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EUCLID OF ALEXANDRIA. 



EUCLID OF ALEXANDRIA. 



126 



allowing the Greeks to have received the merest rudiments either from 

 Egypt or India, or any other country, it is impossible to name any 

 quarter from which we can with a shadow of probability imagine them 

 to have received a deductive system, to ever BO small an extent. That 

 their geometry, or any of it, came direct from India, is a supposition 

 of some difficulty : those who brought it could hardly have tailed to 

 bring with it the decimal notation of arithmetic. That Pythagoras 

 travelled into India, is (according to Stanley) only the assertion ol 

 Apuleiua and Clemens Alexandrinus, though rendered probable by 

 several of his tenets. 



Thales (B.C. 600) and Pythagoras (B.C. 540) founded the earliest 

 tchools of geometry. The latter is said to have sacrificed a hecatomb 

 when he discovered the property of the hypothenuse before alluded 

 to, and this silly story is repeated whenever the early history ol 

 geometry is given. A large collection of miscellanies mi^ht easily be 

 made from the works of writers who were not themselves acquainted 

 with geometry ; but, rejecting such authorities, we shall content our- 

 selves with citing Pappus and Proclus, both geometers, who, living in 

 the 4th and 5th centuries after Christ, had abundant opportunities of 

 hearing the stories to which we allude, and of receiving or rejecting 

 them. 



According to Proclus (book ii. ch. 4, ' C'omm. in Eucl.'), Pythagoras 

 was the first who gave geometry the form of a science, after whom 

 came Auaxagoras, (Enopides, Hippocrates of Chios (who invented the 

 well-known quadrature of the lunules), and Theodorua of Cyrene. 

 Plato was the next great advancer of the science, with whom were 

 contemporary Leodamas, Archytas, and Thesetetua of Thasus, 

 Tarentum, and Athens. After Laodamas came Neoclides, whose 

 disciple Lso made many discoveries, added to the accuracy of the 

 elements, and gave a method of deciding upon the possibility or 

 impossibility of a problem. After Leo came Eudoxus, the friend of 

 1'lato, who generalised various results which came from the school of 

 the latter. Aiayclas, another friend of Plato, and the brothers 

 iMtnacchmus and Diuostratus, made geometry more perfect. Theudius 

 wrote excellent elements, and generalised various theorems. Cyzicinus 

 of Athens cultivated other parts of mathematics, but particularly 

 geometry, HermotimuH enlarged the results of Eudoxus and These- 

 tetus, and wrote on ' loci.' Next is mentioned Philippus, and after 

 him Euclid, " who was not much younger than those mentioned, and 

 who put together elements, and arranged many things of Eudoxus, 

 and gave unanswerable demonstrations of many things which had been 

 loosely demonstrated before him." He lived under the first Ptolemy, 

 by whom he was asked for an easy method of learning geometry ; to 

 which he made the celebrated answer, that there was no royal road. 

 He was younger than the time of Plato, and older than Eratosthenes 

 and Archimedes. He was of the Platonic sect. 



Such is, very nearly entire, the account which Proclus gives of the 

 rise of geometry in Greece. 



Before the time of Euclid demonstration had been introduced, 

 about the time, perhaps by the instrumentality, of Pythagoras ; pure 

 geometry had been restricted to the right line and circle, but by 

 whom is not at all known : the geometrical analysis, and the study of 

 the conic sections, as al.-o the considerations of the problems of the 

 duplication of the cube, the finding of two mean proportionals, and 

 the trisection of the angle, had been cultivated by the school of Plato ; 

 the quadrature of a certain circular space had been attained, and the 

 general problem suggested and attempted by Hippocrates and others; 

 a curve of double curvature bad been imagined and used by 

 Archytas ; writings existed both on the elements, and on conic 

 uectious, loci, and detached subjects. It is in this part of the present 

 article that we have judged it best to introduce what would otherwise 

 have formed the article EUCLID. 



It is not known where EUCLID or ALEXANDRIA was born. He opened 

 a school of mathematics at Alexandria, in the reign of Ptolemaeua the 

 son of Lagus (323 284 B.C.), from which school came Eratosthenes, 

 Archimede*, Apollonius, Ptolemams, the Theons, &c., &c., so that from 

 and after Euclid the hiatDry of the school of Alexandria is that of 

 Greek geometry. He was, according to Pappus, of a mild and gentle 

 temper, particularly towards those who studied the mathematical 

 sciences : but Pappus is too late an authority for the personal demea- 

 nour of Euclid, and moreover may have been incited to praise him 

 for the purpose of depreciating Apollonius, of whom he is then 

 speaking, and against whom he several times expresses himself. 

 Besides the Element*, Euclid wrote, or is supposed to hare written, 

 the following works : 



1, Siryypo/ifia YcvSafxW, a treatise on 'Fallacies,' preparatory to 

 geometrical reasoning. This book, mentioned by Proclus, does not 

 now exiat, and there is no Greek work of which we so much regret 

 the loss. Had it survived, mathematical students would not have 

 been thrown directly upon the Elements, without any previous 

 exercise in reasoning. 



