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 or… - Pappus of Alexandria

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 (άνάπαλɩν λὐσɩν).

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.

He <nowiki>[</nowiki>Apollonius of Perga<nowiki>]</nowiki> spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought.