and Callistratus. He sometimes ridicules classes of persons, as the Thebans in his 'Avτión. His language is simple, elegant, and generally pure, containing few words which are not found in writers of the best period. Like Antiphanes, he was extensively pillaged by later poets, as, for example, by Alexis, Ophelion, and Ephippus. Suidas gives the number of the plays of Eubulus at 104, of which there are extant more than 50 titles, namely, Αγκυλίων, Αγχίσης, Αμάλθεια, Ανασωζόμενοι, Αντιόπη, Αστυτοι, Αὔγη, Βελλεροφόντης, Γανυμήδης, Γλαύκος, Δαίδαλος, Δαμαλίας is a corrupt title (Suid. s. v. 'Aσкwλide), for which Meineke would read Aaμarías, Aeukaλíwv, AιovýLos, in which he appears to have ridiculed the confusion which prevailed in all the arrangements of the palace of Dionysius (Schol. ad Aristoph. Thesm. 136), Aióvuσos, or, according to the fuller title (Athen. xi. p. 460, e.), Zeμéλn Aióvvoos, Δόλων, Εἰρήνη, Ευρώπη, Ηχώ, Ιξίων, Ιων, Καλα- | θηφόροι, Καμπυλίων (doubtful), Κατακολλώμενος | (doubtful), Κερκώπες, Κλεψύδρα, Κορυδαλός, Κυβευταί, Λάκωνες ή Λήδα, Μήδεια, Μυλωθρίς, Μυσοί, Νάννιον, Ναυσικάα, Νεοττίς, Ξούθος, Ὀδυσσεύς, ἢ Πανόπται, Οἰδίπους, Οινόμαος ἢ Πέλοψ, Ολβία, | Ορθάνης, Πάμφιλος, Παννυχίς, Παρμενίσκος, Πλαγγών, Πορνοβοσκός, Προκρίς, Προσουσία ἢ Κύκνος, Στεφανοπώλιδες, Σφιγγοκαρίων, Τιτθαί, Τιτάνες, Φοίνιξ, Χάριτες, Χρυσίλλα, Ψάλτρια. (Meineke, Frag. Com. Graec. vol. i. pp. 355-367, vol. iii. pp. 203-272; Clinton, Fust. Hell. sub ann. B. C. 375; Fabric. Bibl. Graec. vol. iv. pp. 442— 444.) [P.S.] EUCADMUS (Evкadμos), an Athenian sculptor, the teacher of ANDROSTHENES. (Paus. x. 19. $ 3.) [P. S.] EUCA'MPIDAS (Edkauídas), less properly EUCA'LPIDAS (Eukaπídas), an Arcadian of Maenalus, is mentioned by Demosthenes as one of those who, for the sake of private gain, became the instruments of Philip of Macedon in sapping the independence of their country. Polybius censures Demosthenes for his injustice in bringing so sweeping a charge against a number of distinguished men, and defends the Arcadians and Messenians in particular for their connexion with Philip. At the worst, he says, they are chargeable only with an error of judgment, in not seeing what was best for their country; and he thinks that, even in this point, they were justified by the result, as if the result might not have been different, had they taken a different course. (Dem. de Cor. pp. 245, 324; Polyb. xvii. 14.) [CINEAS.] Eucampidas is mentioned by Pausanias (viii. 27) as one of those who led the Maenalian settlers to Megalopolis, to form part of the population of the new city, B. c. 371. [E. E.] EUCHEIR (Exep), is one of those names of Grecian artists, which are first used in the mythological period, on account of their significancy, but which were afterwards given to real persons. [CHEIRISOPHUS.] 1. Eucheir, a relation of Daedalus, and the inventor of painting in Greece, according to Aristotle, is no doubt only a mythical personage. (Plin. vii. 56.) | 43, comp. xxxv. 5; Thiersch, Epochen, pp. 163 Tu 3. Eucheirus (Euxeipos, for so Pausanias giva the name) of Corinth, a statuary, was the pupil Syadras and Chartas, of Sparta, and the teacher Clearchus of Rhegium. (Paus. vi. 4. § 2.) H must therefore have fourished about the 65th 66th Olympiad, B. c. 520 or 516. [CHARTAS PYTHAGORAS OF RHEGIUM.] This is probabi the Euchir whom Pliny mentions among those who made statues of athletes, &c. (H. N. xxxiv. & s. 19, § 34.) 4. Eucher, the son of Eubulides, of Athens, a sculptor, made the marble statue of Hermes, in his temple at Pheneus in Arcadia. (Paus. viii. 14. § 7.) Something more is known of him through inscriptions discovered at Athens, in reference which see EUBULIDES. [P.S.] EUCHEIRUS, statuary. [EUCHEIR, No. 3.] EUCHE'NOR (Evxvwp), a son of Coeranus and grandson of Polyidus of Megara. He took part in the Trojan war, and was killed. (Paus.i. 43. § 5.) In Homer (Il. xii. 663) he is called son of the seer Polyidus of Corinth. There are two other mythical personages of this name. (Apollod ii. 1. § 5 ; Eustath. ad Hom. p. 1839.) [L. S.] EUCHE'RIA, the authoress of sixteen elegiac couplets, in which she gives vent to the indignation excited by the proposals of an unworthy suitorstringing together a long series of the most absurd and unnatural combinations, all of which are to be considered as fitting and appropriate in comparison with such an union. The idea of the piece was evidently suggested by the Virgilian lines Mopso Nisa datur; quid non speremus amantes? Jungentur jam grypes equis; aevoque sequenti Cum canibus timidi venient ad pocula damae, while in tone and spirit it bears some resemblance to the Ibis ascribed to Ovid, and to the Dirae of Valerius Cato. The presumptuous wooer is called a rusticus servus, by which we must clearly understand, not a slave in the Roman acceptation of the term, but one of those villani or serfe who, according to the ancient practice in Germany and Gaul, were considered as