Of Euclid's Elements, the first four books treat of the properties of plane figures ; the fifth contains the theory of proportion ; and the sixth its application to plane figures ; the seventh, eighth, ninth, and tenth, relate to arithmetic, and the doctrine of incommensurables ; the eleventh and twelfth contain the elements of the geometry of solids; and the thirteenth treats of the five regular solids, or Platonic bodies, so called because they were studied in that cele brated school : two books more, viz. the fourteenth and fifteenth, on regular solids, have been attributed to Euclid, but these rather appear to have been written by Hypsicles of Alexandria.
It is only the first six, and the eleventh and twelfth books, that are now commonly taught in the schools; for the books on arithmetic have been superseded by the modern theories of algebra, and the regular solids have long ceased to be particularly interesting : they may be comnared to mines u lab have been abandoned, because the produce was not equal to the expence of working them. Euclid's Ele ments have had a number of commentators ; the earliest was Theo') of Alexandria, who lived about the middle of the fourth century. Proclus also has gi \ en a commentary on the first book, which i s only valuable on account of the information it contains respecting the history and metaphy sics of geometry. After the revival of learning, the Ele ments of Euclid were first known in Europe, through the medium of an Arabic translation ; from this it was deci phered and translated into Latin, by Athelard in England, and Campanus in Italy, about the same time, in the 12th or 13th centuries. Athelard's translation exists only in manuscript, in some libraries; that of Campanus served as the basis of the greater part of the Latin translations, made about the end of the 15th and the beginning of the 16th cen turies. The editio prince/is is that which Ratdolt of Augs burg, a celebrated printer, gave in 1482, at Venice, in fo lio; the Greek text did not appear until 1533, when it was printed at Basle, by J. liervage, under the care of J. Gry wens. The earliest English edition is that of Billingsley, in 1570 : But the history of the various editions of this work, either in whole or in part, that have been published in all countries, in which science has been cultivated, is far too extensive to find a place here. The curious reader may find a copious list in the second volume of the Biblio theca Mathematica, by Murhard. At present, the edition of Euclid most esteemed in this country, is that of the late Dr Simon of Glasgow, which contains the first six and the eleventh and twelfth books, and the book of Euclid's Data. We have lately seen the first volume of an edition in the original Greek, accompanied with a Latin and French translation by Peyrard, a French professor of mathematics, and author of a French translation of Archimedes; it gives the original text as exhibited in a great number of manuscripts, and on this account it must be extremely va luable.
Besides the Elements, the only other entire geometrical work of Euclid that has come down to the present times, is his Data. This is the first in order of the books writ ten by the ancient geometers to facilitate the method of resolution or analysis. In general, a thing is said to be given, which is actually exhibited, or can be found ; and the propositions in the book of Euclid's Data, show what things can be found from those which by hypothesis are al ready known.
\Ve learn from Pappus of Alexandria, that there exist ed four books by Euclid on Conic Sections, and two con cerning Loci ad Superficiem; these were curves of double curvature. But his most profound work, and that of which the loss is most regretted, was his three books on Porisms, which Pappus says were a most artful collection of many things that relate to the analysis of the more difficult and general problems. We shall explain this subject under the word PORISM. Proclus cites another work of Euclid's, which he entitles, De Divisionibus. This probably treated of the division of figures. These are all the known geo metrical writings of Euclid :—his other works do not be long to this place. See EUCLID.
In the order of time, Archimedes is the next of the an cient geometers that has drawn the attention of the mo derns. He was born at Syracuse, about the year 287 A. C. He cultivated all the parts of mathematics, and in particu lar geometry. The most difficult part of the science is that which relates to the areas of curve lines, curve sum-faces. Archimedes applied his fine genius to this sub ject, and Ile laid the foundation of all the subsequent dis coveries relating to it. His writings on geometry are nu merous. We have, in the first place, two books on the sphere and cylinder ; these contain the beautiful discovery, that the sphere is two-thirds of the circumscribing cylin der, whether we compare their sum-Faces, or their solidi fies, observing that the two ends of the cylinder are consi dered as forming a part of its surface. He likewise shews, that the curve surface of any segment of the be tween two planes perpendicular to its axis, is equal to the curve surface of the corresponding segment of the sphere. Archimedes was so much pleased with these discoveries, that he requested after his death that his tomb might be inscribed with a sphere and cylinder.