Borelli also claims attention on account of his editions of Euclid, A pollonius, and Archimedes, works remarkable for their brevity and perspicuity ; and also because of his efforts in translating from the Arabic three hooks of the Conics of Apollonius, which were then supposed to have been lost. See CONIC SECTIONS.
Viviani, another disciple of Galileo, must here also be noticed. His geometrical writings were of the most ele gant and valuable kind. We have spoken, in our treatise on CONIC SECTIONS, of his restoration of the Conics of Apollonius ; and in our treatise on FLuxioNs, (art. of his beautiful problem concerning the quadrature of a por tion of the surface of a sphere.
We have already noticed some of Descartes's geometri cal labours, but his main effort, for which his name will be banded down to posterity with honour, was his application of algebra to geometry ; an invention by which the proper ties of geometrical figures were represented by equations. His Geometry, which contains his views on this subject, was published first in 1637. The union of geometry and algebra promoted very much the discovery of the new cal culus, the germ of which lay concealed in the method of cxhaustions of the ancients, was partly evolved by Caval lerius, and still farther in the arithmetic of infinites of Dr Wallis, and, lastly, fully expanded by Newton and Leibnitz. The history of geometry becomes now interwoven with that of the modern analysis, and is chiefly interesting by the extent to which the science has been carried by that powerful instrument of invention.
Although the ancient geometry was thus in a manner supplanted by the modern theories, the science by no means lost its interest. Sir Isaac Newton held it in such esteem, that he delivered his sublime discoveries in its lan guage, and established their truth by its peculiar mode of de monstration. The conic sections, one of its earliest and most profound theories, acquired a value by its application to as tronomy, far beyond what it ever had when considered merely as an object of intellectual speculation ; and the learning and genius of Halley and Gregory were employ ed in restoring and bringing into view the precious re mains of Euclid and Apollonius.
For upwards of a century, the physico-mathematical sciences have very much engaged the attention of mathe maticians; but in these, the ancient geometry affords a very limited degree of aid in comparison to the modern : hence no doubt it has happened, that the venerable theo ries of the ancients have heen less noticed. There have, however, been some who have sedulously cultivated them, and endeavoured to restore them to their former impor tance; and this spirit has been particulatly manifested in Britain. it is a curious circumstance, that when the sub tile reasonings of the ingenious Bishop of Cloyne had rais ed doubts as to the justness of the high claims of the doc trine of fluxions, the boast of the modern analysis, the in genious Maclaurin thought the safest course he could fol low was, to call geometry to its aid, and to explain its prin ciples in the clear style, although circuitous manner, of the ancients.
Passing over several foreigners who have cultivated geo metry in the 18th century, as well as natives of Britain, for whose particular labours we cannot find room, we must notice the celebrated Scottish geometer Dr Robert Sim son. To him we arc indebted for a treatise on conic sec
tions, composed on the model of the ancients ; also for re• storations of the Plane Loci of Apollonius, and the books of determinate sections ; but more especially for his resti tution of thd Porisms of Euclid: a task which we are not certain was ever accomplished by any geometer before his time ; although Albert Girard, as we have already observ ed, claimed a like honour. His edition of the first six, and the eleventh and twelfth books of Euclid's Elements, has now in a manner superseded all others in this country, and is almost universally taught in our schools.
Dr Mathew Stewart, the friend of Simson, was another geometer, whose writings and example have greatly con tributed to establish n col rect taste for this study in Scot land. Isis first essay was, a Collection of General Theo rems; the most elegant of any that are known in the whole compass of mathematics. They were given without de monstrations, except a few of the more simple, which he has proved with a degree of elegance that renders them the finest models of geometrical reasoning. His tracts, physical and mathematical, his essay on the sun's distance, and his solution of Kepler's problem, are attempts to ap ply pure geometry to the sublimest inquiries in astronomy. His partial success has shewn what may be accomplished by the force of genius ; and the points in which he has failed serve also to spew, that even in such a masterly hand, the geometrical method is still limited in its application, and can by no means be compared in power with the mo dern geometry. This excellent geometer published also a work entitled Propositiones Geometricx, Afore Veterum demonstrate, ad Geometriam 4ntiquam illustrandam et pro movendam idonex, which we reckon one of the most valua ble that could be put into the hands of a student that is previously acquainted with the elements, and is desirous of learning the true spirit of the ancient geometry. To such of our readers as wish to appreciate the high merit of these two geometers, we recommend the Rev. Dr Traill's excellent life of Dr Simson (1812), and an elegant biogra phical account of Dr Stewart, composed by Mr Playfair, and read before the Royal Society of Edinburgh, (Edin. Phil. Trans. vol. i.) See also the articles Simsos and STEW ART in our Work.
Regretting that our limits oblige us to omit many Brit ish geometers, whose names deserve preservation, we shall yet mention two ; the Rev. Mr Lawson, author of a Dis sertation on the Geometrical Analysis of the Ancients, and English editions of the Tangencies and Determinate Section of Apollonius ; and Dr Horsley, Bishop of St Asaph. This learned prelate has given a restoration of Apollonius' work on Inclinations, also a neat edition, in Latin, of Euclid's Elements, besides other works on geo metry. For farther information relative to the history of geometry, the reader may consult the articles ANALYSIS,