compasses, the solutions of Menechmus would have all the advantage of geometrical construction in the sense which the ancients applied to the term. But at present no instruments have been made that will describe the conic sections in this manner. I cannot, however, pass over the problem of the trisection of an angle, which is of the same kind with that of doubling the cube, both of which were equally agitated in the school of Plato; and although a solution was not to be attained by means of the rule and compasses only, yet it was reduced to a very neat and simple proposition. This consists in drawing a right line from a given point to the semi-periphery of a circle, which line shall cut this periphery, and the prolongation of the diameter that forms its base, so that the part of the line comprised between the two points of intersection shall be equal to the radius; a result that gives rise to many simple constructions, two of which may be seen in Bonnycastle's Elements of Geometry, page 282. Most of the ancients, however, were so possessed with the hope of solving these two problems with the rule and compasses only, that they could not be persuaded to give it up; they made many fruitless endeavours; and this anxiety raged like an epidemic disease, which has been transmitted from age to age down to the present day; but after they have baffled the attempts of such illustrious characters as Archimedes among the ancients, and Newton and Maclaurin among the moderns, it would surely árgue a want of discretion in a young mathematician to waste his time in such ill-fated speculations. It may not be improper to mention the celebrated Aristotle, the successor of Plato, and preceptor to Alexander the Great; but though, in other respects, he may be regarded as one of the greatest men of his or indeed of any other time, yet we are not told that he made any improvements in the mathematical sciences. After attending the lectures of Plato, he opened a school himself in the Lyceum of Macedonia, which was assigned him by the magistrates, and was the founder of the sect called the Peripatetics. Among the number of his disciples were Theophrastus and Eudemus, who particularly applied themselves to the study of the mathematics. The former wrote a History of the Mathematics in eleven books, from their origin to his own time, four of which treated on Geometry, six on Astronomy, and one on Arithmetic. The latter also wrote a work of a similar kind, consisting of six books, on the History of Geometry, and another of the same number of books on that of Astronomy. But these, which would have been so useful to the modern scientific inquirer, which would have assisted him so much in his researches after the precise origin of the various mathematical sciences, and their progress in those times, have not been transmitted to the present age. Notwithstanding the ancients were not successful in the object they sought to attain, yet Geometry received additional splendor from the researches they were continually making; new theories were introduced, and some ingenious instruments for solving the two problems in question, so as to approximate the truth near enough for practical purposes; most of these methods are now lost, but those of four eminent geometricians, viz. Dinostratus, Nicomedes, Pappus, and Diocles, deserve particular praise for their merit; but the reader must excuse my not entering into an explanation, or exhibiting to him a view of their several plans, as such would swell this introduction much beyond the limits which I intend it should oc cupy. Next after the period of Plato, and his disciples here mentioned (passing over Euclid for the present), may be reckoned Archimedes of Syracuse, who was born about 280 years before Christ. In his youth, he devoted himself to the study of Geometry; and in his maturer years, he travelled into Egypt, where the Greeks usually resorted in the pursuit of science. After an absence of several years, which he spent in the society of Conon and other eminent men, and during which time he gave very promising indications of his future fame, he returned into his own country, and then continued his studies with the greatest zeal and assiduity. Such, indeed, were the intenseness and ardour of his application to mathematical sciences, that he prosecuted his studies to the neglect both of food and sleep, and improved the minutest circumstance that occurred into an occasion of making very important and useful discoveries. His active and comprehensive genius led him to the study of every branch of science then known; Geometry, Arithmetic, Optics, &c. equally engaged his attention, and alike experienced the powerful effects of his superior talents, talents which placed him with such distinguished lustre in the view of the world as to render him both the honour of his own age, and the admiration of posterity. He was, indeed, the prince of the ancient mathematicians, being to them what Newton is to the moderns, to whom, in his genius and character, he bears a very near resemblance. He was the first who squared a curvilineal space, excepting Hippocrates, on account of his lunulæ. He applied himself closely to the measuring of conic sections, as well as other figures. He determined the relations of spheres, spheroids, and conoids, to cylinders and cones, and the relations of parabolas to rectilineal planes, whose quadratures had long before been determined in geometry. He also proved that a circle is equal to a right angled triangle, whose base is equal to the circumference, and its altitude equal to the radius. Being unable to determine the exact quadrature of the circle, for want of the rectification of its circumference, which all the methods he devised would not effect, he proceeded to assign a useful approximation to it: this he effected by the numeral calculation of the perimeters of the inscribed and circumscribed polygons; from which calculation it appears, that the perimeter of the circumscribed regular polygon of 192 sides is to b its diameter in a less ratio than 34 to 1, and that the perimeter of the inscribed polygon of 96 sides is to the diameter in a greater ratio than that of 3 to 1: therefore the ratio of the circumference to its diameter must be between these two ratios. But that which has rendered him most famous in the eyes of posterity is the fabrication of such admirable engines for the defence of Syracuse when besieged by the Roman consul Marcellus, showering upon the enemy sometimes long darts, and stones of vast weight, and in great quantities; at other times lifting their ships up into the air, that had come near the walls, and dashing them to pieces, by letting them fall down again; nor could they find their safety in removing out of the reach of his cranes and levers, for there he continued to fire them with the rays of the sun reflected from burning glasses. However, Syracuse was at length taken ; "what gave Marcellus the greatest concern," says Plutarch," was the unhappy fate of Archimedes, who was at that time in the museum; so intent was his mind, as well as his eye, upon some geometrical figures, that he heard not the clashing of arms, nor the invasion of the city; in this state of abstraction, a soldier came suddenly upon him, and commanded him to follow him to Marcellus; but he refusing to stir till he had finished his problem so much enraged the soldier that he ran his sword through his body." Livy says, that Marcellus was so much grieved that he took care of his funeral, |