This might be postponed until after the problem to bisect an angle, since it merely requires the bisection of a straight angle; but considering the immaturity of the average pupil, it is better given independently. The usual case considers the point not at the extremity of the line, and the solution is essentially that of Euclid. In practice, however, as for example in surveying, the point may be at the extremity, and it may not be convenient to produce the line. [Illustration] Surveyors sometimes measure _PB_ = 3 ft., and then take 9 ft. of tape, the ends being held at _B_ and _P_, and the tape being stretched to _A_, so that _PA_ = 4 ft. and _AB_ = 5 ft. Then _P_ is a right angle by the Pythagorean Theorem. This theorem not having yet been proved, it cannot be used at this time. A solution for the problem of erecting a perpendicular from the extremity of a line that cannot be produced, depending, however, on the problem of bisecting an angle, and therefore to be given after that problem, is attributed by Al-Nair[=i]z[=i] (tenth century A.D.) to Heron of Alexandria. It is also given by Proclus. [Illustration] Required to draw from _P_ a perpendicular to _AP_. Take _X_ anywhere on the line and erect _XY_ [perp] to _AP_ in the usual manner. Bisect [L]_PXY_ by the line _XM_. On _XY_ take _XN_ = _XP_, and draw _NM_ [perp] to _XY_. Then draw _PM_. The proof is evident. These may at the proper time be given as interesting variants of the usual solution. PROBLEM. _To bisect a given line._ Euclid said "finite straight line," but this wording is not commonly followed, because it will be inferred that the line is finite if it is to be bisected, and we use "line" alone to mean a straight line. Euclid's plan was to construct an equilateral triangle (by his Proposition 1 of Book I) on the line as a base, and then to bisect the vertical angle. Proclus tells us that Apollonius of Perga, who wrote the first great work on conic sections, used a plan which is substantially that which is commonly found in textbooks to-day,--constructing two isosceles triangles upon the line as a common base, and connecting their vertices. PROBLEM. _To bisect a given angle._