“The second part of the book… contains an exposition of the first principles of the theory of complex quantities; hitherto, the very elements of this theory have not been easily accessible to the English student, except recently in Prof. Chrystal's excellent treatise on Algebra. The subject of Analytical Trigonometry has been too frequently presented to the student in the state in which it was left by Euler, before the researches of Cauchy, Abel, Gauss, and others, had placed the use of imaginary quantities, and especially the theory of infinite series and products, where real or complex quantities are involved, on a firm scientific basis. In the Chapter on the exponential theorem and logarithms, I have ventured to introduce the term "generalized logarithm" for the doubly infinite series of values of the logarithm of a quantity.”

A Treatise on Plane Trigonometry https://books.google.com/books?id=_ktLAAAAMAAJ (1891) Preface

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E. W. Hobson20

British mathematician 1856–1933

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“My specific… object has been to contain, within the prescribed limits, the whole of the student's course, from the confines of elementary algebra and trigonometry, to the entrance of the highest works on mathematical physics. A learner who has a good knowledge of the subjects just named, and who can master the present treatise, taking up elementary works on conic sections, application of algebra to geometry, and the theory of equations, as he wants them, will, I am perfectly sure, find himself able to conquer the difficulties of anything he may meet with; and need not close any book of Laplace, Lagrange, Legendre, Poisson, Fourier, Cauchy, Gauss, Abel, Hindenburgh and his followers. or of any one of our English mathematicians, under the idea that it is too hard for him.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

Preface, p. iii
The Differential and Integral Calculus (1836)

“Quite distinct from the theoretical question of the manner in which mathematics will rescue itself from the perils to which it is exposed by its own prolific nature is the practical problem of finding means of rendering available for the student the results which have been already accumulated, and making it possible for the learner to obtain some idea of the present state of the various departments of mathematics…. The great mass of mathematical literature will be always contained in Journals and Transactions, but there is no reason why it should not be rendered far more useful and accessible than at present by means of treatises or higher text-books. The whole science suffers from want of avenues of approach, and many beautiful branches of mathematics are regarded as difficult and technical merely because they are not easily accessible…. I feel very strongly that any introduction to a new subject written by a competent person confers a real benefit on the whole science. The number of excellent text-books of an elementary kind that are published in this country makes it all the more to be regretted that we have so few that are intended for the advanced student. As an example of the higher kind of text-book, the want of which is so badly felt in many subjects, I may mention the second part of Prof. Chrystal’s “Algebra” published last year, which in a small compass gives a great mass of valuable and fundamental knowledge that has hitherto been beyond the reach of an ordinary student, though in reality lying so close at hand. I may add that in any treatise or higher text-book it is always desirable that references to the original memoirs should be given, and, if possible, short historic notices also. I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history.”

James Whitbread Lee Glaisher (1848–1928) English mathematician and astronomer

Source: "Presidential Address British Association for the Advancement of Science," 1890, p. 466 : On the need of text-books on higher mathematics

“This work… was designed in the first instance to be a second edition of a Treatise on Algebra, published in 1830, and which has been long out of print; but I have found it necessary, in carrying out the principles developed in that work, to present the subject in so novel a form, that I could not with propriety consider it in any other light than as an entirely new treatise.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. iii
A Treatise on Algebra (1842)

“Many of the greatest masters of the mathematical sciences were first attracted to mathematical inquiry by problems relating to numbers, and no one can glance at the periodicals of the present day which contain questions for solution without noticing how singular a charm such problems still continue to exert. The interest in numbers seems implanted in the human mind, and it is a pity that it should not have freer scope in this country. The methods of the theory of numbers 271 are peculiar to itself, and are not readily acquired by a student whose mind has for years been familiarized with the very different treatment which is appropriate to the theory of continuous magnitude; it is therefore extremely desirable that some portion of the theory should be included in the ordinary course of mathematical instruction at our University. From the moment that Gauss, in his wonderful treatise of 1801, laid down the true lines of the theory, it entered upon a new day, and no one is likely to be able to do useful work in any part of the subject who is unacquainted with the principles and conceptions with which he endowed it.”

