Thales, Anaxagoras, Pythagoras, Hippocrates, Plato, Apollonius, Ptolemy, with other ancient mathematicians, have given methods for approximating to the area of the circle; and many also among the moderns. In 1775, the Paris Academy of Science determined to discourage papers devoted to this subject, and their course in this respect was soon after adopted also by The Royal Society, it being found that there was among certain geometers a complete mania for settling this and similar problems, the solution of which was either unattainable, or if attained of very questionable value. DUPLICATION OF THE CUBE. The Duplication of the Cube it is asserted can readily be demonstrated. It is usually called the Delian Problem, from its having been suggested by the oracle of Apollo at Delphos, requiring that Apollo's cubical altar should be doubled. It is something in its favour to say that the enquiry has had the attention of Newton and of Huygens. TRISECTION OF AN ANGLE. Lastly, we shall notice among problems of this class--the Trisection of an Angle, which it is asserted can only be accomplished by means of the conic sections and some other curves. A rule for the cubic equation by which the problem of trisection is solved has been given by Cardan. The difficulty only arises when we attempt the trisection of any other than a right angle, its trisection being easily effected with a pair of compasses. On this subject it has been observed that, "there is no more trouble in trisecting an angle, not a right angle, than in finding a cube root." * * * * * These three celebrated problems have received the attention of mathematicians in every age and country, and led to many learned discussions, and controversial writings. But in point of litigiousness the Squarers of the Circle most decidedly carry off the palm, having frequently laid and lost heavy wagers, and even appeared in a Court of Justice to settle their monetary disputes. They are renowned for their pamphlets, in which philosophers of every class are charged with prejudice, conceit, and ignorance, and denounced for their want of candour and consistency in not giving audience to the projector of the last best demonstration. PERPETUUM MOBILE. To conclude this Lecture we shall offer a few remarks on Perpetuum Mobile, or the search for a means of obtaining a mechanical perpetual motion. As a mathematical problem it dates back some 2000 years or more, but we know nothing of any actual attempt earlier than the 14th century to construct a machine intended to be self motive, by containing within itself the means of continually overbalancing. External motive agency such as the tides, magnetism, and the like are not included; the only admitted agent being gravity.