part of the live stock indissolu bly bound to the soil which they cultivated. From this circumstance, from the introduction here and there of a barbarous word, from the fact that most of the original MSS. of these verses were found in France, and that the name of Eucherius was common in that country in the fifth and sixth centuries, we may form a guess as to the period when this poetess flourished, and as to the land of her nativity; but we possess no evidence which can entitle us to speak with any degree of confidence. (Wernsdorf, Poet. Lat. Min. vol. iii. p. lxv. and p. 97, vol. iv. pt. ii. p. 827, vol. v. pt. iii. p. 1458; Burmann, Anthol. Lat. v. 133, or n. 385, ed. Meyer.) [W. R.] EUCHE'RIUS, bishop of Lyons, was born, during the latter half of the fourth century, of an illustrious family. His father Valerianus is by 2. Eucheir, of Corinth, who, with Eugrammus, many believed to be the Valerianus who about this followed Demaratus into Italy (B. c. 664), and period held the office of Praefectus Galliae, and introduced the plastic art into Italy, should proba- was a near relation of the emperor Avitus. bly be considered also a mythical personage, desig-cherius married Gallia, a lady not inferior to himnating the period of Etruscan art to which the self in station, by whom he had two sons, Salonius earliest painted vases belong. (Plin. xxxv. 12. s. and Veranius, and two daughters, Corsortia and Eu fr S de to D t Tullia. About the year a. D. 410, while still in the separate tracts are carefully enumerated by he vigcur of his age, he determined to retire from Schönemann, and the greater number of them will he world, and accordingly betook himself, with be found in the "Chronologia S. insulae Lerinenis wife and family, first to Lerins (Lerinum), and sis," by Vincentius Barralis, Lugdun. 4to. 1613; from thence to the neighbouring island of Lero or in "D. Eucherii Lug. Episc. doctiss. Lucubrationes St. Margaret, where he lived the life of a hermit, cura Joannis Alexandri Brassicani," Basil. fol. evoting himself to the education of his children, 1531; in the Bibliotheca Patrum, Colon. fol. 1618, to literature, and to the exercises of religion. vol. v. p. 1; and in the Bibl. Pat. Max. Lugdun. During his retirement in this secluded spot, he ac- fol. 1677, vol. vi. p. 822. (Gennad. de Viris. Ill. quired so high a reputation for learning and sanc- c. 63; Schoenemann, Bibl. Patrum. Lat. ii. § 36.) tity, that he was chosen bishop of Lyons about This Eucherius must not be confounded with A. D. 434, a dignity enjoyed by him until his another Gaulish prelate of the same name who death, which is believed to have happened in 450, flourished during the early part of the sixth cenunder the emperors Valentinianus III. and Marci-tury, and was a member of ecclesiastical councils anus. Veranius was appointed his successor in the episcopal chair, while Salonius became the head Sof the church at Geneva. 1726. There is yet another Eucherius who was bishop of Orleans in the eighth century. [W. R.] held in Gaul during the years A. D. 524, 527, 529. The latter, although a bishop, was certainly not bishop of Lyons. See Jos. Antelmius, Assertio pro The following works bear the name of this pre-unico S. Eucherio Lugdunensi episcopo, Paris, 4to. late: I. De laude Eremi, written about the year A. D. 428, in the form of an epistle to Hilarius of P. Arles. It would appear that Eucherius, in his passion for a solitary life, had at one time formed the project of visiting Egypt, that he might profit by the bright example of the anchorets who thronged the deserts near the Nile. He requested information from Cassianus [CASSIANUS], who replied by addressing to him some of those collationes in which are painted in such lively colours the habits and rules pursued by the monks and eremites of the Thebaid. The enthusiasm excited by these details called forth the letter bearing the above title. 29 2. Epistola paraenetica ad Valerianum cognatum de contemtu mundi et secularis philosophiae, composed about A. D. 432, in which the author endeavours to detach his wealthy and magnificent kinsman from the pomps and vanities of the world. An edition with scholia was published by Erasmus at Basle in 1520. 3. Liber formularum spiritalis intelligentiae ad Veranium filium, or, as the title sometimes appears, De forma spiritalis intellectus, divided into eleven chapters, containing an exposition of many phrases and texts in Scripture upon allegorical, typical, and mystical principles. 4. Instructionum Libri II. ad Salonium filium. The first book treats " 'De Quaestionibus difficilioribus Veteris et Novi Testamenti," the second contains "Explicationes nominum Hebraicorum." R 5. Homiliae. Those, namely, published by Li1 vineius at the end of the "Sermones Catechetici Theodori Studitae," Antverp., 8vo. 1602. 201 ། of The authenticity of the following is very doubtful. 6. Historia Passionis S. Mauritii et Sociorum Martyrum Legionis Felicis Thebaeae Agaunensium. 