“[T]he application of algebra to geometry… far more than any of his metaphysical speculations, has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences.”

— John Stuart Mill

An Examination of Sir William Hamilton's Philosophy (1865) as quoted in 5th ed. (1878) p. 617. https://books.google.com/books?id=ojQNAQAAMAAJ&pg=PA617

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John Stuart Mill 179

British philosopher and political economist 1806–1873

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“[T]he sciences that are expressed by numbers or by other small signs, are easily learned; and… this facility rather than its demonstrability is what has made the fortune of algebra.”

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“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.”

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“More than any of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundations of the science, insisting upon accurate definitions, clear assumptions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic is… clear. …That Plato should hold the view… is not a cause for surprise. The world's thinkers have always held it. No man has ever created a mathematical theory for practical purposes alone. The applications of mathematics have generally been an afterthought.”

David Eugene Smith (1860–1944) American mathematician

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“The absolute requisites for the study of this work… are a knowledge of algebra to the binomial at least, plane and solid geometry, plane trigonometry, and the most simple part of the usual applications of algebra to geometry.
…A. De Morgan. London July 1, 1836”

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

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“As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection.”

Joseph Louis Lagrange (1736–1813) Italian mathematician and mathematical physicist

Dans Les Leçons Élémentaires sur les Mathématiques (1795) Leçon cinquiéme,Tr. McCormack, cited in Robert Edouard Moritz, Memorabilia mathematica or, The philomath's quotation-book (1914) Ch. V The teaching of mathematics, p. 81. https://archive.org/stream/memorabiliamathe00moriiala#page/80/mode/2up

“It is odd that the last twenty-five years which have witnessed the greatest progress ever made in physical science—the greatest victories ever achieved by mind over matter—should have produced hardly a volume that will be remembered in 1900.”

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“The modern poet has no essential alliance with regular schemes of any sorts. He reserves the right to adapt his rhythm to his mood, to modulate his metre as he progresses. Far from seeking freedom and irresponsibility (implied by the unfortunate term free verse) he seeks a stricter discipline of exact concord of thought and feeling.”

Herbert Read (1893–1968) English anarchist, poet, and critic of literature and art

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Literary Quotes

“It is the business of the production engineer to know every single item that constitutes his finished product, and every step involved in the handling of every piece. He must know what is the most advantageous manufacturing quantity of every single item so as to secure uniformity of flow as well as economy of manufacture. He must know how long each step ought to take under the best attainable working conditions. He must be able to tell at any time the exact condition as regards quantity and state of finishedness of every part involved in his manufacturing process.”

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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)

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“The method of the science not being practiced much nowadays, except what logic prescribes to all sciences generally, that fitted for the peculiar nature of metaphysics being simply ignored, it is no wonder that those who everlastingly turn the Sisyphean stone of this inquiry do not seem so far to have made much progress. Though here I neither can nor will expatiate upon so important and extensive a subject, I shall briefly shadow forth what constitutes no despicable part of this method, namely, the infection between sensuous and intellectual cognition, not only as creeping in on those incautious in the application of principles, but even producing spurious principles under the appearance of axioms.”

Immanuel Kant (1724–1804) German philosopher

Kant's Inaugural Dissertation (1770), Section V On The Method Respecting The Sensuous And The Intellectual In Metaphysics

“[T]he application of algebra to geometry… far more than any of his metaphysical speculations, has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences.”

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John Stuart Mill 179

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