(29 



EUCLID OF ALEXANDRIA. 



EUCLID OF MEGARA. 



if ix TWI/ 0nroi avvovauev, from 1 which it was inferred that the 

 compilation of the elements was the work of Theon, from the mate- 

 rials left by Euclid. It is certain that Theon, in his commentary on 

 the Almagest, speaks of his edition (exSocris) of Euclid, and mentions 

 that the part of the last proposition which relates to the sectors was 

 added by himself. On looking at that proposition, it is found that the 

 demonstration relative to the sectors comes after the ' mep e5ei 5eT|ai,' 

 with which Euclid usually ends his propositions. And Alexander, the 

 commentator on AristotK, who lived before Theon, calls that the 

 'fourth' proposition of the tenth book which is the 'fifth' in all the 

 manuscripts. We can then distinctly trace the hand of Theon as a 

 commentator, and may suspect that he performed the duty of a revis- 

 ing editor to the work of Euclid as it now appears ; but there is not 

 the smallest reason to suppose that Tbeon actually digested the work 

 into the form which it now has. These remarks relative to the claims 

 of Theon were first made by Sir Henry Savile, who opened the chair of 

 geometry which he founded at Oxford by thirteen lectures on the 

 fundamental parts of the first book of Euclid, which were delivered 

 in 1620, and published in 1621. 



We now give a short summary of the early editions of Euclid, 

 which have appeared in Greek or Latin. It is unnecessary to specify 

 the common editions of Simson, Playfair, &c., which confine them- 

 selves to the first six books, and the eleventh and twelfth, and are 

 generally knowu. 



I. Editions of the whole of Euclid's works : 1. An imperfect Latin 

 edition, by Bartholomew Zamberti, Venice, 1505. 2. A Latin edition, 

 printed at Basel, marked ' Banileto apud Johannem Hervagium,' 1537, 

 1546, and 1558. 3. Greek edition, with Scholia, Basel, 1539. But the 

 principal edition of all the works of Euclid is that published by the 

 Oxford press in 1703, under the care of David Gregory, then Savilian 

 professor. 



IL Greek editions of the Elements only : 1. An edition cura Simonia 

 Gryntei, Basel, 1530. 2. Another, with the commentary of Proclus, 

 'Basileie apud Johannem Hervagium,' 1533. 3. Greek and Italian, 

 by Angeli Cajaui, Rome, 1545. 4. At Strasburg, 1559. 5. Greti and 

 Latin, wita Scholia, by Conrad Dasypodius, Strasburg, 1564. 



III. Latin editions of the Elements only : 1. That of Campanus, 

 the first Eucli,] printed, Ratdolt, Venice, 1482. 2. A reprint of the 

 preceding, marked ' Vincentiie, anno adutis 1491.' 3. An edition con- 

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

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

 1505 ; aud again in 1516. This edition is very commodious for a 

 general coinpari-on of the Greek and Arabic. 4. Edition of Lucas de 

 Burgo, Venice, 1509, according to Murhard, and 1489 according to 

 Hfilbronncr, who appears to be the authority for the existence of this 

 edition, and is doubted (with reason, we think) by Harles, in his Fabri- 

 cius. 5. Edition of Stephen Qracilis, Paris, 1657, 1573, nnd 1578. The 

 first edition of Clavius is that of Rome, 1574 ; of Commandine, Pesaro, 

 1572. [CLAVIUS; COMMANDINE.] 



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

 ' The Elements of Geometry of the most antient philosopher Euclid 

 of Megara, lie.,' 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 Philosophe, tc.,' per Nicolo Tartalea, Venice, 1543. 

 Dutch 'De se erste boecken Euclidis, &c.,' dor Jan Pieterszoon Dou, 

 Amsterdam, 1608 (or 1BOS). Swedith' De sex Forsta, 4c.,' by Marten 

 Stromer, Upsal, 1753. Spanitk By Joseph Saragoza, Valentia, 1673. 

 Murhard (compared with Fabricius) is the authority (on all of these, 

 except the first. 



It has long ceased to be usual to reaj more of Euclid than the 

 first MX books and the eleventh. Those who wish to see more of the 

 ' Elements ' will probably most easily obtain those of Williamson 

 (London, 1788, two vols. 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 Ccmmandine and Gregory), Oxford, 1802. 

