Meanwhile, Viviani was advised by his friends not to lose the fruit of his investigations, and accordingly, with out being made acquainted with the contents of the books that had been found, he proceeded and published the result of his labours, in 1659. The translation made by Borelli, accompanied by learned notes, was published in 1661 ; and it is remarkable, that in the Arabic manu script he had found, as well as in that of Golius, and in the abridged version of Abdolmelec which Ravius had brought from the East, and published in 1669, the eighth book is entirely wanting, so that it is now, in all proba. bility, lost for ever. Dr Halley, however, attempted to restore it from the hints afforded by Pappus, and pub lished the fruit of his researches along with the other seven books, and two books on the sections of cylinders and cones, written by Serenus, a geometer who lived in some of the early centuries of the Christian era. It is commonly supposed, that the restoration is so excellent, as to leave but little reason to regret the loss of the original.
The conics of Apollonius procured him the appellation of the Great Geometer : a character to which he appears to have been justly entitled, whether WO consider the difficulty of the subjects on which he wrote, or the subtlety of his investigations and the shill and success with which he has conducted them. Among the im provements which he introduced into the mode of treat ing the subject, there is one particularly worthy of re mark, because it is one of many instances in the history of science, of the slow progress of the human mind in passing from particular to more general truths. Before his time, the different curves were defined, by suppos ing right cones to be cut by planes perpendicular to their sides. By this method, three different cones were required to produce the three sections: a right angled cone for the parabola ; an acute angled cone for the ellipse ; and an obtuse angled cone for the hyperbola : But Apollonius shelved how all the three sections might be formed by any one cone, whether right or oblique.
In the early ages of science, the conic sections were cultivated merely as a geometrical theory, that might afford an agreeable subject of contemplation to the mind, but without a prospect of its ever being applicable to the explanation of the phenomena of nature. The discoveries of modern times, however, have greatly extended its utility, and rendered it by far the most interesting specu lation in pure geometry. Galileo spewed, that the path of a body projected obliquely is a parabola, and Kepler discovered that the planetary orbits are ellipses these facts alone were sufficient to enhance greatly the value of the theory of the conic sections ; but the numerous discoveries of Newton that followed went much farther, and incorporated it with those of astronomy and the other branches of natural philosophy.
In explaining the nature and properties of geometrical figures, it becomes a question how they are to be defined. A figure may be defined from any one of its properties, which distinguishes it from all other figures of a dif ferent kind ; but that ought to be chosen which is simple, which shows how the figure may be readily constructed, and which naturally leads to its other properties. The ancients defined the class of curves we are about to con sider, by supposing a cone to be cut by a plane ; and their example has been followed by several modern writers, who have, upon this principle, composed elaborate and valuable treatises. As the doctrine of solids is, how ever, a more intricate branch of geometry than that of plane figures, other modern writers, of whom Dr \Vallis was the first, have thought it better to define time curves, by shewing how they may be described on a plane, with out any reference to a cone ; and treatises not less valua ble, and (in our opinion) in some respects more simple, have been written on this plan. We propose to follow this second method, believing it to be the best adapted to our work.
If our system of GEOMETRY were written, we should refer to it as often as we had occasion to quote a propo sition in the elements of geometry ; as, however, from our mode of publication, that article is not vet ready, we shall, in the mean time, refer to the propositions in Euclid's Elements of Geometry, as contained in Mr Pro fessor Playfair's edition : and in our article GEOMETRY give a table, shewing to what proposition, there, each of Euclid's corresponds.
We shall conclude this Introduction with a catalogue of some of the more curious and valuable works on this branch of geometry.
Among the works of Archimedes which have been preserved, there is a treatise, On the- Quadrature of the Parabola, and another On Conoids and Spheroids. See Barrow's edition, London, 1675; or Torelli's edition in Greek and Latin, Oxford, 1792 ; or Peyrard's Ft ench translation of Archimedes' works, Paris, 1808.
Pergei Conicorum libri oat), et Sereni An tissensis de Sectionc Cylindri et Cozzi libri duo. Tills is Dr Valley's edition. The lirst four books, together with the Lemmata of Pappus, and the commentaries of Eutocius, have been published from Greek manuscripts, accompanied with a Latin translation ; the 5th, 6th, and 7th books, which also contain the Lemmata, have been translated from Arabic into Latin ; the eighth has been restored by Dr Malley. The books of Severn's are in Greek and Latin.