CLASS V. ECLECTIC PHILOSOPHERS, OTHERWISE CALLED THE ALEXANDRIAN SECT, OR LATER PLATONISTS, Flourished during the Second, Third, and Fourth Centuries. 68. The preceding enumeration of the principal sects of Grecian, Roman, and Alexandrian philosophers, both scientific and speculative, may, it is presumed, be useful to the juvenile reader of the followng essays, by enabling him, as he proceeds, to determine the school of philosophy to which each of those individuals belonged, who will be more distinctly noticed hereafter; and by exhibiting, at one view, the ramifications of ancient philosophy, from when the scion was first transplanted from the East, to the period in which its sapless trunk mouldered into decay in its native soil. It would extend this historical review of ancient literature far beyond its assigned limits, and render the work extremely dry and uninteresting, were but a brief summary given of the distinguishing tenets of each sect, or of the systems advocated by the most celebrated of these ancient philosophers. Instead of this, the retrospect will be pursued in the following order. I. The history of the abstract sciences, or ancient mathematics. II. The history of physical science, both speculative and practical. III. Ancient dialectics, or logic. IV. Metaphysics, or pneumatology; and, V. Ethics, or moral science. CHAPTER III. THE ABSTRACT SCIENCES. SECT. I. HISTORY OF MATHEMATICS, 69. GEOMETRY is the only branch of mathematical science with which the ancients were acquainted, if we except a slight and imperfect knowledge of arithmetic, said to have been derived from the Phonicians. It will be remembered, that, in the sketch of Egyptian literature, attempted in a former section, a reference was made to the origin of geometry in that country, and the circumstances were mentioned which first led to the study and practice of this important science. In its beginning it was considered but as a more convenient and accurate mode of mensuration than any previously known. Its practical utility therefore recommended it to the Egyptians, since it enabled them, with the utmost accuracy, to determine the rights of landed proprietors, after the annual inundation of the Nile had obliterated all traces of former occupation. Thence it was transferred to Greece, where it early assumed the form of an abstract science, and was cultivated far more successfully than in its native soil. Thales and Pythagoras, the celebrated founders of the two principal sects of philosophy, are universally acknowledged to have been the first Grecian geometricians; but the degree of their mathematical knowledge, or the exact amount of their attainments in this department of science, it is by no means easy to determine. It is probable that many of the propositions found in the elementary works of later geometricians were first discovered by them, though others, by whom they have been preserved and transmitted to posterity, may have obtained the credit of their invention. 70. THALES, who flourished between A. c. 640 and 600, during his residence in Egypt, either taught the Egyptians, or, as is more probable, learned from them, the method of measuring the Pyramids of Memphis by the extent of their shadows. In either case, the fact proves some practical acquaintance with geometry. Several of the fundamental propositions subsequently incorporated into the Elements of Euclid have been attributed to him, particularly those in which it is proved that a "circle is bisected by its diameter; that the angles at the base of an isosceles triangle are equal; that vertical angles are equal; and that the angle in a semicircle is a right angle." These, though amongst the simplest theorems, were no inconsiderable discoveries in the infancy of mathematical science. All the ancient writers speak of him as a learned geometrician, and, among other important discoveries, ascribe to him the first employment of a circle for the measurement of angles. PYTHAGORAS, who flourished about half a century later, pushed forward the discoveries of Thales, and made more important discoveries. His name is rendered immortal among geometricians, by his well-known discovery," that the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the other two sides;" a discovery which is said to have occasioned such an ecstacy of joy, as to induce him to offer, as an expression of gratitude, an hecatomb to the gods. This problem is classed among the most important of geometrical truths, both from the singularity of its result, and the number of cases to which it may be applied in every department of mathematical science. To this philosopher belongs also the praise of having reduced geometry to a more regular system, by his demonstrations and inductive reasonings. Diogenes Laertius informs us, that it was he "who first abstracted geometry from matter, and elevated it from a scheme of mensuration to a truly philosophical science." 71. Next to the above-mentioned philosophers, the names of Enopidus, Zenodorus, Democritus, and Hippocrates, occur in the list of ancient mathematicians. The former of these was a native of Chios, who flourished about A. c. 450, and of whom Plato speaks, as "one who obtained mathematical glory," though it is to be regretted that no traces of his works remain. Among the simple but important problems attributed to him, are those by which a perpendicular is let fall on a right line from a given point; an angle made equal to a given angle; and by which it is bisected. Of Zenodorus, |