377 



GEOMETRY. 



GEOPONIKA. 



373 



containing the text and comment of Campanus, from the Arabic ; also 

 the text and comment of Zamberti, from the Greek ; Paris, Henry 

 Stephens, 1505 ; and again in 1516. This edition is very commodious 

 for a general comparison of the Greek and Arabic. (4.) Edition 

 of Lucas Pacioli, Venice, 1509. (5.) Edition of Henry Stephens, 

 Paris, 1516. These five editions are in folio; the second and fourth 

 are very scarce. The first edition of Clavius is that of Rome, 

 1574; of Commandine, Pesaro, 1572. [CLAVIUS; COMMANDINE, in 

 Bioo. Drv.] 



IV. Earliest editions of the Elements in modern tongues. English : 

 ' The Elements of Geometry of the most ancient philosopher Euclid of 

 Megara, &c.,' by H. Billingsley, with a preface by John Dee, London, 

 1570, and again in 1661. French : ' Les quinze livres des Elements, 

 &c., &c.,' Par D. Henrion, Mathematicum, first edition, Paris, 1565 ? 

 second, 1623, with various others. According to Fabricius, there was 

 an edition by Peter Forcadel, in 1565. German : ' Die sechs ersten 

 bucher, &c.,' by William Holtzmann, Augsburg, 1562. Scheubelius 

 had previously given the 7th, 8th, and 9th books, in 1555. Italian : 

 ' Euclide Megarense Philosopho, &c.,' per Nicolo Tartalea, Venice, 1543. 

 Dutch: ' De ses erste boecken Euclidis, &c.,' dor Jan Pieterszoon 

 Don, Amsterdam, 1608 (or 1606). Stcedith :' De sex forsta, &c.,' by 

 Marten Stromer, Upsal, 1753. Spanish : By Joseph Saragoza, Valentia, 

 1673. Murhard (compared with Fabricius) is the authority for all 

 of these, except the first. 



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 (Lon- 

 don, 1788, two volumes 4to.), the translation of which is very literal. 

 Those who prefer the Latin may find all the twelve books in the edition 

 of Horsley (from Commandine and Gregory), Oxford, 1802. The 

 number of editors of Euclid is extremely great, but our limits will not I 

 allow of further recapitulation. 



The progress of geometry is connected with the names of Archi- 

 medes, Apollonius, Theon, &c., and it continued to flourish at Alex- 

 andria tiil 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 most, original writers of no great note. In the articles 

 Locus, POBISM, DUPLICATION, PROPORTIONAL, will be found some of 

 the details of the Greek geometrical analysis. Spherical trigonometry, 

 or rather that portion of their geometry which supplied its place in 

 astronomy, is connected with the names of Hipparchus, Meuelaus, 

 Theodosius, Ptolemy. The following list contains the names of the 

 most 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 : 



Thales, B.C. 600 ; Ameristus ? Pythagoras, 550 ; Anaxagoras ; CEuo- 

 pides ; Hippocrates, 450 ; Theodoras ; Archytas ? preceptor of Plato ; 

 Leodamas; Theajtetus; Aristaeus, 350; Perseus? Plato, 310; Me- 

 nxchmua, Dinostratus, Eudoxus, contemporaries of Plato : Neoclides ; 

 Leon ; Amyclas ; Theudius ; Cyzicinus ; Hermotimus ; Philippus : 

 Euclid, 285 ; Archimedes, 240 ; Apollonius, 240 ; Eratosthenes, 240 ; 

 Nicomedes, 150; Hipparchus, 150; Hypsicles, 130? Geminus, 100; 

 Theodosius, 100; Menelaus, A.D. 80; Ptolemy, 125; Pappus, 390; 

 Serenus, 390 ; Diocles ? Proclus, 440 ; Marinus ? Isidorus ? Eutocius, 

 540. 



The age of Diophantus is not sufficiently well known even for so 

 rough a summary as the preceding ; he is usually given to the 2nd 

 century, we place him at the end of the 6th century. 



The following is the summary of books of geometrical analysis (qui 

 ad resplutum locum pertinent), given by Pappus as extant in his time ; 

 of Euclid, the Data, three books of porisms, and two books loconim 

 ad luperjiciem ; of Apollonius, two books de propartionit tectione, two 

 de spatii sectione, two de tactionibui, two de inclinatimibiw, two piano- 

 rum l'ii:nrnm, and eight on conic sections; of Aristanis, five books 

 I'K-ni -II.M gtjlvlorum ; of Erastosthenes, two books on finding mean pro- 

 portionals. But besides these he describes a book (of Apollonius) 

 which treats de determinatd sectwur. 



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 Pappus or others ; and 

 from Vieta to Robert Simson, 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, Magini, Pitiscus, Snell, Napier, Guldinus, Cavalieri, 

 Roberval, Fermat, Pascal, Descartes, Kepler, &c., &c., in the Bioo. Div. 

 and also the article QUAIJIIATCUE OK THE CIRCLE. 



