who improved and made additions to the science. Among his cotemporaries may be named Leodamas of Thasus, Archytas of Tarentum, and Theætetus of Athens, by whom some theorems were discovered, and some improvements made in the methods of demonstration. Diogenes Laertius reports of Archytas, that he was the first who brought Mechanics into method by the use of mechanical principles, and the first who applied organic motions to Geometrical figures, and found out the duplication of the cube in Geometry. (Book VIII.) His solution of the duplication of the cube is given by Eutocius, in his commentary on the sphere and cylinder of Archimedes. Archytas was also the writer of a work on the elements of Geometry, which is not extant. The same writer relates, that Leodamas, by means of the Geometrical Analysis, which he had learned from Plato, discovered the solution of many problems, and made many discoveries. Theætetus was celebrated for a treatise on Geometry, which is lost. Plato honoured this disciple by giving to one of his dialogues the title of Theætetus; and in another, entitled Phædrus, he ascribes the origin of Geometry to Thoth, an Egyptian divinity. Aristeas, a disciple and friend of Plato, composed five books on the Conic Sections, which were highly esteemed. Another of the scholars of Plato was Neoclides, whose name is celebrated by Proclus. Leon was a scholar of Neoclides (B. c. 368), and was the author of several discoveries in Geometry. To Leon is attributed the invention of a method for discriminating the possibility or impossibility of a problem. He is also mentioned as the author of a work on the elements of Geometry. None of his discoveries or writings have descended to posterity. Amyclas was another friend of Plato; and the brothers Menæchmus and Dinostratus are both celebrated; the former for his application of the Conic Sections to solve the problem of the duplication of the cube, and the latter, for the discovery of a curve known by the name of the quadratrix. They are also said to have made some other additions to the science of Geometry, and to have rendered the whole more perfect. Theudias appears to have excelled both in mathematics and philosophy, and is said to have composed a work on Geometry, and to have generalized some theorems. At the same period, Cyzicenus of Athens, besides other branches of the mathematics, successfully cultivated Geometry. These friends and disciples of Plato used to resort to the Academy, and employ themselves in proposing, by turns, questions for solution. Eudoxus, a native of Cnidus, a town of Caria in Asia Minor, was one of the most intimate of the friends of Plato. He is reported to have written on the Elements, and to have generalized many results which had originated in the school of Plato, and to have advanced the science of Geometry by many important discoveries. To him is attributed the invention of the doctrine of proportion as treated in the 5th book of Euclid's Elements. He is said to have been the first who discovered that the volume of a cone or pyramid is equal to one third of its circumscribing cylinder or prism; that is, of a cylinder or prism having the same base and altitude. He is also reported to have advanced the knowledge of the higher Geometry, by the discovery of several important properties of the Conic Sections. He died B. c. 368, at 53 years of age. Diogenes Laertius, in his short memoir of Eudoxus, describes him as an astrologer, a physician, a legislator, and a geometer. Though none of his writings have descended to modern times, his name has been celebrated by the

eminent men, both of Greece and of Rome. Hermotimus wrote on Loci, and is said to have extended the results of Eudoxus and Theætetus. Philippus the Mendean, another disciple of Plato, is reported to have discovered problems, and to have proposed questions, being a great lover of the mathematical sciences. All these were either disciples of Plato, or attached themselves to the school he founded at Athens, and are mentioned by Proclus as having advanced or improved the mathematical sciences. Xenocrates also was a hearer of Plato, and is said to have been one of the instructors of Aristotle; he was renowned for his knowledge of the mathematical sciences. Aristotle, though originally a disciple of Plato, and attached to his school for a period of twenty years, became the founder of a new sect of philosophers-the Peripatetics, and opened a school, B. c. 341, at the Lyceum on the banks of the Ilissus. There he continued twelve years, till the false accusation of Eurymedon obliged him to flee to Chalcis, where he died at the age of sixty-three.

On this division of the Platonic school, the two sects-both the Academics and the Peripatetics-continued to hold the same opinion on the utility of Geometry, as the necessary introductory knowledge for all who were desirous of proceeding with the study of philosophy. Thus the science of Geometry continued to be cultivated, and to make advancement. Among the numerous writings of Aristotle, there is a treatise on Mechanics, and a collection of Problems in 38 divisions; the 15th consists of mathematical questions. There is also a tract on indivisible lines, which has been ascribed by some ancient commentators to Theophrastus, as we learn from Simplicius. (Rev. J. W. Blakesley's Life of Aristotle.)

Two of the Peripatetic school are especially celebrated, Theophrastus and Eudemus, who devoted themselves chiefly to mathematical studies. Theophrastus was the author of the first history of the mathematical sciences, from the earliest times to his own. The work consisted of eleven books, of which there were four on Geometry, six on Astronomy, and one on Arithmetic. Eudemus also wrote a history of Astronomy in six books, and a history of Geometry in six books, from which Proclus acknowledges that most of his facts were taken. None of these writings have been preserved.

