“The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations.”

— Niels Henrik Abel

A Memoir on Algebraic Equations, Proving the Impossibility of a Solution of the General Equation of the Fifth Degree (1824) Tr. W. H. Langdon, as quote in A Source Book in Mathematics (1929) ed. David Eugene Smith

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Niels Henrik Abel 6

Norwegian mathematician 1802–1829

Lloyd Alexander (1924–2007) American children's writer

"The Flat-Heeled Muse", Horn Book Magazine (1 April 1965)

“Someone told me that each equation I included in the book would halve the sales. I therefore resolved not to have any equations at all. In the end, however, I did put in one equation, Einstein's famous equation, E = mc^2. I hope that this will not scare off half of my potential readers.”

Stephen Hawking book A Brief History of Time

A Brief History of Time (1988)

“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)

“I taught myself decimals, equations, the square, cube, and biquadrate roots. I got some knowledge of logarithms, and some of algebra. I readily got through a small schoolbook of geometry; and having an odd volume, the first, of Williamson's ' Euclid,' I attacked it vigorously and perseveringly. Williamson's is by no means the best book on the subject, yet I am still of opinion that it is the best book I could have had for the purpose of teaching myself.”

Francis Place (1771–1854) English social reformer

Source: The life of Francis Place, 1771-1854, 1898, p. 18

“Ideas and techniques known in quantum electrodynamics have been applied to the Bardeen-Cooper-Schrieffer theory of superconductivity. In an approximation which corresponds to a generalization of the Hartree-Fock fields, one can write down an integral equation defining the self-energy of an electron in an electron gas with phonon and Coulomb interaction. The form of the equation implies the existence of a particular solution which does not follow from perturbation theory, and which leads to the energy gap equation and the quasi-particle picture analogous to Bogolyubov's.”

Yoichiro Nambu (1921–2015) American physicist

[Quasi-particles and gauge invariance in the theory of superconductivity, Physical Review, 117, 3, February 1960, 648–663, 10.1103/PhysRev.117.648]

“We have already various treatises on Mechanics, but the plan of this one is entirely new. I intend to reduce the theory of this Science, and the art of solving problems relating to it, to general formulae, the simple development of which provides all the equations necessary for the solution of each problem. I hope that the manner in which I have tried to attain this object will leave nothing to be desired. No diagrams will be found in this work. The methods that I explain require neither geometrical, nor mechanical, constructions or reasoning, but only algebraical operations in accordance with regular and uniform procedure. Those who love Analysis will see with pleasure that Mechanics has become a branch of it, and will be grateful to me for having thus extended its domain.”

Joseph Louis Lagrange book Mécanique analytique

Mécanique analytique (1788) as quoted by E. W. Hobson, Mathematics, from the points of view of the Mathematician and of the Physicist (1912) an address delivered to the Mathematical and Physical Society of University College London, p.13. https://books.google.com/books?id=H7Y_AQAAIAAJ&pg=PA13

“You see, some mathematicians have clear and far-ranging. "programs". For instance, Grothendieck had such a program for algebraic geometry; now Langlands has one for representation theory, in relation to modular forms and arithmetic. I never had such a program, not even a small size one.”

Jean-Pierre Serre (1926) French mathematician

An Interview with Jean-Pierre Serre - Singapore Mathematical Society https://sms.math.nus.edu.sg/smsmedley/Vol-13-1/An%20interview%20with%20Jean-Pierre%20Serre(CT%20Chong%20&%20YK%20Leong).pdf

“The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration ofjkbni semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form.”

Leonid Kantorovich (1912–1986) Russian mathematician

"On one class of functional equations" (1936), as cited in: O'Connor, John J.; Robertson, Edmund F., " Leonid Kantorovich http://www-history.mcs.st-andrews.ac.uk/Biographies/Kantorovich.html", MacTutor History of Mathematics archive, University of St Andrews

“The same equations have the same solutions”

Richard Feynman (1918–1988) American theoretical physicist

volume II; lecture 12, "Electrostatic Analogs"; p. 12-1
The Feynman Lectures on Physics (1964)

“When an equation…clearly leads to two negative or imaginary roots, [Diophantus] retraces his steps and shows by how by altering the equation, he can get a new one that has rational roots. …Diophantus is a pure algebraist; and since algebra in his time did not recognize irrational, negative, and complex numbers, he rejected equations with such solutions.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 143.

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Niels Henrik Abel 6

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