(The application of algebra to geometry, of which some instances had 

 been given by Bombelli, and many more by Vieta, 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 and grasping 

 character of the ancient reasoning did not accompany that of the new 

 method : algebra was rather a half-understood art than a science, and 

 all who valued strictness of demonstration adhered as close as possible 

 to t'je ancient geometry. Thin was particularly the case in our own 

 country, aiid unfortunately thu usual attendants of rigour were mis- 



taken for rigour itself, and i-ice versd. The algebraical symbols and 

 methods were by many reputed inaccurate, while the same processes, 

 conducted on the same principles, in a geometrical form, were pre- 

 ferred and even advanced as more correct. Newton, an admirer of the 

 Greek geometry, clothed his Principia in a dress which was meant to 

 make it look (so far as mathematical methods were concerned) 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 unexception- 

 able, yet the method of exhaustions must be applied to most of the 

 lemmas of the first section, before the Greek geometer would own 

 them. 



The methods of algebra, so far as expressions of the first and second 

 degrees are concerned, apply with great facility to many large classes 

 of questions connected with straight lines, circles, and other sections 

 of the cone. Practical facility was gained by them, frequently at the 

 expense of reasoning : the time came when a new Descartes showed 

 how to return to geometrical construction with means superior to those 

 of algebra, in many matters connected with practice. This was Monge, 

 the inventor of descriptive geumetry. The science of perspective and 

 many other applications of geometry to the arts had previously required 

 isolated methods of obtaining lines, angles, or areas, described under 

 laws not readily admitting of the application of algebra, and its con- 

 sequence, the construction of tables. The descriptive geometry is a 

 systematized form of the method by which a ground-plan and an eleva- 

 tion are made to give the form and dimensions of a building. The 

 projections of a point upon two planes at right angles to one another 

 being given, the position of the point itself is given. From this it is 

 possible, knowing the projections of any solid figure upon two such 

 planes, to lay down on either of those planes a figure similar and equal 

 to any plane section of the solid. In the case . where the section 

 is a curve it is constructed by laying down a large number of con- 

 secutive contiguous points. The methods by which such an object 

 is to be attained were generalised and simplified by Monge, whose 

 ' Ge'ome'trie Descriptive ' (the second edition of which was published 

 in 1820) is one of the most elegant and lucid elementary works in 

 existence. 



The methods of descriptive geometry recalled the attention of 

 geometers to the properties of projections in general, of which such 

 only had been particularly noticed as could be applied in the arts of 

 design or in the investigation of primary properties of the conic 

 sections. From the time of Monge to the present this subject has 

 been cultivated with a vigour which has produced most remarkable 

 results, and promises more. Pure geometry has made no advance since 

 the time of the Greeks which gives greater help to its means of inven- 

 tion than that which the labours of what we must call the school of 

 Monge have effected. One of the most distinguished pupils of this 

 great master, M. Chasles, published an ' Apercu historique des rne- 

 thodes en Ge'ome'trie,' forming the eleventh volume of the ' Me'inoires 

 Coin-onus's ' of the Academy of Brussels, a work of great importance 

 in the historical point of view. 



On the history of geometry, as distinguished from other parts of 

 mathematics, there is very little to cite. The references in the article 

 MATHTMATICS may be consulted. 

 GEOMETRY OF THE GREEKS. [GEOMETRY.] 

 GEOPO'NIKA (or, a ' Treatise on Agriculture '), is the title of a 

 compilation, in Greek, of precepts on rural economy, extracted from 

 ancient writers. The compiler, in his proemium, shows that he was 

 living at Constantinople, and dedicated his work to the Emperor 

 Constantine, "a successor of Constantine, the first Christian emperor," 

 stating that he wrote it in compliance with his desire, and praising him 

 for his zeal for science and philosophy, and for his philanthropy. This 

 emperor is supposed by some to have been Constantine Porphyrogenitus, 

 and the compilation is generally ascribed to Cassianus Bassus, a native of 

 Bithynia, who however is stated by others to have lived some centuries 

 before the time of Porphyrogenitus. The question of the author- 

 ship of the ' Geoponika ' has excited much discussion. Needham, in 

 his Greek and Latin edition of the ' Geoponika,' Cambridge, 1704, has 

 treated the subject at great length. The work is divided into twenty 

 books, which are subdivided into short chapters, explaining the various 

 processes of cultivation adapted to various soils and crops, and the 

 rural labours suited to the different seasons of the year ; with directions 

 for the sowing of the various kinds of corn and pulse ; for the training 

 of the vine, and the art of wine-making, upon which the author is very 

 diffuse. He also treats of olive plantations and oil-making, of orchards 

 and fruit-trees, of evergreens, of kitchen-gardens, of the insects and 

 reptiles that are injurious to plants, of the economy of the poultry- 

 yard, of the horse, the ass, and the camel ; of horned cattle, sheep, 

 goats, pigs, &c., and the care they require ; of the method of salting 

 meat ; and, lastly, of the various kinds of fishes. Every chapter is 

 inscribed with the name of the author from whom it is taken, and the 

 compiler gives at the beginning of the first *book a list of his principal 

 authorities, who are Atricanus, Anatolius, Apuleius, Berytius, Damo- 

 geron, Dernoeritus, Didyuuis, Dionysius Uticensis (the translator of 

 Mago, the Carthaginian writer on agriculture), Diophaues, Florentiuus, 

 Leontius, Pamphilus, Paxamus, the Quintilii, Sotiou, Varro, Vinda- 

 nonius, and Zoroaster. Other authors besides these are quoted in the 

 course of the work. Two or three chapters are inscribed with the 

 name of C:n-iamw, -,vho speaks of himself in them as a native of 



