PROBLEM (epgi)pnea) means simply a thing put forward or pro posed. In mathematical language it is anything which is required to be done, and in the earlier writers is distinguished from a theorem, or assertion to be proved, in that the latter does not require any specific object to be effected. Thus," all the angles of a triangle are together equal to two right angles" is to be shown or made evident, and is a theorem ; but "to draw a circle through three given points" presents an object to be effected, and is a problem. But it must be remembered that this difference lies more in the nature of the result than in the method ; for the solution of this problem, so called, is an intuitive corollary from the theorem that if three points be joined, and perpen diculars be drawn bisecting two of the joining lines, the intersection of these perpendiculars is equidistant from the three points. It is also to be noted that this distinction of theorem and problem appears neither in the Greek of Euclid, Apollonius, nor Archimedes, the general term employed by all three, and in all cases, being wp6racris, which is trans lated by proposition. The distinction then is of a later date, and is
the work of annotators ; it appears in Pappus, according to the Latin of Commandinc. It does not appear in the translation of Euclid by Athelard (which goes by the name of Campanus); and the first edition of the Elements in which we find it is the subsequent edition of.Zambertus. If we leave the modern followers of the old geometers, we fiud the word problem used in its simple etymological sense of something proposed ; but for the most part employed when the some thing proposed contains, or has contained, a remarkable difficulty. Thus to this day we talk of the problem of three bodies, as being one of the methods which are hoped to be found capable of decided improvement. In algebra the word is variously used, though, accord ing to the ancient distinction, the solution of any equation of condition should be called a problem, and the establishment of any identity a theorem.