James Whitbread Lee Glaisher (1848–1928) English mathematician and astronomer

Source: "Presidential Address British Association for the Advancement of Science," 1890, p. 467 : On the theory of numbers

“Formally this work is divided into four parts, but in a sense it consists of two short treatises. One is an exposition of a theory of cooperation and organization and constitutes the first half of the book. The second is a study of the functions and of the methods of operation of executives in formal organizations.”

Chester Barnard book The Functions of the Executive

Preface
The Functions of the Executive (1938)

“I have separated arithmetical from symbolical algebra, and I have devoted the present volume entirely to the exposition of the principles of the former science and their application to the theory of numbers and of arithmetical processes: the second volume, which is now in the press, will embrace the principles of symbolical algebra: it will be followed, if other and higher duties should allow me the leisure to complete them, by other works, embracing all the more important departments of analysis, with the view of presenting their principles in such a form, as may make them component parts of one uniform and connected system.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. iii
A Treatise on Algebra (1842)

“The first demonstration (Disq. Arith., Arts. 125-145) which is presented by Gauss in a form very repulsive to any but the most laborious students, has been resumed by Lejeune Dirichlet in a memoir in Crelle's Journal… and has been developed by him with that luminous perspicuity by which his mathematical writings are distinguished.”

Henry John Stephen Smith (1826–1883) mathematician

Report on the Theory of Numbers (1859) Part I, p. 59.
The Collected Mathematical Papers of Henry John Stephen Smith (1894) Vol. 1

“A work of art that contains theories is like an object on which the price tag has been left.”

Alexander Pope (1688–1744) eighteenth century English poet

Une oeuvre où il y a des théories est comme un objet sur lequel on laisse la marque du prix. <br class="br">Marcel Proust, In Search of Lost Time, part VII: Time Regained, chapter III, "An Afternoon Party at the House of the Princesse de Guermantes" ( French version http://web.archive.org/web/20010708070436/http://gallica.bnf.fr/proust/TempsRetrouve.htm and English translation http://etext.library.adelaide.edu.au/p/proust/marcel/p96t/chapter3.html). <br class="br">Misattributed

“Legendre's Law of Quadratic Reciprocity' … is …the most important general truth in the science of integral numbers which has been discovered since the time of Fermat. It has been called by Gauss 'the gem of the higher arithmetic,' and is equally remarkable whether we consider the simplicity of its enunciation, the difficulties which for a long time attended its demonstration, or the number and variety of the results which have been obtained by its means. …[W]e find in the 'Opuscula Analytica' of Euler… a memoir… which contains a general and very elegant theorem from which the Law of Reciprocity is immediately deducible, and which is, vice versâ, deducible from that law. But Euler… expressly observes that the theorem is undemonstrated; and this would seem to be the only place in which he mentions it in connexion with the theory of the Residues of Powers; though in other researches he has frequently developed results which are consequences of the theorem, and which relate to the linear forms of the divisors of quadratic formulae. But here also his conclusions repose on induction only; though in one memoir he seems to have imagined… that he had obtained a satisfactory demonstration.”

Henry John Stephen Smith (1826–1883) mathematician

Report on the Theory of Numbers (1859) Part I, pp. 56-57.
The Collected Mathematical Papers of Henry John Stephen Smith (1894) Vol. 1

“[W]hen a body has experienced any changes, and when after a certain number of transformations it returns to precisely its original state, that is, to that state considered in respect to density, to temperature, to mode of aggregation—let us suppose, I say, that this body is found to contain the same quantity of heat that it contained at first, or else that the quantities of heat absorbed or set free in these different transformations are exactly compensated. This fact has never been called in question. It was first admitted without reflection, and verified afterwards in many cases by experiments with the calorimeter. To deny it would be to overthrow the whole theory of heat to which it serves as a basis. For the rest, we may say in passing, the main principles on which the theory of heat rests require the most careful examination. Many experimental facts appear almost inexplicable in the present state of this theory.”

Nicolas Léonard Sadi Carnot (1796–1832) French physicist, the "father of thermodynamics" (1796–1832)

Reflections on the Motive Power of Heat (1824)

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E. W. Hobson20

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