mathematical force made to bear upon the subject, and then wondered to see that the subject stood it so well.
[Letter to James Clerk Maxwell, 25 March 1857, commenting on Maxwell's paper titled "On Faraday's Lines of Force"]"