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APO reign of Ptolemy Philopator, from 222 to 205. Amid that constellation of illustrious geometers, which comprehends the names of Euclid, Archimedes, Apollonius, and Ptolemy, the great writer, whose achievements we shall briefly explain, occupies no second place. Unfortunately, it is of his works only that we can speak; for, concerning the man Apollonius, or any incidents of his life, nothing whatever is known.—I. The distinctive place occupied by Apollonius may be easily defined. Geometry in its largest signification consists of two parts. The first concerns itself with the measurement or evaluation of enclosed or finite spaces; a subject of remarkable simplicity, so long as its inquiries are limited to spaces enclosed by straight lines, but at once difficult and complex when the bounding lines are curves. The glory of having effected the transition from the simple to the arduous class of such problems, belongs unquestionably to Archimedes, whose genius indeed still illumines all this region of inquiry. His memorable discovery of the method of Exhaustions, contains the germ of our potent Infinitesimal Calculus, or at least the fundamental idea of that calculus as it presented itself to the mind of Newton. With this portion of geometrical science, the labours of Apollonius had little to do. But there is a great division of the subject; and if we do not allege that Apollonius stands towards that division, altogether as Archimedes to the first, it is only because it is impossible to forget the Data of, and those fragments of his treatise on Porisms. The large and imposing section of geometrical inquiry now referred to, is that which the Stagyrite must have had in his eye, when he asks (Metaphysics, Book II., chap. 3,) "With what conceptions are mathematicians engrossed, if not such as relate to order and proportion?" It is occupied not with the measurement of spaces (unless incidentally), but with consideration of the properties of Form, as Form—with the properties of enclosed spaces, in so far as such properties result from their figure and position; and with no great injustice to the well-known author of the "Elements," this division of the science may rightly be termed "The Geometry of Apollonius." Here also, the acknowledged Head and Leader of future inquiry is pre-eminently distinguished, if not by the inauguration, at least by the confirmation and development of a most fertile and powerful method. It is known to every reader of the Elements that geometers often advance to new theorems by starting from bases or truths already demonstrated; which truths they combine by tentative methods, and reach at length the end aimed at. This method the Greeks termed Synthesis. But synthesis is a difficult and confined method; permitting little freedom to invention, rightly so called. The honour of suggesting an inverse process—the process called Analysis—is attributed to Plato. Geometrical analysis consists in assuming as true, the truth supposed to be true, or assuming as constructed, the problem required to be constructed; and then, by reasoning downwards, in endeavouring to reach some truths already established, or some recognized property. There is no doubt of the vast superiority of this analytic method, or of the amazing freedom within which an expert inquirer feels himself, while he is exercising it. If, from a plain, one is required to ascend to a mountain top, how limited his choice of a pathway—how uncertain the best that he can think of! But, from the top of the mountain, practicable routes may be descried on every side—routes apparently practicable at least,—for in neither case can the adventurer escape deceptions caused by the foreshortenings of perspective. If Plato suggested the process of geometrical analysis, it was reserved for Apollonius to realise it,—to exemplify, in the more arduous regions of inquiry, its consummate freedom, its vast capacity, its power to excite an infinity of tours de force. From causes which cannot now be specified, this method was lost for ages. But it ever suggested itself anew—although in modified and comparatively feeble forms—to men of genius; and its late but confirmed triumph over comparatively barren Synthesis, is at the root of the revival, which, in our modern times, is rapidly restoring geometrical methods to their rightful place. It was an intimate sense of the power inherent in analysis, as well as the feeling of its great beauty, that more than justified the emphatic words of ,—"This is the geometry that was so fertile in the hands of Archimedes, of Hipparchus, of Apollonius; none other was known to Napier, Vieta, Fermat, Descartes, Galileo, Pascal, Huygens, Roberval; and surely a geometry which was cultivated, as if with the force of a predilection, by our Newtons, Halleys, and Maclaurins, cannot be supposed to be without its peculiar advantages."