7. Exhortatio ad Monachos, the first of three printed by Holstenius in his "Codex Regularum," Rom. 1661, p. 89. 8. Epitome Operum Cassiani. The following are certainly spurious: 1. Commentarius in Genesim. 2. Commentariorum in 3 libros Regum Libri IV. 3. Epistola ad Faustinum. 4. Epistola ad Philonem. 5. Regula duplex ad Monachos. 6. Homiliarum Collectio, ascribed in some of the larger collections of the Fathers to Eusebius of Emesa, in others to Gallicanus. Eucherius is, however, known to have composed many homilies; but, with the exception of those mentioned above (5), they are believed to have perished. No complete collection of the works of Eucherius has ever been published. The various editions of 1 ܠܵܐ EUCLEIA (Euкλeía), a divinity who was worshipped at Athens, and to whom a sanctuary was dedicated there out of the spoils which the Athenians had taken in the battle of Marathon. (Paus. i. 14. § 4.) The goddess was only a personification of the glory which the Athenians had reaped in the day of that memorable battle. (Comp. Böckh Corp. Inscript. n. 258.) Eucleia was also used at Athens as a surname of Artemis, and her sanctuary was of an earlier date, for Euchidas died in it. (Plut. Arist. 20; EUCHIDAS.) Plutarch remarks, that many took Eucleia for Artemis, and thus made her the same as Artemis Eucleia, but that others described her as a daughter of Heracles and Myrto, a daughter of Menoetius; and he adds that this Eucleia died as a maiden, and was worshipped in Boeotia and Locris, where she had an altar and a statue in every market-place, on which persons on the point of marrying used to offer sacrifices to her. Whether and what connexion there existed be tween the Attic and Boeotian Eucleia is unknown, though it is probable that the Attic divinity was, as is remarked above, a mere personification, and consequently quite independent of Eucleia, the daughter of Heracles. Artemis Eucleia had also a temple at Thebes. (Paus. ix. 17. § 1.) [L. S.] EUCLEIDES (Evkλeídns) of ALEXANDRRIA. The length of this article will not be blamed by any one who considers that, the sacred writers excepted, no Greek has been so much read or so variously translated as Euclid. To this it may be added, that there is hardly any book in our language in which the young scholar or the young mathematician can find all the information about this name which its celebrity would make him desire to have. Euclid has almost given his own name to the science of geometry, in every country in which his writings are studied; and yet all we know of his private history amounts to very little. He lived, according to Proclus (Comm. in Eucl. ii. 4), in the time of the first Ptolemy, B. c. 323-283. The forty years of Ptolemy's reign are probably those of Euclid's age, not of his youth; for had he been trained in the school of Alexandria formed by Ptolemy, who invited thither men of note, Proclus would probably have given us the name of his. teacher: but tradition rather makes Euclid the founder of the Alexandrian mathematical school than its pupil. This point is very material to the formation of a just opinion of Euclid's writings; he was, we see, a younger contemporary of Aristotle (B. c. 384-322) if we suppose him to have been of mature age when Ptolemy began to patronise literature and on this supposition it is not likely that Aristotle's writings, and his logic in particular, should have been read by Euclid in his youth, if at all. To us it seems almost certain, from the structure of Euclid's writings, that he had not read Aristotle on this supposition, we pass over, as perfectly natural, things which, on the contrary one, would have seemed to shew great want of judgment. Euclid, says Proclus, was younger than Plato, and older than Eratosthenes and Archimedes, the latter of whom mentions him. He was of the Platonic sect, and well read in its doctrines. He collected the Elements, put into order much of what Eudoxus had done, completed many things of Theaetetus, and was the first who reduced to unobjectionable demonstration the imperfect attempts of his predecessors. It was his answer to Ptolemy, who asked if geometry could not be made easier, that there was no royal road (μὴ εἶναι βασιλικὴν ἄτραπον πρὸς γεωμετρίαν).