+ Pappus of Alexandria’s Collection, Book 7, containing geometric problems, both plane and solid ⁂

That Those who wish to investigate the science of geometry original: "γεωμετρίαν"; literally "earth-measuring," though by this period it referred to the broad study of logic and shape. and who desire to make the most powerful and best division of it, see fit to call "theorems" original: "θεωρήματα"; from a root meaning "to look at" or "contemplate." Theorems are propositions stating a truth that can be proved. those matters in which the aim is to investigate the properties of a given hypothesis. On the other hand, they call "problems" original: "προβλήματα"; tasks or challenges posed for solution. those in which it is proposed to perform or construct something, [and] in which the hypothesis is not merely a statement of what is and will certainly be indicated. This distinction is observed among the ancients, some of whom called these "problems," while others called them "theorems." Now, the person who proposes a theorem, having understood the underlying subject in a certain way, seeks only the consequence and does not divide it in an unsuitable manner. But the person who proposes a problem signifies the performance and preparation of something specific and its construction to the best of one's ability, recommending? freedom of expression for the author. For it is the task of the seeker to define both what is possible and what is impossible, and if it is possible, when, and how, and in how many ways it is possible.

He spoke? of someone making the mathematical sciences original: "μαθήματα"; referring to the four mathematical arts: arithmetic, geometry, music, and astronomy. suspended in some way, challenging them because of the deadlines set by those who put forward these studies; yet no one knows any of the aforementioned investigations? that benefit our dispute?, nor is there a proof of similar things in which those that were erroneous were resolved. And the rest of the items in this first part of the Collection original: "συναγωγῇ"; the title of Pappus's great eight-book compendium of mathematics. have been separated by us from the problems, [forming] a brief? arrangement of ten items in total; for the [application] of the theorems from which "two mean proportionals" original: "δύο μέσα ἀνάλογον"; a famous classical problem of finding two values between two others to form a geometric progression, like 2:4:8:16. This was essential for the "doubling of the cube." were shown to be taken in continuous proportion—which were sought through plane theory and likewise devised by us—have until now been left aside regarding the? management? in the nine?, nor because many things [are done] in this manner ⁝

Let there be two given [lines] AB, BC at right angles to each other, and let a certain BD be taken from point B parallel to AB This likely refers to a construction on a diagram that is now lost; parallel lines from the same point usually indicate the extension of a coordinate system., and let [a line] ZH TH HE be drawn? on AC, and let DZ be drawn, and let BE be joined at point E, and from E let a certain HE be parallel to BD, and let the rectangle BZ be completed, and let a certain DH be taken from point D parallel to BE, and let them be placed as if BA were equal to the segments DH, HL, LZ, XK, and through the points H, L, X, K [let there be] parallels to BD, namely HO, LP, XR, KS, and let KR be equal to BA, and let AR and KR be joined at point T, and as KH is to DE, so is TW to AB; but as ED is to BD, so is BT to AF, and those through L, X, O, P...