 As to the Greek, the edition of Gregory is scarce, as is the edition of 

 Peyrard, in Greek, Latin, 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, &c., the reader 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, AD. 640. But its latter day 

 produced only commentators upon the writers of the former, or, at 

 most, original writers of no great note. 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 tlieir lives, but are iu many instances uncertain : 



Tliales, B.C. COO ; Ami-ristns (?) ; Pythagoras, B.C. 550; Anaxagorns; 

 <!.n"|,i<le; Hippocrates, B.C. 460; Theodorus; Archytas (?) preceptor 

 of Plato; Leodamas; Thetetetus; Aristteus, B.C. 350; Perseus (?); 



Plato, B.C. 310; Meusechmus, Dinostratus, Eudoxus, contemporaries 

 of Plato; Neoclides; Leon ; Amyclas ; Theudius; Cyzieinus; Herroo- 

 timus; Philippus; Euclid, B.C. 285; Archimedes, B.C. 240; Apollonius, 

 B.C. 240; Eratosthenes, B.C. 240; Nicomedes, B.C. 150; Hipparchus, 

 B.C. 150; Hypsicles, B.C. 130 (?); Geminus, B.C. 100; Theodosius, B.C. 

 100; Meuelaus, A.D. 80; Ptoleminus, A.D. 125; Pappus, A.D. 390 ; Sere- 

 nus, A.D. 390; Diocles (?), Proclus, A.D. 440; Marinus (?), Isidorus ('!), 

 Eutocius, A.D. 540. 



The age of Diopbantus 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 resolutum locum pertinent), given by Pappus as extant in his 

 time : of Euclid, the ' Data,' three books of porisms, and two books 

 ' Locorum ad Superficiem ; ' of Apollonius, two books 'De Propor- 

 tionis Sectione," two ' De Sputii Sectione,' two ' De Tactionibus,' two 

 ' De Incliuationibus,' two ' Planorum Locorum,' and eight on conic 

 sections ; of Aristaeus, five books ' Locorum Solidorum ; ' of Eras- 

 tosthenes two books on finding mean proportionals. But besides 

 these he describes a book (of Apollonius} which treats 'De Determiuata 

 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 Pappus or others ; 

 and from Vieta to Robert Simson, a long list of names might b 

 collected of those who have endeavoured to repair the lasses of time. 

 On the advance of geometry in general the reader may consult the 

 lives of Vieta, Metius, Mgini, Pitiscus, Snell, Napier, Guldinus, 

 Cavalieri, Robevnl, Fermat, Pascal, Descartes, Kepler, &c. 



The application of algebra to geometry, of which some instances 

 had been given by Boinbelli, 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 a d 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 

 nil who valued strictness of demonstration 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 versd. The algebraical symbols 

 nnd methods were by many reputed inaccurate, while the same pro- 

 OOie, conducted on the same principles, in a geometrical form, were 

 preferred aud 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 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, aud 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 supe- 

 rior to tho.se of algebra, in many matters connected with practice. 

 This was Monge, the inventor of descriptive geometry. The science 

 of perspective and many other applications of geometry to tbe arts 

 had previously reouin-d isolated methods of obtaining lines, angles, 

 or areas, described under laws not readily admitting of the applica- 

 tion of blgebra, and its consequence, the construction of tables. The 

 descriptive geometry is a systematised form of the method by which 

 a ground-plan and an elevation 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 tlie section is a curve it is constructed by laying 

 down a large number of consecutive 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 ono of the most elegant 

 and lucid elementary works in existence. 



The methods of descriptive geometry recalled the attention of 

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

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

 iesign or iu the investigation of primary properties of the conic 

 sections. From the time of Monge to the pre.seut this subject has 

 aeen cultivated with a vigour which has produced most remarkable 

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

 since tlie time of the Greeks which gives greater help to its means of 

 nvention than that which the labours of what we must call the 

 school of Monge have effected. 



EUCLID (EwcAei'S7)s) of Megara is eaid to be a different person from 

 ihe geometrician of the same name. He was a scholar of Socrates, 