* This piece of wit has had many imitators; "Quel diable" said a French nobleman to Rohault, his teacher of geometry, "pourrait entendre cela ?" to which the answer was "Ce serait un diable qui aurait de la patience." A story similar to that of Euclid is related by Seneca (Ep. 91, cited by August) of Alexander. Pappus (lib. vii. in praef.) states that Euclid was distinguished by the fairness and kindness of his disposition, particularly towards those who could do anything to advance the mathematical sciences: but as he is here evidently making a contrast to Apollonius, of whom he more than insinuates a directly contrary character, and as he lived more than four centuries after both, it is difficult to give credence to his means of knowing so much about either. At the same time we are to remember that he had access to many records which are now lost. On the same principle, perhaps, the account of Nasir-eddin and other Easterns is not to be entirely rejected, who state that Euclid was sprung of Greek parents, settled at Tyre; that he lived, at one time, at Damascus ; that his father's name was Naucrates, and grandfather's Zenarchus. (August, who cites Gartz, De Interpr. Eucl. Arab.) It is against this account that Eutocius of Ascalon never hints at it. At one time Euclid was universally confounded with Euclid of Megara, who lived near a century before him, and heard Socrates. Valerius Maximus has a story (viii. 12) that those who came to Plato about the construction of the celebrated Delian altar were referred by him to Euclid the geometer. This story, which must needs be false, since Euclid of Megara, the contemporary of Plato, was not a geometer, is probably the origin of the confusion. This celebrated anecdote breaks off in the middle of the sentence in the Basle edition of Proclus. Barocius, who had better manuscripts, supplies the Latin of it; and Sir Henry Savile, who had manuscripts of all kinds in his own library, quotes it as above, with only eπl for πpòs. August, in his edition of Euclid, has given this chapter of Proclus in Greek, but without saying from whence he has taken it. Harless thinks that Eudoxus should be read fr Euclid in the passage of Valerius. In the frontispiece to Whiston's translation of Tacquet's Euclid there is a bust, which is said to be taken from a brass coin in the possession d Christina of Sweden; but no such coin appears i the published collection of those in the cabinet d the queen of Sweden. Sidonius Apollinaris sap (Epist. xi. 9) that it was the custom to paint Euch with the fingers extended (laxatis), as if in the act of measurement. The history of geometry before the time of Euclid is given by Proclus, in a manner which shews that he is merely making a summary of wel known or at least generally received facts. He begins with the absurd stories so often repeated, that the Aegyptians were obliged to invent geo metry in order to recover the landmarks which the Nile destroyed year by year, and that the Phoenicians were equally obliged to invent arithmetic for the wants of their commerce. Thales, he goes on to say, brought this knowledge into Greece, and added many things, attempting some in a general manner (кabоλiкúтeрov) and some in a perceptive or sensible manner (αισθητικώτερον). Proclus clearly refers to physical discovery in geometry, by measurement of instances. Next is mentioned Ameristus, the brother of Stesichorus the poet. Then Pythagoras changed it into the form of a liberal science (Taideías éλevlépov), took higher views of the subject, and investigated his theorems immaterially and intellectually (dws kal voepws): he also wrote on incommensurable quantities (aλóywv), and on the mundane figures (the five regular solids). Then Barocius, whose Latin edition of Proclus has been generally followed, singularly enough translates aλoya by quae non explicari possunt, and Taylor follows him with such things as cannot be explained." It is strange that two really learned editors of Euclid's commentator should have been ignorant of one of Euclid's technical terms. come Anaxagoras of Clazomenae, and a little after him Oenopides of Chios; then Hippocrates of Chios, who squared the lunule, and then Theodorus of Cyrene. Hippocrates is the first writer of ele ments who is recorded. Plato then did much for geometry by the mathematical character of his writings; then Leodamos of Thasus, Archytas of Tarentum, and Theaetetus of Athens, gave a more scientific basis (ènioтημоVIKWтÉρav σÚσTαow) to various theorems; Neocleides and his disciple Leon came after the preceding, the latter of whom increas ed both the extent and utility of the science, in par ticular by finding a test (diopiouóv) of whether the thing proposed be possible* or impossible. Eudoxus of Cnidus, a little younger than Leon, and the companion of those about Plato [EUDOXUS], increased the number of general theorems, added three proportions to the three already existing, and in the things which concern the section (of the cone, no doubt) which was started by Plato himself, much increased their number, aud employed analyses upon them. Amyclas Heracleotes, the companion of Plato, Menaechmus, the disciple of Eudoxus and of Plato, and his brother Deinostratus, made geometry more perfect. Theudius of Magnesia * We cannot well understand whether by SuvaTóv Proclus means geometrically soluble, or possible in the common sense of the word. 1 generalized many particular propositions. Cyzici- | This history of Proclus has been much kept in 1 VOL. II. script supports him: how, then, did ne know? He saw that there ought to have been such a definition, and he concluded