Autolycus, of Pitane in Æolis, lived about 300 B. C. He was preceptor in mathematics to Arcesilaus, a disciple of Theophrastus the successor of Aristotle, as we learn from Diogenes Laertius. His treatise on the moveable sphere is the earliest written on that subject. The original Greek, with a Latin translation, was published in 1572. He also was the author of another treatise "On the Rising and Setting of the Stars," which is still extant, and has been translated and printed.

Aristeas is said to have composed five books on the Conic Sections, and five books on Solid Loci. He is also said to have been the friend of Euclid, and his instructor in Geometry.

We come next to the time of Euclid. The birth-place and even the country of Euclid are unknown, and he has been very frequently confounded with another philosopher of the same name, who was a native of Megara. He studied at Athens, and became a disciple of the Platonic school. He flourished in the time of Ptolemy Lagus (B.c. 323 to 284), to whom he made the celebrated reply, that "there is no royal road to Geometry." He is said to have successfully cultivated and

taught Geometry and the mathematical sciences at Alexandria, shortly after the school of Philosophy was founded in that city. The school at Alexandria became most distinguished for the eminent mathematicians it produced, both in the lifetime of Euclid and afterwards, until the destruction of the great library at Alexandria, and the subjugation of Egypt by the Arabians. Euclid has become celebrated chiefly by his work on the Elements of Geometry, for which his name has become a synonym. It consists of thirteen books, nine of which are devoted to the subject of Geometry, and four to the properties of numbers, as discussed by the Greek Arithmetic, and applied to Geometry. There are two other books on the five regular solids, usually found appended to the thirteen books of Euclid. These, however, were subsequently added to the Elements by Hypsicles of Alexandria. In some editions a sixteenth book is found, which was added by Flussas. Another geometrical work of Euclid is the Data, which consists of 100 propositions. This is the oldest specimen of the principles auxiliary to the Geometrical Analysis. The object of the several propositions of this book is to shew that, in cases where certain properties or ratios are given, other properties or ratios are also given, or may be found geometrically; and thus pointing out what data are essential in order that Geometrical Problems may be determinate and free from all ambiguity. Three books on Porisms are attributed to Euclid by Pappus and Proclus; the former, in the seventh book of his Mathematical Collections, has given some account of them and the general enunciations of some propositions. This contains all that is known to exist of these three books. Attempts have been made with some success in later times for their restoration. Euclid also wrote a Treatise on Fallacies in geometrical reasoning, and another on Divisions. Pappus makes mention of another work on Geometry attributed to Euclid under the title of TÓT Tрos éπipάveιav (Coll. Math. Lib. VII. Introd.), which his Latin translator Commandine has rendered by "Locorum ad Superficiem." Pappus also states that Euclid composed four books on the Conic Sections, which were afterwards augmented by Apollonius Pergæus. Proclus has made no allusion to this work in his account of the writings of Euclid. Besides these writings on Geometry, Euclid is reported by Proclus to have been the author of a work on Optics and Catoptrics, and a work on Harmonics; but it is very questionable whether the treatise on Harmony which is extant, and attributed to Euclid, was really composed by him. Pappus mentions a work on Astronomy entitled "The Phænomena." This treatise contains some geometrical properties of the sphere.

It has been a question whether Euclid was the author or the compiler of the Elements of Geometry, which bear his name. If Euclid were the discoverer of the propositions contained in the thirteen books of the Elements, and the author of the demonstrations, he would be a phenomenon in the history of science. It is by far more probable that he collected and arranged the books on Geometry in the order in which they have come down to us, and made a more scientific classification of the geometrical truths which were known in his time. Euclid may also have been the discoverer of some new propositions, and may have amended and rendered more conclusive the demonstrations of others. From the slow advances of the human mind in making discoveries, and the general history of the progress of the sciences, it would seem

unreasonable to assign to Euclid a higher place than that of the compiler and improver of the Elements of Geometry. This is in complete accordance with the statement of Proclus, who relates that "Euclid composed Elements of Geometry, and improved and arranged many things of Eudoxus, and perfected many things which had been discovered by Theatetus, and gave invincible demonstrations of many things which had been left loosely or unsatisfactorily demonstrated before him." The Elements of Geometry, thus arranged and improved by Euclid, were acknowledged so far superior in completeness and accuracy to the elementary treatises then existing, that they entirely superseded them, and in course of time all have disappeared. The book of Euclid's Elements is therefore the most ancient work on Geometry known to be extant. The Greek Arithmetical Notation, employed in the arithmetical portion, has yielded to the more perfect system of the Indian; but the geometrical portion, from the time it was first put forth till the present day, a period of upwards of 2000 years, has maintained its high character as an elementary treatise, in all nations wherever the sciences have been cultivated.

