cumstances continuing the same, if the quarter of the circumference, instead of making a complete revolution, make but a given part of one, the spherical space, comprised between the quarter of the circumference in it's initial position, the arc corresponding to the base, and the spherical spiral, is to the square of the radius, as the arc of the base is to a quarter of the circumference. Many great geometricians have treated generally the question of determining spaces that may be squared on a given surface, as will be seen in the fourth period. To the praise of Pappus must likewise be added, that at the end of the preface to his seventh book we find a tolerably distinct idea of the celebrated theorem commonly ascribed to father Guldin, a jesuit: that the superficies or solid, generated by the motion of a line or a plane, is equal to the product of the generating line or plane multiplied by the path described by it's centre of gravity.' Though few of the works of Diocles have come down to us, we have enough to inform us, that he was endowed with great sagacity. Beside his cissoid, he discovered the solution of a problem, which Archimedes had proposed in his treatise on the sphere and cylinder, and which consisted in cutting a sphere by a plane in a given ratio. We know not whether Archimedes himself had resolved this question, at that tíme very difficult, and which leads to an equation of the third order in the modern methods. The solution of Diocles, which is learned and profound, terminates in a geometrical construction by means of two conic sections cutting each other. It has been trans transmitted to us by Eutocius, who was himself a good geometrician, and whose commentaries on part of the works of Archimedes and Apollonius in particular are much esteemed. A. D. 520. Serenus, another learned geometrician, is placed about the time of Diocles. We have of him two books on the section of the cylinder and the cone, which Halley has published in greek and latin at the end of the edition of Apollonius. In his first book Serenus considers the ellipsis as an oblique section of the cylinder, and shows that the curve formed in this manner is the same as the ellipsis of the cone. He likewise teaches us to cut a cylinder and a cone, so that the two sections shall be equal and similar. The second book treats on sections of the right and oblique cone by planes passing through the apex, which produce right lined triangles, and by their comparison give rise to a great number of curious problems and theorems, from the different ratios that may subsist between the axis, the radius of the base, and the angle which the axis makes with the base. The whole work of Serenus is a chain of interesting propositions very perspicuously demonstrated. We know no particulars respecting the author. I must not forget to mention Proclus, the head of the platonic school established at Athens. A. D. 500. He has rendered important service to the sciences; he encouraged those, who embraced their pursuit, by his example, instruction, and acts of kindness; and he has left a commentary on the first book of Euclid, which E 3 which contains many curious observations respecting the history and metaphysics of geometry. His successor was Marinus, the author of a preface or introduction to the Data of Euclid, which is commonly printed at the head of that work. We have none of the writings of Isidorus of Miletus, a disciple of Proclus: but his name must not be omitted here, as he is said to have been very learned in geometry, and mechanics, and was employed in erecting the temple of St. Sophia at Constantinople, under the emperor Justinian, jointly with Anthemius, of whom we have a valuable fragment, on which I shall treat more at large, when I come to speak of the burning mirrors of Archimedes. A. p. 530, Hero the younger, so called to distinguish him from Hero of Alexandria, who will be noticed under the article of hydrostatics, is also mentioned among the ancient geometricians. His Geodesia, a work in other respects of little importance, contains the method of finding the area of a triangle by means of it's three sides, but without a demonstration This proposition is supposed to belong to some preceding and more profound mathematician. It is useless to swell this historical abstract with the names of a few other geometricians, from whom their contemporaries perhaps derived instruction; but who, not having perceptibly contributed to the advancement of science, scarcely merit the notice of posterity. СНАР. CHAP. III. Origin and Progress of Mechanics. As the ancients were unacquainted with the theoretical principles of mechanics till a very late period, it is not a little surprising, that the construction of machines, or the instruments of mechanics, should have been pursued with such industry, and carried to such perfection by them. Vitruvius, in his tenth book, enumerates several ingenious machines, which had then been in use from time immemorial. We find, that for raising or transporting heavy burdens they employed most of the means, which we at present apply to the same purposes; the crane, the inclined plane, culties give birth to resources, Ctesiphon the architect, employed in building the temple of Ephesus*, had procured the pillars, which were to support or adorn that vast edifice, to be fashioned in the quarries, and they were now to be conveyed to Ephesus, he was aware, that, if they were placed on a common waggon, their enormous weight would sink the wheels into the ground, and render them incapable of moving, Accordingly he had recourse to another very simple mode. He fixed such as the capstan, pulley, &c. pulley, &c. Diffi For instance, when We know not the date of the building of this temple; but it was burnt by Herostratus, on the night when Alexander was born, in the year before Christ 356. into the centres of the opposite ends of a pillar two stout iron pins, which turned in holes cut in two long beams of wood joined together by a cross piece. Oxen being then harnessed to this sort of frame easily rolled along the pillar. It is by a similar mechanism, that we smooth our terraces, the gravel walks of our gardens, &c. In like manner Metagenes, the son of Ctesiphon, who succeeded his father in continuing the building, having to transport to Ephesus the stones, which were to form the architraves of the temple, fastened these stones between two wheels of twelve feet diameter, which, from their proximity, formed as it were but one cylinder. I might quote a multitude of other instances of the genius of the ancients in practical mechanics, of which the art of war alone would afford me several; for we know, that with their catapulta, scorpions, balistæ, &c. they produced a part of those terrible effects, which, to man's misfortune, the invention of powder has too much facilitated, In the theory of mechanics the ancients were not so successful. We see by some of the writings of Aristotle, that even he had only confused or errone◄ ous notions concerning the nature of equilibrium and motion, and of course all his predecessors must have been still more deficient. A. C. 320. The true theory of the equilibrium of machines dates no higher than the time of Archimedes, to whom we are indebted for it's elements. In his book of Equiponderants he considers a balance supported on a fulcrum, and having a weight in each basin. Taking as a fundamental principle, that, when the two arms of |