that, therefore, there had been one. Now we by no means uphold Euclid as an all-sufficient guide to geometry, though we feel that it is to himself that we owe the power of amending his writings; and we hope we may protest against the assumption that he could not have erred, whether by omission or commission. Some of the characteristics of the Elements are briefly as follows: is First. There is a total absence of distinction between the various ways in which we know the meaning of terms: certainty, and nothing more, the thing sought. The definition of straightness, an idea which it is impossible to put into simpler words, and which is therefore described by a more difficult circumlocution, comes under the same heading as the explanation of the word "parallel.” Hence disputes about the correctness or incorrectness of many of the definitions. Secondly. There is no distinction between propositions which require demonstration, and those which a logician would see to be nothing but different modes of stating a preceding proposition. When Euclid has proved that everything which is not A is not B, he does not hold himself entitled to infer that every B is A, though the two propositions are identically the same. Thus, having shewn that every point of a circle which is not the centre is not one from which three equal straight lines can be drawn, he cannot infer that any point from which three equal straight lines are drawn is the centre, but has need of a new demonstration. Thus, long before he wants to use book i. prop. 6, he has proved it again, and independently. Thirdly. He has not the smallest notion of admitting any generalized use of a word, or of parting with any ordinary notion attached to it. Setting out with the conception of an angle rather as the sharp corner made by the meeting of two lines than as the magnitude which he afterwards shews how to measure, he never gets rid of that corner, never admits two right angles to make one angle, and still less is able to arrive at the idea of an angle greater than two right angles. And when, in the last proposition of the sixth book, his definition of proportion absolutely requires that he should reason on angles of even more than four right angles, he takes no notice of this necessity, and no one can tell whether it was an oversight, whether Euclid thought the extension one which the student could make for himself, or whether (which has sometimes struck us as not unlikely) the elements were his last work, and he did not live to revise them. In one solitary case, Euclid seems to have made an omission implying that he recognized that natural extension of language by which unity is considered as a number, and Simson has thought it necessary to supply the omission (see his book v. prop. A), and has shewn himself more Euclid than Euclid upon the point of all others in which Euclid's philosophy is defective. Fourthly. There is none of that attention to the forms of accuracy with which translators have endeavoured to invest the Elements, thereby giving them that appearance which has made many teachers think it meritorious to insist upon their pupils remembering the very words of Simson. Theorems are found among the definitions: assump F tions are made which are not formally set down among the postulates. Things which really ought to have been proved are sometimes passed over, and whether this is by mistake, or by intention of supposing them self-evident, cannot now be known: for Euclid never refers to previous propositions by name or number, but only by simple re-assertion without reference; except that occasionally, and chiefly when a negative proposition is referred to, such words as "it has been demonstrated" are employed, without further specification. Fifthly. Euclid never condescends to hint at the reason why he finds himself obliged to adopt any particular course. Be the difficulty ever so great, he removes it without mention of its existence. Accordingly, in many places, the unassisted student can only see that much trouble is taken, without being able to guess why. What, then, it may be asked, is the peculiar merit of the Elements which has caused them to retain their ground to this day? The answer is, that the preceding objections refer to matters which can be easily mended, without any alteration of the main parts of the work, and that no one has ever given so easy and natural a chain of geometrical consequences. There is a never erring truth in the results; and, though there may be here and there a self-evident assumption used in demonstration, but not formally noted, there is never any the smallest departure from the limitations of construction which geometers had, from the time of Plato, imposed upon themselves. The strong inclination of editors, already mentioned, to consider Euclid as perfect, and all negligences as the work of unskilful commentators or interpolators, is in itself a proof of the approximate truth