“I thought fit to… explain in detail in the same book the peculiarity of a certain method, by which it will be possible… to investigate some of the problems in mathematics by means of mechanics. This procedure is… no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards… But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.”

— Archimedes , book The Method of Mechanical Theorems

The Method of Mechanical Theorems

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Archimedes 20

Greek mathematician, physicist, engineer, inventor, and ast… -287–-212 BC

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“I am persuaded that it [The Method of Mechanical Theorems] will be of no little service to mathematics; for I apprehend that some, either of my contemporaries or of my successors, will, by means of the method when once established, be able to discover other theorems in addition, which have not yet occurred to me.”

Archimedes book The Method of Mechanical Theorems

The Method of Mechanical Theorems

“First then I will set out the very first theorem which became known to me by means of mechanics, namely that
Any segment of a section of a right angled cone (i. e., a parabola) is four-thirds of the triangle which has the same base and equal height,
and after this I will give each of the other theorems investigated by the same method. Then at the end of the book I will give the geometrical [proofs of the propositions]…”

Archimedes book The Method of Mechanical Theorems

The Method of Mechanical Theorems

“There are two general Methods made use of in the Mathematicks, viz. Synthesis and Analysis, which we shall explain, after having acquainted the Reader, that the Method we make use of to resolve a Mathematical Problem, is called Zetetick; and that that Method which determines when, and by what way, and how many different ways a Problem may be resolved, is called Poristick. But in treating of Methods, we will first premise, that in general, a Method is the Art of disposing a Train of Arguments or Consequences in a right Order, either to discover the Truth of a Theorem, which we would find out, or to demonstrate it to others, when found.”

Jacques Ozanam (1640–1718) French mathematician

Source: A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms, 1702, p. 26

“On the one hand, the student has been informed by some writers that the only certain way lies in the use of the entropy-function and the thermodynamic potentials; on the other hand, he is told with equal authority that the method used by the original investigators has been the consideration of cyclic processes, and that the former method is nothing but a mathematical (perhaps unnecessary) refinement of the results obtained by the latter. These extreme attitudes appear to me to be unfortunate, and more especially when one observes the physical clearness introduced by the use of cyclic processes, but at the same time remembers that most of the results obtained by separate investigators using cyclic processes had, with a great many more, previously been found by J. Willard Gibbs by means of a purely analytical method.”

J. R. Partington (1886–1965) British chemist

A Text-Book of Thermodynamics with Special Reference to Chemistry (1913)

“A development method may be regarded as a path or a procedure by which the developer proceeds from a problem of a certain class to a solution of a certain class. In trivial cases, the method may be fully algorithmic; for example, there is an algorithmic procedure for obtaining the square root of a nonnegative number to any desired degree of accuracy.”

Michael A. Jackson (1936) British computer scientist

Michael A. Jackson. "A system development method," in: Tools and notions for program construction: An advanced course, Cambridge University Press, 1982. p. 1

“I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof”

Pierre de Fermat (1601–1665) French mathematician and lawyer

Letter to Frénicle (1657) Oeuvres de Fermat Vol.II as quoted by Edward Everett Whitford, The Pell Equation http://books.google.com/books?id=L6QKAAAAYAAJ (1912)
Context: There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square.

“[T]he methods used for measuring Astronomical distances are in some applications absolutely the same as the methods of ordinary theodolite-surveying, and are in other applications equivalent to them…”

George Biddell Airy (1801–1892) English mathematician and astronomer

Introduction
Popular Astronomy: A Series of Lectures Delivered at Ipswich (1868)

“Methods by which engineers stabilise their mechanisms suggest analogous possibilities for stabilising economic systems.”

Arnold Tustin (1899–1994) British engineer

Arnold Tustin (1957) " The mechanism of economic instability http://books.google.com/books?id=Nou8mkjPMPUC&pg=PA8" in: New Scientist, Oct. 27, 1957. p. 8

“Descartes… complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof… What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish…. historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is… paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics for the Nonmathematician (1967), pp. 255-256.

“It was in 1929 that Salvador Dali [Dali is writing about himself] brought his attention to hear upon the internal mechanism of paranoiac phenomena and envisaged the possibility of an experimental method based on the sudden power of the systematic associations proper to paranoia; this method afterwards became the delirio-critical synthesis which hears the name of "paranoiac-critical activity."”

Salvador Dalí (1904–1989) Spanish artist

Paranoia: delirium of interpretive association bearing a systematic structure. Paranoiac-critical activity: spontaneous method of irrational knowledge based upon the interpretive-critical association of delirious phenomena.
Source: Quotes of Salvador Dali, 1931 - 1940, My Pictorial Struggle', S. Dali, 1935, Chapter: 'My Pictorial Struggle', p. 15

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