2. Four books of ' Conic Sections,' afterwards amplified and appro- 

 priated by Apollonius, who added four others. So says Pappus, as 

 already mentioned in APOLLONIDS PERO*US. That Euclid did not 

 write these book", appears to ua more tbau probable from the silence 

 of Proclus the Platonist, who, eulogising Euclid the Platonist, and 

 stating that he wrote on the regular solids (a part of geometry culti- 

 vated by the Platonists), being led thereto by Platonism, never 



woo. Div. VOL, ii. 



mentions his writing on the still more Platonic subject of the conic 

 sections. But that Aristseus had written on the subject is known, 

 and that Euclid taught it cannot be doubted, any more thau that 

 Apollonius, like other writers, prefixed to his own discoveries all 

 that he judged fit out of what was previously knowu on the subject. 



3. Ucpi AiaipcVew, on 'Divisions.' This work is mentioned by 

 Proclus in two words. John Dee imagined the book of Mohammed 

 of Bagdad (which is annexed to the English edition of Euclid herein- 

 after cited) on the division of surfaces to be that of Euclid now under 

 consideration ; but there seems to bo no ground for this notion. The 

 Latin of this work (from the Arabic) is given at the end of Gregory's 

 Euclid, together with a fragment ' De Levi et Ponderoso,' attributed, 

 without any foundation, to Euclid. 



4. Ilepi nop/oytaTajc, on ' Porisms,' in three books. This is men- 

 tioned both by Pappus and Proclus, the former of whom gives the 

 enunciations of various propositions in it, but the text is so corrupt 

 that they can hardly be understood. 



5. Tfaraiv irp&s trriittdixiai', ' Locorum ad Superficiem,' which we cannot 

 translate. It is mentioned by Pappus, but has not come down to us. 



The preceding works are either lost or doubtful; those which follow 

 all exist, and are contained in Gregory's edition, in the order inverse to 

 that in which they are here mentioned. 



6. "OirriKi KO! KaTOTrrpuid, on 'Optics and Catoptrics.' These books 

 are attributed to Euclid by Proclus, and by Marinus in the preface to 

 the ' Data,' or rather books on these subjects. Savile, Gregory, and 

 others doubt that the books which have come down to us are those of 

 Euclid, and Gregory gives his reasons in the preface, which are that 

 Pappus, though he demonstrates propositions in optics and also in 

 astronomy, and mentions the ' Phenomena ' of Euclid with reference 

 to the latter, does not mention the ' Optics ' with reference to the 

 former ; and that there are many errors in the works in question, such 

 as it is not likely Euclid would have made, Proceediug on the sup- 

 position that rays of light are carried from the eye to (Ac object, the 

 first of these books demonstrates some relations of apparent magni- 

 tude, and shows how to measure an unknown height by the well-known 

 law of reflected light In the second an imperfect theory of convex 

 and concave mirrors is given. 



7. turf/urn, on ' Astronomical Appearances,' mentioned by Pappus 

 and Philoponus (cited by Gregory). It contains a geometrical doctrine 

 of the sphere, and, though probably much corrupted by time, is 

 undoubtedly Euclid's. 



8. KoToTo/tJ) navtims and EiVcryaiy)) itpuwiK-fi, the ' Division of the 

 Scale ' and ' Introduction to Harmony.' Proclus mentions that Euclid 

 wrote on harmony, but the first of these treatises is a distinct geo- 

 metrical refutation of the principles laid down in the second, which 

 renders it unlikely that Euclid should have written both. The second 

 treatise is Aristoxenian [ARISTOXENUS], while the first proceeds on 

 principles of which Gregory states he never found a vestige in any 

 other writer who was reputed anterior to Ptolemacus (to whom he attri- 

 butes it). The second treatise is not geometrical, but is purely a 

 description of the system mentioned, and as this treatise is not alluded 

 to by Ptolemaeus nor by any previous writer on the subject, it is very 

 probable that Euclid did not write it. 



9. AeSo^tcVo, a ' Book of Data.' This is the most valuable specimen 

 which we have left of the rudiments of the geometrical analysis of the 

 Greeks, Before a result can be fouud, it should be known whether 

 the given hypotheses are sufficient to determine it. The application 

 of algebra settles both points ; that ia, ascertains whether one or 

 more definite results can be determined, and determines them. But 

 in geometry it is possible to propose a question which is really inde- 

 terminate, and in a determinate form, while at the same time the 

 methods of geometry which give one answer may not give the means 

 of ascertaining whether the answer thus obtained is the only one. 

 Thus the two following questions seem equally to require one specific 

 answer, to one not versed in geometry : 



Given, the area of a parallelogram, and the ratio of its sides; 

 required, the lengths of those sides : and 



Given, the area of a parallelogram, the ratio of its sides and one of 

 its angles ; required, the lengths of the sides. 



The first question admits of an intinite number of answers, and the 

 second of only one ; or, iu the language of Euclid, if the area, ratio of 

 sides, and an angle of a parallelogram be given, the sides themselves 

 are given. The same process by which it may be shown that they are 

 given serves to find them ; so that the Data of Euclid may be looked 

 upon as a collection of geometrical problems, in which the attention of 

 the reader is directed more to the question of the sufficiency of the 

 hypothesis to produce one result, and one only, than to tho method 

 of obtaining the result. 



A preface to this" book was written by one Marinus, the disciple and 

 successor of Proclus, explaining at tedious length the distinction of 

 'given' and 'not given.' 



10. 2Toix7a, the 'Elements' (of Geometry). For a long time 

 writers hardly considered it necessary to state whose ' Elements ' they 

 referred to, since a certain book of the elements always signified that 

 jook of Euclid ; and it was customary in England to call each book 

 an element; thus in Billingsley's old translation tho sixth book is 

 called the sixth element. 



The reason why tho ' Elements ' have maintained their ground ia 



3n 