of the character they give the work; to which it may be added that editors in general prefer Euclid as he stands to the alterations of other editors. The Elements consist of thirteen books written by Euclid, and two of which it is supposed that Hypsicles is the author. The first four and the sixth are on plane geometry; the fifth is on the theory of proportion, and applies to magnitude in general; the seventh, eighth, and ninth, are on arithmetic; the tenth is on the arithmetical characteristics of the divisions of a straight line; the eleventh and twelfth are on the elements of solid geometry; the thirteenth (and also the fourteenth and fifteenth) are on the regular solids, which were so much studied among the Platonists as to bear the name of Platonic, and which, according to Proclus, were the objects on which the Elements were really meant to be written. At the commencement of the first book, under the name of definitions (Spot), are contained the assumption of such notions as the point, line, &c.. and a number of verbal explanations. Then follow, under the name of postulates or demands (airhuara), all that it is thought necessary to state as assumed in geometry. There are six postulates, three of which restrict the amount of construction granted to the joining two points oy a straight line, the indefinite lengthening of a terminated straight line, and the drawing of a circle with a given centre, and a given distance measured from that centre as a radius; the other three assume the equality of all right angles, the much disputed property of two lines, which meet a third at angles less than two right angles (we mean, of course, much disputed as to its propriety as an assumption, not as to its truth), and that two straight lines cannot inclose a space. Lastly, under the name of common notions (koival ěvvola) are given, either as common to all men or to all sciences, such assertions as that-things equal to the same are equal to one another-the whole is greater than its part-&c. Modern editors have put the last three postulates at the end of the common notions, and applied the term axiom (which was not used till after Euclid) to them all. The intention of Euclid seems to have been, to distinguish between that which his reader must grant, or seek another system, whatever may be his opinion as to the propriety of the assumption, and that which there is no question every one will grant. The modern editor merely distinguishes the assumed problem (or construction) from the assumed theorem. Now there is no such distinc tion in Euclid as that of problem and theorem; the common term póraσis, translated proposition, includes both, and is the only one used. An immense preponderance of manuscripts, the testimony of Proclus, the Arabic translations, the summary of Boethius, place the assumptions about right angles and parallels (and most of them, that about two straight lines) among the postulates; and this seems most reasonable, for it is certain that the first two assumptions can have no claim to rank among common notions or to be placed in the same list with the whole is greater than its part." Without describing minutely the contents of the first book of the Elements, we may observe that there is an arrangement of the propositions, which will enable any teacher to divide it into sections. Thus propp. 1-3 extend the power of construction to the drawing of a circle with any centre and any radius; 4-8 are the basis of the theory of equal triangles; 9-12 increase the power of construction; 13-15 are solely on relations of angles; 16-21 examine the relations of parts of one triangle; 22-23 are additional con structions; 23-26 augment the doctrine of equal triangles; 27-31 contain the theory of parallels;* 32 stands alone, and gives the relation between the angles of a triangle; 33-34 give the first properties of a parallelogram; 35-41 consider parallelograms and triangles of equal areas, but different forms; 42-46 apply what precedes to augmenting power of construction; 47-48 give the celebrated property of a right angled triangle and its converse. The other books are all capable of a similar species of subdivision. The second book shews those properties of the rectangles contained by the parts of divided straight lines, which are so closely connected with the common arithmetical operations of multiplication and division, that a student or a teacher who is not fully alive to the existence and difficulty of incommensurables is apt to think that common arithmetic would be as rigorous as geometry. Euclid knew better. The third book is devoted to the consideration of the properties of the circle, and is much cramped in several places by the imperfect idea already al luded to, which Euclid took of an angle. There are some places in which he clearly drew upon experimental knowledge of the form of a circle, *See Penny Cyclopaedia, art.“ Parallels," for some account of this well-worn subject. |