“The so called άναλυόμϵνος ('Treasury of Analysis') is… a special body of doctrine provided for the use of those who, after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving (the construction of) lines, and it is useful for this alone. It is the work of three men, Euclid the author of the Elements, Apollonius of Perga, and Aristaeus the elder, and proceeds by way of analysis and synthesis.”

Source: The Thirteen Books of Euclid's Elements (1908), Ch. IX. §6

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Pappus of Alexandria5

Greek mathematician of Antiquity 290–350

James Gow (scholar) (1854–1923) scholar

A Short History of Greek Mathematics (1884)

“Analysis and synthesis, though commonly treated as two different methods, are, if properly understood, only the two necessary parts of the same method. Each is the relative and correlative of the other. Analysis, without a subsequent synthesis, is incomplete; it is a mean cut of from its end. Synthesis, without a previous analysis, is baseless; for synthesis receives from analysis the elements which it recomposes.”

Sir William Hamilton, 9th Baronet (1788–1856) Scottish metaphysician (1788–1856)

Lectures on Metaphysics and Logic: "6th Lecture on Metaphysics", p. 69, ed. 1871, Boston; partly reported in Austin Allibone ed. Prose Quotations from Socrates to Macaulay. (1903), p. 34

“Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.”

Richard Courant (1888–1972) German American mathematician (1888-1972)

Richard Courant in: The Australian Mathematics Teacher, Volumes 39-40 http://books.google.co.in/books?id=CofxAAAAMAAJ, Australian Association of Mathematics Teachers, 1983, p. 3

“Either one or the other [ analysis or synthesis ] may be direct or indirect. The direct procedure is when the point of departure is known-direct synthesis in the elements of geometry. By combining at random simple truths with each other, more complicated ones are deduced from them. This is the method of discovery, the special method of inventions, contrary to popular opinion.”

André-Marie Ampère (1775–1836) French physicist and mathematician

André-Marie Ampè, in André-Marie Ampère: Enlightenment and Electrodynamics http://books.google.co.in/books?id=QWZKQWB-sbQC&pg=PA159, Cambridge University Press, 1996, p. 159

“It seems clear that [set theory] violates against the essence of the continuum, which, by its very nature, cannot at all be battered into a single set of elements. Not the relationship of an element to a set, but of a part to a whole ought to be taken as a basis for the analysis of a continuum.”

Hermann Weyl (1885–1955) German mathematician

Riemanns geometrische Ideen, ihre Auswirkungen und ihre Verknüpfung mit der Gruppentheorie (1925), as quoted/translated by Erhard Scholz, "Philosophy as a Cultural Resource and Medium of Reflection for Hermann Weyl" (2004)

“But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what were before antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought; and this we call synthesis.”

Pappus of Alexandria (290–350) Greek mathematician of Antiquity

Source: The Thirteen Books of Euclid's Elements (1908), Ch. IX. §6