At this part of our subject we cannot forbear making a few remarks on the comparative claims of the Egyptians and Greeks to the merit of being the authors of Scientific Geometry. The learned Sir Gardiner Wilkinson, in his profound work on the Ancient Egyptians, unhesitatingly assigns this honour to the Egyptians. In page 342 of Volume I. he expresses his judgment in the following terms: "Anticlides pretends that Moeris was the first to lay down the elements of that science, which, he says, was perfected by Pythagoras; but the latter observation is merely the result of the vanity of the Greeks, who claimed for their countrymen the credit of enlightening a people on the very subjects which they had visited Egypt for the purpose of studying." The vanity both of the later and the earlier Greeks may be readily admitted, without allowing that it suggests the true answer to this question. Diogenes Laertius, (Book VIII.) in his life of Pythagoras, writes thus: "Anticlides reports that Maris was the first who invented Geometry, and Pythagoras brought his imperfect notions to perfection." Moris was an early king of Egypt (Herod. 11. 13), the son of Amenophis, and lived before the age of Sesostris. Whatever we learn from Herodotus respecting Geometry in Egypt, is referred to the age of Sesostris, and does not carry us beyond such processes as might exist without any attempts at science. The Geometry which was brought from Egypt to Greece, appears to have been in its infancy; and all that Pythagoras and others borrowed from the Egyptians could not have exceeded some practical rules and their applications. In the learned work referred to, any positive information as to the existence of Scientific Geometry among the early Egyptians, has been sought in vain; nor do ancient writers exhibit any remains of the Scientific Geometry of the Egyptians, or even make any claims in their favor to the merit of its invention. The general tenor of all the traditional and probable evidence tends to shew, that the scientific form of the Elements of Geometry is due to the acute intellect of the Greeks. And this presumption is reduced almost to historical certainty by the existing remains of the Greek Geometry. We may further observe, that in the very brief review we have given of the earliest commencement of the Greek Geometry and Mathematics, their simplicity is such as might be expected to characterize the original

attempts of an acute people. The advances were so gradual that any supposition of the sudden introduction of more perfect science from foreign sources is completely removed. Moreover, at the time that the Greek Geometrical science was rapidly advancing towards perfection in the school of Plato, there is a total absence of any accounts that the Geometry of the Egyptians or of any other nation had proceeded so far in its development. Plato truly ascribes to the Egyptians and Phonicians a certain commercial activity, but distinguishes their native character from that of the Greeks, which he represents as remarkable for its curiosity and desire of knowledge. (De Repub. IV.) Five centuries afterwards the same characteristic of that people is remarked by St Paul in his first epistle to the Corinthians.

Archimedes was born at Syracuse, B.C. 287, about the period of the death of Euclid, and became the most eminent of all the Greek mathematicians. His discoveries in Geometry, Mechanics, and Hydrostatics, form a distinguished epoch in the history of mathematical science; and his remaining writings on the pure Mathematics are the most valuable portion of the ancient Geometry. He was the first who discovered that the volume of a sphere is two thirds of its circumscribing cylinder, or of a cylinder having the same diameter and altitude as the sphere, and that the curved surface of each is equal to four great circles of the sphere. He also found the relation of the volumes and surfaces of a hemisphere and cone upon the same base. These and other properties are investigated in two books, entitled, "On the Sphere and Cylinder," which have descended to our times in the original Doric Greek, together with the Commentary of Eutocius. Another work is still extant on the measurement of the circle, in which he shews that the area of a circle is equal to that of a right-angled triangle whose altitude is equal to the radius and whose base is equal to the circumference. Though he failed in his attempts to discover the exact proportion of the circumference to the diameter of a circle, he discovered an useful approximation to that ratio. He found, by numerical calculation, that the perimeter of a regular polygon of 192 sides, circumscribing a circle, is to the diameter in a less ratio than 3% to 1; and that the perimeter of the inscribed polygon of 96 sides is to the diameter in a greater ratio than 3 to 1: whence he concluded that the ratio of the circumference to the diameter of the circle must lie between these two ratios. The book of Lemmas, a collection of Problems and Theorems on Plane Geometry, attributed to Archimedes, is not known to be extant in the original Greek. It comes to us from the Arabic, of which a translation in Latin was published, for the first time, in 1659. On the Higher Geometry, three tracts of Archimedes are still extant. 1. On the Quadrature of the Parabola, in which he proves that the area included between the curve and two ordinates is equal to two thirds of the circumscribing parallelogram. This is the first instance known of the discovery of the quadrature of a figure bounded partly by a curved line, if we except that of the lunes of Hippocrates. 2. His Treatise on Conoids and Spheroids, which contains many discoveries: : among them may be named, the ratio of the area of an ellipse to a circle having the same diameter as the axis major of the ellipse; and that the sections of conoids and spheroids are conic sections. He also first proved that the volume of the cone and parabolic conoid of the same base and altitude are in the proportion of 2 to 3. This tract also contains the demonstrations of several discoveries