“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.”

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

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“Analysis… takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we assume that which is sought as if it were (already) done (ɣϵɣονός) and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards”

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

άνάπαλɩν λὐσɩν
The Thirteen Books of Euclid's Elements (1908)

“There is a certain way of searching for the truth in mathematics that Plato is said first to have discovered; Theon named it analysis, and defined it as the assumption of that which is sought as if it were admitted and working through its consequences to what is admitted to be true. This is opposed to synthesis, which is the assuming what is admitted and working through its consequences to arrive at and to understand that which is sought.”

François Viète (1540–1603) French mathematician

Source: In artem analyticem Isagoge (1591), Ch. 1 as quoted by Douglas M. Jesseph, Squaring the Circle: The War Between Hobbes and Wallis (1999) p. 225

“Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematical. (1) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted: then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (b), if we come upon something admittedly false, that which is sought will also be false. (2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (a) what is admitted is possible and obtainable, that is, what mathematicians call given, what was originally proposed will also be possible, and the proof will again correspond in reverse order to the analysis, but if (b) we come upon something admittedly impossible, the problem will also be impossible.”

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

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

“Philosophy is based on speculation, on logic, on thought, on the synthesis of what we know and on the analysis of what we do not know. Philosophy must include within its confines the whole content of science, religion and art.”

P. D. Ouspensky (1878–1947) Russian esotericist

Source: A New Model of the Universe (1932), p. 33
Context: Philosophy is based on speculation, on logic, on thought, on the synthesis of what we know and on the analysis of what we do not know. Philosophy must include within its confines the whole content of science, religion and art. But where can such a philosophy be found? All that we know in our times by the name of philosophy is not philosophy, but merely "critical literature" or the expression of personal opinions, mainly with the aim of overthrowing and destroying other personal opinions. Or, which is still worse, philosophy is nothing but self-satisfied dialectic surrounding itself with an impenetrable barrier of terminology unintelligible to the uninitiated and solving for itself all the problems of the universe without any possibility of proving these explanations or making them intelligible to ordinary mortals.

“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

“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