“The goal of deriving all the phenomena of nature from a few basic physical laws and the axioms of mathematics had been set by Galileo…
In studying curvilinear motions on the earth Galileo had found the parabola to be the basic curve. In the heavens… Kepler… had found the ellipse to be the basic curve. Why this difference? …since parabola and ellipse are both conic sections there was the provocative suggestion that perhaps some physical law unified these related paths of motion. …
It has often happened in the history of mathematics and science that major problems remained outstanding… great minds… succeeded only in revealing the true difficulties… and in generating an atmosphere of dispair… Then a genius appeared… with ideas that seemed remarkably simple once propounded, clarified the entire situation, dispelled the confusion, restored order, and produced a new synthesis that embraced far more even than the phenomena under consideration. The genius who… picked up the torch of science dropped by Galileo, was Isaac Newton.”

— Morris Kline

Source: Mathematics and the Physical World (1959), p. 225

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Morris Kline 42

American mathematician 1908–1992

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), pp. 224-225

“To the scientists of 1850, Hamilton's principle was the realization of a dream. …from the time of Galileo scientists had been striving to deduce as many phenomena of nature as possible from a few fundamental physical principles. …they made striking progress …But even before these successes were achieved Descartes had already expressed the hope and expectation that all the laws of science would be derivable from a single basic law of the universe. This hope became a driving force in the late eighteenth century after Maupertuis's and Euler's work showed that optics and mechanics could very likely be unified under one principle. Hamilton's achievement in encompassing the most developed and largest branches of physical science, mechanics, optics, electricity, and magnetism under one principle was therefore regarded as the pinnacle of mathematical physics.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 441.

“It has been observed that missiles and projectiles describe a curved path of some sort; however no one has pointed out the fact that this path is a parabola. But this and other facts”

Galileo Galilei (1564–1642) Italian mathematician, physicist, philosopher and astronomer

Author, Third Day. Change of Position
Dialogues and Mathematical Demonstrations Concerning Two New Sciences (1638)
Context: It has been observed that missiles and projectiles describe a curved path of some sort; however no one has pointed out the fact that this path is a parabola. But this and other facts, not few in number or less worth knowing, I have succeeded in proving; and what I consider more important, there have been opened up to this vast and most excellent science, of which my work is merely the beginning, ways and means by which other minds more acute than mine will explore its remote corners.

“Galileo had raised the concepts of space and time to the status of fundamental categories by directing attention to the mathematical description of motion.”

Gerald James Whitrow (1912–2000) British mathematician

The Structure of the Universe: An Introduction to Cosmology (1949)
Context: Galileo had raised the concepts of space and time to the status of fundamental categories by directing attention to the mathematical description of motion. The midiaevel qualitative method had made these concepts relatively unimportant, but in the new mathematical philosophy the external world became a world of bodies moving in space and time. In the Timaeus Plato had expounded a theory that outside the universe, which he regarded as bounded and spherical, there was an infinite empty space. The ideas of Plato were much discussed in the middle of the seventeenth century by the Cambridge Platonists, and Newton's views were greatly influenced thereby. He regarded space as the 'sensorium of God' and hence endowed it with objective existence, although he confessed that it could not be observed. Similarly, he believed that time had an objective existence independent of the particular processes which can be used for measuring it.<!--p.46

“The outlook of a Bruno, a Kepler, or a Galileo is determined by the idea of a comprehensive, all-embracing unity of the physical world. The same law governs the courses of the stars, the falling of stones, and the flight of birds. This homogenization of the physical world with respect to the validity of law deprives the division of physical objects into rigid abstractly defined classes of the critical significance it had for Aristotelian physics, in which membership in a certain conceptual class was considered to determine the physical nature of an object.”

Kurt Lewin (1890–1947) German-American psychologist

Source: 1930s, A Dynamic Theory of Personality, 1935, p. 10 as cited in: Coleman Roberts Griffith (1943) Principles of systematic psychology. p. 215.

“The Ellipse is the most simple of the Conic Sections, most known, and nearest of Kin to a Circle, and easiest describ'd by the Hand in plano. Though many prefer the Parabola before it, for the Simplicity of the Æquation by which it is express'd. But by this Reason the Parabola ought to be preferr'd before the Circle it self, which it never is. Therefore the reasoning from the Simplicity of the Æquation will not hold. The modern Geometers are too fond of the Speculation of Æquations.”

Isaac Newton book Arithmetica Universalis

p, 125
Arithmetica Universalis (1707)

“The great Cartesian invention had its roots in those famous problems of antiquity which originated in the days of Plato. In endeavoring to solve the problems of the trisection of an angle, of the duplication of the cube and of the squaring of the circle, the ruler and compass having failed them, the Greek geometers sought new curves. They stumbled on the conic sections…There we find the nucleus of the method which Descartes later erected into a principle. Thus Apollonius referred the parabola to its axis and principal tangent, and showed that the semichord was the mean propotional between the latus rectum and the height of the segment. Today we express this relation by x2 = Ly, calling the height the ordinate (y) and the semichord the abscissa (x); the latus rectum being… L. …the Greeks named these curves and many others… loci… Thus the ellipse was the locus of a point the sum of the distances of which from two fixed points was constant. Such a description was a rhetorical equation of the curve…”

Tobias Dantzig (1884–1956) American mathematician

Number: The Language of Science (1930)

“The Ellipse is the most simple of the Conic Sections, most known, and nearest of Kin to a Circle, and easiest describ'd by the Hand in plano.”

Isaac Newton book Arithmetica Universalis

Though many prefer the Parabola before it, for the Simplicity of the Æquation by which it is express'd. But by this Reason the Parabola ought to be preferr'd before the Circle it self, which it never is. Therefore the reasoning from the Simplicity of the Æquation will not hold. The modern Geometers are too fond of the Speculation of Æquations.
Arithmetica Universalis (1707)

“To appreciate the nature of fractals, recall Galileo's splendid manifesto that "Philosophy is written in the language of mathematics and its characters are triangles, circles and other geometric figures, without which one wanders about in a dark labyrinth." Observe that circles, ellipses, and parabolas are very smooth shapes and that a triangle has a small number of points of irregularity. Galileo was absolutely right to assert that in science those shapes are necessary. But they have turned out not to be sufficient, "merely" because most of the world is of infinitely great roughness and complexity. However, the infinite sea of complexity includes two islands: one of Euclidean simplicity, and also a second of relative simplicity in which roughness is present, but is the same at all scales.”

Benoît Mandelbrot (1924–2010) Polish-born, French and American mathematician

A Theory of Roughness (2004)

“In addition, the teaching of theories from axioms, or some close imitation of them such as the basic laws of an algebra, is usually an educational disaster.”

Ivor Grattan-Guinness (1941–2014) Historian of mathematics and logic

Source: The Rainbow of Mathematics: A History of the Mathematical Sciences (2000), p. 739.

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