—II. But passing from general considerations, let us glance at the specific works of Apollonius. These were arduous and numerous; but owing to the loss and destruction of monuments of literature and science that accompanied the fall of the Empire, we should have known little of them, unless for the indications contained in those most precious "Collections of Pappus." In the first place, as to his smaller treatises. These, according to Pappus, who has given us the heads of their contents, although in a most enigmatical form, were entitled, "De Sectione Rationis," "De Sectione Spatii," "De Sectione Determinata," "De Tactionibus," "De Inclinationibus," "De Locis Planis." Of these six treatises, in which the great geometer appears to have unfolded all the resources of analysis in the resolution and discovery of problems and theorems, one alone survived the ravages of time, viz. the "Section of Ratio," a translation of which from an Arabic manuscript we owe to our admirable Halley. The hints given by Pappus, however, have enabled skilful men so far to recover the others, that we may safely account ourselves to be in possession of most of the methods of Apollonius. The treatise on "Plani Loci" was, after the failure of many previous geometers,—amongst whom we must place even Fermat,—ultimately reproduced by Robert Simson of Glasgow, in a work that will ever be accounted a model of geometrical rigour and elegance. The character and contents of the "Section of Space" were divined by Halley. The "Determinate Section" was revived also by Simson. The two books on Inclination's we owe anew to Marinus Ghetaldus; and the two books on Tangencies, to Vieta, Ghetaldus, and Anderson of Aberdeen. Several of these treatises were published in English by the late Mr. Lawson. Whoever refers to the scantiness and obscurity of those hints of Pappus which led to so successful a reconstruction of the works of Apollonius, will not consider it inappropriate to recall the triumphs of the modern comparative anatomist, which, through effect of determinate laws, have enabled him to reconstruct out of a few scattered bones, races of animals that for ages have been extinct. But the grand production of the ancient geometer is unquestionably his "Conic Sections," a work in eight books, which we may venture to say are now possessed by us, almost if not wholly, in their original purity and completeness. Our magnificent edition of the Conics, printed by the liberality of Oxford, is one of the greatest of those achievements which make science so profound a debtor to Dr. Halley. The first four books are given in their original form, accompanied by a Latin equivalent; three are translated from an Arabic manuscript; and the eighth as it was restored after Pappus, by the unsurpassed sagacity of the editor. It were not easy to overrate the merits of this memorable work. Not only is every logical geometrical artifice, known until quite modern times, developed and employed in it, but that remarkable power of generalization—of finding his way from single propositions towards new and large fields of thought—which pre-eminently distinguished Apollonius, inspires it all. In the first four books all the finest properties of conics are brought out with the penetration, and co-ordinated with the skill of a master. In the third book, for instance, we find a proposition that is in reality the basis of the modern theory of reciprocal polars—the proposition, viz.:—"If, from the points of meeting of two tangents to a conic section, a transverse be drawn cutting the curve in two points, and the chord joining the points of contact of the two tangents in a third point, then this third point and the point at which the two tangents meet, will be harmonic conjugates, in reference to the first two points." It is in the fifth book, however, that the genius of Apollonius chiefly appears. In that book, for the first time, we find researches concerning Maxima and Minima; and no initiated reader can fail to detect the germ of our modern theory of centres of osculation, and therefore of a perfect determination of the evolute of a conic.—Our space limits us to one general reflection. It were vain, of course, to expect, in the Greek geometry, resources in any way comparable to those of the science in the posture it has assumed since the times of Carnot and Monge. At length its procedures have escaped all limitation; so soon as one new property of figurate space is established, we can discern that property penetrating a wide range of forms, modified by fixed laws. Pure geometry, indeed, which has nothing of the obscurity of the algebraic methods of the middle ages, now surpasses them in facility and generality. Nevertheless, the rudiments of the greater part of these recent and