It has long ceased to be usual to read more of Euclid than the first six books and the eleventh. Those who wish to see more of the ' Elements' will probably most easily obtain those of Williamson (London, 1798, two vols. 4to), the trepidation of which is very literal. Those who prefer the Latin may find all the twelve books in the edition of Horsley (from Ccmmandine and Gregory), Oxford, 1802. As to the Greek, the edition of Gregory is scarce, ft8 is the edition of Peyrard, in Greek, Latio, and French, Paris, 1814 ; that of Camerer and Hauber, Berlin, 1824, contains the first six books in Greek and Latin, with valuable notes. The number of editors of Euclid is extremely great, but our limits will not allow of further recapitu lation.
Under the names of Archimedes, Apollonius, Pappus, Proclus, Theon, tte., the render will find further details upon the progress of Greek geometry, which continued to flourish at Alexandria till the taking of that town by the Saracens, A.D. 640. But its latter day produced only commentators upon the writers of the former, or, at moat, original writers of no great note. The following list contains the names of the moat celebrated geometers who lived before the decline of the Greek language : the dates represent nearly the middle of their lives, but are in many instances uncertain : Thaler, B.C. C00; Ameristus (1); Pythagoras, B.O. 550; ADflarlgOrtla ; (Enopides; Hippocrates, B.O. 450; Theodorue; Archytas (1) preceptor of Plato ; Leodamas; Themtetus; Arietreus, B.O. 350; Perseus (1); Plato, B.C. 310; Mensechmus, Dinostratue, Eudoxue, contemporaries of Plato ; Neoclides; Leon ; Amyclas; Theudius ; Cyzicinue; Hermo thous ; Philippus ; Euclid, n.c. 285 ; Archimedes, B.C. 240 ; Apollonius, B.C. 240; Eratostheues, B.C. 240; Nicomedea, B.C. 150; Hipparchus, B.C. 150 ; Hypsiclea, B.O. 130 (?); Geminus, B.C. 100 ; Theodosius, to. 100 ; Menelaus, A.D. 80; Pt0/01:09300, A.D. 125 ; Pappus, A.n. 390 ; Sere. Dna, A.D. 390; Diodes (1), Proclus, A.n. 440; Marinus (?), Isidorus (1), Eutocius, A.D. 540.
The age of Diophantus is not sufficiently well known even for so rough a summary as the preceding.
The following is the summary of books of geometrical analysis (qui ad resolutuni locum pertinent), given by Pappue as extant in his time : of Euclid, the Data,' three books of poriems, and two books Locorum ad Superficiem ;' of Apollonius, two books 'De Proper tionia Sectione,' two 'De Spatii Sections,' two 'De Tactionihus,' two De Inclinationibus,' two Planer= L9corum,' and eight on conic sections; of Aristmus, five books ' Locorurn Solidorum ; ' of Eras tosthenea two books on finding mean proportionate. But besides
these he describes a book (of Apollonius) which treats 'De Determinati. Sectione.' The manifold beauties of the ' Elements' of Euclid secured their universal reception, and it was not long before geometers began to extend their results. It became frequent to attempt the restitution of a lost book by the description given of it by Pappue or others ; and from Vieta to Robert Simeon, a long list of names might be collected of those who have endeavoured to repair the losses of time. On the advance of geometry in general the reader may consult the lives of Vieta, Metius, Pitiscus, Snell, Napier, Guldinus, Cavalieri, Robeval, Fermat, Pascal, Descartes, Kepler, &c.
The application of algebra to geometry, of which some instances had been given by Bombelli, and many more by Vista. grew into a science in the hands of Descartes (1596-1650). It drew the attention of mathematicians completely away from the methods of the ancient geometry, and considering the latter as a method of discovery, the change was very much for the better. But the close a •d grasping character of the ancient reasoning did not accompany that of the new method : algebra was rather a half-underetood art than a science, and all who valued strictness of denuocstration adhered as close as possible to the ancient geometry. This was particularly the case iu our own country, and unfortunately the usual attendants of rigour were mistaken for rigour itself, and vice vend. The algebraical symbols and methods were by many reputed inaccurate, while the same pro cesses, conducted on the same principles, in a geometrical form, were preferred and even advanced as more correct. Newton, an admirer of the Greek geometry, clothed his Priucipia in a dress which was meant to make it look (so far as mathematical methods were con cerned) like the child of Archimedes, and not of Vieta or Descartes; but the end was not attained in reality, for though the reasoning is really unexceptionable, yet the method of exhauetions roust be applied to most of the lemmas of the first section, before the Greek geometer would own them.